aci structural journal january-february 2013 v. 110 no. 1 complete

VOL. 110, NO. 1JANUARY-FEBRUARY 2013

ACISTRUCTURAL

J O U R N A L

A J O U R N A L O F T H E A M E R I C A N C O N C R E T E I N S T I T U T E

111 Rapid repair procedure

ACI Structural Journal/January-February 2013 1

Discussion is welcomed for all materials published in this issue and will appear in the November-December 2013 issue if received by July 1, 2013. Discussion of material received after specified dates will be considered individually for publication or private response.ACI Standards published in ACI Journals for public comment have discussion due dates printed with the Standard.Annual index published online at www.concrete.org/pubs/journals/sjhome.asp.ACI Structural JournalCopyright © 2013 American Concrete Institute. Printed in the United States of America.

The ACI Structural Journal (ISSN 0889-3241) is published bimonthly by the American Concrete Institute. Publica-tion office: 38800 Country Club Drive, Farmington Hills, MI 48331. Periodicals postage paid at Farmington, MI, and at additional mailing offices. Subscription rates: $161 per year (U.S. and possessions), $170 (elsewhere), payable in advance. POSTMASTER: Send address changes to: ACI Structural Journal, 38800 Country Club Drive, Farmington Hills, MI 48331.

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CONTENTSBoard of Direction

PresidentJames K. Wight

Vice PresidentsAnne M. EllisWilliam E. Rushing Jr.

DirectorsNeal S. AndersonKhaled AwadRoger J. BeckerJeffrey W. ColemanRobert J. FroschJames R. HarrisCecil L. JonesSteven H. KosmatkaDavid A. LangeDenis MitchellJack P. MoehleDavid H. Sanders

Past President Board MembersKenneth C. HoverFlorian G. BarthLuis E. García

Executive Vice PresidentRon Burg

Technical Activities CommitteeDavid A. Lange, ChairDaniel W. Falconer, SecretarySergio M. AlcocerJoAnn P. BrowningChiara F. FerrarisCatherine E. FrenchTrey HamiltonRonald JanowiakKevin A. MacDonaldAntonio NanniJan OlekMichael SprinkelPericles C. StivarosEldon G. Tipping

StaffExecutive Vice PresidentRon Burg

EngineeringManaging DirectorDaniel W. Falconer

Managing EditorKhaled Nahlawi

Staff EngineersMatthew R. SenecalGregory Zeisler

Publishing ServicesManagerBarry M. Bergin

EditorsCarl R. BischofKaren CzedikKelli R. SlaydenDenise E. Wolber

Editorial AssistantAshley Poirier

ACI StruCturAl JournAl

JAnuAry-FebruAry 2013, V. 110, no. 1a journal of the american concrete institutean international technical society

3 Ultimate Strength Domain of Reinforced Concrete Sections under Biaxial Bending and Axial Load, by Francesco Vinciprova and Giuseppe Oliveto

15 Design Considerations for Shear Bolts in Punching Shear Retrofit of Reinforced Concrete Slabs, by Maria Anna Polak and Wensheng Bu

27 Shortening Estimation for Post-Tensioned Concrete Floors—Part I: Model Selection, by Guohui Guo and Leonard M. Joseph

35 Shortening Estimation for Post-Tensioned Concrete Floors—Part II: Calculations, by Guohui Guo and Leonard M. Joseph

43 Shear Behavior of Reinforced High-Strength Concrete Beams, by S. V. T. Janaka Perera and Hiroshi Mutsuyoshi

53 Design of Anchor Reinforcement for Seismic Shear Loads, by Derek Petersen and Jian Zhao

63 Innovative Flexural Strengthening of Reinforced Concrete Columns Using Carbon-Fiber Anchors, by Ioannis Vrettos, Efstathia Kefala, and Thanasis C. Triantafillou

71 Adaptive Stress Field Models: Formulation and Validation, by Miguel S. Lourenço and João F. Almeida

83 Adaptive Stress Field Models: Assessment of Design Models, by Miguel S. Lourenço and João F. Almeida

95 Flexural Drift Capacity of Reinforced Concrete Wall with Limited Confinement, by S. Takahashi, K. Yoshida, T. Ichinose, Y. Sanada, K. Matsumoto, H. Fukuyama, and H. Suwada

105 Shake-Table Studies of Repaired Reinforced Concrete Bridge Columns Using Carbon Fiber-Reinforced Polymer Fabrics, by Ashkan Vosooghi and M. Saiid Saiidi

115 Cyclic Loading Test for Reinforced-Concrete-Emulated Beam-Column Connection of Precast Concrete Moment Frame, by Hyeong-Ju Im, Hong-Gun Park, and Tae-Sung Eom

127 Unified Calculation Method for Symmetrically Reinforced Concrete Section Subjected to Combined Loading, by Liang Huang, Yiqiu Lu, and Chuxian Shi

137 Cyclic Behavior of Substandard Reinforced Concrete Beam-Column Joints with Plain Bars, by Catarina Fernandes, José Melo, Humberto Varum, and Aníbal Costa

149 Discussion

Effective Capacity of Diagonal Strut for Shear Strength of Reinforced Concrete Beams without Shear Reinforcement. Paper by Sung-Gul Hong and Taehun Ha

Unbonded Tendon Stresses in Post-Tensioned Concrete Walls at Nominal Flexural Strength. Paper by Richard S. Henry, Sri Sritharan, and Jason M. Ingham

Contents cont. on next page

2 ACI Structural Journal/January-February 2013

MEETINGS

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The Institute is not responsible for statements or opinions expressed in its publications. Institute publications are not able to, nor intend to, supplant individual training, responsibility, or judgment of the user, or the supplier, of the information presented.

Papers appearing in the ACI Structural Journal are reviewed according to the Institute’s Publication Policy by individuals expert in the subject area of the papers.

Contributions to ACI Structural Journal

The ACI Structural Journal is an open forum on concrete technology and papers related to this field are always welcome. All material submitted for possible publi-cation must meet the requirements of the “American Concrete Institute Publi-cation Policy” and “Author Guidelines and Submission Procedures.” Prospective authors should request a copy of the Policy and Guidelines from ACI or visit ACI’s website at www.concrete.org prior to submitting contributions.

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All materials conforming to the Policy requirements will be reviewed for editorial quality and technical content, and every effort will be made to put all acceptable papers into the information channel. However, potentially good papers may be returned to authors when it is not possible to publish them in a reasonable time.

Discussion

All technical material appearing in the ACI Structural Journal may be discussed. If the deadline indicated on the contents page is observed, discussion can appear in the designated issue. Discussion should be complete and ready for publication, including finished, reproducible illustra-tions. Discussion must be confined to the scope of the paper and meet the ACI Publi-cation Policy.

Follow the style of the current issue. Be brief—1800 words of double spaced, typewritten copy, including illustrations and tables, is maximum. Count illustrations and tables as 300 words each and submit them on individual sheets. As an approxi-mation, 1 page of text is about 300 words. Submit one original typescript on 8-1/2 x 11 plain white paper, use 1 in. margins, and include two good quality copies of the entire discussion. References should be complete. Do not repeat references cited in original paper; cite them by original number. Closures responding to a single discussion should not exceed 1800-word equivalents in length, and to multiple discussions, approximately one half of the combined lengths of all discussions. Closures are published together with the discussions.

Discuss the paper, not some new or outside work on the same subject. Use references wherever possible instead of repeating available information.

Discussion offered for publication should offer some benefit to the general reader. Discussion which does not meet this requirement will be returned or referred to the author for private reply.

Send manuscripts to:http://mc.manuscriptcentral.com/aci

Send discussions to:[email protected]

2013

JANUARY

11-13—The Precast Show 2013, Indianapolis, IN, www.precast.org/theprecastshow

28-30—ACCTA 2013, Johannesburg, South Africa, www.spin.bam.de/en/accta_2013/index.htm

FEBRUARY

4-8—World of Concrete, Las Vegas, NV, www.worldofconcrete.com

15-16—CEMCON 2013, Pune, India, www.icipunecentre.org/cemcon2013.aspx

14-15—IABSE Workshop on Safety, Failures, and Robustness of Large Structures, Helsinki, Finland, www.iabse2013helsinki.org

FEBRUARY/MARCH

28-2—CSDA Annual Convention and Tech Fair, Duck Key, FL, www.csda.org

MARCH

10-14—FraMCoS-8, Toledo, Spain, www.framcos8.org

20-22—ICRI Spring Convention, St. Pete Beach, FL, www.icri.org

APRIL

17-20—14th International Congress on Polymers in Concrete, Shanghai, China, www.rilem.net

22-24—2013 fib Symposium, Tel Aviv, Israel, www.fib2013tel-aviv.co.il

MAY

6-8—International IABSE Spring Conference, Rotterdam, the Netherlands, www.iabse2013rotterdam.nl

13-15—2013 APWA Sustainability in Public Works Conference, San Diego, CA, www.apwa.net/sustainability

26-29—Twin International Conferences on Civil Engineering Towards a Better Environment and The Concrete Future, Covilhã, Portugal, www.uc.pt/en/iii/novidades/2012/twinconferencesucubi

27-29—International Conference on Concrete Sustainability (ICCS13), Tokyo, Japan, www.jci-iccs13.jp

UPCOMING ACI CONVENTIONSThe following is a list of scheduled ACI conventions:

2013—April 14-18, Hilton & Minneapolis Convention Center, Minneapolis, MN2013—October 20-24, Hyatt Regency & Phoenix Convention Center, Phoenix, AZ2014—March 23-27, Grand Sierra Resort, Reno, NV2014—October 26-30, Hilton Washington, Washington, DC

For additional information, contact:Event Services, ACI38800 Country Club DriveFarmington Hills, MI 48331Telephone: (248) 848-3795e-mail: [email protected]

Behavior of Steel Fiber-Reinforced Concrete Deep Beams with Large Opening. Paper by Dipti R. Sahoo, Carlos A. Flores, and Shih-Ho Chao

Nonlinear Cyclic Truss Model for Reinforced Concrete Walls. Paper by Marios Panagiotou, José I. Restrepo, Matthew Schoettler, and Geonwoo Kim

Behavior of Lap-Spliced Plain Steel Bars. Paper by M. Nazmul Hassan and Lisa R. Feldman

162 In ACI Materials Journal

Contents cont.


Title no. 110-S01

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2010-301.R3 received March 22, 2012, and reviewed under Institute

publication policies. Copyright © 2013, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the November-December 2013 ACI Structural Journal if the discussion is received by July 1, 2013.

Ultimate Strength Domain of Reinforced Concrete Sections under Biaxial Bending and Axial Loadby Francesco Vinciprova and Giuseppe Oliveto

A direct method is provided for the construction of the ultimate strength domain of reinforced and/or prestressed concrete sections under biaxial bending and axial force. The method, based on the principle of plane sections, only requires the specification of the stress-strain relationships for each component material, the preten-sion strains, and possibly any other applied distortion. The results may be used for safety checks in new designs and in the rehabilita-tion of vulnerable or deteriorated structures. A few examples are used to demonstrate the performance of the method and its useful-ness in practical applications.

Keywords: analytical techniques; biaxial bending; reinforced concrete sections; strength domain.

INTRODUCTIONThe construction of the strength domains of cross sections

of structural members has a long history. It is related to the problem of designing safe structures and is in fact to some degree an extension to the cross section of the strength criteria for materials such as Rankine, Tresca, and the Von Mises criteria, to quote only a few of the most popular. However, as is well-known, the strength criteria are often not in agree-ment with each other and some work is better than others, depending on the situation considered. Strength domains or failure surfaces in terms of stress resultants for a cross section should be the natural extension of strength criteria for materials. As there are six independent stress components for the Cauchy stress tensor, there are six stress resultants to be considered for the cross section—that is, an axial force, two shear forces, two bending moments, and a torque.

The task of producing a complete yield or resistance or rupture surface for a cross section is so daunting, however, that so far, none actually exist, even though in several branches of mechanics they are very much needed. The first complication with respect to a strength criterion is that there is the need for integration of the stress components over the cross section and the result is obviously dependent on the shape of the cross section. Moreover, the cross section is often of a composite nature, where different materials are used, such as concrete, reinforcing steel, prestressing steel, and fiber-reinforced polymer (FRP), to mention only widely used materials and technologies.

Although the six-dimensional domain is difficult to obtain exactly, several cross sections of the actual surface have been obtained with the development in the last century of metal and concrete structural plasticity. The results are available in text-books on steel structures and on concrete structures.1-3 Well-known, for instance, are the bending-moment-axial-force-interaction domains that are plane curves if the bending is direct. Even when neglecting both shear forces and torque, it has been recognized that in many instances the interac-tion of biaxial bending and axial force cannot be ignored. This has been recognized by most design codes in the

world, which provide simplified methods to account for the abovementioned interaction. Among these are ACI 318,4 AS 3600,5 Eurocode 2 (EN 1992-1-1),6 Eurocode 4 (EN-1994-1-1),7 and Eurocode 8 (EN 1998-3).8 In this paper, the authors are concerned with the development of a simple numerical algorithm for the construction of the interaction domain of reinforced concrete sections, possibly prestressed and/or including FRP in the form of bars or plies, or of steel sections encased in concrete or even tubular steel and FRP sections filled with concrete.

An overview of the state of the art in the field may be found in the paper by Furlong et al.9 In this paper, only essential literature will be reviewed. All code developments are supported by scientific research and for the problem being considered, the relevant research is provided in a paper by Bresler,10 where two empirical methods for the construc-tion of the interaction surface are introduced—namely, the reciprocal load method (RLM) and the load contour method (LCM). Because these methods are well-known and avail-able in the literature, it suffices to say that these are semi-analytical methods in the sense that they provide equations that, on the basis of empirically evaluated parameters, allow for the construction of the interaction surface.

Following this seminal paper, three main research lines have been pursued to deepen understanding of this topic. One line has consisted of experimental research aimed at checking the assumptions on the basis of the two methods mentioned in the previous paragraph and at evaluating the accuracy that can be obtained. Noteworthy among this research is the work by Ramamurthy.11 Another line of research has been aimed at providing more accurate analytical expressions for the construction of the failure surface. The works by Ferguson et al.,12 Wight and Mac Gregor,13 Silva et al.,14 and Bonet et al.15 follow this line. However, a considerable effort has been devoted to a third line of research, which has tackled the problem of providing the failure surface using numerical algorithms based on the integration of the governing equilibrium equations and constitutive laws for the constituent materials. Early basic work is considered in textbooks by Park and Pauley,16 Nielsen,17 and MacGregor and Bartlett.18 Also in this line of research are the works by Kawakami et al.19; Landonio and Perego20; Hulse and Mosley21; Contaldo and Faella22; Spiegel and Limbrunner23; Bousias et al.24; De Vivo and Rosati25; Rodriguez-Gutierrez and Aristizabal-Ochoa26; Rodriguez-Gutierrez and Aris-


Francesco Vinciprova is a Lecturer of physics and mathematics at the Bonaventura Secusio High School, Caltagirone, Sicily, Italy. He received his MS in civil engineering and his PhD in structural engineering at the University of Catania, Sicily, Italy. His research interests include the field of application of boundary element methods in structural mechanics and structural dynamics.

Giuseppe Oliveto is Full Professor of structural engineering at the University of Catania. His research interests include structural mechanics, structural dynamics, and earthquake engineering.

tizabal-Ochoa,27 who also considered prestressed concrete sections; Fafitis,28 who transforms the double equilibrium integrals into line integrals; Sfakianakis29; Consolazio et al.30; Hock and Cheong31; Charalampakis and Koumousis32; and Di Ludovico et al.33 Other works in this line study the integration methods of the stresses over the cross section. Among these are studies by Bonet et al.34,35

The reference by Di Ludovico et al.33 is notable for a state-of-the-art treatment of the topic and because it provides a good basis for the work that will be presented herein. The paper develops a numerical algorithm based on a finite element discretization of the cross section. The applied load is increased monotonically until the ultimate limit state of the cross section is reached. For a given value of the axial force acting on the cross section, the loading axis in the cross section is provided. For a given value of the bending moment, which is perpendicular to the loading axis, a nonlinear system of equations must be solved to find the position of the neutral axis and subsequently strains, stresses, and stress resultants. This procedure is iterated several times for successive increments of the bending moment or curvature until the ultimate limit state for the cross section is reached. Having completed this task, one point of the failure surface of the cross section under axial force and biaxial bending is found. Other points are obtained by slightly changing the ratio between bending moments and repeating the previous

task. In the end, a cross section of the failure surface for constant axial force is generated.

RESEARCH SIGNIFICANCEThe construction of the entire failure surface with the

method described at the end of the introduction requires the repetition of the previously described tasks for several values of the axial force, from the failure value in compres-sion to the failure value in tension. It is obvious that the construction of the entire failure surface in this way is a very time-consuming task, but a much more efficient method is available. This method, already known in literature and recommended by Marín36 but somewhat neglected, will be formulated and exploited in the following sections.

GENERAL FORMULATION OF THE PROBLEMGiven a cross section of general shape and material compo-

sition (Fig. 1), the problem is to solve is the construction of the failure surface under axial force and biaxial bending in the quickest way possible. To make the problem as general as possible, assume that the cross section does not admit axes of symmetry. Let a Cartesian orthogonal reference frame O, X, Y be defined within the plane of the cross section, to which the axial force and the bending moments are referred. For a given position of the neutral axis, a rotated Cartesian orthog-onal reference system O, x, y is defined such that the x-axis is parallel to the neutral axis n, and a is the anti-clockwise rotation of the x-axis with respect to the X-axis. The Bernoulli hypothesis, which states that initially plane cross sections remain plane after deformation, allows for the deformation shown graphically in Fig. 1 to be given by the following mathematical expression

e e k= +0 y (1)

Fig. 1—General cross section, global and local reference frames, neutral axis, and strain and stress diagrams.


where e0 is the axial strain on the fiber defined by the equa-tion y = 0; and k is the curvature.

At this point, one can only assume that the axial stress is related to the axial strain via a general nonlinear relation-ship—that is, s = f(e)—where only one value of the stress corresponds to a given value of the strain, while more than one strain may correspond to a given stress.

The three equations given as follows must be satisfied to ensure equilibrium between internal stresses and externally applied load resultants

s e

s e

s e

dA f dA N

y dA yf dA M

x dA xf dA M

A A

A Ax

A Ay

∫ = ( )∫ =

∫ = ( )∫ =

∫ = ( )∫ =

(2)

The transformation of coordinates

x

y

X

Y

=

−

⋅

cos sin

sin cos

a aa a

(3)

allows for Eq. (2) to be written with reference to the fixed reference frame O, X, Y, providing the following system of equations

f dA N

X f dA Y f dA M M

X

A

A AX Y

e

a e a e a a

a

( )∫ =

⋅ ( )∫ + ⋅ ( )∫ = −sin cos cos sin

cos ⋅⋅ ( ) ⋅∫ − ⋅ ( ) ⋅∫ = +f dA Y f dA M MA A

X Ye a e a asin sin cos

(4)

The system of Eq. (4) provides the proper basis for the problem. In most of the literature, the problem is approached in the following way. Given the three stress resultants (N, MX, MY), find the position of the neutral axis—that is, the angle a and the ordinate y0 = e0/k—and check whether the ultimate strain has been exceeded in the most distant fibers from the neutral axis. Generally, a loading process is assigned by which the external stress resultants are increased step by step according to a given law until a failure condition is reached—that is, one of the component materials in the cross section reaches the rupture strain. It is evident that the governing system of equations is highly nonlinear because of the nonlin-earity of the constitutive laws of the constituent materials and because of the geometrical nonlinearity associated to the position of the neutral axis. This system must be solved many times just to obtain a single point of the failure surface. It is obvious that this is not a very efficient method if one wants just to construct the failure surface of a given cross section. However, the method may be justified if one has just one set of stress resultants and wants to check whether this is safe or not, or what the safety coefficient is for a given loading path.

Before a much more efficient method of producing the failure surface is formulated, it may be worth noting that the given system of Eq. (4) is strictly sufficient for the evaluation of the three kinematical unknowns a, e0, and k, which solve the problem.

MARÍN’S METHODThe following describes a much more efficient way

of solving the problem of constructing the failure surface

of a cross section under axial force and biaxial bending. Although the method has often been used by engineers to solve problems elegantly that otherwise would be very hard to solve, it was recommended in the present context by Marín36 in 1979 to satisfy the ACI 318-71 requirements. Rather than starting from a set of stress resultants and then trying to find a point of the failure surface by following a given step-by-step loading process, the method starts from a given position of the neutral axis—that is, angle a and the equation of the neutral axis are known

y y= 0 (5)

Equation (1) and Eq. (5) provide the following results

e k e k0 0 0= − ⋅ = ⋅ −( )y y y (6)

For each material that constitutes the cross section, the maximum strain et is related to the maximum distance yi

from the neutral axis via the relationship

e ki iy y i N= ⋅ −( ) =0 1 2, , ,

(7)

The failure of the cross section occurs as soon as a material reaches the rupture or breaking strain—namely ef = ef

r—and the curvature that leads the cross section to failure, for the given position of the neutral axis, is given by the equation

ke

r

jr

jy y=

− 0

(8)

At this point, the strain diagram at failure is completely known if the result from Eq. (8) is used in Eq. (6). The corre-sponding stress resultants may be obtained by using Eq. (2), while a coordinate transformation provides the results in the fixed coordinate system O, X, Y. The set of stress resultants thus evaluated provides a point of the failure surface of the cross section. The most important thing that must be observed is that there are no nonlinear equations to be solved with this method and the only operations needed are the integral calcu-lations for the evaluation of the stress resultants. As will be shown later, even those can be performed in closed form.

The construction of the failure surface proceeds with the calculation of a family of isogonic lines, each of them associ-ated with a given angle a for the neutral axis. By translating the neutral axis slowly, a set of failure points on a isogonic line may be generated. These points may be as close as one wishes because their distance depends only on the amount by which the neutral axis is translated. The isogonic lines are not generally plane curves, but are certainly plane if the neutral axis is parallel to one of the principal axes of the cross section. Once an isogonic line has been completed, the next one is constructed by rotating the neutral axis by a conve-nient amount. It is obvious that the family of isogonic lines thus constructed allows for a complete characterization of the failure surface of a given cross section. The entire surface may be constructed or simply areas of the entire surface of specific interest. The accuracy of the entire surface or of a required area can be made as good as required by reducing the transla-tion and rotational steps of the neutral axis.


IMPLEMENTATION OF METHODThe implementation of the model requires some basic

operations that are listed in the following. The first step is the description of the cross section with the definition of exterior and interior boundaries, position of reinforcement and pretensioned bars, position of FRP bars and/or plies, and position of structural steel. The second step consists of the definition of the constitutive laws of the constituent mate-rials. The third step consists of the calculation of the stress resultants as specified by Eq. (2) and the final step consists of their transformation into the fixed reference frame O, X, Y. The various phases that have been listed previously will be described in some detail as follows.

Description of cross sectionIt is assumed that all boundaries of the cross section

are straight line segments; if a curved boundary exists, it is assumed that this can be approximated by a polygonal, which can be made as close as possible to the given boundary as the number of vertices of the approximating polygonal increases. All geometrical parameters are referred to the fixed reference frame O, X, Y as follows:• Xi

c and Yic are the coordinates of the vertices of the

various boundaries of the cross section;• Xi

s and Yis are the coordinates of the centroid of the ordi-

nary reinforcing bars;• Xi

sp and Yisp are the coordinates of the centroid of the

prestressing bars; and• Xi

ss and Yiss are the coordinates of the vertices of the

structural steel section encased in the concrete section.The corresponding coordinates in the local reference

frame O, x, y are obtained via the coordinate transformation provided by Eq. (3). The equation of the boundary segment j between vertices i and i + 1 takes the expression

x m y qj j= + (9)

where

mx x

y yq x

x x

y yyj

iC

iC

iC

iC j i

C iC

iC

iC

iC i

C=−−

= −−−

+

+

+

+

1

1

1

1

(10)

provided that the segment is not parallel to the x-axis, in which case the equation becomes

y y yiC

iC= = +1 (11)

Constitutive lawsAny of the constitutive laws used in the literature are

allowed within the context of this work; for instance, the constitutive laws listed by Di Ludovico et al.33 However, for illustrative purposes, some constitutive laws recommended by Eurocode 26 and Eurocode 47 are used herein, along with the softening Kent and Park37 law for unconfined concrete.

According to the considered Eurocodes, the constitutive law for concrete assumes that the concrete does not respond in tension while a parabola-rectangle stress diagram is asso-ciated to compression strains. In mathematical terms, this may be expressed as follows

s e e

s e ae

ee

ee e

s e a

c

c cdc c

c

c

f

( ) = ∀ ≥

( ) = − ⋅ −

∀ ∈[ ]( ) = − ⋅

0 0

2 01 1

1,

ffcd cu c∀ ∈[ ]e e e, 1

(12)

When the Kent and Park37 law is used, the first and second equations of Eq. (12) still hold while the third equation is replaced by Eq. (13), with stress varying linearly from –a·fcd to –b·fcd.

s e ae e

e ea b e e e( ) = − +

−−

−( )

∀ ∈[ ]c

cu ccd cu cf1

11, (13)

By setting

t t tc c c c c

00

1

0

11

1

0

12

2

12

22

1= −

= −

= −ee

ee

ke

ee

ke

(14)

the second equation of Eq. (12) may be written as follows

s ac cdy f t t y t y( ) = − ⋅ + +( )0 1 22 (15)

By setting

t tLc cu

cu c

L

cu c0

0 1

11

1

1 1=

−

− +

−=

−

−

ba

e ba

e e

e e

k ba

e e (16)

Equation (13) may be written as follows

s ac cdL Ly f t t y( ) = − ⋅ +( )0 1

(17)

The parameters appearing in the previous equations are specified by Eurocodes 2 and 4.

The constitutive law for reinforcing steel assumes elastic-perfectly plastic behavior with limited tensile and compres-sion strain. In mathematical terms, this is expressed by the following equations

s e e e e e

s e e e e e e

s s yd yd

s s yd su yd

E

E

( ) = ∀ ∈ −

( ) = ( ) ∀ ∈ − − ∪

,

sgn , ee eyd su, (18)

For prestressing steel, Eurocode 2 specifies a bilinear stress-strain relationship

se

ee e

sγ

e ee e

e

p pdyd

yd

p pd

pk

spd

yd

ud yd

f

ff

f

= ∀ ∈

= + −

−−

∀

0,

∈∈ e eyd ud,

(19)

Stress resultantsThe cross section is divided into a finite number of trap-

ezoids with bases parallel to the neutral axis and the two oblique sides belonging to its boundary, as is shown in


Fig. 1. The partition lines parallel to the neutral axis, besides the neutral axis itself defined by the equation y = y0 and the straight line y = yc1 for which it is e(yc1) = ec1, all pass through vertices of the boundary of the cross section. Axial force N and bending moments Mx and My may be given by the expressions

N N N N N NiS

i

n

iP

i

n

iSS

i

n

iC

i

n

iC

i

nSpS SS

tC

cPC

= ∑ + ∑ + ∑ + ∑ + ∑ += = = = =1 1 1 1 1

NN N

M M M M

iC

i

n

iC

i

n

x xiS

i

n

xiP

i

n

xiSS

i

cLC

cRC

SpS

= =

= = =

∑ + ∑

= ∑ + ∑ +

1 1

1 1 1

nn

xiC

i

n

xiC

i

n

xiC

i

n

xiC

i

n

y

SStC

cPC

cLC

cRC

M M M M

M

∑ + ∑ + ∑ + ∑ + ∑= = = =1 1 1 1

== ∑ + ∑ + ∑ + ∑ += = = = =

M M M M MyiS

i

n

yiP

i

n

yiSS

i

n

yiC

i

n

yiC

i

nSpS SS

tC

1 1 1 1 1

ccPC

cLC

cRC

M MyiC

i

n

yiC

i

n

∑ + ∑ + ∑= =1 1

(20)

where nS is the total number of ordinary reinforcing steel rods; nS

p is the total number of pretensioned steel bars; nSS is the number of trapezoids into which the structural steel is divided; nC

t is the number of trapezoids into which the part of cross section in tension is divided; nC

cp is the number of trapezoids in the compression part of the cross section where the stress has a parabolic distribution; nC

cL is the number of trapezoids in the compression part of the cross section where the stress has a linear distribution; and nC

cR is the number of trapezoids in the compression part of the cross section where the stress is constant.

Let us now consider the i-th trapezoid whose oblique sides are the j-th and the k-th sides of the polygon, which defines the boundary of the cross section, and let ysup and yinf be the ordinates of the two bases of the trapezoid.

The contributions to the stress resultants of the trapezoid considered are calculated as follows

N x y dxdy x y dxiC

cx y

x y

y

y

cm y q

m y q

k

j

k k

j j

= ( )∫∫ = ( )∫( )

( )

+

+

s s, ,inf

sup

ddy

M x y ydxdy x y y

y

y

xC

cx y

x y

y

y

cik

j

inf

sup

inf

sup

, ,

∫

= ( )∫∫ = ( )( )

( )s s ddxdy

M x y xdxdy

m y q

m y q

y

y

yC

cx y

x y

y

k k

j j

ik

j

+

+

( )

( )

∫∫

= ( )∫

inf

sup

in

,sff

sup

inf

sup

,y

cm y q

m y q

y

y

x y xdxdyk k

j j

∫ = ( )∫∫+

+

s

(21)

By using Eq. (9), (11), (15), and (17), the aforementioned integrals lead to the following results if the trapezoid belongs to the region where the stress has a parabolic distribution

N f t q q y y t m m t q qy

iC

cd j k j k j k= − −( ) −( ) + −( ) + −( ) a 0 0 1

2

sup inf

sup −−+

+ −( ) + −( ) −

+ −

y

t m m t q qy y

t m mj k j k j k

inf

sup inf

2

1 2

3 3

2

2

3(( ) −

= − −( ) −+ −

y y

M f t q qy y

t m mxiC

cd j k j

sup inf

sup inf

4 4

0

2 2

0

4

2a kk j k

j k j k

t q qy y

t m m t q q

( ) + −( ) −

+

+ −( ) + −( )

1

3 3

1 2

3sup inf

−

+ −( ) −

= −

y yt m m

y y

M f t q

j k

yiC

cd j

sup inf sup inf4 4

2

5 5

0

4 5

1

2a 22 2

0 12 2

21

2−( ) −( ) + −( ) + −( )

q y y t m q m q t q qy

k j j k k j ksup inf

sup −−+

+ −( ) + −( ) + −( )

y

t m m t m q m q t q qj k j j k k j k

inf2

02 2

1 22 2

2

1

2

1

2

−+

+ −( ) + −( )

−

y y

t m m t m q m qy

j k j j k k

sup inf

sup

3 3

12 2

2

4

3

1

2

yyt m m

y yj k

inf sup inf4

22 2

5 5

4

1

2 5+ −( ) −

(22)

If the trapezoid belongs to the region where the stress is linear, the following results are found

N f t q q y y t m m t q qy

iC

cdL

j kL

j kL

j k= − −( ) −( ) + −( ) + −( ) a 0 0 1sup inf

suup inf sup inf2 2

1

3 3

0

2 3

−+ −( ) −

= −

yt m m

y y

M f t q

Lj k

xiC

cdLa jj k

Lj k

Lj kq

y yt m m t q q

y y−( ) −

+ −( ) + −( ) −

+sup inf sup inf2 2

0 1

3 3

2 3tt m m

y y

M f t q q y

Lj k

yiC

cdL

j k

1

4 4

02 2

4

1

2

−( ) −

= − −( )

sup inf

a ssup inf

sup inf−( ) + −( ) + −( )

−y t m q m q t q q

y yL

j j k kL

j k0 12 2

2 21

2 22

1

2 3

10

2 21

3 3

+

+ −( ) + −( )

−+t m m t m q m q

y yL

j kL

j j k k

sup inf

22 412 2

4 4

t m my y

Lj k−( ) −

sup inf

(23)

If, instead, the trapezoid belongs to the region where the stress is constant, it results

N f q q y y m my y

M

iC

cd j k j k

xi

= − −( ) −( ) + −( ) −

a sup inf

sup inf2 2

2

CCcd j k j k

yi

f q qy y

m my y

M

= − −( ) −+ −( ) −

a sup inf sup inf2 2 3 3

2 3

CCcd j k j j k kf q q y y m q m q

y ym= − −( ) −( ) + −( ) −

+a1

2 2

1

22 2

2 2

sup inf

sup inf

jj kmy y

2 2

3 3

3−( ) −

sup inf

(24)

Equations (22) through (24) provide closed-form contribu-tions to the stress resultants and are only valid for the consti-tutive laws for concrete defined in the appropriate section above. However, corresponding results may be easily found for several other common constitutive laws.

EXAMPLES AND COMPARISON WITH RESULTS IN LITERATURE

In this section, results obtained using the presented method are compared with those available in the literature. The results are also compared to those obtained using fiber models, as implemented in the OpenSees38 software. Prob-lems, which may arise when using those methods or similar ones, are highlighted and the usefulness of the proposed method is reaffirmed. Strength domains for some typical cross sections are also shown.

Comparison of results for square cross sectionThe simplest cross section that can be considered is the

rectangular one. Some results for this type of cross section are shown by Di Ludovico et al.33 The 250 x 250 mm (9.84 x 9.84 in.) square cross section shown in Fig. 2 is provided with four reinforcement bars, each 12 mm (0.472 in.) in diameter, with a concrete cover of 30 mm (1.18 in.).

The material properties considered, also shown in Fig. 2, are steel design yield stress fyd = 278 MPa (40.32 ksi) and concrete design strength afcd = 13.28 MPa (1.93 ksi).

The strength domain constructed by using the method presented in the previous section is shown in Fig. 3(a). In the construction of this domain, the rotation of the neutral axis from one position to the adjacent one was chosen as 10 degrees or p/18 radians. Therefore, 18 isogonic curves were generated. The number of points on each of these curves was established by requiring 25 points for each character-istic rupture mode of the cross section, defined by a strain interval ]emin, emax[. Because there are six rupture modes, the total number of points on each isogonic curve turns out to be equal to 150. Therefore, the surface, or boundary, of the strength domain shown in Fig. 3 is identified by an irreg-ular grid of 18 x 150 points. The quadrilaterals visible in Fig. 3 each correspond to two adjacent isogonic curves and two adjacent strain values.

The intersection of the strength domain with planes of constant axial force, as shown in Fig. 3(b), provides the


plane curves given in Fig. 4(a). These were produced to compare the results obtained by the presented method with corresponding results available in the literature. In fact, the results by Di Ludovico et al.33 were given in the form of discrete points in the MX-Mr plane, for various levels of axial force. In Fig. 4(a), the continuous lines refer to the results of the present method, while the symbols show the results by Di Ludovico et al.,33 each line being characterized by a given ratio n of the actual axial force to the ultimate value in compression. The agreement is excellent, showing that both methods lead to the same results. However, it should be noticed that no iterations are needed to produce the present results and the entire strength domain can be derived with little computational effort.

To have a quantitative measure of the accuracy of the results obtained and to be able to make a comparison with

other methods, the following definition is adopted for the error. If P – O is the vector denoting the solution obtained by the present method and Q – O denotes the solution obtained by another method, the relative error may be defined by the following formula

eQ P Q P

P O P O=

−( )⋅ −( )−( ) ⋅ −( )

(25)

The graph of the error of the results provided by Di Ludovico et al.33 is shown in Fig. 4(b). It may be seen that the maximum error is less than 0.01.

The software OpenSees (Mazzoni et al.39) has been used to implement a fiber model for the evaluation of some points of the boundary of the strength domain (failure surface),

Fig. 2—Reinforced concrete square cross section considered by Di Ludovico et al.33

Fig. 3—Strength domain for reinforced concrete section shown in Fig. 2: (a) domain; and (b) planes of constant axial force.

Fig. 4—Cross sections of strength domain for constant axial force: (a) comparison with results by Di Ludovico et al.33; and (b) relative error.


much in the same way as was done by Di Ludovico et al.33 A specific procedure for the evaluation of the strength domain is not available in OpenSees. Users must therefore apply their skills to develop a method that may serve the purpose, taking advantage of the available procedures. The present authors have used the three following methods.

First method—Given the values of the axial force N and of one bending moment (for example, MX), the value of the moment MY that leads to the ultimate curvature kYu—that is, the one derived on the basis of the constitutive model—is obtained via a step-by-step procedure in which the required bending moment is increased gradually until the failure conditions are reached. It should be noticed, however, that the ultimate curvature kYu must be provided to terminate the procedure. The authors used the method proposed in the present work to provide the ultimate curvature. In this case, results appear to be fairly good for low and moderately large values of the compression axial force. A maximum error equal to 0.04 is found for an axial force ratio n = 0 at a moment ratio MY/MX equal to 0.2. Alternatively, the present procedure can be applied by prescribing the axial force N and the bending moment MY and by increasing the curvature kX up to the ultimate value kXu which, as before, has been evaluated by using the method proposed in this work.

The largest error is again equal to 0.04 for an axial force ratio n = 0, but this time it occurs for a bending moment ratio MX/MY ≅ 0.2. Although the error incurred by using the proce-dure described herein is generally rather small, it is clear that there can be instances when it can be noticeable. The results discussed previously were derived by using the OpenSees procedure MomentCurvature3D.tcl.

Second method—Rather than prescribing the axial force N and one bending moment as in the previous method, here, only the axial force N is provided. Then two incre-mental analyses are performed in sequence; in the first one, the largest between the two curvature components kX and kY is increased to the ultimate value max{kXu, kYu}, while in the second the smallest between the two components is increased to the ultimate value min{kXu, kYu}.

The results obtained by OpenSees tend to cluster around the axes of the graphs, leaving the central part uncovered, and are not shown for the sake of brevity.

This results in a relatively large error which, in some cases, can reach nearly 0.15 (or 15%) in the region where the two bending moments are almost equal.

One might now wonder why the two incremental analyses were performed in the sequence previously described. As a matter of fact, if this ordering had not been imposed, the succession would have simply been kX followed by kY. Very large errors, in the range of 10 to 40%, would have been obtained in most cases. Again, the graphs are not shown for space limitations.

The results referring to the second method described previ-ously were obtained by a simple modification of the example of the OpenSees procedure MomentCurvature3D.tcl.

Third method—In this method, the axial force N is prescribed just as in the two previous methods. The two components kXu and kYu of the ultimate curvature are assigned together with the prescribed value of the axial force N as a loading condition using the OpenSees Command “sp.”

It is important to realize that even in the present case, the method proposed in this work for the evaluation of the ultimate curvature is of paramount importance. The results obtained are shown in the graphs of Fig. 5(a) and in terms of relative error in the graphs of Fig. 5(b).

It is interesting to notice that in this case, the results are similar to those obtained by Di Ludovico et al.,33 and the rela-tive error is also of the same order of magnitude. A variant of the present method consists in assigning as a loading condi-tion the triplet {eG, kXu, kYu} corresponding to {N, kXu, kYu}. In this case, the error has been calculated starting from dimen-sionless expressions of the stress resultants (stress resultant divided by its ultimate value in the absence of the other two). Although the error is almost everywhere below 0.01, there are a few instances where it approaches 0.03. In conclusion, it appears that OpenSees can be advantageously used for the calculation of the strength domain, but the procedure for its calculation should be carefully selected.

From the analyses presented herein, it appears that the third method, where the triplet {N, kXu, kYu} is prescribed as a load condition, provides the best results, which are also comparable in terms of accuracy with those provided in the literature by using specifically designed procedures. However, the fact that any procedure activated within

Fig. 5—Cross sections of strength domain for constant axial force: (a) comparison with results obtained by using OpenSees software, given values of axial force N and of two curvatures; and (b) relative error.


OpenSees takes advantage of the method proposed herein should not be overlooked.

This application shows one of the potential uses of the present method which is to check that numerical algorithms for axial force and biaxial bending interaction provide accu-rate results.

Comparison of results for L cross sectionsA series of results concerning an L-shaped section is due to

Fafitis.28 The cross section considered is shown in Fig. 6 and consists of two unequal concrete flanges 21 in. (533.4 mm) thick and 135 in. (3429 mm) and 175 in. (4445 mm) long, respectively. The reinforcement, placed on the centerline, consists of 29#10@10”cc with a design yield stress fyd = 60,000 psi (413.69 MPa). The stated concrete strength is fcd = 9000 psi (62.05 MPa).

The comparison with the results obtained by the present method is given in Fig. 7(a) and the results provided by Fafitis28 are generally in excellent agreement. As may be seen from Fig. 7(a), the results are provided in the form of graphs MX-MY for three different values of the axial force: N = –9000 kips (–40,034 kN), N = –24,000 kips (–106,757 kN), and N = –44,000 kips (–195,722 kN). At the bottom of the curve corresponding to N = –9000 kips (–40,034 kN); a few of the points provided by Fafitis28 show a rather erratic behavior. This may indicate that the method used by Fafitis28 can occasionally have accuracy problems.

This is confirmed by the relative error curves shown in Fig. 7(b), where a maximum error of nearly 8% is found for an axial force of –9000 kips (–40,034 kN). However, for the remaining values of the axial force considered, the maximum error never exceeds 2%.

The comparison with the results obtained by OpenSees assigning the two limit curvature components and the corre-sponding limit axial deformation show good agreement, with the relative error never exceeding 0.02.

All of the examples given previously show that the present method is capable of producing the strength domain for any type of reinforced concrete cross section in an efficient way. The way the domain is constructed avoids all problems related to accuracy or convergence in the numerical solution of the nonlinear algebraic equations. The only issues that have to be addressed are the discretization of the boundary of the cross section and the choice of the number of points used for the representation of the boundary surface of the strength domain. The first is, in most cases, not a real issue because the reinforced concrete sections are generally of polygonal form while the second can be easily addressed by increasing the number of points as necessary.

It should be noted that the strength domains obtained by the present method can be replicated by using fiber models in standard computer programs such as OpenSees, provided that a suitable procedure is implemented that accurately simulates the method proposed herein. For instance, the best results with OpenSees are obtained if the limit defor-mations {eG, kXu, kYu} are evaluated by the present method and assigned by means of the “sp” command. Slightly better results were found in the authors’ comparisons by assigning the alternative triplet {N, kXu, kYu}. However, the first approach should be preferable because in a strength domain, the axial force N is a natural unknown.

Comparison of results for prestressed cross sectionsSeveral results concerning prestressed cross sections have

been presented recently by Marmo et al.40 Although all the cross sections considered by the quoted authors can be easily analyzed by the present method, for the sake of brevity, only the Y section shown in Fig. 8 will be discussed herein. The material properties, areas, and positions of the ordinary and prestressing reinforcement they used are not shown herein for the sake of brevity but may be found in Tables 4 through 6 on Fig. 6—L-section considered by Fafitis.28

Fig. 7—Cross sections of strength domain for reinforced concrete section of Fig. 6: (a) comparison with results by Fafitis28; and (b) relative error.


page 98 of their paper. Obviously, the strength domain for the considered cross section is a three-dimensional (3-D) surface (Fig. 9), but only the 12 cross sections at constant N given by the authors are shown here to simplify the comparison (Fig. 10). The figure is split into two parts to avoid exces-sive curve crossing and to maintain each curve as distinc-tive as possible. The continuous curves refer to the results obtained by the present method while the discrete dots are values provided by the quoted authors. The agreement can be considered good, albeit not perfect, but the limited number of points provided by Marmo et al.40 is indicative of the minor computational effort required by the present method.

STEEL SECTIONS ENCASED IN CONCRETEThe purpose of this section is twofold. On one hand, it

wants to show how the presented method can be used to deal with common technological situations such as steel sections encased in concrete or even tubular steel41-43 and FRP sections filled with concrete44; on the other hand, it wants to show how relevant the appropriateness of the constitutive law for concrete is on the obtained results. To this purpose, a cross section is used (Fig. 11), previously considered by Leon and Hajjar,42 to illustrate the application of the 2005 AISC Specification. However, in the applica-tion presented herein, the specifications of Eurocode 4 for composite steel and concrete structures are used for the material parameters instead of the AISC ones. Eight cross sections of the 3-D strength domain with constant axial force planes are shown with solid lines in Fig. 12. The constitutive law prescribed by Eurocode 47 has then been changed by replacing the rectangular part of the stress diagram (–afcd = –20.68 MPa [–3 ksi]) by a softening branch with ultimate strain ecu = 0.004 and ultimate stress scu = –bfcd = –12.7 MPa (–1.84 ksi). These parameters were evaluated using the Kent and Park37 law for unconfined concrete. The results are shown with broken lines in Fig. 12 and are significantly different from the previous ones, leading to the conclusion that the constitutive law should be consistent with the appli-cation considered. Furthermore, it appears that for tensile axial forces, no significant difference appears between the two constitutive laws if not in the case of bending around the minor axis (MX = 0), due to the fact that the contribution of the encased steel section is less effective. As the compres-sive axial force becomes larger and larger, the strength domain tends to shrink more and more. Most interesting is the fact that the strength reduction is more pronounced along the principal axes of bending and is minimal along lines of biaxial bending of nearly equal components.

Besides the Kent and Park37 law for confined concrete, another constitutive law has been formulated for concrete confined by FRP45,46 and applied to the derivation of strength domains.47 Such laws and similar ones can be handled effi-ciently by the present method.

STRENGTH REDUCTION FACTORS AND INSTABILITY EFFECTS

The strength domain constructed with the method described in the previous paragraphs is dependent on the geometry and the material properties of the cross section considered. Its use for design purposes requires modifications that account for different levels of uncertainty associated with failure modes (compression-controlled, intermediate, tensile-controlled) and for local and global geometrical effects related to the member and the structure to which the cross section belongs. Briefly, the different uncertainties related to the failure modes are accounted for, in codes such as ACI 318-084 (Chapter 9), by the so-called strength reduction factors, while the geomet-rical effects are considered via the moment magnifier design procedure (ACI 318-08,4 Chapter 10). This is a method that enables to account indirectly for geometric nonlinear effects by empirically increasing the bending moments obtained by a linear analysis. This is a difficult and controversial topic because such empirical amplifications could be avoided altogether if member forces were to be determined through a complete material and geometric nonlinear analysis. However, such analyses are usually expensive both in terms

Fig. 8—Y-section considered by Marmo et al.40

Fig. 9—Three-dimensional strength domain for reinforced concrete section of Fig. 8.


of human expertise and computational effort, making the empirical relations embedded in the codes the most effective and reliable alternative.

CONCLUSIONSAn exact method for the construction of the strength

domain of reinforced concrete and of prestressed concrete sections has been presented. The method is direct and does not require iterative procedures. Therefore, it does not exhibit accuracy and convergence problems often shown by methods attempting to solve nonlinear algebraic equa-tions. The strength domain can be used to check the safety conditions of the cross section, especially to see whether code requirements are satisfied. Therefore, it appears to be an extremely useful and simple tool to be used in design. The present method makes the construction extremely simple, quick, and accurate, therefore encouraging its application. If in the design process it is found that one or more loading conditions violate the code requirements, the section must be redesigned with a consequent change of the strength domain; this, however, can be reconstructed with little effort via the

present method and the optimal design may be achieved via a trial-and-error procedure.

The strength domain evaluated with the present method has been checked against several results available in the literature. In some cases, the agreement has been excellent; in other cases, the agreement has been generally good, but with some exceptions suggesting numerical problems in the methods used in the literature to derive those results. In a few other cases, the comparison has shown some significant disagreement, probably due to the preliminary nature of the results considered and/or the inappropriate setting of the accuracy and convergence parameters. The present method can be implemented in any software that uses fiber models for the discretization of the cross section. The method can be used efficiently to provide the deformation triplet, which corresponds to a point on the boundary of the strength domain in the stress resultants’ space. The evaluation of the corresponding triplet of strength resultants requires only the integration over the cross section of the stresses specified in terms of appropriate constitutive laws from the prescribed limit deformations. This has been applied in the present work by using OpenSees.

Fig. 10—Cross sections of strength domain for reinforced concrete section of Fig. 8 and comparison with results by Marmo et al.40

Fig. 11—Reinforced concrete square cross section with embedded W14 x 48 section.

Fig. 12—Cross sections of strength domain for reinforced concrete section of Fig. 11. Compar-ison between parabola-rectangle constitutive law and Kent and Park37 constitutive law.


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NOTES:


Title no. 110-S02


ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2010-333.R1 received January 4, 2012, and reviewed under Institute


Design Considerations for Shear Bolts in Punching Shear Retrofit of Reinforced Concrete Slabsby Maria Anna Polak and Wensheng Bu

This study addresses design aspects for the punching shear bolt retrofitting system for reinforced concrete flat slabs supported on columns. Shear bolts are an external type of reinforcement installed in existing slabs by first drilling small holes through the slab thick-ness, installing the bolts into them, and tightening the bolt nuts. Previous experimental research at the University of Waterloo showed the effectiveness of this system in increasing strength and ductility of existing slabs, with little changes in the slab’s appear-ance. The procedures for the design of the shear bolts and their installation in slabs are proposed in the paper. Recommendations are provided regarding the stem diameter, the head area and thick-ness, and the spacing and the layout of the bolts in a plane of a slab.

Keywords: design; punching shear; reinforced concrete; retrofit; shear bolts; slabs.

INTRODUCTIONPunching shear retrofit of reinforced concrete (RC) flat

slabs supported on columns is often a necessity due to inadequate strength and ductility of existing slab-column connections. Many of these slabs were built without any shear reinforcement and rely on concrete strength alone to carry punching shear stresses. Changes in a building’s use, construction of openings next to columns (for example, for ventilation ducts), errors in original designs, corrosion of reinforcement, and cracking of concrete are the primary reasons why the existing slabs can be deemed inadequate in terms of punching shear strength.

Flat concrete slab-on-column construction is common for buildings and parking garages. Such structural systems are easy to build, economical, and allow for the construc-tion of flat ceilings, simplified formwork, and reduced story height. The main design issue for these slabs is high trans-verse stresses concentrated at the slab-column connection that result in high diagonal tension stresses. If the slab is unreinforced for shear, this can lead to a sudden and brittle punching shear failure (Fig. 1 [Wood 2003]), and because the connection failing in this mode has very little post-failure rotational capacity and ductility, it creates a possibility of a catastrophic progressive collapse. The most efficient method for preventing punching shear failures is to provide properly designed transverse reinforcement; however, many of these slabs were constructed without it.

Several researchers recognized the need for the develop-ment of the punching shear strengthening method. Ghali et al. (1974) provided shear retrofit by introducing transverse prestressing to the slab area. High-strength prestressing steel bolts and large anchor plates were used in this application. Ebead and Marzouk (2002) used solid steel plates on both sides of the slabs, which were then bolted together through the slab and glued to the slab surfaces. Ruiz et al. (2010) used inclined post-installed anchors together with the adhesive for concrete. Sissakis and Sheikh (2007) strengthened the

slabs using carbon fiber-reinforced polymer (CFRP) loop stirrups and adhesives. Stark et al. (2005) and Lawler and Polak (2011) used fiber composites for seismic retrofit of slab-column connections. Stark et al. used CFRP stirrups inserted in a form of loops in drilled holes and anchored using an adhesive, and Lawler and Polak used glass fiber-reinforced polymer (GFRP) bolts anchored by crimping the bar ends with aluminum fittings.

Shear bolt reinforcement is another method for punching shear retrofit, which was developed and tested at the Univer-sity of Waterloo (El-Salakawy et al. 2003; Adetifa and Polak 2005; Bu and Polak 2009). It does not require prestressing, large anchor plates, and application of adhesives. A shear bolt is manufactured from normal-strength steel and consists of a bolt stem with a head at one end and a washer and nut at the other, threaded end (Fig. 2). Previous research at the University of Waterloo provides evidence of the effective-ness, and at the same time simplicity, of this retrofit method in both static and seismic loads scenarios. New design criteria for the shear bolts are discussed in this paper with appropriate recommendations for sizing of bolts, drilling of the holes, and placement of the bolts in the plan of a slab. The process of the design of the retrofit system consists of the following steps aimed at the determination of:

1. The required cross-sectional area of the stem to achieve the required punching shear strengthening.

2. The hole sizes to be drilled in the slab for bolt installation.

Fig. 1—Example of punching shear failure: PiperRow parking garage (Wood 2003).


Maria Anna Polak, FACI, is a Professor in the Department of Civil and Environ-mental Engineering, University of Waterloo, Waterloo, ON, Canada. She received her PhD from the University of Toronto, Toronto, ON, Canada. She is a member of ACI Committee 435, Deflection of Concrete Building Structures, and Joint ACI-ASCE Committee 445, Shear and Torsion. Her research interests includes experi-mental and numerical work related to shear and torsion in concrete members, nonlinear finite element analysis, and material modeling.

Wensheng Bu is a Structural Engineer at TRL and Associates Ltd., Calgary, AB, Canada. He received his PhD from the Department of Civil and Environmental Engineering at the University of Waterloo. His research interests include experimental testing, theoretical analysis, and seismic retrofit methods for reinforced concrete structures.

3. The spacing of the bolts in the plane of a slab (this step is linked with Step 1).

4. The number of peripheral rows of bolts to be installed around the column.

5. Patterns of the bolts in the plane of a slab.6. The area and the thickness of the head. Because a bolt

has a head at one end and a washer at the other end, the same requirements apply to the area and the thickness of the washer.

7. Construction procedures.This study includes all of the aforementioned steps. To

better explain the derivation process, however, the description of the design process is organized in the following sections:• Design of the shear bolt head (Step 6);• Design of the retrofit of a slab with shear bolts (Steps 1

through 5); and• Construction procedures (Step 7).

RESEARCH SIGNIFICANCEThis paper presents design procedures for retrofitting

slabs using shear bolts. Punching shear in flat concrete slabs can lead to dangerous and brittle failures, possibly causing progressive collapse of a structure. Many existing RC slabs do not have any shear reinforcing elements. For these slabs, a punching shear retrofit method is often the best solution for increasing their strength. Shear bolt retrofit has been exten-sively and successfully tested in the laboratory; however, to use the method in the field, the design recommendations are needed. The method is simple and yet allows preventing punching failures and increases in ductility in slabs that need retrofitting due to changes in building use, the need

of installing new services, deterioration due to corrosion of reinforcement, and construction or design errors.

TESTS ON SLABS STRENGTHENED WITH SHEAR BOLTS

The behavior of RC slabs strengthened with shear bolts has been experimentally verified by tests on isolated speci-mens representing both edge and interior RC slab-column connections, under static and static plus reversed cyclic horizontal loads (El-Salakawy et al. 2003; Adetifa and Polak 2005; Bu 2008; Bu and Polak 2009, 2011). Figure 3 shows the example of the test results for slabs with and without shear bolts, but are otherwise identical. These three slabs (SW4, SW5, and SW8) were subject to a constant gravity plus a cyclic, increasing lateral displacement-controlled load applied through the column stub. These were isolated slab-on-column specimens, simply supported along the outer edge perimeter and loaded through the column stubs extending above and below the slab’s surfaces (Bu and Polak 2009). It was concluded that the application of the shear bolts to a nonreinforced-for-shear slab enhances its strength and ductility. Slabs SW4 and SW9 had the same quantity and spacing of the shear bolts but with different bolt patterns in the plane of the slabs—SW4 orthogonal and SW9 radial. Their behaviors were very similar. Also, it should be added that the behavior of a retrofitted slab is similar to the behavior of an identical but properly reinforced-for-shear slab (Polak 2005). It should be made clear, however, that shear bolts were developed for retrofit and strengthening; new slabs should be properly designed and reinforced for shear during construction.

DESIGN OF SHEAR BOLTSThe design of a shear bolt starts with the appropriate sizing

of the bolt stem and its minor diameter (the diameter of the core inside the tread). The minor diameter of the bolt stem is determined based on a slab’s strength considerations, and this problem will be discussed later in the paper. The size of the thread and the nut must follow the rules for bolt thread design (for example, Shigley et al. [2004]).

Thickness of bolt headThe bolts described in this paper have a short length of

thread at one end of a round stem (Fig. 2 and 4). The rest

Fig. 2—Slab strengthened with shear bolts and a shear bolt.


of the bolt is a solid shaft attached at the other end to the head. The design of the bolt’s head is based on the radius of the solid part of the stem’s cross-sectional area (major radius/diameter of the bolt), which is attached to the head. Constant loading distributed over the cross section of the stem is acting on the head, which is assumed to be simply supported at the hole’s edge. The loading is assumed to be equal to the yield stress of the steel of the stem. This can be considered conservative because the yielding should occur first in the threaded part of the stem, which has the reduced (with respect to the full stem) cross-sectional area.

The design of the head involves the design of the head thickness and the head area. The required head thickness is

the largest for the location next to the stem. Depending on the manufacturing process of the bolt (heat forged, welded head, or using washers on both sides of the bolt), this thick-ness can remain constant throughout the head or decrease with the distance from the stem. The allowable decrease of the head thickness is also determined in the paper. The design is completed assuming that the cross-sectional areas of the bolt stem and the drilled hole are known.

It should also be noted that in the previous tests conducted at the University of Waterloo, the bolts were manufactured from typical shear studs with one end being a stud’s head and the other end was threaded and anchored by means of a large washer (larger and thicker than the calculations herein indicate) and a nut. The head area manufactured for a stud was too small for bolt application and thus the additional washer under the stud’s head was used to enlarge the contact area between the head and the concrete (Fig. 2). The sizes of both the washer at the threaded end and the washer under the head were based on strength calculations. In these first tests, however, the washers were larger than needed to avoid problems with failure of the bolts during testing. This was in keeping with the primary purpose of these tests, which was to observe slab behavior.

Flexural strength—The bolt and its head are shown in Fig. 4. The calculations to determine the bolt head thickness are done using the elastic thin plate theory to determine the flexural strength.

The bolt head and the bolt stem can be considered an axisymmetric elastic body with axisymmetric loading. For such an element, the internal bending moments Mr and Mq around axisymmetric axes r and q are shown in Fig. 5. The head is considered a circular plate of constant thickness t and supported on the edge of the concrete hole (Fig. 4). The flex-ural strength of the head is checked at the extreme (bottom or top) fibers of the plate, where sr and sq are principal stresses. Using the Von Mises Criterion, one acquires

2 2y r rq qs = s + s − s s (1)

where sy is the yield stress of the steel, and the other stresses are

Fig. 3—Moment versus lateral drift ratio of specimens tested by Bu and Polak (2011). All specimens were subject to constant gravity plus increasing reversed cyclic lateral displacements.

Fig. 4—Schematic diagram of shear bolt and loading on bolt head.


2 2 2( ) ( ) 6t tr

r rz z

Mt= =−s = − s = (2)

2 2 2( ) ( ) 6t tz z

Mt

qq q= =−s = − s = (3)

By substituting Eq. (2) and (3) into Eq. (1), the equation for plate thickness t is obtained

2 24 2

36 ( )r ry

t M M M Mq q= + −s

(4)

The round stem applies an evenly distributed circular load to the head (Fig. 4). The values of the internal moments in the head are calculated using the elastic thin plate theory (Pilkey 2005), assuming that the plate is simply supported on the

Fig. 5—Internal moments Mr and Mq for axisymmetric element under axisymmetric loading and boundary conditions.

Fig. 6—Normalized bolt head thickness versus normalized distance from bolt stem (for all stem diameters).

edge of the hole (at R) and loaded by a constant circular load (stress) of radius r0 and equal to sy. These are

( ) ( )

2 2 2

4 2 22

1 1(3 )(1 )16 16

1 3 1 4 1 ln

r y yM R R= s + n − a − s

− n × + n − b − a + + n b a a

(5)

( ) ( ) ( )

2 2 2

24 2 2 22

1 1(3 ) (1 3 )16 16

1 1 3 1 4 1 ln 2(1 ) 1

y yM R Rq = s + n − + n a − s

− n × + n + b − a + + n b a + − n − b a

(6)

where sy is the uniformly distributed circular load (stress)—in this case, equal to fyv, which is the yield stress of the steel of the bolt. R is the radius of the circular head plate, which is equal to the radius of the hole drilled in the concrete slab because the head is assumed to be simply supported at the edge. r0 is the radius of the loading area on the head, which is equal to the cross-sectional area of the bolt stem area attached to the head). r is the distance from the center of the stem, and a = r/R. b = r0/R is the ratio of stem radius to the drilled hole radius.

Figure 6 shows the graph in normalized coordinates. The horizontal axis represents a normalized distance from the bolt edge to the hole edge: x/R(1 – b), and the vertical axis is the ratio of head thickness over the bolt stem diameter: t/r0. The relation curves are drawn for different b values. Based on the flexural strength check, for the largest investigated drilled hole (b = 0.5), the maximum thickness at the stem for x = 0, should be 1.37r0. Also, based on the slope of the curve for b = 0.5 (Fig. 6), the required thickness decreases at a ratio of approximately 0.5 mm (0.02 in.) per 1 mm (0.0394 in.) distance from the stem.

Shear strength—The shear strength of the bolt head cross section is checked at the extreme location—at the stem. Using the von-Mises theory, the shear strength is

/ 3y yt = s (7)

For the rectangular cross section

32max

Qbt

t = (8)

where Q is the applied shear force; b is width of the cross section; and t is depth of the cross section. For the bolt, the applied force is equal to the stress from the stem times the stem’s cross-sectional area: Q = sy × pr0

2, and the width of the cross section is taken as the perimeter of the stem: b = 2pr0. Therefore, the depth of the cross section, t, corresponds to the required thickness of the bolt’s head, and is

0 03 3 1.34

t r r= = (9)

Based on the aforementioned calculations, the following recommendations for the bolt head thickness are proposed:


1. The minimum required head thickness at the stem should be t = 1.3r0, assuming that the diameter of drilling holes is usually 2 mm (0.08 in.) larger than bolt stem diameter, 0.6 ≤ b < 1. (If the hole diameter is much larger than the bolt stem, head thickness is controlled by the flexural strength).

2. The thickness can remain constant throughout the head (preferred in practice) or it can decrease with the distance from the stem at the ratio of 0.5 mm (0.02 in.) per 1 mm (0.0394 in.) distance. If the head thickness is decreasing, concrete underneath it should be shaped accordingly or a washer might have to be placed between the head and the concrete to accommodate the curved part and provide uniform stress on concrete.

These recommendations are based on the assumption that the head and the stem are made from the same strength steel: sy = fyv. In the case when the head or the washer is made from a steel material of strength fy and the stem is made from steel with the yield strength of fyv, the required thickness should be taken as

01.3 yv

y

ft r

f= (10)

Surface area of bolt headThe shear bolt strength relies on the proper bearing

strength of concrete under the head and the washer. The main consideration for the bolt head area is to check the bearing resistance of the concrete under the head. The prin-ciple in this design is that the bolt should yield without crushing of concrete under the head or washer. ACI 318-08 (ACI Committee 318 2008) specifies that the factored bearing resistance of the concrete can be taken as 0.85ffc′Ac, and when the supporting surface is wider than the loaded area, the resistance can be multiplied by a magnifying factor of up to 2, where f = 0.65 is the strength reduction factor, fc′ is the concrete compressive strength, and Ac is the loaded area, which is the area of the bolt head. Thus, the maximum nominal concrete resistance of 1.7 ffc′Ac = 1.1fc′Ac was used in the calculations. Assuming the bolt stem yields at failure and equating the yield load Fbolt and the bearing resistance Fn, Fbolt = Fn, the following equations can be obtained

20bolt yvF f r= p (11)

22 2 2 00 0 21.1 ( ) 1.1n c c

rF f R R f R

= p − = p −′ ′ b

(12)

The bolt head area Ac is pR02 and the stem section area

Ab is pr02. R0 and r0 are the radii of the head and bolt stem

cross section, respectively. The radius of the stem is the outer radius including the threads, which is a conservative assumption in these calculations. R is the hole radius, fyv is the yield strength of the bolt’s stem. b is equal to r0/R. From Eq. (11) and (12), the ratio of the bolt head area over the stem cross-sectional area is

202 2

0

11.1

yvc

b c

fA RA fr

= = +b′

(13)

In Equation (13), the ratio of the bolt head area over the bolt stem area is related to three parameters: the ratio b of bolt stem radius r0 over the radius of the hole, the yield strength of the bolts (fvy), and the concrete strength fc′. Figure 7 gives the ratio of bolt head area over bolt stem section area for different values of b and different concrete compressive strengths. The yield strength of bolts, fyv, was taken as 400 MPa (58,015 psi) in these calculations. For different yield strength of bolts, fy, the values from the graph should be added to a value of

f f fy yv c−( ) ′( )1 1. (14)

It can be seen that for most typical cases, the area of the head should be approximately 15 to 20 times larger than the cross-sectional area of the stem.

DESIGN OF SLAB RETROFIT USING SHEAR BOLTS

Bolt stem designThe minor (inner) stem diameter of the bolt must provide

adequate strength to the slab. The punching shear strength of the slab strengthened with shear bolts can be calculated using the code equations for headed shear reinforcements. The additional requirements specific for bolts are:

1. Consideration of the reduced cross-sectional area of the bolt stem due to threading.

2. Consideration of the reduced concrete transverse area contributing to shear-carrying capacity of the slab due to drilling of holes for bolt insertion.

The presented procedure is based on the ACI 318-08 recom-mendations. For other codes of practice, analogous proce-dures can be defined using the appropriate code design procedures for punching shear reinforcement plus the addi-tional requirements for bolts stated previously.

According to ACI 318-08, the slab’s punching shear strength vn is calculated as

v v vn c s= + (15)

Fig. 7—Ratio of head area and stem cross-sectional area for different concrete compressive strengths. Calculations made using bolt yield strength of 400 MPa (58,015 psi) and stem major diameter.


v vn uf ≥ (16)

where vn is the nominal shear strength of a slab; vc is the shear resistance from concrete; vs is the shear resistance from shear reinforcement; vu is factored shear stress at the critical perimeter; and f = 0.75 is the resistance factor in shear.

The punching shear strength contribution for the shear bolts is

0

v vys

A fv

b s′

= (17)

where b0 is the length of the critical perimeter (d/2 from the column perimeter); Av′ is the effective area of the bolt cross section called the tensile stress area, used to calculate bolt strength (Shigley et al. 2004); fvy is the yield strength of the bolts; and s is the radial spacing of the shear rein-forcement. In the case of bolts, which are normally not uniformly spaced due to drilling restrictions imposed by flexural reinforcements, s is an average spacing of the bolts. Av′ can be taken conservatively as 0.7Av, where Av is the sum of cross-sectional areas of all shear bolts (including threads) in each peripheral line parallel to the column faces.

Shear bolt retrofit requires drilling small holes in concen-tric rows around the column. Because these holes would have some influence on the concrete shear-carrying capacity, their effect should be accounted for in the design. It is proposed that to account for the reduction in the concrete critical section area due to the drilled holes, the critical perimeter length should be reduced by the length equal to the number of holes in the concentric rows times the diameter of the holes. This results in the following effective critical section perimeter length b0′

′ = − ×( )b b n dh0 0 (18)

where n is the number of holes drilled along critical section perimeter; and dh is the diameter of the drilled hole (for bolt installation).

Equation (18) can then be used to calculate the concrete contribution to carrying punching shear. According to ACI 421.1R-99 (Joint ACI-ASCE Committee 421), for slabs strengthened with headed stud reinforcement, concrete contribution to carrying shear is vc = 0.25√fc′. Therefore, the effective concrete contribution for punching shear for the slabs with the shear bolts can be taken as

0

0

0.25c c

bv f

b′

= ×′ (19)

Bolt spacing in radial directionThe definition of radial and tangential spacing of the

bolts is shown in Fig. 8. Radial spacing of the shear bolts, S0 and S1, should be treated as maximum allowable spacings because flexural reinforcement location can enforce smaller spacings (drilling cannot be done through the flexural rein-forcement). S0 and S1 must ensure that the bolts cross the punching shear cracks inside the slab. For the slabs with shear reinforcement, the shear cracks in the zone with the shear reinforcement have a steeper inclined angle (q1) than that of cracks in the non-shear-reinforced zones (q2). Dilger and Ghali (1981) found that in the concrete slab with headed shear studs, angle q1 can be approximately 40 to 50 degrees, whereas angle q2 is usually around 20 to 30 degrees.

The tests on slabs in this research showed similar crack angles as those previously mentioned (Bu 2008). The exis-tence of small drilled holes around the column perimeter did not visibly affect the crack angles. The tested slabs with openings next to the columns allowed observation of the crack angles on the opening edges (Fig. 9). The tested slabs were subjected to a constant gravity and increasing lateral cyclic loads, and the openings were located along the line of application of lateral loads (Bu and Polak 2011). For the slab without the shear reinforcement (SW6), the shear crack was formed at an angle of about 30 degrees (Fig. 9(a)). For Slab SW8 (with shear reinforcement in the radial pattern), the shear crack formed at approximately 45 degrees and for Slab SW7 (Fig. 9(b)) (with shear reinforcement in the orthogonal pattern), the crack formed at an angle of approximately 40 degrees to the horizontal (Fig. 9(c)). All shear cracks started from the column faces. Because the distance between the first shear bolt and the column face, S0, should be within the inclined crack, and assuming the bolt crosses the crack in the middle, the following can be calcu-lated for the two extreme values of the crack angle (taken as 40 and 50 degrees)

For q1 = 40°, S0 =0.5d/(tan40°) = 0.59d (20a)

For q2 = 50°, S0 =0.5d/(tan50°) = 0.42d (20b)

Based on the aforementioned calculations, the first bolt should be placed at S0 = 0.45d. For very thin slabs of 120 to 150 mm (5 to 6 in.), this distance can be slightly larger but no more than S0 = 0.55d. This special consideration for thin slabs accounts for the fact that drilling of the holes requires a certain distance from the column to allow space for the drill. Considering that the bolt head has a confining effect on the concrete underneath it, the increased limit on S0 should provide a safe design.

Fig. 8—Spacing S0, S1, and S2 of shear bolts. (Note: S0 and S1 are radial direction spacing; S2 is tangential direc-tion spacing.)


Figure 10 shows the section of the concrete slab strength-ened with the shear bolts. The crack angle is q1. The crack is assumed going across from the bolt edge at the bottom of the slab to the top flexural reinforcing bar on the adja-cent bolt outer edge. The crack angle q1 is assumed to be between 40 and 50 degrees. With the maximum spacing of bolts assumed as in Fig. 10 with at least one bolt crossing the crack inside the reinforcement layers

tanq1 = d x (21)

where d is the effective depth of the flexural reinforcement; and x is the spacing between the two adjacent bolt outer stem edges

x S r= +1 02 (22)

where 2r0 is the major diameter (including thread) of the shear bolts, and S1 is the spacing of the bolts

0 11

1

2 tantan

d rS

− q=

q(23)

Table 1 shows the calculations performed to determine the S1/d for slabs of thickness 120 to 500 mm (5 to 20 in.). Two different crack angles are assumed: 40 degrees for normal loads, and 50 degrees for heavy loads. Calculations are done assuming certain logical values for concrete cover, flex-ural reinforcement diameter, and shear bolt diameter. The maximum calculated ratios S1/d are shown. Considering that steeper cracks are likely to form under heavy vertical loads, it is recommended that the maximum shear bolt radial spacing S1 be

1. For normal loads: vu ≤ 0.5f√fc′

for slab thickness of 250 mm (10 in.) or less:

S d1 0 9= . (24a)

Fig. 9—Shear cracks in opening edges of: (a) Slab SW6 without shear bolts; (b) Slab SW8 with shear bolts of radial layout; and (c) Slab SW7 in orthogonal patterns (Bu 2008).

Table 1—Bolt radial spacing calculations

Slab depth h, mm

Assumed Normal loads, q = 40° Heavy loads, q = 50°

Bolt diameter 2r, mm

Flexural bar diameter df,

mmCover,

mm

Effective depth d,

mm

Bolt spacing versus effective depth of slabs S1/d

Confining effect of head D/d (Fig. 11)

Bolt spacing versus effective depth of slabs S1/d

Confining effect of head D/d (Fig. 11)

Calculated (Fig. 10, Eq. (23))

For design

Calculated (Fig. 10, Eq. (23)) For design

120 9.5 11.5 20 91 1.06 0.90 0.84 0.71 0.70 1.12

150 9.5 11.5 30 111 1.09 0.90 0.79 0.73 0.70 1.06

200 12.5 16 30 158 1.09 0.90 0.68 0.74 0.70 0.97

250 12.5 16 30 208 1.11 0.90 0.54 0.76 0.70 0.91

300 12.5 16 30 258 1.13 0.70 0.65 0.78 0.50 0.91

350 12.5 20 30 308 1.13 0.70 0.60 0.77 0.50 0.89

400 12.5 20 30 358 1.14 0.70 0.56 0.78 0.50 0.84

500 15 25 50 435 1.13 0.70 0.58 0.78 0.50 0.87

Note: 1 mm = 0.0394 in.


for slab thickness greater than 250 mm (10 in.):

S d1 0 7= .(24b)

2. For heavy loads: vu > 0.5f√fc′

for slab thickness of 250 mm (10 in.) or less:

S d1 0 7= .(25a)

for slab thickness greater than 250 mm (10 in.):

S d1 0 5= . (25b)

The normal and heavy loads are defined through the calcu-lation of vu, which is the factored applied shear stress at the critical perimeter.

The above calculations do not include any consideration for the confining effect of the bolt’s head. To account for this effect, the additional check was completed assuming that compressive stresses under the head follow a 45-degree angle (Bu 2008). Figure 11 shows the assumed compressive stress distribution in the slab concrete due to the shear bolt heads. The overlap of the compression zones created by the heads is calculated as

D = − −( )h S R1 02 (26)

These check calculations were done assuming bolt spacing recommended in this section (Eq. (24) and (25)) and the head area 15 times larger than the stem cross-sectional area (therefore, R0 = 3.9r0). In all cases, the ratio of D/d is larger than 0.5 (Table 1), which means that the heads provide bene-ficial compressive stresses over the large portion of flexural tensile zone in the slab.

Bolt layout and spacing in tangential directionDifferent codes of practice specify different layouts (for

example, orthogonal, radial) of shear reinforcing elements in the plan of a slab. When shear bolts are installed, inter-ference from the longitudinal reinforcing elements often prevents installing bolts in a very regular pattern; therefore, more general rules are needed for guidelines on how the bolts should be placed around the column. In the experi-mental research at the University of Waterloo (Bu and Polak 2011), specimens were tested with the same number and spacing of shear bolts but with either orthogonal or radial patterns (Fig. 12). The slabs with the orthogonal and radial patterns of shear bolts had very similar capacities and ductil-ities (Fig. 3). Therefore, a practical rule for placing the shear bolts should be to maintain the rotational symmetry of the installed bolts around the column. With this in mind, the following rules for the placement of the bolt in a tangential direction are proposed:

1. The bolt tangential spacing S2 in the first row from the column should be between d and 2d (ACI 318-08).

2. The shear bolt installation requires drilling holes around the flexural reinforcement mats. Therefore, due to the inter-ference from the existing reinforcements, it can be diffi-cult (if not impossible) to achieve perfectly orthogonal or radial shear bolt patterns. The bolts should be placed in the following manner:

a. Follow as closely as possible the chosen regular pattern (orthogonal or radial);

b. Place the same number of bolts in each of the quadrants of the slab defined by:

i. the x-y axes of the slab system; andii. the x′-y′ axes of the slab system (Fig. 8).

c. Place the bolts uniformly distributed in each of the x-y and x′-y′ quadrants; and

d. The outermost row of shear bolts shall be placed at a distance not greater than 0.5d within the perimeter at which no shear bolts are required. At least four peripheral rows of bolts should be installed. This is based on the previous testing done in Waterloo; only the first two rows of bolts experience any substantial straining, and specimens with four rows of bolts shows very good ductility (Bu and Polak 2009; Adetifa and Polak 2005). Based on ACI 318-08, the perimeter where no shear bolts are required is located where the factored applied shear stress vu is less than

0.17u c cv v f≤ f = f × ′ (27)

where f = 0.75; and fc′ is concrete compressive strength.

CONSTRUCTION PROCEDURESThe retrofit procedure starts by defining the location of

the holes to be drilled. To avoid drilling through the slab flexural reinforcing bar, it is important to determine the loca-tion of the longitudinal reinforcing bar from the construction drawing and using nondestructive testing techniques. If the

Fig. 10—Crack angle q1 in slab strengthened with shear bolts.

Fig. 11—Assumed pressure in slab concrete by shear bolt heads used to calculate ratio of S1/d; d is effective depth of slab (not shown for clarity).


drilling results in hitting the reinforcing bar, drilling should be stopped, the created hole patched, and the neighboring location used instead.

Before drilling, it is important to check the capacity of the slabs with the holes. All live and removable dead load should be removed and only a few holes should be drilled at a time so as not to excessively weaken the slabs. The number of holes that can be drilled at one time must be checked by appropriate calculations for each slab-column connection. The nominal strength of the slab with drilled holes should be taken as

00.33n c

bv f

b′

= ×′ (28)

where b0′ is the length of the critical perimeter considering drilled holes. The bolts can be installed in the holes and tightened to ensure a close fit between the head, washer, and the slab surfaces. In this research, the bolts were tightened using a torque wrench, imposing strains of approximately 10% of the yield strength of the bolt steel. It should be noted that this bolt small strain can be assumed to have had no effect on the slab response.

Fire and corrosion protection of bolts need further inves-tigation; however, some recommendations can be offered. Surface coatings can be applied to the bolts and the washers before installation to slow down their corrosion. It should be noted that, in the case of excessive corrosion, the bolts can be easily replaced by the new bolts installed in the same holes. Fire protection should follow appropriate specifica-tions for the building and the fire protection methods for the steel elements.

DESIGN EXAMPLEA flat slab of 200 mm (7.8 in.) depth is supported on

columns that are 400 x 400 mm (15.7 x 15.7 in.) in cross section. The load transmitted by the slab to the column has been increased to 480 kN (107,908 lb) due to a change in building use. Check the slab and, if needed, retrofit the internal slab column connection for punching shear. The concrete compressive strength is 25 MPa (3626 psi), the longitudinal reinforcement consists of 15M (16 mm [0.63 in.] diameter) bars, and the concrete cover is 25 mm (1 in.).

1. Check the strength of the slab without shear bolts.

Use effective depth: d = 200 – 25 – 16 = 159 mm (6.26 in.)

Critical perimeter length b0 = (400 + 159) × 4 = 2236 mm (88.03 in.)

a. Factored applied stress:

3

0

480 10 1.35 MPa (195 psi)2236 159u

Pvb d

×= = =

× ×

b. Nominal strength of slabs without shear reinforcement:

0.33 1.65 MPa (239 psi)n cv f= =′

c. fvn = 0.75 × 1.65 = 1.24 MPa (179.8 psi) < 1.34 MPa (194.4 psi) = vu; therefore, punching shear strength is inadequate.

2. Design strengthening using shear bolts (Steps 1, 2, and 3).Use 12.5 mm (1/2 in.) diameter bolts, 13.5 mm (0.53 in.)

diameter drilled holes, bolt spacing in radial direction equal to the effective depth 0.9d, cross-sectional area of the bolt As = 122.7 mm2 (0.19 in.2), the effective bolt area reduc-tion factor of 0.7, eight shear bolts in the concentric row around the column, yield strength of the bolts fy = 370 MPa (53,664 psi), and critical perimeter length b0 = (400 + 159) × 4 = 2236 mm (88.03 in.).

a. Steel contribution:

0

122.7 8 0.7 370 0.79 MPa (115 psi)2236 159 0.9

v vys

A fv

b s′ × × ×

= = =× ×

Effective length of the critical perimeter considering the drilled holes, b0′ = 2236 – 8 × 13.5 mm (83.78 in.)

b. Concrete contribution:

Fig. 12—Orthogonal and radial placement of bolts in slabs tested by Bu (2008). Slabs were tested under constant gravity load plus reversed cyclic lateral displacements. Final crack pattern on compression face (under gravity load) of Speci-mens SW4 and SW9.


0

0

21280.25 0.25 25 1.19 MPa (173 psi)2236c c

bv f

b′

= × = × =′

c. Nominal shear strength of the strengthened slab:

vn = f(vc + vs) = 0.75(1.19 + 0.79) = 1.49 MPa (216 psi) > vu = 1.35 MPa (196 psi)

3. Bolt spacing in radial direction (Step 3):In the strength calculations, the maximum spacing of 0.9d

was assumed. For practical reasons (drilling around flexural bars), the bolts spacing should be maintained between 0.7d = 111 mm (0.43 in.) and 0.9d = 143 mm (5.63 in.).

4. Number of peripheral rows (Step 4 and 5).a. Assuming four rows of orthogonally placed bolts with

an average radial spacing of 0.8d and the first row at 0.5d from the columns, the distance of the last bolt from the columns: L = (0.5 + 3 × 0.8)d = 461 mm (18.15 in.)

b. Strength should be checked at 0.5d from the last bolts

L + 0.5d = 461 + 159/2 = 540.5 mm (21.28 in.)

The critical perimeter is (orthogonal pattern of bolts):

20 (400 2 540.5 ) 4 4658 mm (183 in.)b = + × × =

3

0

480 10 0.65 MPa (94 psi) 4658 159

1.24 MPa (180 psi)

u

n

Pvb d

v

×= = =

× ×< f =

four rows of bolts is adequate5. Design the bolt head and the washer (Step 6)

12.5 0.913.5

rR

b = = =

a. Head thickness at the bolt stem: t = 1.3 × (12.5/2) = 8 mm (0.31 in.).

The thickness can decrease with the distance from the stem at the rate of 0.5 mm (0.02 in.) thickness per 1 mm (0.0394 in.) distance. Assuming the same or higher strength of the steel is used to produce the washer, use 8 mm (0.315 in.) thick washers.

b. Head area: for b = 0.9 and 25 MPa (3626 psi) concrete, the head area/stem cross sectional ratio is 17 (Fig. 7). The steel yield strength is 370 MPa (53,664 psi) and Fig. 7 was created for 400 MPa (58,015 psi). There, using Eq. (14), the adjusted ratio is

17 + (370 – 400)/1.1 × 25 = 17 – 1 = 16

Therefore

22 16 12.5 = 50 mm (2 in.)R = ×

The diameter of the head and the washer should be at least 50 mm (2 in.).

6. Construction procedure (Step 7).a. After removing all live load, the remaining factored

dead load on the slab is 200 kN (44,961 lb), resulting in the factored applied stress

3

0

200 10 = 0.56 MPa (81 psi)2236 159u

Pvb d

×= =

× ×

b. Nominal strength of slabs without shear reinforcement and with eight drilled holes around the critical perimeter

00.33 = 1.57 MPa (228 psi)n c

bv f

b′

= ×′

fvn = 0.75 × 1.57 = 1.18 MPa (171 psi) > 0.56 MPa (81 psi) = vu

Punching shear strength of the slab without live load and with all eight holes drilled around the critical perimeter is adequate. Drilling of the holes should be done in a construc-tion sequence that minimizes the existence of open drilled holes without the bolts. For example, the holes from one row can be drilled at the same time but before drilling the subse-quent rows, the bolts should be installed first.

SUMMARY AND CONCLUSIONSThe design recommendations for the punching shear

retrofit of flat RC slabs using shear bolts are presented in this paper. The shear bolts allow an easy solution for punching shear strengthening of slabs without changing the slabs’ aesthetic appearance. Extensive laboratory testing showed that slabs retrofitted with shear bolts have higher punching shear strength and ductility than the slabs without the shear reinforcement. The presented methodology for design includes design of the bolts and their heads and design of slabs retrofitted with the shear bolts. Recommendations for construction procedures are given. The design example that follows the steps described in the paper is presented.

ACKNOWLEDGMENTSThis research was funded by a grant from the Natural Sciences and Engi-

neering Council (NSERC) of Canada. The shear bolts were manufactured and donated by Decon Inc. Canada. The ready mixed concrete was donated by Hogg Fuel and Supply Ltd. Ready Mix Division in Kitchener, ON, Canada. The authors wish to express their gratitude for the received support.

REFERENCESACI Committee 318, 2008, “Building Code Requirements for Structural

Concrete (ACI 318-08) and Commentary,” American Concrete Institute, Farminton Hills, MI, 473 pp.

Adetifa, B., and Polak, M. A., 2005, “Retrofit of Interior Slab-Column Connections For Punching Using Shear Bolts,” ACI Structural Journal, V. 102, No. 2, Mar.-Apr., pp. 268-274.

Bu, W., 2008, “Punching Shear Retrofit using Shear Bolts for Reinforced Concrete Slabs under Seismic Loads,” PhD thesis, University of Waterloo, Waterloo, ON, Canada.

Bu, W., and Polak, M. A., 2009, “Seismic Retrofit of RC Slab-Column Connections using Shear Bolts,” ACI Structural Journal, V. 106, No. 4, July-Aug., pp. 514-522.

Bu, W., and Polak, M. A., 2011, “Effect of Openings and Shear Bolt Pattern in Seismic Retrofit of Reinforced Concrete Slab-Column Connec-tions,” Engineering Structures, V. 33, No. 12, pp. 3329-3340.

Dilger, W. H., and Ghali, A., 1981, “Shear Reinforcement for Concrete Slabs,” Journal of the Structural Division, ASCE, V. 107, No. 12, pp. 2403-2420.


Ebead, U., and Marzouk, H., 2002, “Strengthening of Two-Way Slabs Using Steel Plates,” ACI Structural Journal, V. 99, No. 1, Jan.-Feb., pp. 23-31.

El-Salakawy, E.; Polak, M. A.; and Soudki, K., 2003, “New Shear Strengthening Technique for Concrete Slabs,” ACI Structural Journal, V. 100, No. 3, May-June, pp. 297-304.

Ghali, A.; Sargious, M. A.; and Huizer, A., 1974, “Vertical Prestressing of Flat Plates Around Columns,” Shear in Reinforced Concrete, SP-42, V. 2, American Concrete Institute, Farmington Hills, MI, pp. 905-920.

Joint ACI-ASCE Committee 421, 1999, “Shear Reinforcement for Slabs (ACI 421.1R-99),” American Concrete Institute, Farmington Hills, MI, 15 pp.

Lawler, N., and Polak, M. A., 2011, “Development of FRP Shear Bolts for Punching Shear Retrofit of Reinforced Concrete Slabs,” Journal of Composites in Construction, ASCE, V. 15, No. 4, pp. 591-601.

Pilkey, W. D., 2005, Formulas for Stress, Strain, and Structural Matrices, second edition, John Wiley & Sons, Inc., Hoboken, NJ, 1536 pp.

Polak, M. A., 2005, “Shell Finite Element Analysis of Reinforced Concrete Plates Supported on Columns,” Engineering Computations:

International Journal of Computer Aided Engineering and Software, V. 22, No. 3/4, pp. 409-429.

Ruiz, M. F.; Muttoni, A.; and Kunz, J., 2010, “Strengthening of Flat Slabs Against Punching Shear Using Post-Installed Shear Reinforcement,” ACI Structural Journal, V. 107, No. 4, July-Aug., pp. 434-442.

Shigley, J. E.; Mischke, C. R.; and Budynas, R. G., 2004, Mechanical Engineering Design, seventh edition, McGraw-Hill, pp. 396-448.

Sissakis, K., and Sheikh, S., 2007, “Strengthening Concrete Slabs for Punching Shear with Carbon Fiber-Reinforced Polymer Laminates,” ACI Structural Journal, V. 104, No. 1, Jan.-Feb., pp. 49-59.

Stark, A.; Binici, B.; and Bayrak, O., 2005, “Seismic Upgrade of Reinforced Concrete Slab-Column Connections Using Carbon Fiber-Reinforced Polymers,” ACI Structural Journal, V. 102, No. 2, Mar.-Apr., pp. 324-333.

Wood, J. G. M., 2003, “Pipers Row Car Park, Wolverhampton, Quantita-tive Study of the Causes of the Partial Collapse on March 20, 1997,” Health and Safety Executive Report, UK.

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ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2011-028 received January 27, 2011, and reviewed under Institute


Shortening Estimation for Post-Tensioned Concrete Floors—Part I: Model Selectionby Guohui Guo and Leonard M. Joseph

For post-tensioned (PT) concrete floors, shortening occurs due to elastic compression, concrete creep, concrete shrinkage, and temperature change. The longer the continuous concrete slab, the greater the cumulative shortening effect. More commonly, some restraints to shortening exist, such as shear walls and stocky columns. At best, restraint to shortening (RTS) can induce cracks and spalls that affect aesthetics and durability. At worst, it can affect structural safety. Therefore, appropriate models and proce-dures that can predict shortening are quite necessary. In the first part of this study, five concrete creep and shrinkage models are compared for both ultimate values and relative shortening rates. The B3 model is considered to be the most appropriate model. Adjustments to the B3 model are introduced by using equivalent concrete age to account for high early concrete strength frequently used in PT construction. Part II of this study covers detailed proce-dures to estimate PT concrete floor shortening.

Keywords: creep; equivalent concrete age; floor shortening; post-tensioned concrete; pour strip; relative humidity; shrinkage; volume-to-surface ratio.

INTRODUCTIONShrinkage shortens concrete floors in both post-tensioned

(PT) and non-prestressed reinforced concrete (RC) struc-tures. However, PT and RC structures respond to floor shortening in different ways.1-3 As shrinkage takes place in an RC structure with typical quantities of bonded mild reinforcement, numerous, well-distributed, closely spaced narrow cracks typically occur. As a result, concrete floors may become flexurally “softer” but exhibit minimal hori-zontal movement. In contrast, PT structures with unbonded tendons have much less bonded reinforcement, and concrete floor shortening usually results in fewer, widely spaced cracks. Once tendons are stressed, concrete precompres-sion tends to close most cracks, directly causing PT slabs to shorten overall. In addition, concrete elastic and creep effects typically associated only with PT structures shorten concrete floors even more. Due to the nature of PT slabs and the amount of shortening that would exist if allowed to occur freely, restraint to shortening (RTS) can be a major source of cracking and distress for both structural and nonstructural elements of PT structures4 if not adequately addressed during design and construction. Understanding the magnitudes and development rates of shortening behaviors is essential to anticipate and design for RTS conditions.

It is very important for engineers to understand the effect of floor shortening on the various components of the structure and appropriately account for it in the design.4,5 RTS prob-lems have been recognized for many years, and many mitiga-tion measures were studied and discussed for PT buildings6; however, there are currently no generally accepted proce-dures to estimate PT floor shortening. It has been recom-mended that to assess the movement of slabs at expansion or contraction joints from the time of placing concrete, 650 me

should be considered normal.7 The drying effect of air condi-tioning can increase this to 1000 me. This recommendation is only empirical and should be adjusted for specific situations. The estimation of concrete floor shortening should take multiple factors into consideration, including concrete prop-erties, slab volume-to-surface ratio (v/s), relative humidity, slab average compressive stress from tendons, concrete age when drying begins, and concrete age when tendons are stressed. Once the estimated floor shortening is calculated, structural elements can be designed and detailed to absorb the resulting movements or resist the forces that result from restraints to movement.

In this paper, five models that are frequently used to predict concrete creep and shrinkage are compared for both ultimate shortening values and relative shortening rates. Detailed procedures are then outlined to estimate concrete floor shortening for PT structures with and without the use of pour strips. The purpose of this two-part study focuses on shortening estimation for a concrete floor without restraints. Thus, shortening effects on structures with restraints and the design of structures considering these effects are beyond the scope of this study.

RESEARCH SIGNIFICANCERTS problems frequently occur in PT structures as a result

of concrete floor shortening from several sources. This paper compares five models that are frequently used to predict concrete creep and shrinkage based on both ultimate short-ening values and relative shortening rates. The B3 model8-10 is determined to be the most appropriate model for shortening estimation. High-early-strength concrete commonly used in the PT industry is appropriately considered through the use of equivalent concrete age when drying begins and tendons are stressed.

CONCRETE CREEP AND SHRINKAGE MODELSThere are many factors affecting the prediction of concrete

floor shortening, such as concrete properties, weather, age at loading, concrete curing method and period, and slab v/s. A model would be excellent if it could predict shortening within 15% of the test results and adequate if the prediction results are within 20% of the test results.11 According to the authors’ experience, error ranges from 30 to 50% are quite normal. Many concrete creep and shrinkage models have been developed to date. This study reviews several commonly


Guohui Guo is a Senior Project Engineer with Haris Engineering, Inc., Kansas City, KS. He received his PhD of science in structural engineering from the University of Kansas, Lawrence, KS. His research interests include the analysis and design of a wide range of structures, including high-rise buildings; bridges; and sports, health-care, parking, post-tensioned, and long-span structures.

Leonard M. Joseph is a Principal with Thornton Tomasetti, Inc., Irvine, CA. He received his BS in civil engineering from Cornell University, Ithaca, NY, and his MS from Stanford University, Stanford, CA. His research interests include analysis, design, and investigation of projects throughout the world, ranging from the Petronas Towers in Malaysia to parking decks throughout the United States.

used models, including B3,8-10 CEB-FIP,12 ACI 209,13,14 and GL2000.15 In addition, creep and shrinkage data from the PCI Handbook16 based on prestressed concrete members are included for comparison.

The required parameters for each of the prediction models14 are tabulated in Table 1(a). For Models B3, CEB-FIP, ACI 209, and GL2000, concrete creep and shrinkage at different ages can be easily obtained using prediction functions. For concrete creep and shrinkage data published in the PCI Handbook,16 no prediction functions are available and correction factors are provided to consider the effects of concrete strength, average concrete compres-sive stress from tendons, relative humidity, and slab v/s.

The five models are compared using the concrete proper-ties and other parameters shown in Table 1(b). A concrete strength of 5000 psi (34.5 MPa), based on standard cylinder tests at 28 days, is typically used in PT concrete structures, with a slab average compressive stress of 200 psi (1.4 MPa). Currently, concrete mixtures exhibiting high early concrete strength are commonly used for PT buildings, especially for high-rise buildings, to permit earlier tendon stressing and shorten the construction schedule. It is not uncommon today for tendons to be stressed as soon as 1 day after concrete placement. In this study, it is assumed that concrete will be wet-cured for 3 days, with tendons being stressed at the

end of concrete curing. While values vary, those shown in Table 1(b) are representative of typical PT structures.

It is well-known that concrete member v/s and rela-tive humidity have significant effects on both creep and shrinkage. In this study, slabs with thicknesses varying from 5 to 12 in. (127 to 305 mm) and ambient relative humidity values varying from 40 to 90% are considered to investigate their effects on PT concrete floor shortening. The v/s for slabs with different thicknesses are tabulated in Table 2, based on a slab with plan dimensions of 100 x 300 ft (30.5 x 91.5 m).

Compared with concrete elastic shortening, both concrete creep and shrinkage are slow, long-term processes. For regular concrete structural elements of typical proportions under typical conditions, designers often estimate that more than 90% of combined concrete creep and shrinkage takes place within the first 15 years of use. It is common in the industry to declare the values predicted by each model at a concrete age of 75 years as “ultimate” values even if they have not approached the predicted asymptote. The term “ultimate” will be used the same way in this paper for all figures, comparisons, and descriptions. The ultimate creep and shrinkage values based on the PCI Handbook are final values, as shown in Table 3.12.3 of that publication.16 This is because the PCI Handbook16 provides final concrete creep and shrinkage values, along with correction factions to consider the effects of concrete strength, v/s, average prestress, and relative humidity. For the other four models, both concrete creep and shrinkage predictions are presented as a function of time.

CONCRETE CREEP AND SHRINKAGE COMPARISONS

For PT concrete structures, slabs with a thickness of 5 to 10 in. (127 to 254 mm) are widely used in both one- and two-way floor systems. Occasionally, thicker slabs are

Table 1(a)—Parameters included in different prediction models

Parameters B3 Model CEB-FIP GL2000 ACI 209 PCI

Age of concrete when drying begins √ √ √ √ —

Age of concrete at tendon stressing √ √ √ √ —

Aggregate content √ — — — —

Cement content √ — — √ —

Water content √ — — — —

Cement type √ √ — √ —

Concrete mean strength at 28 days √ √ √ — √

Concrete mean strength at loading — — √ — —

Modulus of elasticity at 28 days √ — √ — —

Modulus of elasticity at loading — — √ — —

Curing condition √ — — √ —

Relative humidity √ √ √ √ √

Shape of specimen √ — — — —

v/s √ √ √ √ √

Concrete slump — — — √ —

Fine aggregate percentage — — — √ —

Air content — — — √ —


used to carry heavier live loads, span longer distances, or transfer offset column loads; therefore, slabs with thick-nesses from 5 to 12 in. (127 to 305 mm) are considered in this paper to study both the ultimate values and relative rates of creep and shrinkage development.

Effects of v /s on ultimate shorteningIn PT structures, it is common practice to use pour strips or

closure strips to reduce RTS problems and achieve long floor dimension without the use of expansion joints. Pour strips are left open for a certain period of time to allow for the free move-ment of the two sections adjacent to a pour strip. Specified pour strip open periods are typically between 30 and 90 days, although there are cases of pour strips being filled much later after the structure is complete. To estimate concrete floor shortening, both ultimate values and relative rates of creep and shrinkage are required.

Predicted ultimate shrinkage and creep strains using the five models previously described are shown in Fig. 1 and 2, respectively. The figures are obtained using the param-eters shown in Table 1(b), but with relative humidity fixed at 50%. As shown in Fig. 1, Models B3, GL2000, and CEB-FIP predict nearly the same ultimate shrinkage value regardless of v/s. The ultimate shrinkage for a 12 in. (305 mm) slab is approximately 96% of that for a 5 in. (127 mm) slab for the GL200 and B3 models, and 90% of that for a 5 in. (127 mm) slab for the CEB-FIP model. For Models PCI and ACI 209, the predicted ultimate shrinkage for a 12 in. (305 mm) slab is less than 70% of that for a 5 in. (127 mm) slab. It is gener-ally expected that shrinkage for members with different v/s should be the same once the entire member thickness has reached equilibrium with ambient relative humidity—their “truly ultimate shrinkage” value. Members with a higher v/s should exhibit a much lower shrinkage rate, taking longer to completely dry out. From this standpoint, Models B3, GL2000, and CEB-FIP appear more credible than the other three models. Among Models B3, GL2000, and CEB-FIP, ultimate shrinkage is highest for GL2000 and lowest for CEB-FIP, but it should be noted that different parameters are considered in different models, as shown in Table 1(a). Because assumed parameters in Table 1(b) are used to obtain Fig. 1 and 2, the difference between Models B3, GL2000, and CEB-FIP might change if actual field data are used. Therefore, it is strongly recommended that specified values for each model be replaced by field or laboratory test data for a specific project.

As shown in Fig. 2, ultimate creep predicted by the B3, GL2000, and CEB-FIP models is nearly constant for different v/s, with 12 in. (305 mm) slab values more than 90% of 5 in. (127 mm) slab values. The other two models show 12 in. (305 mm) slab values to be proportionately lower than 5 in. (127 mm) slab values. This observation further strengthens the impression that Models B3, GL2000, and CEB-FIP are more likely to provide credible predictions of ultimate shortening. Models B3, GL2000, and CEB-FIP have higher ultimate creep values than the other two models. This is partly because concrete creep predicted by these three models is significantly affected by concrete age at loading, and this study assumes that slabs are wet-cured for 3 days and then tendons are stressed. The ACI 209 model includes a correction factor to consider the age of concrete at loading later than 7 days. No creep correction factor is considered, however, because the loading age is only 3 days in this study.

It should be noted that none of the aforementioned models are based on creep data for high-early-strength concrete and they do not reflect early concrete strength gain. For param-eters shown in Table 1(b), predicted modulus of elasticity development is based on Eq. (1) to (3) for different models.

ACI 209 and B3 models: ( ) (28)4 0.85

tE t Et

=+

(1)

Table 1(b)—Parameters used to predict concrete creep and shrinkageConcrete strength, psi (MPa)

5000 (34.5)

Slab average stress, psi (MPa)

200 (1.4)

Cement, lb/ft3 (kg/m3)

25.6 (410)

Slab thickness, in. (mm)

5 to12 (127 to 305)

Water-cement ratio (w/c) 0.45 Relative humidity, % 40 to 90

Water, lb/ft3 (kg/m3)11.5 (184)

Age when drying begins, days

3

Aggregate-cement ratio 4.28Age at tendon stressing, days

3

Cement type Type I Shape of specimen Infinite slab

Table 2—v /s for different slab thicknessesSlab thickness, in. 5 6 7 8 9 10 11 12

v/s, in. 2.6 3.0 3.5 4.0 4.5 5.0 5.4 5.9

Note: 1 in. = 25.4 mm.

Fig. 1—Ultimate shrinkage strain: various v/s. (Note: 1 in. = 25.4 mm.)

Fig. 2—Ultimate creep strain: various v/s. (Note: 1 in. = 25.4 mm.)


concrete compressive strength. To use any of the models for concrete floor shortening, either field or laboratory test results should be used to reflect high-early-strength concrete commonly used in PT structures.

Effects of v /s on shortening rateTo reduce RTS problems that frequently occur in PT struc-

tures, relative rates of creep and shrinkage are required; they are as important as the ultimate values when pour strips are used and the appropriate “open time” for the strip must be determined. In this section, the relative rate of shortening is simply defined as the ratio of shortening at a certain concrete age with ultimate shortening. Predicted relative shortening rates versus slab thickness—a stand-in for v/s—at concrete ages of 30 and 60 days for the five models are shown in Fig. 3 for shrinkage and Fig. 4 for creep.

As shown in Fig. 3, the shrinkage rate according to the ACI 209 model is independent of v/s. This contradicts the physical reality that shrinkage rate reduces as v/s increases. The other four models exhibit this behavior. Among the other four models, the PCI model shows the highest shrinkage rate at all concrete ages. The B3, GL2000, and CEB-FIP models exhibit lower, nearly identical shrinkage rates. By the age of 60 days, the shrinkage rate curves for all models except the ACI 209 model are roughly parallel to each other, with the PCI model on the high side and the B3 and CEB-FIP models on the low side.

Figure 4 shows relative creep-rate curves of the five models for different v/s. Relative creep rates exhibited by the PCI model are much lower than those for other models and generally decrease as the v/s increases. The other four models exhibit a nearly constant creep rate, which agrees well with the fact that increasing v/s affects the shrinkage

Table 3—Concrete modulus of elasticity development with time relative to 28-day value

Age of concrete, days

Model 1 2 3 4 5 6 7 14 28

ACI 209/B3 0.45 0.59 0.68 0.74 0.78 0.81 0.84 0.94 1

GL2000 0.27 0.40 0.49 0.56 0.61 0.66 0.69 0.86 1

CEB-FIP 0.58 0.71 0.77 0.81 0.84 0.86 0.88 0.95 1

Fig. 3—(a) Percentage of shrinkage at 30 days with various v/s; and (b) percentage of shrinkage at 60 days with various v/s. (Note: 1 in. = 25.4 mm.)

Fig. 4—(a) Percentage of creep at 30 days with various v/s; and (b) percentage of creep at 60 days with various v/s. (Note: 1 in. = 25.4 mm.)

=+

3/4

3/4GL2000 model: ( ) (28)2.8 0.77

tE t Et

(2)

0.25 28CEB-FIP model: ( ) (28)exp 1

2E t E

t

= −

(3)

Table 3 shows predicted modulus of elasticity develop-ment with time relative to 28-day values, concentrating on the first week after concrete placement. As shown in Table 3, the predicted modulus of elasticity development varies, and the CEB-FIP and GL2000 models represent the upper and lower bound, respectively. For high-early-strength concrete with a 28-day compressive strength of 5000 psi (34.5 MPa), it is not uncommon to achieve 3000 psi (20.7 MPa) within 1 day. The concrete modulus of elasticity after 1 day of curing is approximately 77% of that at 28 days because the concrete modulus of elasticity is proportional to the square root of


process significantly but has very marginal effects on the creep process.

It has been stated that concrete creep and shrinkage at 60 days account for more than 50% of the ultimate values.1-3 This conclusion seems to be based on the creep and shrinkage prediction data in the PCI Handbook’s Table 3.12.316 without considering any correction factors.The data in Table 3.12.3 are based on a concrete strength of 3500 psi (24.2 MPa), a relative humidity of 70%, an average concrete compressive stress of 600 psi (4.1 MPa), and a v/s of 1.5. It is not clear when tendons are stressed and how test specimens are cured. As shown in Fig. 3 and 4 for the PCI model, the shortening at 60 days is somewhere between 38 and 62% of the ultimate shortening values for shrinkage and 27 and 44% of the ultimate values for creep. In contrast, the B3, GL2000, and CEB-FIP models show much lower shrinkage rates and higher creep rates than the PCI model. It should be noted that only these three prediction models have the capacity to consider the effects of concrete age at loading (tendon stressing).

Effects of relative humidity on ultimate shorteningAs stated previously, relative humidity is another impor-

tant factor that may affect concrete shrinkage and creep. To compare the effects of relative humidity on concrete creep and shrinkage, a slab thickness of 8 in. (203 mm) was selected with all other parameters the same as those shown in Table 1(b).

Figures 5 and 6 respectively show ultimate shrinkage and creep predictions for relative humidity values ranging from 40 to 90%. In Fig. 5, as relative humidity increases, ultimate shrinkage decreases in a similar trend for all models with the exception of the GL2000 model, which shows a steeper drop. In Fig. 6, ultimate creep curves for all models are roughly parallel and decrease almost linearly as relative humidity increases.

Effects of relative humidity on shortening rateBecause it is well-known that relative humidity does

not affect the relative shrinkage rate, no graphs are shown for shrinkage rate. The creep rates at concrete ages of 30 and 60 days are shown in Fig. 7. The PCI and ACI 209 model creep rates remain the same as relative humidity increases. The CEB-FIP model creep rate shows a slight decrease for relative humidity between 60 and 90%. Models B3 and GL2000 show creep rates increasing as rela-tive humidity increases.

Discussion of resultsFor actual concrete behavior, it is expected that ultimate

creep and shrinkage should be similar for different slab thicknesses; only the timing should differ. However, in Fig. 1 and 2, the ACI 209 and PCI models show less ulti-mate creep and shrinkage for thicker slabs. Based on these observations, the B3, GL2000, and CEB-FIP models seem to be more appropriate for estimating the amount and timing of concrete floor shortening in PT structures than the other three models.

Among the B3, GL2000, and CEB-FIP models, the CEB-FIP model shows the least creep and shrinkage (less conservative) in Fig. 1 and 2. In addition, the CEB-FIP model does not allow separate calculation of drying creep due to the drying process. Thus, this model cannot be used to evaluate the effects of different relative humidity values

Fig. 5—Ultimate shrinkage strain at various relative humidities.

Fig. 6—Ultimate creep strain at various relative humidities.

Fig. 7—(a) Percentage of creep at 30 days at various relative humidities; and (b) percentage of creep at 60 days at various relative humidities.


during and after construction on concrete drying creep. Both the B3 and GL2000 models predict drying creep separately from basic creep, which makes the consideration of different relative humidity values over time possible. As shown in Fig. 2, the concrete ultimate creep values predicted by the B3 model are substantially higher than those predicted by the other models. The author of the B3 model, Bažant, pointed out that some prediction models, including the ACI 209 and CEB-FIP models, grossly underestimate multi-decade creep when compared to the B3 model.17 The reason is that these models assume the creep to terminate at some fixed upper bound for which no experimental support exists.

Based on the previous discussion, the authors consider the B3 model to be the most appropriate prediction model to estimate concrete floor shortening in PT buildings. Among the nine creep prediction models, Fanourakis and Ballim18 pointed out that the B3 model was the most accu-rate model based on the overall coefficient of variation. The B3 model can also be calibrated using short-term test data on the actual concrete mixture to greatly reduce the range of uncertainty in shortening predictions. Accuracy is improved even if the B3 model is calibrated with short-term tests of a 1- to 3-month duration.19 Improved accuracy is extremely important for studying shortening effects in sensitive struc-tures. However, the effects of high-early-strength concrete are not considered in the B3 model, so a workaround adjust-ment is proposed; otherwise, concrete creep predicted by the B3 model may be higher than in reality.

SHRINKAGE AND CREEP BASED ON B3 MODELTo reflect the effects of high early strength on concrete

creep, it is proposed to use an equivalent age when drying

begins and tendons are stressed. In this study, the age when drying begins and tendons are stressed is assumed to be 3 days, which reflects typical construction practice for PT structures using unbonded tendons. Tendons are typically stressed when concrete reaches a minimum strength of 3000 psi (20.7 MPa), mainly because tendon anchorages are designed to distribute tendon forces to 3000 psi (20.7 MPa) concrete. Therefore, concrete mixtures for PT structures are designed to provide a minimum strength of 3000 psi (20.7 MPa) at the end of a 3-day wet cure. For the B3 model, Eq. (4) shows the concrete strength development with time.

( ) (28)4 0.85

tf t ft

=+

(4)

From Eq. (4), if one assumes f(28) is 5000 psi (34.5 MPa), concrete strength reaches 3000 psi (20.7 MPa) in 5 days rather than in 3 days. For this reason, the authors propose using the equivalent concrete age when drying begins and load is applied to be approximately 5 days. Throughout the balance of this paper, a 5-day equivalent concrete age is used rather than a 3-day age at loading.

Figures 8 and 9 show shrinkage and creep versus time for slabs of different thicknesses, using an equivalent concrete age when drying begins and tendons are stressed. The parameters in Table 1(b) were used in producing the figures. As shown in Fig. 8(b), approximately 15 to 30% of the ulti-mate shrinkage occurred by 60 days. Figure 9(b) shows that approximately 50% of the ultimate creep is anticipated to take place within 60 days regardless of v/s.

Fig. 8—(a) Shrinkage versus time based on B3 model; and (b) percentage of shrinkage versus time based on B3 model. (Note: 1 in. = 25.4 mm.)

Fig. 9—(a) Creep versus time based on B3 model; and (b) percentage of creep versus time based on B3 model. (Note: 1 in. = 25.4 mm.)


Table 4—Ultimate shrinkage for Cases A through C, me

Relative humidity

Case A Case B Case C

Shrinkage ShrinkageReduction,

% ShrinkageReduction,

%

0.4 629 629 — 601 4

0.5 588 588 — 561 5

0.6 527 527 — 503 5

0.7 442 442 — 422 5

0.8 328 328 — 313 5

0.9 182 182 — 174 4

Table 5—Ultimate creep for Cases A through C, me

Relative humidity

Case A Case B Case C

Creep Creep Reduction, % Creep Reduction, %

0.4 321 296 8 270 16

0.5 281 255 9 238 15

0.6 253 227 10 217 14

0.7 234 209 11 202 14

0.8 221 195 12 192 13

0.9 211 186 12 184 13

Another strategy would be to adjust the 28-day strength. If concrete achieves 3000 psi (20.7 MPa) in 3 days, the 28-day concrete strength will be 6550 psi (45.2 MPa) according to Eq. (4). To determine the impact of different approaches, the following three cases are considered in this study for comparison: • Case A: A concrete strength of 5000 psi (34.5 MPa) and

an age of 3 days at loading;• Case B: A concrete strength of 5000 psi (34.5 MPa) and

an age of 5 days at loading; and• Case C: A concrete strength of 6550 psi (45.2 MPa) and

an age of 3 days at loading.Case A simply ignores the effects of high early concrete

strength and Case B corresponds to concrete with only high early strength, but with a 28-day strength of 5000 psi (34.7 MPa).

Tables 4 and 5 show ultimate shrinkage and creep values for different relative humidity values for the three cases. Using a 28-day strength of 6550 psi (45.2 MPa) (Case C) reduces predicted shortening by approximately 4 to 5% for shrinkage and 13 to 16% for creep. Using an equivalent concrete age of 5 days (Case B) at loading, however, has no effect on shrinkage and reduces creep by approximately 8 to 12%. Based on the results in Tables 4 and 5, if the actual 28-day concrete strength is not available, it is conservative to use the specified 28-day strength and determine an equiv-alent concrete age at the tendon stressing approach (Case B).

In addition to the concrete age when drying begins and tendons are stressed, other parameters—including v/s, concrete strength, and relative humidity—have major effects on the development of concrete creep and shrinkage. This is covered in Part II of this study.

CONCLUSIONSBased on the results of this study, the following conclu-

sions are drawn:1. Ultimate concrete shrinkage and creep are approxi-

mately the same for slabs with different v/s using the B3, GL2000, and CEB-FIP models. This appears to distinguish them as more accurate and appropriate for estimating PT slab shortening than the PCI and ACI 209 models.

2. Among the B3, GL2000, and CEB-FIP models, the B3 model is determined to be the most appropriate model for shortening estimation because: 1) it can be calibrated with short-term creep and shrinkage test results; 2) it has the capability to calculate drying creep for different rela-tive humidity values during and after construction (covered in detail in Part II of this study); and 3) all other models included in the study seem to underestimate concrete creep.

3. As relative humidity increases, both ultimate shrinkage and creep show significant drops, with ultimate shrinkage decreasing much faster than ultimate creep. Thus, consid-ering different relative humidity values during and after construction might be necessary for practical reasons for PT concrete structures.

4. None of the models are capable of reflecting high early concrete strength commonly used in the PT industry. To make prediction models suitable for this purpose, an equivalent concrete age when drying begins and tendons are stressed is proposed.

ACKNOWLEDGMENTSThe authors wish to express their gratitude to Z. P. Bažant at Northwestern

University, Evanston, IL, for his valuable comments and suggestions.

NOTATIONE(t) = concrete modulus of elasticity at t daysE(28) = concrete modulus of elasticity at 28 daysf(t) = concrete compressive strength at t daysf(28) = concrete compressive strength at 28 dayst = age of concrete, daysv/s = volume-to-surface ratio

REFERENCES1. Bondy, K. B., “Shortening Problem in Post-Tensioned Concrete Build-

ings,” Design Review and Inspection of Prestressed Concrete Building Proj-ects, SEAOC Seminar Proceedings, Jan. 1989, pp. 1-19.

2. Aalami, B. O., and Barth, F. G., “Restraint Cracks and Their Mitiga-tion in Unbonded Post-Tensioned Building Structures (PTI DC20.2-88),” Post-Tensioning Institute, Farmington Hills, MI, 1988, 49 pp.

3. PTI Committee DC-20, “Design, Construction and Maintenance of Cast-in-Place Post-Tensioned Concrete Parking Structures (PTI DC20.7-01),” Post-Tensioning Institute, Farmington Hills, MI, 2001, 173 pp.

4. PTI Technical Activities Board (TAB), Post-Tensioning Manual (PTI TAB.1-06), sixth edition, Post-Tensioning Institute, Farmington Hills, MI, 2006, 354 pp.

5. Stevenson, A. M., “Post-Tensioned Concrete Floor Design in Multi-Story Buildings,” British Cement Association, Growthorne, Berkshire, UK, 1994, 20 pp.

6. Guo, G.; Joseph, M. L.; and Bieberly, G. E., “Restraint Design of Cast-in-Place Post-Tensioned Underground Parking Structures,” PTI Journal, V. 7, No. 1, Aug. 2009, pp. 5-18.

7. PTI Technical Activities Board (TAB), Post-Tensioned Concrete Floors: Design Handbook (PTI TAB.1-06), second edition, Technical Report No. 43, The Concrete Society, Camberley, UK, 2005, 110 pp.

8. Bažant, Z. P., “Creep and Shrinkage Prediction Model for Analysis and Design of Concrete Structures—Model B3,” Materials and Structures, V. 28, 1995, pp. 357-365.

9. Bažant, Z. P., and Baweja, S., “Justification and Refinements of Model B3 for Concrete Creep and Shrinkage 1. Statistics and Sensitivity,” Mate-rials and Structures, V. 28, 1995, pp. 415-430.

10. Bažant, Z. P., and Baweja, S., “Justification and Refinements of Model B3 for Concrete Creep and Shrinkage 2. Updating and Theoretical Basis,” Materials and Structures, V. 28, 1995, pp. 488-495.

11. Gardner, N. J., and Lockman, M. J., “Design Provisions for Drying Shrinkage and Creep of Normal-Strength Concrete,” ACI Materials Journal, V. 98, No. 2, Mar.-Apr. 2001, pp. 159-167.


12. “CEB-FIP Model Code 1990,” CEB Bulletin d’Information No. 213/214, Comité Euro-International du Beton, Lausanne, Switzerland, 1993, pp. 33-41.

13. ACI Committee 209, “Prediction of Creep, Shrinkage, and Tempera-ture Effects in Concrete Structures (ACI 209R-92),” American Concrete Institute, Farmington Hills, MI, 1992, 47 pp.

14. ACI Committee 209, “Guide for Modeling and Calculating Shrinkage and Creep in Hardened Concrete (ACI 209.2R-08),” American Concrete Institute, Farmington Hills, MI, 2008, 45 pp.

15. Gardner, N. J., “Comparison of Prediction Provisions for Drying Shrinkage and Creep of Normal Strength Concretes,” Canadian Journal of Civil Engineering, V. 31, No. 5, Sept.-Oct. 2004, pp. 767-775.

16. PCI Design Handbook—Precast and Prestressed Concrete (MNL-120-99), fifth edition, Precast/Prestressed Concrete Institute, Chicago, IL, 1999, 690 pp.

17. Bažant, Z. P.; Li, G. H.; and Yu, Q., “Excessive Long-Time Deflec-tions of Prestressed Box Girders,” Structural Engineering Report No. 09-12/ITI, Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL, July 2010, 44 pp.

18. Fanourakis, G. C., and Ballim, Y., “Predicting Creep Deformation of Concrete: A Comparison of Results from Different Investigations,” Proceedings, 11th FIG Symposium on Deformation Measurements, Santo-rini, Greece, 2003, 8 pp.

19. Bažant, Z. P., and Baweja, S., “Creep and Shrinkage Prediction Model for Analysis and Design of Concrete Structures: Model B3,” Adam Neville Symposium: Creep and Shrinkage—Structural Design Effects, SP-194, ACI Committee 209, ed., American Concrete Institute, Farmington Hills, MI, 2000, pp. 1-83.


Title no. 110-S04


ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2011-029 received January 27, 2011, and reviewed under Institute


Shortening Estimation for Post-Tensioned Concrete Floors—Part II: Calculationsby Guohui Guo and Leonard M. Joseph

Part I of the study compares five concrete shortening prediction models, and the B3 model is considered to be the most appropriate model. Part II of the study provides detailed procedures to estimate post-tensioned (PT) concrete floor shortening. Different relative humidity values during and after construction are used to simulate the actual conditions concrete floors experience. For typical PT construction, concrete floor shortening due to elastic compression, concrete creep, and shrinkage can be estimated as approximately 1 in. (25 mm) per 100 ft (30.5 m). If different relative humidity values during and after construction are considered, the construc-tion period can affect total floor shortening by more than 15%. Total shortening is minimized by longer construction exposure in regions with high relative humidity and shorter construction expo-sure in regions with low relative humidity. The helpful effect of pour strips in reducing PT floor shortening is studied and appears rather limited for typical construction.

Keywords: creep; equivalent concrete age; floor shortening; post-tensioned concrete; pour slip; relative humidity; shrinkage; volume-to-surface ratio.

INTRODUCTIONDue to a growing awareness of the advantages of post-

tensioning among engineers, the use of post-tensioning grows at a rapid speed, with an average annual growth rate of 8.5% between 1985 and 2004.1 At the same time, modern post-tensioned (PT) concrete structures are being built with ever-longer dimensions without the use of expan-sion joints. For example, a 663 ft (202 m) long hospital using a PT concrete frame system was built with the use of two pour strips.2 Li and Sha3 reported that an eight-story PT building with a length of 534 ft (165.6 m) was success-fully built without the use of expansion joints. This was largely attributed to the use of two middle pour strips and a shrinkage-reducing admixture that cut concrete shrinkage roughly in half.

Although pour strips are widely used to reduce concrete floor restraint to shortening (RTS) problems,4 the realistic contribution of pour strips to the reduction of floor move-ment has not been investigated. Over the years, some engi-neers have mistakenly thought that once they specify the use of pour strips, PT floor RTS should not be a problem. This might be true for PT concrete floors with relatively short dimensions and moderate (flexible) restraints. For long building dimensions and severe (stiff) restraints, however, RTS problems could still be serious and deserve a system-atic approach to account for them during design.5,6 It is very important to understand that pour strips can help to reduce concrete floor shortening to some degree, but the help could be very limited, especially where stiff supports occur at the far ends of a design strip.

The second part of this study focuses on detailed proce-dures to estimate shortening for PT concrete floors with and without the use of pour strips using the B3 model. Different

relative humidity values are used during and after construc-tion to evaluate the effects on floor shortening of realistic environmental conditions a building may experience. Proce-dures are presented to estimate PT concrete floor shortening when pour strips are used. The relationship between the open period of a delayed pour strip and its effect on floor shortening is studied as well.

RESEARCH SIGNIFICANCEThe second part of this study provides detailed procedures

to estimate PT concrete floor shortening considering volume-to-surface ratio (v/s), relative humidity, and concrete proper-ties. The use of pour strips to reduce concrete floor shortening is illustrated using different pour strip open periods. The effects on PT floor shortening of different relative humidity values during and after construction are studied.

ESTIMATING CONCRETE FLOOR SHORTENINGAccording to Bažant,7 the prediction using the material

parameters of the B3 model is restricted to portland-cement concrete with the following parameter ranges: concrete strength at 28 days between 2500 and 10,000 psi (17.3 to 69 MPa), water-cement ratio (w/c) between 0.3 and 0.85, cement content between 10 to 45 lb/ft3 (160 to 720 kg/m3), and aggregate-cement ratio between 2.5 and 13.5. In addition, the formulas are valid for concrete cured for at least 1 day, and for service stress below 45% of the concrete strength at 28 days. The following section considers the use of different relative humidity values in the B3 model.

Effects of relative humidityBecause relative humidity greatly affects concrete creep

and shrinkage, it is necessary to consider the effects on floor shortening of different relative humidity values during and after construction, especially for a project with a long construction period. According to the B3 model,7 the effects of relative humidity can be seen in Eq. (1) for concrete shrinkage and in Eq. (2) for concrete drying creep. The defi-nition of each parameter is presented in Reference 7.

e esh sh ht t k S t, 0( ) = − ( )∞ (1)


Guohui Guo is a Senior Project Engineer with Haris Engineering, Inc., Kansas City, KS. He received his PhD of science in structural engineering from the University of Kansas, Lawrence, KS. His research interests include the analysis and design of a wide range of structures, including high-rise buildings; bridges; and sports, healthcare, parking, post-tensioned, and long-span structures.

Leonard M. Joseph is a Principal with Thornton Tomasetti, Inc., Irvine, CA. He received his BS in civil engineering from Cornell University, Ithaca, NY, and his MS from Stanford University, Stanford, CA. His research interests include analysis, design, and investigation of projects throughout the world, ranging from the Petronas Towers in Malaysia to parking decks across the United States.

C t t t q H t H td , , exp exp′( ) = − ( ){ } − − ′( ){ } 0 5

1 28 8 (2)

where H(t) and creep at drying, q5, are defined in Eq. (3) and (4), respectively.

H t h S t( ) = − −( ) ( )1 1 (3)

5 1 0.65 7.57 10 c shq f − −

∞= × e′ (4)

For typical buildings with conditioned air, relative humidity is maintained at approximately 50% during service life. However, ambient outdoor relative humidity varying from 20 to 90% affects shortening during construc-tion. The aforementioned prediction formulas do not appear to specifically address relative humidity values varying over time. To evaluate floor shortening considering different relative humidity values, the following approach is proposed to “splice together” shortening from two different humidity levels:

Step 1—Determine concrete shrinkage and drying creep values with time using the average annual relative humidity during normal service life (50%, for example) based on Eq. (1) and (2). Record the calculated shrinkage and drying creep curves as f′(t) and, herein, relative humidity during building normal service life is used.

Step 2—Denote n as the construction period, in days, and t as the concrete age, in days. At a concrete age after construc-tion (t > n), the change in shrinkage or drying creep since the end of construction (start of occupancy) is f′(t) – f′(n).

Step 3—Determine concrete shrinkage and drying creep using the actual average relative humidity over the construc-tion period based on Eq. (1) and (2). Record the calculated shrinkage or drying creep curve using Eq. (1) and (2) as f(t).

Step 4—Create composite concrete shrinkage and drying creep curves reflecting different relative humidity values. At a concrete age of t days, the shrinkage or drying creep can be calculated as follows

shrinkage or drying creep =( ) for

( ) ( ( ) ( ) for

t t nn t n t n

f ≤f + f − f >′ ′

(5)

It should be noted that basic creep is still calculated according to the B3 model without any change because it is not affected by relative humidity.

Elastic shorteningConcrete floor slabs shorten due to elastic deforma-

tion, concrete shrinkage, concrete creep, and temperature changes. For PT buildings, elastic shortening8,9 can be calcu-lated using Eq. (6) as follows

cpaes

f LK

ED = (6)

In the following calculations, Kes = 0.5 is used because tendons are typically stressed in sequential order for PT buildings with unbonded tendons. For Tables 1 to 3, the elastic shortening is calculated with a concrete strength of 3000 psi (20.7 MPa) at tendon stressing. For typical PT concrete floors, the average compressive stress immediately after the tendon stressing, fcpa, is approximately 8% higher than the average compressive stress after all long-term stress losses. In this study, based on the average compres-

Table 1—Concrete floor shortening per 100 ft (30.5 m), by slab thicknessSlab thickness, in. 5 6 7 8 9 10 11 12

Elastic shortening, in. 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04

Shrinkage shortening, in.

0.71 0.71 0.71 0.71 0.71 0.71 0.70 0.70

Creep shortening, in. 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31

Total shortening, in. 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05

Elastic shortening, % of total

4

Shrinkage shortening, % of total

67

Creep shortening, % of total

29

Ultimate shrinkage, me 588 588 588 588 588 588 587 586

Ultimate creep, me 255 255 255 255 255 255 255 254

Total ultimate, me 843 843 843 843 843 843 842 841

Note: 1 in. = 25.4 mm.

Table 2—Concrete floor shortening per 100 ft (30.5 m), by concrete strength28-day concrete strength, psi (MPa)

4000 (27.6)

4500 (31.1)

5000 (34.5)

6000 (41.4)

7000 (48.3)

8000 (55.2)

Elastic shortening, in. 0.04 0.04 0.04 0.04 0.04 0.04

Shrinkage shortening, in.

0.73 0.72 0.71 0.69 0.67 0.65

Creep shortening, in. 0.35 0.33 0.31 0.28 0.25 0.24

Total shortening, in. 1.12 1.08 1.05 1.00 0.96 0.93


3 4 4 4 4 4


65 66 67 69 70 70


31 30 29 28 26 25

Ultimate shrinkage, me 611 599 588 571 557 545

Ultimate creep, me 292 272 255 230 211 197

Total ultimate, me 903 871 843 801 768 742

Note: 1 in. = 25.4 mm.


sive stress of 200 psi (1.4 MPa) as shown in Table 1(b) of Part I, an fcpa value of 216 psi (1.5 MPa) is used to calculate elastic shortening.

The shrinkage and creep shortening values are calculated according to the B3 model. To reflect the effects of high early concrete strength commonly used in PT structures, an equivalent concrete age is used when drying begins and tendons are stressed: 5 days instead of the actual concrete age of 3 days, as discussed in Part I of this paper.

Floor shortening for different v /sThe B3 model predicts that, consistent with material

science, concrete ultimate shrinkage and creep values do not change as the v/s increases. For parameters shown in Table 1(b) of Part I and with 50% fixed relative humidity, predicted floor shortening per 100 ft (30.5 m) is shown in Table 1, excluding thermal shortening effects.

As shown in Table 1, shrinkage is the biggest contrib-utor, accounting for approximately 67% of the total floor shortening. Elastic and creep shortening account for the remainder. Total shrinkage and creep shortening strain is approximately 840 me, or approximately 1 in. (25 mm) of unrestrained slab shortening per 100 ft (30.5 m).

Floor shortening for different concrete strengthsTable 2 shows concrete floor shortening per 100 ft

(30.5 m) of slab length for different concrete strengths, using parameters shown in Table 1(b) of Part I with a 50% fixed relative humidity and 8 in. (203 mm) slab thick-ness. Table 2 shows that as concrete strength increases from 4000 to 8000 psi (27.6 to 55.2 MPa), the total short-ening decreases from 903 to 742 me. The shrinkage is still the biggest contributor, accounting for approximately 65 to 70% of the total shortening. However, this ratio may change significantly if the average precompression from PT tendons is much more than the typical value of 200 psi (1.4 MPa), as assumed in this study.

Floor shortening for different relative humidity valuesTable 3 shows the effects of relative humidity on concrete

shrinkage and creep. The table is developed using param-eters in Table 1(b) of Part I with an 8 in. (203 mm) slab thick-ness. As relative humidity increases from 40 to 90%, total shortening drops rapidly from 925 to 368 me, and slab short-ening per 100 ft (30.5 m) reduces from 1.15 in. (29 mm) to approximately 0.5 in. (13 mm). Shrinkage drops more than creep, as shown in the differing percentage of total values.

Discussion of resultsAs shown in Tables 1 to 3, ultimate shrinkage is insensi-

tive to v/s and concrete strength, and very sensitive to relative humidity. Concrete creep, however, is insensitive to v/s and is sensitive to both concrete strength and relative humidity. Therefore, total concrete shortening due to elastic, creep, and shrinkage could vary. The suggested value of 650 me in the PTI Handbook10—intended to include elastic, shrinkage, and creep effects—does not appear conservative in low-strength and low-humidity situations.

Results based on the B3 model also show that short-ening due to concrete creep is at least five times that due to elastic deformation. This ratio is much higher than the ratio of 1.6 reported in PTI DC20.2-885 and the recom-mended ratio of 2.5 in the PT concrete floor design hand-book.10 The ratio of 1.6 was based on estimation and no

detailed calculation or testing was performed. The ratio of 2.5 does not consider elastic shortening reduced by the sequential stressing reduction factor that is very typical for PT concrete buildings with unbonded tendons. Without that factor, elastic shortening would be doubled and the recom-mended ratio of 2.5 would be consistent with lower-bound findings presented previously for higher concrete strengths and higher relative humidity values.

Tables 1 to 3 also show that concrete shrinkage is the largest contributor for floor shortening and accounts for at least 60% of the total shortening for relative humidity less than 80%.

SHORTENING OF PT CONCRETE FLOOR WITH POUR STRIPS

Concrete creep and shrinkage shortening predicted by the B3 model can be used to estimate unrestrained PT slab shortening. The shortening data can be used by structural engineers in analyses as an imposed strain, or “artificial cooling,” to evaluate the effects of floor shortening on struc-tural members such as columns, walls, and slabs.

The current design trend for PT structures with long plan dimensions is to use more pour strips and reduce or elimi-nate permanent expansion joints. This is particularly useful in parking structures where expansion joints are suscep-tible to damage that can reduce durability of the concrete structure.11 Thus, detailed estimation of slab shortening is required to properly evaluate the effects of RTS on struc-tural elements.

Shortening effectsThis section figuratively presents the use of pour strips

to help reduce the effects of concrete floor shortening on the overall structure. Figure 1(a) shows an 11-span concrete floor with 30 ft (9.1 m) spans supported on one-story columns. Figure 1(b) to (d) shows frame-deflected shapes due to concrete floor shortening with and without a pour strip. Shortening is exaggerated, but drawn in the same scale on all figures so that rough comparisons can be made between figures. In Fig. 1(a), columns are numbered

Table 3—Concrete floor shortening per 100 ft (30.5 m), by relative humidityRelative humidity, % 40 50 60 70 80 90

Elastic shortening, in. 0.04 0.04 0.04 0.04 0.04 0.04

Shrinkage shortening, in. 0.75 0.71 0.63 0.53 0.39 0.22

Creep shortening, in. 0.36 0.31 0.27 0.25 0.23 0.22

Total shortening, in. 1.15 1.05 0.94 0.82 0.67 0.48


3 4 4 5 6 8


67 67 67 65 60 45


31 29 29 31 35 46

Ultimate shrinkage, me 629 588 527 442 328 182

Ultimate creep, me 296 255 227 209 195 186

Total ultimate, me 925 843 754 651 523 368

Note: 1 in. = 25.4 mm.


Fig. 1—Shortening effects for concrete floors with and without pour strips. (Note: 1 ft = 3.28 m.)

Step 2—Determine slab strain corresponding to floor shortening from elastic shortening, concrete shrinkage, and creep at the end of the pour strip open period. The bottom half of Fig. 2 shows concrete shrinkage shortening strain on the left and creep shortening strain on the right. From both curves, concrete shrinkage and creep values are found at the end of a pour strip open period. Elastic shortening is calculated using Eq. (6), based on the concrete strength and modulus of elasticity when tendons are stressed. Frame deformations are conservatively estimated by multiplying unrestrained shortening strains by distances to the point of zero movement. The deformed shape during this stage is shown in Fig. 1(c).

Step 3—After the pour strip is filled, the two locations of zero movement, as shown in Fig. 1(c), disappear and the new location of zero movement is at the midlength pour strip location. The entire concrete floor acts as one unit for the balance of concrete creep and shrinkage movements. Deter-mine additional building frame deformations by applying floor shortening strains from the end of the open period to the age of interest using concrete shrinkage and creep values in the bottom half of Fig. 2. Conservatively multiply strains by distances to the new location of zero movement.

Step 4—Add Step 2 and Step 3 deformations to find the final deformed shape. Based on the specific situation, columns close to the center could end up displaced toward or away from the final location of zero movement.

C1 to C6 for easy identification. Without a pour strip, the deflected frame after all total shortening takes place is shown in Fig. 1(b). If all columns have the same stiffness, the loca-tion of zero movement is at the midpoint of the floor length. Floor shortening D1 at the end Column C1 is half of the total floor shortening. When a pour strip is incorporated in the middle span and kept open for 60 days, the two segments initially shorten independently toward their own locations of zero movement, as shown in Fig. 1(c). The final deflected shape of the concrete frame is shown in Fig. 1(d), where D2 represents the shortening at end Column C1. The differ-ence between D1 and D2 represents the beneficial effect of a pour strip on floor shortening.

Shortening calculation proceduresFigure 2 shows the procedure that is used in this paper

to determine horizontal movements at individual column locations due to concrete floor shortening predicted by the B3 model:

Step 1—Determine concrete properties, slab thickness, average precompression, relative humidity, and concrete ages when drying begins and tendons are stressed. Although not shown in Fig. 2, concrete aggregate information is required as well. On an actual project, concrete field or laboratory test data (if available) should be used to improve concrete creep and shrinkage predictions.


Deformations are calculated using distances to the point of zero movement that can be determined based on stiffness of all vertical elements. Stiff shear walls and columns of differing stiffness will affect the location of zero movement.

The aforementioned procedure calculates frame deforma-tions from total shortening of unrestrained concrete floors. In reality, concrete floors may shorten much less due to the presence of restraint from columns and walls. At the same time, significant forces may be developed in both horizontal and vertical structural elements. The effects of concrete floor shortening can generally be evaluated by imposing calcu-lated unrestrained slab shortening strains within a model of the complete structure. The effects on the complete struc-ture due to concrete slab creep or shrinkage shortening are similar to those due to a slab temperature drop as presented by Mohammad,12 where end columns experience the highest bending and shear and the slab near the midpoint of a design strip experiences the largest tension due to temperature drop. Restraint forces that reduce in-slab axial forces can also reduce creep strain, and sustained forces in columns and walls restraining slab movement can generate creep in those members, reducing their effective stiffness and RTS. Detailed procedures regarding evaluation of these interre-lated effects, and their consideration in design, are beyond the scope of this study.

Effects of pour strip on floor shorteningWhen a pour strip is present, the pour strip open period

is a key parameter. Using the procedures presented previ-ously, unrestrained horizontal movement at each column location due to floor shortening can be calculated for two situations: with and without a pour strip. A separate calcu-lation is required for each different pour strip open period being considered. Based on the parameters shown in Fig. 2, Table 4 tabulates horizontal movement at the top of each column due to floor shortening for different pour strip open periods. As the pour strip open period increases, horizontal movement at the slab ends gradually decreases. A positive movement value indicates the column top is bending toward

the center of the frame, whereas a negative sign indicates movement away from the center.

Table 5 shows the percentage reduction of movement from shortening at each column for different pour strip open periods when compared with movement from floor shortening without the use of a pour strip. The sign “—” simply means that the reduction is not meaningful because

Table 4—Movement from floor shortening at column locations, in.

Columns C1 C2 C3 C4 C5 C6

Without pour strip 1.73 1.42 1.10 0.79 0.47 0.16

Pour strip open period, days

7 1.55 1.24 0.92 0.61 0.29 –0.02

14 1.52 1.21 0.89 0.58 0.26 –0.05

21 1.50 1.19 0.87 0.56 0.24 –0.07

30 1.48 1.17 0.85 0.54 0.22 –0.09

60 1.43 1.12 0.80 0.49 0.17 –0.14

90 1.39 1.08 0.76 0.45 0.13 –0.18

180 1.32 1.00 0.69 0.37 0.06 –0.26

365 1.22 0.90 0.59 0.27 –0.04 –0.36

Note: 1 in. = 25.4 mm.

Table 5—Percentage reduction of movement from floor shortening by column location

Columns C1 C2 C3 C4 C5 C6


7 10% 13% 16% 23% 38% —

14 12% 15% 19% 27% 45% —

21 13% 16% 21% 29% 49% —

30 14% 18% 23% 32% 53% —

60 17% 21% 27% 38% 64% —

90 20% 24% 31% 43% 72% —

180 24% 30% 37% 53% 87% —

365 29% 37% 46% 66% — —

Fig. 2—Example of floor shortening calculation.


pour strips result in these columns bending away from the final center of zero movement. As shown in Fig. 1(c) and (d), shortening at Columns C1 to C3 is additive before and after filling the pour strip, while the opposite applies for Columns C4 to C6. The percentage of the shortening reductions is different for exterior and interior columns. For example, with a pour strip open period of 30 days, the percentage of short-ening reduction is approximately 14% and 53% for Columns C1 and C5, respectively. As a practical matter, reductions at C1 and C2 are most meaningful, representing typical end and interior columns, respectively.

Pour strip open time can be a point of contention between designers and contractors. The aformentioned example clearly illustrates several pour strip key points:

1. There is a definite structural benefit from a longer pour strip open time. However, that can impact the contractor’s ability to enclose and fit out the affected floors around the pour strip.

2. Changing the open time by a few days has little effect on the structural benefit. A long open time is required to make a significant difference in structural behavior.

3. The largest effects of slab shortening occur at the locations of greatest restraint—the fixed foundation in this example. It may be practical to locate pour strips or request long open times only at highly restrained floors, such as the lowest framed levels. At higher superstructure levels, adja-cent floors shorten similarly with minimal restraint, so it may be possible to either avoid pour strips or, if they are needed for PT tendon stressing, to use short open times that do not impact the fit-out schedule.

A long open time needed to achieve a particular goal should be stated to the contractor in the construction documents. Pour strips are typically left open between 30 and 90 days; however, a much longer opening period may be required when a structure is long and an expansion joint is not acceptable. In a 25-story PT frame building with dimen-sions of 663 x 646 ft (202 x 197 m), the two basement levels were constructed with two pour strips in each direction.2 To reduce the effects of RTS cracking problems, concrete was

not placed at pour strip locations until the structure’s roof was topped out. It was estimated that at least 60% of the concrete creep and shrinkage had taken place at the time when pour strips were filled.

Effects of relative humidity during constructionPart I of this study and Table 3 of this part showed

the large effect that relative humidity has on concrete creep and shrinkage. As shown in Fig. 5 and 6 in Part I, and in Table 3 of this part, as relative humidity increases from 40 to 90%, concrete shrinkage is reduced at a much faster rate than concrete creep. Thus, a realistic estimate of concrete floor shortening should consider different rela-tive humidity values during and after construction, espe-cially when the construction period is long and ambient relative humidity during construction greatly differs from that in normal service. For most buildings with condi-tioned air, a relative humidity of approximately 50% can be assumed during normal service life. However, ambient relative humidity varies greatly by region. In this study, two scenarios are considered: one for a low relative humidity area such as Las Vegas, NV, and the other for a high relative humidity area such as Houston, TX. Average annual rela-tive humidity is taken as approximately 30% for Las Vegas and 75% for Houston. To demonstrate the effect of the construction period on concrete shortening, construction periods varying from 3 months to 2 years are considered.

The values in Tables 6 to 9 are based on the parameters in Table 1(b) of Part I, with a slab thickness of 8 in. (203 mm) and an equivalent concrete age of 5 days when drying begins and tendons are stressed. In Table 6, for Las Vegas, 30% rela-tive humidity during construction increases both ultimate creep and shrinkage. As the construction period is increased from 3 months to 2 years, the total shortening increases from 2.7 to 8.4% compared to a constant 50% relative humidity scenario. Thus, in low relative humidity areas, if construc-tion takes less than 2 years, increasing total shortening by 8% can conservatively account for the effects of low relative humidity during construction.

Table 6—Shortening in Las Vegas, based on different construction periods

Construction period, days Shrinkage, me

Increase versus 50% RH, % Creep, me

Increase versus 50% RH, % Total shortening, me

Increase versus 50% RH, %

Normal* 588 — 255 — 843 —

90 606 3.1 260 2.0 866 2.7

180 612 4.1 264 3.5 876 3.9

365 621 5.6 270 5.9 891 5.7

730 631 7.3 283 11.0 914 8.4*Constant relative humidity of 50% is used for ultimate shrinkage and creep calculation.Note: RH is relative humidity.

Table 7—Shortening in Houston, based on different construction periods

Construction period, days Shrinkage, me

Reduction versus 50% RH, % Creep, me

Reduction versus 50% RH, % Total shortening, me

Reduction versus 50% RH, %

Normal* 588 — 255 — 843 —

90 536 8.8 249 2.4 785 6.9

180 516 12.2 246 3.5 762 9.6

365 489 16.8 242 5.1 731 13.3

730 459 21.9 234 8.2 693 17.8*Constant relative humidity of 50% is used for ultimate shrinkage and creep calculation.Note: RH is relative humidity.


Table 7 shows the change of concrete shrinkage and creep shortening for Houston for different relative humidity values during and after construction. Longer construction periods reduce both concrete shrinkage and creep. Shrinkage decreases much faster than creep. If construction takes more than a year, total shortening can be easily reduced by more than 10%. Therefore, in high relative humidity areas, consider reducing total concrete shortening by 10% if construction takes longer than a year.

Table 8 tabulates lateral movement at Column C1 from unrestrained slab shortening, for floors with and without pour strips, considering different pour strip open periods and different construction periods. Looking at any one pour strip open period, it is clear that in low humidity areas, the longer the construction period, the larger the movement at Column C1. The opposite holds true for high humidity areas. The need for pour strips with longer open times in dry climates is also clear: to meet the column movement value for no-strip Houston 730-day construction, the Las Vegas project would require a pour strip with a 180-day open time. To meet the same movement value as a 7-day open time on Houston 90-day construction, Las Vegas requires a 60-day open time.

Table 9 shows the percentage of floor shortening reduc-tion at Column C1 for different construction periods in both low- and high-humidity areas. Both Tables 4 and 9 show similar results: pour strips are able to reduce shortening for

floors with long dimensions, but the effects seem to be very limited. Table 9 also reveals something hidden in Table 8 but is consistent with material behavior: for the same construc-tion period and pour strip open period, the percentage of movement reduction due to the use of pour strips is greater in a dry climate, until the construction period is so long that overall shortening overwhelms the benefit.

Discussion of resultsIn view of the aforementioned data, the longer pour strip

open period may or may not be worth the increased construc-tion costs due to a generally longer construction schedule and other construction inconveniences. If the construction schedule allows, fill the pour strip as late as possible, or when the entire structure is completed. Typically, the most significant restraints are realized between ground and the first elevated floor, where the elevated floor tends to shorten and the foundation tends to prevent shortening. Thus, the first-level columns, especially edge columns, may suffer severe bending and shear due to the RTS process. It is often practical to leave pour strips open at the first elevated floors for a period much longer than those at upper levels, where the first floor slabs are in parking areas or are scheduled to receive late, high-end lobby finishes. In such cases, the longer open period at the first elevated levels may not be on the construction schedule critical path.

Table 8—Movement at Column C1 for different construction periods, in.

Construction period, days 90 180 360 730


Location LV Hou LV Hou LV Hou LV Hou

No strip 1.78 1.61 1.80 1.57 1.83 1.51 1.87 1.44

7 1.59 1.46 1.61 1.41 1.64 1.35 1.69 1.27

14 1.56 1.43 1.58 1.39 1.61 1.33 1.65 1.25

21 1.54 1.42 1.56 1.37 1.59 1.31 1.63 1.24

30 1.51 1.4 1.53 1.36 1.56 1.3 1.61 1.22

60 1.46 1.37 1.48 1.32 1.51 1.26 1.55 1.19

90 1.41 1.34 1.43 1.30 1.46 1.23 1.51 1.16

180 — — 1.35 1.24 1.38 1.18 1.42 1.11

365 — — — — 1.26 1.12 1.3 1.04

730 — — — — — — 1.16 0.97

Notes: 1 in. = 25.4 mm; LV is Las Vegas; Hou is Houston.

Table 9—Percentage reduction of movement from floor shortening at Column C1

Construction period, days 90 180 360 730


Location LV Hou LV Hou LV Hou LV Hou

7 10.6% 9.6% 10.4% 10.3% 10.3% 10.6% 9.8% 11.6%

14 12.3% 11.4% 12.1% 11.6% 11.9% 12.0% 11.9% 13.0%

21 13.4% 12.0% 13.2% 12.9% 13.0% 13.3% 13.0% 13.7%

30 15.1% 13.3% 14.9% 13.5% 14.7% 14.0% 14.0% 15.1%

60 17.9% 15.1% 17.7% 16.0% 17.4% 16.6% 17.2% 17.2%

90 20.7% 17.0% 20.4% 17.3% 20.1% 18.6% 19.4% 19.3%

180 — — 24.9% 21.1% 24.5% 21.9% 24.2% 22.7%

365 — — — — 31.1% 25.9% 30.6% 27.6%

730 — — — — — — 38.1% 32.5%

Notes: LV is Las Vegas; Hou is Houston.


In the authors’ opinion, a pour strip open period of at least 30 days is recommended to achieve a useful degree of shortening relief. For PT concrete floors with long plan dimensions, effects of RTS should always be investigated— especially for supports between foundations and the first elevated floor. As a general approach, apply calculated unrestrained member strains to a full structural model with appropriate boundary conditions. Because both concrete creep and shrinkage occur slowly over time, restraints will also creep, reducing their effective stiffness, so the afore-mentioned estimated strains are typically reduced to more realistically represent their eventual effects on structure members. For example, the PCI Handbook13 divides strains by a factor of 4.0 for concrete creep and shrinkage. Others suggest dividing concrete creep and shrinkage by a factor of 2.14 However, a detailed discussion of the interaction of support restraints and slab shortening, and the effects of shortening on structural member design, is beyond the scope of this study.

CONCLUSIONSIn the second part of this study, the B3 model is used to

predict concrete creep and shrinkage considering the effects of v/s, concrete strength, and different relative humidity values during and after construction. Detailed procedures are provided to estimate shortening of PT floors with or without the use of delayed-pour strips. Based on the results of this study, the following conclusions can be drawn:

1. For the concrete properties and parameters shown in Table 1(b) of Part I and relative humidity between 40 and 70%, the B3 model predicts shrinkage, creep, and elastic shortening to account for approximately 67%, 29%, and 4%, respectively, of the total shortening.

2. Concrete floor total shortening decreases as concrete strength and relative humidity increase. For commonly used concrete with strength between 4000 and 7000 psi (27.6 and 48.3 MPa) and loaded at 3 days after concrete placement, at a relative humidity less than 60%, total shrinkage and creep shortening is between 750 and 950 me.

3. As a rule of thumb, total unrestrained concrete floor shortening can be taken as 1 in. (25.4 mm) per 100 ft (30.5 m) of floor plan length for typical PT concrete structures. Total unrestrained floor shortening decreases for longer construc-tion periods in humid regions, and for shorter construction periods in dry regions. For a construction period of 2 years, the estimated value could decrease 18% in high-humidity areas, but increase 8% in low-humidity areas.

4. Delayed pour strips provide a rather limited benefit in reducing long-term movements from slab shortening, espe-cially for movements at the far ends of a design strip. A delayed-pour strip is not equivalent to a permanent move-ment joint. For PT concrete floors with long plan dimen-sions, or short dimensions but stiff restraints far from the slab center, RTS should always be investigated.

5. The percentage reduction of movement from floor short-ening provided by pour strips is not uniform for all column locations. The closer a support to a pour strip, the greater the reduction in total movement from floor shortening.

6. A pour strip open period of at least 30 days is recom-mended for some degree of floor shortening relief.

7. Exercise caution when using the aforementioned esti-mated values for situations that are dramatically different from those as assumed in Table 1(b) of Part I. The two most important factors are water content and average precompres-

sion. Ultimate shrinkage may vary with water content to the power of 2.1, as shown in the B3 model prediction function. Concrete creep is proportional to PT floor average compres-sive stress from tendons. Both factors may affect total unre-strained floor slab shortening by a large margin.

ACKNOWLEDGMENTSThe authors wish to express their gratitude to Z. P. Bažant at Northwestern

University, Evanston, IL, for his valuable comments and suggestions.

NOTATIONCd(t,t′,t0) = compliance function for creep due to dryingE = concrete modulus of elasticity at loadingfc′ = concrete strength at 28 daysfcpa = average compressive stress in concrete along member at

center of gravity of tendons immediately after stressingh = relative humidityKes = 0.5 when tendons are tensioned in sequential order to

same tensionkh = humidity dependenceL = concrete floor lengthn = construction period, daysP = concrete average precompression forceq5 = creep at dryingS(t) = time function for shrinkaget = time, dayst′ = age at loading, dayst0 = age when drying begins, daysv/s = volume-to-surface ratioD = elastic shorteningesh(t,t0) = shrinkage strainesh∞ = ultimate shrinkage strainj(t) = shrinkage or drying creep calculated using actual relative

humidity during constructionj′(t) = shrinkage or drying creep calculated using relative

humidity during normal service

REFERENCES1. Post-Tensioning Manual, sixth edition, Post-Tensioning Institute,

Farmington Hills, MI, 2006, 354 pp.2. Liu, B., and Hu, E., “Construction Technology of Long Post-Tensioned

Concrete Floors in a New Hospital Building,” Chinese and Overseas Archi-tecture, No. 3, 2007, pp. 94-95.

3. Li, K., and Sha, A., “Application of Unbonded Prestressing Technology in Super Long Structures,” Quality of Civil Engineering and Construction, No. 8, 2004, pp. 41-43.

4. Guo, G.; Lin, K.; and Joseph, M. L., “Pour Strip Design in Post-Tensioned Buildings Using Unbonded Tendons,” PTI Journal, V. 6, No. 2, Aug. 2008, pp. 21-28.

5. “Restraint Cracks and Their Mitigation in Unbonded Post-Tensioned Building Structures (PTI DC20.2-88),” Post-Tensioning Institute, Farm-ington Hills, MI, 1988, 49 pp.

6. Bondy, K. B., “Shortening Problem in Post-Tensioned Concrete Build-ings,” Design Review and Inspection of Prestressed Concrete Building Proj-ects, SEAOC Seminar Proceedings, Los Angeles, CA, Jan. 1989, pp. 1-19.

7. Bažant, Z. P., “Creep and Shrinkage Prediction Model for Analysis and Design of Concrete Structures—Model B3,” Materials and Structures, V. 28, 1995, pp. 357-365.

8. Kelley, G. S., “Prestress Losses in Post-Tensioned Structures,” PTI Technical Notes, Issue 10, Sept. 2000, 6 pp.

9. Aalami, B., “Prestressing Losses and Elongation Calculations,” ADAPT Technical Note, Issue T9-04, Oct. 1998, 16 pp.

10. Post-Tensioned Concrete Floor Design Handbook, Concrete Society Technical Report No. 43, second edition, The Concrete Society, Camberley, UK, 2005, 110 pp.

11. ACI Committee 362, “Guide for the Design of Durable Parking Struc-tures (ACI 362.1R-97) (Reapproved 2002),” American Concrete Institute, Farmington Hills, MI, 1998, 24 pp.

12. Mohammad, I., “Thermal Movement in Parking Structures,” ACI Structural Journal, V. 104, No. 5, Sept.-Oct. 2007, pp. 542-548.

13. PCI Design Handbook—Precast and Prestressed Concrete (MNL-120-99), fifth edition, Precast/Prestressed Concrete Institute, Chicago, IL, 690 pp.

14. “Design, Construction and Maintenance of Cast-in-Place Post-Tensioned Concrete Parking Structures (PTI DC20.7-01),” Post-Tensioning Institute, Farmington Hills, MI, 2001, 173 pp.


Title no. 110-S05


ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2011-043.R1 received February 14, 2011, and reviewed under Institute


Shear Behavior of Reinforced High-Strength Concrete Beamsby S. V. T. Janaka Perera and Hiroshi Mutsuyoshi

This paper describes the shear behavior of reinforced high-strength concrete (RHSC) beams (fc′ > 100 MPa [14,500 psi]) without web reinforcement. The use of high-strength concrete (HSC) has led to some concerns about its shear strength because of its brittleness, smooth fracture surface, and high early-age shrinkage. Test results indicated that the ratio of uniaxial compressive strength to tensile strength (the ductility number) of the concrete relative to that of the aggregate governs the shear strength of HSC. When the ductility number of the concrete coincided with that of the aggregate, the shear strength remained constant, irrespective of concrete strength. When the ductility number of the concrete was higher than that of the aggregate, however, shear strength started to decrease due to the smooth fracture surface and brittleness. By introducing early-age shrinkage and a suitable aggregate size, the modified compres-sion field theory (MCFT) was found to accurately predict the shear strength of RHSC beams.

Keywords: brittleness; ductility number; fracture surface; high-strength concrete; shear capacity.

INTRODUCTIONHigh-strength concrete (HSC) with a strength fc′

exceeding 100 MPa (14,500 psi) is being increasingly used in buildings and prestressed concrete bridges in Japan because it enables the use of smaller cross sections, longer spans, and reduced girder height while improving dura-bility.1 According to the ACI 318-052 and Japan Society of Civil Engineers (JSCE) Code design equations,3 the diagonal cracking shear strength of reinforced concrete (RC) beams without web reinforcement increases as the concrete strength increases (Fig. 1). However, the diagonal cracking shear strength of reinforced high-strength concrete (RHSC) beams does not increase as expected with the concrete compressive strength.4 Further, increasing the compressive strength of concrete results in greater early-age shrinkage (autogenous shrinkage) due to self-desiccation, brittleness, and smooth-ness of crack fracture surfaces. These limitations have led to some concerns about the shear strength of RHSC beams.

For slender RC beams without web reinforcement, where the shear span-depth ratio (a/d) is greater than 2.5, the diag-onal cracking shear force is carried by: 1) the shear resis-tance of uncracked concrete in the compression zone; 2) the interlocking action of aggregate along the rough concrete surfaces on each side of a crack; and 3) the dowel action of the longitudinal reinforcement. In rectangular beams, the proportions of the shear force carried by these mechanisms are approximately as follows: 20 to 40% by the uncracked concrete of the compression zone, 33 to 50% by aggregate interlocking, and 15 to 25% by dowel action.5

According to past studies,6 the shear resistance of uncracked concrete in the compression zone is lower with HSC as a result of its brittleness. The crack surface of HSC beams is relatively smoother than normal-strength concrete (NSC) because cracks penetrate through the aggregates. The smooth crack surface reduces aggregate interlock and lowers the shear strength of RHSC beams.4 Until now, no

research has attempted to quantitatively evaluate the rough-ness of concrete fracture surfaces. Also, early-age shrinkage causes deterioration in shear strength at diagonal cracking of RHSC beams. Maruyama et al.7 detected cracking around reinforcing bars due to early-age shrinkage of HSC and, by comparing self-induced stresses in RC prisms with different early-age shrinkages, concluded that such cracking degrades the bond stiffness. This means that the dowel action of the longitudinal reinforcement is affected by early-age shrinkage. Previous studies7,8 have also shown that the use of admix-tures, such as expansive additives and shrinkage-reducing agents, is effective in reducing early-age shrinkage. It has also been shown that an increase in the a/d results in a reduction in shear strength.4 However, most of the studies mentioned have been carried out using concrete with a strength of less than 100 MPa (14,500 psi) due to design limitations.

Against this background, the objectives of this study are: 1) to quantitatively explain the effect of concrete compressive strength, brittleness, fracture surface rough-ness, aggregate strength, and a/d on the diagonal cracking shear behavior of RHSC beams where the concrete strength exceeds 100 MPa (14,500 psi); and 2) to accurately predict the shear capacity of RHSC beams using the modified compression field theory (MCFT).9

RESEARCH SIGNIFICANCEThe rapidly increasing use of HSC is outpacing the devel-

opment of appropriate recommendations for its application.

Fig. 1—Comparison of design code formulas. (Note: 1 MPa = 145 psi.)


ACI member S. V. T. Janaka Perera is a Postgraduate Student in the Department of Civil and Environmental Engineering at Saitama University, Saitama, Japan. He received his BSc and MPhil in civil engineering from the University of Mora-tuwa, Moratuwa, Sri Lanka. His research interests include high-strength concrete, prestressed concrete, and composite construction.

ACI member Hiroshi Mutsuyoshi is a Professor in the Department of Civil and Envi-ronmental Engineering at Saitama University. He received his Doctorate in engineering from the University of Tokyo, Tokyo, Japan. He is a member of ACI Committee 440, Fiber-Reinforced Polymer Reinforcement, and the ACI International Partnerships & Publications IC Subcommittee. His research interests include high-strength concrete, prestressed concrete, and fiber-reinforced polymer reinforcement.

There is currently no research that quantitatively explains why the shear strength of RHSC beams (fc′ > 100 MPa [14,500 psi]) without web reinforcement does not increase with increasing concrete compressive strength. This study aims to fill this information gap. The shear behavior of RHSC is described using the ductility number of concrete and aggregate. Experimental data are compared with MCFT predictions to assess their accuracy. The results presented in this paper will lead to improved recommendations for the use of HSC.

EXPERIMENTAL INVESTIGATIONTo investigate the influence of concrete compressive strength,

early-age shrinkage, brittleness, fracture surface roughness, aggregate strength, and a/d on the diagonal cracking shear behavior of RHSC beams, 12 beams without web reinforce-ment were fabricated for study. The test variables are summa-rized in Table 1. A brief summary of the experimental inves-tigation and an introduction to the concept of the brittleness index and fracture surface roughness index follows.

MaterialsTable 2 indicates the concrete mixture proportions. The

materials consist of silica fume cement, in powder form, with a 5720 cm2/g (2791 ft2/lb) specific surface area (SSA), as measured by Blaine’s method; an expansive lime-based additive (EX) with an SSA of 3480 cm2/g (1698 ft2/lb); a polycarboxylic acid-based high-range water-reducing admixture (HRWRA); and an air-reducing admixture (DA). Table 3 lists the properties of the coarse aggregates used. Three types of reinforcing bar were used; refer to Fig. 2 for their mechanical properties.

Table 1—Test variables and beam test results

Specimen Concrete type a/d fc′, MPa ft, MPa Vfl, kN Vc, kN Vu, kN Failure mode

NSC40-I NSC40 3.0 38 3.2 23.5 75.0 75.0 DT

NSC40-II NSC40 3.5 38 3.4 20.5 78.0 78.0 DT

NSC40-III NSC40 4.0 36 3.1 17.5 76.5 76.5 DT

HA100-I HA100 3.0 133 6.1 25.0 85.5 142.0 SC

HA100-II HA100 3.5 116 5.4 25.0 85.0 93.0 SC

HA100-III HA100 4.0 114 5.2 26.5 85.0 85.0 DT

HA120 HA120 4.0 138 7.2 28.0 82.5 82.5 DT

HA160-I HA160 3.0 165 7.4 27.5 81.0 226.0 SC

HA160-II HA160 3.5 194 6.8 28.0 77.0 80.0 SC

HA160-III HA160 4.0 183 7.4 27.5 75.0 105.5 SC

LA120 LA120 4.0 155 8.3 34.5 85.0 120.5 SC

LA160 LA160 4.0 175 8.5 35.2 67.0 153.0 SC

Notes: Vfl is flexural cracking load; Vc is shear force at diagonal cracking; Vu is shear force at failure; DT is diagonal tension failure; SC is shear compression failure; longitudinal tensile reinforcement ratio rw is 3.04%; 1 kN = 225 lbf; 1 MPa = 145 psi.

Table 2—Mixture proportions of concrete

Concrete type W/B, %

Unit weight, kg/m3

W OPC SF S G EX HRWRA DA

NSC40 46 168 363 — 1017 757 — 3.45 —

HA80 30 160 — 534 873 795 — 6.94 0.53

HA100 23 165 717 — 608 940 — 12.9 0.78

HA120 20 155 — 775 703 792 — 11.63 0.78

HA160 17 155 — 912 592 792 — 14.59 0.91

LA120 20 155 — 750 703 792 25 11.63 0.78

LA160 17 155 — 882 595 792 30 14.59 0.91

Notes: B is binder; B = OPC + SF + EX; W is water; OPC is ordinary portland cement; SF is silica fume cement; S is sand; G is gravel; EX is expansive additive; HRWRA is high-range water-reducing admixture; DA is air-reducing admixture; 1 kg/m3 = 0.062 lb/ft3.

Table 3—Properties of aggregates

Type Density, g/cm3 Absorption, % Fineness modulus sc, MPa st, MPa sc/st Maximum size

Crushed granite 2.53 0.42 6.64 190 to 285 8.9 to 15.3 18.6 to 21.3 19 mm

Notes: sc is uniaxial compressive strength; st is tensile strength; 1 mm = 0.039 in.; 1 g/cm3 = 0.036 lb/in.3; 1 MPa = 145 psi.


Specimen designThe cross sections and layout of the test beams are

shown in Fig. 2. In the experiments, the concrete compres-sive strength, early-age shrinkage, and a/d were varied, as shown in Table 1. All beams were 200 mm (7.9 in.) wide and had an effective depth of 250 mm (9.8 in.). The value of fc′ was varied from 38 to 194 MPa (5511 to 28,137 psi). Due to experimental limitations, there was no beam test with Mixture HA80. However, its shear behavior is already well-documented in existing literature.3,4 The value of a/d was varied from 3.0 to 4.0. Three high-strength steel bars (fy = 750 MPa [109 ksi]) were used as longitudinal reinforcement so shear failure would precede flexural failure.

Concrete prisms made of high early-age shrinkage (HA) and low early-age shrinkage (LA) concretes with a cross section of 100 x 100 mm (3.9 x 3.9 in.) and a length of 400 mm (15.7 in.) were prepared to measure shrinkage of the concrete (Fig. 3). Prisms of the same dimensions were prepared for fracture energy measurements (with a 30 mm [1.18 in.] notch height) (Fig. 4). Cylinder speci-mens measuring 100 mm (3.9 in.) in diameter and 200 mm (7.9 in.) in height were prepared for the compressive strength and modulus of elasticity tests; others measuring 150 mm (5.9 in.) in diameter and 150 mm (5.9 in.) in height were prepared for the splitting tensile strength tests.

Immediately after casting, all specimens, including the beams, cylinders, and square prism specimens, were covered with polyethylene sheeting to reduce concrete drying. All beams and specimens were moist-cured for approximately 28 days and were tested 28 to 30 days after casting.

To determine the aggregate strengths, uniaxial compres-sive strength sc and tensile strength st tests of rock cylin-ders were measured. Cylinder specimens measuring 50 mm (2.0 in.) in diameter and 100 mm (3.9 in.) in height were prepared for the uniaxial compressive strength tests; others measuring 50 mm (2.0 in.) in diameter and 50 mm (2.0 in.) in height were prepared for the tensile strength tests (measured using the Brazilian test). Refer to Table 3 for the results of these tests.

Roughness index Rs of fracture surfaceThe interlocking action of aggregate along a crack can be

described using post-failure evidence from the fracture surface. It is commonly recognized that the roughness of the fracture surface can vary depending on the concrete mixture design. Until now, however, this has not been quantitatively explained.

For the surface roughness test, fractured splitting tensile strength test specimens were tested, as they were used to measure the tensile capacity of concrete. The fracture surfaces of these specimens were not damaged because they failed in Mode I. A laser light confocal microscope was used

to scan the fractured surface three-dimensionally. The rough-ness index Rs was calculated from the directly measured surface area,10 as shown by Eq. (1) (Fig. 5).

sAiRA

∑=∑

(1)

Fig. 2—Details of RC beam. (Note: Dimensions in mm; area of D6 steel bar is 31.7 mm2 with yield strength fy = 360 MPa and Young’s modulus of steel Es = 187 GPa; area of D19 steel bar is 286.5 mm2 with fy = 384 MPa and Es = 200 GPa; area of D25 steel bar is 506.7 mm2 with fy = 750 MPa and Es = 201 GPa; a/d = 4.0; 1 mm = 0.039 in.; 1 mm2 = 0.00155 in.2; 1 MPa = 145 psi; 1 GPa = 145 ksi.)

Fig. 3—Details of early-age shrinkage specimen. (Note: Dimensions in mm; 1 mm = 0.039 in.; t is thickness.)

Fig. 4—Experimental load-crack mouth opening displace-ment curves from fracture energy tests. (Note: 1 kN = 225 lbf; 1 mm = 0.039 in.)


where Ai is the fractured surface area; and A is the projected surface area.

Brittleness index BVarious parameters have been proposed to characterize the

brittleness of concrete. The characteristic length lch = EGF/ft2,

as proposed by Hillerborg,11 has been used to characterize the brittleness of concrete, rock, and glass. The normal-ized shear strength vc/ft of geometrically similar beams is governed by the dimensionless ratio between absolute struc-ture size D and lch.12 This ratio is regarded as a measure of the brittleness of structural elements that are sensitive to tensile stress-induced fracture. A higher value of B corresponds to a more brittle structural element.

2t

F

f DB

EG= (2)

where ft is tensile strength; E is the Young’s modulus; GF is the fracture energy in Mode I; D is the absolute structural element size (in the case of a beam, equal to the effective depth of the beam)12,13; and vc is shear strength. The tensile strength and modulus of elasticity of concrete are dependent on compressive strength, and fracture energy is dependent on aggregate size and compressive strength.

Test procedureThe test beams were simply supported and loaded symmet-

rically with two equal concentrated loads. For the entire

test program, the distance between the two point loads was kept constant at 300 mm (11.8 in.). The dimensions of the loading and reaction plates were 25 x 80 x 200 mm (1 x 3.1 x 7.9 in.). At each load increment, the vertical deflection and the strains at the top and bottom of the beam were measured. Visual observations of any cracks were made during the test and all cracks were marked. The failure of the beams was brittle and accompanied by a loud noise. Each beam test was followed by tests of compressive strength, splitting tensile strength, modulus of elasticity, and fracture energy.

Concrete shrinkage was measured immediately after placement. A strain gauge with a reference length of 100 mm (3.9 in.) was embedded at midheight in the center of the 100 x 100 x 400 mm (3.9 x 3.9 x 15.7 in.) prisms. A fracture energy test of the concrete was carried out at almost the same age as the beams when subjected to loading tests, as outlined in a Japan Concrete Institute (JCI) technical report published in 2003.14

To calculate the roughness index, the fracture surface of the splitting tensile strength test specimen was scanned as described previously. Due to instrument limitations, a 75 x 75 mm (3.0 x 3.0 in.) area of the fractured surface at the center of the specimen was scanned with a 250 mm (9842 min.) pixel size and a resolution of 0.01 mm (0.4 min.).

EXPERIMENTAL RESULTS AND DISCUSSIONThe test results are presented in Tables 1 and 4. The results

in Table 1 include the observed values for the diagonal cracking load, failure load, and mode of failure observed. Information on the properties of concrete is presented in Table 4. The experimental results and discussion are presented in four categories: the properties of concrete, load-deflection relationship, effect of compressive strength, and comparison of shear behavior of RHSC beams.

Properties of concreteThe compressive strength, splitting tensile strength,

Young’s modulus Ec, fracture energy, and shrinkage strain of the concrete at the time of the beam test are tabulated in Table 4. Figure 4 shows the effect of experimental load on crack mouth opening displacement (CMOD) observed in fracture energy tests. According to the test results, a 15% greater ft was observed in LA concrete compared with HA concrete. However, Ec and GF were not significantly higher in LA concrete than in HA concrete. In other words, the effect of early-age shrinkage on the mechanical properties of HSC was not noticeable other than in the value of ft.

The roughness index Rs of the fracture surface is described using the ductility number DN. In rock mechanics, the DN (or brittleness number) is the ratio of uniaxial compressive strength to tensile strength. It is the most widely used index for the quantification of rock brittleness.15,16 The higher the DN, the more brittle the material. As the DN increases, the size of the crushed zone caused by loading increases, as well as the number and length of main cracks outside the crushed zone.16 The two strength measures (sc and st) have maximum and minimum values that depend on the orientation of grains in the rock.17 According to Table 3, the aggregate type (crushed granite) used in this study has a DN in the region of 18 to 22. It should be noted that the aggregate content of the concrete mixture proportions used in this study is approximately equal, except in the case of Mixture HA100 (Table 2). The aggregate content of Mixture

Fig. 5—Schematic view of roughness parameter (Rs = SAi/SA).

Table 4—Properties of concrete

Concrete type fc′, MPa ft, MPa Ec, GPa

GF, N/mm esh × 10–6 lch, mm

NSC40 36 3.1 32.1 0.200 –73 676

HA80 81 4.9 34.1 0.223 –95 258

HA100 114 5.2 36.5 0.220 –322 347

HA120 138 7.2 39.4 0.229 –412 176

HA160 183 7.4 43.5 0.250 –511 200

LA120 155 8.3 41.0 0.248 –168 166

LA160 175 8.5 44.7 0.259 –225 161

Notes: Ec is Young’s modulus of concrete; esh is shrinkage strain in concrete; 1 MPa = 145 psi; 1 N/mm = 0.175 lbf/in.


HA100 was different from other mixtures to investigate the effect of aggregate content on the Rs of HSC.

Test results indicated that the DN of NSC was between 11 and 13 and it was more ductile than the aggregate (Table 3). Inspection of the failure surfaces of NSC speci-mens indicated that more than four-fifths of the aggregate particles had fractured with sharp edges. Therefore, Rs has a maximum value (1.269) in NSC (fc′ = 36 MPa [5221 psi]).

The DN of HSC with a strength between 114 and 155 MPa (16,534 and 22,481 psi) was between 18 and 22, coinciding with that of the aggregate (refer to Fig. 6). Inspection of the failure surfaces of HSC specimens in this strength region indicated that approximately one-eighth of the aggregate particles had fractured with sharp edges. There was a 14% reduction in Rs as the strength of the concrete increased from 36 to 114 MPa (5221 to 16,534 psi) (Fig. 7). However, as concrete strength further increased from 114 to 155 MPa (16,534 to 22,481 psi), the change in Rs was minimal (Fig. 8). This was due to the similar brittle behavior of both HSC and aggregate. Also, the value of Rs of Mixture LA160 with a strength of 175 MPa (25,382 psi) was the same as that of Mixture LA120 with a strength of 155 MPa (22,481 psi). This behavior was a result of brittleness and it will be discussed in a later section of this paper. Moreover, as concrete strength further increased from 155 to 183 MPa (22,481 to 26,542 psi), DN rose from 22 to 29; in this concrete strength region, the concrete is more brittle than the aggregate. Inspection of the failure surfaces indicated that approximately all of the aggre-gate particles had fractured with a flat and smooth fracture surface. As a result, Rs was further reduced, as shown in Fig. 8. The aggregate content of all concrete mixtures was approximately the same, except that of Mixture HA100. Irre-spective of aggregate content, however, the Rs of HSC was highly dependent on the DN of concrete.

Fig. 6—Effect of compressive strength of concrete on ductility number DN and diagonal cracking shear capacity of RC beams. (Note: Aggregate used in this study was crushed granite; 1 MPa = 145 psi; 1 kN = 225 lbf.)

Fig. 7—Fractured surfaces of splitting tensile strength specimens. (Note: Surface elevation in mm; 1 mm = 0.039 in.; 1 MPa = 145 psi.)


Load-deflection relationshipFigure 9 shows the load-deflection curves of tested

beams with a/d = 4.0. All beams exhibit similar behavior; Beam HA160-III is described herein as an example. In the HA160-III load-deflection curve, flexural cracks first appeared at an early stage of loading. The load dropped slightly after formation of the first flexural crack and then continued to rise. The flexural cracking load at which the first flexural cracking formed was 43 to 102% greater in RHSC beams than NSC beams (Table 1). This was mainly due to the higher tensile strength of HSC.

The diagonal crack then occurred in the shear span and the load dropped sharply. The load soon continued to increase, however, dropping slightly once again with the formation of another crack. Thus, even though diagonal cracking took place, the beam was still able to bear the applied load through arch action. Finally, the beam failed in shear compression when the diagonal cracks in the shear span widened and the concrete near the crack tip in the compression zone was crushed. Beams HA100-I, HA100-II, LA120, HA160-I, HA160-II, HA160-III, and LA160 all failed in shear compression, while all other beams, including HA120 and HA100-III, failed in diagonal tension. Diagonal tension failure occurred just after the occurrence of critical diagonal cracking.

In RC beams with a concrete strength exceeding 100 MPa (14,500 psi), when the a/d was greater than 3.0, failure occurred after diagonal cracking and the beams failed in shear compression, as described previously. In the case of RHSC beams (HA100-III and HA120), however, when the

a/d was 4.0, diagonal cracking became unstable and the beams failed in diagonal tension. For RHSC beams (fc′ > 100 MPa [14,500 psi]), the transition point between shear compression failure and diagonal tension failure shifted to a/d greater than 4.0, while for NSC beams it was 3.013 (Table 1).

LA beams tended to fail at higher loads in shear compres-sion after the formation of an arch mechanism as compared with HA beams (Table 1). This behavior could be due to the strength of the compression strut and is closely related to the compressive strength of the concrete. Also, the reduced early-age shrinkage (by 55 to 60%) in LA beams improved both bond stiffness between the reinforcement and concrete,6 as well as the stiffness of the compression zone. As a result, the failure loads were increased. Due to the small sample size, however, more tests are needed to confirm this.

Effect of compressive strengthAfter cracking, shear is resisted by aggregate interlock,

the dowel action of tension reinforcement bars, and resis-tance provided by uncracked concrete in the compression zone of the beam.5 The percentage carried by aggregate interlock depends strongly on the surface roughness at the crack. Examination of failure surfaces of the beams and splitting tensile strength specimens revealed that the fracture surfaces were similar and the crack surfaces were smooth in HSC beams (Fig. 7 and 8), indicating that the shear force carried by aggregate interlock decreases with increasing fc′. There was an 8% decrease in Rs between concrete with a strength of 36 and 81 MPa (5221 and 11,748 psi). A further 8% decrease was observed as the strength of the concrete increased from 81 to 138 MPa (11,748 to 20,015 psi). However, as concrete strength further increased from 138 to 183 MPa (20,015 to 26,542 psi), the change in Rs was minimal. Test results indicated that the diagonal cracking shear strength of the RHSC beam with a concrete strength of 114 MPa (16,534 psi) (Beam HA100-III) was 11% higher than that with a concrete strength of 36 MPa (5221 psi) (Beam NSC40-III). This increase was due to the ductility of concrete and the 67% increase in ft. The shear strength of RHSC beams was constant, with a maximum value for concrete strengths between 114 MPa (16,534 psi) (Beam HA100-III) and 155 MPa (22,481 psi) (Beam LA120). This behavior was due to the DN of the concrete and aggre-gate being approximately equal (Fig. 6). However, the shear strength of Beam HA160-III (fc′= 183 MPa [26,542 psi]) was 12% lower than that with a concrete strength of Beam LA120 (fc′ = 155 MPa [22,481 psi]) (Fig. 6). This reduction was due to the increase in the DN of the concrete rather than any effect of the aggregate. However, the shear strength of Beam LA160 (fc′ = 175 MPa [25,382 psi]) was lower than that of other RHSC beams, although its DN value was main-tained using LA concrete. This behavior was due to the brittleness of Mixture LA160 compared to other concrete types; the RHSC beam with a concrete strength of more than 155 MPa (22,481 psi) did not stay constant as expected with increasing fc′ (Table 1).

Comparison of shear behavior of RHSC beamsMeasuring the diagonal cracking load experimentally is

not easy. In this study, shear strength is defined as the shear stress at the point when the diagonal crack that causes failure becomes inclined and extends beyond the neutral axis.

To analyze the brittleness of beams and the diagonal cracking load relationship in detail, the normalized shear

Fig. 8—Fracture surface roughness of splitting tensile strength specimens. (Note: 1 MPa = 145 psi.)

Fig. 9—Comparison of load-deflection relationship of RC beams (a/d = 4.0). (Note: 1 kN = 225 lbf; 1 mm = 0.039 in.)


strength vc/ft was used. The brittleness index B is inversely proportional to material shear strength. Therefore, for a better understanding, 1/B was analyzed with respect to vc/ft. To show the continuity of the shear behavior of RHSC beams, data reported previously by Fujita et al.18 and Sato and Kawakane8 were used. All data were for an effective depth of 250 mm (9.8 in.), an a/d of 3.0, and a longitudinal reinforcement ratio of 1.53%.

As can be seen in Fig. 8 and 10, the values of Rs and 1/B decreased by 15% and 74%, respectively, with increasing concrete strength from NSC (Beam NSC40-III with a strength of 36 MPa [5221 psi]) to HSC (Beam HA120 with a strength of 138 MPa [20,015 psi]). The value of vc/ft (Table 1 and Fig. 11) decreased from 0.49 to 0.23. In Beam HA120, the thickness of the uncracked compression zone also decreased with increasing concrete strength. Because the beam was very brittle, this thin uncracked compression zone was reduced further by flexural cracking. As a result of these factors, the vc/ft value of Beam HA120 was reduced by 53%.

The value of Rs changed slightly (approximately 2%) when concrete strength increased from 138 to 175 MPa (20,015 to 25,382 psi). Also, vc/ft and 1/B fell slightly to 0.16 and 0.64 at a concrete strength of 175 MPa (25,382 psi), respectively. In this region of concrete strength, the normal-ized shear strength of both Beams LA120 and LA160 was lower than that of Beams HA120 and HA160, respectively. This is because of the lower 1/B value in LA beams compared to HA beams (LA beams were more brittle).

At a concrete strength of 183 MPa (26,542 psi), an increase in both vc/ft (0.20; a 33% increase) and 1/B (0.8; a 25% increase) was seen. At the same time, the change in Rs was minimal at a concrete strength of 183 MPa (26,542 psi). Therefore, it is possible to conclude that the behavior of vc/ft and 1/B in HSC beams is proportional (Fig. 10 and 11). Also, the normalized shear strength of HSC beams greatly depended on the brittleness index of the beam.

ANALYTICAL INVESTIGATIONThe diagonal cracking shear strength results obtained in

the tests described previously are compared with values calculated on the basis of the recommendations given in ACI 318-05,2 the JSCE Code,3 equations proposed by Fujita et al.18 and Khuntia and Stojadinovic,4 and the modified compression field theory (MCFT).9,19 (Note: Equations in this section are in SI units; 1 MPa = 145 psi.)

ACI 318-052

ACI 318-05,2 Eq. (11-5)

1 20.158 17.2 (MPa)uc c w

u

V dv f

M= + r′ (3)

In this code, the recommended highest compressive strength of concrete is 70 MPa (10,000 psi). (Note: This equation is given in U.S. Customary units in Reference 8.)

JSCE Code3

Eq. 6.3.3

( ) ( )1 31 4 1 3100 0.2 (MPa)c w cv d f−= r ′ (4)

According to this code, when the characteristic compres-sive strength of concrete exceeds 80 MPa (11,600 psi), there may not be a significant increase in vc.

Equation proposed by Fujita et al.18

( ) ( )( )1 31 2 1 2180 100 0.75 1.4 (MPa)c c wv f d a d− −= r +′ (5)

In this code, the recommended compressive strength of concrete is between 80 and 125 MPa (11,600 and 18,130 psi).

Equation proposed by Khuntia and Stojadinovic4

0.5

30.537 (MPa)cc w c

u

V dv f

M

= r ′ (6)

where vc is diagonal cracking shear stress; Vc is the diagonal cracking shear force; Vu is the shear force at the section considered; Mu is the bending moment at the section consid-ered; d is effective depth; and rw is the longitudinal reinforce-ment ratio.

This equation was found to accurately predict shear strength in concrete with a compressive strength between 28 and

Fig. 10—Behavior of brittleness index. (Note: 1 MPa = 145 psi.)

Fig. 11—Behavior of normalized shear strength with compressive strength of concrete. (Note: 1 MPa = 145 psi.)


83 MPa (4000 and 12,000 psi).4 (Note: This equation is given in U.S. Customary units in Reference 4.)

MCFTThe MCFT9 relates the shear stress that is transmitted

across a crack to the crack width, the maximum aggregate size, and the concrete strength as

0.18 (MPa, mm)

240.3116

cci

g

fv

wa

′=

++

(7)

where ag is the maximum aggregate size; and w is the crack width.

1 (mm)w sq= e (8)

where e1 is the principal tensile strain in the cracked concrete; and sq is the crack spacing in the q direction. (Note: Equations (6) and (7) are given in U.S. Customary units in Reference 9.)

Aggregate size does not influence aggregate interlock capacity in HSC because of the smooth fracture surface. To account for this, Bentz19 suggested that an effective maximum aggregate size be calculated by reducing ag to zero as fc′ increases from 60 to 80 MPa (8500 to 11,603 psi).

Comparison of predictions and experimental resultsAccording to the ACI 318-052 and the JSCE Code3 shear

prediction equations without limitations, the shear strength of RC beams increases as concrete strength increases. However, the experimental results obtained in this study showed that with increasing concrete strength, the shear strength of RHSC beams did not increase (Fig. 12). There-fore, for all RHSC beams, ACI 318-052 and the JSCE Code3 overpredicted the shear strength by between 9 and 73% (ACI 318-052) and 18 to 77% (JSCE3) (Table 5). However, the predictions made by Fujita et al.18 and Khuntia and Stojadinovic4 were found to be conservative: the averages of the ratio of the tested-to-predicted shear strength were 1.12 and 1.05 with standard deviations of 0.08 and 0.11, respectively. In the equation by Fujita et al.,18 the material properties factor in the shear strength of HSC was evaluated using fc′–1/2, unlike NSC, which is evaluated using fc′1/2 or fc′1/3. Among the aforementioned methods, those proposed by Fujita et al.18 and Khuntia and Stojadinovic4 were found to be the most reliable for design purposes.

According to the MCFT,9,19 an increase in ag improves the ability of cracked concrete to transfer shear. Therefore, in

Fig. 12—Comparison of design code formulas with experi-mental results (a/d = 4.0). (Note: 1 MPa = 145 psi.)

Table 5—Comparison of experimental results with predicted values of diagonal cracking shear strength

Specimen

Vn,test/Vn,predicted

Eq. (3) by ACI 318-052 Eq. (11-5)

Eq. (4) byJSCE3

Eq. (5) byFujita et al.18

Eq. (6) by Khuntia andStojadinovic4

ByMCFT9,19

ByMCFT*

NSC40-I 1.21 1.09 (0.49) 1.18 1.12 1.12

NSC40-II 1.31 1.13 (0.54) 1.27 1.21 1.21

NSC40-III 1.36 1.13 (0.54) 1.30 1.27 1.27

HA100-I† (0.82) (0.82) (1.05) (1.09) (0.91) 1.06

HA100-II† (0.89) (0.85) 1.03 (1.15) 1.00 1.09

HA100-III† (0.91) (0.86) 1.06 (1.19) 1.07 1.13

HA120† (0.81) (0.78) (1.14) (1.12) (0.97) 1.08

HA160-I† (0.71) (0.72) (1.10) (1.00) (0.81) 0.97

HA160-II† (0.64) (0.65) (1.20) (0.96) (0.77) 0.91

HA160-III† (0.65) (0.64) (1.19) (0.97) (0.80) 0.94

LA120† (0.79) (0.77) (1.24) (1.13) (0.97) 1.02

LA160† (0.59) (0.58) (1.04) (0.87) (0.73) 0.78

Average (RHSC beams) 0.76 0.74 1.12 1.05 0.89 1.00

SD (RHSC beams) 0.11 0.10 0.08 0.11 0.12 0.11

Average (all beams) 0.89 0.84 0.97 1.10 0.97 1.05

SD (all beams) 0.26 0.19 0.28 0.13 0.17 0.13*MCFT9,19 with authors’ recommended modifications. †RHSC beam. Notes: Vn,test is experimental shear strength; Vn,predicted is predicted shear strength; SD is standard deviation; values in parentheses are predictions outside recommended concrete strength.


this study, the value of ag was varied to represent the influ-ence of fracture surface roughness. As Bentz19 suggested, ag was taken as zero for concrete with a strength greater than 80 MPa (11,603 psi). According to the analysis, the average of the ratio of the tested shear strength to the predicted shear strength of RHSC beams was 0.89 with a standard deviation of 0.12.

According to this study, however, the surface roughness of NSC specimens with a strength of 36 MPa (5221 psi) was approximately 8% and 20% greater than that of HSC specimens with a strength of 81 and 155 MPa (11,748 and 22,481 psi), respectively. Therefore, during the analysis, ag was reduced linearly from ag = 19 mm (0.748 in.) to zero as fc′ increased from 36 to 155 MPa (5221 to 22,481 psi) (Fig. 13). Additionally, because crack width increases with an increase of e1—thereby reducing shear transfer capacity—in this analysis, early-age shrinkage strain was uniformly added to the concrete strain along the longitudinal axis of a beam. This was to represent the degradation of bond strength between reinforcing bars and HSC. The early-age shrinkage strain of each RC beam was taken from Table 4 with respect to its concrete type.

The test results compared with predictions based on the MCFT using these recommended modifications are reported in Table 5. According to the results, the average ratio of the tested-to-predicted shear strength of RHSC beams was 1.00 with a standard deviation of 0.11 (Table 5). There is a clear need to modify the MCFT to better evaluate aggre-gate interlocking and the brittleness of HSC. More testing is currently required to study this problem further.

In brief, the MCFT9,19 with a recommended modifica-tion was found to be reliable for design purposes. On the other hand, the predictions given by ACI 318-052 and the JSCE Code3 were found to overestimate the shear capacity of RHSC beams.

CONCLUSIONSThe shear behavior of RHSC beams (fc′ > 100 MPa

[14,500 psi]) without web reinforcement was investigated. The results of a series of tests on 12 beams are presented and analyzed. Based on these results, the following conclusions are drawn:

1. By considering concrete brittleness and fracture surface roughness in conjunction with the ductility numbers of concrete and coarse aggregate, designers’ understanding of shear behavior will be enhanced.

2. The ductility number of the aggregate relative to that of concrete governs the fracture surface roughness of concrete and the shear strength of HSC. When the ductility number of concrete was lower than that of the aggregate, the diag-onal cracking shear strength increased with the increase of concrete strength due to the rough fracture surface and increased tensile strength. When the ductility numbers of the concrete and aggregate were equal, shear strength stayed constant at the maximum value. When concrete had a higher ductility number than the aggregate, however, shear strength decreased due the smooth fracture surface and high brittle-ness of the concrete. In this study, however, the maximum coarse aggregate size was 19 mm (0.748 in.) and the rock type was crushed granite. Therefore, further studies on different aggregate sizes and rock types are essential.

3. The ductility number of the aggregate (crushed granite) used in this study ranged from 18 to 22. The ductility number of the NSC was between 11 and 13.

Therefore, diagonal cracking shear strength increased by approximately 9 to 14% as concrete strength increased from 36 to 114 MPa (5221 to 16,534 psi). Concrete with a strength between 114 and 155 MPa (16,534 and 22,481 psi) has the same ductility number as the aggregate. In this strength region, shear strength is not dependent on concrete strength. The ductility number of concrete with a strength greater than 155 MPa (22,481 psi) was more than 22. There-fore, shear strength started to decrease due to the smooth fracture surface and high brittleness of the concrete.

4. The change in the fracture surface roughness index of beams with a concrete strength between 155 and 183 MPa (22,481 and 26,542 psi) was minimal, while the beam with a concrete strength of 155 MPa (20,015 psi) was approximately 4% lower than the beam with a concrete strength of 114 MPa (16,534 psi). The change in the fracture surface roughness index of the beam with a concrete strength of 114 MPa (16,534 psi) was approximately 14% lower than that of the beam with a concrete strength of 36 MPa (5221 psi).

5. The ACI 318-052 and JSCE Code3 equations for evalu-ating the shear strength of HSC beams need to be modified according to the suggestions made in this paper.

6. The average ratio of the tested-to-predicted shear strength of RHSC beams using the MCFT and including the authors’ recommendations was 1.00 with a standard devia-tion of 0.11. This compares with the previous result without the authors’ modifications, where the average of the ratio of tested-to-predicted shear strength was 0.89 with a standard deviation of 0.12.

7. The MCFT can predict the effect of fracture surface roughness and early-age shrinkage of concrete on diagonal shear strength but should be improved to include the effects of aggregate strength and concrete brittleness.

ACKNOWLEDGMENTSSupport for this research by the Grant-in Aid for Scientific Research

provided by the Ministry of Education, Culture, Sports, Science and Tech-nology in Japan is greatly appreciated. Special thanks to R. Takeda and H. Watanabe from the graduate school at Saitama University for assisting with the experiments.

REFERENCES1. Mutsuyoshi, H.; Ohtsuka, K.; Ichinomiya, T.; and Sakurada, M.,

“Outline of Guidelines for Design and Construction of High Strength Concrete for Prestressed Concrete Structures,” Proceedings of JSCE of 8th International Symposium on Utilization of High Strength and High Perfor-mance Concrete, Tokyo, Japan, Oct. 2008, pp. 111-117.

Fig. 13—Recommended modification to MCFT. (Note: 1 MPa = 145 psi; 1 mm = 0.039 in.)


2. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2005, 430 pp.

3. JSCE Guidelines for Concrete No. 3, “Structural Performance Veri-fication,” Standard Specification for Concrete Structures—2002, Japan Society of Civil Engineers (JSCE), Tokyo, Japan, 2002.

4. Khuntia, M., and Stojadinovic, B., “Shear Strength of Reinforced Concrete Beams without Transverse Reinforcement,” ACI Structural Journal, V. 98, No. 5, Sept.-Oct. 2001, pp. 648-656.

5. Taylor, R., and Brewer, R. S., “The Effect of the Type of Aggregate on the Diagonal Cracking of Concrete Beams,” Magazine of Concrete Research, V. 115, No. 44, July 1963, pp. 87-92.

6. Gettu, R.; Bažant, Z. P.; and Karr, M. E., “Fracture Properties and Brit-tleness of High-Strength Concrete,” ACI Materials Journal, V. 87, No. 6, Nov.-Dec. 1990, pp. 608-618.

7. Maruyama, I.; Kameta, S.; Suzuki, M.; and Sato, R., “Cracking of High Strength Concrete around Deformed Reinforcing Bar due to Shrinkage,” International RILEM-JCI Seminar on Concrete Durability and Service Life Planning, K. Kovler, ed., Ein-Bokek, Israel, 2006, pp. 104-111.

8. Sato, R., and Kawakane, H., “A New Concept for the Early Age Shrinkage Effect on Diagonal Cracking Strength of Reinforced HSC Beams,” Journal of Advanced Concrete Technology, V. 6, No. 1, Feb. 2008, pp. 45-67.

9. Vecchio, F. J., and Collins, M. P., “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear,” ACI Journal, Proceedings V. 83, No. 2, Mar.-Apr. 1986, pp. 219-321.

10. Lange, D. A.; Jenings, H. M.; and Shah, S. P., “Relationship between Fracture Surface Roughness and Fracture Behavior of Cement Paste and Mortar,” Journal of the American Ceramic Society, V. 76, No. 3, 1993, pp. 589-597.

11. Hillerborg, A., “Results of Three Comparative Test Series for Deter-mining the Fracture Energy GF of Concrete,” Materials and Structures, V. 18, No. 107, Sept.-Oct. 1985, pp. 407-413.

12. Gustafsson, P. J., and Hillerborg, A., “Sensitivity in Shear Strength of Longitudinal Reinforced Concrete Beams to Fracture Energy of Concrete,” ACI Structural Journal, V. 85, No. 3, May-June 1988, pp. 286-294.

13. Kim, J. K., and Park, Y. D., “Prediction of Shear Strength of Reinforced Concrete Beams without Web Reinforcement,” ACI Materials Journal, V. 93, No. 3, May-June 1996, pp. 213-222.

14. JCI-S-00-2003, “Method of Test for Fracture Energy of Concrete by Use of Notched Beam,” Japan Concrete Institute (JCI), Tokyo, Japan, 2003.

15. Goktan, R. M., and Yilmaz, N. G., “A New Methodology for the Analysis of the Relationship between Rock Brittleness Index and Drag Pick Cutting Efficiency,” Journal of the South African Institute of Mining and Metallurgy, V. 105, Nov. 2005, pp. 727-733.

16. Gong, Q. M., and Zhao, J., “Influence of Rock Brittleness on TBM Penetration Rate in Singapore Granite,” Tunnelling and Underground Space Technology, V. 22, 2007, pp. 317-324.

17. Prikryl, R., “Some Microstructural Aspects of Strength Variation in Rocks,” International Journal of Rock Mechanics and Mining Sciences, V. 28, 2001, pp. 671-682.

18. Fujita, M.; Sato, R.; Matsumoto, K.; and Takaki, Y., “Size Effect on Shear Strength of RC Beams Using HSC without Shear Reinforcement,” Translation from Proceeding of JSCE, V. 56(711), Aug. 2002, pp. 113-128.

19. Bentz, E. C., “Sectional Analysis of Reinforced Concrete Members,” PhD dissertation, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 2000.

20. Angelakos, D.; Bentz, E. C.; and Collins, M. P., “Effect of Concrete Strength and Minimum Stirrups on Shear Strength of Large Members,” ACI Structural Journal, V. 98, No. 3, May-June 2001, pp. 290-300.


Title no. 110-S06


ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2011-048.R1 received August 1, 2011, and reviewed under Institute


Design of Anchor Reinforcement for Seismic Shear Loadsby Derek Petersen and Jian Zhao

Existing design codes recommend hairpins and surface reinforce-ment consisting of hooked bars encasing an edge reinforcement to improve the behavior of anchor connections in shear. Concrete breakout is assumed to occur before anchor reinforcement takes effect in the current design methods. This paper presents an alter-native design method for anchor shear reinforcement. The proposed anchor shear reinforcement consists of a group of closed stirrups proportioned to resist the code-specified anchor steel capacity in shear and placed within a distance from the anchor bolt equal to the front-edge distance. Steel fracture was achieved in the tests of twenty 25 mm (1 in.) reinforced anchors with a front-edge distance of 152 mm (6 in.). Meanwhile, the observed anchor capacities were smaller than the code-specified anchor steel capacity in shear because concrete cover spalling caused combined bending and shear action in the anchor bolts. Reinforcing bars are needed along all concrete surfaces to minimize concrete damage in front of reinforced anchors for consistent seismic behavior in shear.

Keywords: anchor connections; anchor reinforcement; cast-in anchors; composite construction; fastening; headed studs; seismic design.

INTRODUCTIONConcrete anchor connections are a critical component of

load transfer between steel and concrete members, affecting structural performance during earthquake events. Observa-tions of damage in recent major earthquakes have raised concerns about the seismic performance of anchor connec-tions.1-4 Cast-in-place anchors may experience steel fracture or concrete breakout failure when subjected to a shear force toward a free edge.5 The failure modes are mainly depen-dent on the front-edge distance ca1 when the anchor bolt is placed in plain concrete. Concrete breakout cones, such as the one shown in Fig. 1, vary in shape, while an idealized breakout cone6 (encased in the dashed lines) is generally assumed in calculating the anchor breakout capacity. With the breakout cone partially formed, the anchor bolt may lose concrete support when subjected to reversed cyclic shear loads, leading to unreliable seismic performance.

Building codes5 and design guidelines7,8 allow engineers to use steel reinforcement to increase the shear capacity of anchors placed near an edge. The recommended anchor shear reinforcement usually consists of horizontal hairpins that wrap around the anchor shaft or hooked bars along the direc-tion of the shear force close to the top concrete surface, as illustrated in Fig. 2. The existing design methods5,7,8 assume that the concrete breakout similar to that observed for anchors in plain concrete occurs before steel reinforcement takes effect. With this assumption, the shear resistance of the anchor is exclusively provided by the anchor reinforcement. Anchor reinforcement in terms of hooked bars is required to be fully developed in the assumed breakout cone5 or the contribution from each bar is calculated according to its development length in the assumed breakout cone.7,8 The development length requirements limit the distance from the anchor bolt within which the reinforcement can be deemed effective, as illustrated in Fig. 2.

RESEARCH SIGNIFICANCESignificant efforts have been invested in testing anchors

reinforced with hairpins. Laboratory tests of anchors reinforced with other types of reinforcement is scarce, especially for anchors under cyclic shear loading. This paper presents tests of cast-in-place anchors reinforced using closed stir-rups under both monotonic and cyclic shear loading. Closed stirrups encasing bars placed at the corners and distributed along concrete surfaces can restrain concrete breakout such that the shear load is transferred to the structure through the

Fig. 1—Concrete breakout failure under shear.

Fig. 2—Schematics of existing anchor shear reinforcement.


Derek Petersen is a Structural Engineer at Osmose Railroad Services Inc., Madison, WI. He received his MS in civil/structural engineering from the University of Wisconsin at Milwaukee (UWM), Milwaukee, WI.

ACI member Jian Zhao is an Assistant Professor in the UWM Department of Civil Engineering and Mechanics. He received his PhD from the University of Minnesota, Minneapolis, MN. He is a member of ACI Committee 355, Anchorage to Concrete, and Joint ACI-ASCE Committee 447, Finite Element Analysis of Reinforced Concrete Structures. His research interests include the behavior of reinforced concrete struc-tures, concrete-steel connections, and earthquake engineering.

confined concrete. A design method is proposed for anchor shear reinforcement based on the observed anchor behavior.

BACKGROUNDVarious types of anchors have been developed over the

past 40 years. Numerous studies have been performed to develop the design method of anchors in plain concrete corresponding to various identified failure modes.6 The behavior of anchors and headed studs in plain concrete has been discussed at length7,9-12 and the tests have been summa-rized in several databases.13-15 On the other hand, studies are limited on the performance of anchors with reinforce-ment. The existing studies on anchor shear reinforcement are reviewed and the available design methods are summarized in the following.

Previous studiesThe most investigated anchor reinforcement for resisting

shear forces is horizontal hairpins that wrap around the anchor, as illustrated in Fig. 2. Swirsky et al.16 tested 24 cast-in-place anchors consisting of 25 and 51 mm (1 and 2 in.) diameter ASTM A307 carbon steel or ASTM A449 medium carbon or alloy steel bolts reinforced with No. 4 or No. 5 hair-pins under monotonic and cyclic loading. The hairpins had a 120-degree bend wrapping around the bolts 51 mm (2 in.) below the concrete surface. A capacity increase of 15 to 87% was observed at a displacement of approximately 25 mm (1 in.). Only six anchors were reported to fail with anchor shaft fracture, in part because the development length for the hairpins was 20db (db is the diameter of the hairpins), which is not sufficient. Many tests were terminated after bond failure of hairpins was observed. Two additional tests were conducted with two No. 4 vertical stirrups placed 51 mm (2 in.) away from the bolt. The use of stirrups is similar to the anchor reinforcement proposed in this paper; however, the amount of the reinforcement was not sufficient, and both tests stopped after the concrete cracked and a large displace-ment was observed.

The behavior of anchor bolts reinforced with hairpins was further studied by Klingner et al.17 through 12 monotonic tests and 16 cyclic tests of 19 mm (0.75 in.) diameter A307 bolts. A No. 5 hairpin with a 180-degree bend and a develop-ment length of approximately 37db was placed 19 or 51 mm (0.75 or 2 in.) below the top surface. The tests showed that the most effective way to transfer anchor shear force to the hairpin is through the contact between the anchor shaft and the hairpin near the surface. Hairpins that were not in contact with the anchor shaft were found to be effective in monotonic tests but unreliable under cyclic loading. The No. 5 hairpin provided sufficient shear resistance compared to the anchor steel capacity; however, most tests were terminated before anchor fracture was achieved—likely because a capacity drop was observed during the tests.

Lee et al.18 conducted 10 tests of 64 mm (2.5 in.) diam-eter anchor bolts with a 381 mm (15 in.) edge distance and a 635 mm (25 in.) embedment depth reinforced with U-shaped hairpins and hooked reinforcing bars. The reinforce-ment was proportioned to carry the shear capacity of anchor steel, resulting in a combination of No. 6 hairpins and No. 8 hooked bars dispersed within 381 mm (15 in.) from the anchor bolt with a spacing of 152 mm (6 in.). Three layers of No. 8 hairpins were used in some specimens. Most tests were terminated before a peak load was observed due to the limited stroke of the loading device. The unfinished tests were not able to fully demonstrate the effectiveness of the various anchor reinforcement designs.

In Europe, as documented by Schmid,19 Paschen and Schönhoff20 examined 10 types of anchor reinforcement layouts. Hairpins touching anchor shafts and reinforcing bars distributed near the top surface, as illustrated in Fig. 2, were found to be the most effective. Similar conclusions were made by Ramm and Greiner21 based on their tests of anchors reinforced with five types of reinforcement. Randl and John22 observed a capacity increase of 300% in their tests of post-installed anchor bolts with hairpins. It was concluded that the thickness of concrete cover affected the effectiveness of hairpins as anchor shear reinforcement. Recently, Schmid19 conducted tests on five types of anchors with hooked reinforcing bars, which simulated the reinforcing bars in an existing concrete element. A model was proposed for determining the shear capacity of reinforced anchors, which can be obtained from the summation of the contri-butions from all reinforcing bars bridging the assumed 35-degree breakout crack. The contribution from each reinforcing bar included the bearing force of the bent leg and the bond force of the straight part within the breakout cone. Schmid’s19 equation for the capacity of reinforced anchors in shear is a refined version of the equation proposed by Fuchs and Eligehausen,23 who clearly defined the assumption that a concrete cone must form before steel reinforcement takes effect. On the other hand, many of Schmid’s19 tests were terminated after the spalling of the concrete cover, which might have not indicated the final failure of the specimens.

Existing design recommendationsThe methods for proportioning anchor shear reinforcement

are summarized in Table 1. Note that many design methods that focused on the capacity calculation for anchors with a known configuration of anchor reinforcement, such as that proposed by Schmid,19 are not included in Table 1. In summary, most existing design methods require the reinforcement to provide more resistance than the anchor steel capacity in shear. This is achieved by either increasing the design force5 or reducing the effectiveness of anchor reinforcement based on their relative vertical locations.7,8 Note that there are few tests with such overdesigned reinforcement, and many such tests were terminated before a true ultimate load was achieved.

Hairpins are deemed effective as anchor shear reinforce-ment because they can be placed close to the anchor shaft using a small bending radius on the hairpin.17,18 The transfer of shear load to surface reinforcement shown in Fig. 2 is usually visualized using a strut-and-tie model (STM).23,25 STMs permit large-sized reinforcing bars located at a large distance from the anchor bolt as anchor reinforcement as long as the angle between the concrete strut and the applied shear force is small (for example, less than 55 degrees); however, tests18,26 have indicated that reinforcing bars placed closer to


the anchor are more effective. As a result, the existing design guidelines5,7,8 require the anchor reinforcement to be within a distance equal to half of the front-edge distance (0.5ca1), as illustrated in Fig. 2. Such requirements leave a small window of applicability for practical implementations of the anchor reinforcement. Oftentimes, the front-edge distance needs to be increased to accommodate the anchor reinforce-ment, which in turn increases the concrete breakout capacity such that the anchor reinforcement may no longer be needed.

Anchor reinforcement design for shear in this study considered the following four aspects: 1) an effective reinforcement layout that restrains concrete breakout failure; 2) a proper design force for proportioning the anchor reinforcement; 3) a reason-able distance on each side of the anchor bolt within which the anchor reinforcement is deemed effective; and 4) an accurate estimation of shear capacity of reinforced anchors.

PROPOSED ANCHOR SHEAR REINFORCEMENT DESIGN

The proposed anchor reinforcement is shown in Fig. 3 for anchors with both unlimited and limited side-edge distances. The goal of the proposed design for anchor shear reinforce-ment is to prevent concrete breakout using closely spaced stirrups placed parallel to the plane of the applied shear force and the anchor. With the concrete confined around the anchor, it is expected that the concrete will restrain the anchor shaft and provide shear resistance. The stirrups should be propor-tioned using the anchor steel capacity in shear, as specified by the equation in the last row of Table 1. The nominal yield strength of reinforcing steel should be used in the calcula-tion. Two stirrups should be placed next to the anchor shaft, where the breakout crack in concrete may initiate under a shear load. The rest of the required stirrups should be placed with a center-on-center spacing of 51 to 76 mm (2 to 3 in.). A smaller spacing may be used, provided that the clear spacing requirements, such as those in ACI 318-11,5 are satisfied. The stirrups can be distributed within a distance of ca1, as shown in Fig. 3. Note that the horizontal legs of the closed stirrups are used as anchor shear reinforcement, while the vertical legs close to the anchor shaft25 may be used as anchor tension reinforcement, as shown in the Phase III

tests of this study. For this purpose, the depth of the stirrups should be large enough such that the vertical legs are fully developed for the tension load.

The development length requirements for the horizontal legs of the closed stirrups are satisfied similar to the trans-verse reinforcement in a flexural member, where the stirrups are fully developed at both sides of a shear crack through the interaction between the closed stirrups and longitu-dinal bars at all four corners.27 Meanwhile, reinforcing bar pullout tests, in which both legs of No. 4 U-shaped bars embedded 38 and 76 mm (1.5 and 3 in.) in concrete were loaded in tension, indicated that a minimum embedment depth of 6db was needed to develop a No. 4 stirrup through the interaction. Therefore, the length of the horizontal legs of the vertical closed stirrups should be at least 8db on both sides of the anchor, as shown in Fig. 3. This requirement

Fig. 3—Proposed anchor shear reinforcement layout.

Table 1—Summary of design equations for anchor shear reinforcement

Reference Design equation for Asa given load Vsd Development in cone Actual shear capacity Vs Notes

Shipp and Haninger24 ,

1.85cos 45uta se N

ys sa

F AF A =

°

Not neededDesign based on

equivalent tensionHairpins

Klingner et al.17 FysAsa = FutaAse,V Not needed Vs = 0.6FutaAse,V Hairpins

CEB7 0.5FysAsa = 1.15VsdConsidered in capacity

calculation2s dh bdV l uf= ∑

Bars within 0.5ca1

ACI 318-115 0.75FysAsa = Vsd

*

Vs = FysAsaBars within 0.5ca1 or

0.3ca2

Widianto et al.25 ssAsa = FutaAse,V or 2.5Vsd

ss reduced for not fully developed barsNot considered in STM Vs = Vsd

Stirrups, ties, and J-hooks

fib design guide8 0.5FysAsa = Vsd([es/z] + 1)Considered in capacity

calculationbd

s dhre

fV l u= ∑a

Bars within 0.5ca1

Proposed FysAsa = 0.6FutaAse,V 8db on both sides Vs = 0.45FutaAse,V Closed stirrups within ca1

*Refer to Chapter 12 of ACI 318-115 for details. Notes: Asa is area of anchor reinforcement; Fys is yield strength of reinforcement; Ase,V, Ase,N is effective cross-sectional area of anchor; ca1, ca2 are edge distances of anchor; es is distance from shear to reinforcement; fbd is design bond strength; Futa is ultimate strength of anchor; Ldh, ldh is development length of hooked bar in breakout cone; u is circumfer-ence of reinforcing bar; Vsd is design shear force; z is reinforcement position; are is modification factor; ss is stress in anchor reinforcement.

0.02 ysdh b

c

FL d

fψ

=λ ′


results in a minimum edge distance of 8db plus the concrete cover. The design of reinforced anchors should also satisfy other edge distance requirements, such as those in Section D.8 of ACI 318-11.5

Bars at all four corners of the closed stirrups (referred to as “corner bars” hereafter) restrain splitting cracks, as well as other bars distributed along the concrete surfaces (referred to as “crack-controlling bars” hereafter). Therefore, the corner bars and crack-controlling bars need to be fully developed at both sides of the anchor bolt, and a 90-degree bend (as shown in dashed lines) in Fig. 3, may be needed. The selection of corner bars may follow the common practices in selecting longitudinal corner bars for reinforced concrete beams, such as those specified in Section 11.5.6 of ACI 318-11.5 Crack-controlling bars were not provided in the tests and the split-ting cracks were observed, as presented in the following. Crack-controlling bars are therefore recommended as shown in Fig. 3, and the determination of these bars can be based on the well-recognized STMs.23,25

EXPERIMENTAL INVESTIGATIONSpecimens

This group of experimental tests is part of a research program that focused on the behavior and design of cast-in-place anchors under simulated seismic loads.28 Sixteen tests were conducted using 25 mm (1 in.) diameter anchors consisting of an ASTM A193 Grade B7 threaded rod (fy = 724 MPa [105 ksi] and fut = 1069 MPa [131 ksi]) and a heavy hex nut welded to the end. Another four tests using 19 mm (0.75 in.) diameter ASTM F1554 Grade 55 anchors (fy = 434 MPa [63 ksi] and fut = 524 MPa [76 ksi]) were conducted with two tests, each under monotonic shear and cyclic shear loading. Ready mixed concrete with a targeted strength of 27.6 MPa (4000 psi) was used, and cylinder tests using three batches of three 100 x 200 mm (4 x 8 in.) cylinders tested throughout the anchor test period showed an average compressive strength of 24.3 MPa (3525 psi).

The dimensions of the test blocks containing four anchors each are illustrated in Fig. 4. One block was prepared for

Type 19-150-100 specimens, and two blocks were prepared for Type 25-150-150 and Type 25-150-150H specimens. Another block similar to that for Type 25-150-150 speci-mens was used for Type 25-150-150SG specimens. Strain gauges were installed on the reinforcing bars of the two anchors in this block. All anchors had an embedment depth of 152 mm (6 in.). The width and depth of the test blocks were selected such that the spacing between the anchors was larger than two times their front-edge distances. Anchors in Type 25-150-150H specimens had two limited side-edge distances equal to 1.5 times their front-edge distance. The height of the blocks was 432 mm (17 in.), similar to all other anchor tests in the study.28

The anchor shear reinforcement was proportioned to carry the maximum capacity of the anchor bolts in shear: 68 kN (15.3 kips) for the 19 mm (0.75 in.) anchors and 209 kN (47 kips) for the 25 mm (1 in.) anchors. Using the nominal yield strength of Grade 60 steel, the required anchor reinforce-ment was found to be 164 mm2 (0.25 in.2) for the 19 mm (0.75 in.) anchors and 503 mm2 (0.78 in.2) for the 25 mm (1 in.) anchors. Therefore, two No. 4 bars were provided for Type 19-150-100 specimens, as shown in Fig. 4. The required anchor reinforcement for the 25 mm (1 in.) anchors was provided using four No. 4 bars with a spacing of 51 mm (2 in.) for Type 25-150-150 specimens, two No. 4 and four No. 3 bars for Type 25-150-150H specimens with a spacing of 76 mm (3 in.), and eight No. 3 bars for Type 25-150-150SG specimens with a spacing of 51 mm (2 in.). Two addi-tional No. 3 J-hooks were added beside the outermost bars in Type 25-150-150SG specimens, as shown in Fig. 4, to host two more strain gauges, which were approximately 250 mm (10 in.) away from the anchor bolt. One straight bar was provided at each corner of the closed stirrups. Note that some specimens had several narrow stirrups placed behind the anchors—the vertical legs of which were intended to be anchor tension reinforcement—in which case one additional corner bar was provided along the top surface. However, the planned tension tests were not performed because the concrete blocks were not sufficient for the large tension load that would be carried by the reinforced anchors. The additional stirrups did not affect the shear behavior of the anchors because they were placed behind the anchor bolts. All reinforcing bars were placed with a cover of 38 mm (1.5 in.).

Test setupThe loading frame, actuator placement, and instrumenta-

tion setup used for the tests are shown in Fig. 5. Instead of a self-balanced load frame, a tie-down rod 381 mm (15 in.) behind the test anchor was used to fix the test block to the strong floor. In addition, the concrete block was wedged against the strong floor to minimize the slip of the test block under cyclic loads, as shown in Fig. 5. A 245 kN (55 kip) actuator was used to apply shear loading to the anchor bolt through a loading plate. The actuator body was braced against the floor to eliminate the downward motion of the actuator swivel head and the rotation of the loading plate. To minimize the friction between the loading plate and the concrete top surface, a net tension force of 0.8 kN (0.2 kips) was applied to the loading plate by a 489 kN (110 kip) actuator, which was used for applying tension loads in other tests. The nut fixing the loading plate to the anchor bolt was first hand-tightened and then loosened one-eighth of a turn to allow slight vertical movement of the loading plate

Fig. 4—Configurations of anchor specimens.


when the 0.8 kN (0.2 kip) tension force was applied at the beginning of a test. The test anchors were inserted through a standard 3 mm (0.125 in.) oversized hole in the loading plate, and a steel sleeve shim was inserted between the anchor and the hole to eliminate the clearance and prevent damage to the loading plate.

Loading protocolMonotonic shear tests were performed first to determine

the typical actuator displacement at failure, and the tests indicated a failure displacement of approximately 35 mm (1.4 in.). Hence, the cyclic displacement steps for each three-cycle group were chosen as 2, 3, 4 (failure displacements for typical unreinforced anchors), 8, 16, and 32 mm (0.08, 0.12, 0.16, 0.32, 0.64, and 1.28 in.), as shown in Fig. 5. The loading rate for the displacement cycles at or below 4 mm (0.16 in.) was kept at 2 mm/min (0.08 in./min), while the load rate was increased to 10 mm/min (0.4 in./min) for the 8, 16, and 32 mm (0.32, 0.64, and 1.28 in.) cycles to reduce test time. Most reversed cyclic shear tests were conducted following Loading Pattern C1 shown in Fig. 5, in which the maximum displacement was set as 4 mm (0.16 in.) when the shear loading was applied opposite to the front edge. This was to prevent early anchor fracture under reversed loads and observe the cyclic behavior over a full displace-ment range. Cyclic tests following Loading Pattern C2 in Fig. 5 with equal peak displacements in both directions of shear loading were conducted for two Type 25-150-150H specimens. Note that the control of the actuator was based on the actuator piston motion instead of anchor displace-ment; hence, the actual anchor displacements were smaller than the aforementioned target displacements.

InstrumentationString pots and linear variable differential transformers

(LVDTs) were used to measure the anchor displacements, as illustrated in Fig. 5. The displacements of the load plate were actually used as the anchor displacement because the anchor shaft just above the concrete surface was not assess-able. A data acquisition system was used to collect data from all sensors, as well as the force and displacement outputs from the actuators. The sampling frequency was 5 Hz and the collected data were filtered using an in-house program with a cutoff frequency of 0.1 Hz. The observed anchor behavior is discussed in the following.

EXPERIMENTAL RESULTS AND DISCUSSIONBehavior of anchors under monotonic loading

The configuration and loading types of the anchor speci-mens are summarized in Table 2 along with the measured shear capacities. The load-versus-displacement behavior is shown in Fig. 6 for the reinforced anchors subjected to monotonic shear along with selected images of failed speci-mens. For comparison purposes, the load-versus-displace-ment behavior for a 19 mm (0.75 in.) anchor with a front-edge distance of 100 mm (4 in.) in plain concrete is shown in Fig. 6(a) and the result of another anchor with a front-edge

Fig. 5—Experimental test setup. (Note: 1 mm = 0.0394 in.)

Table 2—Summary of reinforced anchor tests in shear

Specimen ID Block type da, in. ca1, in. Load type Peak load, kips

9132010 — 0.75 4 M 22.19

9132010_2 — 0.75 4 M 22.47

9172010 — 0.75 4 C1 16.69

9202010 — 0.75 4 C1 15.50

9282010 — 1.0 6 M 39.18

9292010 — 1.0 6 M 44.11

9302010 — 1.0 6 C1 38.71

10042010 — 1.0 6 C1 35.92

10052010 — 1.0 6 C1 34.35

10062010 H 1.0 6 M 38.40

10062010_2 H 1.0 6 M 34.71

10072010 H 1.0 6 M 33.40

10082010 H 1.0 6 C1 33.62

10082010_2 H 1.0 6 C1 31.77

10122010 H 1.0 6 C1 33.88

10132010 H 1.0 6 C2 –42.68*

10142010 H 1.0 6 C2 –47.79*

10292010 SG 1.0 6 M 36.13

11192010 SG 1.0 6 M 39.33*Anchor fracture occurred when shear was applied opposite to front edge. Notes: 1 in. = 25.4 mm; 1 kip = 4.45 kN.


distance of 150 mm (6 in.) is shown in the rest of Fig. 6. The unreinforced anchors were tested with a concrete strength of 39 MPa (5656 psi), whereas the reinforced anchor tests had a concrete strength of 24.3 MPa (3525 psi); therefore, the load values for the unreinforced anchors were normal-

ized using a factor of 24.3 39 in Fig. 6. In general, the reinforced anchors failed by anchor shaft fracture, while the unreinforced anchors with similar edge distances failed by concrete breakout. The failure loads for the reinforced anchors increased by approximately 100% and the displace-ments corresponding to the peak loads increased more than six times compared with those of the unreinforced anchors.

The load-displacement behavior of 19 mm (0.75 in.) anchors in reinforced concrete did not show much difference

from that in plain concrete (Fig. 6(a)) before a crack was observed at the top surface at a load of approximately 45 kN (10 kips). Rather than propagating vertically along the anchor shaft, as observed in the tests of unreinforced anchors as represented by Fig. 1, the crack propagated around the corner of the stirrups (refer to the inserted figure in Fig. 6(a)). The loss of the 38 mm (1.5 in.) thick concrete cover in front of the anchor caused a small capacity loss for the 19 mm (0.75 in.) anchors, as shown in Fig. 6(a). Because the 19 mm (0.75 in.) anchor only mobilized the top concrete before cracking, similar to that suggested by Randl and John22 (approximately 2da deep), the anchor shaft in bending was not able to resist the same amount of load until a larger displacement was applied. Such a post-spalling load drop has been observed in other tests of anchors reinforced with hairpins.17,18 The failure load exceeded the code-specified anchor shear capacity because the failure was caused by the fracture of the anchor shaft largely under tension, as shown in Fig. 7(a), although the fracture may have started from a flexural crack.

The shear load did not drop noticeably after the concrete cover spalled in the tests of 25 mm (1 in.) anchors, as shown in Fig. 6(b) through (d). The 25 mm (1 in.) anchors mobi-lized deeper concrete such that the loss of bearing support from the cover concrete was immediately resisted by lower concrete restrained by the anchor reinforcement. Another contributing factor is that the 25 mm (1 in.) anchors had a larger bending stiffness such that a small displacement was needed to mobilize their load-carrying capacities. The 25 mm (1 in.) anchors failed at loads lower than the code-specified anchor steel capacity in shear. The fractured 25 mm (1 in.) anchors in Fig. 7(c) showed a different failure mode from that of the 19 mm (0.75 in.) anchors; the anchor shaft cracked under a bending moment and the rest of the anchor shaft then fractured in shear. For the shear-dominant failure mode, the flexural cracking reduced the cross-sectional area, thus leading to a lower ultimate shear capacity.

Fig. 6—Monotonic shear test results of reinforced anchors.

Fig. 7—Typical fractured shape of anchor bolts.


Anchor steel failure was achieved in all 25 mm (1 in.) diam-eter anchors, indicating that reinforcing bars placed outside the code-specified effective distance—such as 0.5ca1 in Type 25-150-150SG and 0.3ca2 in Type 25-150-150H—can be effective as anchor shear reinforcement. However, rein-forcing bars must be evenly distributed with a small spacing for outside bars to be mobilized. The effective distance was verified by the measured strains in the reinforcing bars in Type 25-150-150SG specimens, as shown in Fig. 8. The anchor reinforcement consisted of eight No. 3 stirrups at a spacing of 51 mm (2 in.) and two additional No. 3 J-hooks. The thin dashed lines in Fig. 8 indicate the assumed breakout crack at the concrete surface, and the strain gauges were installed 25 mm (1 in.) behind the assumed breakout crack line on the inside face of the stirrups. In general, larger strains were observed in the bars closer to the anchor bolt. Mean-while, the outside bars, as indicated by Gauges 4S and 4N located 170 mm (6.7 in.) from the anchor bolt, also devel-oped significant strains, especially after the surface crack formed. Note that the gauge positions relative to a crack should be considered to interpret the measured strains. For example, the strains by Gauge 2N may have been affected by the crack passing the gauge location, as shown in Fig. 8. More importantly, smaller strains measured by the gauges on the outside bars may have been due to the fact that the gauges were away from the actual crack. In addition, the measured strains indicated that none of the Grade 60 bars yielded at the peak load; hence, the shear capacity of reinforced anchors may not be calculated as the summation of the yield forces of the anchor reinforcement. The shear force was actually transferred to the supports (for example, the tie-down rods on the back and the steel wedging tube at the bottom, in this case) through the concrete confined by the closed stirrups.

Anchors in Type 25-150-150H specimens had a lower ulti-mate capacity, as shown in Fig. 6(c). This might have been due to the poor confinement of concrete in front of the anchor bolt; additional splitting cracks were observed and deeper concrete crushed in these tests, leading to a longer portion of exposed and unsupported anchor bolts (for example, up to 0.5da larger than those in Type 25-150-150 specimens). Finite element analyses indicated that the anchor capacity controlled by shear fracture can be affected by anchor diam-eter and concrete cover depth.29 It is thus envisioned that the following measures, as illustrated in Fig. 3, can be effec-tive in improving the post-spalling behavior and capacity of reinforced anchors in shear: 1) corner bars should be fully developed; 2) crack-controlling bars should be provided along both the top and front surfaces of concrete; and 3) a separate bar can be placed directly in front of the anchor bolt to alleviate the large local compressive stress in concrete.

Anchor shear capacityMost anchor bolts in this group of tests failed by shear frac-

ture of a reduced anchor shaft cross section, as shown by the typical fractured sections in Fig. 7. This failure mode occurred when a short portion of the anchor bolt was exposed and a lever arm developed in the anchors after the cover concrete spalled. The effect of lever arms in anchor bolts is recognized in the existing design codes.5,8 For example, ACI 318-115 stip-ulates that the design capacity of anchor connections having grout leveling pads should be reduced by a factor of 0.8 for the anchor steel strength in shear. Such capacity reduction considers the combined bending and shear in the anchor shaft but does not consider the thickness of the grout pads,

which is similar to the exposed length at the ultimate load. Eligehausen et al.12 proposed an equation for predicting the strength of an exposed anchor, assuming that the anchor fails by pure bending. This equation was not found to be appli-cable for predicting the capacity of the anchors in this study, likely due to the fact that the anchor failure was controlled by shear fracture. Lin et al.29 improved the equation by Elige-hausen et al.12 by considering the contributions from flexural, shear, and tensile resistance of an exposed anchor shaft to the shear capacity of exposed anchors; however, the equation was based on double shear tests and finite element analyses of threaded rods, and the lateral support to the actual anchor shaft from partially damaged concrete was not considered. Therefore, the equation may provide lower-bound estimates of the actual anchor capacities.

The capacity of anchor bolts with a lever arm was instead examined using the test data available in the literature, as shown in Fig. 9. The measured anchor capacities were normalized by the design capacity of anchor bolts in shear specified in ACI 318-11.5 The exposed depth of the anchors in other tests16,26 was defined as the distance between the

Fig. 8—Strains in anchor shear reinforcement (Type 25-150-150SG1). (Note: 1 mm = 0.0394 in.)

Fig. 9—Capacity of anchor bolt with lever arm.


bottom face of a base plate and the lowest solid concrete surface. The anchor steel capacity observed in this study is low compared with other available tests. This might have been due to the fact that friction between the load plate and the concrete surface was minimized, as previously described in the test setup section.

The statistical analysis of the limited data in Fig. 9 did not follow the procedures of predictive inference,30,31 which are usually used to predict future occurrences based on the existing observed data. Instead, a 5-percentile value of 0.73 was obtained using a descriptive statistical analysis of the 22 collected data points. Considering the afore-mentioned reasons for the low observed capacities in this study, it is proposed that the shear strength of reinforced anchors can be estimated as 75% of the code-specified steel capacity for anchors without a lever arm. This is slightly lower than the reduction factor in ACI 318-115 because of two data points observed in specimens with limited side-edge distances (Type 25-150-150H). It is envisioned that as more data points become available in future tests with the recommended anchor shear reinforcement shown in Fig. 3, the statistical importance of these two data points can be reduced. Using the suggested capacity reduction for exposed anchors should be limited to those with an exposed length less than three times the anchor diameter (3da). Beyond this limit, the anchor steel failure in shear needs further study.

Behavior of anchors under cyclic loadingSeismic actions on structural components are mostly

simulated in laboratories using quasi-static cyclic tests with reversed loading.32 Therefore, displacement-controlled loading33 was used in this study, although many cyclic tests of anchors have been conducted with load-controlled loading.16,17,34 The load-versus-displacement behavior of two 19 mm (0.75 in.) anchors subject to Type C1 cyclic shear loading is plotted in Fig. 10(a). The monotonic curve was closely followed by cyclic curves until a displacement of 10 mm (0.4 in.), beyond which the cyclic loads were lower than that of the monotonic test. The slope of the

cyclic curves again had a sudden change at a displacement of approximately 2 mm (0.16 in.), indicating the concrete cover spalling. The difference in the observed loads at this displacement may have been due to variations in the speci-mens, such as the actual edge distances and cover depths. The first three displacement cycles did not see significant degradation in loads with successive cycles to the same displacement, while the degradation was obvious at the larger-displacement cycles. This was because the displaced cover concrete during the first cycle of each three-cycle group was not able to recover, leading to reduced restraint to the anchor shaft in the successive cycles. An average capacity reduction of 28% was observed in the cyclic shear capacity for the 19 mm (0.75 in.) anchors. This reduction was partly attributed to the change of failure modes, as shown by the fractured shape of the anchor in Fig. 7(a) and (b); the anchor failure was controlled by the shear fracture under cyclic loading, while the tensile fracture controlled the anchor failure in the monotonic test. Note that the reduced cyclic shear capacities of the 19 mm (0.75 in.) anchors were higher than the proposed capacity of exposed anchors under monotonic loading because of the monotonic failure mode.

The behaviors of Type 25-150-150 specimens are compared in Fig. 10(b). The monotonic load-displacement curve nicely envelopes the cyclic curves represented by the first loading cycle in each three-cycle group. The load degra-dations during the successive two cycles were again due to the irreversible crushing of the concrete cover in front of the anchors. No capacity drop was observed in the tests of Type 25-150-150 specimens. An average capacity drop of 6.8% was observed for Type 25-150-150H anchors with a limited side-edge distance, as shown in Fig. 10(c). In this group of three cyclic tests, concrete deeper than the 38 mm (1.5 in.) cover crushed, likely due to poor confinement condi-tions, as indicated by splitting cracks. The larger exposed length led to a larger moment under the same shear load and thus a lower shear capacity. Note that the poor confine-ment conditions can be improved by the crack-controlling bars recommended in Fig. 3. In addition, a bar placed just

Fig. 10—Cyclic behavior of reinforced anchor bolts.


in front of the anchor shaft can help distribute the localized high compressive stresses such that the exposed length of the anchors would not be affected by the cyclic loading. Finally, the tests of two Type 25-150-150H anchors with fully reversed cyclic loading (Type C2 in Fig. 5) ended with anchor fractured under a shear load applied opposite to the front edge. The ultimate load capacities were, on average, 5% lower than the code-specified anchor steel capacity, as shown in Fig. 10(d). Hence, it is reasonable to ignore the reduction of steel capacities for reinforced anchors in cyclic shear, considering that the monotonic capacity of reinforced anchors has already been reduced by 25%, as proposed previously.

CONCLUSIONSA design method for anchor shear reinforcement was

proposed and verified using experimental tests of single cast-in-place anchors. With a goal to prevent concrete breakout and confine concrete in front of an anchor bolt, the proposed anchor shear reinforcement consisted of closely spaced stir-rups, corner bars, and crack-controlling bars distributed along all concrete faces. The horizontal legs close to the concrete surface of the closed stirrups were proportioned to carry a force equal to the code-specified anchor steel capacity in shear. The needed reinforcement was provided by closely spaced, small-sized stirrups distributed within a distance from the anchor equal to its front-edge distance. Although not specifically tested in the study, the selec-tion of corner bars should follow the practices specified in Section 11.5.6.2 of ACI 318-115 for corner bars in beams, and crack-controlling bars may be determined following the well-recognized STMs.

With the proposed anchor shear reinforcement, concrete breakout was prevented and anchor shaft fracture was observed in all the tests of single anchors in this study. Cover concrete in front of the anchor bolts spalled, causing the top portion of the anchor shaft close to the concrete surface to become exposed. The full anchor steel capacity in shear was not achieved because the exposed anchors were subjected to a combination of shear, bending, and tension at failure. An analysis of the test results of exposed anchors in the literature indicated that a reduction factor of 0.75, which is slightly lower than that in ACI 318-115 on anchors with a grout pad, can be used to determine the shear capacity of reinforced anchors. In addition, quasi-static cyclic tests of the rein-forced anchors in shear showed insignificant capacity reduc-tion, which is comparable to other displacement-controlled cyclic tests. Although large capacity reductions were observed in load-controlled cyclic tests in the literature, no further capacity reduction is recommended in this study for reinforced anchors subjected to cyclic shear loading.

ACKNOWLEDGMENTSThe study reported in this paper is from a project supported by the

National Science Foundation (NSF) under Grant No. 0724097. The authors gratefully acknowledge the support of J. Pauschke, who served as the Program Director for this grant. The authors also thank their colleagues in ACI Committee 355 for their valuable input. Any opinions, findings, and recommendations or conclusions expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

NOTATIONAsa = area of anchor reinforcementAse,V, Ase,N = effective cross-sectional area of anchor in shear and tensionca1 = front-edge distance of anchorca2 = side-edge distance of anchor

da = anchor diameterdb = reinforcement diameteres = distance from shear force to surface reinforcementFys = yield strength of steel reinforcementfbd = design bond strength of anchor reinforcement in breakout conefc′ = concrete compressive strengthfuta = ultimate tensile strength of anchor steelfy = yield strength of anchor steelldh = development length of hooked bar in breakout coneu = circumference of reinforcing barVs = actual shear capacity of exposed anchorVsd = design shear capacity of anchorz = vertical reinforcement positionss = stress in anchor reinforcement

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3. Grauvilardell, J.; Lee, D.; Hajjar, J.; and Dexter, R., “Synthesis of Design, Testing and Analysis Research on Steel Column Base Plate Connections in High-Seismic Zones,” Structural Engineering Report No. ST-04-02, University of Minnesota, Minneapolis, MN, 2005.

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11. Cannon, R., “Straight Talk about Anchorage to Concrete—Part II,” ACI Structural Journal, V. 92, No. 6, Nov.-Dec. 1995, pp. 724-734.

12. Eligehausen, R.; Mallée, R.; and Silva, J., Anchorage in Concrete Construction, Wilhelm Ernst & Sohn, Berlin, Germany, 2006, 391 pp.

13. Muratli, H., “Behavior of Shear Anchors in Concrete: Statistical Analysis and Design Recommendations,” MS thesis, University of Texas at Austin, Austin, TX, 1998, 181 pp.

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15. Pallarés, L., and Hajjar, J., “Headed Steel Stud Anchors in Composite Structures, Part I: Shear,” Journal of Constructional Steel Research, V. 66, 2009, pp. 198-212.

16. Swirsky, R.; Dusel, J.; Crozier, W.; Stoker, J.; and Nordlin, E., “Lateral Resistance of Anchor Bolts Installed in Concrete,” Report No. FHWA-CA-ST-4167-77-12, California Department of Transportation, Sacramento, CA, 1978, 100 pp.

17. Klingner, R.; Mendonca, J.; and Malik J., “Effect of Reinforcing Details on the Shear Resistance of Anchor Bolts under Reversed Cyclic Loading,” ACI JOURNAL, Proceedings V. 79, No. 1, Jan.-Feb. 1982, pp. 471-479.

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Title no. 110-S07


ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2011-053.R1 received March 7, 2011, and reviewed under Institute


Innovative Flexural Strengthening of Reinforced Concrete Columns Using Carbon-Fiber Anchorsby Ioannis Vrettos, Efstathia Kefala, and Thanasis C. Triantafillou

This paper presents the results of an experimental program that aimed to study the behavior of reinforced concrete (RC) columns under simulated seismic loading strengthened in flexure with anchored carbon-fiber sheets. The role of different parameters is examined by comparing the lateral load-versus-displacement response characteristics (peak force, drift ratios, energy dissi-pation, and stiffness). These parameters included the number of anchors and the volume of fibers in each anchor. The results were combined with a simple analytical model to yield values for the effective strain in the anchors at failure. It is concluded that carbon-fiber anchors provide a viable solution toward enhancing the flexural resistance of RC columns subjected to seismic loads, especially if they are made of a substantial amount of fibers.

Keywords: carbon-fiber anchors; columns; flexure; seismic retrofitting; strengthening.

INTRODUCTION AND BACKGROUNDEarthquakes worldwide have proven the vulnerability of

existing reinforced concrete (RC) columns to seismic loading. Poorly detailed columns are the most critical structural elements, which may fail due to shear, compressive crushing of concrete, reinforcing bar buckling, bond at lap splices, and flexure. Seismic retrofitting of RC columns is a challenging task that may be addressed successfully today using exter-nally bonded composite materials (fiber-reinforced polymers [FRPs]) for all of the aforementioned failure mechanisms except flexure. FRPs in the form of jackets with the fibers typically in the columns’ circumferential direction are quite effective in carrying shear and providing confinement, thus increasing the shear resistance and deformation capacity of existing RC columns. However, effective strengthening of columns in flexure—often needed, for instance, to satisfy capacity design requirements (that is, the elimination of weakness in strong-beam, weak-column situations) or when existing reinforcing bars have been affected by corrosion—calls for the continuation of longitudinal reinforcement. This reinforcement should extend beyond the end cross sections, where moments are typically at a maximum. Therefore, the placement of externally bonded FRP is not applicable. As a result, flexural strengthening of RC columns is currently typically achieved by using RC jackets or some form of steel jackets—namely, steel “cages”—also followed by shotcreting. RC jackets or steel cages covered by shotcrete require intensive labor and artful detailing; they increase the dimensions and weight of columns and result in substantial obstruction of occupancy. Moreover, increasing the stiffness of the column will attract a higher force because forces are distributed according to the relative stiffness of the elements. Therefore, the implementation of a low-labor and minimal obstruction flexural strengthening technique for RC columns is a challenging task that was addressed for the first time in

a systematic way by Bournas and Triantafillou1 through the use of near-surface-mounted (NSM) reinforcement.

In this study, the authors investigated flexural strength-ening of columns with externally bonded FRP sheets that are anchored at the columns’ end sections with fiber anchors in the form of spikes. Fiber anchors have received the attention of some investigators in applications related to shear strengthening of columns,2,3 shear strengthening of beams,4 and flexural strengthening of beams.5,6 Some studies have also focused on specific bond aspects of fiber anchors7,8 or tensile properties.9

The only study reported in the international literature on flexural strengthening of columns with anchored FRP sheets is that of Prota et al.,10 who used steel spikes at the base of cantilever-type RC columns in combination with glass FRP confining jackets. The specimens were tested under mono-tonic lateral load in combination with constant axial load. A comparison of the strength results for unstrengthened and strengthened columns shows an increase in the range of 33 to 54%.

This paper presents a study on the combination of FRP sheets and fiber anchors for flexural strengthening of RC columns under simulated seismic loading. Details are provided in the following sections.

RESEARCH SIGNIFICANCEColumns, the most critical structural elements in RC struc-

tures, are often in need of flexural strengthening to satisfy capacity design requirements (relocation of plastic hinges from columns to beams) or when longitudinal reinforcing bars have been affected by corrosion. The implementation of a low-labor and minimal-obstruction flexural strengthening technique for RC columns still remains a challenging task, which is addressed in this study for the first time through the use of longitudinal carbon FRP (CFRP) sheets combined with carbon-fiber anchors.

EXPERIMENTAL PROGRAMTest specimens and experimental parameters

The experimental program aimed to study the flex-ural strengthening of old-type nonseismically detailed RC columns with externally bonded FRP sheets, which are anchored at the columns’ end sections with fiber anchors in the form of spikes, and compare the effectiveness of different anchor schemes. A total of four large-scale RC column


Ioannis Vrettos received his Diploma in civil engineering and his MSc in seismic design of structures from the University of Patras, Patras, Greece, in 2007 and 2009, respectively. His research interests include advanced materials and seismic retrofitting of reinforced concrete structures.

Efstathia Kefala received her Diploma in civil engineering and her MSc in seismic design of structures from the University of Patras in 2007 and 2009, respectively. Her research interests include advanced materials and seismic retrofitting of reinforced concrete structures.

ACI member Thanasis C. Triantafillou is a Professor of civil engineering and Director of the Structural Materials Laboratory at the University of Patras. He received his Diploma in civil engineering from the University of Patras in 1985 and his MSc and PhD from the Massachusetts Institute of Technology, Cambridge, MA, in 1987 and 1989, respectively. He is a member of ACI Committee 440, Fiber-Reinforced Polymer Reinforcement. His research interests include the application of advanced polymer- or cement-based composites in combination with concrete, masonry, and timber with an emphasis on strengthening and seismic retrofitting.

specimens with the same geometry were constructed and tested under cyclic uniaxial flexure with constant axial load (Fig. 1(a)). The specimens were flexure-dominated cantile-vers (that is, slender and designed to fail by yielding of the longitudinal reinforcing bars) with a height to the point of application of the load (shear span) of 1.6 m (63 in.) (half a typical story height) and a cross section of 250 x 250 mm (9.84 x 9.84 in.). To represent old-type columns, speci-mens were reinforced longitudinally with four deformed bars 14 mm (0.55 in.) in diameter and 8 mm (0.32 in.) diam-eter deformed stirrups, closed with 90-degree hooks at both ends, at a spacing of 200 mm (7.87 in.). The geometry of a typical cross section is shown in Fig. 1(b).

The specimens were designed such that the effect of two basic parameters on the effectiveness of anchors—the number of anchors and the amount of fibers in each anchor—could be investigated. The specimens are described in the following, supported by Fig. 2.• One specimen was tested without flexural strengthening

as the control specimen. As in all strengthened speci-mens, however, longitudinal fiber sheets were confined at the base of the column with an FRP jacket so buckling of those fibers could be prevented; the same confining jacket was also used in the control specimen (Fig. 2(a)).

As a result, the only difference between the control specimen and any other specimen was due to the imple-mentation of flexural strengthening through the use of longitudinal sheets in combination with anchors. The jacket was made of a CFRP sheet that extended from the column base to a height of 600 mm (23.62 in.).

• Specimen 2_1.5 was strengthened with a 200 mm (7.87 in.) wide epoxy-impregnated carbon-fiber sheet on each of the two opposite sides of the column (those with the highest tension/compression). The CFRP sheet extended from the column base to a height of 1.4 m (55.12 in.) and was anchored at the base block with two carbon-fiber spike anchors on each side (Fig. 2(b)). The cross-sectional area of the fibers in each anchor was equal to 0.75 times the cross-sectional area of the fibers in the CFRP sheet; hence, the total cross-sectional area of the fibers in the two anchors was equal to 1.5 times that of the CFRP sheet. Finally, the column was confined with a jacket identical to that used in the control specimen.

• Specimen 3_1.5 was strengthened the same as 2_1.5 but with three instead of two anchors per side (Fig. 2(c)). Those anchors were 33% lighter than those in 2_1.5: each one had a cross-sectional area equal to 0.50 times the cross-sectional area of the fibers in the CFRP sheet; hence, the total cross-sectional area of the fibers in the three anchors was again equal to 1.5 times that of the CFRP sheet.

• Specimen 2_1.0 (Fig. 2(d)) was strengthened the same as 2_1.5 but with the light anchors used in Specimen 3_1.5.

In summary, except for the control specimen, the speci-mens’ notation is as follows: the first number denotes the number of anchors on each side at the base of the column and the second number denotes the ratio of the fiber cross section in the anchors to that in the CFRP sheet.

Strengthening procedureOne unidirectional carbon-fiber sheet 1.4 m (55.12 in.)

long and 200 mm (7.87 in.) wide was bonded on a properly prepared concrete surface on each of the two opposite sides

Fig. 1—(a) Schematic of test setup; and (b) cross section of columns. (Note: Dimensions in mm [in.].)


of the strengthened columns. The sheet was placed with fibers in a vertical configuration and was terminated at the column base.

Fiber anchor spikes were applied on top of the CFRP sheet at a spacing of 100 mm (3.94 in.) or 67 mm (2.64 in.) for columns with two or three anchors per side, respectively (Fig. 2). Spikes were formed from dry carbon fibers (half dry and half coated with epoxy). Holes were drilled into the

base of the column with a depth of 250 mm (9.84 in.) and a diameter of 14 or 16 mm (0.55 or 0.63 in.) for Specimens 3_1.5 and 2_1.0 or 2_1.5, respectively. The holes were filled with epoxy (Fig. 3(a)) to half of their depths. Each anchor spike was inserted into the holes after applying the CFRP sheets on the two opposite sides of the columns (Fig. 3(b)) and the protruding dry fibers were fanned out over the CFRP sheet (Fig. 3(c)). This method of anchoring was selected on

Fig. 2—Four columns tested: (a) control; (b) Specimen 2_1.5; (c) Specimen 3_1.5; and (d) Specimen 2_1.0. (Note: Dimensions in mm [in.].)

Fig. 3—(a) Filling of holes in anchorage region with epoxy resin; (b) place-ment of carbon-fiber anchor; (c) fanning out of fiber anchors over CFRP sheet; (d) local jacketing with CFRP; and (e) position of displacement transducers. (Note: Dimensions in mm [in.].)


the basis of transferring the tension forces from the CFRP sheet terminating at the bottom of each column into the concrete base. Finally, all columns received jacketing by wrapping a single layer of a 600 mm (23.62 in.) wide carbon sheet, identical to that used in the columns’ longitudinal direction (Fig. 3(d)). The effectiveness of confinement was improved by rounding the four corners near the base of each column to a radius equal to 25 mm (0.98 in.).

Test setup and materialsThe columns were fixed into a heavily reinforced 0.5 m

(19.68 in.) deep base block 1.2 x 0.5 m (47 x 19.7 in.) in plan, within which the longitudinal bars were anchored with 50 mm (1.97 in.) radius hooks at the bottom. The longitudinal bars 14 mm (0.55 in.) in diameter had a yield stress of 545 MPa (79.0 ksi), a tensile strength of 652 MPa (94.5 ksi), and an ultimate strain equal to 13.7% (average values from six specimens). The corresponding values for the steel used for the stirrups were 351 MPa (50.9 ksi), 444 MPa (64.4 ksi), and 19.5%. To simulate field conditions, the base blocks and the columns were cast with separate batches of ready mixed concrete (on 2 consecutive days). Casting of the columns was also made with separate batches due to the unavailability of a large number of molds. The average compressive strength and standard deviation on the day of testing the columns—measured on 150 x 150 mm (5.9 x 5.9 in.) cubes (average values from three specimens)—were equal to 17.1 and 0.95 MPa (2478 and 138 psi), respectively, suggesting that the variability in concrete strength would not affect the column test results. Cylinders with a diam-eter of 150 mm (5.9 in.) and a height of 300 mm (11.81 in.) were also used to obtain the splitting tensile strength of the concrete; the average tensile strength that was obtained from six specimens on the day of testing was equal to 2.2 MPa (319 psi).

The carbon-fiber sheet used as both longitudinal reinforce-ment (vertical fibers) and confinement (horizontal fibers) was a commercial unidirectional fiber product with a weight of 644 g/m2 (2.62 × 10–6 lb/in.2) and a nominal thickness (based on the equivalent smeared distribution of fibers) of 0.37 mm (0.0146 in.). The mean tensile strength and elastic modulus of the fibers (as well as of the sheet when the nominal thickness is used) was taken from data sheets equal to 3790 MPa (549.27 ksi) and 230 GPa (33,333 ksi), respectively. The carbon-fiber sheet was impregnated with a commercial low-viscosity structural adhesive (two-part epoxy resin with a mixing ratio of 3:1 by weight) with a tensile strength of 70 MPa (10.15 ksi) and an elastic modulus of 3.2 GPa (464 ksi) (cured for 7 days at 23°C [73°F]). The values of the tensile strength and elastic modulus for the epoxy-impregnated sheet were taken from data sheets equal to 986 MPa (142.9 ksi) and 95.8 GPa (13,884 ksi), respectively, corresponding to a thickness equal to 1 mm (0.039 in.).

Each anchor comprised a tow of carbon fibers of the same type used in the unidirectional sheets. The weight of the fibers for the anchors used in Specimens 3_1.5 and 2_1.0 was 63 g/m (0.0035 lb/in.); the anchors used in Specimen 2_1.5 were 50% heavier and the respective weight of the fibers was 94.5 g/m (0.0053 lb/in.). Impregnation and bonding of the fiber anchors was done using the same epoxy adhesive used for the impregnation of the carbon sheets.

The columns were subjected to lateral cyclic loading, which consisted of successive cycles progressively increasing

by 5 mm (0.20 in.) of displacement amplitudes in each direc-tion. The loading rate was in the range of 0.2 to 1.1 mm/s (0.008 to 0.043 in./s)—the higher rate corresponding to a higher displacement amplitude—all in displacement-control mode. At the same time, a constant axial load was applied to the columns, corresponding to 25.4% of the members’ compressive strength, which was calculated by multiplying the gross section area by the strength of the concrete. The lateral load was applied using a horizontally positioned 250 kN (56.2 kip) MTS actuator. The axial load was exerted by a set of four hydraulic cylinders with auto-mated pressure self-adjustment acting against two vertical rods connected to the strong floor of the testing frame through a hinge (Fig. 1(a)). As a result of this loading scheme, the variation of axial load during each test was negligible. With this setup, the P-D moment at the base section of the column is equal to the axial load times the tip displacement (that is, at the piston fixing position) of the column times the ratio of the hinge distance from the base (0.25 m [9.84 in.]) and the top (0.25 + 1.60 = 1.85 m [72.83 in.]) of the column (that is, times 0.25/1.85 = 0.135).

The displacements and axial strains at the plastic hinge region were monitored using six displacement transducers (three on each side, perpendicular to the loading direction) fixed at the cross sections 130, 260, and 450 mm (5.12, 10.24, and 17.72 in.) from the column base, as shown in Fig. 1(a) and 3(e).

EXPERIMENTAL RESULTSStrength, failure modes, and deformations

The response of all columns tested is given in Fig. 4 in the form of the load-drift ratio (obtained by dividing the tip displacement by the column’s height) loops. The corre-sponding envelope curves are given in Fig. 5; key results are also presented in Table 1. They include: 1) the peak resistance in the two directions of loading; 2) the degree of strength-ening—that is, the peak resistance normalized with respect to the peak load sustained by the control specimen in the two directions of loading; and 3) the observed failure mode.

The performance and failure mode of all tested specimens was controlled by flexure, as expected due to their design characteristics (a high value of the shear span ratio L/h = 6.4 and a relatively low ratio of longitudinal reinforcement). This was an important requirement, as the main objective in this study was to evaluate the effectiveness of spike anchors as a means of flexural strengthening of RC columns. The control specimen attained a peak load of approximately 37.8 kN (8.45 kips). After yielding of the longitudinal reinforcement, the load remained nearly constant up to a large drift ratio of 8%, corresponding to the termination of the test. The confinement provided by the CFRP jacket prevented spalling of the concrete cover and potential buck-ling of the longitudinal reinforcing bars.

All strengthened specimens displayed higher flexural resistance (from 17% up to approximately 35%) compared to the control specimen. The response of strengthened columns was not in all cases completely symmetrical in the two directions of loading due to slight differences in the internal reinforcement’s effective depth and the configura-tion of anchors in each strengthened side. Flexural cracking at the column base started at the early stages of loading and increased substantially with increasing drift ratios due to slip of the internal bars. Failure in all strengthened columns was due to tensile rupture of the anchors at the cross section of


maximum moment (column base). No other failure, such as debonding, was observed, indicating that the epoxy adhesive performed well and the anchor length was sufficient.

Rupture of the anchors resulted in a sudden drop of the applied force when the drift ratio was approximately 2.5%. As in the case of the control specimen, the confinement provided by the CFRP jacket prevented spalling of the concrete cover and potential buckling of the longitudinal reinforcing bars. The pinching observed in the hysteresis loops shown in Fig. 4 is attributed to slip of the internal bars and the nonyielding response of the fiber anchors.

Figure 6 gives the relation between the drift ratio and the slip rotation qslip of the cross section at the interface between the column and the base. The latter was measured using the data from the displacement transducers in two cross sections at a distance l1 = 130 mm (5.12 in.) and l2 = 260 mm (10.24 in.) from the base as follows: qslip = q2 – fl2 = q1 – fl1, where f is the mean curvature at the column base equal to (q2 – q1)/(l2 – l1). This assumption of a constant mean curva-ture is applicable if this distance l2 – l1 is small in the order of the typical distance of two adjacent flexural cracks if the behavior prior to yielding is of interest or the length within which concrete is expected to spall or crush and reinforcing bars may buckle or even break. In experiments, values of l2 – l1 in the range of h/2 to h are commonly selected. In this way, it is possible to estimate the contribution of the slip rotation to the overall column deformation. The qslip-drift ratio relation is nearly bilinear for all columns, with a first branch up to approximately the peak lateral load and a second one with a higher slope beyond that. The contri-bution of slip rotation to the columns’ overall behavior was prevalent, as it comprised the major part of their deformation capacity (drift ratio).

By comparing the degree of strengthening for all tests, some useful conclusions concerning the relative effective-ness of different anchor configurations can be made. Speci-mens 2_1.5 and 3_1.5 had anchors with the same total amount of fibers distributed in two and three anchors per side, respectively, with Specimen 2_1.5 displaying a higher degree of strengthening (1.35 versus 1.25). Hence, it may be concluded that two “heavier” anchors per side are more effective than three “lighter” ones. This is counterintuitive, but it may be explained by the higher probability of poor anchor installation as the number of anchors increases. More test results should clarify this observation further.

Specimens 2_1.5 and 2_1.0 had the same number of anchors; those in Specimen 2_1.5 were 50% heavier. Yet, the increase in flexural resistance due to strengthening in

Fig. 4—Load-versus-drift ratio curves for tested specimens.

Fig. 5—Load-versus-drift ratio envelope curves.

Table 1—Summary of results

Specimen

Peak force Pmax, kN (kips)

Column base moment at peak

force, kN·m (kip·ft)Drift ratio at peak

force, %Degree of strengthening (Pmax,Specimen/Pmax,Control)

Concrete cover in tension steel, mm (in.)

Effective strain in tension anchors, %

Push Pull Push Pull Push Pull Push Pull Push Pull Push Pull

Control37.91 (8.52)

–37.73 (–8.48)

62.53 (4.29)

–62.35 (–4.27)

3.67 –3.88 1.00 1.00 33 33 — —

2_1.551.15

(11.50)–50.66

(–11.39)83.06 (5.69)

–82.25 (–5.64)

2.40 –2.35 1.35 1.34 25 36 0.53 0.52

3_1.547.49

(10.68)–42.11* (–9.47)

77.40 (5.30)

— 2.79 — 1.25 — 32 31 0.43 —

2_1.045.04

(10.12)–43.33 (–9.74)

73.32 (7.25)

–70.56 (–4.84)

2.48 –2.42 1.19 1.15 35 26 0.49 0.43

*Unreliable result (not used in further calculations) due to wrong positioning of one anchor.

Fig. 6—Slip rotation at base in terms of drift ratio.


Fig. 7—(a) Cumulative dissipated energy during test; and (b) stiffness versus drift ratio.

Fig. 8—(a) Stresses at column base; and (b) approximate bilinear stress-strain curve for FRP-confined concrete.

Specimen 2_1.5 (34.5% on average in both directions of loading) was nearly double that in Specimen 2_1.0 (17% on average in both directions of loading), indicating that heavier anchors are more effective. On the basis of the limited test results presented in this study, it is concluded that anchors should be as few and as heavy as possible.

Stiffness and energy dissipationTo further evaluate the effectiveness of the various anchor

configurations, the stiffness and cumulative dissipated energy—computed by summing up the area enclosed within the load-versus-piston displacement curves—were recorded for each loading cycle and are plotted in Fig. 7. Overall, the use of anchors results in higher stiffness (in the order of 10 to 40%, depending on anchor configuration), up to the drift corresponding to anchor rupture, whereas the increase in energy dissipation is marginal.

It should be noted at this point that the increased stiffness of the strengthened columns may result in increased seismic forces. However, this is not of concern and should not lead to the conclusion that the positive effect of strengthening is counterbalanced by the negative effect of stiffening. What is of crucial importance in capacity design is the higher strength of columns versus that of beams, which is typically the reason

why flexural strengthening of columns is a demand—a fact that was verified experimentally in this study.

Effective strain of fiber anchorsOf crucial importance in the design of an FRP-based

strengthening system is the so-called “effective strain,” defined herein as the average tensile strain in the fiber anchors at failure. This value was calculated by performing an analysis of the cross section at the column base through the use of standard—in RC—force equilibrium, strain compatibility, and material constitutive conditions corre-sponding to the maximum bending moment at the cross section (Fig. 8(a)). In this analysis, the spike anchors are modeled as linear elastic tension elements. Note that the analysis was performed using the “exact” values of concrete cover as measured after each test and not the nominal values shown in Fig. 1(b); these values are listed in Table 1.

To account for the effect of FRP confinement, the compressive stress-strain behavior of concrete was modeled as bilinear (Fig. 8(b)) in agreement with extensive experi-mental evidence.11 According to the typical approach toward modeling confinement of concrete by FRP,12-14 the confined strength fcc and ultimate strain eccu depend on the confining stress at failure (fracture of the jacket in the circumferential direction) slu as follows15


11 ( )mcc co u cof f k f= + s l (1)

21 ( )nccu co u cok fe e = + s l (2)

The confining stress sl is, in general, nonuniform, espe-cially near the corners of rectangular cross sections. As an average for sl in a cross section with dimensions of b and h, one may write (Fig. 9(a) through (c))

( )

, , 2 212 2

j jh be j j j j

e j j j

t tk E E

h b

b hk t E

bh

s + s s = = e + e

+= e

l ll

(3)

where Ej and ej are the elastic modulus and strain, respec-tively, of the FRP jacket in the lateral direction; tj is the jacket thickness; and ke is an effectiveness coefficient which, for continuous jackets with fibers in the direction perpendicular to the member axis, is defined as the ratio of the effectively confined area (Ae in Fig. 9(d)) to the total cross-sectional area Ag as follows16

2 2

13e

g

b hkA+′ ′= − (4)

Hence, the confining stress at failure slu is given by Eq. (3) with Ejej replaced by the effective jacket strength in the lateral direction fje

( )u e j je

b hk t f

bh+

s =l (5)

The literature on the precise form of confinement models for concrete is vast. Some of these models, especially the older ones, are based on the assumption that the relation-ship between confined strength and ultimate strain and their unconfined counterparts is linear—that is, m and n are both equal to 1. In other models, especially in some of the most recent ones, m and n are taken as less than—but still close to—1. Whereas the main advantage of the former approach is simplicity, the disadvantage is that linear rela-tionships between fcc-slu and eccu-slu tend to overpredict both the confined strength and the confined ultimate strain for high confining stresses. As the authors’ objective in this paper is not to elaborate on confinement models for concrete but, rather, to perform a simple cross-section analysis with FRP confinement taken into account, the authors also make the assumption of linearity—that is, that m = 1 and n = 1. Moreover, in agreement with the typical CFRP confinement models for concrete,11 the authors take k1 = 2.15 and k2 = 10.

The aforementioned procedure was implemented in a computer program that performs equilibrium iterations in an automated way and yields the tensile strain in the spike anchors at failure of the cross section. It should be noted that failure was always reached when the spike anchors developed their strength and fractured in tension, while the maximum compressive strain in the concrete was less than the confined

strength fcc. This failure mode is in perfect agreement with experimental observations for all three FRP-strengthened columns. The resulting values of strain in the anchors at failure, summarized in Table 1, indicate that the effective strain in the spike anchors at failure—on average, equal to 0.0047—is well below the theoretical deformation capacity of the (carbon) fibers comprising the anchors. The main reasons for this difference are the stress concentrations at the anchor bend and the cyclic nature of stresses in the anchors.

CONCLUSIONSThis paper presents the first study on the combination of

FRP sheets and fiber anchors for flexural strengthening of RC columns under simulated seismic loading. The design of specimens allowed for an investigation of the number of anchors and the volume of fibers in each anchor. The results were combined with a simple analytical model to yield values for the effective strain in the anchors at failure. The main conclusions are summarized as follows:• Carbon-fiber anchors provide a viable solution toward

enhancing the flexural resistance of RC columns subjected to seismic loading.

• The effectiveness of anchors increases almost linearly with their weight.

• A fixed amount of fibers placed in the form of anchors is more effective when two heavier anchors are used instead of three but are lighter. This may be attributed to the increased probability of poor installation as the number of anchors increases and should be investi-gated further.

• On the basis of standard cross-section analysis, which accounts for the effect of confinement, the effec-tive strain in carbon-fiber anchors subjected to cyclic loading is in the order of 0.5%.

In view of the limited number of tests performed in this study, the aforementioned results should be considered as rather preliminary. Future research should be directed toward providing a better understanding of the parameters, including other amounts of fibers in the anchors, the level of axial load, initial column damage, different shear spans,

Fig. 9—(a) to (c) Approximate average confining stresses; and (d) effectively confined area in columns with rectan-gular cross section.


different loading histories, other cross sections, and other types of fibers.

ACKNOWLEDGMENTSThe authors wish to thank C. Papanicolaou, P. Apostolopoulou, and

K. Giannakopoulos for their assistance in the experimental program. The study reported in this paper was partially funded by FYFE EUROPE SA.

NOTATIONAg = gross section areaEj = elastic modulus of jacket in lateral directionfcc = compressive strength of confined concretefco = compressive strength of unconfined concretefje = effective strength of jacket in lateral directionh = cross-section heightk1, k2 = empirical constantske = confinement effectiveness coefficientL = lengthli = distance of cross section i from column base; i = 1, 2, 3m = empirical constantn = empirical constantPmax = peak forcerc = radius at corners of rectangular sectionstj = thickness of jacketec = compressive strain in concreteeccu = ultimate strain of confined concreteeco = strain at failure of unconfined concreteej = jacket strain in lateral directionf = mean curvature at column baseqi = rotation of cross section i; i = 1, 2qslip = slip rotation at column basesa = tensile stress in anchorssc = compressive stress in concretescc = maximum compressive stress in concretesl = lateral stress due to jacketingsl,b = lateral stress perpendicular to side bsl,h = lateral stress perpendicular to side hslu = ultimate lateral stress due to jacketingss = tensile stress in longitudinal steel reinforcement

REFERENCES1. Bournas, D. A., and Triantafillou, T. C., “Flexural Strengthening of

RC Columns with NSM FRP or Stainless Steel,” ACI Structural Journal, V. 106, No. 4, July-Aug. 2009, pp. 495-505.

2. Nagai, H.; Kanakubo, T.; Jinno, Y.; Matsuzaki, Y.; and Morita, S., “Study on Structural Performance of Reinforced Concrete Columns with Waist-High Walls Strengthened by Carbon Fiber Reinforced Plastic Sheets,” Proceedings of 4th International Symposium on Fiber Reinforced Polymer Reinforcement for Reinforced Concrete Structures, SP-188, C. W. Dolan, S. H. Rizkalla, and A. Nanni, eds., American Concrete Institute, Farmington Hills, MI, 1999, pp. 255-267.

3. Kobayashi, K.; Fujii, S.; Yabe, Y.; Tsukagoshi, H.; and Sugiyama, T., “Advanced Wrapping System with CF-Anchor—Stress Transfer Mecha-nism of CF-Anchor,” Proceedings of the 5th International Conference on Fibre-Reinforced Plastics for Reinforced Concrete Structures, C. J. Burgoyne, ed., Thomas Telford, London, UK, 2001, pp. 379-388.

4. Jinno, Y.; Tsukagoshi, H.; and Yabe, Y., “RC Beams with Slabs Strengthened by CF Sheets and Bundles of CF Strands,” Proceedings of the 5th International Conference on Fibre-Reinforced Plastics for Reinforced Concrete Structures, C. J. Burgoyne, ed., Thomas Telford, London, UK, 2001, pp. 981-988.

5. Ekenel, M.; Rizzo, A.; Myers, J. J.; and Nanni, A., “Flexural Fatigue Behavior of Reinforced Concrete Beams Strengthened with FRP Fabric and Precured Laminate Systems,” Journal of Composites for Construction, ASCE, V. 10, No. 5, 2006, pp. 433-442.

6. Orton, S.; Jirsa, J. O.; and Bayrak, O., “Design Considerations of Carbon Fiber Anchors,” Journal of Composites for Construction, ASCE, V. 12, No. 6, 2008, pp. 608-616.

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8. Niemitz, C. W.; James, R.; and Breña, S. F., “Experimental Behavior of Carbon Fiber-Reinforced Polymer (CFRP) Sheets Attached to Concrete Surfaces Using CFRP Anchors,” Journal of Composites for Construction, ASCE, V. 14, No. 2, 2010, pp. 185-194.

9. Ozbakkaloglu, T., and Saatcioglu, M., “Tensile Behavior of FRP Anchors in Concrete,” Journal of Composites for Construction, ASCE, V. 13, No. 2, 2009, pp. 82-92.

10. Prota, A.; Manfredi, G.; Balsamo, A.; Nanni, A.; and Cosenza, E., “Innovative Technique for Seismic Upgrade of RC Square Columns,” Proceedings of the 7th International Symposium on Fiber-Reinforced Polymer (FRP) Reinforcement for Concrete Structures, SP-230, K. Shield, J. P. Busel, S. L. Walkup, and D. D. Gremel, eds., American Concrete Insti-tute, Farmington Hills, MI, 2005, pp. 1289-1304.

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Title no. 110-S08


ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2011-060.R2 received April 22, 2011, and reviewed under Institute


Adaptive Stress Field Models: Formulation and Validationby Miguel S. Lourenço and João F. Almeida

Stress field models or strut-and-tie models are commonly recog-nized as powerful tools for the development of consistent design methods for structural concrete discontinuity regions. Service-ability and ductility topics are usually indirectly checked through an adequate model selection, together with appropriate reinforce-ment details. Stress field models are an alternative to finite element analyses, which are frequently referred to as the main tool for the assessment of the aforementioned topics. The proposed approach, called Adaptive Stress Field Models, extends the application of stress-field-based models for the nonlinear analysis of struc-tural concrete discontinuity regions, allowing a consistent study of service behavior, ductility and, more generally, model assess-ment topics. The numerical results are compared with the test and nonlinear finite element analysis results.

Keywords: adaptive structures; design models; discontinuity regions; nonlinear analysis; stress field models; structural concrete.

INTRODUCTIONIn recent times, the rapid increase of computational capa-

bilities has allowed numerical models to be widely used in the analysis process, leading to a further level of complexity in structural analysis. It must be kept in mind that whereas a scientific theory is formulated in precise mathematical terms and must cover all the details of a physical phenomenon, engineering models are derived from the former, but with a simpler approach that allows them to be applicable. These streamlined models must, however, preserve the proper eval-uation of the physical evidence and not disregard the main phenomena behavior (Marti 2005). An approach based on stress field models—stated in terms of equilibrium condi-tions—provides a simpler yet reliable framework to consis-tently evaluate a structural concrete region. It reiterates the idea of understanding the flow of forces within the elements, providing an unprecedented awareness of structural behavior, which is the hallmark of good design and detailing.

Since the beginning of the 1990s, several researchers have worked on automatic tools for the development of strut-and-tie models (Schlaich 1989; Kuchma and Tjhin 2002; Kostic 2006), on the application of stress field models for the nonlinear analysis of structural concrete regions (Rückert 1991), and on approaches based on the automatic development of stress fields using the finite element method (Fernández Ruiz and Muttoni 2007).

This study is meant to provide a step forward in the appli-cation of stress field models for the analysis and design of structural concrete. The proposed technique, Adaptive Stress Field Models, employs the convenient simplifica-tions inherent to the stress field method to develop a tool for the nonlinear analysis of discontinuity regions. From the authors’ point of view, the interpretation of the flow of forces provided by the method offers a unique awareness of the structural behavior throughout the loading process, which can be considered essential for an adequate judgment of the output. The internal stress redistributions due to the nonlinear

behavior of the materials are accomplished by the introduction of the adaptive structures concept to stress field systems—in particular, model follows energy. This means that the model configuration at each load step is thus obtained following the least complementary energy. It intends to consistently follow the stress field model concept by setting an initial stress field distribution and the appropriate variables that will be adjusted (geometry and/or forces of the model).

The mechanical properties of the compression and tension elements are obtained directly from the geometry of the stress fields, accounting for the nonlinear constitu-tive relationships of the materials. A detailed study of the behavior of the elements is developed based on the consti-tutive relationships of the materials for monotonic loading. Special attention is given to the behavior of reinforced ties because the global energy is deeply influenced by the tension stress fields (Schlaich et al. 1987; Rückert 1991; Sundermann 1994). The well-known constitutive relation-ship proposed in technical documents (CEB-FIP MC90 1993; ACI Committee 318 2008), mainly defined for stabi-lized cracking, can be inappropriate for some discontinuity regions. The cracking pattern is sometimes characterized by a main crack that deeply influences global structural behavior. It was assumed that this phenomenon was related to the tie stress distribution, preventing the formation of a stabilized cracking. Therefore, the reinforced concrete tie element’s behavior was obtained following the Tension Chord Model concept (Marti et al. 1998).

The proposed technique is validated by comparing the numerical results with the outcome from the test and nonlinear finite element results. Several discontinuity regions are analyzed—namely, simply supported deep beams with top and suspended loads and continuous deep beams. In all cases, it is shown that the main aspects of the structural behavior were well-simulated, showing the capa-bility of the methodology for the nonlinear analysis of struc-tural concrete regions.

RESEARCH SIGNIFICANCEDespite all its unquestionable advantages, the applica-

tion of stress field models in the design process is not fully exploited. One of the frequently mentioned aspects is the non-uniqueness of the design models, which raises the discussion of the validity of the models, mainly concerning ductility and service behavior. The proposed approach intends to be a contribution to overcome the aforementioned limitations and extends the application of stress field models to the predic-


Miguel S. Lourenço is a Partner and Technical Director of JSJ Consulting Ltd., Lisbon, Portugal. He received his civil engineering degree, his MSc in structural engi-neering, and his PhD from Instituto Superior Técnico (IST-ICIST), Lisbon, Portugal, in 1995, 2000, and 2010, respectively. His research interests include design models and nonlinear behavior of structural concrete.

João F. Almeida is an Associate Professor of Concrete Structures at IST-ICIST and a Partner and Technical Director of JSJ Consulting, Ltd. He received his civil engi-neering degree in 1981 and his MSc in structural engineering and PhD in civil engi-neering from the Technical University of Lisbon, Lisbon, Portugal, in 1985 and 1990, respectively. His research interests include design models and nonlinear behavior of structural concrete.

tion of nonlinear behavior of structural concrete disconti-nuity regions. The development and practical application of such a specific design tool will improve the consistency and clearness of design methods, improving the knowledge of structural concrete behavior.

STRESS FIELD MODELSThe knowledge of stress field modeling was considerably

developed by several authors during the 1980s and 1990s, essentially in Zurich, Stuttgart, and Lausanne. Schlaich et al. (1987) provided guidelines for the uniform design of every part of any structure regarding safety and serviceability, “replacing empirical procedures, rules of thumb and guess work by a rational method.” Reineck (1996, 2005) increased the knowledge by applying the method to shear design and the design of D-regions of prefabricated members with several numerical examples. An excellent overview of this technique can be found in Muttoni et al. (1998). The divul-gation in the United States soon began with a survey of the technique presented by Marti (1985) for the ACI commu-nity. Later, Mitchell and Cook (1991) presented examples of the application of strut-and-tie models and nonlinear finite element analysis. Joint ACI-ASCE Committee 445 (1997) published a bibliography of the strut-and-tie method, and Kuchma and Tjhin (2002) published “Computer-Based Tools for Design by Strut-and-Tie Method: Advances and Challenges.” The collection of examples in ACI SP-208 (Reineck and ACI Committee 445 2002) should also be mentioned, which included a justification of Appendix A of ACI 318-02 presented by MacGregor. Finally, ACI SP-273 (Reineck and Novak 2010) was recently published, with several examples of design with strut-and-tie models based on Appendix A of ACI 318-08 (ACI Committee 318 2008). Referring to CEB-FIP MC90 (1993), design with strut-and-

tie models was very well-described in Schäfer (1999), and it is worth mentioning the FIP Recommendations (1999), a technical recommendation fully based on design with strut-and-tie models.

Stress field models intend to reproduce the compression and tension fields due to the load path deviation within any structural concrete region. They present a simple design tool that allows the visualization of the flow of forces throughout a structural concrete region, wherein every element clearly has a physical meaning. Strut-and-tie models, along with a draft representation of stress fields, are very useful for prac-tical design applications. Stress field models indicate the necessary amount and correct distribution of reinforcement, as well as the node geometry definition for the design of support widths and anchorage lengths.

A plane and uniaxial stress state is assumed for each of the three types of elements (refer to Fig. 1): ties, fan-shaped struts, and prismatic-shaped struts. In this particular case, to establish the compression and tension field widths, the following considerations were adopted: prismatic struts are defined by the compression widths at the region boundaries; diagonal compression field spreads within the region and field widths are evaluated essentially by vertical reinforce-ment distribution widths and the anchorage length of the horizontal reinforcement. Prismatic struts are assumed to have constant compressive stresses—simply obtained by dividing the resultant force by the stress field area—whereas fan struts have compressive radial stresses with a hyperbol-ical variation along their length. On a reinforced tie element, the bond constitutive relationship leads to different steel and concrete stresses along the length of the tie. The stress field resultants are obtained by nodal equilibrium and the stress state in each element is described in the following.

FORMULATIONThe mechanical properties of the stress fields are based on

their geometry and stress state, as mentioned previously, and current structural analysis techniques are applied. The stress redistribution is simulated following the adaptive structures concept—in particular, form follows energy. This means that the model configuration is adjusted according to the least global energy—obtained by the sum of the total computed energy of each element of the model—and at each load step.

Adaptive structures can be defined as systems whose mechanical or geometrical characteristics are adapted according to a specific optimization process. Teuffel (2004)

Fig. 1—Stress field model of dapped beam (Muttoni et al. 1998).


presented several optimization criteria for truss structures according to different principles: minimize structure weight by changing the element section and/or structural geom-etry, optimize axial forces on bars by adjusting the element cross section, and adapt structural geometry according to minimum energy criteria (form follows energy).

This last principle is somehow related to the energetic principles stated by Koiter (1960) for elastic-plastic solids: “Minimum principle for the stress state rates—For all stati-cally admissible stress rate distribution, the exact solution is attained by minimizing complementary energy in each load increment.”

The model-follows-energy concept consists of minimizing complementary energy U = ∫V∫sedsdV satisfying equilibrium and compatibility conditions, where U should, in particular, depend on the structure configuration, V is the integration domain, and s and e are the element’s stress and strain, respectively. The node position or angles between bars are set as variables and will be called adaptive variables.

For truss structures, the application of this methodology is computationally effortless because each element is subjected only to an axial force. The equilibrium equations can be obtained from nodal equilibrium in the global referential (SFx = 0, SFy = 0) and are condensed in Eq. (1) (refer to Fig. 2).

0+ =TN F (1)

where TT = [TcTs] with Tc = ATDc and Ts = ATDs; matrix AT (the transpose of matrix A) represents the node incidences of elements. The term Aij is 1 or –1 (according to the direc-tion of the elements) if node i connects element j; otherwise, it is 0; Dc and Ds are diagonal matrixes with cosines and sines, respectively, of each bar angle with horizontal direc-tion; vector N includes the axial forces of the elements NT = [N1...Nn]; and vector F represents the applied forces at nodes in the global referential FT = [Fx1...Fxm|Fy1...Fym].

Compatibility equations are obtained by establishing a relationship between nodal displacement and bar elongation, as defined by Eq. (2).

0 0 + = ⇔ + = D d D dT T Tc sA D TD A (2)

where Vector D contains the total axial elongation of the

elements: 10 1 0

nLLTnds ds = ∫ e ∫ e D ; and vector d contains

the node displacements in the global referential dT = [dx1...dxm|dy1...dym].

Finally, the geometry adaptation and/or stress distribution of the models is obtained by minimizing the global comple-mentary energy U subjected to equilibrium and compatibility constraints (refer to Eq. (3)). The minimization process is transformed into a set of nonlinear equations (Eq. (4)) with the same number of equations as adaptive variables vi, which may be solved iteratively by numerical methods such as Newton’s method.

min , subject to the constraints 0 and 0+ = + =TU TN F TD d (3)

min 0i

Uv

∂= =

∂U (4)

COMPRESSION STRESS FIELDSIn prismatic or parallel stress fields, the stresses are

obtained by simply dividing the resultant compressive force by the area perpendicular to the force. Fan-shaped stress fields are, in general, noncentered and the definition of the node boundary geometry is established according to equi-librium conditions of an infinitesimal strut within the fan (Baumann 1988). The node boundary curve is defined by a second-order polynomial (refer to Fig. 3(a)) and the radial stresses at each boundary are defined according to Eq. (5).

2

2

1 tan

tanr II

II

I

+ as

+s

s = sa

(5a)

2

2

1 tan

tanr II

II

I

+ a= s′ s′ +

s′

s′a

(5b)

The stresses along each chord within the fan vary hyper-bolically—from the value sr into sr′ at each radial angle a—and are obtained according to the geometrical and equilib-rium conditions of an infinitesimal width of a specific chord (Eq. (6)). For practical and computational implementation, it is feasible and reliable to consider a linear stress distribution to simulate the behavior of the fan. In this case, the stresses are calculated according to the node’s dimensions, as shown in Fig. 3(a).

( ) ( ) ( )0, with rr r o

r r

rr r

ra s′

s a = s a =s − s′

l (6)

The concrete strain distribution is obtained by the common hyperbolical sc-ec curve (Eq. (7)). The factor h in Eq. (7) is meant to predict the reduction of the concrete strength caused by the presence of transversal strains. This reduction usually occurs in compression stress fields crossing tensioned reinforcement and is mainly due to the bond induced by the reinforcement between the cracks and the friction stresses, leading to transversal tensile stresses and strains in concrete (Reineck 1995, 2002). Other reasons for this reduction are also the smaller effective width of the crack surface and the

Fig. 2—Forces and displacements of element.


disturbances induced by the reinforcement. This effect was evaluated by several authors (Schlaich and Schäfer 1983; Schäfer et al. 1990; Eibl and Neuroth 1988; Kollegger and Mehlhorn 1990; Vecchio and Collins 1993) and it is usually simulated by a decrease in the concrete compressive peak value, depending on the applied transversal tensile strains h(e1) (Vecchio and Collins 1986).

( )2

1c cc cc

c ccfs ae − e

=+ be′

(7)

with

11 1 1 1

; 2; ; c c c cc cc

c c c c

E E fE

E Ee

a = b = − = e =e e

The complementary energy is defined by Eq. (8), illus-trated in Fig. 3(b), and computed by Gauss integration within the fan area.

2

1 0; c

Lcompl compl A complu d U u dAdss

s= ∫ e s = ∫ ∫ (8)

TENSION STRESS FIELDSThe global internal energy is mainly influenced by ties,

especially after cracking of the concrete. To properly simu-late the behavior of a structural concrete region under service loads and for post-yielding phases, special atten-tion of tension stress fields is required. In general, technical documents define constitutive relationships for reinforced concrete ties assuming a stabilized crack pattern, which is not suitable for some discontinuity regions. In fact, situa-tions of nonstabilized cracking are commonly observed—for example, the load near the support and dapped-end beam. In such cases, the crack pattern is usually characterized by a main crack that has a relevant effect on the global structural behavior. This phenomenon can be related with the stress distribution along the tie length, in which the length of the tie with constant stresses may not be enough to allow the formation of a stabilized crack pattern. The length of the tie with variable stress distribution is obtained directly from the stress field definition, consistently following the stress field

model. For prismatic and fan compression fields intersecting the tension-compression node, a linear and a parabolic stress variation are respectively obtained; however, it was always assumed to be a linear distribution because minor differ-ences in the global tie behavior are expected.

To properly simulate a reinforced concrete tie element, the approach presented in the Tension Chord Model by Marti et al. (1998) is applied. This model establishes equilib-rium conditions for an infinitesimal reinforced concrete tie (Fig. 4(a)), allowing the determination of concrete and steel stress variation along the element length. The concrete in tension has a linear sc-ec relation until it reaches its tensile strength fct with the Young’s modulus of Ec. The reinforced bars have a bilinear ss-es relation with the stiffness Es in the elastic branch and, according to the steel ductility, with hard-ening Et after yielding. A rigid perfect relationship for bond stresses was shown to be adequate (Marti et al. 1998) to simu-late a reinforced concrete tie with tb0 = 2fct and tb1 = fct before and after yielding of the reinforcement steel, respectively.

Compatibility conditions and the constitutive relation-ships for concrete in tension, steel, and bond, together with suitable strain integration, allow the calculation of the crack widths, the total elongation, and the element mean strain.

Figure 4(b) and (c) shows stress, strain, and slip varia-tion along the length of the element of two identical ties with different anchorage lengths. Quite different crack patterns, steel stresses, and concrete stresses are obtained. As expected, the tie with larger lengths of variable axial load reveals a stiffer behavior, evident in the stress-mean strains curves shown in Fig. 4(d). It is worth mentioning that a reliable model for stabilized and nonstabilized cracking is essential for the application of the Adaptive Stress Field Model in structural concrete regions because substantially different complementary energies and thus different global behavior are computed.

s c s s c cN N N A A= + = s + s (9)

4 bc

c

qdA

t rs = +

f(10a)

Fig. 3—(a) Radial stresses of noncentered fan; and (b) stress-strain relationship of concrete and complementary energy definition.


4 bsd

ts =

f (10b)

( )2

1 s c dxssd = ∫ e − e (11a)

sm L

de = (11b)

Similar to the compression stress fields, the complemen-tary energy is defined by Eq. (12) and represents the area “behind” the stress-mean strain curve.

2

1 0; Lcompl sm compl A complu d U u dAdss

s= ∫ e s = ∫ ∫ (12)

TEST EXAMPLETo illustrate the behavior of an adaptive stress field model,

a simple example of a statically indeterminate problem with

three bars is presented. In the following example (Fig. 5), Element 1 is assumed to have a nonlinear behavior with a bilinear s-e constitutive relationship, and Elements 2 and 3 are considered linear elastic. The main goal is to compute and minimize the complementary energy at each load step and show the model adjustment given by the angle q. Note that the compatibility conditions in Node N2 are fulfilled only by setting the vertical reaction in Node N3 as an adaptive vari-able—that is, imposing equilibrium conditions in the mini-mization process without explicitly imposing compatibility constraints. Thus, two adaptive variables—q and Fy3—were selected and the complementary energy is calculated as shown in Eq. (13). The model geometry q and the statically indetermi-nate variable Fy3 are obtained by correspondingly computing the derivatives of the complementary energy. Equations (14) and (15) are obtained by assuming that all the bars have the same E—that is, the structure is in the elastic or “uncracked” branch. For this phase, the model configuration that minimizes the complementary energy is given for q = 60 degrees and the statically indeterminate variable is Fy3 = 3/13F. Note that Eq. (15) represents the compatibility conditions for the vertical displacement d (e2L2 = –e1L1cosq + e3L3sinq), following the energetic principle of the minimum stress state rate mentioned previously. Developing similar analysis for the “post-yielding” branch—that is, assuming Element 1 with Et—a model config-uration of q ≈ 80 degrees is obtained.

( )2 23 1

11

cot12

yF F LU

EA

− q= (13a)

( )2 23 2

22

sin12

yF F LU

EA

− q= (13b)

23

33

3

12

yF LU

EA= (13c)

3 20 2cos 3cos 1 0 603

U∂ p= ⇒ q + q − = ⇒ q = = °

∂q (14)

Fig. 4—(a) Stresses of infinitesimal reinforced concrete tie. Steel and concrete stresses and strains, slip, and crack width of reinforced concrete tie; (b) stabilized cracking; (c) nonsta-bilized cracking; and (d) reinforced concrete tie constitutive relationships for stabilized and nonstabilized cracking.

Fig. 5—Test example: geometry and mechanical properties. (Note: 100 MPa = 14.5 ksi.)


( )

( )

− q∂= ⇒ − −

∂

−+ = ⇒ =

q

23 1

3 1

3 2 3 332

32

cot0

1 302 13sin

y

y

y yy

F F LUF EA

F F L F LF F

EAEA

(15)

To better understand the adaptive analysis process, the force-displacement curves F-d (Fig. 6(a)) are calculated, assuming that the geometry remains unchanged with q = 60 degrees and q ≈ 80 degrees. These results are further compared with an incremental nonlinear adaptive analysis (refer also to Fig. 6(a)). Figure 6(b) shows the geometry variation q along the load incremental process. Finally, it is possible to observe that the adaptive nonlinear analysis adjusts structure geometry to follow the different “linearized stages.” In the “uncracked” stage, the structure geometry remains unchanged at q = 60 degrees. After cracking, steel stresses and geometry adjustment tend to the “post-yielding linearized stage.”

VALIDATIONIntroduction

The validation of the proposed formulation is performed by comparing the obtained numerical results with the test and nonlinear finite element results for some basic but

typical discontinuity regions: simply supported deep beams with top and suspended loads, and continuous deep beams.

Deep beams—top loadThe presented methodology is compared with the well-

known Leonhardt and Walther (1966) deep beam tests and with nonlinear finite element analysis using the ATENA program. Two simply supported beams (WT2 and WT3) in particular were analyzed. These results were previously presented by Lourenço et al. (2006) and later by Nunes (2008).

The geometrical dimensions, loads, and steel reinforce-ment layouts of Deep Beams WT2 and WT3 are represented in Fig. 7. Tests have shown considerable redistribution of internal stresses due to cracking. Experimental ultimate loads reached Pu = 1.28pu = 1195 kN (268.6 kips) and Pu = 1290 kN (290.0 kips) for Deep Beams WT2 and WT3, respectively. At rupture, Deep Beam WT2 exhibited yielding of the bottom reinforcement and maximum support pressures of approxi-mately 1.06fcu. With regard to Deep Beam WT3, bottom steel stresses of approximately 370 MPa (53.7 ksi) and support pressures of approximately 1.19fcu were measured (fcu—compression strength of 200 x 200 x 200 mm3 [7.87 x 7.87 x 7.87 in.3] cubes; f1c = 0.89fcu according to ACI 318-08). Steel reinforcing detailing at the node region, where U-loops were adopted, improved concrete confinement and thus a higher local concrete strength was achieved.

Each developed stress field model (Fig. 8(a) and (c)) simulates all horizontal deep beam reinforcements, and the adaptive variables are node horizontal coordinates of the web reinforcement.

The numerical results agree quite well with tests at all load stages (Fig. 8(b) and (d)). As mentioned previously, the reinforcing detailing at the node region, where U-loops were adopted, improved concrete confinement and increased concrete strength at the supports. This aspect was disre-garded in the numerical model, leading to ultimate load values slightly lower than the experimental results (Pu = 1100 kN [247.3 kips]). Concerning the slight deviations of the mean steel strains between the numerical analysis and experimental results, it must be kept in mind that the steel strains were measured by strain gauges glued on bars, although the test outputs depend on the crack location and strain gauge positions.

It should be pointed out that the stiffness reduction related to cracking of the main tie induces significant internal stress redistributions, evident in the stress field model configu-rations (Fig. 8(a) and (c)) or in the internal lever arm (z/l)

Fig. 6—Adaptive analysis of test example: (a) force versus displacement; (b) force versus stresses at Element 1; and (c) force versus model configuration. (Note: 100 MN = 22,481 kips; 0.1 m = 3.94 in.; 100 MPa = 14.5 ksi.)

Fig. 7—Deep beam geometry and reinforcement layout (Leonhardt and Walther 1966): (a) WT2; and (b) WT3. (Note: 1 m = 39.4 in.; 100 mm2 = 0.155 in.2)


Fig. 8—(a) Deep Beam WT2 adaptive stress field model for several load steps; (b) test, nonlinear finite element, and adaptive stress field results for Deep Beam WT2; (c) Deep Beam WT3 adaptive stress field model for several load steps; and (d) test, nonlinear finite element, and adaptive stress field results for Deep Beam WT3. (Note: 100 kN = 22.5 kips; 1 mm = 0.039 in.)


Fig. 9—Deep Beams WT2 and WT3 numerical and test crack pattern for ultimate load: (a) WT2: test (Leonhardt and Walther 1966); (b) WT2: adaptive stress field model; (c) WT2: finite element analysis; (d) WT3: test (Leonhardt and Walther 1966); (e) WT3: adaptive stress field model; and (f) WT3: finite element analysis.

variation. These aspects are particularly relevant for the Deep Beam WT2 test, where the available ductility allowed full use of the beam effective depth. Figure 9 shows the agreement of the numerical and test crack patterns for ultimate load.

In general, the numerical results agreed quite well with those obtained experimentally. The force-strain and force-crack width curves obtained by the adaptive stress field analysis and the finite element method closely follow the test at different stages—namely, uncracked, cracked, and post-yielding. In this study, the finite element method provides slightly stiffer results, evident in the force-displacement curves; however, higher crack widths were obtained. The Adaptive Stress Field Model approach was revealed to be quite appropriate and, furthermore, the graphical output allowed an excellent visualization of structural behavior for all load stages.

Deep beams—suspended loadTwo simply supported deep beams with suspended

loads (WT6 and WT7) were also tested by Leonhardt and Walther (1966) and numerically compared with the obtained results. Deep beam geometrical dimensions, loads, and steel reinforcement layouts are represented in Fig. 10(a) and (b). Identical dimensions and reinforcement layouts of Deep Beams WT2 and WT3 were adopted. The web vertical reinforcement suspends the applied loads at the bottom surface of the deep beams.

Three layers of reinforcement were considered in the stress field models. The top distributed reinforcement layers were discarded because little influence on the global analysis was observed. The adaptive variables are the node horizontal position of the bottom web reinforcement and the node vertical position of one of the vertical reinforcements.

Such as the deep beams with top loads, the stiffness reduc-tion related to cracking of the bottom tie leads to internal

stress redistribution, clearly shown in the stress field model configurations illustrated in Fig. 10(c) and (d). Again, the available ductility allowed the deep beam with less bottom reinforcement to fully exploit the effective depth. The numer-ical mean strain results showed good agreement with the experimental output, especially in the uncracked stage and after cracking of the deep beam bottom surface (Fig. 11(a) and (b)). However, a significant discrepancy was obtained for the numerical and test crack widths because the observed test crack widths were measured in the deep beam front surface not effectively controlled by the main reinforcement.

Figure 11(c) to (f) shows the agreement of the numerical stress field distribution and test crack pattern for ultimate load.

Continuous deep beamA two-span continuous Deep Beam DWT2 (Leonhardt

and Walther 1966) was also analyzed, and the numerical and test results were compared. Figure 12(a) presents the beam test geometry and loads. The main and distributed reinforce-ments of the deep beam were simulated in the adaptive stress field analysis. The adaptive variables were the horizontal location of the nodes leading to tension or compression in horizontal elements.

Figure 12(b) illustrates the comparison between the numerical and experimental results. A reasonable agreement should, in general, be noticed. The model thickness enlarge-ment of the central support region all along the beam height was not simulated in the numerical model, which seems to be more flexible; however, the initial elastic range of the numerical model closely follows the linear results obtained with the finite element analysis.

Figure 12(c) and (d) shows the good agreement between the test crack pattern and the numerically obtained stress field distribution. Because the thickness enlargement of the


Fig. 10—Deep beam geometry and reinforcement layout (Leonhardt and Walther 1966): (a) WT6; and (b) WT7. Adaptive stress field model for several load steps: (c) WT6; and (d) WT7. (Note: 1 m = 39.4 in.; 100 mm2 = 0.155 in.2; 100 kN = 22.5 kips.)

Fig. 11—Comparison of numerical and test results: (a) WT6; and (b) WT7. Stress field configuration and test ultimate load crack pattern (Leonhardt and Walther 1966): (c) and d) WT6; and (e) and (f) WT7. (Note: 100 kN = 22.5 kips.)


Fig. 12—(a) Deep Beam DWT2 geometry and reinforcement layout (Leonhardt and Walther 1966); (b) Deep Beam DWT2 tests and adaptive stress field results; (c) Deep Beam DWT2 test crack pattern (Leonhardt and Walther 1966); and (d) adaptive stress field model. (Note: 1 m = 39.4 in.; 100 mm2 = 0.155 in.2; 100 kN = 22.5 kips; 100 MPa = 14.5 ksi.)

middle support was not simulated, the numerically obtained stress field inclination is slightly flatter.

CONCLUSIONSThis study aimed to provide a step forward in the applica-

tion of the stress field models for the analysis and design of structural concrete. The proposed technique, Adaptive Stress Field Models, employed the convenient simplifications inherent to the stress field models to develop a tailor-made tool for structural concrete. The method was implemented in a computer program; however, the idea was not to build up a fully automatic tool, even less a “black box.” One of the main aspects is the visualization of the flow of forces provided by the method, which offered a unique awareness of the struc-tural behavior along the loading process—essential for an adequate judgment of the outputs. The main features of the proposed approach are as follows:• The internal stress redistributions due to the nonlinear

behavior of the materials were accomplished by the incorporation of the adaptive structures concept to stress field systems—in particular, model follows energy.

• The model configuration at each load step was thus obtained following the least complementary energy. It intends to consistently follow the stress field model concept by setting an initial stress field distribution and selecting the appropriate variables that will be adjusted: geometry and/or forces of the model. The need to select a model to start the tool should not be considered a drawback; rather, it should be considered a support for the engineer to understand the structure’s main behavior before starting a more complex analysis.

• The mechanical properties of the compression and tension elements were directly obtained from the geom-etry of the stress fields, accounting for the nonlinear constitutive relationships of the materials.

• Special attention was given to the behavior of reinforced ties: the well-known constitutive relationship proposed in technical documents, mainly defined for stabilized cracking, can be inappropriate for some discontinuity regions. The cracking pattern is sometimes character-ized by a main crack that deeply influences global struc-tural behavior. It was assumed that this phenomenon was related to the tie stress distribution, preventing the formation of a stabilized cracking.

• The decrease in the concrete compressive strength due to its transverse strain state is also considered in the formu-lation to predict an eventual premature concrete crushing.

The technique was validated by comparing the numerical results with the outcomes from the tests and nonlinear finite element analysis. The following aspects should be pointed out:• The internal stress redistributions after cracking

observed in several of the presented tests were well-simulated in the numerical model. In fact, the stiffness reduction related to cracking of the main ties induced significant stress redistributions—evident in the stress field model configurations.

• The force-strain and force-crack width curves obtained by the adaptive stress field analysis closely followed the test results at different stages (cracked and post-yielding).

• The model approach was revealed to be quite appro-priate and, furthermore, the graphical output allowed an excellent visualization of structural behavior for all load stages.

• Finally, in all cases, it was considered that the main struc-tural behavior aspects were well-simulated, showing the capability of the methodology for the nonlinear analysis of structural concrete regions.

NOTATION(matrixes and vectors in bold; subscript notation follows primary notation)A = area


AT = transpose of matrix A that represents node incidences of elementsa, b = widthsDc, Ds = matrixes with cosines and sines, respectively, of each bar angleE = elastic modulusF = applied forcesF = vector of applied forcesf = material strengthl = lengthM = momentN = axial forceN = vector of axial forcesP = applied forcesq = distributed load r = radiusT = condensation of matrixes Tc and Ts

Tc, Ts = condensation of matrixes ATDc and ATDs, respectivelyt = thicknessu, U = complementary energyV = domainv = variablew = crack widthx, y = coordinatesz = inner level arma, b, h = coefficientsD = elongationd = displacemente = strainq = angler = reinforcement ratio (r = As/Ac)s = stresst = tangential stressØ = diameter

subscripts1c = uniaxial compression strength of concreteb = bondc = concrete, compressiond = designi, j = constantk = characteristicm = mean, mediumr = radius, radial s = steelt = tension, tangentialu = ultimatew = web, wedgey = yielding



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Sundermann, W., 1994, “Tragfähigkeit und Tragverhalten von Stahl-beton-Scheibentragwerken,” PhD thesis, Institut für Tragwerksentwurf und -konstruktion, University of Stuttgart, Stuttgart, Germany, 1994. (in German)

Teuffel, P., 2004, “Entwerfen Adaptiver Strukturen,” PhD thesis, Insti-tute for Lightweight Structures and Conceptual Design (ILEK), Universitat Stuttgart, Stuttgart, Germany, 2004. (in German)

Vecchio, F. J., and Collins, M. P., 1986, “The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear,” ACI JOURNAL, Proceedings V. 83, No. 2, Mar.-Apr., pp. 219-231.

Vecchio, F. J., and Collins, M. P., 1993, “Compression Response of Cracked Reinforced Concrete,” Journal of Structural Engineering, ASCE, V. 119, No. 12, Dec., pp. 3590-3611.

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Title no. 110-S09




Adaptive Stress Field Models: Assessment of Design Modelsby Miguel S. Lourenço and João F. Almeida

When using strut-and-tie or stress field models for design, different engineers may propose different models for the same situation, which may sometimes lead to quite different amounts of required reinforcement. This can lead to concerns regarding the suitability of the chosen model. This study intends to provide a step forward for the assessment of strut-and-tie models, applying the Adap-tive Stress Field Models concept. Several representative disconti-nuity regions are studied and different design models and corre-sponding reinforcement layouts are analyzed for each of them. The main purpose is to discuss to what extent the design model can be adjusted, preventing deficient service behavior and/or prema-ture failure. For several cases, the suitable redistributions from the reference models (elastic-based solution) were assessed, providing guidance for the development and application of strut-and-tie models for the design of discontinuity regions.

Keywords: adaptive structures; discontinuity regions; nonlinear analysis; stress field models; structural concrete; strut-and-tie models.

INTRODUCTIONDuctility and service behavior topics are frequently

referred to as limitations of the strut-and-tie method appli-cation. Several authors (Schlaich et al. 1987; Muttoni et al. 1998; Reineck 2002; Fernández Ruiz and Muttoni 2007) addressed this aspect of choosing the proper design model for a particular discontinuity region. The general and simplified energetic criteria mentioned by Schlaich et al. (1987) to choose a model out of several possible ones has shown to be suitable for most practical cases: “In selecting the model, it is helpful to realize that loads try to use the path with the least forces and deformations. Since reinforced ties are much more deformable than concrete struts, the model with the least and shortest ties, is the best.” Other researchers presented additional proposals: Muttoni et al. (1998) provided some guidelines for establishing stress field models by selecting load-path mechanisms that prevent the opening of large cracks and indirectly control service behavior; Sigrist et al. (1995) applied the upper-bound methods of limit analysis considering failure mecha-nisms to complement strut-and-tie-based models.

The general and consensus rule initially referred to by Schlaich et al. (1987) and Schlaich and Schäfer (1991) was further included in technical documents (FIP Recom-mendations 1999; EN 1992-1-1:2004): if the strut-and-tie model’s main compression and tension fields are orien-tated according to the elastic trajectories, one can indirectly ensure a good service behavior. Furthermore, because small internal stress redistributions are to be expected, the same model can be used for ultimate and service loads. The model orientation through the elastic solution should not be inter-preted as a “perfect mapping” of the elastic trajectories but merely to ensure that the main compression and tension fields are close to the principal stress trajectories. In some discontinuity regions, however, following strictly elastic stress trajectories may lead to detailing difficulties and, in

particular cases, improper reinforcement layout. In prac-tice, the modeling must take into consideration the practical reinforcement layout and the strut-and-tie model should, in general, be used with an appropriately distributed minimum reinforcement where no main reinforcement is required. The minimum steel area must be able to equilibrate a signifi-cant fraction of the concrete tensile strength that overcomes locally after cracking. Special attention should also be given to regions where compression stress fields cross over impor-tant transversal strains because a significant reduction of the peak compression strength of the concrete is obtained. It is worth mentioning that, in most cases, the widths of the compression stress fields are determined by the static condi-tions (for example, dimensions of the loading plates), and the concrete effective strength is rarely attained inside the region. Finally, the selection of a strut-and-tie model for design requires some engineering judgment to ensure a good service behavior and avoid a premature failure.

In the following, several discontinuity regions will be analyzed and their model assessments discussed. The appli-cation of the Adaptive Stress Field Models methodology is exceptionally useful for these purposes because it allows a fundamental guidance for the selection of strut-and-tie models and extends the application of stress-field-based models to the prediction of nonlinear behavior of struc-tural concrete discontinuity regions. The main objective is to discuss within which limits the designer can choose the model, preventing deficient service behavior and/or prema-ture failure. It should be noted that the models that will be analyzed in the following may be considered as representa-tive of several discontinuity regions. They can simulate the general model of a particular discontinuity region or possibly be included in more complex models. For this reason, the rules presented in the following may provide guidance for the proper design and detailing of most of the structural concrete regions.

RESEARCH SIGNIFICANCESeveral studies for systematizing and validating the

strut-and-tie method have been developed; however, very few regard the model’s assessment topics. The Adaptive Stress Field Model provides a preferential and reliable tool for the selection and validation of strut-and-tie models for design purposes because it consistently follows the stress field model concept for the nonlinear analysis of structural concrete discontinuity regions. Furthermore, the devel-


Miguel S. Lourenço is a Partner and Technical Director of JSJ Consulting Ltd., Lisbon, Portugal. He received his degree in civil engineering, his MSc in structural engineering, and his PhD from Instituto Superior Técnico (IST-ICIST), Lisbon, Portugal, in 1995, 2000, and 2010, respectively. His research interests include design models and nonlinear behavior of structural concrete.

João F. Almeida is an Associate Professor of Concrete Structures at IST-ICIST and a Partner and Technical Director of JSJ Consulting, Ltd. He received his degree in civil engineering from IST-ICIST in 1981 and his MSc in structural engineering and PhD in civil engineering from the Technical University of Lisbon, Lisbon, Portugal, in 1985 and 1990, respectively. His research interests include design models and nonlinear behavior of structural concrete.

opment of model assessment topics is a key issue for the dissemination and application of strut-and-tie models.

ASSESSMENT OF DESIGN MODELSTypical models

An engineer with experience in the application of strut-and-tie models for the design of discontinuity regions easily realizes that some typical models occur repeatedly in different practical cases. Some strut-and-tie models with different levels of complexity are presented in the following; the main questions that could arise during the model’s construction will be discussed for each case.

Deep beam models often occur in a discontinuity region or parts of it. For example, Fig. 1(a) shows the global design model for a pier of a viaduct. The load path defined by Model M1 is identical to a common deep beam model with top loads, and it is easy to observe the similarity of Model M2 to a deep beam with suspended loads. In these particular cases, to ensure appropriate service behavior, the solution was based on the elastic trajectories and the inner level arms z were set to approximately 0.7L for both models. However, to what extent can the designer adjust the inner level arm of each model to guarantee an appropriate struc-tural behavior for service and ultimate loads? Even in rather complex models, such as the deep beam of a building shown in Fig. 1(b), one can see several similarities with continuous deep beams with top and indirect loads. The same doubts concerning the proper inner level arm of each deep beam model can arise. Usually, in these cases, to develop a simpler and clear model, it is useful to set the position of the nodes

of the horizontal ties coincident to the nodes of the applied loads at the floor levels. The external reactions are another relevant topic in continuous deep beams. In fact, the chosen internal load path influences the external reactions, wherein slight variations may halve or double the tie forces in the span and over the supports. Along with the choice of the appropriate inner level arm, another possible question could arise: Within which limits can the designer select the load path because it influences the external reactions?

The strut-and-tie model shown in Fig. 1(b) also illustrates a different typical model: the suspension of concentrated loads. In this particular case, this model arises due to the indirect load of an out-of-plane deep beam located near the right opening. The main question may be: Is it appropriate to suspend the load with vertical reinforcement or must it be provided with inclined reinforcement to ensure adequate service behavior?

Another D-region to be analyzed is the re-entrant corner, which frequently occurs in openings in walls (refer to Fig. 1(c)) and in dapped beam ends. It is recognized that, in several cases, the adoption of an orthogonal reinforcement layout at the re-entrant corner does not effectively control crack widths. Thus, technical documentation recommends the combination of two different load paths, especially for highly stressed regions, in which part of the load is trans-ferred by the model associated with the orthogonal reinforce-ment and the remaining part is transferred by inclined reinforcement. As in the aforementioned typical models, the designer may have doubts concerning the necessary amount of inclined reinforcement that ensures adequate service behavior. This aspect is well-illustrated in the strut-and-tie model proposed by Schlaich et al. (1987) for a deep beam with an opening in which 50% of the load is transferred by inclined reinforcement and the remaining is transferred by orthogonal reinforcement.

Finally, concerning concentrated forces, point loads near supports cover all pairs of concentrated loads with a ratio of the distance between the applied forces a and inner level arm z of 0.5 ≤ a/z ≤ 2 and usually occur for point loads near supports—for example, corbels and pile caps. For these cases, part of the load must be hung up by stirrups, and the remaining part is transferred directly to the support. The force

Fig. 1—(a) Design strut-and-tie model for pier of viaduct for vertical loads; (b) design strut-and-tie model of deep beam within building; and (c) design strut-and-tie model for deep beam with opening (Schlaich et al. 1987). (Note: L and L1 are in m (in.); numbers adjacent to bars are forces (kN); 1 kN = 0.224 kips; 1 m = 39.37 in.)


in the stirrups is, based on semi-empirical rules, proportional to a/z, as defined in several technical and normative publica-tions. Usually, these regions have very high stresses and no ability to develop alternative load paths. The current ques-tion is: How far can the designer be from the reference solu-tion while avoiding deficient service behavior?

Deep beams with top loadsLeonhardt and Walther (1966) performed tests of two

simply supported deep beams with equal dimensions, in which one has half of the main reinforcement of the other. Similar experimental ultimate loads were attained for both deep beams, and the rupture was governed by the compres-sion in the node region. At the ultimate load, the deep beam with a smaller steel area exhibits yielding of the bottom reinforcement; for the second deep beam, steel stresses in the bottom reinforcement of approximately 370 MPa (63.1 ksi) were obtained from the measured strains. Signifi-cant internal stress redistributions were observed imme-diately after cracking, especially for the deep beam with the lower bottom reinforcement ratio, where the available ductility allowed the use of the full beam’s effective depth. The important internal redistribution led to steel stresses and crack width values for service loads at a quite appropriate level for both deep beams.

Following these results, two different statically admissible strut-and-tie models were considered (Fig. 2(a) and (b)). The first model was based on the elastic stress distribution (qd = 68.2 degrees); the second assumed a higher inner level arm

where the compression zone is fully used (qd = 75.1 degrees). For a design load Psd = 1000 kN (224.8 kips), consider-ably different amounts of reinforcement were obtained—As = 460 mm2 (0.713 in.2) and As = 306 mm2 (0.414 in.2) (fsyd = 435 MPa [63.1 ksi]). As usually recommended for deep beams, an additional minimum web reinforcement of approximately r = 0.2% was adopted (Fig. 2(c)). To under-stand the structural behavior of each deep beam and to eval-uate the steel stresses along the loading process, an adap-tive stress field analysis for each reinforcement layout was developed in which the main and distributed reinforcement were simulated (Fig. 2(d)). Setting the horizontal coordi-nates of the nodes as the adaptive variables, it is possible to simulate either the tension or the compression in the hori-zontal elements and therefore evaluate the increase of the inner level arm. The analysis outputs include the steel stress in the main and distributed reinforcement, as well as the concrete stresses within the compression fields, throughout the loading process. However, only the main results are presented and discussed.

To clearly understand the application of the adaptive stress field analysis for this particular case, some comments concerning the interpretation of the results and its relation with the developed analysis are mentioned. Figure 2(e) shows the variation of the inner level arm, defined by the distance between the resultant of all horizontal compres-sion fields and the bottom reinforcement. For the uncracked stage, the relation between the stiffness of the tension and compression elements is relatively small, and the minimiza-

Fig. 2—(a) Elastic stress trajectory-based model (z/L = 0.69; qd = 68.2 degrees); (b) design (z/L = 1.04; qd = 75.1 degrees); (c) deep beam reinforcement layouts; (d) Adaptive Stress Field Model and adaptive variables; (e) adaptive stress field analysis: numerical inner level arm; and (f) adaptive stress field analysis: steel stresses and crack widths throughout loading process. (Note: 100 MPa = 14.5 ksi; 1000 kN = 224.8 kips; 1 mm2 = 0.00155 in.2; 1 mm = 0.0394 in.)


tion of the global complementary energy is obtained for a model configuration in which the inner level arm is close to that obtained by the elastic theory. After cracking, the crack stiffness of the bottom reinforcement decreases; thus, its contribution to the global energy is higher than for the uncracked stage. Therefore, the model configuration that “spends” less energy is obtained for an increase of the inner level arm and, consequently, a decrease in the force of the bottom tie. Obviously, this effect is more significant for the deep beam with a lower reinforcement ratio (As = 306 mm2 [0.474 in.2]). With increasing loading, the available ductility allows for the effective deep beam height to be fully used in both deep beams, although it happens for quite different load steps.

Concerning service behavior, for loads between 350 and 650 kN (78.7 and 146.1 kips), steel stresses between 150 and 200 MPa (21.8 and 29.0 ksi) (for the model As = 460 mm2 [0.713 in.2]) and 200 and 250 MPa (29.0 and 36.3 ksi) (for the model As = 306 mm2 [0.414 in.2]) are obtained. These results lead to appropriate service behavior in both cases, given that crack widths less than 0.20 mm (0.008 in.) are obtained (refer to Fig. 2(f)). Because reinforced ties are modeled following the Tension Chord Model (Marti et al. 1998), crack widths are calculated accordingly. Further-more, steel stresses calculated with the strut-and-tie model used for the ultimate limit state will give slightly conser-vative results (continuous line in Fig. 2(f)). In fact, for an applied load of P = 500 kN (112.4 kips), steel stresses of ss = 218 MPa (31.6 ksi) are obtained in both cases. On the other hand, if steel stresses are calculated according to the elastic-based model for both deep beams, significantly higher steel stresses for the deep beam with As = 306 mm2 (0.474 in.2) are obtained (hidden line in Fig. 2(f)); ss ≈ 250/tg(68.2°)/306 × 10–6 = 327 MPa (47.4 ksi).

Finally, for the deep beam models, the analysis demonstrates that the designer has plenty of freedom for choosing the design strut-and-tie model and, moreover, in spite of the considerable redistributions assumed for the design model (from the elastic-based model), the model used at the ultimate limit state may be applied for checking service behavior.

Deep beams with suspended loadsThe consideration of different static conditions for simply

supported deep beams, such as indirect loading, is essential to cover the most practical cases of deep beams. Herein, the same questions concerning the proper inner level arm may arise. For this reason, three different strut-and-tie models (refer to Fig. 3(a)) are considered. In the first model, an inner level arm z/L = 0.69 is considered, similar to that obtained from the elastic stress trajectories. In the remaining models, different levels of redistributions are assumed, setting z/L = 0.90 and z/L = 1.04, leading to different amounts of reinforcement for the bottom tie. For each design model and corresponding reinforcement layout, an adaptive stress field analysis was developed. Again, the main goal is to find out if there is the same freedom for choosing the design model as in deep beams with top loads.

The reinforcement layout for each design model is shown in Fig. 3(b). Figure 3(c) presents the adaptive variables considered in the nonlinear stress field analysis. The nodes’ horizontal positions lead to either compression or tension in the horizontal elements (which define the reinforcement layers). The variation of the nodes’ vertical positions intends to simulate an increase or decrease on the resultant inner level arm.

Figure 3(d) shows the obtained variation of the resultant inner level arm z/L for each load level and for each reinforce-ment layout. After cracking of the horizontal bottom tie, an important stress redistribution occurs, related to an increase

Fig. 3—(a) Different design models for deep beam with suspended loads; (b) reinforcement layout for each design model; (c) Adaptive Stress Field Model; (d) inner level arm variation; and (e) steel stresses of bottom reinforcement. (Note: 100 MPa = 14.5 ksi; 1000 kN = 224.8 kips; 1 m = 39.37 in.; 1 mm2 = 0.00155 in.2; 100 kN/m = 22.48 kip/ft.)


of the inner level arm, most relevant for the deep beam with a lower bottom reinforcement ratio (z/L = 1.04). When the vertical tie that holds up the bottom load reaches the cracking load, the ratio z/L decreases similarly to that obtained from the uncracked stage. It should be mentioned that the analysis for deep beams with top loads showed different behavior because the inner level arm kept increasing with increasing loading. For the indirect loads, the increase of the inner level arm z/L is observed only after yielding of the bottom reinforcement. This leads to moderate-to-high steel stresses and corresponding crack widths at service loads. In fact, between P = 350 kN (78.7 kips) and P = 650 kN (146.1 kips), steel stresses of 110 to 220 MPa (15.95 to 31.9 ksi) (z/L = 0.69), 140 to 290 MPa (20.3 to 42.1 ksi) (z/L = 0.90), and 150 to 330 MPa (21.8 to 47.9 ksi) (z/L = 1.04) are obtained (refer to Fig. 3(e)). A final remark concerning service behavior: checking steel stresses with the same model used at the ultimate limit state is particularly suitable for the elastic-based model z/L = 0.69. For the other models, the obtained values slightly underestimate steel stresses (refer to the hidden lines in Fig. 3(e)). Alternatively, checking steel stresses based on the elastic inner level arm for the model with z/L = 1.04 would be clearly conservative (refer to the continuous line in Fig. 3(e)).

Continuous deep beamsTo assess the design model for continuous deep beams,

several strut-and-tie models were developed. The reference strut-and-tie model is based on the elastic stress trajectories. The remaining models (refer to Fig. 4) are defined to simu-late different inner level arms, as well as different support and midspan reinforcement ratios (continuous deep beams are statically redundant systems; thus, the external reac-tions also depend on the reinforcement layout). As usual, a minimum reinforcement of r = 0.2% is adopted in all cases.

Models A and B were based on the elastic inner level arm, but the steel reinforcement areas over the support were set to approximately half and double those of the reference solution, respectively. Model 1 intends to simulate a typical simplification for design purposes, in which the tension over the support is located at the same level as the adjacent compression (z = 0.7L). In Model 2, the effective height of the continuous deep beam is fully used and the reinforcement layout is placed at the top face, similar to a slender beam. In Model 3, an extreme redistribution is assumed by setting two independent simply supported deep beam models with no reinforcement over the support region.

An adaptive stress field analysis was developed for each reinforcement layout in which three levels of reinforcement were considered. The adaptive variables are the horizontal

Fig. 4—Continuous deep beam design strut-and-tie models and subsequent reinforcement layout. (Note: Dimensions in m; 100 kN = 22.48 kips; 1 m = 39.37 in.; 1 mm = 0.0394 in.; 100 kN/m = 22.48 kip/ft.)


node location and the external reaction R1 (Fig. 5(a)). The design load was q = 800 kN/m (54.8 kip/ft) and the analysis was developed for load steps of 100 kN/m (6.85 kip/ft).

Figure 5(b) to (e) shows the obtained steel stresses in all analyzed reinforcement layers for each deep beam, where smid span is the steel stresses at the bottom midspan reinforce-ment, stop supp is the steel stresses at the top reinforcement over the support, smid supp is the steel stresses at the reinforce-ment placed at middle height over the support, and sbottom

supp is the steel stresses at the reinforcement placed at bottom height over the support.

For all reinforcement layouts, the predicted design load was attained, which denotes the capacity of the structure to adapt itself to the chosen design model. After cracking of the bottom reinforcement, the support steel stresses increase at all levels and the bottom reinforcement steel stresses remain approximately constant. Increasing applied loads, cracking of the concrete region over the support occurs and, as expected, an increase of the bottom steel stresses is observed. Curiously, this increase has similar slopes for all deep beams in spite of the different amounts of rein-forcement. Moreover, at the design load level, yielding of

steel was not observed in all analyzed cases. This effect is mainly due to the increase of the inner level arm (similar to the behavior that was observed for the simply supported deep beams with top loads). Concerning service behavior, for deep beams with low reinforcement ratios over the support—Models A, 2, and 3—the steel stresses can lead to moderate crack widths. In the upper bound of the service load range q = 550 kN/m (37.7 kip/ft), steel stresses of approximately 350 MPa (50.8 ksi) are obtained (refer to Fig. 5(e)). The other models presented suitable steel stresses at service loads for all reinforcement levels.

To clearly understand the level of redistributions tested in the analyzed cases, when compared with slender beams, redistributions of the “resultant moments” up to 70% of the support and midspan moment did not significantly affect service behavior (refer to Models R, A, and B, where the elastic inner level arm was adopted). It should be noted that the adopted internal redistributions led to variations from the reference model up to 20% of the hyperstatic external reaction R1. Moreover, the evaluation of steel stresses at service loads with the same model used for the ultimate load provided, in general, conservative results (refer to the

Fig. 5—(a) Continuous deep beam adaptive stress field configuration and identification of adaptive variables. Continuous deep beam steel stresses for different reinforcement layouts: (b) bottom reinforcement; (c) top support reinforcement; (d) middle support reinforcement; and (e) bottom support reinforcement. (Note: Dimensions in (a) are in m; 100 MPa = 14.5 ksi; 100 kN/m = 224.8 kip/ft; 1 m = 39.37 in.)


dotted lines in Fig. 5(b) to (e)). As shown for Models 2 and 3, however, increases on the inner level arm higher than 50% can lead to inappropriate steel stresses at service loads.

This example shows that there is plenty of freedom for choosing the design model for continuous deep beams and the assumed redistributions can be considerably higher than those allowed for slender beams. Nevertheless, to prevent deficient service behavior, extreme redistributions, such as those illustrated by Models 2 and 3, should be avoided.

Suspension of concentrated loadingThe strut-and-tie model for a corbel subject to point load

was well-established many years ago, but if the same corbel suspends the load, a few concerns could arise. The elastic stress trajectories suggest that part of the indirect load may be carried by stirrups and the remaining may be carried by inclined reinforcement (Fig. 6(b)). Strictly following the elastic theory, however, it may not be easy to calculate the

load fraction that should be equilibrated by each load path. To assess the design model for the suspension of concen-trated loads, three distinct strut-and-tie models with different values of the part of the load suspended by vertical stirrups (defined by the variable kd) are studied (Fig. 6(a)). As in the other examples, for each reinforcement layout, an adaptive stress field analysis is developed following the model shown in Fig. 6(c). The only adaptive variable is k, which repre-sents the fraction of the load that is suspended by stirrups. To analyze an eventual premature failure, it was considered the reduction of the concrete strength (Vecchio and Collins 1986) for the diagonal compression (Element 8) because it crosses significant transversal strains induced by the diag-onal tie (Element 2).

Figure 6(d) to (g) shows the main numerical results obtained from the analysis: the load fraction carried by vertical stirrups, the steel stresses in both ties, and the diagonal compression stresses. As expected, in the uncracked stage, a

Fig. 6—(a) Different corbel design strut-and-tie models and corresponding reinforcement layouts; (b) elastic stress trajec-tories; (c) corbel adaptive stress field analysis. Corbel numerical results: (d) load fraction transmitted by vertical stirrups; (e) vertical reinforcement steel stresses; (f) inclined reinforcement steel stresses; and (g) diagonal compression stresses. (Note: Dimensions in (a) are in m; 100 MPa = 14.5 ksi; 100 kN = 22.48 kips; 1 m = 39.37 in.; 1 mm = 0.0394 in.)


value of k ≈ 0.42 for all models was obtained, leading to the conclusion that both load paths had similar stiffness and, thus, their contribution to the global complementary energy was identical. Concrete cracking at vertical and inclined ties occurred at the same load step V = 100 kN (22.5 kips). For all design models, the structure has a tendency to follow the initial strut-and-tie model. This is intimately related with the stiffness relation of Ties 1 and 2, wherein the computed energies are highly influenced by the provided steel area and its length; thus, the stiffer element carries a higher load. Obviously, this also happens in the model (kd = 0.5), but the identical steel areas adopted lead to similar resultant forces in both ties. Concerning service behavior, the stress redistribution that occurred immediately after cracking kept steel stresses at both vertical and inclined reinforce-ments at appropriate levels, suggesting low-to-moderate crack widths. It is worth mentioning that the superposition of both load paths—assuming 50% of the load equilibrated by vertical stirrups (kd = 0.90)—provides quite lower steel stresses. The ultimate load was achieved in all cases and low-to-moderate compression stresses were obtained. As expected, the diagonal compression stresses (Fig. 6(g)) are higher for the model that provides more vertical stirrups; however, the concrete stresses remain lower than the effec-

tive concrete strength. Finally, for practical purposes, the designer could choose one of the possible load paths and provide a distributed reinforcement of approximately 15% of the main reinforcement to improve service behavior. Note that the inclined reinforcement can be replaced by an orthogonal reinforcement layout with approximately the same ratio in both directions. Additionally, at service loads, the calculation of the steel stresses in the main reinforce-ment, following the strut-and-tie model used for design, will lead to slightly conservative values.

Re-entrant cornersThe re-entrant corner model can be characterized by a

strut-and-tie model of a dapped beam. The model’s unique-ness for dapped beams has been discussed by several authors (Cook and Mitchell 1988; Reineck 2002). One of the frequently asked questions concerns the required inclined reinforcement to adequately control the diagonal crack in the re-entrant corner. It is known that the elastic stress trajectories suggest that the load should be mainly equilibrated by an inclined reinforcement (approximately 75% of the total applied load; refer to Fig. 7(a)). For prac-tical reasons, however, it is sometimes useful to provide an orthogonal reinforcement layout, and the elastic stresses do

Fig. 7—(a) Dapped-end beam elastic stress trajectories; (b) design strut-and-tie model assuming 25% of load is equilibrated by vertical reinforcement; (c) design strut-and-tie model assuming 50% of load is equilibrated by vertical reinforcement; and (d) design strut-and-tie model assuming 90% of load is equilibrated by vertical reinforcement. (Note: Bar diameters are in mm; 100 kN = 22.48 kips; 1 mm = 0.0394 in.)


Fig. 8—Steel stresses variation: (a) horizontal reinforcement; (b) inclined reinforcement; and (c) vertical reinforcement. (Note: 100 MPa = 14.5 ksi; 100 kN = 22.48 kips.)

not fully reveal all the information needed for the design—for example, the calculation of the anchorage length of the horizontal reinforcement over the support (refer to Fig. 7(a), where horizontal and vertical stress distributions are plotted). It is recognized that modeling singular zones with the finite element method is usually complex and turns out to be more complicated if doing so along with the current and prac-tical reinforcement layouts. For these reasons, some tech-nical documents (FIP Recommendations 1999) suggest the strut-and-tie method for the design of these kinds of regions, taking into account the superposition of two possible models to efficiently control the cracks in the re-entrant corner: the first considering an orthogonal reinforcement and the other reflecting an inclined reinforcement. It is common, especially for openings in walls, to provide an orthogonal reinforce-ment layout designed for the total load; for moderate-to-high loads, an additional (usually not explicitly calculated) inclined reinforcement at the opening corners is provided.

The main concern is related to the amount of inclined reinforcement that should be provided to adequately control the diagonal crack width at service loads. For the evaluation of the region behavior under different reinforcement layouts, three different design models were developed (Fig. 7(b) to (d)), where kd is the part of the vertical applied force that is equilibrated by the orthogonal reinforcement model. Defining k as the adaptive variable, the developed adap-tive stress field analysis allows the calculation of compres-sion and tension stresses in all elements. Figure 8 shows the k variation throughout the loading process and the steel stresses obtained for each analysis.

The following aspects should be pointed out:• Before cracking, the part of the vertical load V that is

equilibrated by the orthogonal reinforcement follows the elastic theory because values of approximately k = 0.30 for all models were numerically obtained. Imme-diately after cracking, high stress redistribution occurs and the load path tends to follow the initial design strut-

and-tie model. This behavior, together with the fact that the concrete compressive stresses did not reach its effec-tive strength in any situation, allowed for the design load to be achieved in all cases.

• Concerning service behavior and assuming service loads between V = 200 kN (45.0 kips) and V = 300 kN (67.4 kips), steel stresses from 150 to 300 MPa (21.8 to 43.5 ksi) were obtained, leading to acceptable service behavior. The diagonal reinforcement achieved steel stresses close to yielding stresses only for the design strut-and-tie model (kd = 0.90), which may be related to unacceptable crack widths. This result is also in accor-dance with common practice and test outputs, where it is known that the orthogonal reinforcement does not effectively control the diagonal crack.

Loads near supportAlmeida and Lourenço (2005) and, later, Lobo (2008),

developed an adaptive stress field analysis based on the model presented in Fig. 9(a). The adaptive variable is the part of load F transmitted by the stirrups, k, which is asso-ciated with the length zd. For a given force distribution, all variables are established by equilibrium conditions—the length zd and the compression and tension model resultants, as well as the stress field widths. The main objective is to assess the region’s structural behavior if the provided vertical reinforcement is different from the recommended values of the FIP Recommendations (1999), based on the equation

1 2 13

wF aF z

= −

For these purposes, an adaptive stress field analysis was developed for different ratios of provided and reference


vertical reinforcement h = As/As,ref and different a/z values. The main results of the analysis are presented in Fig. 9; for more detailed information, refer to Lobo (2008). Stirrup steel stresses are presented in Fig. 9(b) to (e), demonstrating that it is reasonable to have a slight reduction of the required reference stirrups without significantly affecting service behavior (reductions up to 25%). The ultimate design load was achieved in all cases.

CONCLUSIONSThe Adaptive Stress Field Model extends the application

of stress-field-based models for the nonlinear analysis of structural concrete discontinuity regions, allowing a consis-tent study of service behavior, ductility and, more generally, the model’s assessment topics. Several representative models for discontinuity regions were studied, providing guidance

for adequate modeling and detailing in many practical situa-tions. The following general conclusions can be drawn:• The practical rule of orientating the model at the elastic

stress fields is a simple and conservative recommen-dation; however, the selected model does not reflect cracking and thus does not consider the redistribution of the inner flow of forces up to the service load and ultimate load levels.

• In general, the engineer has plenty of freedom for choosing the model for the design of discontinuity regions, and the engineer’s judgment based on previous experience should not be underestimated. The defor-mation capacity of the materials after yielding allows considerable stress redistributions and successive formation of new static systems. In practical applica-tions, strut-and-tie models should be used with a well-

Fig. 9—(a) Adaptive Stress Field Model for point load near support. Stirrup steel stresses for different ratios h = As/As,ref (Lobo 2008): (b) a/z = 0.75; (c) a/z = 1.0; (d) a/z = 1.75; and (e) a/z = 2.0. (Note: 100 MPa = 14.5 ksi; 100 kN = 22.48 kips; 1 cm2 = 0.155 in.2)


distributed minimum reinforcement, which should also prevent premature failures at the cracking load.

• An unpredicted concrete failure is mainly related to the reduction of the compressive concrete strength due to the presence of high transverse strains. In fact, in most practical cases, this effect is not particularly relevant because compressive stresses within the regions are usually lower than the effective strength due to the moderate concrete stresses imposed by the strength verification of the contiguous nodal regions. However, a more detailed analysis should be performed when important compressive stresses cross diagonal ties with significant transverse strains induced.

• The assessment of models at the service load level has shown that quite large deviations from the elastic solution can be tolerated without greatly affecting the service behavior. In fact, despite such important internal redistributions observed in some of the studied cases, the steel stresses and crack widths at the service load level remained at quite appropriate levels.

• This study confirmed that, in general, the same model selected for the ultimate load level was seen to be valid for checking cracking at service loads, even for moderate deviations from the elastic-based solution.

• For several cases, the suitable redistributions from the reference models were assessed, providing guidance for the development and application of strut-and-tie models to the design of discontinuity regions. Furthermore, following the indications mentioned in the following, the design model can be used to check service behavior: ◦ Deep beams: The results showed that deviations of

the inner level arm lower than 50% or deviations of the external reactions lower than 20% (for statically indeterminate deep beams) relative to the reference model (elastic-based) are acceptable. Note that these variations can halve or double the reinforce-ment area.

◦ Suspension of concentrated loads: The suspen-sion of the loads due to an indirect support may be equilibrated by a combination of inclined and vertical reinforcement. The designer is free to choose one of the possible load paths. To improve service behavior, however, it is recommended to provide a distributed reinforcement of at least 15% of the main reinforcement.

◦ Re-entrant corners: It is recognized that, in several cases, the adoption of an orthogonal reinforcement layout near the re-entrant corner does not effectively control crack widths. The reference model requires 75% of the applied load to be equilibrated by inclined reinforcement. The analysis results showed that assigning 50% of the load at each load path will provide good service behavior. For higher redistri-butions, the inclined crack close to the re-entrant corner can present unsatisfactory crack widths.

◦ Loads near supports: Loads near support models cover all pairs of point loads within a ratio of the distance between the applied forces and inner level arm of 0.5 ≤ a/z ≤ 2. For these cases, part of the

load must be hung by stirrups and the remaining load must be directly transferred to the support. The obtained results pointed out that redistributions of 25% of the reference solution do not significantly influence service behavior.

NOTATIONa, b = widthsb (subscript) = bottlec = concrete, compressiond = designE = elastic modulusF = applied forcesf = material strengthk = coefficientL, l = lengthM = bending momentp = pressure, distributed loadq = distributed loads = steelV = vertical force, shearw = crack widthw (subscript) = web, wedgey = yieldingz = inner level arme = strainq = angler = reinforcement ratios = stressw = mechanical ratio

REFERENCESAlmeida, J., and Lourenço, M., 2005, “Stress Field Models for Struc-

tural Concrete,” Keep Concrete Attractive, fib Symposium, V. 1, Budapest, Hungary, pp. 525-531.

Cook, W. D., and Mitchell, D., 1988, “Studies of Disturbed Regions Near Discontinuities in Reinforced Concrete Members,” ACI Structural Journal, V. 85, No. 2, Mar.-Apr., pp. 206-216.

EN 1992-1-1:2004, 2004, “Eurocode 2: Design of Concrete Structures—Part 1-1: General Rules and Rules for Buildings,” British Standards Institu-tion, London, UK, Dec., 225 pp.

Fernández Ruiz, M., and Muttoni, A., 2007, “On Development of Suit-able Stress Fields for Structural Concrete,” ACI Structural Journal, V. 104, No. 4, July-Aug., pp. 495-502.

FIP Recommendations, 1999, “Practical Design of Structural Concrete,” Practical Design, FIP Commission 3, SETO, London, UK, 113 pp.

Leonhardt, F., and Walther, R., 1966, Wandartiger Träger, DAfStb Heft 178, Wilhelm Ernst & Sohn, Berlin, Germany, 159 pp. (in German)

Lobo, P., 2008, “Cargas Próximas dos Apoios em Elementos de Betão Estrutural,” PhD thesis, Universidade Técnica de Lisboa, Instituto Superior Técnico, Lisbon, Portugal, July, pp. 1-103. (in Portuguese)

Marti, P.; Alvarez, M.; Kaufmann, W.; and Sigrist, V., 1998, “Tension Chord Model for Structural Concrete,” Structural Engineering Interna-tional, V. 8, No. 4, Nov., pp. 287-298.

Muttoni, A.; Schwartz, J.; and Thürlimann, B., 1998, Design of Concrete Structures with Stress Fields, Birkhäuser, Basel, Switzerland.

Reineck, K.-H., ed., 2002, Examples for the Design of Structural Concrete with Strut-and-Tie Models, SP-208, American Concrete Institute, Farmington Hills, MI, 242 pp.

Schlaich, J., and Schäfer, K., 1991, “Design and Detailing of Structural Concrete Using Strut-and-Tie Models,” The Structural Engineer, V. 69, No. 6.

Schlaich, J.; Schäfer, K; and Jennewein, M., 1987, “Toward a Consistent Design for Structural Concrete,” PCI Journal, V. 32, No. 3, pp. 75-150.

Sigrist, V.; Alvarez, M.; and Kaufmann, W., 1995, “Shear and Flexure in Structural Concrete Beams,” Ultimate Limit State Design Models, CEB Bulletin 223, June, pp. 7-49.

Vecchio, F. J., and Collins, M. P., 1986, “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear,” ACI JOURNAL, Proceedings V. 83, No. 2, Mar.-Apr., pp. 219-231.


NOTES:


Title no. 110-S10


ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2011-062 received March 2, 2011, and reviewed under Institute


Flexural Drift Capacity of Reinforced Concrete Wall with Limited Confinementby S. Takahashi, K. Yoshida, T. Ichinose, Y. Sanada, K. Matsumoto, H. Fukuyama, and H. Suwada

The flexural drift capacity of reinforced concrete (RC) walls is discussed in this study based on the test results of 10 specimens. The test parameters were wall length, thickness, detailing, and axial force. The detailing of the ties did not satisfy the ACI 318-08 requirements. Each specimen had a column at one end where an axial force was applied. All specimens failed in flexural compres-sion after yielding of the longitudinal bars. The observed flexural drift capacity was between 0.4 and 1.2%. A set of equations to predict the drift capacity is proposed wherein the hinge zone length is assumed to be 2.5 times that of the wall thickness.

Keywords: boundary element; compressive failure; confinement; drift capacity; plastic hinge; reinforced concrete wall.

INTRODUCTIONACI 318-081 requires special detailing for boundary

elements of reinforced concrete (RC) walls to prevent flex-ural compressive failure under seismic forces. One of the approaches to the detailing is based on the displacement-based concept.2 If the compressive strain of concrete is expected to be larger than 0.003, the compressive zone is required to be reinforced according to the requirement of the special boundary element for confinement. This requirement is verified by Thomsen and Wallace,3 who tested walls with rectangular- and T-shaped cross sections.

The Japanese Code4 prescribes the ductility of RC walls based mainly on the ratio of the neutral axis depth to the wall thickness. This prescription is based on several experimental studies, including those by Tabata et al.,5 who tested RC walls with rectangular cross sections and large shear-span ratios.

The plastic hinge length Lp is important for estimating drift capacity. Researchers have proposed various approaches. In the study by Wallace and Orakcal,2 which was the basis of the seismic requirements of ACI 318-08,1 the plastic hinge length was assumed as one half of the wall length (Lp = lw/2). Tabata et al.5 assumed Lp = 0.3lw. On the other hand, Kabeyasawa et al.6 idealized the wall, assuming that the strain of each boundary element is uniform within each story; this idealization is almost equivalent to the assumption of Lp = h, where h is story height. Orakcal and Wallace7 divided the wall into eight segments in the direc-tion of the height; this idealization is almost equivalent to the assumption of Lp = h/8. Paulay and Priestley8 assumed that Lp = 0.20lw + 0.044a from the test results of canti-lever walls, where a is shear span length. Takahashi et al.9 showed that the prescription of the Japanese Code4 is implicitly based on the assumption of Lp = 2.5t, where t is wall thickness.

The objective of this paper is to propose a set of equations to predict the drift capacity of RC walls based on the assump-tion of Lp = 2.5t. To verify this assumption, 10 specimens were tested. The detailing of these specimens does not satisfy the seismic requirements of ACI 318-08,1 but such detailing

may be preferred for ease of construction. The test param-eters were wall length, thickness, detailing, and axial force. The details of this experiment are available elsewhere.10,11

RESEARCH SIGNIFICANCEThere are many RC buildings that do not satisfy the

requirements of ACI 318-08,1 including those in Chile. Most of the wall damage caused by the 2010 Chile earthquake was related to the configuration and reinforcement detailing of wall boundary elements.12 The damage indicated that the performance of these walls was brittle, as expected. On the other hand, there may have been many buildings that resisted the earthquake, although they did not satisfy the require-ments of ACI 318-08.1

The ACI 318-081 requirements are quite strict about the boundary element; it may be sufficient to ensure large ductility of walls. However, the findings of this research may lead to simpler detailing for walls where a relatively smaller compressive strain is expected. The findings of this research may also be used to evaluate the seismic capacity of buildings with walls that do not satisfy the ACI 318-081 requirements.

EXPERIMENTAL PROGRAMSpecimens

Ten RC wall specimens with a boundary column on only one side were prepared to investigate the differences in deformation capacity. Figure 1 shows the cross sections of the specimens used in this research. Each specimen is named as follows:

1. Perpendicular end wall: The first letter of the speci-men’s name—“P” or “N”—means with or without a perpen-dicular wall, respectively. For example, Specimen PM5 in Fig. 1(g) has a perpendicular end wall 130 mm (5.1 in.) long and 60 mm (2.4 in.) thick. All the perpendicular end walls have single-layer reinforcement (D4 at 80 mm [3.15 in.], where D4 is a deformed bar with a nominal diameter of 4 mm [0.16 in.]) and do not have confinement.

2. The ratio of wall panel length to wall thickness (lwp/t): The second letter of the specimen’s name—“S,” “M,” or “L”—expresses that the lwp/t ratio is 6, 12, or 18, respec-tively, where the wall panel length lwp is defined, excluding the column. For example, the ratio of Specimen PM5 in Fig. 1(g) is 1200/100 = 12.

3. The ratio of neutral axis depth to wall thickness (c/t): The vertical arrows in Fig. 1 indicate the location of the


Susumu Takahashi is a PhD Student of architectural engineering at Nagoya Institute of Technology, Nagoya, Japan.

Kazuya Yoshida is a Master’s Student of architectural engineering at Nagoya Insti-tute of Technology.

ACI member Toshikatsu Ichinose is a Professor of architectural engineering at Nagoya Institute of Technology.

ACI member Yasushi Sanada is an Associate Professor of architecture and civil engi-neering at Toyohashi University of Technology, Toyohashi, Japan.

Kenki Matsumoto is a Master’s Student of architectural engineering at Nagoya Insti-tute of Technology.

ACI member Hiroshi Fukuyama is a Chief Research Engineer at the Building Research Institute, Tsukuba, Japan.

Haruhiko Suwada is a Research Engineer at the Building Research Institute.

neutral axis, whose computation method will be shown in a later section of the paper. The number of the specimen’s name expresses the approximate ratio of c/t. For example, the ratio of Specimen PM5 in Fig. 1(g) is 493/100 = 4.9. Although the sections of Specimens NM5 and NM4 are identical, as shown in Fig. 1(a), the locations of the neutral axis are different because of the difference of axial forces, as shown in a later section.

4. With or without crosstie in boundary element: Figure 2 shows the detail of the boundary elements of the specimens, except Specimens NM5, NM4, and NM2′. The horizontal bars have a 135-degree hook, as shown in Fig. 2(a). The cap bars have a 90-degree hook at both ends, as shown in Fig. 2(b), and the cap bars’ vertical spacing is 35 mm (1.4 in.), as shown in Fig. 2(c). The crossties have 90- and 135-degree hooks, as shown in Fig. 2(b), and are staggered with a spacing of 70 mm (2.8 in.), as shown in Fig. 2(c). The crossties in Specimens NM5 and NM4 are located at a spacing of 35 mm (1.4 in.), as shown in Fig. 3(a).

Specimen NM2′ does not have crossties, as shown in Fig. 3(b).

The detailing of the wall reinforcement for Specimen NM3 is shown in Fig. 4. The spacings of the horizontal and vertical bars are 35 and 100 mm (1.4 and 4.0 in.), respectively. The reinforcement details of the wall panels of the other speci-mens are the same as those in Fig. 4. Because the wall thick-ness of each specimen is different, the lateral and vertical wall reinforcement ratios vary from 0.54% to 0.84% and from 0.19% to 0.30%, respectively. The lengths of boundary elements (220 mm [8.7 in.] in Fig. 2(a)) are longer than half the length of the neutral axis in most specimens, as specified by the seismic requirements of ACI 318-08.1 In the vertical direction, crossties are provided from the bottom to one-third of the clear height h, as shown in Fig. 4. This value of h/3 is much shorter than the requirement of ACI 318-08.1 The spacing of the crossties (70 mm [2.8 in.] in most specimens) does not satisfy the requirements of ACI 318-081 either. The cross-sectional areas of the crossties vary from 15 to 31% of ACI 318-08.1

The clear heights of Specimens NM4 and NM5 are 1000 mm (3.3 ft), whereas those of the other specimens are 1200 mm (4.0 ft), as shown in Fig. 4. Eight No. 3 (D10) longitudinal bars are provided in the boundary element of all specimens (Fig. 2(a)). Twelve No. 5 (D16) longitudinal bars are provided in the boundary column, except that of Specimen NL2, where eight No. 4 (D13) bars are provided (Fig. 1(e)) so the longitudinal reinforcement ratio (2.5%) is similar to that of the other specimens.

All specimens are designed to fail in flexure; the shear-to-flexural-capacity ratios vary from 1.2 to 2.3, where the flexural and shear capacities are calculated based on the Architectural Institute of Japan (AIJ) standards13 and ACI 318-08,1 respectively. The material properties of the steel bars are indicated in Table 1, where fy is the yield strength, fu is the tensile strength, and Es is the elastic modulus. The material properties of concrete are indicated

Fig. 1—Specimen sections. (Note: Dimensions in mm; 1 mm = 0.039 in.; No. 4 is D13; No. 5 is D16.)


in Table 2, where fc′ is the compressive strength, Ec is the elastic modulus, and fr is the modulus of rupture.

Test setupFigure 5 shows the test setup. Lateral force was applied by

a hydraulic jack to a stiff loading steel beam fastened to the specimen. All specimens had stiff RC stubs at both the top and bottom for fixing with the loading frame. No axial force was applied for Specimen NM4. For the other specimens, two vertical hydraulic jacks were force-controlled so the moment around the center of the boundary column is zero, as shown in Fig. 5. The amount of the axial force was approximately 20% of the axial capacity of the boundary column (fc′Ag), where Ag is the gross cross-sectional area of the column. The applied axial load was approximately 240 kN (54 kips) for Specimen NL2, 400 kN (90 kips) for NM5, and 540 kN (121 kips) for the other specimens. Horizontal load was applied 2425 mm (8.0 ft) above the bottom of the wall panel for NM5 and NM4 (Fig. 5). The height of the horizontal load was 2525 mm (8.3 ft) for the other specimens. The shear-span ratio of NL2 is 2525/2000 = 1.26, which is the smallest. The shear span ratio of NS3 is 2525/1020 = 2.48, which is the largest.

OBSERVED DAMAGE AND DEFLECTION COMPONENT

Figure 6 shows the lateral load-drift relationship of Specimen NL2. Lateral drift R is defined as the ratio of measured lateral displacement D to specimen height h. The displacement was measured at the top of the clear height in all specimens. During the positive loading (column in tension), the maximum strength (530 kN [119 kips]) was observed at a +1.2% drift level. The strength was 1.1 times the analytical flexural strength. Between the drift levels of +2 and +3%, strength decreased rapidly. During the negative loading direction (column in compression), the maximum strength

Fig. 2—Boundary element except for Specimens NM5, NM4, and NM2′. (Note: 1 mm = 0.039 in.; No. 3 is D10.)

Fig. 3—Boundary element of Specimens NM5, NM4, and NM2′.

Fig. 4—Elevation of Specimen NM3. (Note: Dimensions in mm; 1 mm = 0.039 in.)

Table 1—Material properties of reinforcing bars

Bar fy, MPa fu, MPa Es, GPa

D4 411 (351)* 521 (544)* 173 (192)*

No. 3 (D10) 391 (376)* 469 (520)* 199 (188)*

No. 4 (D13) 367 503 183

No. 5 (D16) 389 (387)* 559 (563)* 180 (180)*

*Numbers in parentheses indicate material properties for Specimens NM4 and NM5.Notes: 1 MPa = 145 psi; 1 GPa = 145 ksi.

Table 2—Material properties of concrete

Specimen fc′, MPa Ec, GPa fr, MPa

NS338.3 28.4 2.65

NM3

NM237.8 27.8 2.45

NM2′

NL2

37.6 28.6 3.07PL6

PM5

PM3

NM533.4 23.9 2.55

NM4

Notes: 1 MPa = 145 psi; 1 GPa = 145 ksi.


(280 kN [63 kips]) was observed at a –1.5% drift level. The strength was 1.2 times the analytical flexural strength.

Figure 7 shows Specimen NL2 at the end of the experi-ment at a 5% drift level. The spalling of concrete started at the bottom right corner of the wall panel at a +2% drift level.

The spalled zone extended toward the column until a +3% drift level. On the other hand, the concrete of the boundary column slightly spalled during the negative loading but not during the positive loading, even at a +5% drift level. Figure 8 shows the buckling of the longitudinal bars (eight No. 3 [D10]) in the boundary element at a 3% drift level. The buckled bars fractured in tension between the drift levels of –2 and –3%.

Figure 9 shows the linear variable differential transducer (LVDT) used to evaluate the flexural drifts.7 Figure 10 shows

Fig. 5—Loading setup.

Fig. 6—Load versus drift of Specimen NL2.

Fig. 7—Specimen NL2 at maximum drift.

Fig. 8—Buckling of reinforcement.

Fig. 9—Instrumentation of specimens.


the load-versus-flexural-drift relationship of Specimen NL2. Flexural drift at 80% of maximum strength Vmax (the black circle in Fig. 10) is defined as flexural drift capacity in this paper. The hysteresis loops are more spindle-shaped than those in Fig. 6. The strength degradation in Fig. 10 is milder than that in Fig. 6, which will be discussed in the following.

Figure 11 shows the load-versus-shear-drift relationship of Specimen NL2. Shear drift was obtained by subtracting flexural drift from total drift. Shear drift increased after flex-ural yielding. The black triangles in Fig. 10 and 11 show the load step at the onset of strength degradation. The shear drift just before the strength degradation (1.25%) was larger than the corresponding flexural drift (0.75%). Note that the maximum applied shear force is much smaller than the shear strength computed according to ACI 318-081 (the top broken line in Fig. 11). The shear drift did not increase during the degradation. This is the reason why the strength degradation in Fig. 10 is milder than that in Fig. 6.

Figure 12 shows the horizontal slip along one of the flexural cracks near the center of the wall when the shear drift was 1.25% or the shear deformation was 15 mm (0.59 in.) (the black triangle in Fig. 11). The observed slip was 8.5 mm (0.33 in.), which was more than one half of the total shear deformation (15 mm [0.59 in.]). The damage and overall behavior of the other specimens were similar to Specimen NL2, except that the slips along the flexural cracks were smaller than those in NL2. To discuss the cause of the slip, the compressive force of concrete C is defined in Eq. (1).

st y sc yC N A f A f= + −∑ ∑ (1)

where N is applied axial load; Ast is the gross sectional area of longitudinal bars in tension; Asc is the gross sectional area of longitudinal bars in compression; and fy is the yield strength of the longitudinal bar.

The variable Vmax in the horizontal axis of Fig. 13 indi-cates the maximum applied shear force. The vertical axis of Fig. 13 shows the slip drift angle, which is defined as the sum of the observed slips (Ss) just before the strength degrada-tion (the black triangle in Fig. 11) divided by the clear height h. Slip was larger in the specimens with larger Vmax/C ratios.

Such a correlation is not obtained between the slip and the average shear stress (Vmax/Ag, where Ag is the gross sectional area of the specimen). Because the slip is not the focus of this study, only flexural drift is discussed in the following.

The load-versus-flexural-drift relationships of Specimens NM3 and PM3 are shown in Fig. 14(a) and 15(a) to inves-

Fig. 10—Load versus flexural drift of Specimen NL2. Fig. 11—Load versus shear drift (deformation) of Specimen NL2. (Note: 1 mm = 0.039 in.)

Fig. 12—Slip along flexural crack of Specimen NL2. (Note: 1 mm = 0.039 in.)

Fig. 13—Lateral slip drift Ss/h versus Vmax/C.


tigate the effect of a perpendicular wall on flexural drift capacity. The difference between these two specimens is the existence of a perpendicular wall. The drift capacities of NM3 (0.57%) and PM3 (0.61%) were similar. The compres-sive failure of the perpendicular wall of PM3 occurred before the lateral strength degradation started. Because the perpen-dicular wall is provided with no confinement (Fig. 1), the

compressive ductility of this wall was very limited at the ulti-mate drift. Therefore, the contribution of the perpendicular wall should be ignored in evaluating the drift capacity. The vertical axes of Fig. 14(b) and 15(b) show the average strain at the compression edge (strain between Points E and F). The plastic hinge lengths of these two specimens, which will be evaluated later as 2.5 times the wall thickness (300 mm [11.81 in.]), are similar to the length between Points E and F (400 mm [1.3 ft]). The strains at the ultimate drifts (the black circles in the figures) were approximately 0.008, which agrees with the ultimate strain of concrete eu computed in the following considering the confinement effect.

FLEXURAL DRIFT CAPACITYSimplification of plastic deformation

In this paper, flexural drift capacity is decomposed into elastic and plastic components (Ru = Ry + Rp), as shown in Fig. 16(a). The curvature at yielding fy is computed based on the yield strain of longitudinal reinforcement (Fig. 16(b)).

yy d c

ef =

− (2)

where ey is yield strain of reinforcement; d is effective depth, defined as the distance between the compression edge and the center of the boundary column; and c is the neutral axis depth computed from Eq. (3).

10.85 c

Ccf t

=b ′

(3)

where C is the compressive force of concrete computed from Eq. (1); b1 is the reduction factor to determine the neutral axis; fc′ is the compressive strength of concrete; and t is wall thickness.

Theoretically, Eq. (3) is effective at the ultimate state and ineffective at yielding; however, this difference may be negligible because the amount of the wall reinforcements is much smaller than that in the boundary column.

In this study, linear distribution was used for elastic curva-ture (Fig. 16(a)). Therefore, the elastic drift Ry is computed from Eq. (4).

2

2 6fy

y yh hR

h aD

= = − ⋅f (4)

Fig. 14—Load and strain versus flexural drift of Specimen NM3.

Fig. 15—Load and strain versus flexural drift of Specimen PM3.

Fig. 16—Elastic and plastic deformations.


where Dfy is flexural displacement at yielding; a is shear span length; and h is the specimen’s clear height.

The ultimate curvature fu is computed based on the ulti-mate strain of concrete (Fig. 16(c)), where eu is the ultimate compressive strain of concrete defined in a later section

uu c

ef = (5)

The plastic drift is computed using plastic hinge length Lp.

( )p p u yR L= ⋅ f − f (6)

Substituting Eq. (2) and (5) into Eq. (6) leads to the equa-tion to compute the plastic drift.

pp u y

L cRc d c

= e − e − (7)

Figure 17 shows the strain distribution measured using LVDTs in Fig. 9 along the clear height of Specimen NM4 when the lateral force decreased to 80% of the maximum strength. On the compressive side (the right edge), the strain localized between Points E and F, whereas the strain between Points F and G or G and H was quite small. On the tensile side (the left edge), even the strain between Points C and D exceeded the yield strain (0.0029 > ey = 0.002). It is concluded that, for plastic deformation, the compressed area is limited near the bottom of the wall, as indicated by the shaded rectangle (compressed area) in Fig. 17 and 18, whereas the area in tension is trapezoidal. The hatched area in Fig. 18 is assumed as rigid. The strain at the compressed edge in Fig. 18 is assumed to be uniformly ep, which equals the term inside the parentheses of Eq. (7)

p u yc

d ce = e − e

− (8)

where the second term is the strain caused by the elastic deformation. In this paper, ep is called the plastic component of ultimate strain. Note that the rigid area in Fig. 18 rotates

pp p

LR

c= e (9)

around the neutral axis. Therefore, the height of the compressed area in Fig. 18 can be regarded as the plastic hinge length Lp. The deformation shown in Fig. 18 agrees with the observed crack patterns and is similar to that proposed by Hiraishi.14

There are two unknown parameters—Lp and ep—in Eq. (9). In the following sections, these parameters are examined using the tested specimens. Specimens tested by Wallace and Orakcal2 and Tabata et al.5 are used because flexural deformations of specimens are reported.

Plastic component of ultimate strainThe broken lines in Fig. 19 show the stress-strain relation-

ships of confined and unconfined concrete in Specimen NS3 evaluated by the Saatcioglu and Razvi15 model, which is applicable to rectangular sections. Figure 20(a) shows the boundary element of NS3, where the shaded zone is assumed to be confined. The confining pressure on the shaded zone in the x-direction is computed assuming that the horizontal bars are 100% effective. The confining pressure in the y-direction is computed assuming that the

Fig. 17—Measured strain distribution of Specimen NM4. (Note: NA is neutral axis.)

Fig. 18—Simplified model for plastic deformation.

Fig. 19—Assumed stress-strain model for concrete of Specimen NS3.


crossties are 100% effective, while the cap bars at the wall end with 90-degree hooks are 50% effective because the observed strain of the cap bars was approximately one-half of the yield strain. Figure 20(b) shows the boundary element of Specimen NM2′. Although only cap bars are provided in NM2′, the shaded area is again assumed to be confined. The confining pressure in the y-direction is computed assuming that the cap bars are 50% effective.

The solid line in Fig. 19 shows the average stress-strain relationship of the boundary element calculated as the weighted average of confined and unconfined concrete.

c cc u

t t tt t

−s = s ⋅ + s ⋅ (10)

where sc is the stress of confined concrete; tc is the center-to center distance between the horizontal wall bars (Fig. 20(a)); and su is the stress of unconfined concrete.

The ultimate concrete strain eu is defined as the strain when the average stress of concrete decreases to 80% of the maximum strength. The ultimate strain of Specimen NM2′ without crossties is 0.0066, while the ultimate strains of the other specimens are between 0.0078 and 0.0084. The ulti-mate strains of the specimens of Wallace and Orakcal2 are approximately 0.008, except for Specimen TW2, which had a strain of 0.0112. Figure 20(c) shows the boundary elements of Specimens No. 2 and 3 of Tabata et al.5; although the confinement ratio is higher than that of Fig. 20(a), its ulti-mate strain is 0.008 because its concrete strength was high (70 MPa [10.15 ksi]). The strain at the compressive edge when tensile reinforcement bars yield shown in Fig. 16(b) (the second term in Eq. (8)) is approximately 0.001 in most specimens. Therefore, ep in Eq. (8) is approximately 0.008 – 0.001 = 0.007 for all specimens except NM2′ and TW2.

Plastic hinge lengthAs discussed in the Introduction, Wallace and Orakcal2 and

Tabata et al.5 assumed that Lp is equal to 0.5lw and 0.3lw, respectively, where lw is wall length. If Lp is proportional to lw, based on Eq. (9), Rp must be proportional to lw/c because ep is similar for most specimens. The relationship between Rp and lw/c is examined in Fig. 21. The variable Rp is the plastic drift, which is the flexural drift minus the elastic drift Ry computed by Eq. (4). The broken line in Fig. 21 is the regression line, which is imposed to pass the origin. The correlation coefficient is 0.70, where the results of Specimens TW2 and NM2′ are neglected because the ep values of these specimens are very different from those of the other specimens. The solid lines show the assump-tions of Lp = 0.5lw and 0.3lw with ep = 0.007. They do not agree with the regression line. The circles in Fig. 21 show two of the data, which have different trends from the other specimens. Specimen NS3, whose lw/t ratio is the smallest (=8.5), exhibited a drift capacity twice that expected by the regression line. On the other hand, Specimen NL2, whose lw/t ratio is the largest (=20), exhibited a drift capacity 60% of that expected by the regression line. Similarly, specimens with small or large lw/t ratios are located above or below the regression line, respectively. This tendency indicates that hinge length is not simply proportional to wall length.

ACI 318-081 requires that the special boundary region shall be longer than a/4, where a is shear span length. This requirement implies that Lp in Fig. 18 equals a/4. To investi-gate whether the hinge length is related to shear span length a, the relationship between Rp and the a/c ratio is shown in Fig. 22. The solid line shows the assumption of Lp = a/4 with ep = 0.007, which does not agree with the regression line (the broken line). The correlation coefficient is 0.84 and is better than that in Fig. 21. However, it is noted that the results of Specimens No. 2 and 3 of Reference 5, whose a/t ratios (=50) are much larger than those of the other speci-mens (=18 to 28), are located quite lower than the regression line. This tendency again indicates that hinge length is not simply proportional to shear span length.

Markeset and Hillerborg16 conducted uniaxial compres-sion tests of plain concrete prisms with various lengths and sectional dimensions. They observed that compressive failure is quite limited within a certain length. They concluded that the failure length was 2.5 times the shorter side length of the compressed section (Fig. 23). In this study, wall thick-ness t is shorter than neutral axis depth c in all specimens. Figure 24 shows the damage of Specimen NS3 at a 2% drift

Fig. 20—Assumed confined regions. (Note: 1 mm = 0.039 in.)

Fig. 21—Relationship between Rp and lw/c.


Fig. 22—Relationship between Rp and a/c.

level (right after the strength degradation). The damage length seems to be almost 2.5 times the wall thickness. Therefore, 2.5t is used for plastic hinge length Lp in this study. It should also be noted that even in Specimen NM2 with the largest thickness (t = 140 mm [5.5 in.]), the damage of concrete was limited within the height of confinement (400 mm [1.3 ft]). Therefore, it is concluded that the height of confinement may be limited to 3t if the expected compressive strain is not greater than 0.008.

Figure 25 shows the relation between Rp and t/c. The corre-lation coefficient is 0.94 and larger than the correlation coef-ficients in Fig. 21 and 22 (0.70 and 0.84). The broken line in Fig. 25 shows the regression line. The solid line shows Rp estimated from Lp = 2.5t and ep = 0.007. The solid line reasonably agrees with the regression line, which implies that the assumption of Lp = 2.5t is appropriate.

Figure 26 compares the observed and estimated flex-ural drift capacities (Ru = Ry + Rp). All test data except Specimen TW2 are within 30% from the estimated drift capacities. The observed drift of TW2 is much larger than the estimated value. The observed compressive failure zone of TW2 was also much wider than Lp = 2.5t. Similar

Fig. 23—Compression localization.16

Fig. 24—Observed failure of Specimen NS3 at 2% drift.

Fig. 25—Relationship between Rp and t/c.

Fig. 26—Comparison between estimated and observed flex-ural drift capacities.


tendencies are observed for the test results of Paulay and Priestley,8 which are not plotted in Fig. 26 because their flexural drift capacities are not reported. These discrepancies may be attributable to the difference of confinement. Recall that the strain localization depicted in Fig. 23 is observed in plain concrete. In the case of well-confined concrete columns subjected to uniaxial compression, large plastic strain shall be distributed uniformly over its entire length until strength degradation starts (emax in Fig. 19). If the concrete column is long enough, buckling occurs17 before the strain reaches emax. The strain at buckling depends on the length of the column and the tangential stiffness at the strain.17 The boundary elements of Specimen TW2 and the specimens of Paulay and Priestley8 almost satisfied the requirements of ACI 318-08.1 If such a wall would be subjected to pure bending, uniform curvature corresponding to emax or less would be observed in its clear height when out-of-plane buckling would occur. Therefore, for such a wall, eu in Eq. (7) should be replaced with the strain at buckling and Lp should be a function of shear span length, which would be much longer than 2.5t. On the other hand, the cross-sectional areas of the hori-zontal bars and the crossties of the specimens tested by the authors are 40% to 52% and 6% to 31% of those required by ACI 318-08,1 respectively. The cross-sectional areas of the confining bars of Specimens No. 2 and 3 tested by Tabata et al.5 and Specimens RW1 and RW2 tested by Wallace and Orakcal2 are 24 to 63% of those required by ACI 318-08.1 It is concluded that Lp = 2.5t is valid if the confinement of the boundary element is less than half of that required by the seismic provisions of ACI 318-08.1 Otherwise, the equation tends to underestimate the capacity.

CONCLUSIONSThe test results of 10 RC walls are described in this

study. Based on the experimental results and analytical work presented in this paper, the following conclusions are obtained:

1. All tested RC walls with limited confinement in the boundary element failed in compression after flexural yielding. The observed flexural drift capacity was between 0.4 and 1.2%.

2. Shear drift caused by lateral slip along the flexural crack may be large if the ratio of the maximum shear force to the compressive force of concrete is large (Fig. 13).

3. Plastic components of flexural drift can be modeled as shown in Fig. 18, where the length of the compression zone Lp is 2.5 times the wall thickness (Lp = 2.5t) if the depth of the neutral axis is longer than the wall thickness and the confinement of the boundary element is half of that required by the seismic provisions of ACI 318-08.1

4. Ultimate flexural drift is defined as the drift when the lateral force decreases to 80% of maximum lateral strength. Ultimate flexural drift can be computed as the sum of Eq. (4) and (7), where eu shall be determined as shown in Fig. 19 considering the confinement effect.

5. The detailing of the boundary element shown in Fig. 2(c) with the height of the confined area 3t is sufficient to obtain an ultimate strain of eu = 0.008.

6. The effect of a perpendicular wall on ultimate drift is negligible if the wall is not confined.

ACKNOWLEDGMENTSThis study was financially supported by the Ministry of Land and Trans-

portation. The authors thank J. Wallace of the University of California, Los Angeles (UCLA), who independently conceived the idea that hinge length may be proportional to wall thickness, for valuable discussions. Data provided by T. Tabata, H. Nishihara, and H. Suzuki of Ando Corporation are greatly appreciated. The authors also thank H. Sezen of The Ohio State University for critically reading the manuscript.

REFERENCES1. ACI Committee 318, “Building Code Requirements for Structural


2. Wallace, J. W., and Orakcal, K., “ACI 318-99 Provisions for Seismic Design of Structural Walls,” ACI Structural Journal, V. 99, No. 4, July-Aug. 2002, pp. 499-508.

3. Thomsen, J. H. IV, and Wallace, J. W., “Displacement-Based Design of Slender Reinforced Concrete Structural Walls—Experimental Verification,” Journal of Structural Engineering, ASCE, V. 130, No. 4, 2004, pp. 618-630.

4. “Design Guidelines for High-Rise RC Frame-Wall Structures,” National Institute for Land and Infrastructure Management, Kaibundo, Tokyo, Japan, 2003, pp. 53-85.

5. Tabata, T.; Nishihara, H.; and Suzuki, H., “Ductility of Reinforced Concrete Shear Walls without Column Shape,” Proceedings of the Japan Concrete Institute, V. 25, No. 2, 2003, pp. 625-630. (in Japanese)

6. Kabeyasawa, T.; Shiohara, H.; Otani, S.; and Aoyama, H., “Analysis of the Full-Scale Seven-Story Reinforced Concrete Test Structure,” Journal of the Faculty of Engineering, V. 37, No. 2, 1983, pp. 431-478.

7. Orakcal, K., and Wallace, J. W., “Flexural Modeling of Reinforced Concrete Walls—Experimental Verification,” ACI Structural Journal, V. 103, No. 2, Mar.-Apr. 2006, pp. 196-206.

8. Paulay, T., and Priestley, M. J. N., “Stability of Ductile Structural Walls,” ACI Structural Journal, V. 90, No. 4, July-Aug. 1993, pp. 385-392.

9. Takahashi, S.; Yoshida, K.; Ichinose, T.; Sanada, Y.; Matsumoto, K.; Fukuyama, H.; and Suwada, H., “Flexural Deformation Capacity of RC Shear Walls without Column on Compressive Side,” Journal of Structural and Construction Engineering: Transactions of AIJ, V. 76, No. 660, 2011, pp. 371-377. (in Japanese)

10. Takahashi, S., “Modeling for RC Shear Walls with or without Boundary Column,” PhD thesis, Department of Civil Engineering, Nagoya Institute of Technology, Nagoya, Japan, 2011. (in Japanese)

11. Nagoya Institute of Technology, “Flexural Deformation of RC Wall with One Side Column,” http://kitten.ace.nitech.ac.jp/ichilab/research/flex-uralwall2010. (last accessed Nov. 5, 2012)

12. National Earthquake Hazards Reduction Program, “Executive Summary on Chile Earthquake Reconnaissance Meeting,” NEHARP Library, http://www.nehrp.gov/library/ChileMeeting.htm. (last accessed Nov. 5, 2012)

13. Architectural Institute of Japan, “AIJ Standard for Structural Calcu-lation of Reinforced Concrete Structures,” Maruzen, Tokyo, Japan, 2010, pp. 484-491.

14. Hiraishi, H., “Evaluation of Shear and Flexural Deformations of Flexural Type Shear Walls,” Bulletin of the New Zealand National Society for Earthquake Engineering, V. 17, No. 2, 1984, pp. 135-144.

15. Saatcioglu, M., and Razvi, S. R., “Strength and Ductility of Confined Concrete,” Journal of Structural Engineering, ASCE, V. 118, No. 6, 1992, pp. 1590-1607.

16. Markeset, G., and Hillerborg, A., “Softening of Concrete in Compres-sion Localization and Size Effects,” Cement and Concrete Research, V. 25, No. 4, 1995, pp. 702-708.

17. Shanley, F. R., “Inelastic Column Theory,” Journal of the Aeronau-tical Sciences, V. 14, No. 5, 1947, pp. 261-268.


Title no. 110-S11


ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2011-066.R4 received November 18, 2011, and reviewed under Institute


Shake-Table Studies of Repaired Reinforced Concrete Bridge Columns Using Carbon Fiber-Reinforced Polymer Fabricsby Ashkan Vosooghi and M. Saiid Saiidi

The main objective of this study was to develop a rapid and effec-tive repair method using carbon fiber-reinforced polymer (CFRP) fabrics for earthquake-damaged reinforced concrete (RC) bridge columns. Shake-table studies, repair methods, and test results are discussed in this paper. One standard bent of a two-span bridge model with two low-shear columns, two standard high-shear columns, and one low-shear and one high-shear substandard column were tested on shake tables. The column models were repaired using CFRP fabrics; fast-setting, nonshrink repair mortar; and epoxy injection and retested on shake tables to evaluate the performance of the repair procedure. The results indicated that the strength and displacement capacity of the standard columns were successfully restored and those of the substandard columns were upgraded to meet current seismic standards after the repair. However, the stiffness was not restored due to material degradation during the original tests.

Keywords: carbon fiber-reinforced polymer fabric; earthquake damage; emergency repair; reinforced concrete bridge column; repair; shake-table test.

INTRODUCTIONThe current earthquake engineering design practice for

ordinary bridges allows for damage to bridge columns during moderate and strong earthquakes. The target response under the maximum considered earthquake is “no-collapse,” real-izing that the structure would undergo considerable nonlin-earity associated with extensive concrete damage, yielding of bars, or even rupture of a limited number of the bars. For the more frequent earthquakes, the target response is repair-able damage that would allow for relatively rapid restora-tion of the bridge. The level of damage to different columns of a bridge varies depending on the intensity of the ground shaking, type of earthquake, and the force and deformation demand on individual members. Based on the inspection of the damaged columns, engineers have to determine whether the bridge is sufficiently safe to be kept open to traffic without repair, whether it is repairable within a reasonable time frame, or if it needs to be replaced. This study was aimed at developing a reliable and efficient repair procedure for earthquake-damaged reinforced concrete (RC) bridge columns using carbon fiber-reinforced polymers (CFRPs).

Although there are numerous studies on seismic retrofit of RC columns (Saadatmanesh et al. 1996; Seible et al. 1997; Haroun and Elsanadedy 2005; Laplace et al. 2005), only a few studies have focused on seismic repair (Priestley and Seible 1993; Saadatmanesh et al. 1997; Lehman et al. 2001; Li and Sung 2003; Saiidi and Cheng 2004; Belarbi et al. 2008). In these studies, the damaged concrete was replaced with new concrete and the cracks were epoxy-injected. The buckled or fractured bars were replaced with new bars (Lehman et al. 2001) or replaced with equivalent fiber-reinforced polymer (FRP) fabrics (Saiidi and Cheng 2004; Belarbi et al. 2008). The repaired columns were serviceable after full curing of

new concrete in at least 28 days. In addition, replacing frac-tured bars was complicated and time-consuming because it required removing a significant amount of concrete from the damaged zone and adjacent footing. The techniques could not be considered “rapid repair.” In this study, fast-setting repair mortar and accelerated CFRP jacket curing were used to restore service in less than 1 week. This type of repair may be labeled as “emergency” repair due to its urgency and the speed of repair work.

Shake-table studies of repaired bridge column models are presented in this paper. Original column models were tested on a shake table until reaching the highest target repairable damage state. They were subsequently repaired using unidirectional CFRP fabrics with fibers in the hoop direction and retested on the shake table until failure to evaluate the repair performance.

RESEARCH SIGNIFICANCEDelay in opening an earthquake-damaged bridge to traffic

can have severe consequences on the passage of emergency response vehicles, detour lengths, and traffic congestion. Rapid and effective repair methods are needed to enable quick opening of the bridge to minimize impact on the community and beyond. In this study, a rapid repair procedure using CFRP fabrics was developed and evaluated for RC bridge columns. The experimental studies indicated that a damaged column can be repaired in only a few days using CFRP fabrics. The proposed repair method using CFRP fabrics can be very useful in emergency repair of earthquake-damaged bridges.

DESCRIPTION OF TEST MODELSOne standard bent consisting of two low-shear columns

(Bent-2), two standard high-shear columns (NHS1 and NHS2), one low-shear substandard column (OLS), and one high-shear substandard column (OHS) were studied. Columns meeting current seismic design requirements are referred to as “standard” columns. Other columns are labeled as “substandard.”

Two-column bentChoi et al. (2007) tested a one-fourth-scale, two-span

bridge model supported on three two-column piers using three shake tables at the University of Nevada, Reno (Fig. 1(a)). The seismic design of the bridge was based on recent seismic design guidelines (Johnson et al. 2008). The


ACI member Ashkan Vosooghi is a Postdoctoral Fellow in the Department of Civil and Environmental Engineering at the University of Nevada, Reno, Reno, NV (UNR). He received his BSC from Isfahan University of Technology, Isfahan, Iran; his MSCE in earthquake engineering from the International Institute of Earthquake Engineering and Seismology, Tehran, Iran; and his PhD in structural/earthquake engineering from UNR with an emphasis on bridge engineering in 1999, 2002, and 2010, respectively. His research interests include seismic design, retrofit, and repair of structures.

M. Saiid Saiidi, FACI, is a Professor of Civil & Environmental Engineering and the Director of the Center for Advanced Technology in Bridges and Infrastructure at UNR. He is the founding and former Chair and a current member of ACI Committee 341, Earthquake-Resistant Concrete Bridges. He is also a member of Joint ACI-ASCE Committee 352, Joints and Connections in Monolithic Concrete Structures, and ACI Subcommittee 318-D, Flexure and Axial Loads: Beams, Slabs, and Columns.

measured yield stress of the longitudinal bars and spirals was 67.9 and 55.8 ksi (468 and 385 MPa), respectively, and the measured concrete compressive strength was 6.47 ksi (44.6 MPa) on the test day. Because the middle bent was the most severely damaged pier, it was used in the emer-gency repair study. The original middle bent was designated as Bent-2 and the repaired bent was designated as Bent-2R. Bent-2 was composed of two columns that spanned a distance of 75 in. (1.9 m) center to center. The details of the column sections are shown in Fig. 1(d).

The axial load index is defined as the compressive axial force due to gravity loads divided by the product of the cross-section area of the column and the specified concrete compressive strength. Based on Caltrans design practice, this index typically varies from 5 to 10% in bridge columns,

and it was 8.2% in the two-span bridge model (Johnson et al. 2008). The average shear stress is calculated as the ratio of the plastic shear divided by the effective shear area. The effective shear area is taken as 80% of the gross section area Ag in circular columns (Caltrans 2006). The shear stress index is calculated by dividing the average shear stress by √fc′ (psi) (0.083√fc′ [MPa]). This index is used to deter-mine the level of shear stress in columns. In this study, an index smaller than 4 was treated as the low shear level in the column. The columns in Bent-2 were flexure-dominated with a shear index of 2.3.

Columns NHS1 and NHS2As part of this study, two similar standard high-shear RC

bridge columns were studied (Vosooghi and Saiidi 2010a). NHS1 and NHS2 (new design high shear) are the designa-tions of the two original column models (Fig. 1(b) and (d)). The repaired models were labeled NHS1-R and NHS2-R. The designation of “new” indicates that the column meets current seismic code requirements. The models were flexure-dominated, as the current codes do not allow shear-domi-nated columns. However, the level of shear was relatively high in NHS1 and NHS2 with the shear index being 6.12. NHS1 and NHS2 were constructed at different times because the study of NHS2 was found to be necessary after NHS1-R testing and analysis of the data. The latest Caltrans Seismic Design Criteria (SDC) (version 1.4) (Caltrans 2006) and Bridge Design Specifications (Caltrans 2003) were used to

Fig. 1—Shake-table setup and column model details: (a) two-span bridge; (b) double-curvature column; (c) cantilever column; and (d) column sections.


design the columns. The axial load index was 10% for the column models. The measured yield stresses of the longi-tudinal bars and spirals of NHS1 were 73.5 and 60.6 ksi (507 and 418 MPa), and those of NHS2 were 66.5 and 67.0 ksi (458 and 462 MPa), respectively. The measured concrete compressive strengths in NHS1 and NHS2 were 7.29 and 6.17 ksi (50.3 and 42.5 MPa), respectively, on the test day.

Columns OLS and OHSRC bridge columns designed prior to the 1970s were not

adequately detailed to resist seismic loads and are considered to be substandard. They have insufficient lateral reinforce-ment and their longitudinal bars are lap-spliced at the base.

As part of this study, one substandard low-shear and one substandard high-shear RC bridge column were studied (Vosooghi and Saiidi 2010a). OLS and OHS are the designa-tions used for the old design low shear and the old design high shear column models, respectively, and OLS-R and OHS-R are the designations used for the repaired columns. The designation of “old” indicates that the columns do not meet current seismic code requirements. The details of the columns are shown in Fig. 1(b) through (d). The axial load index was 7.5% for both columns and the shear indexes were 1.9 and 5.2 for OLS and OHS, respectively.

Prior to the 1970s, Grade 40 and 50 steel was used in RC construction. Due to the unavailability of Grade 40 bars, Grade 60 steel was used in the column models and the steel ratio was modified proportionally. The measured yield stresses of the longitudinal bars in OLS and OHS were 64.5 and 66.5 ksi (445 and 458 MPa), respectively. The measured yield stress of hoops was 60.0 ksi (414 MPa) in both columns. The measured concrete compressive strengths in OLS and OHS were 4.94 and 4.97 ksi (34.0 and 34.3 MPa), respectively, on the test day. The lap-splice length varies from 20 to 30 times the longitudinal bar diameter db in substandard columns. The splice length of 24db was selected (Laplace et al. 2005). Because the required length of the splice is proportional to steel yield stress, it was scaled up by the factor of 3/2, which is the ratio of the specified yield stresses of Grade 60 and Grade 40 steel.

SHAKE-TABLE TEST SETUP AND TEST PROCEDUREDifferent setups were used for the two-span bridge tests and

single-column tests. The two-span bridge was supported on three shake tables. The superstructure consisted of six girders and was post-tensioned laterally and longitudinally to form a rigid slab. To produce the target axial load in the columns, concrete blocks and lead pallets were placed on the bridge deck (Fig. 1(a)). The original bridge model was subjected to fault-parallel near-field motions with amplitudes increasing gradually to simulate fault rupture (Choi et al. 2007).

The single-column models (NHS1, NHS2, OLS, and OHS) were tested on one of the shake tables at the Univer-sity of Nevada, Reno. The inertia mass system designed by Laplace et al. (1999) was used to apply the lateral inertial force to the columns (Fig. 1(b) and (c)). A single swiveled link system or a double-link system was used to transmit the lateral inertial load from the mass rig, depending on the column. These configurations allow the columns to be tested in single or double curvature. The high shear columns were tested under double curvature (Fig. 1(b)) and the low shear column was tested under single-curvature loading (Fig. 1(c)). The footing and head of the columns were designed so they remained elastic during the shake-table tests.

The 1994 Northridge Sylmar Hospital ground motion record with peak ground acceleration (PGA) of 0.61g was selected for earthquake simulation of the single columns. This motion induced high ductility demands and residual drifts in the columns that made the repair challenging. In each shake-table test, the column was subjected to multiple simulated earthquakes—each referred to as a “run”—with gradually increasing amplitudes. In the shake-table tests of the original bent and columns, the amplitude of each run was determined such that no steel bars fractured during the tests. The number of runs was kept as low as possible to reduce low-cycle-fatigue rupture of the longitudinal bars. The maximum strains in critical bars in the plastic hinge zone were carefully monitored for each shake-table run before deciding the amplitude of the subsequent run. In the shake-table test of the repaired bent and columns, the input motion and the loading protocols were similar to those used in the original column tests, but additional runs with higher acceleration amplitudes were applied until failure. All the models were extensively instrumented to measure strains, curvatures, displacements, forces, and accelerations.

EXPERIMENTAL RESULTS FOR ORIGINAL COLUMNS

In a previous study (Vosooghi and Saiidi 2010b), five repairable apparent damage states were identified for RC columns subjected to earthquakes. The damage states excluded failure due to bar fracture because it was believed that repair of columns with fractured bars could not be done rapidly. The damage states were defined as DS-1: flexural cracks; DS-2: minimal spalling and possible shear cracks; DS-3: extensive cracks and spalling; DS-4: visible lateral and/or longitudinal reinforcing bars; and DS-5: compressive failure of the concrete core edge (imminent failure). The five damage states are applicable to columns meeting current design codes. Substandard columns do not necessarily reach higher damage states because they are brittle.

The models were tested to reach the highest repairable damage state. The standard columns reached DS-5. At this damage state, many spirals and longitudinal bars are visible, some of the longitudinal bars begin to buckle, and the edge of the concrete core is damaged (Fig. 2). Due to severely inadequate transverse steel and longitudinal bar lap splice, the substandard columns did not undergo significant plastic deformations. Testing of Columns OLS and OHS was stopped at DS-3 and DS-2, respectively (Fig. 2), to avoid complete failure of the columns. Shear cracks covering a large area of OLS and OHS were formed during the last run. Considering the very low amount of transverse steel in these columns, it was felt that additional motions would lead to total failure of the columns, thus preventing repair.

The cumulative measured force-displacement hysteresis curves of Bent-2, NHS2, and OHS are shown in Fig. 3(a) to (c), respectively. Other test models had comparable hyster-esis curves, although energy dissipation (area enclosed by the hysteresis hoops) in Column OLS was larger than that of OHS. The envelope of each hysteresis curve was determined and idealized by an elasto-plastic curve (Vosooghi and Saiidi 2010a). Using idealized force-displacement curves, the yield drift ratio, maximum drift ratio, displacement ductility, and strength of the columns were determined and are listed in Table 1. Note that the ductilities in the table are not the ductility capacities because the original columns were not tested to failure. Generally, the low shear columns (Bent-2 and OLS)


Fig. 2—Apparent damage after original shake-table tests.

Fig. 3—Force-displacement relationships.


reached approximately two times the displacement ductility of the high shear columns at the highest repairable damage states. Because high shear induced relatively large strains in concrete, the high-shear columns reached the target damage level under smaller lateral displacements than the low-shear columns. According to Table 1, the substandard columns (OLS and OHS) reached approximately one-third of the displacement ductility of the standard columns (Bent-2 and NHS1 or NHS2) with the same level of shear because the substandard columns had severely insufficient transverse steel. For instance, OHS reached a displacement ductility of 1.7 but NHS1 and NHS2 reached a displacement ductility of 4.8 and 5.1, respectively.

The peak measured strains in the longitudinal and trans-verse steel of the plastic hinge zone of the columns are also listed in Table 1. The measured strains indicate that spirals in standard high-shear columns yielded during the shake-table tests; however, the spirals in standard low-shear columns remained elastic. High shears resulted in extensive shear cracks that induced large strains in the transverse steel. In substandard columns, the strains were measured in the spliced bars. In the case of splice degradation, the measured strains remain constant or decrease with increasing column lateral displacement. This was not observed; therefore, it was concluded that no slippage occurred during the shake-table tests, even though the lap-splice length was too short to develop the yield stress by 51% (ACI Committee 318 2008).

CFRP JACKET DESIGN FOR STANDARD COLUMNSThe repair of the standard column models was designed

with the objective of restoring the lateral load strength and displacement capacity of the column. Unidirectional CFRP fabrics were used for this purpose. The study of other types of jackets was not within the scope of this study. The CFRP fabrics had a nominal thickness of 0.04 in. (1 mm) per layer and the fibers were in the hoop direction of the columns in all repaired models.

The CFRP jacket was designed so the repaired column could reach the plastic flexural capacity. The radial dilating strain of the jacket was limited to 4000 me (1 me = 10–6 in./in.) to avoid degradation in concrete aggregate interlock (Priestley et al. 1996). The contribution of concrete and spirals to shear strength was treated differently among the test models as data became available in the course of the study. Bent-2 was the first model that was tested, repaired, and retested. Due to lack of information about the contribu-tion of concrete and spirals to the shear strength in repaired columns, their contributions were neglected conservatively along the entire column height. In the plastic hinge region, because some of the thin cracks in the core were not repair-able, the shear strength of concrete was neglected in NHS1-R and NHS2-R. The spirals in NHS1 experienced a maximum

strain of approximately 1.6 times the yield strain. As a result, the shear strength of the spirals was assumed to be zero in NHS1-R. Subsequent to testing, the strain data indicated that the spirals and CFRP jacket in NHS1-R contributed to the shear strength equally. A second similar column (NHS2-R) was designed, neglecting the concrete shear strength in the plastic hinge, but the jacket was designed for one-half of the column shear demand, and the other half was assumed to be resisted by the spirals. Outside the plastic hinge region, because the spirals did not yield, the entire shear strength of the spirals was used in NHS1-R and NHS2-R. Although some shear cracks occurred outside the plastic hinge, the level of damage was much lower than that of the plastic hinges. As a result, 50% of the concrete shear strength was assumed to exist outside the plastic hinges.

Because there were no seismic repair design guidelines, the Caltrans (2007) seismic retrofit guidelines were used to restore confinement and the ductility capacity of the columns using a CFRP jacket. This document requires a confinement pressure of 300 psi (2.07 MPa) at a radial dilating strain of 4000 me in the plastic hinge regions. The confinement pressure can be reduced to 150 psi (1.03 MPa) at the same dilating strain outside the plastic hinges.

The required and actual thicknesses of the CFRP jackets are listed in Table 2. In Bent-2R, the jacket consisted of two layers of CFRP at the plastic hinge regions and one layer of CFRP elsewhere. The Caltrans (2007) confinement require-ments governed the jacket design in the plastic hinge regions and shear strength governed the jacket design outside the plastic hinge regions. In Bent-2R, the end 18 in. (457 mm) (one-and-a-half-column diameter) was assumed to be the primary plastic hinge zone and the adjacent 12 in. (304.8 mm) (one-column diameter) was treated as the secondary plastic hinge zone. There was a concern that plastic deformation could extend beyond the primary plastic hinge zone; hence, the number of CFRP layers in the primary plastic hinge was maintained in the secondary plastic hinge (Seible et al. 1997).

In NHS1-R, the jacket consisted of four layers of CFRP at the plastic hinge regions and one layer of CFRP elsewhere. The shear strength governed the jacket design in the plastic hinge regions and the Caltrans (2007) confinement require-ments controlled the jacket design outside the plastic hinge regions. In NHS2-R, the jacket consisted of two layers of CFRP at the plastic hinge regions and one layer of CFRP elsewhere. Table 2 shows that the Caltrans (2007) confine-ment requirements governed the jacket design inside and outside the plastic hinge regions of NHS2-R. In NHS1-R and NHS2-R, a length of 24 in. (610 mm) was used for the plastic hinge regions (one-and-a-half-column diameter). The test results for Bent-2R indicated that plastic hinging did not extend into the secondary plastic hinge regions. Conse-quently, no secondary plastic hinge region was considered in

Table 1—Measured responses of original columns

Model Yield drift, % Maximum drift, % Displacement ductility Strength, kips (kN)

Peak strain, me

Longitudinal steel Transverse steel

Bent-2 1.06 10.41 9.8 33.2 (148) 69,868 (30ey) 1227 (0.31ey)

NHS1 1.58 7.54 4.8 90.5 (402) 77,522 (17.2ey) 6510 (1.6ey)

NHS2 1.26 6.42 5.1 75.7 (337) 54,283 (23.6ey) 3108 (1.35ey)

OLS 1.14 3.97 3.5 23.9 (106) 33,096 (15ey) 6292 (1.57ey)

OHS 0.82 1.43 1.7 45.0 (200) 23,552 (10ey) 1340 (0.3ey)

Note: ey is yield strain.


NHS1-R and NHS2-R. A jacket gap of 0.75 in. (19 mm) was specified at the ends of the columns to prevent jacket bearing against the footing or the cap beam under large rotations.

CFRP JACKET DESIGN FOR SUBSTANDARD COLUMNS

The repair of substandard column models was designed with the objective of upgrading shear strength, preventing lap-splice slippage, and upgrading confinement of the columns by using unidirectional CFRP fabrics. The thickness of CFRP for shear strength was calculated using the method described previously. The existing steel hoop contribution to the shear strength was neglected in the repaired substandard column models because the amount of the transverse steel was minimal. Due to reasons discussed previously, the shear strength of the concrete inside the plastic hinge region was neglected, but 50% of the concrete shear strength outside the plastic hinge region was accounted for in the repair design.

The method proposed by Priestley et al. (1996) was used to design the required CFRP jacket thickness to prevent splice failure. They showed that the propensity for splice failure could be predicted by assessment of the concrete tensile capacity across a potential splitting failure surface. After cracking develops on this surface, splice failure can be inhibited with sufficient clamping pressure provided by a CFRP jacket with a radial dilation strain limit of 1500 me across the fracture surface.

In the absence of repair methods, the seismic retrofit guide-lines of Caltrans (2007) were used to design for confinement provided by the CFRP jacket.

The required and actual thicknesses of the CFRP jackets are listed in Table 2. The jacket for OHS-R consisted of two layers of CFRP along the entire column height and the jacket for OLS-R consisted of two layers of CFRP at the plastic hinge region and one layer of CFRP elsewhere. The results show that inhibiting lap-splice failure governed the jacket design in the plastic hinge regions of both columns. Caltrans (2007) confinement requirements controlled the jacket design outside the plastic hinge regions in OLS-R, and shear strength requirements governed the jacket design outside the plastic hinge regions of OHS-R. According to Table 2, a CFRP thickness of 0.091 in. (2.31 mm) was required to inhibit lap-splice failure in OHS-R. Two layers of CFRP with a thickness of 0.08 in. (2.0 mm) were used in OHS-R instead of a three-layer jacket with a thickness of 0.12 in. (3.0 mm) to prevent an overly conservative jacket design.

Similar to the standard columns, a 0.75 in. (19 mm) gap was specified at the ends of the jackets.

REPAIR PROCEDUREThe entire repair work took 3 to 4 days for each column

and consisted of the following steps:

Straightening columnsThe residual drift ratio in Bent-2, NHS1, and NHS2 at the

end of the test was 10.4%, 3.35%, and 2.0%, respectively. Prior to repair, the bent and the columns were straightened to a near-vertical position (1% or less drift ratio) by adjusting the shake tables. The residual drift ratios in OLS and OHS were relatively small at 0.55% and 0.21%, respectively, and the columns were not straightened. In practice, straightening would vary, depending on the bridge, extent of residual displacement, and the bridge surroundings. Pulling the bridge using heavy-duty construction equipment may be an option.

Removal of loose concreteThe loose concrete was removed by an impact hammer

with a chisel head (Fig. 4). The area was cleaned using compressed air to remove dust and the remaining concrete particles after chipping the concrete. No loose concrete was observed in the original substandard columns after the tests. Therefore, this step was not exercised for these columns.

Concrete repairTwo different types of mortar and placement methods were

used. In NHS1-R, a low-shrinkage repair mortar with 1-day and 3-day specified compressive strengths of 2.5 and 4 ksi (17.2 and 27.6 MPa], respectively, was used. A thick mortar was made and applied to the spalled area by hand and consol-idated by thumb pressure. The compressive strength of the mortar was 4.05 ksi (27.9 MPa) on the test day when the mortar was 3 days old. In Bent-2R, NHSR-2, and OLS-R, a low-shrinkage, fast-setting repair mortar with 3-hour and 1-day specified compressive strengths of 3 and 4 ksi (20.7 and 27.6 MPa), respectively, was used. The specified Young’s modulus of this mortar was 3800 ksi (26.2 GPa). Due to the relatively high 1-day compressive strength for the second mortar, it was decided to make a fluid mortar and cast it into a mold instead of patching it in the spalled area (Fig. 4). The mortar was consolidated using a small vibrator. The compressive strength of the mortar used in NHS2-R was 7.87 ksi (54.3 MPa) on the test day at the age of 4 days.

Table 2—Thickness of CFRP jackets in repaired columns

Test model Location Shear strength, in. (mm) Lap splice, in. (mm) Confinement, in. (mm) Actual, in. (mm)

Bent-2ROPHR 0.0345 (0.88) NA (NA) 0.0281 (0.71) 0.04 (1.0)

IPHR 0.0345 (0.88) NA (NA) 0.0563 (1.43) 0.08 (2.0)

NHS1-ROPHR 0.024 (0.61) NA (NA) 0.038 (0.97) 0.04 (1.0)

IPHR 0.138 (3.51) NA (NA) 0.075 (1.91) 0.16 (4.0)

NHS2-ROPHR 0.002 (0.05) NA (NA) 0.03 (0.76) 0.04 (1.0)

IPHR 0.049 (1.24) NA (NA) 0.06 (1.52) 0.08 (2.0)

OLS-ROPHR 0.008 (0.20) 0 (0) 0.025 (0.64) 0.04 (1.0)

IPHR 0.022 (0.56) 0.080 (2.0) 0.05 (1.27) 0.08 (2.0)

OHS-ROPHR 0.047 (1.19) 0 (0) 0.025 (0.64)

0.08 (2.0)IPHR 0.061 (1.55) 0.091 (2.31) 0.05 (1.27)

Notes: OPHR is outside plastic hinge region; IPHR is inside plastic hinge region; NA is not available.


Epoxy injectionTo provide integrity and stiffness for the damaged

columns, the cracks were injected with epoxy at a pres-sure of 40 to 50 psi (0.28 to 0.34 MPa). To inject the epoxy, several ports were attached on all the visible cracks and the crack surfaces were sealed with a removable sealer. Epoxy was injected into a given crack through one port and injection was continued until bleeding from another port occurred. The epoxy injection process is shown in Fig. 4.

Surface preparation and CFRP wrappingAfter the concrete was repaired and epoxy was injected,

the column surface was smoothened slightly by a grinder to remove any surface roughness and any injected material resi-dues from the column surface. A layer of epoxy was applied to prime the column surfaces (Fig. 4). Subsequently, a thick-ened epoxy was applied directly on the columns to smooth out imperfections. After preparing the surface, the epoxy was applied to CFRP layers using a paint roller and the sheets were wrapped around the columns manually (Fig. 4).

Accelerated curing of jacketThe entire curing of the jacket took approximately 48 hours

for each column and consisted of accelerated curing for the first 24 hours, followed by curing under the labora-tory ambient condition. Note that specifications call for a minimum of 7 days of curing for CFRP jackets in the ambient condition. During accelerated curing, the temperature was elevated to 110°F (43°C) and the relative humidity was reduced to 15% by covering the area around the models with plastic sheets (Fig. 4), using heat lamps directed away from the columns and electric heaters, and a fan for circulation. This condition was maintained for approximately 24 hours. The plastic sheet was subsequently removed to allow for installation of strain gauges and linear variable differential transformer (LVDT) displacement transducers. The jackets were cured at the ambient temperature in the laboratory for an additional 24 hours.

CFRP MATERIAL PROPERTIESThe design and measured modulus of elasticity and

measured rupture strain of the CFRP material are listed in Table 3. CFRP properties recommended by Caltrans (Steckel

et al. 1999) were used in the jacket design of Bent-2R. For other columns, the measured properties of the CFRP from previous tests—but limited to the specified properties—were used in the jacket design. The measured properties were determined based on coupon tests. The coupons were cured under the same conditions as those of the column jackets. The specified modulus of elasticity and rupture strain of the CFRP after full curing were 11,900 ksi (82.0 GPa) and 10,000 me, respectively. In all columns, the measured modulus of elas-ticity exceeded the design value, indicating that the accel-erated curing was effective. The average measured rupture strain was comparable to the specified value.

EXPERIMENTAL RESULTS FOR REPAIRED COLUMNS

The repaired bent and columns were tested on shake tables under generally the same loading protocols as those used in the original bent and column tests. The main difference was that the loading protocols for the repaired columns included additional runs with increasing amplitudes until failure.

In Bent-2R, no damage was observed during the shake-table runs until the drift ratio of 9.0%. At this drift ratio, the first CFRP rupture occurred in the column under compressive force due to the overturning moment (west column). This rupture was extended during subsequent runs. The second CFRP rupture was observed in the east column during the last run at a 13.1% drift ratio. The ruptured jackets in both columns are shown in Fig. 5. After the shake-table test, the CFRP jacket and some concrete were removed from the plastic hinge regions and no ruptured bar was observed (Fig. 5).

Fig. 4—Rapid repair procedure.

Table 3—CFRP material properties

Model

Modulus of elasticity, ksi (GPa)

Measured rupture strain, meDesign Measured

Bent-2R 8000 (55.2) 8215 (56.6) 10,562

NHS1-R 8000 (55.2) 10,306 (71.1) 11,440

NHS2-R 10,000 (69.0) 13,468 (92.9) 8415

OLS-R 12,000 (82.7) 12,310 (84.9) 8699

OHS-R 12,000 (82.7) 14,453 (99.7) 9907


In NHS1-R, no damage was observed during the test. During the last run, a sound of steel rupture was heard. After removal of the jacket and some concrete, two broken longitudinal bars were found at the column base (Fig. 5). Removal of the jacket and concrete at the column top did not reveal any ruptured bars.

In NHS2-R, no damage was observed during the test until a drift ratio of 9.94%. The jacket ruptured over an approxi-mately 0.25 in. (6 mm) area at the column base at this drift ratio. This rupture was extended during the subsequent run, which was the last one (Fig. 5). The maximum measured drift ratio was 13.3% during this run. During the last run, the CFRP rupture was accompanied by a sound of steel rupture. After removal of the jacket and some concrete, two broken longitudinal bars were found at the column base. No bar fractures were noted at the top plastic hinge after removal of the jacket and concrete in NHS2-R.

No damage was observed in OLS-R and OHS-R during the tests. During the last run, the sound of steel rupture was heard. After removal of the jacket and some concrete, a broken longitudinal bar was found in the primary tension side of each column base. Removal of the jacket and concrete at the top of OHS-R did not reveal any ruptured bars.

The CFRP jackets were extensively instrumented with strain gauges in the plastic hinge regions. The jacket ruptured in Bent-2R and NHS2-R. The maximum measured strain in the NHS2-R jacket was 9410 me prior to the failure. The measured jacket strain capacity in the coupon test (Table 3) was smaller due to strain concentrations at the grips of the test machine. In NHS1-R, the maximum measured jacket

Fig. 5—Repaired columns after shake-table test.

strain was 4932 me. The peak strain was developed at the top plastic hinge on the compressive side of the column, where the role of the CFRP jacket was to provide confine-ment. The peak strain was comparable with the design strain of 4000 me. In OLS-R, a maximum jacket strain of 3597 me was developed at the column base on the primary compres-sion side. This strain was smaller than the design strain of 4000 me. In OLS-R, the maximum measured strain due to the clamping force in the lap splice was 1191 me, which was smaller than the design strain of 1500 me. In OHS-R, the maximum recorded CFRP strain was 3811 me at the column base. The peak strain was on the side of the column where strains due to column shear were developed and was smaller than the design strain of 4000 me.

In the substandard repaired columns, the longitudinal bar strains were measured along the lap splice. Using the approach discussed previously, it was concluded that no slip-page occurred in the lap splices based on the measured strains. This indicates that the jacket provided sufficient confinement to prevent splice failure, even under large deformations.

The yield drift ratio, ultimate drift ratio, and lateral load strength of the repaired columns were determined using idealized elasto-plastic envelope curves and are listed in Table 4. The data show that the confinement and lateral load strength of the substandard columns were upgraded effec-tively because the repaired columns underwent a reasonable plastic deformation before failure.

EVALUATION OF REPAIR PERFORMANCEThe force-displacement response of the original and

repaired column models were used to evaluate the repair performance. The measured force-displacement envelopes of all column models are shown in Fig. 3(d) to (h). It should be noted that the end points in the original models do not indicate failure because these columns were not tested to failure. The ultimate points were estimated using a method described in the following sections (displacement capacity index) and marked on the graphs. The envelopes indicate that the strength and displacement capacity of the columns were fully restored and the stiffness of the models was not restored by the repair. The lower stiffness of the repaired

Table 4—Measured responses of repaired columns

Model Yield drift, % Ultimate drift, % Strength, kips (kN)

Bent-2R 1.52 13.11 33.5 (149)

NHS1-R 4.16 13.10 90.3 (402)

NHS2-R 2.56 13.31 85.7 (381)

OLS-R 1.80 5.64 26.6 (118)

OHS-R 1.29 4.57 63.8 (284)


columns is attributed to the residual plastic strains in longi-tudinal bars and core concrete degradation.

To quantify the comparison between the original and repaired models, three nondimensionalized response indexes were developed in terms of strength, stiffness, and displace-ment capacity. The response indexes reveal whether the residual strength, stiffness, and displacement capacity in the damaged column models were restored by the repair. Gener-ally, the residual strength, stiffness, and displacement capacity are smaller than those of the original column due to damage.

Strength indexThe column strength was defined as the plastic lateral load

capacity of the column (Fp) that was determined using the idealized elasto-plastic force-displacement curves. The ratio between the measured strength of the repaired column and the original column was defined as the strength index. This index is shown as Is and is calculated as follows

ps

p

FI

F′

= (1)

where Fp′ and Fp are the lateral strengths of the repaired and original columns, respectively. A strength index equal to or greater than 1 indicates that the column strength was fully restored through the repair. The strength index is plotted in Fig. 6 for all columns. It can be seen that the strength index is greater than 1 for all repaired columns, thus indicating that the repairs were successful. Due to insufficient transverse steel, the shear strength of OHS was significantly lower than its plastic flexural capacity. After repair, the shear strength was increased and OHS-R reached the plastic lateral load capacity. Consequently, a considerably high-strength index is observed for this column.

Service stiffness indexThe serviceability of a structure is addressed based on the

elastic stiffness of the structure. The ratio between the elastic stiffness of the repaired column and the original column was defined as the service stiffness index. This index is shown as Iss and is calculated as follows

ssKIK

′= (2)

where K′ and K are the elastic stiffnesses of the repaired and original columns, respectively. The elastic stiffness of the columns was defined as the initial slope in the idealized elasto-plastic force-displacement relationship. Indexes equal to or greater than 1 indicate that the column service stiffness was fully restored by the repair. The index was calculated for all column models and is plotted in Fig. 6. The plots show that all of the indexes are smaller than 1, meaning that the stiffness of the repaired columns was smaller than that of the original columns due to the reasons discussed previously. In addition, the fact that the epoxy injection of the cracks could not fill the relatively thin cracks in the original column led to stiffness degradation of concrete.

The service stiffness index in the substandard columns was higher than that of the standard columns because the damage level in the original substandard columns was lower than that of the standard columns. Therefore, the stiffness

Fig. 6—Nondimensionalized response indexes.

deterioration in the original substandard columns was less significant—particularly in OHS, where the maximum damage state in the original column was DS-2.

Among the standard columns, NHS1 had the smallest service stiffness index because the repair mortar was not of high quality and its consolidation method was less effective in NHS1 than those of other columns. As discussed previously, the water-cement ratio (w/c) in the repair mortar of NHS1 was reduced due to the relatively low strength of the mortar, and the mortar was placed by hand because of its low workability. Furthermore, the mortar was consolidated by thumb pressure. In Bent-2 and NHS2, due to the high quality of the repair mortar, a fluid mortar was prepared and cast into the mold around the damaged zone. This was followed by consolida-tion using a small vibrator, which was more effective.

Displacement capacity indexThe ratio between the measured displacement capacity

of the repaired and original columns was defined as the displacement capacity index. This index is shown as Id and is calculated as follows

cd

c

ID′

=D

(3)

where Dc′ and Dc are the displacement capacities of the repaired and original models, respectively. The original columns were tested on the shake table up to the highest target repairable damage state, which does not consti-tute failure. This damage state is referred to as “imminent failure.” Therefore, the ultimate displacement capacity for the original columns needed to be estimated before Id could be found. In a previous study (Vosooghi and Saiidi 2010b), it was shown that at a damage state of “imminent failure,” standard low-shear and high-shear columns reach 0.74 and 0.85 of their plastic displacement capacity, respectively. Consequently, the ultimate displacement of Bent-2 was estimated by increasing the maximum measured plastic displacement by 35% and that of NHS1 and NHS2 was esti-mated by increasing the maximum measured displacement by 18%. Plastic displacement was calculated based on ideal-ized elasto-plastic curves.

Indexes equal to or greater than 1 indicate that the column displacement capacity was fully restored by the repair. As mentioned previously, the substandard columns were


repaired and retrofitted simultaneously. The objective of the retrofit was to satisfy the current seismic design codes. Therefore, instead of using the displacement capacity of the original columns (Dc), the displacement corresponding to a target displacement ductility of 5 was used to calculate the displacement capacity index for the substandard columns.

The displacement capacity indexes for all column models are plotted in Fig. 6. It can be seen that the index was close to 1 in Bent-2 and OLS and was greater than 1 in the remaining column models. This indicates that the displacement capacity of the column models was fully restored by the repair.

Figure 6 shows that the indexes of NHS2 were generally higher than those of NHS1, even though the number of CFRP layers in NHS2 was lower. The results demonstrate that the repair procedure in terms of quality and the application method of the repair mortar has a significant effect on the perfor-mance of the repaired column. It is recommended that only high-early-strength, low-shrinkage grout be used in repair. The improved performance of NHS2-R clearly suggests that the relatively high number of CFRP layers in NHS1-R was unnecessary and counting on 50% of the spiral shear-resisting force in NHS2-R was a reasonable assumption.

Generally, even though the repair process was done rapidly and was treated as “emergency” repair with the implication that it was a temporary measure, it can be treated as a perma-nent repair as long as the stiffness of the repaired columns is sufficient under nonseismic loads.

CONCLUSIONSThe following conclusions were drawn based on the

results presented in this paper:• The proposed accelerated curing method for the CFRP

jacket was effective and reduced the required repair time significantly.

• The proposed rapid repair procedure using CFRP for earthquake-damaged standard RC bridge columns to the highest damage state with no bar rupture was effec-tive in restoring the shear strength and displacement ductility capacity.

• The repair procedure in terms of quality and the applica-tion method of the repair mortar had a significant effect on the performance of the repaired columns. High-early-strength, low-shrinkage grout should be used in repair.

• Counting on 50% of the shear strength of transverse steel in the high-shear columns was a reasonable assumption in CFRP jacket design and led to a significant reduction in the required jacket thickness.

• The proposed rapid repair procedure using CFRP for earthquake-damaged substandard RC bridge columns to the highest damage state with no shear and/or splice failure was effective in upgrading the shear strength and displace-ment ductility capacity and inhibiting splice failure.

• Due to stiffness degradation of the steel and the concrete during the original model tests and uninjected micro-cracks, the stiffness of the columns could not be fully restored by the repair.

• Even though the repair process was done rapidly and was treated as “emergency” repair with the implication that it was a temporary measure, it can be treated as a permanent repair as long as the stiffness of the repaired columns is sufficient under nonseismic loads.

ACKNOWLEDGMENTSThis research project was funded by the California Department of Trans-

portation (Caltrans) through Grant No. 59A0543. The conclusions and

recommendations are those of the authors and do not necessarily present the views of the sponsor. The authors wish to thank S. El-Azazy for his helpful comments and support and J. Gutierrez for his instrumental role in the appli-cation of repair work and his close interaction. The authors also wish to thank S. Arnold of the Fyfe Company for arranging for material donation.


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Title no. 110-S12


ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2011-070 received March 8, 2011, and reviewed under Institute


Cyclic Loading Test for Reinforced-Concrete-Emulated Beam-Column Connection of Precast Concrete Moment Frameby Hyeong-Ju Im, Hong-Gun Park, and Tae-Sung Eom

An experimental study was performed to investigate the earthquake resistance of beam-column connections developed for a precast concrete (PC) moment frame. In the moment frame, PC is used for the columns and U-shaped beam shells, and cast-in-place concrete is used for the beam-column joint and the beam core. Six full-scale cruciform beam-column connections, including a conventional reinforced concrete (RC) connection, were tested under cyclic loading. The test parameters were the reinforcement ratio of the beams and the interface details between the PC column and the PC beam shell. The test results showed that regardless of the test parameters, the RC-emulated beam-column connections exhibited good deformation capacities, which were comparable to that of the conventional RC connection. Because of the diagonal shear cracking and reinforcing bar bond slip at the beam-column joint, however, the stiffness and hysteretic energy dissipation significantly decreased. To prevent the degradation of the stiffness and energy dissipation, a strengthening method using headed reinforcing bars was proposed and tested. The performances of the test specimens were evaluated according to the requirements of ACI 374.1-05. On the basis of the test results, design considerations for the beam-column connection were recommended.

Keywords: connections; ductility; energy dissipation; precast concrete; seismic tests; stiffness.

INTRODUCTIONPrecast concrete (PC) structures are popular for a variety

of building facilities because of the advantages of better concrete quality and savings in construction time and cost. Under strong earthquake conditions, however, the beam-column connections in PC moment frames are susceptible to brittle shear failure, which might cause the failure of the overall structure. Thus, to use PC construction for buildings in high and moderate earthquake zones, the earthquake resis-tance of the beam-column connections needs to be enhanced.

To achieve satisfactory seismic performances (that is, stiffness, strength, ductility, and energy dissipation), various beam-column connection methods have been studied.1-6 Figure 1 shows one of the promising methods: a reinforced concrete (RC)-emulated beam-column connection developed for U-shaped PC beam shell construction. The PC moment frame is constructed by erecting PC columns; seating the PC beam shells on the cover concrete of the PC column; placing flexural reinforcements inside the PC beam shell through the beam-column joint; and then pouring concrete to integrate the beam, the column, and the joint. The PC beam shell is used as a formwork that resists construction load. Park and Bull4 tested exterior beam-column connections, as shown in Fig. 1(a). The beam-column connection showed good ductility and energy dissipation. As also shown in Fig. 1(a), the PC beam shell was not integrated with the cast-in-place concrete core. Thus, only the concrete core was considered for the flexural capacity of the beam.

Kim et al.6 tested a cruciform beam-column connec-tion. The details differed from those tested by Park and Bull4 in three aspects (Fig. 1(b)). First, multi-story, one-piece PC columns without concrete at the beam-column

Fig. 1—Beam-column connections incorporated with PC beam shell construction.


Hyeong-Ju Im is a PhD Candidate in the Department of Architecture & Architectural Engineering at Seoul National University, Seoul, South Korea. He received his BE in architectural engineering from Hanyang University, Seoul, South Korea, and his MS in architectural engineering from Seoul National University.

ACI member Hong-Gun Park is a Professor in the Department of Architecture & Archi-tectural Engineering at Seoul National University. He received his BE and MS in archi-tectural engineering from Seoul National University and his PhD in civil engineering from the University of Texas at Austin, Austin, TX. His research interests include the inelastic analysis and seismic design of reinforced concrete and composite structures.

Tae-Sung Eom is an Assistant Professor in the Department of Architecture at the Catholic University of Daegu, Gyeongsan, South Korea. He received his BE, MS, and PhD in architectural engineering from Seoul National University. His research inter-ests include inelastic analysis and the seismic design of reinforced concrete structures.

joint were used to reduce the overall erection time during construction. Second, instead of using hooked bars, straight reinforcing bars were placed inside the PC beam shell to avoid reinforcing bar congestion at the interior beam-column joint. Third, the stirrups of the PC beam shell were anchored to the cast-in-place concrete core to enhance the structural integrity of the beam. Thus, the full beam depth, including the beam shell, can be used for the negative moment capacity of the beam. Although the details were developed to enhance the constructibility and structural performance of the connection, the test results showed that in comparison with the conventional RC beam-column connection, the PC beam-column connection suffered severe diagonal shear cracking, and the hysteretic energy dissipa-tion capacity significantly decreased. In this study, a cyclic loading test was performed for full-scale beam-column connections developed for the PC beam shell construction shown in Fig. 1(b). From the test results, the failure mode and structural performances (strength, stiffness, energy dissipation, and deformability) of the specimens were eval-uated. On the basis of the results, the causes of the capacity degradation were clarified, and design considerations for the beam-column connection were recommended.

RESEARCH SIGNIFICANCERecently, construction methods using U-beam shells and/

or multi-story, one-piece PC columns have become popular in several countries. Because the beam-column connection is integrated by using cast-in-place concrete, its earthquake resistance is intended to be equivalent to that of the conven-tional RC beam-column connection. However, the test results showed that under cyclic loading, the stiffness and energy dissipation of the RC-emulated connection using the U-beam shell significantly decreased. The capacity degrada-tion was attributed to the reduced beam-column joint depth and the increased joint shear force, which were caused by the use of the PC beam shell. To improve the structural capacity of the RC-emulated connection, a strengthening method using headed bars was proposed and its validity was verified by testing.

TEST PROGRAMFull-scale cruciform beam-column connections were tested:

five beam-column connections (Specimens SP1 to SP5) for the PC beam shell construction and a conventional RC beam-column connection (Specimen CP). The test specimens were designed according to the strong-column, weak-beam concept, satisfying the requirements of ACI 550.1R-097 and ACI 318-08.8 The properties and test parameters of the speci-mens are presented in Table 1.

The configurations, dimensions, and reinforcing bar details of Specimen SP1 are shown in Fig. 2(a). The net height of the column from the bottom hinge to the lateral loading point was h = 2700 mm (8.9 ft). The beam length between the two vertical supports was l = 4762 mm (15.6 ft). The cross sections were 400 x 700 mm (15.7 x 27.6 in.) for the beam and 600 x 750 mm (23.6 x 29.5 in.) for the column. The reinforcing bar details for the special moment frame (ACI 318-088) were used for the design of the column and beam. The beam core was filled with cast-in-place concrete and was reinforced with four D32 (As = 794 mm2 [1.27 in.2])

Table 1—Properties of test specimens

Specimen Test parameter

Beam reinforcing bars

Seating length of PC shell, mm

Steel angle for cover concrete‡

Beam-column joint

Top bars r*, %

Bottom bars r†, % Hoops rv, % DDR§

Shear demand, kN||

Shear capacity, kN

SP1 PC control4D32# (1.23)

4D32 (1.66)D13# at 120

(0.30)50 × 20.3 2525.4 2879.9**

SP2Steel angle for cover concrete

4D32 (1.23)

4D32 (1.66)D13 at 120

(0.30)50 ° 20.3 2525.4 2879.9**

SP3Increased

seating length4D32 (1.23)

4D32 (1.66)D13 at 120

(0.30)65 × 19.4 2530.2 2746.9**

SP4Increased

beam strength4D35# (1.49)

4D35 (1.99)D13 at 120

(0.30)50 ° 18.6 3017.6 2879.9**

SP5 Headed bars6D25# (1.21)

4D25# (0.78)D13 at 70

(0.63)50 × 26.0 2387.3 2879.9**

CP RC4D32 (1.23)

2D25 and 2D29# (0.89)

D13 at 160 (0.22) — — 23.4 2485.3 3322.9††

*r = As/bd2; d2 is effective depth for negative moment; refer to Fig. 2.†r = As/bd1; d1 is effective depth for positive moment; refer to Fig. 2.‡L-50 x 50 x 4 (mm).§Column-depth-to-bar-diameter ratio (hc – 2s)/db for Specimens SP1 to SP5 and hc/db for Specimen CP (refer to Fig. 7).||Joint shear demand calculated using plastic moments of beams.#D13 (As = 127 mm2; fy = 503 MPa); D25 (As = 507 mm2; fy = 484 MPa; D29 (As = 642 mm2; fy = 514 MPa); D32 (As = 794 mm2; fy = 412 MPa); D35 (As = 957 mm2; fy = 493 MPa).**Joint shear capacity based on reduced depth (1.2√fc′Aj′; Aj′ = bj(hc – 2s); bj is effective joint width; (hc – 2s) is reduced joint depth).††Joint shear capacity specified by ACI 318-088 (1.2√fc′Aj′ based on full depth hc; Aj = bjhc; bj is effective joint width; hc is effective joint depth).Notes: 1 mm = 0.0394 in.; 1 mm2 = 0.00155 in.2; 1 kN = 0.225 kips; 1 kN = 145 psi.


deformed bars at the top and four D32 deformed bars at the bottom. To avoid reinforcing bar congestion, a large-diam-eter bar (D32) was used. The spacing of the D13 hoops at the beam plastic hinge zones was 120 mm (4.72 in. ≈ d1/4; d1 = 480 mm [18.9 in.]; Fig. 2(a)) based on the effective depth d1 (excluding the beam shell) for the positive moment. The seating length of the PC beam shell on the PC column was 50 mm (1.97 in.). Three D25 (As = 507 mm2 [0.79 in.2]) and two D13 (As = 127 mm2 [0.20 in.2]) deformed bars were used at the bottom and top of the PC beam shell, respec-tively (refer to Section A-A in Fig. 2(a)). However, these reinforcing bars were not considered in the calculation of the flexural capacity of the beam because they were not connected to the beam-column joint. At the beam-column joint, six layers of D16 rectangular hoops were used for the requirements of concrete confinement and joint shear resis-tance, as specified by ACI 318-088 (refer to Section C-C in Fig. 2(a)). Table 1 presents the joint shear capacity and demand, which were calculated using the actual strengths of the concrete and reinforcing bars.

Figure 2(b) to (d) shows the details of Specimens SP2 to SP4, respectively. In SP2 and SP3, the dimensions and

reinforcing bar details of the PC column and beam shell were identical to those of SP1. One of the critical issues in this construction method is the early fracture and crushing failure of the interface between the PC column and beam shell, both during and after construction. In particular, the fracture at the corner of the concrete cover during erec-tion and construction may lead to a catastrophic failure of the structure. Thus, in SP2 (Fig. 2(b)), the concrete cover of the column was strengthened with steel angles (L-50 x 50 x 4 [mm] with a length of 400 mm [15.7 in.]). The steel angle was anchored by 2-D10 inclined bars. The bearing strength of the steel angle strengthening predicted by using the shear-friction mechanism was 79 kN (15.74 kips). It should be noted that the steel angle was used to strengthen the concrete cover in the column. The majority of the beam shear force was transferred by the core concrete in the beam. In SP3 (Fig. 2(c)), the seating length of the PC beam shell was increased to 65 mm (2.56 in.) instead of using steel angles. In SP4 (Fig. 2(d)), four D35 (As = 957 mm2 [1.53 in.2]) bars were used at the top and bottom of the beam core to investigate the effect of the increased flexural strength on the connection behavior.

Fig. 2—Configurations and reinforcing bar details of test specimens. (Note: Dimensions in mm [in.].)


In SP5 (Fig. 2(e)), special details were used to avoid severe damage at the beam-column joint. Four headed bars were used to displace the plastic hinge zone toward the beam by strengthening the beam-column joint. To reduce bond slip at the beam-column joint, a smaller-diameter bar—D25 (As = 507 mm2 [0.79 in.2])—was used for the flexural reinforcing bars of the beam: six D25 and four D25 at the top and bottom of the beam core, respectively. The spacing of the hoops at the beam plastic hinge zone was reduced to 70 mm (2.76 in.) to avoid early crushing and shear failure due to the anchorage force of the headed bars (Fig. 2(e)). Other details were the same as those of SP1. The design calculations for the headed bars are presented in the Appendix.*

Figure 2(f) shows the details of the conventional RC beam-column connection, Specimen CP. The configurations and dimensions of the column and beam were identical to those of SP1. Four D32 bars were used at the top of the beam, and two D25 plus two D29 (As = 642 mm2 [1.00 in.2]) bars were used at the bottom of the beam. In comparison with the PC specimens, the area of the bottom bars decreased because a larger beam depth can be used for the positive moment.

The compressive strength of the precast concrete was fc′ = 35.1 MPa (5.09 ksi) for the PC beam shell and fc′ = 47.5 MPa

(6.89 ksi) for the PC column. The compressive strength of the cast-in-place concrete was fc′ = 34.9 MPa (5.06 ksi). The yield and ultimate strengths of the reinforcing bars, fy and fu, are presented in Table 2.

Figure 3(a) shows the test setup. The column was supported by the bottom hinge. The beam ends were vertically supported, allowing lateral movement. Lateral cyclic loading was applied at the top of the column and was controlled by the lateral displacement of the actuator. Figure 3(b) shows the loading history.

*The Appendix is available at www.concrete.org in PDF format as an addendum to the published paper. It is also available in hard copy from ACI headquarters for a fee equal to the cost of reproduction plus handling at the time of the request.

TEST RESULTSLateral load-drift ratio relationships

Figure 4(a) to (f) shows the lateral load-drift ratio relation-ships. The lateral drift ratio was calculated by dividing the net lateral displacement at the loading point by the net length of the column (h = 2700 mm [8.86 ft]). Table 3 summa-rizes the maximum strength Pu, maximum displacement Du (maximum drift ratio du), ductility m (=Du/Dy), yielding displacement Dy, yield stiffness ky, post-yield stiffness kp, and failure mode of the specimens. The yield displacement Dy was defined by using the equal energy principle (refer to Fig. 5(b)). The maximum displacement Du was defined as the post-peak displacement corresponding to 75% of the maximum strength.

In the conventional RC Specimen CP (Fig. 4(a)), flexural yielding of the beam and the maximum strength of the spec-imen occurred at 0.9% and 2.5% drift ratios, respectively. The maximum drift ratio du was 3.3%. Specimen CP showed relatively large energy dissipation.

As shown in Fig. 4(b) to (e), the lateral load-drift ratio relationships of the PC Specimens SP1 to SP4 were similar, regardless of the effects of the test parameters. Flexural yielding of the beam and the maximum strength of the speci-mens occurred at 0.9% to 1.1% drift ratios and 1.5% to 2.5% drift ratios, respectively. After the maximum strength, the load-carrying capacity gradually decreased. The specimens showed low energy dissipation due to severe pinching. Under cyclic loading repeated at a given lateral displacement, the load-carrying capacities at the second and third load cycles were less than that at the first load cycle. The maximum drift ratio du of the specimens was 3.4 to 5.1%.

The ratio between the measured-to-predicted load-carrying capacities for Specimens SP1 to SP4 and CP (Pu/Pn) ranged from 1.03 to 1.14. In particular, SP3 had the lowest Pu/Pn (1.03) because the seating length of the PC beam shell increased to 65 mm (2.56 in.). The load-carrying capacities of the PC Specimens SP1 to SP3 were 5 to 15% less than that of Specimen CP. In SP4, where a larger-diameter bar D35 was used in the beam, the load-carrying capacity was 24.4% greater than that of SP1.

Figure 4(f) shows the behavior of Specimen SP5 strength-ened by headed bars. Specimen SP5 showed relatively large energy dissipation and the overall performance was similar to that of the RC Specimen CP.

Damage patterns and failure modesFigure 6 shows the failure modes of the specimens. In the

RC Specimen CP (Fig. 6(a)), damage was concentrated at

Fig. 3—Test setup and loading history. (Note: 1 kN = 0.225 kips; LVDT is linear variable differential transformer.)

Table 2—Yield and tensile strengths of reinforcing barsBar size, mm D13 D16 D25 D29 D32 D35

Area, mm2 127 199 507 642 794 957

Yield strength, MPa 503 434 463 514 468 493

Tensile strength, MPa 583 585 630 651 599 605

Notes: 1 mm = 0.0394 in.; 1 mm2 = 0.00155 in.2; 1 MPa = 145 psi.


Fig. 4—Lateral load-drift ratio relationships of test specimens. (Note: 1 kN = 0.225 kips.)

Fig. 5—Envelope curves of test specimens. (Note: 1 kN = 0.225 kips.)

Table 3—Summary of test results

Specimen

Load-carrying capacity* Deformation capacity* Stiffness*

Failure modeTest result

Pu, kNPrediction

Pn, kN Pu/Pn

Yield displacement Dy, mm (dy, %)

Maximum displacement Du, mm (du, %)

Ductility m (=Du/Dy) ky, kN/mm |kp/ky|, %

SP1 745.0 650.6 1.14 24.0 (0.89) 98.3 (3.6) 4.09 31.0 5.8 Bond failure†

SP2 722.9 650.6 1.11 24.8 (0.92) 136.9 (5.1) 5.52 29.1 7.9 Bond failure†

SP3 667.8 645.8 1.03 22.7 (0.84) 122.7 (4.5) 5.40 29.4 6.8 Bond failure†

SP4 926.8 810.4 1.14 28.6 (1.06) 118.3 (4.4) 4.13 32.4 5.1 Bond failure†

SP5 696.1 661.1 1.05 25.1 (0.93) 98.9 (3.6) 3.94 27.7 9.1 Flexural failure‡

CP 787.3 690.7 1.14 23.2 (0.86) 90.7 (3.3) 3.91 33.9 — Flexural failure‡

*Results based on positive loading direction.†Bond failure and diagonal shear cracking at beam-column joint followed by crushing failure at beam end.‡Concrete crushing failure at beam plastic hinges.Notes: 1 kN = 0.225 kips; 1 mm = 0.0394 in.; 1 kN/mm = 5.71 kip/in.


the plastic hinges of the two beams, while diagonal shear cracking at the beam-column joint was minimized. Concrete spalling occurred in the overall beam depth. Ultimately, concrete crushing failure occurred at the top and bottom of the critical section.

In the PC Specimens SP1 to SP4 (Fig. 6(b) to (e)), severe diagonal shear cracking occurred at the beam-column joint. Furthermore, the end of the PC beam shell was separated from the beam-column joint by a maximum of 27 mm (1.06 in.) at a 4.0% drift ratio, which indicates that reinforcing bar bond slip occurred at the beam-column joint. Ultimately, the PC specimens failed due to concrete crushing at the top and bottom of the beam end. On the other hand, SP5 (Fig. 6(f)) failed after plastic hinge formation at the beam ends, which was similar to the failure mode of CP.

In SP1 and SP3 without steel angles, fracture of the concrete cover in the column was initiated at a 1.0% drift ratio. On the other hand, in SP2 and SP4, which were strengthened by steel angles at the interface between the column and beam shell, fracture of the concrete cover did not occur until a 3.0% drift ratio. Such delayed fracture of the concrete cover is attributed to the strengthening effect provided by the steel angles. In the test specimens, as shown in Fig. 4, the facture of the concrete cover did not significantly affect the load-carrying capacity and the post-yield behavior because flex-ural damage did not occur in the columns confined by the hoop reinforcing bars. When a significant axial compression force is applied to a column, however, the fracture of the concrete cover may significantly decrease the stiffness and strength of the beam-column connection.

Strains of reinforcing barsFigure 7(a) to (c) shows the measured strains of the beam

reinforcing bars in Specimens SP3, SP5, and CP, respec-tively. In SP5 and CP (Fig. 7(b) and (c)), the reinforcing bars

at the beam plastic hinge zone developed large tensile plastic strains, while the reinforcing bars at the beam-column joint remained elastic. Thus, the majority of the plastic deforma-tion and hysteretic energy dissipation developed at the beam plastic hinge zone. On the other hand, in SP3 (Fig. 7(a)), the plastic strain of the reinforcing bars at the beam was signifi-cantly less than that in SP5 and CP, and the strains did not increase further as the reinforcing bars at the beam-column joint started to yield. This result indicates that the bond slip at the beam-column joint increased with the inelastic lateral displacement of the specimen. The strains measured in SP1, SP2, and SP4 without headed bars were similar to that of SP3.

As shown in Fig. 7(b), the headed bars in SP5 remained elastic during testing. In all test specimens, the hoops at the beam-column joint remained elastic until the maximum displacement of the specimens.

Load-carrying capacity and strength degradationBecause the specimens were designed according to the

strong-column, weak-beam concept, the load-carrying capacity can be calculated by using the beam yielding mechanism. In the PC Specimens SP1 to SP4, the beam was reinforced with extra reinforcing bars in the PC beam shell and longitudinal bars in the beam core. Thus, the crit-ical section of the beam yielding is located at the interface between the PC beam shell and the beam-column joint (refer to Fig. 8). Using the moment capacity of the beam at the critical section, the maximum load-carrying capacity Pn of the specimen can be calculated as follows

( ) ( )2 ( 2 )

bp bn bp bnn

c

P P l M M lP

h h l h s+ +

= =− +

(1)

Fig. 6—Failure modes of test specimens at end of test.


Fig. 7—Measured strains of longitudinal reinforcing bars in beams. (Note: 1 kN = 0.225 kips; 1 mm = 0.0394 in.)

where Pbp (=Mbp/(lb + s)) and Pbn (=Mbn/(lb + s)) are the reaction forces at the beam supports; Mbp and Mbn are the positive and negative moment capacities of the beam at the critical section, respectively; lb is the clear span of the beam

between the column face and the vertical support; s is the seating length of the PC beam shell; and hc is the depth of the column cross section. In the calculation of the nega-tive moment capacity (in the case of top reinforcing bars in


tension), the contribution of the PC beam shell concrete to the compressive zone was considered. On the other hand, for the positive moment capacity (in the case of bottom reinforcing bars in tension), the contribution of the reinforcing bars at the bottom of the beam shell was not considered because they were not connected to the beam-column joint. In the case of Specimen CP, the critical section was located at the column face. In SP5, the critical section was assumed to be located at the end of the headed bars—240 mm (9.45 in.) away from the column face (refer to Fig. 2(e)). Table 3 and Fig. 4 compare the predicted strength Pn with the test result Pu. In the calculation of Pn, the actual material strengths were used. The test result Pu was 3 to 14% greater than the predicted Pn. Such overstrength is attributed to the cyclic strain hardening of the reinforcing bars, which was not considered in the calculation.

Figure 9(a) shows the definition of cyclic strength degra-dation that occurs during cyclic loading repeated at a drift ratio d.9 The cyclic strength ratio ai is defined as the ratio of the load-carrying capacity at the i-th load cycle to the first load cycle capacity: ai = Pi/P1. Figure 9(b) and (c) shows the cyclic strength ratios a2 and a3 of the second and third load cycles, respectively. In Specimen CP, the cyclic strength degradation was negligible (a2 = a3 ≈ 1.0). On the other hand, in the PC Specimens SP1 to SP5, the cyclic strength degradation was greater. The cyclic strength ratios a2 and a3 decreased as the drift ratio d increased. At a 1% drift ratio, the strength ratios a2 and a3 were approxi-mately 0.9. Subsequently, at every 1% increase in the drift ratio, a2 and a3 decreased by 0.75% and 3.6%, respectively.

At a 3.5% story drift ratio, a3 was 0.8 on average; however, a3 = 0.8 was greater than the acceptance criteria specified by ACI 374.1-05,10 where a3 = 0.75 at a 3.5% story drift ratio.

Stiffness and ductilityFigure 5(a) shows the envelope curves of the PC and RC

specimens. After flexural yielding, Specimen CP showed a stable hardening behavior until a 2.5% drift ratio. After the peak strength, sudden strength degradation occurred due to the concrete crushing at the plastic hinge of the beam. On the other hand, in the PC specimens, the load-carrying capacities gradually decreased after the maximum strength (Fig. 5(a) and Fig. 4(b) to (e)). Such post-peak softening behavior was caused by the loss of the bond strength and diagonal shear cracking at the beam-column joint.

In Fig. 5(b), the yield stiffness ky and post-peak stiffness kp can be estimated by using an idealized trilinear curve and the equal energy principle. The envelope curve up to the peak point is idealized as a bilinear curve with zero post-yield stiffness. The yield displacement Dy and the yield stiffness ky are determined so the area enclosed by the idealized bilinear curve is the same as the area enclosed by the envelope curve. In the same way, the post-peak stiffness kp is determined by applying the equal energy principle for the post-peak curve. Table 3 presents the ky and kp values of the specimens. The yield stiffness ky of the PC specimens was 11.7% less than that of Specimen CP. The post-peak stiffness kp was 5.1 to 9.1% of the yield stiffness

approximately 5.1 to 9.1%p yk k =

Table 3 presents the maximum deformation Du and ductility m of the test specimens. The maximum defor-mations of the PC specimens were greater than that of Specimen CP. This is because in addition to the inelastic deformation of the beam, the bond slip and diagonal shear cracking at the beam-column joint contributed to the overall deformation of the specimen. However, the ductility Du/Dy of the PC specimens was equivalent to that of Specimen CP because the yield deformation Dy of CP was smaller.

The acceptance criteria specified by ACI 374.1-0510 requires that in the third load cycle at a story drift ratio of not less than 3.5%, the secant stiffness k3.5 between the –0.35% and +0.35% drift ratios under unloading/reloading should not be less than 0.05 times the initial stiffness ki (refer to Fig. 10). Table 4 presents the initial stiffness ki and the secant stiffness k3.5 of the specimens. The PC specimens, except Specimen SP5, did not satisfy the acceptance criteria.

Fig. 9—Cyclic strength degradation of test specimens.

Fig. 8—Calculation of load-carrying capacity of PC Speci-mens SP1 to SP4.


Energy dissipation capacityFigure 11(a) and (b) compares the energy dissipation per

load cycle and the cumulative energy dissipation of the test specimens, respectively. As shown in Fig. 11(a), the energy dissipation per load cycle is defined as the area enclosed by a load cycle at a given displacement. The energy dissipa-tion of the PC specimens (SP1 to SP4) was only 64% of the energy dissipation of Specimen CP. As shown in Fig. 11(b), the cumulative energy dissipation was also significantly less than that of CP. On the other hand, the energy dissipation of SP5 was equivalent to that of CP.

According to Park and Eom11 and Eom and Park,12 the energy dissipation per load cycle of RC members is devel-oped mainly due to the flexural reinforcing bars experi-encing large plastic strains. The evidence for this can be seen in Fig. 7. The energy dissipation of the reinforcing bars in SP5 and CP (Fig. 7 (b) and (c)) was significantly greater than that of PC Specimen SP3 (Fig. 7(a)).

The acceptance criteria specified by ACI 374.1-0510 requires that in the third load cycle at a story drift ratio of not less than 3.5%, the energy dissipation ratio b should not be less than 0.125. Herein, b is defined as the ratio of the energy dissipation EII for the third load cycle to the energy dissipation EII0

corresponding to an idealized elasto-plastic behavior (refer to Fig. 10). Table 4 presents the b values. The PC specimens satisfied the acceptance criteria, although they showed low energy dissipation.

DISCUSSIONEffects of test parameters

In Specimens SP2 and SP4, the steel angles prevented early fracture of the concrete cover in the columns by increasing the bearing capacity. However, when compared with SP1 and SP3 without the steel angles, the fracture of the concrete cover did not significantly affect the load-carrying capacity and the post-yield behavior because flexural damage did not occur in the columns. When a significant axial compression force is applied to a column, however, early fracture of the concrete cover may significantly decrease the stiffness and strength of the beam-column connection.

In SP3, where the seating length of the PC beam shell was increased to 65 mm (2.56 in.), the maximum strength was 10% less than that of SP1 (with a seating length of 50 mm [1.97 in.]) and the diagonal shear crack damage at the beam-column joint also increased. Although the increased seating length s contributes to the stability of the structure under construction, it shortens the depth of the beam-column joint

Fig. 10—Acceptance criteria for stiffness and energy dissi-pation specified by ACI 374.1-05.10

Fig. 11—Hysteretic energy dissipation of test specimens. (Note: 1 kN·m = 0.738 k-ft.)

Table 4—Evaluation of seismic performance of specimens by ACI 374.1-0510

Specimen SP1 SP2 SP3 SP4 SP5 CP

Loading direction + – + – + – + – + – + –

Stiffness, kN/mm

ki* 32.2 32.0 32.6 33.6 30.6 31.7 37.3 36.0 37.8 32.1 40.5 41.9

0.05ki 1.61 1.60 1.63 1.68 1.53 1.59 1.87 1.80 1.89 1.61 2.03 2.10

k3.5* at 3.5% 0.47 0.57 1.17 0.93 0.55 0.45 1.35 1.39 2.91 2.64 2.81 3.17

k3.5/0.05ki 0.29 0.36 0.72 0.55 0.36 0.28 0.72 0.77 1.54 1.64 1.38 1.51

Energy dissipation per load cycle, kN·mm

Elasto-plastic energy EII0* 182,770 174,314 177,519 246,816 151,955 163,592

Actual energy EII* 31,342 39,821 27,215 49,103 48,494 57,258

Relative energy dissipation ratio b = EII/EII0

0.171 0.228 0.153 0.199 0.319 0.350

*Refer to Fig. 10.Notes: 1 kN/mm = 5.71 kip/in.; 1 kN·mm = 0.00887 kip·in.


to (hc – 2s) (C in Fig. 12). As a result, reinforcing bar bond slip and diagonal shear cracking increased at the beam-column joint (Fig. 6(d)).

In SP4, where a larger-diameter bar D35 was used in the beam, the load-carrying capacity was 24.4% greater than that of SP1. The load-drift ratio relationship was similar to that of SP1, although severe diagonal shear cracking occurred at the beam-column joint.

In SP5, the use of the headed reinforcing bars helped in avoiding the concentration of ductility demand at the interface between the PC beam and the cast-in-place concrete joint by relocating the plastic hinge at the beam end. Therefore, bond slip and diagonal shear cracking at the beam-column joint and fracture of the concrete cover in the column were successfully restrained. The design calculations and recommendations for the headed bars are presented in the Appendix.13,14

Causes of bond slip and diagonal shear cracking at beam-column joint

1. The seating length s (=50 or 65 mm [1.97 or 2.56 in.]) of the PC beam shell shortened the beam-column joint depth (hc – 2s) (C in Fig. 12). The reduced joint depth decreased the shear strength of the beam-column joint. Table 1 shows the shear capacity of the PC specimens based on the reduced joint depth (hc – 2s), which was 14% less than the shear capacity of Specimen CP based on the full depth hc. If the PC beam shell is supported on a temporary support rather than on the column during construction, the effective depth of the beam-column joint may not be decreased.

2. The reduced joint depth (C) also decreased the column depth-to-bar diameter ratio (refer to DDR in Table 1). To prevent excessive bond slip, ACI 352R-0215 and ACI 318-08,8 respectively, require at least (20fy/420 ≥ 20) and 20 for the ratio. In the PC specimens using D32 bars (SP1 to SP4), the ratio (hc – 2s)/db ranged from 18.6 to 20.3, which was 13 to 21% less than the ratio hc/db = 23.4 of CP.

3. Because of the less-effective depth d1 (H in Fig. 12), a greater reinforcing bar force was required to achieve the same positive moment capacity as that of the conventional RC beam, which required a greater shear force at the beam-column joint.

4. The reinforcing bars (3D25 and 2D13; refer to Fig. 2(a)) in the PC beam shell (E in Fig. 12) increased the flexural strength of the beam, although this was not originally intended (the reinforcing bars in the beam shell were used to

resist construction load and prevent concrete cracking under service loading). Because of the discontinuity of the beam shell, yielding of the beam flexural reinforcing bars was concentrated at the interface between the joint and the beam (F in Fig. 12), and it propagated toward the beam-column joint rather than toward the beam having a greater flexural capacity (G in Fig. 12). The yielding of the reinforcing bars inside the beam-column joint causes the bond slip. For this reason, bond slip can occur at the beam-column joint even when the column-depth-to-bar-diameter ratio is greater than the requirements of ACI 352R-0215 and ACI 318-08.8

Ultimate failure mechanism of PC specimensAs discussed in the section on strains of reinforcing bars,

the beam reinforcing bars at the joints of the RC-emulated connections (Specimens SP1 to SP4) were subjected to significant plastic strains greater than the yield strain of the reinforcing bars (refer to Fig. 7(a)). This result indicates that significant bond slip occurred in the beam-column joints. Because of such bond slip, the tension reinforcing bars in the beam were anchored to the compression zone (I in Fig. 12) of the other beam on the opposite side. Therefore, the compres-sion zone at the beam end was subjected to the combined compressive and anchorage forces (J in Fig. 12). Ultimately, the beam-column connection failed due to concrete crushing in the compression zone of the beam end (refer to Fig. 6).

CONCLUSIONSThe results of these tests showed that the maximum defor-

mation and ductility of the PC specimens were compa-rable to those of the conventional RC specimen. In the PC specimens, however, reinforcing bar bond slip and diagonal shear cracking occurred at the beam-column joint, which significantly decreased the stiffness and energy dissipation capacity. The capacity degradation was attributed to the use of the PC beam shell, which reduced the beam-column joint depth and increased the joint shear force. On the basis of the test results, the following design considerations are recommended for the earthquake design of the RC-emulated beam-column connections.

1. It is recommended that the seating length and thickness of the PC beam shell be decreased as much as possible to increase the effective depth of the beam-column joint and the cross-sectional area of the beam core.

2. The shear strength of the beam-column joint, the devel-opment length of the beam reinforcing bars, and the strength of the column at the joint should be designed using the reduced column depth at the beam-column joint.

3. Under repeated cyclic loading, the load-carrying capacity continues to decrease as the number of load cycles increases. Therefore, when a moment frame is designed against a strong earthquake with a long duration, it is recom-mended that the design strength of the beam-column connec-tion be reduced to 80% of the calculated nominal strength.

4. The PC beam shells are not completely integrated with the cast-in-place beam-column joint. Thus, the yield stiff-ness of the beam-column connection needs to be decreased by 10% in comparison with the comparable conventional RC connection.

5. When the energy dissipation capacity is considered in the design or evaluation of a structure, the energy dissipation capacity should be at least 36% less than that of the equiva-lent conventional RC structure.

Fig. 12—Causes of bond slip and diagonal shear cracking.


6. When the column is subjected to high compression and/or flexural yielding, the interface between the PC column and the PC beam shell may need to be strengthened with an appropriate method, such as steel angles, to prevent early fracture of the concrete cover in the column.

7. The use of a strengthening method such as headed bars is recommended to achieve the structural performance equiv-alent to that of the conventional RC connection and satisfy the requirements of ACI 374.1-0510 or the special moment frame in ACI 318-08.8 Further studies on the possible inter-face connection between the beam shell and the cast-in-place concrete at the beam-column joint are required.

ACKNOWLEDGMENTSThis research was financially supported by the Ministry of Construction

and Transportation of Korea (05 R&D D02-01); the authors are grateful to the authorities for their support. The authors would also like to thank R. E. Klingner at the University of Texas at Austin for his valuable advice.

NOTATIONAj = effective shear area at jointb = width of cross sectionbj = effective joint width for Aj

db = diameter of reinforcing bar at jointd1, d2 = effective depths for positive and negative momentsEII = energy dissipation corresponding to third load cycle

at displacementEII0 = energy dissipation corresponding to idealized elasto-

plastic behaviorfc′ = compressive strength of concretefu, fy = tensile and yield strengths of reinforcing barh, l = height and length of beam-column connectionhc = depth of cross section of columnki = initial stiffnessky, kp = yield and post-yield stiffness of beam-column connectionlb = clear span of beam between column face and beam supportMbn, Mbp = negative and positive moment capacities of beam at crit-

ical sectionPbp, Pbn = reaction forces at beam supportsPn, Pu = calculated and measured load-carrying capacity of beam-

column connections = seating length of PC beam shellai = coefficient of cyclic strength degradationb = energy dissipation ratio (=EII/EII0)Du, Dy = maximum and yield lateral displacements of beam-

column connectiond = lateral drift ratiodu = maximum lateral drift ratio

m = ductility ratio (=Du/Dy)r = reinforcement ratio

REFERENCES1. Bhatt, P., and Kirk, D. W., “Test on an Improved Beam Column

Connection for Precast Concrete,” ACI JOURNAL, Proceedings V. 82, No. 6, Nov.-Dec. 1985, pp. 834-843.

2. Mast, R. F., “A Precast Concrete Frame System for Seismic Zone Four,” PCI Journal, V. 37, No. 1, 1992, pp. 50-64.

3. Priestley, M. J. N., and MacRae, G. A., “Seismic Tests of Precast Beam-to-Column Joint Subassemblages with Unbonded Tendons,” PCI Journal, V. 41, No. 1, 1996, pp. 64-81.

4. Park, R., and Bull, D. K., “Seismic Resistance of Frames Incorporating Precast Prestressed Concrete Beam Shells,” PCI Journal, V. 31, No. 4, 1986, pp. 54-93.

5. Englekirk, R. E., “Development and Testing of a Ductile Connector for Assembling Precast Concrete Beams and Columns,” PCI Journal, V. 40, No. 2, 1995, pp. 36-51.

6. Kim, S. H.; Moon, J. H.; and Lee, L. H., “An Experimental Study of the Structural Behavior on the Precast Concrete Beam-Column Interior Joint with Splice Type Reinforcing Bars,” Journal of the Architectural Insti-tute of Korea, V. 20, No. 1, 2004, pp. 53-61.

7. Joint ACI-ASCE Committee 550, “Guide to Emulating Cast-in-Place Detailing for Seismic Design of Precast Concrete Structures (ACI 550.1R-09),” American Concrete Institute, Farmington Hills, MI, 2009, 17 pp.


9. Applied Technology Council (ATC), “Improvement of Nonlinear Static Seismic Analysis Procedures,” FEMA 440 Report, Federal Emer-gency Management Agency, Washington, DC, 2005, 392 pp.

10. ACI Committee 374, “Acceptance Criteria for Moment Frames Based on Structural Testing (ACI 374.1-05) and Commentary,” American Concrete Institute, Farmington Hills, MI, 2005, 9 pp.

11. Park, H. G., and Eom, T. S., “A Simplified Method for Estimating the Amount of Energy Dissipated by Flexure-Dominated Reinforced Concrete Members for Moderate Cyclic Deformations,” Earthquake Spectra, V. 22, No. 3, 2006, pp. 1351-1363.

12. Eom, T. S., and Park, H. G., “Evaluation of Energy Dissipation of Slender Reinforced Concrete Members and Its Applications,” Engineering Structures, V. 32, No. 9, 2010, pp. 2884-2893.

13. Abdel-Fattah, B., and Wight, J. K., “Study of Moving Beam Plastic Hinging Zones for Earthquake-Resistant Design of R/C Building,” ACI Structural Journal, V. 84, No. 1, Jan.-Feb. 1987, pp. 31-39.

14. Chutarat, N., and Aboutaha, R. S., “Cyclic Response of Exterior Reinforced Concrete Beam-Column Joints Reinforced with Headed Bars—Experimental Investigation,” ACI Structural Journal, V. 100, No. 2, Mar.-Apr. 2003, pp. 259-264.

15. Joint ACI-ASCE Committee 352, “Recommendations for Design of Beam-Column Connections in Monolithic Reinforced Concrete Structures (ACI 352R-02) (Reapproved 2010),” American Concrete Institute, Farm-ington Hills, MI, 2002, 38 pp.


NOTES:


Title no. 110-S13


ACI Structural Journal, V. 110, No. 1, January-February 2013.MS No. S-2011-085.R1 received November 29, 2011, and reviewed under Institute


Unified Calculation Method for Symmetrically Reinforced Concrete Section Subjected to Combined Loadingby Liang Huang, Yiqiu Lu, and Chuxian Shi

This study introduces a unified formula for calculating the ultimate state of reinforced concrete (RC) members under combined loading conditions, including axial compression, shear, bending, and torsion. The proposed formula is deduced by using bending-torsion interaction as a bridge connecting shear-torsion and bending-axial compression interaction. Interactions of various types are directly considered in the proposed formula. Their strength formulas are determined by comparison of experimental data. Finally, the appli-cability of the proposed formula is illustrated using a calculation example of a planar frame. It is concluded that that the unified formula can be used as unified failure criteria of RC components to determine the safety of the members under various loading states, and can be used to calculate residual capacity of components under various types of combined loading.

Keywords: combined loading; interaction; reinforced concrete; strength; unified formula.

INTRODUCTIONStructural concrete members such as frames on the perim-

eter of a building, curved bridge decks, and multideck bridge structures, are subjected to complex loading combinations. Consideration of the interaction of various loadings is required, as a complex stress-state can result when they are applied simultaneously to a structure. In the current design provisions,1-3 simple rules are established to equip structural designers with simplified methods for considering these complex interactions. These rules, however, are incom-plete because they do not cover all interactions. Interactions involving three or four force types are not directly consid-ered (Table 1). For example, when members are subjected to combined loading of bending, shear, torsion, and axial compression, the bending-axial compression, shear-torsion, torsion-axial compression, shear-axial compression, torsion-bending, and shear-bending interactions are considered separately, but the overall bending-shear-torsion-axial compression interaction is not directly examined. Given that combined loading is an overall force state in elements, and that each type of force will interact with all others, the overall interaction should be considered.

Complex interaction problems have been studied using various approaches during the last few decades. Important studies on this subject area were published in the 1960s and 1970s. Researchers have proposed several interac-tion curves and surfaces based on semi-empirical methods, including torsion-shear interaction curve,4-6 torsion-bending interaction curve,7-9 and torsion-bending-shear interac-tion surface.6,7,10 These tests of reinforced concrete (RC) members under combined loading conditions indicated that various kinds of forces interact with one another. These previous studies make an important contribution by proposing interaction curves and formulas that are easily implemented into design, and these formulas have been veri-fied by experimental data. In recent years, more experiments

have been conducted and theoretical models have been proposed to deal with interactions of axial force, bending moment, shear force, and torsional moment. Vecchio and Collins11 and Bentz12 proposed sectional models suitable for analyzing prestressed and reinforced concrete sections under axial force, bending moment, and shear force. Guner and Vecchio13 developed a computer-based analytical procedure, VecTor5, for the nonlinear analysis of frame-related struc-tures. The procedure is capable of representing shear-related effects coupled with flexural and axial behaviors. Cocchi and Cappello14 employed the concept of the discrete truss model to analyze RC sections subjected to bending moment, torsion, and axial force. Further research of the formulation as well as a computer implementation for the case of biaxial bending was made later by Cocchi and Volpi.15 Saritas and Filippou16 and Petrangeli et al.17 developed frame models incorporating the interaction effect of bending moment and shear force. These research results are important in inves-tigating complex interaction problems. The aforementioned studies, however, focused primarily on relatively common load combinations. In other research, axial force, bending moment, shear force, and torsional moment are completely considered. For example, Vecchio and Selby18 proposed a three-dimensional (3-D) nonlinear finite element program named SPARCS based on the equations of the modified compression field theory (MCFT), which is a very powerful tool for studying the behavior of reinforced and prestressed concrete structures subjected to bending, shear, torsion, and axial load. Rahal and Collins19 employed the MCFT model to study the shear and torsion interaction as well as a full inter-action between normal and tangential forces within an RC

Table 1—Types of interaction

LevelNumbers of types Types

Double interactions 2

4C = 6

Axial compression-bending, shear-torsion, bending-torsion, shear-axial compression, torsion-axial compression, bending-shear

Three interactions 3

4C = 4

Bending-shear-axial compression, bending-torsion-axial compression, bending-shear-torsion, shear-torsion-axial compression

Four interactions

4

4C = 1 Bending-shear-torsion-axial compression


Liang Huang is an Associate Professor of structural engineering at Hunan University, Changsha, China, where he received his PhD. His research interests include earth-quake resistance design of structures, energy-saving of masonry structures, and design of reinforced concrete structures.

Yiqiu Lu is an MS Candidate in the College of Civil Engineering at Hunan University, where he received his BS. His research interests include earthquake resistance design of structures and design of reinforced concrete structures.

Chuxian Shi is a Professor of civil engineering at Hunan University, where he received his BS. His research interests include earthquake resistance design of structures, energy-saving of masonry structures, and design of reinforced concrete structures.

and prestressed section. The analytical model COMBINED is capable of predicting the behavior of RC sections subjected to biaxial bending, biaxial shear, torsion, and axial load. A general 3-D model based on the displacement formula-tion for the analysis of reinforced and prestressed concrete frame elements was developed by Gregori et al.20 Arbitrary cross section geometries and combined loading conditions, including axial force, biaxial bending moment, torsion, and biaxial shear forces are taken into account. Additionally, Valipour and Foster21 developed an element for nonlinear analysis of RC framed structures subjected to torsion using a forced-based formulation taking into account of the inter-action between tangential and normal forces. Although the loading combinations considered in these studies were complete, there was no simplified interaction formula that can use various loading combinations and can accurately predict the test data. Therefore, establishing a formula that directly considers the overall interaction of load combina-tions and can accurately predict the test data is imperative.

It is noteworthy to mention Ewida and Mcmullen’s work,22 which proposed that the ultimate capacity of the section for axial force, bending moment, shear, and torsion can satisfy

N

N

M

M

V

V

T

T05

06

07

0

1 2 3 4

1

+

+

+

=h h h h

h h h (1)

where h1 to h7 are parameters determined from calibration with experimental observations. The parameters, however, were not determined in this work.

This study assumes the function form of the unified formula that directly considers the overall interactions between various types of forces, and it can be expressed as a more general expression as follows

1 2 3

0 0 0 0

, , , 1N M V TfN M V T

a a a =

(2)

Using a bending-torsion interaction to establish a link between shear-torsion and bending-axial compression inter-actions, the unified function relationship for symmetri-cally reinforced sections of N, M, V, and T is determined. The 11 types of interaction equations among bending, shear, torsion, and axial compression shown in Table 1 can be obtained from this unified formula. The three parameters a1, a2, and a3 of various interaction equations are determined from the calibration with a wide range of experimental obser-vations. Finally, the applicability of the proposed formula is illustrated using a calculation example of a planar frame.

RESEARCH SIGNIFICANCEA unified formula capable of predicting various inter-

actions of symmetrically reinforced sections subjected to combined loading has been developed. The formula directly considers the overall interactions of bending, shear, torsion, and axial compression. Equations that represent the inter-action of various loading combinations between bending, shear, torsion, and axial compression were developed based on a calibration with the experimental data. Moreover, the unified formula can be used as unified failure criteria of RC components in ultimate state to determine the safety of the members under various loading states and can also be used to calculate residual capacity of components.

DERIVATION OF UNIFIED FORMULAErsoy and Ferguson4 proposed a quarter-circle for the

shear-torsion interaction of concrete beams without stir-rups based on their analysis of experimental data. The equa-tion, which is based on plastic theory, can be expressed in a dimensionless form

2 2

0 0

1T VT V

+ =

(3)

Figure 1 shows the comparison of the quarter-circular shear-torsion interaction curve with the experimental results of different researchers.5,22-25 The interaction circle measured these results within ±10% and indicates the quarter-circular is applicable with and without stirrups.

Usually columns are loaded eccentrically; with an increase in eccentricity e, the behavior of a column section changes in the following manner: 1) e = 0: axial load; 2) e < eb: compression failure; 3) e = eb: balanced condition; 4) e > eb: tension failure; and 5) e = ∞: pure bending.

This range of interaction from axial compression to pure bending can be easily visualized by a bending-axial compression interaction curve, shown in Fig. 2. In current codes, different failure modes are computed by solving different equations, and the bending-axial compression interaction curve is divided into several sections. This study establishes a unified formula interconnecting the five failure modes for bending-axial compression interaction. The shape

Fig. 1—Comparison between quarter-arc circular and measured torsion-shear interaction curves.


of the interaction curve is very similar to a parabola allowing the M-N interaction equation, to be assumed as follows

M = aN2 + bN + c (4)

In Fig. 2, the point on the abscissa denoted as M0 repre-sents pure bending, and the point on the ordinate denoted as N0 represents the concentric load. The point representing the balanced condition has a coordinate of Mb and Nb. These three points must be on the M-N interaction curve. By substi-tuting these points—(0, M0), (N0, 0), and (Nb, Mb)—into Eq. (4), the factors a, b, and c can be obtained by solving three equations. The results are as follows

0 00 02 2 2

0 0

1, , M Mn ma b N a c MN n n N

+ −= ⋅ = − + = −

(5)

where n = Nb/N0; and m = Mb/M0.Equation (4) can then be expressed in a dimensionless form

2

0 0

2

2 2

2

0 0

1

4 4 1, , 2( 1) ( 1)

1

, b b

M Np q hM N

k k kp q hkk k

n mkn n

N Mn m

N M

+ − = + = − = = − − + −

= − = =

(6)

Equation (6) is plotted as M/M0 versus N/N0 in Fig. 3. According to the dimension and material properties of the specimens tested by Southwest Jiaotong University,26 the average values of factors p, q, and h are equal 0.49, 2.8, and 0.41, respectively. As shown in Fig. 3, the proposed curve is a continuous, smooth curve interconnecting the aforemen-tioned five-force states. The curve is in good agreement with the test results. Equation (6) is therefore applicable to the M-N interaction relationship.

Lampert and Collins9 concluded that, in symmetrically reinforced sections, the moment-torsion interaction formula is given as follows

2

0 0

1M TM T

+ =

(7)

The equation is plotted in Fig. 4 along with the test results of some symmetrically reinforced beams.27-30 It can be seen from Fig. 4 that the experimental results verify this interac-tion behavior.

At this point, M-T, M-N, and T-V interactions have been obtained. Subsequently, the M-T interaction is used to connect the V-T and M-N interactions. Considering the form of the M-T interaction equation, assume the M-T-N-V inter-action formula as follows

2 2 2

0 0 0 0

1T V M NA B p q hT V M N

+ + + − =

(8)

If T = 0 and V = 0, Eq. (8) becomes

2

0 0

1M NB p q hM N

+ − =

(9)

Fig. 2—Bending-axial compression interaction curve. Fig. 3—Comparison between proposed and measured bending-axial compression interaction curves.

Fig. 4—Comparison between parabola circular and measured bending-torsion interaction curves.


If M = 0 and N = 0, Eq. (8) becomes

2 2

2

0 0

1T VA BqhT V

+ + =

(10)

Equations (9) and (10) should be in agreement with Eq. (6) and (3), respectively. The factors can then be easily calculated as

211

A qhB

= −

= (11)

From Eq. (6), the value of 1 – qh2 equals p. Hence, the unified formula can be expressed as

2 2 2

0 0 0 0

1T V M Np p q hT V M N

+ + + − =

(12)

where the factors p and q are as same as those in Eq. (6). Equation (12) is the relationship between N, M, V, and T. However, whether it can be applicable to various kinds of interactions is unknown. Therefore, the parameters a1, a2, and a3 can be added to the T, V, and M terms in Eq. (12). These three parameters can be determined from a calibra-tion with the experimental results of different members with different load combinations. The undetermined formula can be expressed as follows

2 2 2

1 2 30 0 0 0


a + a + a + − =

(13)

Interaction equations under different combined loading conditions can therefore be obtained from the unified formula.

In the determination of the interaction equation, M0, V0, T0, N0, Mb, and Nb are computed based on ACI provisions.2 They can be expressed by the following equations

10 10.85 ( )

2c w c sc

M f cb d f A d db = b − + −′ ′ ′ ′

(14)

0 2 v y

c w

A fV f b d d

s= λ +′ (15)

0 2 cott yv

o

A fT A

s= q (16)

( )0 0.85 c g st y stN f A A f A= − +′ (17)

10.85b c w y s s sN f b c f A f A= b − +′ ′ ′ (18)

M f b c dc h

d N f A d db c w b s s= ′ −

+ −

+ ′ ′ − ′0 852 211. ( )b

b

(19)

where fs′ is determined by compression steel strain es′. If es′ ≤ ey, then fs′ = Esey; or if es′ > ey, then fs′ = fy′. When the member is subjected to pure bending, fs′ can be determined by the following two equations

e s = ∀e ∈ e e

e − e s = + − ∀e ∈ e e γ e − e

0,

,

p pd ydyd

pk ydp pd pd yd ud

s ud yd

f

ff f

(20)

= = = = = = =

= = = = = = =

= =

= + + + + + +∑ ∑ ∑ ∑ ∑ ∑ ∑

= + + + + + +∑ ∑ ∑ ∑ ∑ ∑ ∑

= + +∑ ∑

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1

S C C C CS SSp t cP cL cR

S C C C CS SSp t cP cL cR

SS p

n n n n nn nS P SS C C C Ci i i i i i i

i i i i i i i

n n n n nn nS P SS C C C C

x xi xi xi xi xi xi xii i i i i i i

nnS P S

y yi yi yii i

N N N N N N N N

M M M M M M M M

M M M M= = = = =

+ + + +∑ ∑ ∑ ∑ ∑1 1 1 1 1

C C C CSSt cP cL cRn n n nn

S C C C Cyi yi yi yi

i i i i iM M M M

(21)

When the member is at the balanced condition, the compression steel can always yield, thus fs′ can get fy′. Also, the factors p, q, and h can be obtained when M0, N0, Mb, and Nb are computed.

EXPERIMENTAL VERIFICATIONPure force

The formula for a pure single force can be obtained by manipulating the other three forces as 0. For example, the formula for pure axial compression can be derived by taking M = 0, V = 0, and T = 0

0

1NN

=

(22)

Equation (22) is the bearing capacity formula for RC under pure axial compression. The other three formulas for pure force can also be derived from the unified formula. The parameters a1, a2, and a3 are all equal to 1 for the formula under pure force.

Shear-torsion, moment-torsion, and moment-compression interaction

Equation (13) is derived by shear-torsion, moment-torsion, and moment-compression interaction; hence, it is in agreement with these interaction equations. Taking M = 0 and N = 0, Eq. (3) can be obtained. Taking T = 0 and V = 0, Eq. (6) can be obtained. Taking V = 0 and N = 0, Eq. (7) can be obtained. These equations have already been verified as suitable for the corresponding interactions. Values of a1, a2, and a3 also equal to 1 for these three conditions. The other interaction equations are discussed in the following.

Shear-bending interactionTaking N = 0 and T = 0, the unified formula (Eq. (13)) becomes

2

2 30 0

1V MV M

a + a =

(23)

which is the shear-bending interaction equation. Table 2 shows the comparison of Eq. (23) and the test results of Mattock and Wang.31 According to the comparison results of Vexp/V0 and


Mexp/M0, a2 and a3 are equal to 0.25 and 1, respectively. Equation (23) becomes

2

0 0

0.25 1V MV M

+ =

(24)

Figure 5 shows the theoretical shear-bending interaction curve of Eq. (24) and experimental interaction point for tests by Mattock and Wang.31 The mean of the experimental data of Eq. (24) is 1.03, and the coefficient of variation is 6.6%. From Fig. 5, it can be concluded that Eq. (24) can reflect the behavior of shear-bending members correctly.

Torsion-axial compression interactionThe torsion-axial compression interaction when M = 0 and

V = 0 can be obtained from the unified formula as

22

10 0

1T Np q hT N

a + − = (25)

where the factors p, q, and h have different values according to different section properties. Therefore, the torsion-axial compression interaction curve varies with different RC components. According to dimension and material prop-erties of the specimens tested by Bishara and Peir,32 the average value of factors p, q, and h equal 0.38, 3.19, and 0.44, respectively. Table 3 shows the comparison of N/N0, T/T0, and the test results of Bishara and Peir.32 From the comparison results in Table 3, it is seen that T/T0 varies regularly with the different N/N0. The parameter a1 can be expressed as a function of N/N0. The torsion-axial compres-sion equation is as follows

22

20 0

0

1 1

4 2

T Np q hT NN

N

+ − = +

(26)

The mean of the experimental value of Eq. (26) is 1.095, and the coefficient of variation is 11.8%, indicating that Eq. (26) can accurately describe torsion-axial compression interac-tion. Figure 6 shows the theoretical torsion-axial compression interaction curve of Eq. (26) and the experimental interaction

point for tests by Bishara and Peir.32 From Fig. 6, it can be concluded that Eq. (26) can reflect the behavior of torsion-axial compression members correctly.

Shear-axial compression interactionThe shear-axial compression interaction when M = 0 and

N = 0 can be obtained from the unified formula as

22

20 0

1V Np q hV N

a + − = (27)

Table 2—Comparison of test data and shear-bending interaction formula

Specimen V/bh Vexp/V0 Mexp/M0 Value of Eq. (24)

Mattock and Wang31 average = 1.03; COV = 6.6%

C205-D10 2 1.590 0.497 1.130

C205-D20 2 1.402 0.481 0.973

C210-DOA 2 1.357 0.633 1.093

C210-S0 1 1.558 0.389 0.995

C305-D0 3 1.212 0.582 0.949

C310-D10 3 1.041 0.718 0.989

C310-D20 3 1.097 0.757 1.058

Fig. 5—Comparison between proposed and measured bending-shear interaction curves.

Table 3—Comparison of test data and torsion-axial compression interaction formula

Specimen Nexp/Texp/in. Nexp/N0 Texp/T0 Value of Eq. (26)

Bishara and Peir32 average = 1.08; COV = 11.8%

17 0 0.000 2.123 1.048

19 0 0.000 2.046 1.018

22 0.312 0.064 3.288 1.260

23 0.312 0.054 2.740 1.061

7 0.416 0.086 3.082 1.078

10 0.416 0.101 3.596 1.252

9 0.625 0.145 3.562 1.019

13 0.625 0.168 4.109 1.161

24 0.625 0.186 4.657 1.318

12 0.832 0.249 4.829 1.101

14 0.832 0.285 5.240 1.168

15 0.937 0.326 5.616 1.137

16 0.937 0.355 5.753 1.139

11 1.11 0.546 7.328 1.227

21 1.11 0.428 6.164 1.052

20 1.11 0.420 6.010 1.023

8 1.25 0.465 5.685 0.856

18 1.25 0.525 6.815 1.066

A 0 0.000 1.781 0.950

B 0 0.000 1.918 0.981

C 0.625 0.164 4.109 0.882

D 0.625 0.175 4.383 0.916


Structural components are seldom subjected to shear only; they are always subjected to combined shear and bending moment. Components subjected to combined shear, bending, and axial compression are common. Bending-shear-axial compression interactions are discussed in the next section.

Shear-bending-axial compression interactionWhen the torsion is zero, the shear-bending-axial

compression interaction equation can be calculated from the unified formula

22

2 30 0 0

1V M Np q hV M N

a + a + − =

(28)

Equation (28) is calibrated by using the experimental data from Mattock and Wang.31 Table 4 summarizes the compar-ison of the proposed interaction equation with the test results of Mattock and Wang.31 Considering the simplicity of a1 and a2, as well as the accuracy of the equation, a1 and a2 valued 0.25 and 1, respectively. Therefore, Eq. (28) becomes

22

0 0 0

0.25 1V M Np q hV M N

+ + − =

(29)

From Table 4, it can be concluded that the experimental results agree with the theoretical equation very well. The ratio of the experimental results to the calculated value has a mean and coefficient of variation of 1.14 ± 14.7% for tests by Mattock and Wang.31

Table 4—Comparison of test data and bending-shear-axial compression interaction formula

Specimen Nexp/N0 Vexp/V0 Mexp/M0 Value of Eq. (29)

Mattock and Wang31 average = 1.14; COV = 0.147

C205-D11 0.104 2.13 0.894 1.436

C205-D13 0.388 2.215 0.93 1.302

C205-D15 0.519 1.874 0.787 1.052

C205-D16 0.698 1.59 0.668 1.042

C205-D21 0.094 1.935 0.848 1.249

C205-D22 0.188 2.356 1.033 1.466

C205-D24 0.377 2.188 0.959 1.195

C205-D26 0.565 2.328 1.021 1.391

C210-D2 0.154 1.583 0.968 1.110

C210-D4 0.307 1.508 0.922 0.920

C210-D6 0.461 1.508 0.922 0.919

C305-D1 0.078 1.377 0.874 1.068

C305-D2 0.156 1.653 1.049 1.189

C305-D4 0.312 1.653 1.049 1.064

C305-D6 0.467 1.653 1.049 1.067

C310-D11 0.085 1.388 1.277 1.304

C310-D13 0.255 1.35 1.241 1.072

C310-D16 0.509 1.234 1.135 0.957

C310-D21 0.086 1.309 1.203 1.225

C310-D23 0.258 1.443 1.327 1.162

C310-D25 0.43 1.27 1.168 0.956

C310-D27 0.547 1.155 1.062 0.913

Table 5—Comparison of test data and torsion-bending-shear interaction formula

Specimen Texp/T0 Vexp/V0 Mexp/M0 Value of Eq. (31)

Osburn et al.33 average = 1.037; COV = 0.127

A1 1.064 0.611 1.159 1.236

A2 1.003 0.595 1.153 1.182

A3 0.944 0.594 1.134 1.142

A4 0.864 0.575 1.109 1.072

A5 0.811 0.569 1.084 1.030

B1 1.403 0.398 1.119 1.210

B2 1.283 0.387 1.061 1.092

B3 1.251 0.395 1.097 1.095

B4 1.123 0.375 1.073 0.992

B5 1.032 0.361 1.043 0.918

C1 1.107 0.475 1.131 1.098

C2 0.876 0.416 0.973 0.852

C3 0.926 0.454 1.048 0.944

C4 0.891 0.461 1.021 0.922

D1 1.195 0.508 1.213 1.222

D2 1.025 0.48 1.123 1.055

D3 0.93 0.461 1.068 0.963

D4 0.882 0.461 1.041 0.928

E1 1.183 0.513 1.217 1.221

E2 0.945 0.442 1.033 0.935

E3 0.897 0.446 1.037 0.919

E4 0.768 0.405 0.932 0.777

Badawy et al.34 average = 1.239; COV = 0.168

S3 1.867 0.233 0.412 1.132

S4 1.614 0.511 1.129 1.477

S6 1.342 0.941 0 1.338

S7 1.751 0.493 0 1.011

Fig. 6—Comparison between proposed and measured torsion-axial compression interaction curves.


Torsion-bending-shear interactionThe torsion-bending-shear interaction equation when

axial compression is zero can be derived from the unified formula. The equation can be expressed as

2 2

1 2 30 0 0

1T V MT V M

a + a + a =

(30)

Table 5 shows the comparison of Eq. (30) and the test results of Osburn et al.33 and Badawy et al.34 From the comparison results of Texp/T0, Vexp/V0, and Mexp/M0, a1, a2, and a3 can be valued 0.25, 1, and 0.5, respectively. The determined equation is as follows

2 2

0 0 0

0.25 0.5 1T V MT V M

+ + =

(31)

The means of Osburn et al.’s33 experimental value of Eq. (31) is 1.04, and the coefficient of variation is 12.7%. The means of Badawy et al.’s34 experimental value of Eq. (31) is 1.239, and the coefficient of variation is 16.8%. Experimental comparison results indicate that Eq. (31) is a good application of torsion-bending-shear interaction.

Torsion-bending-axial compression interactionIn practice, components, such as brackets and top chord

of roof truss (where shear force can be negligible), are often under combined torsion, bending, and axial compression. The unified formula can derive the torsion-bending-axial compression interaction equation

2 2

1 30 0 0

1T M Np q hT M N

a + a + − =

(32)

According to dimension and material properties of the specimens tested by Zhang et al.,35 the factors p, q, and h equal 0.434, 3.07, and 0.429, respectively; thus, the theo-retical value of Eq. (32) can be obtained. Table 6 shows the comparison of Eq. (32) and the test results of Zhang et al.35 The parameters a1, a2, and a3 can all be valued 1. Thus, the torsion-bending-axial compression equation is as follows

2 2

0 0 0

1T M Np q hT M N

+ + − =

(33)

Figure 7 shows that the interaction curve of the proposed equation is divided into two main parts: 1) when N/N0 is rela-tively low, the value of (T/T0)2 + (M/M0) increases with the increase in N/N0, which can be verified by using the experi-mental data tested by Zhang et al.35; and 2) when N/N0 is rela-tively high, the value of (T/T0)2 + (M/M0) decreases with the increase in N/N0. The law of the interaction curve is similar to that of the bending-axial interaction. The experimental data basically agree with the interaction curve of the proposed equa-tion, and the average value of the ratio of the experimental data to the proposed equation is 1.08 and the coefficient of varia-tion is 10.2%, indicating that Eq. (33) is a good application of bending-torsion-axial compression interaction.

Shear-torsion-axial compression interactionWhen the bending moment is zero, the unified formula becomes

2 2 2

1 20 0 0

1T V Np q hT V N

a + a + − =

(34)

which represents shear-torsion-axial compression interac-tion. Structural components are seldom subjected to shear only; they are always subjected to combined shear and bending moment. The torsion-bending-shear-axial compres-sion interactions are discussed in the next section.

Torsion-bending-shear-axial compression interactionEquation (13) is the torsion-bending-shear-axial compres-

sion interaction equation. This equation was calibrated by comparing the proposed equation with the experimental results from the tests on 13 square beams under combined torsion, bending, shear, and axial compression by Zhao et al.36 Three parameters—a1, a2, and a3—can be valued 0.25, 1, and 0.5, respectively. Equation (13) becomes

2 2 2

0 0 0 0

0.25 0.5 1T V M Np p q hT V M N

+ + + − =

(35)

The comparison results are shown in Table 7, which are in good agreement. The average value of the proposed equation

Table 6—Comparison of test data and torsion-bending-axial compression interaction formula

Specimen Nexp/N0 Texp/T0 Mexp/M0 Value of Eq. (33)

Zhang et al.35 average = 1.08; COV = 0.102

M-1-1 0.084 1.105 0.098 0.938

M-1-3 0.084 1.184 0.293 1.101

M-1-4 0.098 0.988 0.453 0.950

M-1-5 0.084 1.19 0.488 1.192

M-1-6 0.088 1.147 0.611 1.182

M-2-1 0.16 1.265 0.186 0.997

M-2-2 0.196 1.175 0.453 0.950

M-2-3 0.168 1.303 0.585 1.200

M-2-5 0.168 1.172 0.975 1.229

M-3-1 0.241 1.399 0.279 1.079

M-4-1 0.307 1.408 0.356 1.061

Fig. 7—Comparison between proposed and measured torsion-bending-axial compression interaction curves.


using the experimental data is 1.321, and the coefficient of variation is 24.5%.

Determined unified formulaAt this point, every interaction equation in Table 1 is

determined according to a calibration with the experimental results. The determined unified formula is summarized in Table 8.

APPLICATION OF UNIFIED FORMULAThe unified formula, as verified by data, can be used as

unified failure criteria of RC components in ultimate state to determine the safety of members under various loading states. If the value of the left side is lesser than that of the right side, the section can be considered sufficiently safe. Otherwise, if the value of the left side is larger than that of the right side, the section is unsafe. This study provides an example of torsion-shear-bending-axial compression inter-action to illustrate this point.

For the planar frame given in Fig. 8, assume that all the columns have the same section details, as do the beams; the sections are all symmetrically reinforced. The section details of the columns and the beams are summarized in Table 9. The value of the concentrated loads at Nodes 3 and 2 in the x-direc-tion are 250 and 125 kN (56.25 and 28.13 kips), respectively. The value of the concentrated loads at Nodes 2 and 8 in the y-direction are 125 and –125 kN (28.13 and –28.13 kips), respectively. The value of the distributed loads at both the first and second floors is 20 kN/m (0.104 kip/in.).

From the planar frame, the maximum value of axial load, shear force, and bending moment of all the components can be calculated. The ultimate pure axial compression capacity, ultimate pure shear capacity, and ultimate bending moment capacity of columns and beams can also be calculated based on the section details. The calculation results of N, M, V, T, N0, M0, V0, and T0 are substituted into Eq. (35) and are summarized in Table 10.

As shown in Table 10, the failure factors of Components 7 and 9 are larger than 1, indicating that these components are

Table 8—Values of three parameters of unified formula2 2 2

1 2 30 0 0 0


a + a + a + − =

Interactions a1 a2 a3 Equation

Pure force 1 1 1

Torsion-shear 1 1 1 (3)

Torsion-bending 1 1 1 (7)

Bending-axial compression

1 1 1 (6)

Shear-bending — 0.25 1 (24)

Torsion-axial compression 1/(4N/N0 + 2)2 — — (26)

Shear-bending-axial compression

— 0.25 1 (29)

Torsion-bending-shear 0.25 1 0.5 (31)

Torsion-bending-axial compression

1 1 1 (33)

Torsion-bending-shear-axial compression

0.25 1 0.5 (35)

Table 7—Comparison of test data and torsion-bending-shear-axial compression interaction formula

Specimen Nexp/N0 Texp/T0 Vexp/V0 Mexp/M0 Value of Eq. (35)

Zhao et al.36 average = 1.321; COV = 0.245

WV4-3-2 0.357 1.901 0.52 0.432 1.042

WV5-3-3 0.446 2.583 0.65 0.896 1.603

WV6-3-1 0.536 2.273 0.78 1.075 1.640

WV3-3-2b 0.303 2.118 0.508 0.433 1.186

WV3-2-2 0.303 1.76 0.339 0.289 0.775

WV4-2-2 0.401 2.292 0.443 0.377 1.235

WV3-5-1 0.26 1.559 0.945 0.534 1.567

WV3-2-2b 0.303 2.012 0.339 0.289 0.929

WV4-2-3 0.401 3.199 0.443 0.668 1.515

WV3-2-3 0.303 2.805 0.339 0.289 1.547

WV4-2-ul 0.401 2.847 0 0.377 1.572

WV4-3-2m 0.371 1.642 0.426 0.467 0.889

WV5-4-2m 0.464 2.484 0.65 1.326 1.671

Fig. 8—Planar frame. (Note: 1 m = 3.3 ft; 1 kN = 0.225 kips; 1 kN.m = 8.87 kip-in.)

Table 9—Section details of columns and beams

Component bw, mm d, mm rx, % rz, % fy, MPa fc, MPa

Column 500 500 1.5 0.25 400 30

Beam 250 500 1.5 0.3 400 30

Notes: 1 mm = 0.0394 in.; 1 MPa = 145 psi.


not sufficiently safe. Given that the failure factors of Compo-nents 3, 4, 6, 8, and 10 are less than 1, these components are safe. Components 1 and 2 are subjected to axial tension, biaxial shear, biaxial bending, and torsion, so the unified formula is not applicable to these two components. There-fore, the safety of Components 1 and 2 cannot be determined by Eq. (35), which is a flaw of the unified formula. Similarly, Component 5 also cannot be determined.

Components whose failure factors are less than 1 can still sustain additional force. The unified formula can determine how much residual force these components can sustain because it can be used to calculate residual force capacity of components under various kinds of combined loading. Given the planar frame as an example, the residual capacity is calculated by the following steps. First, given the shear force as an example, the overall shear force capacity is calculated by

2 2

0 0 00

1 0.5 0.25N M Tq h p pN M T

V Vp

− − − −

=′ (36)

Then, the residual shear force is calculated by

Vs = V′ – V (37)

The results are also shown in Table 10. For torsion, shear force, and bending moment, the residual force is nega-tive when the failure factor is larger than 1, whereas the residual force is positive when the failure factor is less than 1. The negative values mean that the force should be reduced to ensure that the failure factor is less than 1. Given Component 7 as an example, if 194 kN.m (1720.78 kip-in.) of bending moment is reduced and the axial compression, torsion moment, and shear force are unchanged, the failure factor is 1 and the component is under ultimate state. The positive values mean that the components are able to with-stand additional force capacity. Given Component 3 in Table 10 as an example, the component is able to bear an additional 67 kN.m (593.95 kip-in.) of torsion moment if the axial compression, bending moment, and shear force are unchanged, or an additional 200 kN (45.01 kips) of shear force if the axial compression, torsion moment, and bending moment are unchanged, or an additional 244 kN.m (2163.04 kip-in.) of bending moment if the axial compres-sion, torsion moment, and shear force are unchanged. From the table, however, it can be observed that the torsional and

shear residential capacities of Component 7 cannot be calcu-lated. No matter how the values of torsion moment and shear are modified, these components will not be safe. The compo-nents have been already been destroyed even if the torsion and shear are reduced to 0.

CONCLUSIONSA unified formula capable of predicting the ultimate state

of RC sections subjected to combined loading, including bending, shear, torsion, and axial compression, has been developed. The formula, which is based on bending-torsion interaction as a link between shear-torsion and bending-axial compression interactions, directly considers the overall inter-actions of bending, shear, torsion, and axial compression.

Various kinds of interaction equations can be obtained from the unified formula. Interaction formulas have been calibrated by comparisons of experimental results from members loaded in various kinds of load combinations.

Some of the features of the formula are as follows:1. It directly considers the interaction between various

kinds of loading.2. It can accurately predict test data under various loading

combinations.3. It can be used as unified failure criteria of RC compo-

nents in ultimate state to determine the safety of members under various loading states.

4. It can be used to calculate the residual stress capacity of components under various kinds of combined loading.

NOTATIONAg = cross-sectional area of beamAo = gross area enclosed by centerline of shear flow pathAs = area of tension steelAs′ = area of compression steelAst = total area of longitudinal steelAt = cross-sectional area of one leg of torsional stirrupAv = cross-sectional area of one leg of shear stirrupbw = width of beamc = depth of concrete compression zoned = effective depth of beamd′ = effective depth to compression reinforcementEs = modulus of elasticity of steelfc = concrete cubic compression strength (150 x 150 x 150 mm [6 x

6 x 6 in.])fc′ = compressive cylinder strength of concretefs′ = stress of compression steelfy = yield stress of tension steelfy′ = yield stress of compression steelfyv = yield stress of stirrupsM0 = ultimate pure bending capacityMb = ultimate bending capacity in balanced condition when subjected

to eccentric compressionMexp = experimental bending capacity

Table 10—Failure factor and residual capacity

No.

Maximum force of components Ultimate pure force capacity Failure factor

Safe or not

Residual capacity

N, kN T, kN·m M, kN·m V, kN N0, kN T0, kN·m M0, kN·m V0, kN Value of Eq. (35) Tr, kN Mr, kN·m Vr, kN

3 398 90 448 145 6524 96 271 467 0.87 Yes 67 244 200

4 139 68 341 123 6524 96 271 467 0.88 Yes 70 214 196

6 134 47 235 78 6524 96 271 467 0.80 Yes 119 374 318

7 64 24 332 183 3262 27 136 258 1.21 No — -194 —

8 200 14 232 143 3262 27 136 258 0.89 Yes 22 100 69

9 42 24 347 178 3262 27 136 258 1.24 No — -221 —

10 78 14 235 134 3262 27 136 258 0.99 Yes 4 11 9

Notes: Compression considered positive; tension considered negative; 1 kN = 0.225 kips; 1 kN.m = 8.87 kip-in.


Mr = residual bending capacityN0 = ultimate pure axial compression capacityNb = ultimate axial compression capacity in balanced condition when

subjected to eccentric compressionNexp = experimental axial compression capacityNr = residual axial compression capacitypx = longitude reinforcement ratiopz = stirrup reinforcement ratios = spacing of stirrupsT0 = ultimate pure torsion capacityTexp = experimental torsion capacityV0 = ultimate pure shear capacityV′ = overall shear capacity according to unified formulaVexp = experimental shear capacityVr = residual shear capacityb1 = equivalent factor of rectangular compressive stress distributiones′ = strain of compression steeley = yield strain of steel

ACKNOWLEDGMENTSThis research described in this paper has been sponsored by the National

Natural Science Foundation of China (No. 51078132), the Chang Jiang Scholars Program and Innovative Research Team Project by the Ministry of Education of China (Project No. IRT0917). The authors gratefully acknowl-edge the assistance provided by Z. Xu, H. Yi, S. Chen, J. Li, and B. Yan.

REFERENCES1. GB 50010-2002, “Concrete Structure Design Code,” 2002. (in Chinese)2. ACI Committee 318, “Building Code Requirements for Structural


3. American Association of State Highway and Transportation Officials, “AASHTO LRFD Bridge Design Specifications and Commentary,” third edition, Washington, DC, 2004.

4. Ersoy, U., and Ferguson, P. M., “Concrete Beams Subjected to Combined Torsion and Shear—Experimental Trends,” Torsion of Structural Concrete, SP-18, American Concrete Institute, Farmington Hills, MI, 1968, pp. 441-460.

5. Klus, J. P., “Ultimate Strength of Reinforced Concrete Beams in Combined Torsion and Shear,” ACI Journal, Proceedings V. 65, No. 3, Mar. 1968, pp. 210-216.

6. Syamal, P. K.; Mirza, M. S.; and Ray, D. P., “Plain and Reinforced Concrete L-Beams under Combined Flexure, Shear, and Torsion,” ACI Journal, Proceedings V. 68, No. 11, Nov. 1971, pp. 848-860.

7. Hsu, T. T. C., “Torsion of Structural Concrete Interaction Surface for Combined Torsion, Shear, and Bending in Beams without Stirrups,” ACI Journal, Proceedings V. 65, No. 1, Jan. 1968, pp. 51-60.

8. Victor, D. J., and Ferguson, P. M., “Reinforced Concrete T-Beams without Stirrups under Combined Moment and Torsion,” ACI Journal, Proceedings V. 65, No. 1, Jan. 1968, pp. 29-36.

9. Lampert, P., and Collins, M. P., “Torsion, Bending, and Confusion—An Attempt to Establish the Facts,” ACI Journal, Proceedings V. 69, No. 8, Aug. 1972, pp. 500-504.

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11. Vecchio, F. J., and Collins, M. P., “Predicting the Response of Reinforced Concrete Beams Subjected to Shear Using Modified Compres-sion Field Theory,” ACI Structural Journal, V. 85, No. 3, May-June 1988, pp. 258-268.

12. Bentz, E. C., “Section Analysis of Reinforced Concrete Members,” PhD thesis, University of Toronto, Toronto, ON, Canada, 2000.

13. Guner, S., and Vecchio, F. J., “Pushover Analysis of Shear-Critical Frames: Formulation,” ACI Structural Journal, V. 107, No. 1, Jan.-Feb. 2010, pp. 63-71.

14. Cocchi, G. M., and Cappello, F., “Inelastic Analysis of Reinforced Concrete Space Frames Influenced by Axial, Torsional and Bending Inter-action,” Computer & Structures, V. 46, No. 1, Jan. 1993, pp. 83-97.

15. Cocchi, G. M., and Volpi, M., “Inelastic Analysis of Reinforced Concrete Beams Subjected to Combined Torsion, Flexural and Axial Loads,” Computer & Structures, V. 61, No. 3, Mar. 1996, pp. 479-494.

16. Saritas, A., and Filippou, F. C., “A Beam Finite Element for Shear Critical RC Beams,” Finite Element Analysis of Reinforced Concrete Struc-tures, SP-237, L. Lowes and F. Filippou, eds., American Concrete Institute, Farmington Hills, MI, 2006, pp. 295-310.

17. Petrangeli, M.; Pinto, P. E.; and Ciampi, V., “Fiber Element for Cyclic Bending and Shear of RC Structures I: Theory,” Journal of Engineering Mechanics, ASCE, V. 125, No. 9, Sept. 1999, pp. 994-1001.

18. Vecchio, F. J., and Selby, R. G., “Toward Compression-Field Analysis of Reinforced Concrete Solids,” Journal of Structural Engineering, ASCE, V. 117, No. 6, June 1991, pp. 1740-1757.

19. Rahal, K. N., and Collins, M. P., “Combined Torsion and Bending in Reinforced and Prestressed Concrete Beams,” ACI Structural Journal, V. 100, No. 2, Mar.-Apr. 2003, pp. 157-165.

20. Gregori, J. N.; Sosa, P. M.; Fernandez Prada, M. A.; and Filippou, F. C., “A 3D Numerical Model for Reinforced and Prestressed Concrete Elements Subjected to Combined Axial, Bending, Shear and Torsion Loading,” Engineering Structures, V. 29, No. 12, Dec. 2007, pp. 3404-3419.

21. Valipour, H. R., and Foster, S. J., “Nonlinear Reinforced Concrete Frame Element with Torsion,” Engineering Structures, V. 32, No. 4, Apr. 2010, pp. 988-1002.

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23. Pritchard, R. G., “Torsion-Shear Interaction of Reinforced Concrete Beams,” MSc thesis, University of Calgary, Calgary, AB, Canada, 1970.

24. Rahal, K. N., and Collins, M. P., “Effect of the Thickness of Concrete Cover on the Shear-Torsion Interaction—An Experimental Investigation,” ACI Structural Journal, V. 92, No. 3, May-June 1995, pp. 334-342.

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26. Eccentric Compression Strength Group, “Performance of Reinforced Concrete Members Under Eccentric Compression,” Reinforced Concrete Structure Research Reports, No. 9, Sept. 1981, pp. 19-61. (in Chinese)

27. Collins, M. P.; Walsh, P. F.; Archer, F. E.; and Hall, A. S., “Ultimate Strength of Reinforced Concrete Beams Subjected to Combined Torsion and Bending,” Torsion of Structural Concrete, SP-18, American Concrete Institute, Farmington Hills, MI, 1968, pp. 379-402.

28. Lampert, P., “Reinforced Concrete Beams in Bending and Torsion (Torsion und Beigung von Stanhlbetonbalken),” Schweizerische Bauzeitung (Zurich), V. 88, No. 5, Jan. 29, 1970.

29. Mamullen, A. E., and Warwaruk, J., “The Torsional Strength of Rect-angular Reinforced Concrete Beams Subjected to Combined Loading,” Report No. 2, University of Alberta, Edmonton, AB, Canada, July 1967.

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Title no. 110-S14




Cyclic Behavior of Substandard Reinforced Concrete Beam-Column Joints with Plain Barsby Catarina Fernandes, José Melo, Humberto Varum, and Aníbal Costa

An experimental investigation aimed at assessing the cyclic behavior of substandard interior beam-column joints built with plain reinforcing bars is described in this paper. Five specimens with plain reinforcing bars and one with deformed bars were tested under reversed cyclic loading. The influence of bond properties, displacement history, column axial load, and amount of reinforce-ment was investigated. A comparison was established in terms of maximum strength, damage, energy dissipation, ductility, displace-ment components, and rotation capacity. Better bond properties led to a more spread damage distribution and larger energy dissipation. Higher column axial load resulted in larger lateral strength and energy dissipation. A larger amount of longitudinal and transverse reinforcement did not necessarily lead to enhanced behavior. The test results contribute to the characterization of the cyclic behavior of beam-column joints with plain bars and can be used to calibrate numerical models for the simulation of this type of element.

Keywords: beam-column joints; bond slip; cyclic behavior; full-scale tests; plain reinforcing bars; reinforced concrete.

INTRODUCTIONReinforced concrete (RC) structures built until the mid-

1970s, before the introduction of modern seismically oriented codes, were usually designed for gravity loads only. As a consequence of the absence of any capacity design principles in design and poor reinforcement details, a signifi-cant lack of ductility—at both the local and global levels—is expected for these structures, resulting in inadequate struc-tural performance even under moderate seismic excitations.1

Past experimental investigation and damage observed following recent earthquakes indicate that deficiencies in the detailing of beam-column joints often lead to brittle failure of the connections and, consequently, of the entire frame. Different damage or failure modes are expected to occur depending on the typology (exterior or interior joint) and adopted structural details (namely, total lack or presence of a minimum amount of transverse reinforcement in the joint, use of plain reinforcing bars or deformed bars, and alterna-tive anchorage solutions).2

Numerous experimental studies have investigated the cyclic behavior of RC beam-column joints. However, rela-tively few experimental investigations are focused on the behavior of joints with design details typical of pre-1970s RC structures. Among these, the majority refer to joints with deformed bars.3-5 There is a significant lack of information about the cyclic behavior of beam-column joints built with plain reinforcing bars, which were widely used for longitu-dinal reinforcement in structures built before the 1970s and are characterized by low bond properties.

Examples of recent experimental studies on the cyclic behavior of RC beam-column joints with plain reinforcing bars can be found in References 1, 3, and 6 through 10.

Liu and Park6 tested the response of full-scale beam-column joints designed according to the pre-1970s codes. The main test variables were the manner in which the longi-tudinal beam bars were hooked in the joint core and the level of the column axial load. Similar units using deformed bars were also tested. In comparison to the specimens with deformed bars, the specimens with plain bars displayed significantly lower stiffness and strength, less joint shear distortion but high opening of beam bar hooks in tension, and column bar buckling. Pampanin et al.1 tested four exte-rior and two interior 2/3-scaled beam-column joints designed for gravity loads only and characterized by the absence of transverse reinforcement in the joint and poor anchorage detailing. Two different types of anchorage solutions for the beam longitudinal reinforcement through the joint region were considered in the interior joints’ specimens: continuous reinforcement or lapped splices with hooked-end anchorage outside the joint region. A better global joint behavior was obtained for the specimens with lapped splices and hooked-end anchorage. The bar-slip phenomenon was evidenced by the marked pinching of the hysteresis.

Bedirhanoglu et al.3 tested full-scale exterior corner beam-column joints with plain reinforcing bars and low-strength concrete. The sensitivity of the specimens’ behavior to column axial load, displacement history, amount of joint reinforcement, presence of a transverse beam and a trans-verse slab, and conditions of anchorage within the joint was investigated. The test results show that an increase in column axial load led to less pinching of the hysteresis loops and an increase in the dissipated energy. The influence of displacement history was negligible. The use of transverse reinforcement in the joint resulted in larger lateral strength capacities and energy dissipation capacity. The presence of the transverse beam and slab resulted, in general, in larger lateral load.

The experimental work described herein refers to the cyclic tests performed on six full-scale beam-column joints, representative of interior beam-column joints of existing RC building structures designed without adequate reinforce-ment detailing for seismic loading. Five specimens were built with plain reinforcing bars and one specimen was built with deformed bars to allow a performance comparison to be established. The sensitivity of the specimens’ behavior to bond properties, displacement history, column axial load, and amount of steel reinforcement was investigated.


Catarina Fernandes is a PhD Student at the University of Aveiro, Aveiro, Portugal. Her research interests include assessment and strengthening of existing building structures.

José Melo is a PhD student at the University of Aveiro. His research interests include assessment and strengthening of existing building structures.

Humberto Varum is an Associate Professor in the Civil Engineering Department at the University of Aveiro. His research interests include assessment, strengthening, and repair of existing structures; structural testing and modeling; and earth construction.

Aníbal Costa is a Full Professor in the Civil Engineering Department at the Univer-sity of Aveiro. His research interests include rehabilitation and strengthening of struc-tures and seismic engineering.

RESEARCH SIGNIFICANCEThe available data on the cyclic behavior of substandard

RC beam-column joints built with plain reinforcing bars are less rich and detailed when compared to those for joints with deformed bars. The behavior of this type of element under cyclic loading is not yet comprehensively understood. The experimental investigation described in this paper will contribute to enlarging the available database on substan-dard beam-column joints with plain reinforcing bars. The experimental results presented can be used to upgrade and calibrate numerical models for the adequate simulation of the cyclic behavior of this type of element.

EXPERIMENTAL INVESTIGATIONTest specimens

The specimens were designed to represent an interior beam-column connection. Each column element represents a half-story column in a building, and each beam element represents a half-span beam. Five specimens (JPA-1, JPA-2, JPA-3, JPB, and JPC) were built with plain reinforcing bars and one specimen (JD) was built with deformed bars. The geometry, dimensions, and reinforcement detailing of the test specimens are depicted in Fig. 1. In all specimens, beam and column longitudinal reinforcement was continuous, there was no transverse reinforcement in the joint region, and stirrups in the beams and columns had 90-degree hooks. Longitudinal and transverse reinforcement ratios, computed according to Eurocode 2 (EC2),11 are summarized in Table 1, where: 1) rl,beam is the total longitudinal reinforce-ment ratio in the beam; 2) rl,column is the total longitudinal reinforcement ratio in the column; 3) rw,beam is the ratio of the transverse reinforcement in the beam; and 4) rw,column is the ratio of the transverse reinforcement in the column. The steel reinforcement amount and detailing adopted in Speci-mens JPA-1, JPA-2, and JPA-3 are referred to in this work as standard reinforcement.

The flexural and shear capacities of the beams and columns (considering the axial load) were computed according to EC211 and are indicated in Table 2. The flexural capacity of

Fig. 1—Geometry, dimensions, and reinforcement detailing of test specimens.


the elements is indicated in terms of the corresponding lateral load. Note that the formulation included in EC211 considers the plane cross-section assumption and perfect bond condi-tions between steel and concrete. Hence, for elements with plain reinforcing bars, the empirical procedure included in EC211 may not accurately estimate their behavior.

MaterialsAll specimens were cast on the same day and with the

same concrete mixture. Compressive tests on concrete 0.15 x 0.15 x 0.15 m (5.9 x 5.9 x 5.9 in.) cubic samples, cast together with the specimens, were conducted to deter-mine the concrete compressive strength. A mean strength equal to 23.8 MPa (3.45 ksi) was obtained. The character-istic compressive strength estimated is equal to 19.8 MPa (2.87 ksi), corresponding to the C16/C20 concrete class according to the EC211 classification.

The mean mechanical properties of the steel longitudinal reinforcement are indicated in Table 3. The strength of the plain reinforcing bars is higher than the typical values for this type of steel reinforcement in existing buildings. However, considering that the cyclic behavior of the elements is strongly influenced by the concrete-steel bond properties, the steel strength is not expected to influence the specimens’ response significantly.

Test setup and loading patternFigure 2 illustrates the test setup that was adopted, indi-

cating the idealized support and loading conditions, and the schematics adopted for the linear variable displacement transducers (LVDTs) used for measuring the local relative displacements in the beam-joint and column-joint interfaces (Slice 1) and vicinities (Slice 2) and joint. The specimens

were tested in the horizontal position. Four high-load-carrying capacity devices with reduced friction were placed below the specimens to carry their self-weight. Steel reac-tion frames associated to sliding devices at the beam extrem-ities and to a pinned connection at the base of the column were used to simulate the support conditions. The maximum frictional forces in the devices used to carry the specimens’ self-weights and to simulate the supports at beams were less than 2.5% of the corresponding lateral load imposed.

The test was conducted under displacement-controlled conditions. Two hydraulic actuators were arranged at the top of the superior column: one to impose the lateral displacements dc and the other for the axial load N. Two levels of axial load were considered: 200 kN (45 kips) in Specimens JPA-1, JPA-2, and JD, corresponding to a normalized axial load equal to 9.4%, and 450 kN (101 kips) in Specimens JPA-3, JPB, and JPC, corresponding to a normalized axial load equal to 21.3%. A displacement history constituted by a series of push-and-pull cycles (three cycles for each level of drift imposed) with a total of 18 levels up to a 4% drift was imposed on all speci-mens except JPA-2. In the displacement history imposed on JPA-2, only one push-and-pull cycle was performed for each level of drift, with a total of seven levels up to a 4% drift.

Table 1—Steel reinforcement details

Beam Column

Longitudinal reinforcement Transverse reinforcement Longitudinal reinforcement Transverse reinforcement

Specimen SteelDiameter, mm (in.) rl,beam, %

Diameter, mm (in.) rw,beam, %

Diameter, mm (in.) rl,column, %

Diameter, mm (in.) rw,column, %

JPA-1 Plain bars

12 (0.47)

0.6

8 (0.32)

0.17

12 (0.47)

0.5

8 (0.32)

0.13

JPA-2 Plain bars 0.6 0.17 0.5 0.13

JPA-3 Plain bars 0.6 0.17 0.5 0.13

JPB Plain bars 0.6 0.17 1.0 0.13

JPC Plain bars 0.6 0.34 1.0 0.34

JD Deformed bars 0.6 0.17 0.5 0.13

Table 2—Flexural and shear capacities of beams and columns computed according to Eurocode 211

Specimen

Flexural capacity, kN (kips) Shear capacity, kN (kips)

Beam

Column Beam ColumnPositive direction Negative direction

JPA-1 48 (11) 25 (6) 40 (9) 244 (55) 142 (32)

JPA-2 48 (11) 25 (6) 40 (9) 244 (55) 142 (32)

JPA-3 48 (11) 25 (6) 62 (14) 244 (55) 142 (32)

JPB 48 (11) 25 (6) 92 (21) 244 (55) 142 (32)

JPC 48 (11) 25 (6) 92 (21) 488 (110) 354 (80)

JD 36 (8) 19 (4) 40 (9) 178 (40) 103 (23)

Table 3—Steel mechanical properties (mean values)

Characteristics Plain bars Deformed bars

Yielding strength, MPa (ksi) 590 (86) 430 (62)

Ultimate strength, MPa (ksi) 640 (93) 550 (80)

Elastic modulus, GPa (ksi) 198 (28,717) 200 (29,008)


EXPERIMENTAL RESULTS AND DISCUSSIONLateral-load-versus-drift diagrams

Figure 3 presents the experimental results in terms of lateral load versus imposed drift. Each plot represents the curves obtained for two different specimens for a better understanding of the influence of the variable parameter in question on the specimens’ global behavior, namely: bond properties (JPA-1 versus JD), displacement history (JPA-1 versus JPA-2), column axial load (JPA-1 versus JPA-3), and amount of steel reinforcement (JPA-3 versus JPB and JPA-3 versus JPC).

Table 4 summarizes the values of the maximum lateral load Fc,max achieved by each specimen and the corresponding drift, the yielding displacement Dy (computed according to Annex B.3 of Eurocode 8: Part 1 [EC8-112]), and the strength degradation registered at the maximum imposed drift (computed in relation to Fc,max).

For each specimen, the maximum lateral load reached in the positive and negative loading directions is similar. The specimen with deformed bars reached the lateral strength at a lateral drift inferior to that observed for the specimens with plain reinforcing bars. The specimens with plain reinforcing bars reached the lateral strength at similar drift levels. Within the drift range imposed on the specimens, JPA-3 displayed the greatest strength and also the largest strength degrada-tion. In fact, the failure condition usually adopted, corre-sponding to a strength reduction of 20% with respect to the peak resistance, was only registered for Specimen JPA-3 at a

drift between 3.7 and 4%. The differences registered between maximum lateral load and strength degradation are well-depicted by the lateral load-drift peak envelopes in Fig. 3(f). The pinching effect was observed for all specimens, being less important for the specimen with deformed bars and more evident in the responses of Specimens JPB and JPC.

Comparing the maximum moment demand in each element with the corresponding capacity predicted according to EC211 (Table 2), it is concluded that: 1) for the specimen with deformed bars, EC2 provides a good estimate of the elements’ strength, overestimating the beams’ strength by 10% and underestimating the columns’ strength by 2%; and 2) for all the specimens with plain reinforcing bars, the EC2 expressions overestimate the capacity of the elements by approximately 10 to 30%.

Damage observed and damage indexFigure 4 illustrates the crack pattern corresponding to the

final damage state of the specimens. In general terms, the specimens with plain reinforcing bars displayed damage concentrated at the beam-joint and column-joint interfaces within Slice 1. Damage within Slice 2 was, in most cases, negligible. From Slice 2 to the end of the elements’ length, damage was not observed. Specimen JPA-3 displayed signif-icant damage within the joint region with concrete cover spalling. In Specimens JPB and JPC, damage was heavily concentrated at the beam-joint interfaces, while cracking at the column-joint interfaces was minor. Conversely to what

Fig. 2—Test setup.


was observed for the specimens with plain reinforcing bars, the specimen with deformed bars displayed a more spread damage distribution. Cracking was spread along the beam and column lengths and severe cracking with concrete cover spalling was observed within the joint region. Due to the stiffness degradation associated with the damage evolution, a small reduction in the column axial load was registered for all specimens, with a maximum variation ranging from 2.8% (for JPB) to 10.3% (for JPA-3).

Table 5 indicates the drift at which the first flexural cracks and inclined cracks (within the joint region) were registered. In all specimens, the flexural cracks were first observed in the beams at or close to the interface with the joint region at drifts ranging from 0.07 to 0.20%. Cracking onset at the column-joint interfaces was registered for higher drift levels ranging from 0.33 to 2%. The maximum crack widths regis-tered at the beam-column and joint-column interfaces are also indicated in Table 5.

The maximum value estimated for the principal tensile stress developed in the joint region is equal to 0.32fc

0.5 for Specimen JPA-1, 0.35fc

0.5 for JPA-2, 0.31fc0.5 for JPA-3,

0.26fc0.5 for JPB, 0.25fc

0.5 for JPC, and 0.39fc0.5 for JD (fc is

the concrete compressive strength). Although Specimens JPA-1 and JPA-3 were tested with different column axial load, they developed similar maximum stress in the joint.

However, while in JPA-1 (with lower axial load), cracking was not observed in the joint, in JPA-3, diagonal cracking occurred at the drift corresponding to the maximum prin-cipal tensile stress. Larger column longitudinal reinforce-ment (JPB and JPC) reduced the maximum principal tensile stress by 16%.

The Park and Ang (PA) damage index13 was computed for Specimens JPA-1 and JD (plain reinforcing bars versus deformed bars) for each level of imposed drift. The column properties were considered to estimate the parameters involved in the computation. The time evolution of the PA damage index is depicted in Fig. 5 and 6, together with the contribution of the maximum deformation umax/uu. For the first levels of imposed drift—up to 1%—the average contri-bution of the maximum deformation to the PA damage index was equal to 99% for the two specimens. For the last three levels of imposed drift, the average contribution was approxi-mately 82% and 80% for JPA-1 and JD, respectively. The results show the relatively minor contribution of the energy dissipation to the PA damage index, which was also observed by Varum.14

Figures 5 and 6 show the damage state categories suggested by Park et al.15 and the corresponding global damage indexes boundaries (refer to Reference 14). For each specimen, Fig. 5 and 6 also indicate the drift corre-

Fig. 3—Lateral-load-versus-drift responses for specimens: (a) JPA-1 versus JD; (b) JPA-1 versus JPA-2; (c) JPA-1 versus JPA-3; (d) JPA-3 versus JPB; (e) JPA-3 versus JPC; and (f) peak envelopes. (Note: 1 kN = 0.2248 kips.)

Table 4—Maximum lateral load, yielding displacement, and strength degradation

Specimen

Maximum lateral load

Dy, mm (in.) Strength degradation at 4% drift, %Fc,max, kN (kips) Drift, %

JPA-1 34.0 (7.6) 3.3 28.2 (1.11) 4.5

JPA-2 35.8 (8.0) 3.0 28.2 (1.11) 5.8

JPA-3 43.3 (9.7) 2.7 31.1 (1.22) 26.6

JPB 39.5 (8.9) 2.3 26.0 (1.02) 15.8

JPC 38.3 (8.6) 3.3 27.0 (1.06) 10.0

JD 39.0 (8.8) 2.0 23.8 (0.94) 19.0


sponding to the onset of the main types of damage observed in the experimental tests, namely: cracking at the beam-joint and column-joint interfaces, diagonal cracking in the joint region, and concrete cover spalling in the beams and columns. In both specimens, cracking onset occurred within the expected range of the global damage index. Concrete cover spalling was registered for damage indexes larger than those suggested. The drift at which the specimens reach each damage state is, in general, lower for the specimen with plain reinforcing bars than for the specimen with deformed

bars. For example, a damage index equal to 1 is registered for a drift of 3% for JPA-1 and 3.3% for JD.

Energy dissipation and ductility demandsIn Fig. 7, the evolution of the total dissipated energy,

computed as the area under the lateral load-drift diagrams, is plotted for all specimens except JPA-2. Specimen JPA-2 was excluded from this analysis because it was subjected to a displacement history different from the one imposed on the other specimens. The largest energy dissipation was registered for Specimens JPA-3 (with higher column axial load) and JD (with better bond properties). The pronounced pinching effect observed in the lateral load-drift diagrams of Specimens JPB and JPC led to significantly lower energy dissipation in comparison to JPA-3 (subjected to the same displacement history and column axial load).

Figure 8(a) shows a plot of the response of each spec-imen in terms of equivalent damping versus displacement ductility. For Specimens JPA-1, JPA-3, and JD, the curves that best fit the experimental results are also presented. Specimen JPA-2 was also excluded from this analysis. Equivalent damping xeq was computed according to Varum14 as the ratio between the dissipated energy for half the load-displacement cycle and the viscous damping for the corresponding maximum drift. Displacement ductility Fig. 4—Crack pattern corresponding to final damage state.

Fig. 5—Time evolution of PA damage index for Specimen JPA-1.

Table 5—Drift corresponding to cracking onset

Specimen

Flexural cracks

Drift at first inclined crack within joint region, %

Drift at first crack, % Maximum crack width, mm (in.)

Beam Column Beam Column

JPA-1 0.07 0.83 5.9 (0.23) 5.9 (0.23) —

JPA-2 0.13 0.50 4.3 (0.17) 8.2 (0.32) —

JPA-3 0.13 1.33 6.9 (0.27) 3.1 (0.12) 2.67

JPB 0.20 2.00 12.4 (0.49) 0.1 (0.004) —

JPC 0.13 1.67 12.2 (0.48) 0.3 (0.001) 4.00

JD 0.13 0.33 5.4 (0.21) 10.4 (0.41) 0.67


µD corresponds to the ratio between imposed displace-ment dc and yielding displacement Dy (Table 4). Within the drift range under analysis, the largest and lowest ductility demands were imposed on JD (mD = 5.0) and JPA-3 (mD = 3.8). Even if the lowest demand, in terms of ductility, was imposed on JPA-3, it is recalled that this was the only spec-imen that reached the conventional failure condition.

Figure 8(b) shows a comparison between the experimental results and the equivalent damping-versus-displacement-ductility relationships computed from some of the existing equivalent damping equations,16 namely: Eq. (1), proposed by Priestley17 for concrete frames, and Eq. (2), referring to the Takeda model.18 This comparison is made considering that existing RC building structures built without adequate seismic detailing tend to develop soft story mechanisms and therefore the response of the beam-column joints of a weak story may be correlated to the response of the overall structure. In addi-tion, a comparison is also made with the equivalent damping computed from Eq. (3), proposed by Varum,14 based on the results of a series of pseudo-dynamic tests of an RC frame structure built with plain reinforcing bars.

D

x + ⋅ − p m

120 15 1= (1)

D

x x + ⋅ −

m 0

10.2 1= (2)

( )x ⋅ D +11.041 ln 9.9286= (3)

In Eq. (2), x0 is the initial viscous damping (considered equal to 5%). In Eq. (3), D represents the interstory drift.

The equivalent damping-displacement ductility relation-ships determined from Eq. (1) to (3) significantly overes-

timate the experimental results. Equations (1) and (2) are more adequate for structures with larger energy dissipation capacities. This highlights the need for the development of simplified expressions, based on experimental data, for the assessment of existing RC building structures built without specific seismic detailing and, in particular, with plain reinforcing bars. Regarding Eq. (3), the experimental results used for determining the equivalent damping-displacement ductility relationship refer to a particular story. In these tests, the story response was mainly governed by the behavior of a strong column. Conversely to what was observed in the joint specimens with plain reinforcing bars under study, damage in that strong column was not concentrated at the interface with the joint. Instead, it was spread along a larger element’s plastic hinge region. Hence, the associated energy dissipa-tion is expected to be larger than that displayed by the beam-column joints under study.

Drift componentsThis section studies the relative contribution of beam and

column deformations to the specimen deformation. The direct integration method was used for estimating the total

Fig. 6—Time evolution of PA damage index for Specimen JD.

Fig. 7—Evolution of total dissipated energy. (Note: 1 kN.mm = 8.85 lbf·in.)


drift and the contribution of each component. Considering the concentration of damage observed in the specimens with plain reinforcing bars and assuming a linear distribution of moments along the beam and column lengths, the curvature distribution in each element was analytically established. Thus, in the analytical formulation, the following assump-tions are made for the curvatures of each element (refer to the nomenclature in Fig. 2): a linear variation from the element extremity (where the curvature is zero) to the interface between Slices 1 and 2, and a parabolic variation between this section and the interface between Slice 1 and the joint (where the maximum curvature occurs). In the calculations, the mean curvatures measured in Slices 1 and 2 were used as input in the analytical expressions derived. Considering the assumptions previously presented, the direct integration method was used for determining the deflection and rotation along the elements.

Based on the analytical equations for each element, and considering the compatibility conditions in terms of displace-ments and rotation at the joint, as well as the displacement restraints at the supports, the deformation equations for all the specimens were obtained.

With the expressions derived, the lateral displacement at the free end of the superior column was determined at each time step and compared to the corresponding experimental values. In general terms, a good match was found between the experimental and analytical results. The maximum difference registered between the experimental and analyt-ical displacements was: 8% for JPA-1, 7% for JPA-2, 15% for JPA-3, 19% for JPB, and 9% for JPC. Therefore, the analytical equations were used for determining the contribu-tion of each element to the total lateral displacement, which is proportional to the lateral drift. The relative contribution of beams and columns to the total drift imposed on the speci-mens with plain reinforcing bars is represented in Fig. 9.

The curvature distribution adopted for the elements with plain reinforcing bars does not represent the damage displayed by the specimen with deformed bars. Furthermore, the experimental results in terms of curvature (measured in Slices 1 and 2) are not sufficient for a precise definition of the curvature distribution in this specimen; therefore, the spec-imen with deformed bars was excluded from this analysis.

For all specimens, the results show that at lower imposed drift levels, the beam deformation controls the total drift

Fig. 8—Equivalent damping versus displacement ductility: (a) experimental results; and (b) comparison with results of existing equivalent damping equations.

Fig. 9—Relative partial contribution of beams and columns to total drift in specimens: (a) JPA-1; (b) JPA-2; (c) JPA-3; (d) JPB; and (e) JPC.


associated with the deformation of the specimens. For Specimens JPA-1, JPA-2, and JPA-3, as the imposed lateral drift increases, the contribution of column deformation to the total drift rises. Conversely, in Specimens JPB and JPC, the contribution of column deformation decreases with the imposed drift. These results are in accordance with the damage evolution registered during the testing of each spec-imen. At a 3% drift, the relative contribution of the column deformation to the total drift is equal to 51% for JPA-1, 71% for JPA-2, 26% for JPA-3, 5% for JPB, and 9% for JPC.

Ultimate rotation capacityEurocode 8: Part 3 (EC8-3)19 evaluates the deformation

capacity of RC elements in terms of the chord rotation. For elements with plain reinforcing bars, the ultimate rotation capacity is evaluated by applying a correction coefficient (always inferior to 1), based on experimental data, to the capacity formulations calibrated on elements with deformed bars and seismic detailing. For elements without lapping of the longitudinal bars, the correction coefficient is equal to 0.575. According to Verderame et al.,20 some provisions of EC8-3 have been changed according to Reference 21 and the correction coefficient has been increased to 0.80. In either case, according to the code provisions, the ultimate rotation capacity of elements with plain reinforcing bars is smaller than that of elements with deformed bars with equiv-alent structural characteristics and details. However, recent experimental results20,22 indicate the contrary. According to Verderame et al.,23 when plain reinforcing bars are used, chord rotation results from a combined action of the fixed-end rotation at the base and spreading of yielding over the element length. A critical review of the EC8-3 approach for estimating the ultimate rotation capacity of elements with plain reinforcing bars is made by Verderame et al.20 Based on the test results from recent experiments on RC columns with plain reinforcing bars, the authors also propose a new correction coefficient to be applied to the EC8-3 expres-sions. For elements without lapping of longitudinal bars, the proposed correction coefficient is equal to 1.0.

To make a comparison between the ultimate rotation capacity predicted by EC8-3 and the one estimated from the experimental results for the specimens under study, joint rotation had to be subtracted from the total drift imposed on the specimens so that the columns’ chord rotation could be obtained. Joint rotation was estimated by the direct integration method used for determining the displacement components. For the reasons previously stated, Specimen JD was excluded from the analysis. Specimen JPA-3 was also excluded from this analysis because, for drift levels supe-rior to 3%, the direct integration method did not provide a good match between the experimental and analytical results. Hence, maximum column chord rotation could not be deter-mined for this specimen.

Table 6 presents the theoretical values of ultimate rota-tion capacity computed using the EC8-3 expression and multiplied by the correction coefficient. Three correc-tion coefficients were considered: 1) the correction coef-ficient prescribed by EC8-3, equal to 0.575; 2) the new EC8-3 correction coefficient, equal to 0.80 according to Verderame et al.20; and 3) the correction coefficient proposed by Verderame et al.,20 equal to 1.0. Accordingly and respectively, three different theoretical values of ulti-mate rotation capacity are presented for each specimen: qu,EC8, q′u,EC8, and qu,Verd.

Figure 10 shows the lateral load-column chord rotation diagrams obtained for the specimens under analysis, with indication of the failure condition for which the rotational capacity is usually evaluated (corresponding to a strength reduction of 20% with respect to the peak resistance) and the theoretical values of ultimate rotation capacity. A strength reduction equal to or larger than 20%, measured in the force- versus-chord-rotation diagrams, was not registered for either of the specimens analyzed. In fact, the maximum strength reduction recorded was equal to 6% (for Specimen JPA-2). Hence, neither of the specimens reached the ultimate rota-tion capacity predicted by EC8-3. JPA-1 achieves the ulti-mate rotation capacity qu,EC8 for a strength reduction equal to 4%. JPA-2 reaches qu,EC8 for its peak strength and q′u,EC8 for a strength reduction equal to 6%. For JPA-2, and considering the tendency displayed by the force-versus-column-chord-rotation diagram in the last cycles, the approach proposed by Verderame et al.20 seems to give better results.

Influence of bond propertiesAs shown in Fig. 3(a), Specimens JPA-1 (with plain

reinforcing bars) and JD (with deformed bars) displayed similar stiffness until the beginning of cracking. After cracking onset, for larger displacement demands, JPA-1 showed lower unloading stiffness than JD. The maximum lateral load registered for the specimen with

Table 6—Theoretical values of ultimate rotation capacity

Specimen

Ultimate rotation capacity, %

qu,EC8 q′u,EC8 qu,Verd.

JPA-1 1.98 2.75 3.44

JPA-2 1.98 2.75 3.44

JPB 1.72 2.39 2.99

JPC 1.83 2.55 3.18

Fig. 10—Lateral-load-versus-column chord rotation rela-tionships for specimens: (a) JPA-1; (b) JPA-2; (c) JPB; and (d) JPC. (Note: 1 kN = 0.2248 kips.)


deformed bars (at a 2% drift) was approximately 15% higher than that for the specimen with plain reinforcing bars (at a 3.3% drift). Strength degradation at maximum drift was equal to 4.5% for JPA-1 and 19% for JD. The pinching effect was observed for both specimens but was more evident for the specimen with plain reinforcing bars. The effects of bond properties were particularly evidenced by the differences in the damage distribution (Fig. 4). The specimen with plain reinforcing bars displayed damage concentrated at the beam-joint and column-joint interfaces without damage within the joint region. The specimen with deformed bars displayed a more spread damage distribution, with cracking along the elements’ lengths and significant damage within the joint region. The different damage distributions are reflected in the energy dissipation capacities of the specimens (Fig. 7). As expected, better bond properties led to larger energy dissipation. At maximum drift, the total energy dissipated by JD was 8% higher than that for JPA-1. Within the drift range imposed on the specimens, JD displayed larger ductility demands than JPA-1 (Fig. 8).

Influence of displacement historyThe influence of displacement history was minor. Speci-

mens JPA-1 and JPA-2 displayed similar stiffness, before and after cracking, and similar maximum lateral load (reached at similar drift) and strength degradation (Fig. 3(b)). Damage distribution was also alike (Fig. 4).

Influence of column axial loadAs shown in Fig. 3(c), Specimens JPA-1 (with lower

column axial load) and JPA-3 (with higher column axial load) displayed similar stiffness until the beginning of cracking. After cracking onset, JPA-3 exhibited larger stiff-ness than JPA-1. The increase in column axial load led to an increase of approximately 27% in the maximum lateral load, which was reached for lower drift (2.7%) than in the case of JPA-1 (at a 3.3% drift). The specimen with higher axial load also displayed significantly greater strength degrada-tion, reaching the conventional failure condition corre-sponding to a 20% strength reduction. Strength degradation at maximum drift was equal to 4.5% for JPA-1 and 26.6% for JPA-3. The pinching effect was more evident for JPA-3. However, it practically ceases after the specimen reaches the maximum load. Damage in JPA-1 was concentrated at the beam-joint and column-joint interfaces. In JPA-3, significant damage was also observed within the joint region (Fig. 4). Increasing the column axial load resulted in larger energy dissipation (Fig. 7). At maximum drift, the total energy dissipated by JPA-3 is approximately 47% higher than that for JPA-1. Within the drift range imposed on the specimens, JPA-3 displayed lower ductility demands than JPA-1 (Fig. 8).

Influence of amount of steel reinforcementAs shown in Fig. 3(d) and (e), Specimens JPA-3 (with

standard steel reinforcement), JPB, and JPC displayed similar stiffness until cracking onset. Prior to and after reaching maximum strength, JPA-3 displays the greatest stiffness. The maximum lateral load registered for Speci-mens JPB and JPC is approximately 91% (at a 2.3% drift) and 88% (at a 3.3% drift), respectively, of that for JPA-3 (at a 2.7% drift). Increasing the amount of steel reinforcement resulted in lower strength degradation. At maximum drift, strength degradation was equal to 26.6% for JPA-3, 15.8%

for JPB, and 10% for JPC. The pinching effect was particu-larly evident for Specimens JPB and JPC. The energy dissi-pation associated with these two specimens is significantly lower than that registered for JPA-3 (Fig. 7). At maximum drift, the total energy dissipated by JPA-3 is approximately 56% and 41% higher than that for JPB and JPC, respectively. Within the drift range imposed on the specimens, Speci-mens JPB and JPC display a similar equivalent damping-displacement ductility relationship with larger ductility demands in comparison to JPA-3 (Fig. 8).

CONCLUSIONSAn experimental investigation was performed to assess the

cyclic behavior of full-scale RC interior beam-column joints built with plain reinforcing bars and poor reinforcement detailing. Five specimens with plain reinforcing bars and one specimen with deformed bars were tested under reversed cyclic loading imposed under displacement-controlled conditions. The influence of bond properties, displacement history, column axial load, and amount of steel reinforce-ment was investigated. Two levels of column axial load were considered: 200 kN (45 kips) and 450 kN (101 kips). Two displacement histories were considered, both up to a maximum drift of 4%—one with fewer displacement cycles than the other. Among the specimens with plain reinforcing bars, in comparison to the others, one specimen was built with a larger amount of column longitudinal reinforcement and another with larger amounts of column longitudinal reinforcement and transverse reinforcement in the beams and columns.

The influence of bond properties was particularly evidenced by differences observed in the damage distribution of the specimens. In the specimens with plain reinforcing bars, damage was mainly concentrated at the beam-joint and column-joint interfaces. Conversely, a more spread damage distribution was observed in the specimen with deformed bars, with cracking along the elements’ lengths and signifi-cant damage within the joint region. Better properties led to larger energy dissipation.

The effects of the column axial load were observed mainly in terms of lateral strength and damage distribution. Increasing the column axial load led to larger lateral strength and significantly greater strength degradation. In the spec-imen with higher column axial load, significant damage was also observed within the joint region. This specimen displayed considerably more energy dissipation.

The influence of the amount of steel reinforcement was mainly evidenced by the damage distribution. In the two specimens with larger amounts of steel reinforcement, damage was mostly concentrated at the beam-joint inter-faces. Damage in the columns was minor and no damage was observed within the joint region. The energy dissipation associated with these two specimens was significantly lower than that in the specimen with standard steel reinforcement.

ACKNOWLEDGMENTSThis paper reports research developed under financial support provided

by Fundação para a Ciência e Tecnologia (FCT), Portugal, co-funded by the European Social Fund within the scope of the Programa Operacional Potencial Humano (POPH) of the National Strategic Reference Framework, namely through the PhD grants of the first and second authors, with refer-ences SFRH/BD/27406/2006 and SFRH/BD/62110/2009, respectively. The authors would like to acknowledge the following companies: Civilria, for the construction of the test specimens; and Somague, Grupo Meneses, Silva Tavares & Bastos Almeida and Paviútil, for the construction of the


structural reaction systems. The authors also acknowledge H. Pereira and A. Figueiredo for assisting in the preparation of the test.

REFERENCES1. Pampanin, S.; Calvi, G. M.; and Moratti, M., “Seismic Behavior of

RC Beam-Column Joints Designed for Gravity Loads,” Paper No. 726, Proceedings of the 12th European Conference on Earthquake Engineering, London, UK, 2002, 10 pp.

2. Pampanin, S., “Alternative Performance-Based Retrofit Strategies and Solutions for Existing RC Buildings,” Seismic Risk Assessment and Retro-fitting, A. Ilki, F. Karadogan, S. Pala, and E. Yuksel, eds., Springer, Berlin, Germany, 2009, pp. 267-295.

3. Bedirhanoglu, I.; Ilki, A.; Pujol, S.; and Kumbasar, N., “Behavior of Deficient Joints with Plain Bars and Low-Strength Concrete,” ACI Struc-tural Journal, V. 107, No. 3, May-June 2010, pp. 300-310.

4. Clyde, C.; Pantelides, C. P.; and Reaveley, L. D., “Performance-Based Evaluation of Exterior Reinforced Concrete Building Joints for Seismic Exci-tation,” PEER Report 2000/05, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 2000, 61 pp.

5. Pantelides, C. P.; Hansen, J.; Nadaul, J.; and Reaveley, L. D., “Assess-ment of Reinforced Concrete Building Exterior Joints with Substandard Details,” PEER Report 2002/18, Pacific Earthquake Engineering Research Center, University of California, Berkeley, Berkeley, CA, 2002, 114 pp.

6. Liu, A., and Park, R., “Seismic Behaviour and Retrofit of Pre-1970’s As-Built Exterior Beam-Column Joints Reinforced by Plain Round Bars,” Bulletin of the New Zealand Society for Earthquake Engineering, V. 34, No. 1, 2001, pp. 68-81.

7. Park, R., “A Summary of Results of Simulated Seismic Loads on Reinforced Concrete Beam-Column Joints, Beams and Columns with Substandard Reinforcing Details,” Journal of Earthquake Engineering, V. 6, No. 2, 2002, pp. 147-174.

8. Akguzel, U., and Pampanin, S., “Seismic Upgrading of Exterior Beam-Column Joints Using GFRP,” Proceedings of the 14th European Conference on Earthquake Engineering, Paper 760, Ohrid, Republic of Macedonia, 2010. (CD-ROM)

9. Eligehausen, R.; Genesio, G.; Ožbolt, J.; and Pampanin, S., “3D Analysis of Seismic Response of RC Beam-Column Exterior Joints Before and After Retrofit,” Proceedings of the 2nd International Conference on Concrete Repair, Rehabilitation and Retrofitting, Cape Town, South Africa, 2008, pp. 1141-1147.

10. Hertanto, E., “Seismic Assessment of Pre-1970s Reinforced Concrete Structure,” MSc thesis, University of Canterbury, Christchurch, New Zealand, 2005, 247 pp.

11. EN 1992-1-1:2004, “Eurocode 2: Design of Concrete Structures— Part 1-1: General Rules and Rules for Buildings,” European Committee for Standardization, Brussels, Belgium, 2004, pp. 70 and 156.

12. EN 1998-1:2004, “Eurocode 8: Design of Structures for Earthquake Resistance—Part 1: General Rules, Seismic Actions and Rules for Build-ings,” European Committee for Standardization, Brussels, Belgium, 2004, pp. 201-202.

13. Park, Y.-J., and Ang, A. H.-S., “Mechanistic Seismic Damage Model for Reinforced Concrete,” Journal of Structural Engineering, ASCE, V. 111, No. 4, 1985, pp. 722-739.

14. Varum, H., “Seismic Assessment, Strengthening and Repair of Existing Buildings,” PhD thesis, University of Aveiro, Aveiro, Portugal, 2003, 546 pp.

15. Park, Y.-J.; Ang, A. H.-S.; and Wen, Y. K., “Damage-Limiting Aseismic Design of Buildings,” Earthquake Spectra, V. 3, No. 1, 1987, pp. 1-26.

16. Blandon, C., and Priestley, M. N. J., “Equivalent Viscous Damping Equations for Direct Displacement Based Design,” Journal of Earthquake Engineering, V. 9, No. 2, 2005, pp. 257-278.

17. Priestley, M. N. J., Myths and Fallacies in Earthquake Engineering, Revisited. The Mallet-Milne Lecture, IUSS Press, Pavia, Italy, 2003, pp. 93-115.

18. Gulkan, P., and Sozen, M. A., “Inelastic Responses of Reinforced Concrete Structure to Earthquake Motions,” ACI JOURNAL, Proceedings V. 71, No. 12, Dec. 1974, pp. 604-610.

19. EN 1998-3:2005, “Eurocode 8: Design of Structures for Earthquake Resistance—Part 3: Strengthening and Repair of Buildings,” European Committee for Standardization, Brussels, Belgium, 2005, pp. 36-40.

20. Verderame, G. M.; Ricci, P.; Manfredi, G.; and Cosenza, E., “Ulti-mate Chord Rotation of RC Columns with Smooth Bars: Some Consider-ations about EC8 Prescriptions,” Bulletin of Earthquake Engineering, V. 8, No. 6, 2010, pp. 1351-1373.

21. CEN, “Corrigenda to EN 1998-3, Document CEN/TC250/SC8/N437A,” European Committee for Standardization, Brussels, Belgium, 2009.

22. Di Ludovico, M.; Verderame, G. M.; Prota, A.; Manfredi, G.; and Cosenza, E., “Experimental Investigation on Non-Conforming Full Scale RC Columns,” Proceedings of the XIII Conference ANIDIS, Paper S02_09, Bologna, Italy, 2009. (CD-ROM)

23. Verderame, G. M.; Fabbrocino, G.; and Manfredi, G., “Seismic Response of R.C. Columns with Smooth Reinforcement—Part II: Cyclic Tests,” Engineering Structures, V. 30, No. 9, 2008, pp. 2289-2300.


NOTES:


DISCUSSIONDisc. 109-S13/From the March-April 2012 ACI Structural Journal, p. 139


Discussion by Emil de Souza Sánchez Filho, Marta de Souza Lima Velasco, and Júlio J. Holtz Silva FilhoACI member, DSc, Professor at Fluminense Federal University, Rio de Janeiro, Brazil; DSc, Professor at Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil; DSc, Structural Engineer, Rio de Janeiro, Brazil

The authors should be congratulated for their interesting and excellent investigation into the theoretical study of strength of diagonal struts for the shear strength of reinforced concrete (RC) beams without shear reinforcement. They have made an effective contribution to the theoretical study of the concrete effectiveness factor and strut strength and provide a theoretical method that replaces a host of currently used empirical formulas. However, the discussers would like to address the following questions and point out some special aspects in this study. The objective of this discussion is to obtain additional data on the research and give sugges-tions so the authors can improve their analyses.

1. The concrete effectiveness factor n is an essential parameter that needs to be inserted into the development of the plasticity theory applied to structural concrete elements. The best agreement between the theory and experimental data is obtained by the appropriate choice of n.

2. The concrete effectiveness factor depends on the strut type, reinforcement arrangements—longitudinal and transversal (this reinforcement is not studied in this paper)—and dimensions of the beam; however, the authors consider only one type of strut (prism strut) in their modeling.

3. The longitudinal reinforcement has a pronounced influence on the behavior of the strut strength, and its consideration is the great merit of the theoretical formulation.

4. The discussers suggest the analysis of the theoretical values of n for other strut types (fan and bottle struts) adopted in beams with different a/h ratios. The discussers believe that the theoretical value of n for the fan and bottle struts is different from the effectiveness factor for the prism strut because the parameter a/h influences the beam behavior,16 strut type, and strength of the strut (Fig. 14).

5. The authors failed to explain the theoretical value for the angle q given in Eq. (19), which depends on the position of the neutral axis (Fig. 10). Further precise information regarding the model conception is necessary.

6. The authors should take care when analyzing the results obtained with the expressions of Eurocode 214 because fck is a statistical value and is very different from the values of experimental studies analyzed—for example, fck is different

Disc. 109-S13/From the March-April 2012 ACI Structural Journal, p. 139


Discussion by Andor WindischACI member, PhD, Karlsfeld, Germany

In the course of improvements to the strut-and-tie models “as the essence of member behavior failing in shear,” the

authors arrive at a new attribute. Instead of struts with a uniform cross section or bottle shape—both with a reduced

Fig. 14—Beams without shear reinforcement.16 (Note: Dimensions in cm; 1 in. = 2.54 cm.)

from the average concrete strength fc. The statistical values shown in Table 10 for Eq. 6.a (Eurocode 2)14 are controversial.

Therefore, the discussers would greatly appreciate it if the authors could clarify their queries and provide some comple-mentary information about the research.

REFERENCES16. Leonhardt, F., and Walther, R., “Schubversuche an einfeldrigen

Stahlbetonbalken mit und ohne Schubbewehrung,” Deutscher Ausschuss für Stahlbeton, Heft 1551, 1962. (in German)


effective concrete strength (strut efficiency factor) due to the induced transverse tensile stresses—the authors recognize the reduction of strut width due to the diagonal cracks as the main cause of shear failure.

It is not clear why the authors needed a nonlinear finite element model to obtain the compressive stress trajectories for uncracked reinforced concrete (RC) beams. Furthermore, it is not easy to understand how the bond stresses of the longi-tudinal tensile reinforcement make the compressive strut curved. If this thesis is correct, then any further study would be moot, as the longitudinal reinforcement is bonded. Hence, in the case of any impairment of the straight strut through a crack, the strut could work as a curved strut (compare Fig. 2(b) and (c)). The paper is full of target-oriented—never-theless questionable—assumptions and statements.

Figure 3, which should present the fundamental case of reduction of the strut width, shows the interaction of a flexural crack with the compressive force equilibrating the bending moment. No impairment of the flexural capacity is known. The cracks shown in Fig. 4 are impressive but not correct. The first crack in a corbel develops at the corner, whereas the corner of a dapped end is always inclined, as shown quite recently.17

Figure 5 is full of inconsistencies and contradictions. Please address:• In the authors’ compressive strut, the stress distribution

is not uniform. This is a remarkable “innovation”—reduced width + somehow parabolic compressive stress distribution?

• Point B has no validity at all: “The second crack segment then meets the extension line of the neutral axis at Point B, above which the flexural compression exists in the area of the concrete strut.” This statement is confusing and has no mechanical validity.

• “The remaining segment of the diagonal crack follows the original yield line of the concrete strut.” This yield line is something new in the strut-and-tie models.

• Figure 7 and the related comments explain that bond stresses around the longitudinal reinforcement cause the inclined (shear) cracks. The authors should realize that bond stresses develop on both sides of a flexural crack along the beam region with pure bending also. The cracks remain straight.

It is certain that RC beams without shear reinforcement fail along a critical section consisting of a flexural-shear crack and a sliding surface across the compressive zone, but the authors’ proposal does not support the understanding of the behavior of these members at all.

REFERENCES17. Nagrodzka-Godycka, K., and Piotrkowski, P., “Experimental Study

of Dapped-End Beams Subjected to Inclined Load,” ACI Structural Journal, V. 109, No. 1, Jan.-Feb. 2012, pp. 11-20.

AUTHORS’ CLOSUREClosure to discussion by Sánchez Filho et al.

The authors are grateful to the discussers for their interest in the paper. The following items address the points made in the order they appear in the discussion:• The authors agree with the discussers on the importance

of the effectiveness factor for concrete strength. Although not explicitly derived in the paper, the effectiveness factor is represented by the intactness of the strut width, as shown in Eq. (18). These equations involve the effect of the longitudinal reinforcement

ratio, which is verified by the consistent model safety factor, regardless of the variation of the longitudinal reinforcement ratio, as shown in Fig. 13(b). Therefore, the influence of the longitudinal reinforcement ratio on the effectiveness factor is already considered in the development of the theory.

• Regarding the types of struts other than the prism strut, the value of the effectiveness factor could be different, as the discussers believe. However, the interpretation of the mechanism of penetration of the diagonal crack into the bottle-shaped strut or fans would be challenging compared with the case of a simple prism strut due to geometrical complexity. The authors believe that the approach adopted in the paper for evaluating the effective capacity of the strut is not adequate for those types of struts.

• The derivation process of Eq. (21) was edited due to the maximum paper length limit. The inclined angle of compressive strut q is geometrically determined as follows and was inserted in Eq. (19) with Eq. (18) and (20) to give Eq. (21)

( ) ( )

2

2 222

1sin1

n

n

h c

a h c

− − ζq = =

a + − ζ+ − (24)

• For the comparison with the test results, the characteristic compressive cylinder strength of concrete fck was used for Eq. 6.2.a of Eurocode 214 because the equation is for the purpose of design. For the same reasoning, fc′ was used for Eq. (11-3) of ACI 318-024 and the modified compression field theory (MCFT). The authors believe that even the data in Table 1 could be changed slightly by using any other type of concrete strength; the trends in the plotted data in Fig. 12 will not be changed.

AUTHORS’ CLOSUREClosure to discussion by Windisch

The authors are grateful to the discusser for his interest in the paper. Closures to each item of the discussion are listed in the following in the order they appear in the original discussion by Windisch:• Compressive stress trajectories are a clear representation

of the load path from its point of application to that of resistance. The authors believe that using the graphical results of nonlinear finite element analysis are an efficient way of showing the compressive stress trajectories and, therefore, put them in the introductory part of the paper.

• It is evident that the stress condition around the compressive strut is not as ideal as the stress in the strut in the strut-and-tie model in Fig. 2(a). The strut is affected by the flexural behavior of the member and the bond stress is the result of the shear-flexural behavior. There have been a lot of studies to consider this phenomenon in the past3 and some of them modeled the compressive force flow as a curved arch6 or a series of deviated concrete struts.7 This paper did not model the strut as curved but, rather, straight with partial impairment.

• In Fig. 3, the impairment of the flexural capacity is shown as dotted arrows in the region of strut width that are not considered effective, as opposed to the solid arrows above. The degree of impairment is quantified in


calculating the average concrete stress over the reduced width of strut fc in Eq. (20).

• The examples in Fig. 4 serve as strong evidence of tension cracks penetrating into the strut. Although the direction of the crack at the corner of the dapped end is not as shown in Fig. 4(c), the important point is that the cracks in the recent research17 still interfere with the compressive strut whether the cracks are vertical or diagonal.

• The authors intentionally used a combination of reduced strut width and parabolic stress distribution rather than the conventional combination of intact strut width and (reduced) constant stress distribution for their target-oriented research.

• In Fig. 5, the location of Point B is very important and was not determined at random. The authors believe that this can be clarified when the discusser refers to Fig. 9 and the related description in the paper. The yield line above Point B is not new but very common in finding upper-bound solutions in plasticity of concrete.

• In Fig. 7, the authors wanted to explain the phenomenon that the shear stress from the stress gradient of reinforcing bars develops bond stress between the reinforcing bar and surrounding concrete. From this point of view, the reason why there is no deviation of cracks in the pure flexural region in a four-point bending test is that there is no stress gradient of reinforcing bars in that region.

The authors believe that these closures should shed some light on the discusser’s comments.


Unbonded Tendon Stresses in Post-Tensioned Concrete Walls at Nominal Flexural Strength. Paper by Richard S. Henry, Sri Sritharan, and Jason M. Ingham

Discussion by Huanjun Jiang and Bo FuProfessor, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China; Doctoral Candidate, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University

The discussed paper developed an equation to predict the unbonded tendon stresses in post-tensioned concrete walls, which are designed for use in low seismic regions at nominal flexural strength. The authors should be complimented for their meaningful work. Some findings are interesting to the discussers and are worthy of further discussion.

1. The discussers wonder why the influences of mild reinforcements on the unbonded tendon stresses were not considered in this study. In the discussers’ opinion, the mild reinforcements can at least influence the nominal flexural strength, so they cannot be neglected. In addition, the nota-tions, such as As and fy, which denote the mild reinforcing steel area and yield stress of mild reinforcing steel, respec-tively, can be found in the Notation section of the discussed paper; however, it seems that they were not included in the body of the paper.

2. The discussers cannot quite agree with the statement in the Experimental Tests section that “up until nominal flexural strength is reached, the walls typically exhibit a nonlinear elastic response because strains in the wall toe are less than 0.003 and, therefore, cyclic loading was unlikely to modify wall response.” It is widely acknowledged that when the compressive strain of concrete arrives at approximately 0.002, the material will enter into an elastic-plastic descending section; therefore, the previous statement is not exactly correct and cyclic loading might affect the wall responses.

3. The unbonded tendon lengths of the eight specimens were provided in the paper, while the values for the walls in the parametric study were not presented. The authors should supply these to the readers.

4. There are some minor mistakes in the discussed paper. Firstly, in the Review of Existing Equations section, Eq. (1) and (2) used psi units, whereas (fse + 420) and (fse + 210) in the following paragraph used MPa units. The units should be unified. Secondly, the row heading “fse, % fy” in Tables 1 and 2 should be corrected as “fse, % fpy.” Lastly, Reference 20 was cited with a wrong page range; it should be pp. 938-946.

5. Because the authors stated in the beginning of the paper that “the accurate prediction of tendon stresses is a critical step in calculating the nominal flexural strength of unbonded post-tensioned concrete walls,” the authors should provide the corresponding formula of nominal flexural strength using the proposed equation of unbonded tendon stresses. Then, the formula can be further verified by finite element model (FEM) analysis.

AUTHORS’ CLOSUREThe authors thank the discussers for their thoughtful and

thorough comments and would like to respond to several of the points raised.

1. The stated objective of the paper was to investigate the tendon stresses in post-tensioned concrete walls that are suit-able for applications in regions with low seismicity. For this reason, a simple post-tensioned wall panel was considered without additional energy-dissipating elements, such as mild steel bars. The authors agree that mild steel bars would influ-ence the wall response and, in particular, the nominal flex-ural strength; however, the Asfy term could easily be incor-porated into the calculation of both the neutral axis depth and the nominal flexural strength. Validation of the proposed equations for use in post-tensioned walls containing mild steel bars could be investigated in a future study.

2. The authors agree that when subjected to a uniaxial compression force, unconfined concrete reaches its peak strength at a compressive strain of approximately 0.002, followed by the descending section of the stress-strain response. However, in post-tensioned concrete walls, the foundation provides some confinement to the wall toe, which limits the onset of any crushing to well beyond compressive strains of 0.003, as reported by Henry et al.9 Additionally, experimental tests by Perez et al.2 indicated that there was no significant difference between the lateral-load response of identical post-tensioned wall specimens subjected to both monotonic and cyclic loading until well beyond a compres-sive strain of 0.003 in the wall toe. For these reasons, it was


decided to use monotonic loading for both the experimental and numerical investigations reported in the paper.

3. The authors thank the discussers for noting the omis-sion of the unbonded tendon lengths that were used in the parametric study. For all walls in the parametric study, the unbonded tendon length lp was equal to the effective wall height he, plus an additional 500 mm (19.685 in.) to account for the tendon anchorages.

4. The authors thank the discussers for pointing out several typos in the paper. The previously developed equations were reported in the paper using their original units to ensure familiarity to the readers.

5. The focus of the paper was on the development of an equation to predict the unbonded tendon stresses in post-

tensioned walls at nominal flexural strength; due to space limitations, the procedures for calculating the nominal flex-ural strength of such walls were omitted. The calculation of nominal flexural strength can be completed using the equa-tion presented for predicting the unbonded tendon stresses along with a standard rectangular compression stress block assumption. Validation of this method using the experimental and numerical wall data has been published by Henry.26

REFERENCES26. Henry, R. S., “Self-Centering Precast Concrete Walls for Buildings in

Regions with Low to High Seismicity,” PhD thesis, University of Auckland, Auckland, New Zealand, 2011, http://hdl.handle.net/2292/6875.



Discussion by Bhupinder SinghPhD, Assistant Professor, Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

The authors should be complimented for carrying out a meticulously planned and carefully executed experimental investigation; the results are informative and useful. The following observations are made for their consideration:

1. Because a relatively high dosage of steel fibers was used by the authors in their concrete mixtures, a summary of the measured properties of the fresh steel fiber-reinforced concrete (SFRC) would have been useful. It would be inter-esting to know as to how the workability of fresh SFRC was characterized; what were the measured values?

2. The authors mention that the flexural performance of the SFRC mixtures was evaluated by a three-point test, but the test configuration shown in Fig. 4(b) of the paper repre-sents a four-point bending test setup.

It is suggested that the evaluation of SFRC flexural perfor-mance should have been carried out by following the provi-sions in the 2008 ACI Building Code (ACI Committee 318 2008). These provisions, although applicable for the evalua-tion of SFRC intended to be used as replacement of minimum shear reinforcement in reinforced concrete (RC) beams, would have yielded more objective results—as far as the suitability of SFRC—for use in shear-dominated structural applications or tension-controlled response modes.

3. In addition to the maximum crack widths, it would be more interesting to compare the service-load crack widths across all the deep beam specimens, which were not reported in the paper. This would enable a more relevant evaluation of the effect of SFRC on the cracking behavior of the beams.

4. In Fig. 8(b), the formation of what is evidently a split-ting crack approximately oriented along the axis of the SFRC bottle-shaped strut between the load point and the upper left

corner of the opening is likely to seriously compromise the strength of this inclined strut. Such cracking is not evident in the comparable inclined strut formed in the RC specimen of Fig. 8(a). In this context, how can the authors justify the following statement: “…cracking due to splitting of the concrete compressive strut could be delayed due to the better tensile behavior of SFRC”?

5. It is well-established in strut-and-tie model (STM) design that singular nodes, such as the CCT nodes at the supports of the authors’ beam specimens, are bottlenecks of stresses in the structural member. Schlaich et al. (1987) suggested that the entire D-region (the whole of the authors’ deep beam in the present case) would be safe if the pressure under the most heavily loaded bearing plate is controlled and, in addition, suitable reinforcement is provided to resist tensile stresses in the member. The premature crushing of concrete over one of the supports in some of the authors’ deep beam specimens indicates improper design of the support-bearing plates, resulting in an undesirable failure mode. The provision of steel cage confinement in the beams over support locations need not be a standard detailing feature in deep beams, as was suggested by the authors, provided that the support-bearing plates are properly designed.

6. The authors attributed the failure of Specimen RC2 to fracture of longitudinal tension reinforcement. Given the very large failure strains of steel reinforcement, it is not expected that crushing of concrete in some critically stressed region would preempt fracture of steel reinforcement. Failure due to fracture of tension reinforcement is very rare in RC members.





The authors claim that “it was due to the yielding of the bottom longitudinal reinforcing bars in tension.” Neverthe-less, Specimens SFRC 1 and SFRC2 had the same longit-udinal reinforcement, but the load-displacement responses were quite dangerously brittle. Please clarify why the SFRC specimens performed in a brittle manner.

The authors should be complimented for their statements and conclusions, which highlight the severe risks that may develop when a designer—having found a “lattice of full lines and dashed lines”—adapts this STM without criticism as the basis of the reinforcement pattern.

Conclusion 1 should be corrected as follows:The design STM underestimated the ultimate strength

of the test specimens, which had—different from the STM—a redundant continuous tensile reinforcement under the opening. Test specimens without these reinforcing bars would have failed at the appearance of the first cracks under the opening—far below the predicted ultimate strength. It cannot be determined whether test specimens without continuous bottom tensile reinforcement satisfy the require-ment of secondary reinforcements as per ACI 318-08, Appendix A (ACI Committee 318 2008); provisions would have reached the design strength without failure without further tests.

The load-deflection behavior of the SFRC specimens does not support the proposal made in Conclusion 5. Steel fibers serve the control of cracks—for example, due to shrinkage—nevertheless, they do not replace the ordinary reinforcement for the required load-bearing capacity. Further research is needed.

AUTHORS’ CLOSUREClosure to discussion by Singh

The authors appreciate all the comments received related to this paper and hope that this response provides clarifica-tion. Please do not hesitate to contact us if you have further comments. The following are responses provided to the raised issues:

1. The SFRC used in this study was similar to the highly flowable mixture used by the authors in prior research appli-cations (Liao et al. 2006, 2010). Figure 12 shows the state of the SFRC mixture used during the casting of the test speci-mens. No workability tests on the fresh SFRC mixture were carried out during the casting of the specimens, but tests on similar SFRC mixtures can be found elsewhere (Liao et al. 2006, 2010). Note that the flowability was further enhanced by external vibration, as shown in Fig. 12.

2. The bending test should be called the “third-point” rather than the “three-point” test, as mentioned in ASTM C1609. While ASTM C1609 could serve as the performance evaluation method for SFRC application, such as the one presented in this study, more research is needed to develop suitable criteria, similar to those given in ACI 318-08 (ACI Committee 318 2008).

The authors report on very interesting tests that help readers better understand the merits and shortages of strut-and-tie models (STMs).

The design STM (Fig. 2) adopted in this study has one remarkable characteristic: the concrete part below the opening is not needed for the equilibrium of the deep beam. Nevertheless, a deep beam without an opening requires a continuous tie between the two supports, increasing the size of the opening a frame forms—at least a continuous tie is needed there, too. This “famous” deep beam with the opening and the odd STM was published by Schlaich et al. (1987). At developing the reinforcement pattern, the authors continue the two longitudinal bottom reinforcing bars under the opening up to the left support, even if they regard this section as the “anchorage end (ANC)” of the tie only. The crack propagations shown in Fig. 6(a) and (b) reveal that both reinforced concrete (RC) test specimens would have failed at a 25 to 30 kip (112.5 to 135 kN) load—that is, far below even the very modest ultimate strength predicted by the design STM. (Conclusion 1 should be corrected accordingly.)

Apart from the local concrete damages at the supports of the specimens without the steel cages as secondary reinforce-ment, the flexural cracks controlled the failure mechanism and the ultimate strength. This reveals the fundamental char-acteristic of deep beams: they can fail in flexure only, never in compression of any type. (The corner failures in the case of Specimens RC1 and SFRC1 are node [if ever] failures, not failures of any strut.) At deep beams, the beam theory does not apply—that is, the resultant of the compressive force due to flexure is quite far below the top of the deep beam. The STM shown in Fig. 2 does not account for this.

A comparison of the STM (Fig. 2) with the crack patterns shown in Fig. 6 reveals that the system of the assumed compressive struts and the cracks contradict each other. In the case of Specimen RC2, the position and length of the flexural crack do not allow for any bottle-shaped (or even constant-width) strut.

The dots shown in Fig. 8 refer to microcracking events in concrete gathered with acoustic emission measurements. The authors interpret the target-oriented results as signs of formation of bottle-shaped struts. Nevertheless, the crack patterns—especially in the case of Specimen SFRC1 (Fig. 8(b)) over the opening—contradict this interpretation. Moreover, the discusser misses the microcracks preceding the development of cracks. Furthermore, in both specimens, numerous events are shown above the support opposite to the opening; these should precede the strut formation there. These microcracking events should occur before achieving the 65 kip (293 kN) load level. At this load level, the compressive stress in the concrete just above the support is approximately 14 N/mm2 (2 ksi), which is less than 30% of the actual concrete strength. At this stress level, the occur-rence of very few microcracks is presumable. Please clarify.

According to the load-displacement diagrams, Specimen RC2 displayed proper post-yield strain-hardening behavior.


3. The crack widths and propagation in the test specimens were measured at regular intervals during the testing. The measured crack widths at different load levels of the test specimens were reported in the paper (the Overall Cracking section) in addition to the maximum crack width at the failure load.

4. As shown in Fig. 6(a) and (c), cracks oriented along the axis between the loading point and the upper left corner of both RC and SFRC specimens. In the case of the RC specimen, a single crack was oriented along this axis located close to the bottle-shaped strut in the concrete; however, multiple cracks developed along the concrete strut formed in the SFRC specimen, as shown in Fig. 6(c). Further, the crack that developed at various load levels was similar in both RC and SFRC specimens. It should be mentioned that steel reinforcement bars were present in both perpen-dicular directions near the corner of the RC specimen, whereas these bars were completely absent in the case of Specimen SFRC1. Hence, it was inferred that the steel fibers delayed the cracking, which could have been observed much earlier in plain concrete only.

5. Based on the computer analysis (that is, CAST) with a design load of 31.3 kips (139 kN), the reaction at the support that showed premature cracking is approximately 20 kips (90 kN). This gives only a compressive stress of 0.89 ksi (6.14 MPa) with the bearing plate used in this study. As a consequence, in terms of design, the bearing plates should have been appropriate.

6. The fracture of steel, as mentioned in the paper, was observed during the failure of Specimen RC2. It should be noted that the failure was due to shear failure under the opening (Fig. 11) rather than the flexural failure, which commonly occurs in the case of slender RC flexural members. It is well-known that formation of diagonal cracks will lead to greater tensile force in flexural reinforcement (Park and Paulay 1975).

REFERENCESLiao, W.-C.; Chao, S.-H.; Park, S.-Y.; and Naaman, A. E., 2006, “Self-

Consolidating High Performance Fiber Reinforced Concrete (SCHPFRC)—Preliminary Investigation,” Report No. UMCEE 06-02, 76 pp.

Park, R., and Paulay, T., 1975, Reinforced Concrete Structures, John Wiley & Sons, Inc., New York, 769 pp.

AUTHORS’ CLOSUREClosure to discussion by Windisch

The authors appreciate all the comments received related to this paper and hope that this response provides clarifica-

tion. Please do not hesitate to contact us if you have further comments. The following are responses provided to the raised issues:

1. The authors agree that the cracks under the opening could lead to some local damage if the reinforcing bars did not pass through the segment under the opening. However, prior experimental tests had shown that this discontinuity of bars did not significantly reduce the ultimate strength of the deep beam (Breña and Morrison 2007). In Breña and Morrison’s (2007) Specimen 1A, which has exactly the same geometry and dimensions as the beams tested in this study, the longitudinal bars were not continued under the opening. Although the cracks propagated through the segment under the opening, the ultimate strength was approximately 140 kips (622.7 kN), rather than the low values predicted by the discusser.

2. Numerous investigations (for example, Hsu et al. [1963]) showed that microcracks exist in concrete even prior to application of the load. Stress concentrations at these microcracks induced by low stresses can lead to further crack propagation but in a stable manner. This crack development can occur under loads of 25 to 30% of the ultimate strength, as shown by either sonic or acoustic emission (AE) methods (Jones 1952; Brandt 2009). Therefore, it is not surprising that AE events were detected at low loads. The authors did not claim any particular shape of the strut formed in the SFRC specimens but just used the AE events to demonstrate the widespread cracks and considerable internal stress redis-tribution due to the fiber-bridging effect.

3. The SFRC specimens did not perform in a “dangerously brittle” manner. As shown in Fig. 6, the first cracking in the SFRC specimens occurred at approximately 30 kips (135 kN) (very close to the design load of 31.3 kips [139 kN]). Unlike plain concrete, the strength kept increasing up to very high loads before degradation began. This process was accom-panied by the multiple cracks in the specimens. Note that Specimen SFRC2 reached very high strength even with an extremely lower amount of reinforcing bars. The descending parts of the load-displacement responses of the SFRC speci-mens were steep compared to Specimen RC2 (perhaps this is the “dangerously brittle” behavior that the discusser referred to). The quick drop in the load-carrying capacity after signif-icant cracking and deformation (Fig. 11(a)) was due to the complete fiber pullout—thus the loss of tensile capacity of the SFRC at a few critical locations. This descending response can be significantly enhanced if these critical loca-tions are reinforced by reinforcing bars (Pareek 2011).

Fig. 12—State of fresh SFRC mixture.


4. The experimental results are opposite to the discusser’s statement. As shown in Fig. 9, the ultimate loads (96.8 kips [435 kN]) resisted by Specimen SFRC2 were three times the design value (31.3 kips [139 kN]). The analysis shown in the paper also indicated that the strength of the specimen would have been only 14.3 kips (64 kN) without any reinforcement. In addition to the enhancement in structural behavior, fiber reinforcement adds a host of both tangible and intangible benefits to the performance of a concrete structure, including cracking control, ductility, and increased impact/energy absorption capacity, as well as durability. Detailed informa-tion can be found in numerous studies.

REFERENCESBrandt, A. M., 2009, Cement-Based Composites—Materials, Mechanical

Properties and Performance, second edition, Taylor & Francis, London, UK, 526 pp.

Hsu, T. T. C.; Slate, F. O.; Sturman, G. M. M.; and Winter, G., 1963, “Microcracking of Plain Concrete and the Shape of the Stress-Strain Curve,” ACI JOURNAL, Proceedings V. 60, No. 2, Feb., pp. 209-224.

Jones, R., 1952, “A Method of Studying the Formation of Cracks in a Material Subjected to Stresses,” British Journal of Applied Physics, V. 3, No. 7, July, pp. 229-232.

Pareek, T., 2011, “Use of Steel Fiber Reinforced Concrete in Structural Members with Highly Complex Stress Fields,” master’s thesis, Department of Civil Engineering, the University of Texas at Arlington, Arlington, TX, 296 pp.


Nonlinear Cyclic Truss Model for Reinforced Concrete Walls. Paper by Marios Panagiotou, José I. Restrepo, Matthew Schoettler, and Geonwoo Kim


The authors should be complimented for proposing a new nonlinear truss modeling approach to investigate the behavior of reinforced concrete (RC) walls. Two possible limitations of this method, in the discussers’ opinion, are presented in the following.

Firstly, although the proposed model was capable of accu-rately predicting the general responses, such as the ultimate strength and stiffness upon cracking of RC walls under cyclic loadings, stresses and strains of the concrete and reinforcing bars at the locations without truss elements in the actual walls cannot be provided by the model. In addition, even if the internal forces of the truss elements can be calcu-lated, they cannot represent the actual forces of the concrete and reinforcements, as the elements were just substitutes of all the reinforcing bars and surrounding concrete in certain areas. The whole does not always represent the individual parts. For instance, the internal force of a reinforcing bar in an outer vertical truss element could not be exactly the same with the internal force of the outer vertical truss element.

Secondly, it can be easily seen in Fig. 6(b) that the model overestimates the stiffness and strength of the wall after the first crushing of diagonal concrete struts. The possible reasons are as follows. A couple of orthotropic diagonal concrete struts share a great proportion of concrete. When one concrete strut crushes, the corresponding orthotropic strut will be greatly weakened in the meantime and may even be destroyed. However, such a mechanism cannot be reflected by the model, which results in an overestimation of the actual stiffness and strength of the wall.

The research in the paper is meaningful; hence, the authors are encouraged to further improve their model and

modify it to investigate the seismic behavior of innovative wall systems, such as steel plate shear walls, in the future.

AUTHORS’ CLOSUREThe authors would like to thank the discussers for their

interest in this work and their comments. Two points were brought up as possible limitations of the proposed model.

The first was the inability of the model to compute stresses and strains at points of concrete and reinforcing bars different than the location of the elements used in the model. This is the same limitation found in finite element models, where exact stress and strain values are computed only at the inte-gration points; then, through interpolation, using shape func-tions, stresses, and strains at any point of the element, these values are computed. The same procedure can be readily used in the authors’ model.

The second point the discussers presented was that the model overestimates the overall stiffness and strength after the first crushing of the diagonal concrete struts. The discussers suggested that a possible reason for this is that the model does not account for the dependence of the properties of concrete in a diagonal on the maximum level of compres-sion previously reached in the normal direction. This depen-dence may be implicitly or explicitly accounted for by several existing constitutive models for concrete. Because the post-crushing softening behavior is the part of the response with the largest uncertainty, it is the authors’ belief that the overestimation is primarily due to this uncertainty and the associated mesh size effects. What the discussers describe is part of this uncertainty that the authors’ model—nor anyone else’s—cannot consider explicitly.





The discussed paper presented a series of tests concerning the behavior of lap-spliced plain steel bars and compared the test results with the CEB-FIP Model Code.11 Some findings are interesting to the discussers and are worthy of further discussion.

RESEARCH SIGINFICANCEThe discussed paper mentioned that the relationship for

development length as a function of bar size and splice length was provided. A similar statement can be found in the Summary and Conclusions section. However, it seems that the information about development length was not included in the body of the text. In addition, in the Summary and Conclusions section, the authors concluded that “a linear and proportional relationship for maximum load as a function of development length and bar diameter provides a best fit for the test data.” However, it can be concluded from Eq. (2) or (2a) that the maximum load is a function of spliced length and bar diameter. Therefore, the authors might have confused the difference between “development length” and “spliced length.” The differences between them can be obtained from Reference 10:

“Developed length and development length are used interchangeably to represent the bonded length of a bar that is not spliced with another bar, while spliced length and splice length are used to represent the bonded length of bars that are lapped spliced.”

EXPERIMENTAL INVESTIGATION As Eq. (2) in the next section indicated that the bar

surface roughness has, to some extent, some influence on the maximum load, the discussers are interested about the selection principle of the bar surface roughness in this study.

Were the values of Ry in this study acquired from substantial statistical data of historical bars?

EXPERIMENTAL RESULTSThe discussed paper compared the normalized maximum

loads attained by the splice specimens with deformed bars and plain bars and concluded that the splice specimens with plain bars can resist maximum loads, which are approxi-mately 60% of those identical specimens with deformed bars with the same nominal diameter. The discussers wonder why the authors did not conduct more comparisons, as the limited comparisons with only two specimens can hardly convince readers.

The discussed paper showed that the CEB-FIP Model Code11 provisions for average bond stress underestimate Pmax and proposed two empirical formulas. The discussers made a comparison between the predicted efficiency of the CEB-FIP Model Code11 and the two proposed equations. The results are shown in Table 3 and several conclusions can be drawn.

1. The CEB-FIP Model Code11 provision statisti-cally underestimates the test results for all three kinds of bars. Among them, the bars with a diameter of 19 mm (0.75 in.) made the worst prediction; however, it seems that the predicted accuracy increases with the increases of the bar diameter.

2. Both Eq. (2) and (2a) statistically underestimate the experimental results for the bars with diameters of 19 and 25 mm (0.75 and 1 in.), while they overestimate the results for the 32 mm (1.25 in.) bars.

3. For estimating the bars with diameters of 19 mm (0.75 in.), Eq. (2) and (2a) are superior to the CEB-FIP Model Code,11 while for the 25 and 32 mm (1 and 1.25 in.) bars, the superiority is hardly reflected.

Table 3—Comparisons between predicted efficiency of CEB-FIP Code11 and two empirical equations

Specimen ID

Normalized maximum load L = Pmax/√fc′, kN/√MPa Ratios between predicted and experimental results

Test: L1 CEB-FIP Model Code11 provisions: L2 Eq. (2): L3 Eq. (2a): L4 R1 = L2/L1 R2 = L3/L1 R3 = L4/L1

19-305 8.50 5.06 7.57 7.48 0.60 0.89 0.88

19-410 9.14 7.88 10.32 10.05 0.86 1.13 1.10

19-510 9.58 10.70 13.09 12.50 1.12 1.37 1.30

19-610 17.80 14.00 14.99 14.95 0.79 0.84 0.84

Average ratio for bar diameter db = 19 mm (0.75 in.) 0.69 0.87 0.86

25-410 16.20 12.40 12.47 13.22 0.77 0.77 0.82

25-510 18.40 16.00 14.73 16.45 0.87 0.80 0.89

25-610 20.60 19.40 18.20 19.67 0.94 0.88 0.95

25-810 29.70 25.10 26.63 26.12 0.85 0.90 0.88

Average ratio for bar diameter db = 25 mm (1 in.) 0.81 0.83 0.85

32-410 15.60 14.60 17.83 16.92 0.94 1.14 1.08

32-610 25.10 22.30 25.99 25.18 0.89 1.04 1.00

32-810 31.80 28.80 35.87 33.44 0.91 1.13 1.05

Average ratio for bar diameter db = 32 mm (1.25 in.) 0.92 1.14 1.07




Discussion by M. John Robert Prince and Bhupinder SinghResearch Scholar, Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India; PhD, Assistant Professor, Department of Civil Engineering, Indian Institute of Technology Roorkee

to achieving bond failure in splice beam specimens. If the intention is to enforce a bond failure, then it is desirable to restrict the splice lengths to values significantly less than the critical splice lengths. Preferred splice lengths in splice beam tests are typically less than 20 bar diameters. The use of relat-ively large splice lengths in some of the specimens by the authors—as, for example, in Specimens 19-610 and 25-810, each having a splice length of 32db—may be the reason for the measured moment capacities of these specimens being larger than their nominal yield moment capacities.

3. In structural testing, it is common to load beam speci-mens, for example, in the load-controlled mode up until the peak loads and in the displacement-controlled mode after-ward to capture the post-peak response. Load-controlled testing in the ascending branch facilitates detection and recording of cracking behavior, which may be difficult when testing is done in the displacement-controlled mode. Did the authors have any specific purpose for using only displace-ment-controlled loading in their experiments?

4. On the basis of the measured capacities of Specimens 25-410 and 25-610, the authors suggest that “splice speci-mens reinforced with plain bars are capable of resisting peak loads that are approximately 60% of those recorded for identical specimens reinforced with deformed bars of the same diameter.” There is insufficient direct empirical evidence to back up this assertion, which implies that bond strength of plain bars is a significant fraction (60%) of the bond strength of deformed bars. It is well-known that adhe-sion and friction is the dominant bond mechanism in plain bars and these get quickly lost due to slip at early stages of loading. It is reckoned that the structural capacities of Speci-mens 25-410 and 25-610 attributed by the authors to the bond resistance of plain bars has more to do, rather, with the development of an alternate load-resisting mechanism in the form of arch action in these beams, consequent to a more or less complete loss of bond at early stages of loading. Inter-estingly, the authors reported the development of arch action in their beam specimens reinforced with the plain bars.

The authors should be complimented for carrying out a meticulously planned and carefully executed experimental investigation into the bond behavior of plain steel bars. The following observations are made for their consideration and response:

1. In bond studies, for test results to be directly comparable with each other, it is very important that the confinement conditions of the longitudinal reinforcement bars across different test specimens be identical. Concrete confine-ment to the longitudinal reinforcement is usually expressed in terms of the dimensionless parameter c/db, where c may either be the side clear cover cs or the bottom clear cover cb and db is the diameter of the longitudinal reinforcement bar. For most practical beams, c/db lies in the range of 1 to 1.5. The c/db for the authors’ beams reinforced with the 19, 25, and 32 mm (0.75, 1, and 1.25 in.) diameter bars are 2.65, 2.02, and 1.56, respectively, when c is taken as the bottom clear cover cb. These values indicate that the confinement of longitudinal reinforcement in the authors’ beam specimens was neither uniform nor realistic. The practical range of c/db

values mentioned previously usually result in a splitting type of bond failure, whereas the authors reported a pullout type of bond failure in their beam specimens, which is expected, given their relatively high c/db. Because cover requirements were generally less stringent at the time when most of the historical structures were constructed, the use of relatively large c/db by the authors needs clarification.

All the beams tested by the authors had a constant thick-ness of 305 mm (12 in.) and this implies that the cs/db across their specimens varied, depending on the diameter of the longitudinal reinforcement. Ideally, for identical confine-ment conditions, both cs/db and cb/db should be nominally equal to each other and these values should, in turn, be constant across all the test specimens. For this to happen, the effective depth and the width of the beam specimens should vary, depending on the diameter of the longitudinal reinforcing bars. The authors may want to clarify this issue.

2. The upper-bound splice length of 32.4 times the longit-udinal bar diameter adopted by the authors is not conducive




The authors report on tests that should help readers better understand the behavior of lap splices; however, the success is quite limited.

In the introduction, the authors note that the reason for their research is that “the evaluation of historical reinforced concrete structures may reveal the existence of construc-tion details that do not meet current standards … deform-ation patterns that do not conform to current specifications (plain reinforcement).” It should be noted that—at least in

Europe—at the end of anchorages and lap splices of plain reinforcing bars, hooks (except in shells and thin plates) were mandatory; therefore, lap splices with straight ends could not have occurred in historical structures. Moreover, at least in Europe, the highest yield strength for plain reinforcing bars with a diameter of >8 mm (>0.3 in.) was 220 N/mm2 (31.9 ksi) with an ultimate strength of 340 N/mm2 (49.3 ksi); hence, the properties of reinforcing steels chosen for the test specimens were not representative of those in Europe.


The relevant guidelines of the CEB-FIP Model Code11 are misunderstood and misinterpreted. Equation (1) serves for the calculation of the length of the lap splice only. Along with bond characteristics, this length also comprises the errors at cutting and installation. The formula cannot be used for any calculation for splice lengths smaller than the required one. Moreover, the CEB-FIP Model Code11 explicitly notes that splices (with their full length) should possibly be placed at regions with less tensile stresses in the spliced reinforcement.

All results are presented and discussed on the basis of “normalized” values, where the applied and achieved loads are divided by the square root of the concrete compressive strength. Previous investigations7 should have shown that this relationship was valid. A comparison of Fig. 9 with Fig. 3 shows that this is not the case. The physical reason for the independence of the results from the concrete compressive strength (or its square root, which is related to the tensile strength) is that after the adhesion is destroyed,

the main source of the local bond resistance of plain bars is the friction between the bar surface and concrete.

Specimen 32-910 was identified by the authors as an “outlier”—in Fig. 9(c), it fits very well into the trend; nevertheless, Specimen 32-810 turned out to be an outlier. Specimens 19-610 and 25-610 were judged as outliers by the discusser.

The statement that the “CEB-FIP Model Code11 provi-sions for average bond stress underestimate Pmax by 15.7% on average” is not accurate. The same is valid for Eq. (2) and (2a) also. These would only fit for beams with identical geometries and loading patterns as the test specimens.

Figure 9(a) reveals that for reinforcing bars with a diameter of 19 mm (No. 6 [0.75 in.]), a minimum splice length of approximately 23.5db yields the necessary strength. For reinforcing bars with a diameter of 25 mm (No. 8 [1 in.]) (refer to Fig. 9(b)), the minimum required splice length is shown to be 28db. For reinforcing bars with a diameter of 32 mm (No. 10 [1.25 in.]), Fig. 9(c) yields the relative length of 28.4db. This trend is known: for plain bars, the required splice (and anchorage) length increases with increasing reinforcing bar diameter as the relative bonded surface decreases.

Presenting the crack patterns, the authors conclude that “vertical cracks within the shear spans are an indication of a lack of shear stresses and suggest that the load-car-rying mechanism of the specimens tended toward that of a tied arch.” Checking the maximum shear stresses at the maximum applied loads, it becomes clear that the shear stresses were far below the lower fractile value of the concrete tensile strength; thus, there was no reason for the cracks to develop as shear cracks. The shear load-bearing behavior of uncracked concrete beams has nothing to do with the tied arch model.

It would be interesting to learn the sizes of the end slips shown in Fig. 5. The end slips must somehow be equal to the sum of the crack widths measured along the splice length. Does this fit? Please clarify.

It would also be interesting if the authors reported the posi-tion of the cross sections where the test specimens failed. Were these sections within the splice lengths or just outside?

The midspan deflections shown in Fig. 6 only reveal that Response 2000 is dangerously unable to predict the brittle behavior of a simple beam with spliced tensile reinforcement.

The conclusions related to the bond stresses must be considered with caution. It is known that the measured strain/stress depends on the position of the gauge related to the next crack. The stochastic character of the relative posi-tions of the strain gauges versus the flexural cracks along the splice length makes the size and distribution of the calcu-lated bond stresses and the conclusions questionable.

AUTHORS’ CLOSUREClosure to discussion by Jiang and Fu

The authors would like to thank the discussers for their interest and discussion related to the paper.

The discussers have correctly identified that the terms “development length” and “splice length” have been used interchangeably in the paper, as defined by ACI 408R-0310 for design to mean “the length of embedded reinforcement required to develop the design strength of reinforcement.” The results of a past study, reported in Reference 23, concluded that the determination of lap splice length is directly related to development length. It is for this reason,

Fig. 9—Maximum applied load and maximum predicted load according to CEB-FIP Model Code11 versus development length as function of bar diameter: (a) db = 19 mm (No. 6 [0.75 in.]; (b) db = 25 mm (No. 8 [1 in.]); and (c) db = 32 mm (No. 10 [1.25 in.]). (Note: 1 kN = 0.2248 kips.)


along with their relative simplicity of construction and ability to realistically capture the stresses in the concrete surrounding the reinforcement, that lap-splice specimens are commonly used to determine both splice and development lengths and comprise the bulk of the test database used to establish American and Canadian code provisions for bond and development.10

Reference 7 presents the results of an extensive experi-mental investigation, with determining the influence of bar surface roughness on the bond strength of plain reinforce-ment as one of its objectives. This reference presents the findings of limited measurements of the maximum height of profile Ry of reinforcement removed from a historical structure and measurements of as-received modern plain bar stock. Perhaps even more convincing are the results of the experimental investigation as compared to those reported by Abrams2 in his historical 1913 paper. The comparison persuasively shows that the surface roughness of sandblasted bars with an average maximum height of profile of 11.3 mm (4.45 × 10–4 in.) provides a lower bound for the calculation of maximum and residual average bond stresses of historical bars. The average roughness profile of the reinforcing bars in the study is 9.44 mm (3.72 × 10–4 in.), which suggests that the results are a reasonable lower bound of what would be expected for historical plain bars.

The authors were careful to identify in the paper that the comparison with deformed bars was a rather limited one. Additional testing is ideally recommended. That said, the authors constructed specimens reinforced with plain bars only; therefore, comparisons could only be made to splice specimens reinforced with deformed bars, as reported in the available literature with similar geometry to those tested by the authors.

A more detailed evaluation of the results shown in Table 1, similar to that provided by the discussers, has been published elsewhere.24 Conclusion 1 presented by the discussers is taken directly from the paper. The authors agree with Conclusions 2 and 3 presented by the discussers, as they relate to 25 and 32 mm (1 and 1.25 in.) diameter bars only; again, these conclusions are provided in Reference 24. More specifically, Eq. (2) overestimates the maximum load for specimens with 19 and 32 mm (0.75 and 1.25 in.) diameter bars by 2% and 9%, respectively, while the same equation underestimates the maximum load for specimens reinforced with 25 mm (1 in.) bars by 20%. In comparison, Eq. 2(a) overestimates the maximum load for specimens with 32 mm (1.25 in.) diameter bars by 4% and underestimates the maximum load of specimens reinforced with 25 mm (1 in.) diameter bars by 13%. This same equation, used to predict the maximum applied load for specimens with 19 mm (0.75 in.) bars, results in a test-to-predicted ratio of 1.0.

REFERENCES 23. Orangun, C. O.; Jirsa, J. O.; and Breen, J. E., “A Reevaluation of

Test Data on Development Length and Splices,” ACI JOURNAL, Proceedings V. 74, No. 3, Mar. 1977, pp. 114-122.

24. Hassan, M. N., “Splice Tests of Plain Reinforcing Steel,” MSc thesis, University of Saskatchewan, Saskatoon, SK, Canada, 2011, 241 pp.

Closure to discussion by Prince and Singh The authors would like to thank the discussers for their

interest and comments related to the paper. The questions and concerns raised in the discussion are addressed herein.

Confinement conditions and splitting-type bond failures

Side and bottom cover, as well as spacing between the longitudinal bars, are parameters that need to be controlled when testing and comparing the results of splice speci-mens reinforced with deformed bars of varying sizes. The reason for this is that radial stresses are developed in the concrete due to bearing of the lugs of the reinforcement against the concrete upon bar slip. These radial stresses potentially lead to splitting of the surrounding concrete, thus affecting the maximum load attained by splice speci-mens. Unlike deformed bars, mechanical interlock is not expected for plain bars, as they lack lugs or other surface deformations. The results of a previous investigation of pullout specimens reinforced with plain bars presented in Reference 7 suggested that concrete cover was not a signifi-cant factor for plain bars in their as-received condition, but may be more significant—although this could not be conclu-sively shown—for the roughened bars included in this study. The bars in the study had lower roughness values than those reported in Reference 7, and observations of the splice speci-mens following testing did not show any evidence of split-ting cracks, let alone splitting failures. For these reasons, it does not appear that the need to control concrete cover and bar spacing is particularly significant for plain bars. That noted, the concrete cover selected for the specimens was based on requirements provided by ACI 318-6325—the last to include provisions for plain bars—and so are represen-tative of actual structures from that era. The clear spacing between the longitudinal bars was set at greater than twice this clear cover.

Upper bound on lap splice lengths As described in the introductory section of the paper, plain

steel bars transfer bond by two mechanisms: 1) adhesion between the reinforcement and the surrounding concrete; and 2) the wedging action of small particles that break free from the concrete upon slip. Unlike deformed bars, plain bars do not possess lugs or other surface deformations and so cannot transfer bond forces by mechanical interlock. As a result, it is fully expected that the bond performance of plain bars is inferior to that of deformed bars; therefore, longer lap splice and development lengths are required to achieve yield. Any rules of thumb related to lap-splice lengths used when designing splice specimens with deformed bars, there-fore, do not apply herein. The authors intentionally designed some specimens with very high ratios of lap splice length to bar diameter in an attempt to obtain test data for lap-splice lengths close to that which would allow for full development of the reinforcement. As the discussers have identified, the authors obtained their desired objective.

Displacement-controlled testing Displacement-controlled testing is, in fact, quite commonly

conducted, as it offers stability at all load levels and better data capture immediately preceding and following a sudden specimen failure. Similar methods of crack recording can be conducted irrespective of the method used to control the hydraulic jack.

Comparison of splice performance of plain and deformed bars

As identified in the paper, the comparison of results of splice specimens reinforced with plain and deformed bars is


a rather limited one. Further testing is ideally recommended. However, the authors would like to point out that although the adhesion mechanism for bond of plain bars is lost at rela-tively small values of slip, the sliding friction component is not. Reference 8 presents a mechanics-based bond stress-slip model developed for plain bars that shows this and then provides for experimental validation of the model. The 60% fraction as reported is perhaps then not so unexpected.

REFERENCES25. ACI Committee 318, “Building Code Requirements for Reinforced

Concrete (ACI 318-63),” American Concrete Institute, Farmington Hills, MI, 1963, 144 pp.

Closure to discussion by WindischThe authors would like to thank the discusser for his

interest and perspective related to the paper. Items raised in the discussion are addressed herein.

Anchorage and yield strength It was not until the 1951 edition of ACI 31826 that hooks

were made mandatory at the ends of plain bars. Structures built in the United States and Canada prior to this code would then potentially be reinforced with plain bars with straight ends. Furthermore, the advanced age of structures with plain bars in the United States and Canada is such that a number of these structures have, or will, undergo rehabilita-tion. One such case study of the rehabilitation of a structure reinforced with plain bars is described in Reference 27 and provides an example of a situation where there was a need to cut off the hooked ends of the plain bars, thus requiring the project engineers to assess the capacity and behavior of the affected members—a rather difficult task given the lack of literature in this area and current American and Canadian code guidance.

The yield strength of reinforcement used in Canada is a function of the construction era. According to the Cana-dian Highway Bridge Design Code,3 assuming an unknown material grade, nominal yield strengths of 210, 230, 275, and 300 MPa (30.4, 33.4, 39.9, and 43.5 ksi) can be assumed for the reinforcement when construction occurred: prior to 1914, 1914 to 1972, 1973 to 1978, and after 1978, respec-tively. Bond behavior is a function of bar force, bond stress, and slip; therefore, consideration of different yield strengths is a future research objective of the second author. Given that, the initial experimental program as presented in the paper was conducted with the most readily available material and so had a nominal yield strength of 300 MPa (43.5 MPa). The authors believe that this is a reasonable selection, given that it matches the yield strength of structural grade reinforce-ment as used after 1978 and also captures the range of inter-mediate- (otherwise known as medium-) and hard-grade reinforcement used since 1914 in Canada.3

Use of CEB-FIP Model Code11 average bond stress formula to predict failure load and identification of outliers

The authors agree with the discusser that an average bond stress can only be based on knowledge of the maximum and residual bond stresses, the bond stress distribution, and a specific development length; therefore, the use of such an equation, as is provided in the CEB-FIP Model Code,11 can only provide an approximation of failure loads for speci-mens with different lap-splice lengths. It is this limitation, coupled with the knowledge that bond stresses along a

reinforcing bar are very much nonuniform, that led to the change in American and Canadian codes to instead present provisions directly in terms of development length. That being stated, the provisions included in the CEB-FIP Model Code11 were the only provisions available in the literature that could provide an indication of failure load, and it was necessary to have some means of load prediction during the experimental design phase to ensure that all specimens would fail in bond and therefore provide meaningful test data. As the authors stated in the paper and the discusser has shown graphically, the actual-to-predicted ratio of the maximum specimen load tended to decrease with increasing bar diameter and splice length. This is true for the case of specimens included in the paper because the CEB-FIP Model Code11 provisions tend to capture the behavior of bottom-cast round bars quite reasonably.

The discusser is cautioned in his definition and applica-tion of outlying observations: an outlier should be deemed as such only if physical evidence (that is, errors in spec-imen construction or testing) or a statistical evaluation can be used to rationalize anomalous data. Specimens cannot be identified as outliers simply because their resulting data does not fit a desired trend or hypothesis. Given that, the paper clearly describes the errors in testing that led to Specimen 32-910 being identified as an outlier, while no such evidence exists to justify the exclusion of Specimens 19-610, 25-610, and 32-810 from the test database.

Normalizing by square root of concrete compressive strength

As noted by the discusser, evidence substantiating the method of normalizing the maximum load attained by the splice specimens by the square root of the concrete compres-sive strength (√fc′) is provided in Reference 7 rather than the paper under discussion. It is clearly shown in the paper that both the maximum and residual bond stresses as measured in pullout specimens are proportional to √fc′ and therefore take into account bond resistance, both when adhesion is acting and after it is destroyed. Figure 9(a) provided by the discusser includes data from specimens with 19 mm (0.75 in.) diameter bars only and, as shown in Table 1 in the paper, the range of compressive strength for the concrete varies from 17.4 to 21.0 MPa (2520 to 3040 psi); there-fore, √fc′ varies from 4.17 to 4.58 MPa (50.2 to 55.2 psi). Similarly, the range of concrete compressive strengths for specimens cast with 25 mm (1 in.) diameter bars shown in Fig. 9(b) ranged from 19.2 to 24.0 MPa (2780 to 3480 psi), and that for 32 mm (1.25 in.) diameter bars (Fig. 9(c)) ranged from 15.8 to 19.8 MPa (2290 to 2870 psi). All three groups of specimens included a very small range of concrete compressive strengths and hence even smaller ranges of √fc′, and therefore are insufficient to disprove the normalization method for the maximum applied load as used in the paper.

Location of specimen failures and relationship between bar end slip and crack widths along lap-splice length

As noted in the paper, a flexural crack that formed at one end of the lap splice length widened markedly once the maximum load was attained, indicating that this is the posi-tion along the specimen length where failure occurred. This observation was consistent for all specimens.

The authors agree with the discusser that the magnitude of the end slip of the lapped longitudinal bars should be equal to


the summation of all crack widths intercepting the bars. This is a rather simple statement to make in theory, but one that is rather difficult to confirm in practice, given the narrow width of all cracks (save for the one discussed in the previous para-graph), the nonuniform nature of crack widths along their height, and the fact that some cracks are only evident on one specimen side face. Given that only one crack in each spec-imen widened significantly, the magnitude of the end slip of the reinforcement can be assumed to be approximately equal to—but likely slightly greater than—the width of this one crack. This was physically confirmed for all specimens based on crack measurements made following specimen testing.

Orientation of cracks within shear spans Assessing the existence of cracks by calculating and

comparing the resulting shear stress acting along with the concrete tensile stress, as was done by the discusser, is an oversimplification. Rather, the stress state at any point along the beam is quite complex, as it is subject to a state of plane stress that includes normal stresses on the side faces of the element in combination with the resulting shear stress, as was identified by the discusser. The orientation of the crack can then be determined by solving for the orientation of the element when the principal stresses are acting. An analysis of the resulting stress states at different locations along the specimen length suggests that the resulting angles of crack inclination within the shear spans are less than vertical (that is, 90 degrees) and decrease with increasing applied load. As reported in the paper, however, this is contrary to the obser-vations made for all specimens, given that cracks within the

shear spans remained vertical, and resulted because cracking locally reduced the bond stresses between the reinforcing bar and the surrounding concrete and therefore reduced the formation of bond-induced shear stresses.28 The resulting change in the stress state of an element then reduces the angle of crack orientation with respect to the vertical. This reduction in bond-induced shear stresses also suggests that beam action becomes less dominant in the transfer of forces.

Use of strain gauge data The discusser’s comments related to the interpretation and

use of strain gauge data are rather general, as they apply to any experimental program using such measurement devices, rather than the specific work as presented in this paper. Having stated this, the results presented in this paper are encouraging in that similar trends were established for all instrumented specimens as related to strain compatibility, bond stress distribution along the lap-splice length, and the results of the flexural section analysis.

REFERENCES 26. ACI Committee 318, “Building Code Requirements for Reinforced

Concrete (ACI 318-51),” American Concrete Institute, Farmington Hills, MI, 1951, 63 pp.

27. Feldman, L. R.; MacFarlane, D. C.; Kroman, J. A.; and Bartlett, F. M., “Construction Staging of the Centre Street Bridge Rehabilitation to Accommodate Emergency Vehicle Traffic,” 31st Annual Conference of the Canadian Society for Civil Engineering, 2003, 10 pp. (CD-ROM)

28. Lutz, L. A., and Gergely, P., “Mechanics of Bond and Slip of Deformed Bars in Concrete,” ACI JOURNAL, Proceedings V. 64, No. 11, Nov. 1967, pp. 711-721.


IN ACI MATERIALS JOURNALThe American Concrete Institute also publishes the ACI Materials Journal. This section presents brief synopses of papers appearing in the current issue.

From the January-February 2103 issue

PDF versions of these papers are available for download at the ACI website, www.concrete.org, for a nominal fee.

110-M01—Quantifying Stress Development and Remaining Stress Capacity in Restrained, Internally Cured Mortarsby J. L. Schlitter, D. P. Bentz, and W. J. Weiss

Concrete can develop tensile stress when it is restrained from shrinking freely. Standard tests, such as the restrained ring test (ASTM C1581-09), can be used to quantify how likely it is that a mixture will crack due to the stresses developed under constant temperature conditions. The standardized restrained ring test is a passive test where the residual stress that develops due to restraint can be quantified using strains measured on the inner steel ring. The residual stress can then be compared with the concrete’s tensile strength to determine a mixture’s propensity for cracking. A new dual ring test method has been developed to characterize the early-age behavior of mixtures that expand and/or undergo a temperature change. A new testing approach uses this dual ring test to quantify the remaining stress capacity (that is, the additional stress that can be applied before the concrete develops a through crack). The new testing procedure allows stress to develop under constant temperature conditions before rapidly reducing the temperature to induce cracking. To demonstrate this approach, one plain and three internally cured mortar mixtures were tested and the results of these tests are discussed.

110-M02—Fatigue-Life Prediction of Full-Scale Concrete Pavement Overlay over Flexible Pavement: Super-Accelerated Pavement Testing Applicationby BooHyun Nam, Chul Suh, and Moon C. Won

Thin concrete pavement overlay placed on the top of flexible pavements is referred to as a thin whitetopping (TWT) pavement, which is one of the rehabilitation treatments for deteriorated flexible pavements. The primary goal of this study is to evaluate the fatigue performance of the full-scale TWT pavement due to repeated traffic loading. Super-accelerated pave-ment (SAP) tests on full-scale TWT concrete slabs were performed under static and constant cyclic loading. The stationary dynamic deflectometer (a truck-mounted SAP testing device) was used to statically and dynami-cally load the TWT slabs. To monitor the response of the TWT slabs, accel-erometers and linear variable differential transformers were installed, and the dynamic displacements of slabs were recorded during the entire testing period. The test results show that the tested slabs have dynamic displace-ment peaks around the number of load repetitions corresponding to the first visible cracks. The dynamic displacement increased at a higher rate after the occurrences of the first visible crack. In addition, the tested slabs showed stress redistribution phenomenon during the crack propagation. The concepts of stress level and equivalent fatigue life were used to eliminate, in part, influences of other factors (that is, the water-cement ratio [w/c] and aggregate type and gradation) and correct the effect of different stress ratios, respectively. The S-N curve developed from this study was very close to Thompson and Barenburg’s S-N curve after the application of the equiva-lent fatigue-life concept.

110-M03—Effects of Sand Content, High-Range Water-Reducing Admixture Dosage, and Mixing Time on Compressive Strength of Mortarby Virak Han, Soty Ros, and Hiroshi Shima

This study investigated the effects of unit sand volume, mixing time dura-tion, and high-range water-reducing admixture (HRWRA) dosage on mortar compressive strength. The experiments were conducted on many mortar mixtures made of different water-cement ratios (w/c), sand contents, mixing times, and different limestone powder replacement ratios. Consequently, the experimental results suggested that there existed a critical unit sand volume that caused a change in the mortar compressive strength. When the unit sand volume in the mortar exceeded the critical unit sand volume, the strength of

the mortar increased and decreased in the case of HRWRA dosage of 0.3% or 0.6% and 0.8%, respectively. Moreover, the critical unit sand volume was found to be 0.38 and 0.52 for a w/c of 0.3 and 0.6, respectively. Meanwhile, regardless of the w/c, the optimum duration of mixing time was 3 minutes. The optimum HRWRA dosage required for high cement dispersion leading to high compressive strength of mortar was found to be 1.0% and 1.18% for mortar with and without limestone powder, respectively.

110-M04—Self-Healing of Microcracks in High-Volume Fly-Ash-Incorporated Engineered Cementitious Compositesby Erdogan Özbay, Mustafa Sahmaran, Mohamed Lachemi, and Hasan Erhan Yücel

This paper presents the self-healing ability of engineered cementi-tious composites (ECCs) containing high volumes of fly ash (HVFA). Composites containing two different contents of fly ash (FA) (55 and 70% by weight of the total cementitious material) are examined. A splitting tensile strength test was applied to generate microcracks in ECC mixtures, wherein cylindrical specimens were preloaded up to their 85% maximum deformation capacity at 28 days. These specimens were then exposed to the further continuous wet, continuous air, wet/dry cycle curing regimes for up to 60 days. The extent of the damage was determined by using the rapid chloride permeability test (RCPT), splitting tensile tests, and micro-scopic observation. In terms of permeation properties, microcracks induced by mechanical preloading significantly increase the RCPT values of ECC mixtures. Moreover, increasing FA content is shown to have a negative effect, especially on the permeation properties of virgin ECC specimens at an early age. Without self-healing, however, the effect of mechanical preloading on the chloride-ion penetration resistance of ECC with 70% FA is lower compared to the ECC with 55% FA. The test results also indicate that continuous wet and wet/dry cycle curing contribute and speed up the healing process of the cracks, significantly improve mechanical proper-ties, and drastically decrease the RCPT of ECC. The use of HVFA in ECC production is likely to promote self-healing behavior due to tighter crack width and a higher amount of unhydrated cementitious material available for further hydration. Therefore, it appears that the curing conditions and ECC composition significantly influence the self-healing ability.

110-M05—Factors That Affect Color Loss of Concrete Paving Blocksby Federica Lollini and Luca Bertolini

Concrete paving blocks are used for several applications, such as paving of squares, parking garages, and cycling lines. They are usually made of a dual concrete mixture and the surface layer is colored through the addition of pigments. In time, the surface layer of the blocks may fade as a conse-quence of weathering due to climatic and other factors; this results in the loss of aesthetic requirements.

In this paper, several factors that could affect the color and microstruc-ture of concrete paving blocks are examined on specimens with different coloration. Color and microstructural variations were compared with new blocks. Color analysis—carried out by means of spectrophotometry—and macro- and microstructural analysis showed that the major factors that lead to color variation are wear and environmental exposure.

110-M06—Strain Measurement of Steel Fiber-Reinforced Concrete under Multi-Axial Loads with Fiber Bragg Gratingsby Robert Ritter and Manfred Curbach

During multi-axial loading tests, conventional measuring methods for the determination of strains have difficulty showing the material behavior of specimens, especially in the range of small deformations. As a result,


the measured strains are influenced within the entire loading range and a correction might be required. This paper presents a measuring method that uses fiber Bragg gratings (FBGs) to determine strains inside concrete speci-mens and provides test results as expected, even concerning small strains. By applying six measuring points in a tetrahedron-shaped arrangement, axial and shear strains can be determined. Reference measurements with strain gauges during uniaxial loading tests show that this measuring method produces reliable test results. From multi-axial loading tests with compres-sion-compression-tension loads, a realistic determination of the deforma-tion behavior was also possible.

110-M07—Chemo-Mechanical Micromodel for Alkali- Silica Reactionby Wiwat Puatatsananon and Victor Saouma

This paper presents a two-stage numerical model for alkali-silica reac-tion (ASR)/stress analysis in concrete. The coupled analytical chemo-mechanical model developed by Suwito et al. was modified to include the effects of internal moisture and ion concentration on transport properties of concrete. A finite difference model is used to simulate the coupled diffusion of alkali into concrete and subsequent ASR gel into pores surrounding the aggregates; then, a finite element model is subsequently used to perform a nonlinear analysis. This model is invoked from the master finite differ-ence model, resulting in a coupled chemo-mechanical simulation of ASR-affected concrete with aggregates of different shapes and sizes. Throughout this analysis, the authors kept track of the vertical and lateral expansions of the concrete with time which, in turn, are transformed into equivalent aniso-tropic coefficients of ASR expansion. Finally, the accuracy of the model is assessed through comparison with simulated laboratory tests.

110-M08—Water Permeability of Reinforced Concrete Subjected to Cyclic Tensile Loadingby Clélia Desmettre and Jean-Philippe Charron

A large proportion of reinforced concrete structures are cracked and subjected to cyclic loading in service. The presence of cracks enhances the ingress of aggressive agents into the concrete, resulting in faster struc-ture deterioration. On the other hand, crack self-healing and inclusion of fibers in concrete can improve structure durability. Self-healing under constant loading is a well-known phenomenon; however, the possibility for self-healing to occur under cyclic loading has received insufficient attention. This study used an innovative permeability device to investigate the water permeability of reinforced normal-strength concrete (NSC) and fiber-reinforced concrete (FRC) simultaneously subjected to tensile cyclic loading. Complementary mechanical tests were performed under the same loading procedure to assess the crack-pattern evolution. The experimental

results showed that, at an equivalent stress level in the reinforcement, the water permeability was significantly lower in the FRC than in the NSC—both under constant and cyclic loading. Moreover, two opposite phenomena occurred during cyclic loading: crack propagation and self-healing. In the NSC, the self-healing that occurred under cyclic loading compensated for the increase of permeability resulting from the crack propagation. In the FRC, crack growth was minor; the self-healing was not affected by the cyclic loading and may even have been promoted in comparison to self-healing under constant loading. The results emphasize the benefit of using FRC in structures.

110-M09—Acoustic-Emission-Based Characterization of Corrosion Damage in Cracked Concrete with Prestressing Strandby Jesé Mangual, Mohamed K. ElBatanouny, Paul Ziehl, and Fabio Matta

An accelerated corrosion study to assess the feasibility of acoustic emis-sion (AE) for the detection of active corrosion in prestressing strand is described. Concrete prisms with an embedded steel strand were corroded by supplying a constant potential between the strand and a copper plate while the specimens were immersed in a 3% NaCl solution. Corrosion was detected using the half-cell potential (HCP), steel section loss, and visual inspection. The results were compared to AE data. The location of active corrosion was determined experimentally based on the characteristic wave speed. An intensity analysis approach was used to plot the relative signifi-cance of the corrosion damage and a classification chart is presented. The results indicate that AE is a useful, nonintrusive technique for the detection and quantification of corrosion damage and may be developed as a struc-tural prognostic tool for maintenance prioritization.

110-M10—Resistance Model of Lightweight Concrete Members by Anna M. Rakoczy and Andrzej S. Nowak

The objective of this study is to develop a resistance model for light-weight concrete (LWC) beams and slabs and compare it with normalweight concrete (NWC) beams and slabs, using new material and component test data. The focus is on the development of statistical parameters for bending moment capacity and shear capacity. Resistance parameters can then serve as a basis for determination of rational resistance factors for LWC. Load-carrying capacity (resistance) is considered a product of three random vari-ables representing the uncertainty in material properties, dimensions and geometry (fabrication factor), and analytical model (professional factor). In general, it was observed that the compressive strength of LWC is similar to that of NWC; in some cases, it is even better. However, the lab tests of beams and slabs show that the current shear design procedure is less conservative for LWC compared to NWC. Therefore, the resistance factor for LWC needs to be revised.


NOTES:

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