frp-confined rc columns: analysis, · 2020. 6. 29. · a proper design procedure for frp-confined...

FRP-CONFINED RC COLUMNS: ANALYSIS,

BEHAVIOR AND DESIGN

By

Tao JIANG

A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy

Department of Civil and Structural Engineering The Hong Kong Polytechnic University

July 2008

To my wife, Xiao

II

ABSTRACT

A very popular application of FRP composites is to provide confinement to RC

columns to enhance their load carrying capacity and ductility. This method of

strengthening is based on the well-known phenomenon that the axial compressive

strength and ultimate axial compressive strain of concrete can be significantly

increased through lateral confinement. Despite the increasing popularity of this

strengthening technique, relevant design provisions in most of the existing design

guidelines for external strengthening of RC structures using FRP composites are

only applicable to the design of short columns subjected to concentric compression.

A proper design procedure for FRP-confined RC columns is urgently needed to

facilitate wider practical applications.

Against this background, this thesis is concerned with the development of a rational

design procedure for FRP-confined RC columns to correct the deficiency in

existing design guidelines. The thesis presents a systematic study covering the

behavior and modeling of FRP-confined concrete as well as the analysis and design

of FRP-confined RC columns. A series of axial compression tests on

FRP-confined concrete cylinders was conducted first to gain a good understanding

of the stress-strain behavior of FRP-confined concrete, which is fundamental and

essential to the analysis and design of FRP-confined RC columns. Stress-strain

models for FRP-confined concrete of different levels of sophistication were next

developed as a prerequisite for the analysis of FRP-confined RC columns.

Subsequently, a simple but accurate stress-strain model for FRP-confined concrete

was incorporated into a conventional section analysis procedure to develop design

equations for short FRP-confined RC columns with a negligible slenderness effect.

Finally, two theoretical models of different levels of sophistication were

developed to deal with the slenderness effect in slender FRP-confined RC

columns. The rigorous theoretical model was used to develop a slenderness limit

III

expression to differentiate short columns from slender columns while the simple

theoretical model was used to develop design equations for slender columns. The

results of the present study led to a comprehensive design procedure that includes

a set of design equations for short columns, a simple expression to separate short

columns from slender columns, and a set of design equations for slender columns.

The present study is limited to circular columns, but the framework presented in the

present study can be readily extended to FRP-confined rectangular RC columns

when an accurate stress-strain model for FRP-confined concrete in rectangular

columns becomes available. The present study has been partially motivated by the

need to formulate design provisions for the Chinese Code for the Structural Use of

FRP Composites in Construction, which is currently being finalized. This new code

has been developed within the framework of the current Chinese Code for Design

of Concrete Structures (GB-50010 2002). Therefore, some of the considerations in

the present study follow the specifications given in GB-50010 (2002) and these

considerations are highlighted where appropriate throughout the thesis.

IV

LIST OF PUBLICATIONS

Book Chapter

Teng, J.G. and Jiang, T. “Chapter 6: Strengthening of RC columns with FRP

composites”, Strengthening and Rehabilitation of Civil Infrastructures Using FRP

Composites, Woodhead Publishing Limited, UK.

Refereed Journal Papers

Jiang, T. and Teng, J.G. (2007). “Analysis-oriented models for FRP-confined

concrete”, Engineering Structures, 29(11), 2968-2986.

Teng, J.G., Jiang, T., Lam, L. and Luo, Y.Z. (2008). “Refinement of a

design-oriented stress-strain model for FRP-confined concrete”, Journal of

Composites for Construction, ASCE, submitted.

Conference Papers

Teng, J.G., Jiang, T., Lam, L. and Luo, Y.Z. (2007). “Refinement of Lam and Teng’s

design-oriented stress-strain model for FRP-confined concrete”, Proceedings, 3rd

International Conference on Advanced Composites in Construction (ACIC 2007),

2-4 April 2007, University of Bath, UK, 116-121.

Jiang, T. and Teng, J.G. (2006). “Strengthening of short circular RC columns with

FRP jackets: a design proposal”, Proceedings, 3rd International Conference on FRP

Composites in Civil Engineering, 13-15 December 2006, Miami, Florida, USA,

187-192.

Jiang, T. and Teng, J.G. (2006). “Assessment of analysis-oriented stress-strain

models for FRP-confined concrete under axial compression”, Proceedings, 4th

International Specialty Conference on Fibre Reinforced Materials, 29-31 October

2006, Hong Kong, China, 1-12.

V

ACKNOWLEDGEMENTS

The author would like to express his heartfelt gratitude to his supervisor, Professor

Jin-Guang Teng for his enlightening guidance and enthusiastic support throughout

the course of study. Professor Teng’s rigorous approach to academic research,

breadth and depth of knowledge in structural engineering, and creative and unique

insight into many academic problems have demonstrated the essential qualities

that a good researcher should possess and all of these qualities have greatly

benefited the author. The author would also like to express his warmest thanks to

his co-supervisors, Dr. Lik Lam and Professor Yao-Zhi Luo, for their generous

support and valuable suggestions and advice.

The author is grateful to both the Research Grants Council of Hong Kong Special

Administrative Region and The Hong Kong Polytechnic University for their

financial support. The author is also grateful to The Hong Kong Polytechnic

University for providing the research facilities.

The author wishes to thank the technical support from the Heavy Structures

Laboratory of the Department of Civil and Structural Engineering. The author is

very thankful to the technical staff of the laboratory, in particular, Messrs. K.H.

Wong, W.C. Chan, K. Tam, M.C. Ng, C.F. Cheung and T.T. Wai for their valuable

advice and assistance during the experimental program.

Special thanks go to the author’s friends and colleagues in the Department of Civil

and Structural Engineering of The Hong Kong Polytechnic University, in

particular, Drs. Tao Yu, Lei Zhang, Chi-Lun Ng, Hon-Ting Wong and Yuan-Feng

Duan, Messrs. Yue-Ming Hu, Dilum Fernando, Guang-Ming Chen, Shi-Shun

Zhang and Qiong-Guan Xiao, not only for their constructive discussions but also

VI

for their encouragement in times of difficulty during the course of study. The

author would also like to thank Professor Jostein Hellesland of University of Oslo

for answering the author’s queries about several academic problems.

Last but certainly not the least, the author is greatly indebted to his parents,

parents-in-law, and in particular his wife, Ms. Xiao Chen, for their constant

understanding, encouragement and love. Ms. Xiao Chen has been so considerate,

patient and supportive throughout the author’s candidature and it is she who

shared the most difficult time with the author. The author dedicates this thesis to

her with love.

VII

CONTENTS

CERTIFICATE OF ORIGINALITY I

DEDICATION II

ABSTRACT III

LIST OF PUBLICATIONS V

ACKNOWLEDGEMENTS VI

CONTENTS VIII

NOTATION XIV

CHAPTER 1 INTRODUCTION 1

1.1 BACKGROUND 1

1.2 STRENGTHENING OF RC COLUMNS WITH FRP COMPOSITES 2

1.3 OBJECTIVE AND SCOPE 3

1.4 REFERENCES 8

CHAPTER 2 LITERATURE REVIEW 11

2.1 INTRODUCTION 11

2.2 FRP-CONFINED CONCRETE IN CIRCLUAR COLUMNS UNDER

CONCENTRIC COMPRESSION 11

2.2.1 Confining Action of FRP Jacket 11

2.2.2 Dilation Properties 12

2.2.3 Ultimate Condition 13

2.2.4 Stress-Strain Curves 15

2.2.5 Stress-Strain Models 16

2.2.6 Size Effect 17

2.3 FRP-CONFINED CONCRETE IN RECTANGULAR COLUMNS

UNDER CONCENTRIC COMPRESSION 17

2.3.1 Behavior 17

2.3.2 Stress-Strain Models 18

VIII

2.3.3 Shape Modification 20

2.3.4 Size Effect 21

2.4 FRP-CONFINED CONCRETE UNDER ECCENTRIC

COMPRESSION 21

2.5 FRP-CONFINED RC COLUMNS 23

2.5.1 General 23

2.5.2 Short Columns 23

2.5.3 Slender Columns 24

2.6 ANALYTICAL AND DESIGN METHODS FOR RC COLUMNS 25

2.6.1 General 25

2.6.2 Analytical Methods 26

2.6.3 Design Methods 26

2.7 CONCLUDING REMARKS 27

2.8 REFERENCES 29

CHAPTER 3 ANALYSIS-ORIENTED STRESS-STRAIN MODELS FOR

FRP-CONFINED CONCRETE 40

3.1 INTRODUCTION 40

3.2 TEST DATABASE 42

3.2.1 General 42

3.2.2 Specimens and Instrumentation 43

3.2.3 Test Results 45

3.3 EXISTING ANALYSIS-ORIENTED MODELS FOR


3.3.1 General Concept 46

3.3.2 Peak Axial Stress Point 48

3.3.2.1 Peak axial stress 48

3.3.2.2 Axial strain at peak axial stress 49

3.3.2.3 Stress-strain equation 50

3.3.2.4 Lateral-to-axial strain relationship 51

3.4 ASSESSMENT OF EXISTING MODELS 52

3.4.1 Test Data 52

3.4.2 Dilation Properties 53

IX



3.5 REFINEMENT OF TENG ET AL.’S MODEL 57

3.5.1 General 57

3.5.2 Peak Axial Stress in the Base Model 58

3.5.3 Axial Strain at Peak Axial Stress in the Base Model 59

3.6 CONCLUSIONS 62

3.7 REFERENCES 64

CHAPTER 4 DESIGN-ORIENTED STRESS-STRAIN MODELS FOR


4.1 INTRODUCTION 98

4.2 TEST DATABASE 100

4.2.1 General 100



4.3 LAM AND TENG’S STRESS-STRAIN MODEL FOR


4.4 GENERALIZATION OF EQUATIONS 106

4.5 NEW EQUATIONS FOR THE ULTIMATE CONDITION 106

4.5.1 Ultimate Axial Strain 107

4.5.2 Compressive Strength 107

4.6 MODIFICATION TO LAM AND TENG’S MODEL: VERSION (I)

110

4.7 MODIFICATION TO LAM AND TENG’S MODEL: VERSION (II)

111

4.8 CONCLUSIONS 114

4.9 REFERENCES 116

CHAPTER 5 DESIGN OF SHORT FRP-CONFINED RC COLUMNS 131

5.1 INTRODUCTION 131

5.2 SECTION ANALYSIS 132

5.2.1 The Strength of FRP-confined RC Sections 132

X

5.2.2 Moment-Curvature Curves of FRP-confined RC Sections 135

5.2.3 Comparison with Test Results 135

5.3 EQUIVALENT STRESS BLOCK 138

5.4 DESIGN EQUATIONS 140

5.5 PERFORMANCE OF PROPOSED DESIGN EQUATIONS 142

5.6 CONCLUSIONS 145

5.7 REFERENCES 146

CHAPTER 6 ANALYSIS OF ELASTIC COLUMNS 163

6.1 INTROCUCTION 163

6.2 THE PROLBLEM OF COLUMN DESIGN 164

6.3 EXACT SOLUTIONS 164

6.3.1 General 165

6.3.2 Exact Solution for Restrained Columns 165

6.3.2.1 Deflection caused by the first-order moments 165

6.3.2.2 Deflection caused by the axial load 166

6.3.2.1 Final deflections 169

6.3.3 Exact Solution for Hinged Columns 171

6.4 BEHAVIOR OF RESTRAINED COLUMNS 173

6.5 DESIGN OF RESTRATNED COLUMNS 175

6.5.1 General 175

6.5.2 From a Hinged column to a Standard Hinged Column 175

6.5.3 From a Restrained Column to a Hinged Column 179

6.5.4 Proposed Equations 185

6.6 CONCLUSIONS 187

6.7 REFERENCES 189

CHAPTER 7 THEORETICAL MODEL FOR SLENDER FRP-CONFINED RC

COLUMNS 203


7.2 THEORETICAL MODEL 204

7.2.1 General 204

7.2.2 Construction of Axial Load-Moment-Curvature Curves 205

XI

7.2.3 Numerical Integration for the Column Deflection 205

7.2.4 Generation of the Ascending Branch of the Load-Deflection

Curve 206

7.2.5 Generation of the Descending Branch of the Load-Deflection

Curve 208

7.3 VERIFICATION OF THE THEORETICAL MODEL 209

7.3.1 Comparison with Cranston’s Numerical Results 209

7.3.2 Comparison with Experimental Results 210

7.4 CONCLUSIONS 218

7.5 REFERENCES 220

CHAPTER 8 SLENDERNESS LIMIT FOR SHORT FRP-CONFINED RC

COLUMNS 238


8.2 DEFINITION OF SLENDERNESS LIMIT 240

8.3 PARAMETRIC STUDY 242

8.3.1 Parameters Considered 242

8.3.2 Results for RC Columns 243

8.3.3 Results for FRP-confined RC Columns 245

8.4 SLENDERNESS LIMIT EXPRESSIONS FOR DESIGN USE 246

8.4.1 Slenderness Limit for RC Columns 246

8.4.2 Slenderness Limit for FRP-confined RC Columns 248

8.5 CONCLUSIONS 249

8.6 REFERENCES 251

CHAPTER 9 DESIGN OF SLENDER FRP-CONFINED RC COLUMNS 267


9.2 SIMPLE THEORETICAL MODEL 268

9.2.1 General 268

9.2.2 Method of Analysis 269

9.2.3 Accuracy of the Simple Theoretical Model 270

9.3 LIMITS ON THE USE OF FRP 272

9.4 DESIGN METHOD 275

XII

9.4.1 Review of Current Design Methods for RC Columns 275

9.4.2 Nominal Curvature 276

9.4.3 Axial Load at Balanced Failure 278

9.4.4 Factors 1ξ and 2ξ 279

9.4.5 Proposed Design Equations 282

9.5 CONCLUSIONS 285

9.6 REFERENCES 288

CHAPTER 10 CONCLUSIONS 306


10.2 BEHAVIOR OF FRP-CONFINED CONCRETE 307

10.3 MODELING OF FRP-CONFINED CONCRETE 308

10.4 ANALYSIS AND DESIGN OF FRP-CONFINED RC COLUMNS 309

10.5 FURTHER RESEARCH 310

XIII

NOTATION

A gross area of cross section

cA cross-sectional area of concrete

sA total cross-sectional area of longitudinal steel reinforcement

siA cross-sectional area of the th longitudinal steel bar i

0 0 0 0 0 0, , , , ,a b c d e f coefficients

01 02 03, ,a a a coefficients

b width of a rectangular cross section

01 02 03, ,b b b coefficients

cb width of the section at a distance cλ from the reference axis

mC equivalent uniform moment factor

d diameter of the imaginary steel cylinder

01 02 03, ,d d d coefficients

e load eccentricity

01 02 03, ,e e e coefficients

1e , 2e load eccentricities at column ends

mine minimum eccentricity

D diameter of a concrete cylinder/a circular column

E elastic modulus

2E slope of the second portion of the stress-strain curve of

FRP-confined concrete

cE elastic modulus of unconfined concrete

frpE elastic modulus of FRP

sE elastic modulus of steel

f lateral deflection of a column

XIV

midf lateral deflection at mid-height of a column

lf confining pressure provided by FRP at rupture

nomf nominal lateral deflection

yf yield stress of steel reinforcement

'ccf compressive strength of FRP-confined concrete

'*ccf peak axial stress of concrete under a specific constant

confining pressure '

cof compressive strength of unconfined concrete

cuf characteristic cube strength of unconfined concrete

'cuf axial stress at ultimate axial strain of FRP-confined concrete

1G , 2G column-to-beam stiffness ratios

fcG compressive fracture energy of unconfined concrete

sG the smaller of and 1G 2G

0h effective height of a section

H height of a concrete cylinder

I second moment of area

1k confinement effectiveness coefficient

l length of a column

cl characteristic length of a concrete cylinder

effl effective length of a column

eql equivalent length of a column

N axial load

balN axial load corresponding to balanced failure

cN axial load carried by concrete

crN Euler load of a column

sN axial load carried by longitudinal steel reinforcement

uN axial load capacity of a column

uoN section axial load capacity under concentric compression

XV

,u testN experimental axial load capacity of a column

,u theoN theoretical axial load capacity of a column

M bending moment

1M , 2M first-order moments at column ends

1M , 2M moments at column ends

cM bending moment carried by concrete

1eM , 2eM external moments at column ends

eqM equivalent uniform first-order moment

maxM maximum moment of a column

sM bending moment carried by longitudinal steel reinforcement

uM moment capacity of a column

uoM section bending moment capacity under pure bending

m total number of segments a columns is divided into q constant in Popovics’ stress-strain equation

R radius of a concrete cylinder/a circular column

1R , 2R rotational stiffnesses

sR radius of the imaginary steel cylinder

r radius of gyration

t thickness of an FRP jacket

V shear force caused by unequal 1M and 2M

V shear force caused by unequal 1M and 2M

v lateral deflection of a column due to first-order moment

w lateral deflection of a column due to axial load

x coordinate along column height

nx depth of neutral axis

1α mean stress factor

β ratio of the yield strain of steel reinforcement to the strain of

the extreme compression fiber of concrete

1β block depth factor

XVI

l∆ length of a column segment

cε axial strain of concrete

*ccε axial strain at '*

ccf

coε axial strain at the peak stress of unconfined concrete

cfε axial strain of the extreme compression fiber of concrete

cuε ultimate axial strain of FRP-confined concrete

hε hoop strain of an FRP jacket

,h rupε hoop rupture strain of an FRP jacket

lε lateral strain of concrete

siε axial strain of the th longitudinal steel bar i

tε axial strain at the transition point of the stress-strain curve of

FRP-confined concrete

yε yield strain of longitudinal steel reinforcement

θ ratio of central angle corresponding to the depth of the

equivalent stress block to 2π

0θ ratio of central angle corresponding to the depth of the neutral

axis to 2π

1θ ratio of central angle corresponding to the depth of the yielded

compressive steel reinforcement to 2π '2θ ratio of central angle corresponding to the depth of the yielded

tensile steel reinforcement to 2π

λ slenderness ratio

cλ distance from the reference axis

critλ slenderness limit for short RC columns

maxλ column slenderness above which FRP confinement has little

effect on the load-carrying capacity

φ curvature

balφ curvature corresponding to balanced failure

failφ curvature of the critical section at column failure

midφ curvature at mid-height of a column

XVII

nomφ nominal curvature

secφ maximum curvature that a section can sustain under a given

axial load ϕ moment amplification factor

1ϕ moment amplification factor for standard hinged column

Kρ confinement stiffness ratio

sρ volumetric ratio of longitudinal steel reinforcement

ερ strain ratio

cσ axial stress of concrete

hσ tensile stress in the FRP jacket in the hoop direction

lσ confining pressure

siσ axial stress of the th longitudinal steel bar i

µ effective length factor

sµ secant dilation ratio of concrete

tµ tangent dilation ratio of concrete

1ξ factor in the nominal curvature method that reflects the effect

of axial load level

2ξ factor in the nominal curvature method that reflects the effect

of column slenderness

nξ ratio of the neutral axis depth to the effective height of the

section

XVIII

CHAPTER 1

INTRODUCTION

1.1 BACKGROUND

Fiber reinforced polymer (FRP) composites comprise fibers embedded in a resin

matrix. The fibers are generally carbon, glass and aramid fibers while the resins are

generally epoxy, polyester and vinylester resins. These hi-tech materials can be ten

times as strong as mild steel but only a quarter as heavy as steel, and they are

non-corrosive. Because of these advantages and their ease in site handling derived

from their light weight and the use of adhesive bonding techniques, FRP

composites have a tremendous potential for application in the retrofit of existing

structures as well as the construction of new structures. Nevertheless, it was not

until about two decades ago did engineers and researchers begin to explore the use

of FRP composites in construction, although they have been successfully used in

other industries such as aerospace and automotive industries for many decades.

This was due mainly to their high cost (Teng et al. 2002). With their prices falling

down rapidly in recent years, and the needs for maintaining and upgrading essential

infrastructures all over the world, FRP composites have found their increasingly

wide applications in construction over the last decade.

Nowadays, various forms of FRP products, including bars, sheets, plates and

profiles, among others, are commercially available. These products have been used

in construction in many different ways: from new construction to the retrofit of

existing structures and from internal reinforcing to external strengthening (Bank

2006). As motivated by these applications and in turn to advance these applications

to a new level, related research activities have become very active in recent years.

1

Fig. 1.1 shows the results of SCI database searches using a combination of the

keywords “FRP” and “concrete”. The number of SCI journal publications in the

FRP-concrete area has increased rapidly from zero in 1990 to a total of 234 in 2007.

This exponential growth is clear evidence that intensive research has been and is

being conducted worldwide in this area. It should be noted that the publication

statistics shown in Fig. 1.1 exclude many other papers on other applications of FRP

composites, including the strengthening of steel, masonry and timber structures

using FRP composites, all FRP bridge decks, all FRP pultruded sections as beams

and columns and FRP cables.

The intensity of international research activities in this area may also be reflected

by the following events: 1) in 1996, the American Society of Civil Engineers

(ASCE) launched a new journal, the Journal of Composites for Construction,

dedicated to this new material. The impact factor of this young journal has ranked

among the top journals in the structural engineering field over the last few years; 2)

in 2004, the International Institute for FRP in Construction (IIFC) was established,

with members coming from around the world; and 3) a significant number of

international conference series have been launched and attracted numerous

attendees from both academia and industry. These conference series include the

FRPRCS (Fiber Reinforced Polymer Reinforcement for Concrete Structures) series,

the ACMBS (Advanced Composite Material in Bridges and Structures) series and

the CICE (Composites in Civil Engineering) series, among many others.

1.2 STRENGTHENING OF RC COLUMNS WITH FRP COMPOSITES

Among all the areas in construction involving the use of FRP composites, the

strengthening of reinforced concrete (RC) structures has been the most popular due

to their high strength-to-weight ratio, excellent corrosion resistance and ease of

installation.

Within this area, a very popular application of FRP composites is to provide

confinement to RC columns to enhance their load carrying capacity and ductility.

This method of strengthening is based on the well-known phenomenon that the

axial compressive strength and ultimate axial compressive strain of concrete can be

2

significantly increased through lateral confinement. RC columns may be

strengthened through FRP confinement for two purposes: 1) to increase the axial

load capacity of a column; and 2) to increase the ductility of a column. The former

mainly aims for better performance of the column under static loads such as

increases in dead load or live load while the later mainly aims for better

performance of the column under seismic loads. This thesis is limited to the

analysis, behavior and design of RC columns confined with FRP composites to

achieve the former purpose.

Various methods have been used to provide confinement to columns using FRP

composites (Teng et al. 2002). Among them, in-situ FRP wrapping has been the

most commonly used technique, in which fiber sheets or fabrics are impregnated

with resins and wrapped continuously or discretely around columns in a wet lay-up

process, with the main fibers solely or predominantly oriented in the hoop direction.

This strengthening technique is potentially effective for columns of various section

shapes, but is particularly effective for circular columns. Rectangular columns need

to receive rounding of sharp corners or shape modifications before FRP jacketing to

enhance the effectiveness of confinement. For example, a rectangular section may

be modified into an elliptical section before FRP jacketing (Teng et al. 2002). Fig.

1.2 shows the installation of FRP wraps on RC columns.

1.3 OBJECTIVE AND SCOPE

Several design guidelines (fib 2001; ISIS 2001; ACI-440.2R 2002, 2008; JSCE

2002; CNR-DT200 2004; Concrete Society 2004) for external strengthening of RC

structures using FRP composites have been published as a result of extensive

research and enormous practical needs in this field. Nevertheless, relevant design

provisions for FRP-confined RC columns in these design guidelines are only

applicable to the design of short columns with negligible slenderness effects.

Moreover, Only Concrete Society (2004) and ACI-440.2R (2008) have

recommended a procedure to perform section analysis of short FRP-confined RC

columns so that columns subjected to combined bending and axial compression can

be designed accordingly, but they do not specify the corresponding design

equations. Therefore, a proper design procedure for FRP-confined RC columns is

3

urgently needed.

Against this background, the present study aims to develop a rational design

procedure for FRP-confined RC columns to correct the deficiency in existing

design guidelines. To this end, this thesis presents a systematic study covering the

behavior and modeling of FRP-confined concrete as well as modeling of and

development of design equations for FRP-confined RC columns on a combined

experimental and analytical basis. It should be noted that the present study is

limited to circular columns, strengthened with continuous FRP wraps, with the

fibers solely or predominantly oriented in the hoop direction, but the framework

presented in this study can be readily extended to FRP-confined rectangular RC

columns if an accurate stress-strain model for FRP-confined concrete in rectangular

sections is available. Such a stress-strain model is not yet available as revealed by

the review of existing work on FRP-confined concrete in rectangular sections given

in Chapter 2. The present study has been partially motivated by the need to

formulate design provisions for the Chinese Code for the Structural Use of FRP

Composites in Construction, which is currently being finalized. This new code has

been developed within the framework of the current Chinese Code for Design of

Concrete Structures (GB-50010 2002). Therefore, some considerations in the

present study follow the specifications given in GB-50010 (2002) and they are

highlighted where appropriate throughout the thesis. The topics covered in this

thesis may be summarized as follows.

Chapter 2 presents an extensive literature review of topics related to the present

study. It starts with a brief review of existing stress-strain models for FRP-confined

concrete based on tests on small-scale specimens under concentric compression.

The suitability of applying these models in the analysis of large-scale FRP-confined

RC columns subjected to combined bending and axial compression is then

discussed based on existing experimental and analytical evidence. Existing

analytical methods as well as design methods for RC columns are also reviewed.

Chapter 3 is concerned with analysis-oriented stress-strain models for

FRP-confined concrete, and in particular, those models based on the commonly

accepted approach in which a model for actively-confined concrete is used as the

4

base model. This chapter first provides a critical review and assessment of existing

analysis-oriented models for FRP-confined concrete based on a comprehensive

database of axial compression tests on FRP-confined concrete cylinders recently

conducted at The Hong Kong Polytechnic University. This assessment clarifies

how each of the key elements forming such a model affects its accuracy and

identifies the best-performance model. The chapter then presents a refined version

of this model, which provides more accurate predictions for the stress-strain

behavior of FRP-confined concrete, particularly for weakly-confined concrete.

Chapter 4 deals with design-oriented stress-strain models for FRP-confined

concrete, in particular, the refinement of Lam and Teng’s model. More accurate

expressions for the ultimate axial strain and the compressive strength are proposed

for use in this model. These new expressions are based on experimental results from

the same test database presented in Chapter 3 as well as analytical results from a

parametric study conducted using the refined version of an analysis-oriented model

proposed in Chapter 3. The new expressions account for the effect of confinement

stiffness explicitly and can be easily incorporated into Lam and Teng’s model to

provide more accurate predictions of stress-strain curves. Based on these new

expressions, two modified versions of Lam and Teng’s model are presented. The

first version involves only simple modifications to the original model by updating

the ultimate axial strain and compressive strength equation while the second

version attempts to cater for stress-strain curves with a descending branch, which is

not covered by the original model.

Chapter 5 is concerned with the development of design equations for short

FRP-confined RC columns. Section analysis employing the modified Lam and

Teng model proposed in Chapter 4 is presented for constructing the axial

load-bending moment interaction diagram for FRP-confined RC sections. The

section analysis serves as a basis to develop design equations: the contribution of

the confined concrete to the load capacity of the section is approximated by

transforming the stress profile of concrete into an equivalent stress block; the

contribution of the longitudinal steel reinforcing bars to the load capacity of the

section is approximated by smearing the bars into an equivalent steel ring. The

proposed design equations are in a simple form that is familiar to civil engineers

5

and their performance is shown to be very good through a comprehensive

parametric study.

Chapter 6 deals with the analysis of elastic columns with elastic end restraints to lay

the ground work for the analysis of slender FRP-confined RC columns discussed in

Chapters 7 to 9. The exact solution to the lateral deflection of such columns

induced by combined bending and axial compression is derived first. The rationale

behind column design is next explained: a restrained column with unequal end

eccentricities can be transformed into an equivalent hinged column with equal end

eccentricities. Approximate design equations for elastic columns are also

developed, which represent an improvement to the existing ones.

Chapter 7 presents an analytical model for slender FRP-confined RC columns

subjected to either concentric or eccentric compression. This model allows the

initial eccentricities at the two column ends to be unequal and seeks the deflected

shape of a column in an incremental-iterative manner by making use of the axial

load-bending moment-curvature relationships. The proposed model is described in

great detail first and it is then verified against experimental results of both RC

columns and FRP-confined RC columns, with the latter being emphasized.

Chapter 8 deals with the development of a design expression for the slenderness

limit for short FRP-confined RC columns. With this expression defined, short

columns can readily be differentiated from slender columns so that they can be

designed using the equations proposed in Chapter 5. A comprehensive parametric

study is performed to investigate the effects of various parameters on the

slenderness limit using the analytical model presented in Chapter 7. Based on a

careful interpretation of the parametric study, a simple expression is proposed for

the slenderness limit for FRP-confined RC columns for design use. This expression

accounts for the effects of major parameters and possesses reasonable accuracy for

design purposes. It also reduces to its counterpart for RC columns adopted in

current design codes for RC structures when no FRP confinement is provided.

Chapter 9 is concerned with the development of design equations for slender

FRP-confined RC columns. This chapter completes the design procedure for

6

FRP-confined RC columns. A less sophisticated computer program than the one

presented in Chapter 7 but with sufficient accuracy for the analyses presented in

this chapter is described first. Using this computer program, limits on the use of

FRP are proposed for an effective and economic strengthening scheme. Design

equations are then developed following the framework of the nominal curvature

method, based on the numerical results obtained using the computer program. The

design equations are shown to provide accurate perditions through a comprehensive

parametric study.

The thesis closes with Chapter 10, where the conclusions drawn from previous

chapters are reviewed, and areas in need of further research are highlighted.

7

1.4 REFERENCES

ACI-440.2R (2002). Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, American Concrete Institute, Farmington Hills, Michigan, USA.


Bank, L.C. (2006). Composites for Construction: Structural Design with FRP Materials, John Wiley and Sons, Inc., UK.

CNR-DT200 (2004), Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Existing Structures, Advisory Committee on Technical Recommendations For Construction, National Research Council, Rome, Italy.

Concrete Society (2004). Design Guidance for Strengthening Concrete Structures with Fibre Composite Materials, Second Edition, Concrete Society Technical Report No. 55, Crowthorne, Berkshire, UK.

fib (2001). Externally Bonded FRP Reinforcement for RC Structures, International Federation for Structural Concrete, Lausanne, Switzerland.

GB-50010 (2002). Code for Design of Concrete Structures, China Architecture and Building Press, China.

ISIS (2001). Design Manual No. 4: Strengthening Reinforced Concrete Structures with Externally-Bonded Fibre Reinforced Polymers, Intelligent Sensing for Innovative Structures, Canada.

JSCE (2001). Recommendations for Upgrading of Concrete Structures with Use of Continuous Fiber Sheets, Concrete Engineering Series 41, Japan Society of Civil Engineers, Tokyo, Japan.

Teng, J.G., Chen, J.F., Smith, S.T. and Lam, L. (2002). FRP-Strengthened RC Structures, John Wiley and Sons, Inc., UK.

8

90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 070

50

100

150

200

250

Year

Ann

ual N

umbe

r of P

ublic

atio

ns

Fig. 1.1 Growth of number of SCI journal papers on the application of FRP in

concrete structures

9

(a) Circular columns

(b) Rectangular columns

Fig. 1.2 Installation of FRP wraps on RC columns (Bank 2006)

10

CHAPTER 2

LITERATURE REVIEW

2.1 INTRODUCTION

This chapter presents a review of existing knowledge of or related to FRP-confined

RC columns. Although the present study is limited to circular columns, existing

knowledge on rectangular columns is also reviewed in this chapter with the

emphasis on highlighting the uncertainties in the stress-strain behavior of

FRP-confined concrete in rectangular columns.

This chapter starts with a description of the unique behavior of FRP-confined

concrete, including its dilation properties, ultimate condition and stress-strain

behavior, observed from tests on small-scale specimens under concentric

compression. Stress-strain models dedicated to such concrete are next reviewed.

The suitability of applying these models in the analysis of large-scale FRP-confined

RC columns subjected to combined bending and axial compression is then

discussed based on existing experimental and analytical evidence, with particular

attention to the discussion of possible effects of load eccentricity and size. Lastly,

existing analytical methods as well as design methods for RC columns are reviewed

as the starting point for the analysis and design of FRP-confined RC columns.

2.2 FRP-CONFINED CONCRETE IN CIRCLUAR COLUMNS UNDER

CONCENTRIC COMPRESSION

2.2.1 Confining Action of FRP Jacket

11

The lateral confinement provided by an FRP jacket to concrete is passive in nature.

When the concrete is subjected to axial compression, it expands laterally. This

expansion is confined by the FRP jacket which is loaded in tension in the hoop

direction. Different from steel-confined concrete in which the lateral confining

pressure is constant following the yielding of steel, the confining pressure provided

by the FRP jacket increases with the lateral strain of concrete because FRP does not

yield. The confining action in FRP-confined concrete is illustrated in Fig. 2.1. The

lateral confining pressure acting on the concrete core lσ is given by

2 hl

tDσσ = (2.1)

where hσ is the tensile stress in the FRP jacket in the hoop direction, t is the

thickness of the FRP jacket, and is the diameter of the confined concrete core. If

the FRP is loaded in hoop tension only, then the hoop stress in the FRP jacket

D

hσ is

proportional to the hoop strain hε due to the linearity of FRP and is given by

h frpE hσ ε= (2.2)

where frpE is the elastic modulus of FRP in the hoop direction.

The lateral confining pressure reaches its maximum value lf at the rupture of FRP,

with

,2 2h rup frp h rupl

t Ef

D D, tσ ε

= = (2.3)

where ,h rupσ and ,h rupε are the hoop stress and strain of FRP at rupture

respectively, which are generally not the same as the ultimate tensile strength and

the ultimate tensile strain of FRP obtained from material tests as discussed later in

this chapter.

2.2.2 Dilation Properties

12

It has long been accepted that under axial compression, unconfined concrete

experiences a volumetric compaction up to 90% of the peak stress. Thereafter the

concrete shows unstable volumetric dilation due to the rapidly increasing

lateral-to-axial strain ratio. However, this lateral dilation could be effectively

contained by the FRP jacket through the confining action explained in the previous

section. With the consideration of radial displacement compatibility, the lateral

dilation results in a continuously increasing lateral confining pressure provided by

the FRP jacket which gradually reduces the rate of the lateral dilation itself. In

summary, the dilation properties of FRP-confined concrete depend on both force

equilibrium and geometric compatibility, which explains the interaction between

the FRP jacket and the concrete core.

A number of studies have thus been carried out on the dilation properties of

FRP-confined concrete (e.g. Mirmiran and Shahawy 1997; Samaan et al. 1998;

Xiao and Wu 2000; Teng et al. 2007). The secant dilation ratio sµ is commonly

used to characterize the dilation properties. Here, sµ is defined as the absolute

value of the secant slope of the lateral-to-axial strain curve of FRP-confined

concrete and is given by

ls

c

εµε

= (2.4)

where lε and cε are the lateral strain and axial strain of concrete respectively. It

should be noted that lε and hε are equal in magnitude for circular sections. A

typical experimental secant dilation ratio-axial strain curve is shown in Fig. 2.2. At

the initial stage, the secant dilation ratio has a constant value equal to the Poisson’s

ratio, and then it gradually increases as the concrete core begins to dilate. The

confining action is simultaneously activated, especially after the compressive

strength of unconfined concrete is reached. As a result, the secant dilation ratio

eventually decreases due to this continuously increasing confining pressure.

2.2.3 Ultimate Condition

13

The ultimate condition of FRP-confined concrete refers to its compressive strength

and ultimate axial strain. It is evident that the ultimate condition is closely related to

the confining pressure provided by the FRP jacket when it ruptures. Early

researchers (e.g. Samaan et al. 1998; Saffi et al. 1999; Toutanji 1999) commonly

assumed that this confining pressure is equal to the tensile strength of the same FRP

material obtained from tensile coupon tests. However, later experimental evidence

suggested that the material tensile strength of FRP cannot be reached in

FRP-confined concrete as the hoop rupture strains of FRP measured in compression

tests on FRP-confined concrete have been found to be considerably smaller than

those obtained from material tensile tests (e.g. Xiao and Wu 2000; Shahawy et al.

2000, Lam and Teng 2004). The ratio of the FRP rupture strain to the ultimate

material tensile strain is important for a stress-strain model to produce satisfactory

results. Thus, several possible causes that may result in this phenomenon have been

proposed, including the non-uniform deformation of concrete, the effect of

curvature of an FRP jacket, local misalignment or waviness of fibers, residual

strains and a multi-axial stress state, and the existence of the overlapping zone (De

Lorenzis and Tepfers 2003; Lam and Teng 2004).

Lam and Teng (2004) carried out the first carefully designed tests to investigate

these possible causes. Three types of tests, namely, flat coupon tensile tests, ring

splitting tests, and FRP-confined concrete cylinder compression tests were

conducted and compared for both GFRP and CFRP. Based on the fact that the

average FRP hoop rupture strain in all specimens was closer to the ultimate tensile

strain measured from ring splitting tests than that from flat coupon tensile tests, it

was concluded that the curvature of the jacket is a factor that affects the FRP hoop

rupture strain. Another interesting phenomenon is that for the compression tests,

the FRP hoop strains obtained from the overlapping zone fell far below those

measured outside the overlapping zone. The fact that the confining pressure is

basically the same around the whole circumference whereas the FRP jacket is

thicker in the overlapping zone can give rise to this phenomenon. Therefore, it was

concluded that the existence of the overlapping zone is another fact that affects the

average FRP hoop rupture strain. Lastly, it was discovered that even for those

readings from the strain gauges outside the overlapping zone, they fluctuated

around the average value. It was believed that this was caused by the

14

non-uniformity of the deformation of concrete.

More recently, Bisby et al. (2007) extracted the strain readings of FRP jackets

wrapped around concrete cylinders using a digital image correlation technique.

They showed that the FRP hoop strains varied significantly along the specimen

height, even within the mid-height region. Based on the data obtained using their

novel technique, it was shown that the maximum FRP hoop strain could be very

close to that obtained from tensile coupon tests. In addition, in some cases, the

maximum reading was captured at a position close to but not exactly at the

mid-height of a specimen where strain gauges are commonly installed to record the

hoop strain data in most existing tests. As a result, it can be argued that the strain

capacity and thus the material strength of FRP composites may not be significantly

undermined when they are used to confine concrete. The large scatter in the FRP

strain reduction factor reported by existing tests is mainly due to the non-uniform

nature of concrete and the incomplete measurement of FRP strains.

2.2.4 Stress-Strain Curves

It has now become well-known that the stress-strain curve of FRP-confined

concrete features a monotonically ascending bi-linear shape with sharp softening in

a transition zone around the stress level of the unconfined concrete strength (Fig.

2.3a) if the amount of FRP exceeds a certain threshold value. This type of

stress-strain curves (the ascending type) was observed in the vast majority of

existing tests. With this type of stress-strain curves, both the compressive strength

and the ultimate axial strain are reached simultaneously and are significantly

enhanced over those of unconfined concrete. However, existing tests (e.g. Arie

2001; Xiao and Wu 2000, 2003; Harries 2003) have also shown that in some cases

such a bi-linear stress-strain curve cannot be expected. Instead, the stress-strain

curve features a post-peak descending branch and the compressive strength is

reached before the tensile rupture of the FRP jacket (the descending type). It should

be noted that this type of stress-strain curve may end at a stress value (axial stress at

ultimate axial strain) either larger or smaller than the compressive strength of

unconfined concrete, as illustrated in Figs 2.3b and 2.3c respectively.

15

2.2.5 Stress-Strain Models

A significant number of stress-strain models have been proposed for FRP-confined

concrete in circular columns. These models can be classified into two categories

(Teng and Lam 2004): (a) design-oriented models (e.g. Fardis and Khalili 1982;

Karbhari and Gao 1997; Samaan et al. 1998; Miyauchi et al. 1999; Saafi et al. 1999;

Toutanji 1999; Lillistone and Jolly 2000; Xiao and Wu 2000, 2003; Lam and Teng

2003a; Berthet et al. 2006; Harajli 2006; Saenz and Pantelides 2007; Wu et al. 2007;

Youssef et al. 2007), and (b) analysis-oriented models (e.g. Mirmiran and Shahawy

1997; Harmon et al. 1998; Spoelstra and Monti 1999; Fam and Rizkalla 2001; Chun

and Park 2002; Becque et al. 2003; Harries and Kharel 2002; Marques et al. 2004;

Binici 2005; Teng et al. 2007).

Models of the first category generally comprise a closed-form stress-strain equation

and ultimate condition equations derived directly from the interpretation of

experimental results. The accuracy of design-oriented models highly depends on

the definition of the ultimate condition of FRP-confined concrete. The simple form

of design-oriented models makes them convenient for design use. Existing

design-oriented models have been assessed by a number of studies (De Lorenzis

and Tepfers 2003; Teng and Lam 2004; Bisby et al. 2005). Design-oriented models

are dealt with in Chapter 4, with an emphasis on refining Lam and Teng’s (2003a)

model.

In models of the second category, the stress-strain curves of FRP-confined concrete

are generated using an incremental numerical procedure which accounts for the

interaction between the FRP jacket and the concrete core. The accuracy of

analysis-oriented models depends mainly on the modeling of the lateral-to-axial

strain relationship of FRP-confined concrete. Analysis-oriented models are more

suitable for incorporation in computer-based numerical analysis such as nonlinear

finite element analysis. Analysis-oriented models can also be used to produce

numerical results for the development of design-oriented stress-strain models.

Analysis-oriented stress-strain models, particularly those models that employ a

stress-strain model for actively-confined concrete as a base model are assessed in

Chapter 3.

16

2.2.6 Size Effect

It should be noted that the vast majority of existing tests were conducted on

small-scale specimens and these tests served as the experimental basis to develop

various stress-strain models. Therefore, the suitability of applying these

stress-strain models to large-scale columns is still not clear. Recently, a number of

experimental studies have been carried out on large-scale columns (Youssef 2003;

Carey and Harries 2005; Mattys et al. 2005; Rocca et al. 2006; Yeh and Chang

2007). These studies indicated that the behavior of realistically-sized circular

columns could be reasonably well extrapolated from small-scale tests.

2.3 FRP-CONFINED CONCRETE IN RECTANGULAR COLUMNS

UNDER CONCENTRIC COMPRESSION

2.3.1 Behavior

FRP-confined concrete in rectangular columns has also attracted tremendous

research interest recently since rectangular columns are more common in reality. It

is well-known that FRP confinement is less effective for rectangular columns,

because concrete in rectangular columns is non-uniformly confined. As a means to

enhance the confining effect and to reduce the detrimental effect on the tensile

strength of FRP, the sharp corners of rectangular columns should be rounded before

the wrapping of FRP, as shown in Fig. 2.4, in which and are side

dimensions and is the corner radius. However, due to the existence of internal

steel reinforcement, the radius of the corners is limited in practical applications.

b h

cr

A significant number of experimental studies have been carried out on rectangular

columns (e.g. Rochette and Labossiere 2000; Chaallal et al. 2003; Lam and Teng

2003b; Youssef et al. 2007; Wang and Wu 2008). Failure was generally observed to

occur at the corners by FRP tensile rupture, as shown in Fig. 2.5. Besides the

amount of FRP confinement, other key parameters investigated by these studies

include the aspect ratio of the cross section (ratio of the longer side to the shorter

17

side of the cross section) and the radius of the rounded corners. The aspect ratio

investigated generally varied from 1 to 2 while the corner radius was generally kept

at a small value. Recently, Wang and Wu (2008) experimentally investigated the

effect of the corner radius on the confining effect using a series of specimens with

different cross-section shapes varying from square to circular shapes. All these tests

clearly showed that the effectiveness of confinement increases as the amount of

FRP or the corner radius increases and as the aspect ratio of the section reduces. Fig.

2.6 shows two experimental stress-strain curves of FRP-confined concrete in

square columns corresponding to two different amounts of FRP confinement. For

ease of comparison, the axial stress cσ and the axial strain cε are normalized by

the compressive strength of unconfined concrete and its corresponding axial

strain

'cof

coε respectively. Of the two curves, the one for a smaller corner radius and a

smaller amount of confinement features a descending branch with little strength

enhancement. It should be noted that the average axial stress (load divided by

sectional area) is used in Fig. 2.6 due to the fact that the axial stress of concrete in an

FRP-confined rectangular column varies over the section as a result of the

non-uniform confinement.

2.3.2 Stress-Strain Models

Stress-strain models proposed for FRP-confined concrete in circular columns are

not directly applicable to FRP-confined concrete in rectangular columns due to the

non-uniformity of confinement in the latter. Many stress-strain models for

FRP-confined concrete in rectangular columns have been proposed (e.g. Lam and

Teng 2003b; Harajli 2006; Pantelides and Yan 2007; Wu et al. 2007; Youssef et al.

2007). Most of these models are design-oriented, aiming to predict the average

stress-strain curves. Based on the experimental observation that the stress-strain

curves of FRP-confined concrete in rectangular columns feature a very similar

shape to that of the stress-strain curves of its counterpart in circular columns, many

researchers extended their stress-strain models for FRP-confined concrete in

circular columns to FRP-confined concrete in rectangular columns with the original

ultimate condition equations modified to account for the non-uniform confining

effect. The revised ultimate condition equations are generally based on the concepts

18

of effective confinement area and equivalent circular section to transform a

rectangular section into an equivalent circular section so that FRP-confined

concrete in circular columns and rectangular columns can be treated in a unified

manner.

Lam and Teng’s (2003b) model is used as an example herein to illustrate the

commonly accepted approach. This model is an extension of the model proposed by

the same authors (Lam and Teng 2003a) for FRP-confined concrete in circular

columns. In Lam and Teng’s (2003b) model, the effective confinement area is

contained by four parabolas as illustrated in Fig. 2.7, with the initial slopes of the

parabolas being the same as the adjacent diagonal lines. It is assumed that within

the effective confinement area, the distribution of the axial stresses is uniform and

the magnitude of the axial stresses is the average stress of the section being sought.

It is further assumed that the stress state in the effective confinement area is the

same as that in an equivalent circular section. In Lam and Teng’s (2003b) model,

the equivalent circular section circumscribes the original rectangular section (Fig.

2.7).

It is not inaccurate to say that a commonly accepted model has not yet been

identified. This is mainly due to the common and fundamental drawback of all

stress-strain models of this type: all these models have not been based on a rigorous

understanding of the confinement mechanism in rectangular sections. Instead, they

have been based on assumptions made to provide a good fit to the available test data.

As a result, the accuracy of a particular model based on a set of test data in

predicting another set of test data cannot be ensured.

Finite element models are potentially capable of accurately capturing the complex

stress variations in FRP-confined concrete in rectangular columns. Therefore, finite

element modelling offers a powerful tool for studying the confinement mechanism

of FRP-confined concrete. However, the reliability of previous finite element

studies (e.g. Mirmiran et al. 2000; Parvin and Wang 2001) is uncertain due to the

lack of in-depth studies on the constitutive model of concrete and the lack of test

results to verify the finite element model in terms of the distributions of axial stress

over the section and confining pressure around the section perimeter. Yu (2007)

19

recently conducted a critical assessment of existing Drucker-Prager type plasticity

models and proposed a plastic-damage model for concrete based on this assessment.

This plastic-damage model has led to close predictions for concrete in a number of

different stress states, including actively-confined concrete, concrete under biaxial

compression, and FRP-confined concrete in both circular and annular small-scale

columns. This plastic-damage model however needs to be further verified against

tests results of FRP-confined rectangular concrete columns in terms of both the

general behavior (i.e. average stress-strain behavior and lateral expansion behavior),

the axial stress distribution over the section, and the confining stress distribution

around the perimeter. The absence of existing test results of stress variation in

rectangular columns is mainly due to the difficulty in the measurement method. The

emerging technique of pressure films represents a significant development in the

measurement method. These pressure films can record the pressures between two

contacting surfaces and thus capture the stress variation. These films were used by

Yang et al. (2004) to record the pressures between an FRP strip and a steel substrate

with different corner radii to investigate how the distribution of the contacting

stresses vary with the corner radius, however, this technique has not been used in

any experimental work on FRP-confined concrete in the open literature.

2.3.3 Shape Modification

It has been noted that the confining effectiveness of FRP can be significantly

enhanced in rectangular columns if shape modification is implemented before FRP

jacketing (Teng et al. 2002). Among all the possible schemes of shape modification,

the most attractive one is the natural modification of a rectangular section into an

elliptical section. However, FRP-confined concrete in elliptical columns has

received much less research attention so far (Teng and Lam 2002; Yan et al. 2007;

Pantelides and Yan 2007). Yan et al. (2007) and Pantelides and Yan (2007) were

concerned with post-tensioned FRP-confined columns which involved the

application of expansive concrete. Teng and Lam (2002) have presented the only

study concerning the stress-strain behavior of and providing a strength model for

FRP-confined normal strength concrete in elliptical columns, as can be found in

open literature.

20

2.3.4 Size Effect

Again, the existing stress-strain models are mainly based on small-scale tests. Only

a limited number of studies investigated the behavior of large-scale rectangular

columns (Youssef 2003; Rocca et al. 2006). Youssef (2003) did not explicitly

address size effect in rectangular columns. Rocca et al. (2006) stated that it was

difficult to identify size effect in rectangular columns because of the large scatter of

their test results. Therefore, the size effect in rectangular columns is yet to be

clarified.

2.4 FRP-CONFINED CONCRETE UNDER ECCENTRIC

COMPRESSION

It should be noted that the experimental studies as well as the stress-strain models

reviewed so far are only concerned with FRP-confined concrete in columns

subjected to concentric compression. An important issue that needs to be clarified is

whether these models based on concentric compression tests are directly applicable

in the analysis of columns subjected to combined bending and axial compression.

Most existing studies (e.g. Saadatmanesh et al. 1994; Monti et al. 2001; Yuan and

Mirmiran 2001; Cheng et al. 2002; Teng et al. 2002; Binici 2008; Yuan et al. 2008)

adopted the conventional section analysis approach with the assumption that the

stress profile of FRP-confined concrete in the compression zone of a eccentrically

loaded section can be described using the stress-strain curve obtained from

concentric compression tests, although this assumption has not been experimentally

validated due to the scarcity of such test data and the limitations in the measurement

methods. On the experimental side, Fam et al. (2003) tested eccentrically loaded

circular concrete-filled FRP tubes subjected to a broad range of eccentricities. They

showed that the above assumption led to reasonable prediction of the strength of the

tubes they tested. Hadi (2006) tested five small-scale (150 mm in diameter) circular

normal strength concrete columns wrapped with CFRP and subjected to a fixed end

eccentricity of 42.5 mm. Four specimens were not provided with internal steel

reinforcement so they failed by the cracking of concrete on the tensile face of these

columns. Unfortunately, the hoop strains of the FRP jacket were not reported which

21

makes it difficult to use the test results to verify the above approach. More recently,

Ranger and Bisby (2007) conducted tests on small-scale (152 mm in diameter)

FRP-confined circular RC columns with a fixed height-to-diameter ratio of four.

The load eccentricities covered by their tests ranged from 0 mm to 40 mm. These

specimens were subjected to the coupled effects of load eccentricity and

slenderness and the authors did not compare their experimental results with

analytical predictions to validate the above approach. On the theoretical side, Binici

and Mosalam (2007) proposed an analytical model capable of simulating the

non-uniform hoop strain distribution in the FRP jacket wrapped around circular

columns due to eccentric compression. This model is not subjected to the

assumption given above, but the accuracy of this model depends on a number of

parameters such as the thickness of the adhesive layer which is difficult to

determine. Later, the same researcher (Binici 2008) adopted the above assumption

to perform section analysis of FRP-confined RC columns using a simple bi-linear

stress-strain model for FRP-confined concrete and showed that their section

analysis predicted the test results of Sheikh and Yau (2002) reasonably accurately.

It should be noted that Sheikh and Yau’s (2002) tests were on FRP-confined

circular RC columns subjected to constant axial loading and cyclic lateral loading,

but not on columns subjected to eccentric loading. In summary, it is deemed to be

reasonable to adopt this assumption for circular columns based on the evidence

gained so far, although more tests are needed to fully verify this assumption.

The effect of load eccentricity on the stress-strain behavior of FRP-confined

concrete in rectangular columns is uncertain. Indeed, even the stress-strain behavior

of FRP-confined concrete subjected to concentric compression needs much further

work. There is very limited experimental evidence in this area. Parvin and Wang

(2001) tested 4 small-scale square columns with small load eccentricities while Cao

et al. (2006) tested 5 medium-scale (250mm in side dimension) square columns

with various load eccentricities. These studies were concerned with overall column

behavior, so they did not clarify the effect of load eccentricity on the stress-strain

behavior of the confined concrete. In particular, a rectangular column subjected to

eccentric compression may be bent about either the major axis or the minor axis. It

is unlikely that the same stress-strain model developed from studies on

FRP-confined concrete columns under concentric compression can be used for

22

eccentric loading in both directions. Indeed, such a stress-strain model may not be

applicable to eccentric loading in either direction.

Again, finite element analysis has the potential to simulate the behavior of

FRP-confined concrete subjected to eccentric compression, both in circular and

rectangular columns. However, due to the same reasons as given earlier, existing

finite element models (e.g. Mirmiran et al. 2000; Parvin and Wang 2001; Yu 2007)

have not proven their applicability to the problem under consideration. More tests

with advanced measuring techniques to capture the key results (the distributions of

axial stress over the section and confining pressure around the section perimeter)

are needed for the verification of existing models or the development of a more

convincing finite element model.

2.5 FRP-CONFINED RC COLUMNS

2.5.1 General

Stress-strain models as discussed above are needed in the section analysis of

FRP-confined RC columns which forms an important part of a design procedure for

such columns. Relevant design provisions are now available in existing design

guidelines (fib 2001; ISIS 2001; ACI-440.2R 2002, 2008; JSCE 2002;

CNR-DT200 2004; Concrete Society 2004) for FRP-strengthened RC structures.

However, these provisions are generally concerned with the design of

concentrically loaded short FRP-confined RC columns due to the fact that the effect

of eccentricity on confinement effectiveness and the slenderness effect have not

been well understood. The behavior of FRP-confined RC columns and the relevant

provisions given in the above-mentioned design guidelines are reviewed in this

section.

2.5.2 Short Columns

A short column refers to a column with a negligible slenderness effect. The strength

design of such columns is simply a matter of constructing the axial load-bending

moment interaction diagram for the critical column section using a proper

23

stress-strain model for FRP-confined concrete. However, as the effects of load

eccentricity and size on the stress-strain behavior of FRP-confined concrete are still

uncertain, particularly for concrete in rectangular columns, there are a number of

limitations in existing design guidelines. For circular columns, only Concrete

Society (2004) and ACI-440.2R (2008) give information on the design of

eccentrically loaded columns; other guidelines are only concerned with the design

of concentrically loaded columns. For rectangular columns, only ACI-440.2R

(2008) recommends a procedure to perform section analysis; all the other design

guidelines are only applicable to columns subjected to concentric compression. In

addition, Concrete Society (2004) limits the maximum dimension of a rectangular

section to 250mm. Finally, only ISIS (2001) provides an expression to differentiate

short columns from slender columns. It should be noted that this expression is only

intended for columns with no significant bending (i.e. concentric compression or

slightly eccentric compression). Therefore, designers are faced with a number of

difficulties even when they attempt a design of FRP jackets to strengthen short RC

columns.

2.5.3 Slender Columns

No existing design guidelines have included information for the design of slender

FRP-confined RC columns. This is mainly due to the fact that only a very limited

number of studies have investigated the behavior of slender FRP-confined RC

columns, subjected to either concentric or eccentric loading (Mirmiran et al. 2001;

Yuan and Mirmiran 2001; Tao et al. 2004; Fitzwilliam and Bisby 2006). These

studies have all been concerned with small-scale circular columns.

Mirmiran et al. (2001) tested concentrically loaded fixed-fixed concrete-filled FRP

tubes with length-to-diameter ratios ranging from 2.1 to 18.6. Their test results

showed that these hybrid columns had a more dramatic loss in axial load capacity

than conventional RC columns as the slenderness increased. Yuan and Mirmiran

(2001) further developed an analytical model for such columns subjected to

eccentric compression from which they developed the slenderness limit to

distinguish short columns from slender columns and an approximate expression for

the section flexural stiffness to be used in the moment magnifier method for the

24

design of such columns classified as slender columns. These two studies are beyond

the scope of the present study since the FRP tubes used in these two studies had a

significant longitudinal stiffness.

More recently, Tao et al. (2004) and Fitzwilliam and Bisby (2006) carried out tests

on slender FRP-confined circular RC columns. The columns tested by Tao et al.

(2004) were hinged at both ends, 150mm in diameter and had length-to-diameter

ratios of either 8.4 or 20.4. The nominal end eccentricity of the columns varied from

0 to 150mm. The fibers in the FRP jacket were oriented only in the hoop direction.

In Fitzwilliam and Bisby’s (2006) test series, the columns were 152mm in diameter

and had length-to-diameter ratios up to 8. Some columns received longitudinal FRP

wrapping before hoop FRP wrapping. Both of the above two studies found that

FRP-confined RC columns experienced a larger loss in the axial load capacity than

the corresponding RC columns.

Despite the limited existing research, an important conclusion of the existing work

is that FRP-confined RC columns are subjected to a more profound slenderness

effect than their RC counterparts because FRP confinement can lead to a large

increase in the axial load capacity of an RC section but very limited increase in the

flexural rigidity of the RC section.

2.6 ANALYTICAL AND DESIGN METHODS FOR RC COLUMNS

2.6.1 General

The above review has suggested that a rational design procedure for FRP-confined

RC columns is urgently needed for inclusion in future design guidelines or design

codes. This is particularly true for slender columns where the slenderness effect

must be taken into account. The design of FRP-confined RC columns should follow

the general design procedure for RC columns that is familiar to engineers, so the

current methods dealing with the slenderness effect in RC columns adopted in

existing design codes for RC structures (e.g. ENV-1992-1-1 1992; BS-8110 1997;

GB-50010 2002; ACI-318 2005) are briefly reviewed in this section. Existing

analytical methods for slender RC columns which serve as the analytical basis to

25

develop design equations are also briefly examined.

2.6.2 Analytical Methods

Various analytical methods have been proposed for the analysis of slender columns,

as can be found in many textbooks (e.g. Chen and Atsuta 1976). Despite their

different levels of sophistication, they all aim to predict the lateral deflections of

slender columns, which give rise to the second-order moments. Two

well-established analytical methods are briefly reviewed here.

The first one is only applicable to hinged columns with equal end eccentricities. In

this method, the deflected shape of a column is assumed to be a half sine wave and

equilibrium is only checked at the critical section (the section at the mid-height of

the column) where the maximum lateral deflection of the column takes place. This

method has been widely adopted in the analysis of hinged RC columns with equal

end eccentricities (e.g. Bazant et al. 1991) and has been proven to be very

successful. More details of this method are given in Chapter 9.

The second method is more sophisticated and more versatile than the first one. This

method is generally known as the numerical integration method, in which a column

is divided into a reasonable number of segments and the lateral displacement at

each grid point is found from numerical integration by making use of the axial

load-bending moment-curvature relationship of the column. This method allows

the end eccentricities to be unequal and allows the presence of end restraints. This

method was originally proposed by Newmark (1943) and has been widely adopted

in the analysis of RC columns (e.g. Pfrang and Siess 1961; Cranston 1972), steel

columns (e.g. Shen and Lu 1983) and composite columns (e.g. Choo et al. 2006;

Tikka and Mirza 2006). More details of this method are given in Chapter 8.

2.6.3 Design Methods

The current design approach for RC columns is based on the rationale that a column

with end restraints and unequal end eccentricities can be replaced by an equivalent

hinged column with equal end eccentricities (the rationale of the current design

26

approach is explained in detail through the analysis of elastic columns presented in

Chapter 6). As a result, the design equations are for the equivalent hinged column

with the effects of the end restraints and unequal end eccentricities being dealt with

using the effective length factor and the equivalent moment factor.

There are two major methods in the current design codes for RC structures, namely,

the moment magnifier method and the nominal curvature method. Both methods

approximate the second-order moments by an amplification of the first-order

moments in order to make use of the section strength in design. In other words, the

current design approach transforms the design of a slender column into the design

of a section with an equivalent eccentricity consisting of the end eccentricity of the

slender column and a nominal lateral displacement of the critical section. Therefore,

the key in the current design approach is how to evaluate the lateral displacement of

the critical section.

The moment magnifier method is primarily used in North America (e.g. ACI-318

2005). This method originated from the analysis of elastic columns, where the

lateral deflections can be exactly evaluated provided the section flexural stiffness is

known. When this method is used for the design of RC columns, the key is to find

the equivalent section flexural stiffness that explains the effect of the nonlinearity in

material properties. The nominal curvature method is mainly used in Europe and

also China (e.g. ENV-1992-1-1 1992; BS-8110 1997; GB-50010 2002). The

nominal curvature method was originally proposed by Aas-Jakobsen and

Aas-Jakobsen (1968). This method relates the lateral displacement of the critical

section to the curvature of that section. This method is discussed in greater detail in

Chapter 8.

2.7 CONCLUDING REMARKS

This chapter has presented a review of existing studies on or relevant to

FRP-confined concrete and FRP-confined RC columns. This review has indicated

that the stress-strain behavior of FRP-confined concrete in rectangular columns,

under either concentric or eccentric compression, is still highly uncertain.

Therefore, it seems too early at this stage to carry out detailed theoretical studies on

27

the behavior of FRP-confined rectangular RC columns to develop reliable design

equations for such columns. On the other hand, the stress-strain behavior of

FRP-confined concrete in circular columns is much better understood. Existing

research suggests that the stress-strain behavior of circular columns is not

significantly affected by the possible effects of load eccentricity and size, although

more experimental evidence is still needed. The work presented in the remaining

chapters of this thesis is thus limited to circular columns. Nevertheless, the general

framework presented in this thesis for circular columns can be readily extended to

rectangular columns once an accurate stress-strain model for FRP-confined

concrete in rectangular columns becomes available.

28

2.8 REFERENCES



ACI-318 (2005). Building Code Requirements for Structural Concrete and Commentary, ACI Committee 318, American Concrete Institute, Farmington Hills, Michigan, USA..

Aire, C., Gettu, R. and Casas, J.R. (2001). “Study of the compressive behavior of concrete confined by fiber reinforced composites”, In: Figueiras, J., Juvandes, L., Faria, R., Marques, A.T., Ferreira, A., Barros, J., Appleton, J., editors, Composites in Constructions, Proceedings of the International Conference, A.A. Balkema Publishers, Lisse, The Netherlands, 239-243.

Bazant, Z.P., Cedolin, L. and Tabbara, M.R. (1991). “New method of analysis for slender columns”, ACI Structural Journal, 88(4), 391-401.

Becque, J., Patnaik, A.K. and Rizkalla, S.H. (2003). “Analytical models for concrete confined with FRP tubes”, Journal of Composites for Construction, ASCE, 7(1), 31–38.

Berthet, J.F., Ferrier, E. and Hamelin, P. (2006). “Compressive behavior of concrete externally confined by composite jackets - Part B: modeling”, Construction and Building Materials, 20(5), 338-347.

Binici, B. (2005). “An analytical model for stress–strain behavior of confined concrete”, Engineering Structures, 27(7), 1040-1051.

Binici, B. and Mosalam, K.M. (2007). “Analysis of reinforced concrete columns retrofitted with fiber reinforced polymer lamina”, Composites: Part B, 38(2), 265-276.

Binici, B. (2008). “Design of FRPs in circular bridge column retrofits for ductility enhancement”, Engineering Structures, 30(3), 766-776.

Bisby, L.A., Dent, A.J.S. and Green, M.F. (2005). “Comparison of confinement models for fiber-reinforced polymer-wrapped concrete”, ACI Structural Journal, 102(1), 62-72.

Bisby, L.A., Take, W.A. and Caspary, A. (2007). “Quantifying strain variation in FRP confined concrete using digital image correlation: proof-of-concept and initial results”, Proceeding, Asia-Pacific Conference on FRP in Structures (APFIS 2007), December 12-14, Hong Kong, China, 599-604.

29

http://apps.isiknowledge.com.ezp1.bath.ac.uk/WoS/CIW.cgi?SID=N1jGJNkF87keONjJiKn&Func=Links;ServiceName=TransferToWoS;PointOfEntry=FullRecord;request_from=UML;UT=000236311900007


BS 8110 (1997). Structural Use of Concrete, Part 1. Code of Practice for Design and Construction, British Standards Institution, London, U.K.

Cao, S., Jing, D. and Sun, N. (2006). “Experimental study on concrete columns strengthened with CFRP sheets confinement under eccentric loading”, Proceedings, 3rd International Conference on FRP Composites in Civil Engineering, December 13-15, Miami, Florida, USA, 193-196.

Carey, S.A. and Harries, K.A. (2005). “Axial behavior and modeling of confined small-, medium-, and large-scale circular sections with carbon fiber-reinforced polymer jackets”, ACI Structural Journal, 102 (4), 596-604.

Chaallal, O., Shahawy, M. and Hasssan, M. (2003). “Performance of axially loaded short columns strengthened with CFRP wrapping”, Journal of Composites for Construction, ASCE, 7(3), 200-208.

Chen, W.F. and Atsuta, T. (1976). Theory of Beam-Columns, McGraw-Hill, New York.

Cheng, H.L., Sotelino, E.D. and Chen, W.F. (2002). “Strength estimation for FRP wrapped reinforced concrete columns”, Steel and Composite Structures, 2(1), 1-20.

Choo, C.C., Harik, I.E. and Gesund, H. (2006). “Strength of rectangular concrete columns reinforced with fiber-reinforced polymer bars”, ACI Structural Journal, 103(3), 452-459.

Chun, S.S. and Park, H.C. (2002). “Load carrying capacity and ductility of RC columns confined by carbon fiber reinforced polymer” Proceedings, 3rd International Conference on Composites in Infrastructure (CD-Rom), San Francisco.

CNR-DT200 (2004), Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Existing Structures, Advisory Committee on Technical Recommendations For Construction, National Research Council, Rome, Italy.


Cranston, W.B. (1972). Analysis and Design of Reinforced Concrete Columns, Research Report 20, Cement and Concrete Association, UK.

De Lorenzis, L. and Tepfers, R. (2003). “Comparative study of models on confinement of concrete cylinders with fiber-reinforced polymer composites”, Journal of Composites for Construction, ASCE, 7(3), 219-237.

ENV 1992-1-1 (1992). Eurocode 2: Design of Concrete Structures – Part 1: General Rules and Rules for Buildings, European Committee for Standardization, Brussels.

30

Fam, A.Z. and Rizkalla, S.H. (2001). “Confinement model for axially loaded concrete confined by circular fiber-reinforced polymer tubes”, ACI Structural Journal, 98(4), 451-461.

Fam, A., Flisak, B. and Rizkalla, S. (2003). “Experimental and analytical modeling of concrete-filled fiber-reinforced polymer tubes subjected to combined bending and axial loads”, ACI Structural Journal, 100(4), 499-509.

Fardis, M.N. and Khalili, H. (1982). “FRP-encased concrete as a structural material”, Magazine of Concrete Research, 34(122), 191-202.

fib (2001). Externally Bonded FRP Reinforcement for RC Structures, The International Federation for Structural Concrete, Lausanne, Switzerland.

Fitzwilliam, J. and Bisby, L.A. (2006). “Slenderness effects on circular FRP-wrapped reinforced concrete columns”, Proceedings, 3rd International Conference on FRP Composites in Civil Engineering, December 13-15, Miami, Florida, USA, 499-502.


Hadi, M.N.S. (2006). “Behaviour of wrapped normal strength concrete columns under eccentric loading”, Composite Structures, 72(4), 503-511.

Harajli, M.H. (2006). “Axial stress-strain relationship for FRP confined circular and rectangular concrete columns”, Cement & Concrete Composites, 28(10), 938-948.

Harmon, T.G., Ramakrishnan, S. and Wang, E.H. (1998). “Confined concrete subjected to uniaxial monotonic loading.” Journal of Engineering Mechanics, ASCE, 124(12), 1303–1308.

Harries, K.A. and Kharel, G. (2002). “Behavior and modeling of concrete subject to variable confining pressure”, ACI Materials Journal, 99(2), 180-189.

Harries, K.A. and Kharel, G. (2003). “Experimental investigation of the behavior of variably confined concrete”, Cement and Concrete Research, 33(6), 873-880.



Karbhari, V.M. and Gao, Y. (1997). “Composite jacketed concrete under uniaxial compression–verification of simple design equations”, Journal of Materials in Civil Engineering, ASCE, 9(4), 185-193.

31

http://wos01.isiknowledge.com/?SID=gBb1jiI8EEe9Ce@n2hh&Func=Abstract&doc=29/1








Lam, L. and Teng, J.G. (2003a). “Design-oriented stress-strain model for FRP-confined concrete”, Construction and Building Materials,17 (6-7), 471-489.

Lam, L. and Teng, J.G. (2003b). “Design-oriented stress-strain model for FRP-confined concrete in rectangular columns”, Journal of Reinforced Plastics and Composites, 22 (13), 1149-1186.

Lam, L. and Teng, J.G. (2004). “Ultimate condition of fiber reinforced polymer-confined concrete”, Journal of Composites for Construction, ASCE, 8(6), 539-548.

Lillistone, D. and Jolly, C.K. (2000). “An innovative form of reinforcement for concrete columns using advanced composites”, The Structural Engineer, 78(23/24), 20-28.

Marques, S.P.C., Marques, D.C.S.C., da Silva J.L. and Cavalcante, M.A.A. (2004). “Model for analysis of short columns of concrete confined by fiber-reinforced polymer”, Journal of Composites for Construction, ASCE, 8(4), 332-340.

Mattys, S., Toutanji, H., Audenaert, K. and Taerwe, L. (2005). “Axial behavior of large-scale columns confined with fiber-reinforced polymer composites”, ACI Structural Journal, 102(2), 258-267.

Mirmiran, A. and Shahawy, M. (1997). “Dilation characteristics of confined concrete”, Mechanics of Cohesive-Frictional Materials, 2 (3), 237-249.

Mirmiran, A., Zagers, K. and Yuan, W.Q. (2000). “Nonlinear finite element modeling of concrete confined by fiber composites”, Finite Elements in Analysis and Design, 35, 79-96.

Mirmiran, A., Shahawy, M. and Beitleman, T. (2001). “Slenderness limit for hybrid FRP-concrete columns”, Journal of Composites for Construction, ASCE, 5(1), 26-34.

Miyauchi, K., Inoue, S., Kuroda, T. and Kobayashi, (1999). “Strengthening effects of concrete columns with carbon fiber sheet”, Transactions of the Japan Concrete Institute, 21, 143-150.

Monti, G., Nistico, N. and Santini, S. (2001). “Design of FRP jackets for upgrade of circular bridge piers”, Journal of Composites for Construction, ASCE, 5(2), 94-101.

Mosalam, K.M., Talaat, M. and Binici, B. (2007). “A computational model for reinforced concrete members confined with fiber reinforced polymer lamina: implementation and experimental validation”, Composites Part B: Engineering, 38, 598-613.

Newmark, N.M. (1943). “Numerical rocedure for computing deflections, moments, and buckling loads”, ASCE Transactions, 108, 1161-1234.

32

http://wos.isiknowledge.com/?SID=W1GLipIbOIlFBIh3cgj&Func=Abstract&doc=1/2




http://wos01.isiknowledge.com/?SID=59ef3FBCmMLLn55AiDN&Func=Abstract&doc=1/5


http://wos02.isiknowledge.com/CIW.cgi?SID=T2jA26IePK8iF6o6aom&Func=OneClickSearch&field=AU&val=Marques+DCSC&curr_doc=6/1&Form=FullRecordPage&doc=6/1

http://wos02.isiknowledge.com/CIW.cgi?SID=T2jA26IePK8iF6o6aom&Func=OneClickSearch&field=AU&val=da+Silva+JL&curr_doc=6/1&Form=FullRecordPage&doc=6/1

http://wos02.isiknowledge.com/CIW.cgi?SID=T2jA26IePK8iF6o6aom&Func=OneClickSearch&field=AU&val=Cavalcante+MAA&curr_doc=6/1&Form=FullRecordPage&doc=6/1

http://wos01.isiknowledge.com/?SID=AKdB5GJGKC9Fc@lk4Nh&Func=Abstract&doc=3/1


http://apps.isiknowledge.com/WoS/CIW.cgi?SID=P1o7MGm8knnc6aM4LLF&Func=OneClickSearch&field=AU&val=Audenaert+K&ut=000227277100009&auloc=3&curr_doc=9/1&Form=FullRecordPage&doc=9/1

http://apps.isiknowledge.com/WoS/CIW.cgi?SID=P1o7MGm8knnc6aM4LLF&Func=OneClickSearch&field=AU&val=Taerwe+L&ut=000227277100009&auloc=4&curr_doc=9/1&Form=FullRecordPage&doc=9/1



Pantelides, C.P. and Yan, Z.H. (2007). “Confinement model of concrete with externally bonded FRP jackets or posttensioned FRP shells”, Journal of Structrual Engineering, ASCE, 133(9), 1288-1296.

Parvin, A. and Wang, W. (2001). “Behavior of FRP jacketed concrete columns under eccentric loading”, Journal of Composites in Construction, ASCE, 5(3), 146-152.

Pfrang, E.O. and Siess, C.P. (1961). Analytical Study of the Behavior of Long Restrained Reinforced Concrete Columns Subjected to Eccentric Loads, Structural Research Series No. 214, University of Illinois, Urbana, Illinos.

Ranger, M. and Bisby, L.A. (2007). “Effects of load eccentricities on circular FRP-confined reinforced concrete columns”, Proceedings, 8th International Symposium on Fiber Reinforced Polymer Reinforcement for Concrete Structures (FRPRCS-8), University of Patras, Patras, Greece, July 16-18, 2007.

Rocca, S., Galati, N. and Nanni, A. (2006). “Large-size reinforced concrete columns strengthened with carbon FRP: experimental evaluation”, Proceedings, 3rd International Conference on FRP Composites in Civil Engineering, December 13-15 2006, Miami, Florida, USA.

Rochette, P. and Labossiere, P. (2000). “Axial testing of rectangular column models confined with composites”, Journal of composites for Construction, ASCE, 4(3), 129-136.

Saadatmanesh, H., Ehsani, M.R. and Li, M.W. (1994). “Strength and ductility of concrete columns externally reinforced with fiber composites straps”, ACI Structural Journal, 91(4), 434-447.

Saafi, M., Toutanji, H.A. and Li, Z. (1999). “Behavior of concrete columns confined with fiber reinforced polymer tubes”, ACI Materials Journal, 96(4), 500-509.

Saenz, N. and Pantelides, C.P. (2007). “Strain-based confinement model for FRP-confined concrete”, Journal of Structural Engineering, ASCE, 133 (6), 825-833.

Samaan, M., Mirmiran, A. and Shahawy, M. (1998). “Model of concrete confined by fiber composite”, Journal of Structural Engineering, ASCE, 124(9):, 1025-1031.

Shahawy, M., Mirmiran, A. and Beitelman, T. (2000). “Tests and modeling of carbon-wrapped concrete columns”, Composites Part B-Engineering, 31(6-7), 471-480.

Sheikh, S.A. and Yau, G. (2002). “Seismic behavior of concrete columns confined with steel and fiber-reinforced polymers”, ACI Structural Journal, 99(1), 72-80.

Shen, Z.Y. and Lu, L.W. (1983). “Analysis of initially crooked, end restrained steel columns”, Journal of Constructional steel research, 3(1), 10-18.

33





Spoelstra, M.R. and Monti, G. (1999). “FRP-confined concrete model”, Journal of Composites for Construction, ASCE, 3(3), 143-150.

Tao, Z., Teng, J.G., Han, L.H. and Lam, L. (2004). “Experimental behaviour of FRP-confined slender RC columns under eccentric loading”, Proceedings, 2nd International Conference on Advanced Polymer Composites in Construction, University of Surrey, UK, 20-22 April 2004, 203-212.

Teng, J.G. and Lam, L. (2002). “Compressive behavior of carbon fiber reinforced polymer-confined concrete in elliptical columns”, Journal of Structural Engineering, ASCE, 128(12), 1535-1543.

Teng, J.G. and Lam, L. (2004). “Behavior and modeling of fiber reinforced polymer-confined concrete”, Journal of Structural Engineering, ASCE, 130(11), 1713-1723.

Teng, J.G., Chen, J.F., Smith, S.T. and Lam. L. (2002) FRP-Strengthened RC Structures, John Wiley and Sons, Inc., UK.

Teng, J.G., Huang, Y.L. Lam. L and Ye L.P. (2007). “Theoretical model for fiber reinforced polymer-confined concrete”, Journal of Composites for Construction, ASCE, 11(2), 201-210.

Tikka, T.M. and Mirza, S.A. (2006). “Nonlinear equation for flexural stiffness of slender composite columns in major axis bending”, Journal of Structural Engineering, ASCE, 132(3), 387-399.

Toutanji, H.A. (1999). “Stress-strain characteristics of concrete columns externally confined with advanced fiber composite sheets”, ACI Materials Journal, 96(3), 397-404.

Wang, L.M. and Wu, Y.F. (2008). “Effect of corner radius on the performance of CFRP-confined square concrete columns: Test”, Engineering Structures, 30(2), 493-505.

Wu, G., Wu, Z.S. and Lu, Z.T. (2007). “Design-oriented stress-strain model for concrete prisms confined with FRP composites”, Construction and Building Materials, 21(5), 1107-1121.

Xiao, Y. and Wu, H. (2000), “Compressive behavior of concrete confined by carbon fiber composite jackets”, Journal of Materials in Civil Engineering, ASCE, 12(2), 139-146.

Xiao, Y. and Wu, H. (2003), “Compressive behavior of concrete confined by various types of FRP composite jackets’, Journal of Reinforced Plastics and Composites, 22(13), 1187-1201.

Yan, Z.H. and Pantelides, C.P. and Reaveley, L.D. (2007). “Posttensioned FRP composite shells for concrete confinement”, Journal of Composites for Construction, ASCE, 11(1), 81-90.

34







http://wos.isiknowledge.com/?SID=Z2JooPg9@cOi81a@bhI&Func=Abstract&doc=1/1


Yeh, F.Y. and Chang, K.C. (2007). "Confinement efficiency and size effect of FRP confined circular concrete columns", Structural Engineering and Mechanics, 26(2), 127-150.

Youssef, M.N. (2003). Stress-strain Model for Concrete Confined by FRP Composites, Ph.D. Dissertation, University of California, Irvine.

Youssef, M.N., Feng, M.Q. and Mosallam, A.S. (2007). “Stress-strain model for concrete confined by FRP composites”, Composites Part B – Engineering, 38(5-6), 614-628.

Yu, T. (2007). Structural Behavior of Hybrid FRP-Concrete-Steel Double-Skin Tubular Columns, Ph.D. thesis, The Hong Kong Polytechnic University.

Yuan, W. and Mirmiran, A. (2001). “Buckling analysis of concrete-filled FRP tubes”, International Journal of Structural Stability and Dynamics, 1(3), 367-383.

Yuan, X.F., Xia, S.H., Lam, L. and Smith, S.T. (2008). “Analysis and behaviour of FRP-confined short concrete columns subjected to eccentric loading”, Journal of Zhejiang University Science A, 9(1), 38-49.

35



t

frp hE tε frp hE tε

lσ

D

Fig. 2.1 Confining action of FRP jacket

0 0.005 0.01 0.015 0.02 0.025 0.030

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Axial Strain εc

Sec

ant D

ilatio

n R

atio

µs

Fig. 2.2 Typical secant dilation ratio curve

36

cuεAxial Strain εc ε

'ccf

Axi

al S

tress

σc

'cof

(a) Ascending type

cuεccε

'cuf

'ccf

'cof

Axi

al S

tress

σc

Axial Strain εc

(b) Descending type with ' 'cu cof f≥

cuε

'cof

'ccf

'cuf

ccε

Axi

al S

tress

σc

Axial Strain εc

(c) Descending type with ' 'cu cof f<

Fig. 2.3 Classification of stress-strain curves of FRP-confined concrete

37

h

b

cr

Effective confinement area

Fig. 2.4 Effective confinement area of a rectangular section

(a) Square column (b) Rectangular column

Fig. 2.5 Failure of FRP-confined square and rectangular concrete columns with

rounded corners by FRP rupture

38

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

Normalized Axial Strain εc/εco

Nor

mal

ized

Axi

al S

tress

σc/

f ′ co

Fig. 2.6 Typical stress-strain curves of FRP-confined concrete in square columns

h

( )arctan b h

b

cr

Effective confinement area

D

Fig. 2.7 Lam and Teng’s model for FRP-confined concrete in rectangular columns

39

CHAPTER 3

ANALYSIS-ORIENTED STRESS-STRAIN MODELS

FOR FRP-CONFINED CONCRETE

3.1 INTRODUCTION

As reviewed in Chapter 2, existing stress-strain models fall into two main

categories: (1) design-oriented models and (2) analysis-oriented models. The

former models are generally in closed-form equations directly derived from test

results, treating FRP-confined concrete as a single “composite” material, and are

thus simple and convenient to apply in design. By contrast, the latter models treat

the FRP jacket and the concrete core separately, and predict the behavior of

FRP-confined concrete by an explicit account of the interaction between the FRP

jacket and the confined concrete core via radial displacement compatibility and

equilibrium conditions. Analysis-oriented models are more versatile and accurate

in general, are often the preferred choice for use in more involved analysis than is

required in design (e.g. nonlinear finite element analysis of concrete structures

with FRP confinement), and are applicable/easily extendible to concrete confined

with materials other than FRP. They can also be employed to generate numerical

results for use in the development of a design-oriented model. This chapter deals

with analysis-oriented models; design-oriented models are discussed in Chapter 4.

The confinement provided by an FRP jacket to a concrete core is passive rather

than active, as the confining pressure from the jacket is induced by and increases

with the expansion of the concrete core. In most existing analysis-oriented models

for FRP-confined concrete, a theoretical model for actively-confined concrete (i.e.

the confining pressure is externally applied and remains constant as the axial

40

stress increases), which is referred to as an active-confinement model for brevity

hereafter, is employed as the base model; the axial stress-axial strain curve

(simply referred to as the axial stress-strain curve or the stress-strain curve

hereafter) of FRP-confined concrete is then generated through an incremental

process, with the resulting stress-strain curve crossing a family of stress-strain

curves for the same concrete under different levels of active confinement (Teng

and Lam 2004). Models of Mirmiran and Shahawy (1996, 1997a), Spoelstra and

Monti (1999), Fam and Rizkalla (2001), Chun and Park (2002), Harries and

Kharel (2002), Marques et al. (2004), Binici (2005) and Teng et al. (2007a) are all

of this type. Two other models (Harmon et al. 1998; Becque et al. 2003) adopted

alternative approaches for modelling the concrete (Teng and Lam 2004). The first

approach, in which an active-confinement model is used, has been much more

popular than the other approaches as it leads to conceptually simple yet effective

models. This chapter is limited to analysis-oriented models developed through this

approach. For a brief discussion of the models by Harmon et al. (1998) and

Becque et al. (2003), the reader is referred to Teng and Lam (2004).

It should be noted that among the models with an active-confinement base model,

the models of Spoelstra and Monti (1999), Fam and Rizkalla (2001), Chun and

Park (2002), and Harries and Kharel (2002) have been briefly summarized and

assessed through comparisons with test stress-strain curves of FRP-confined

concrete with a significant level of confinement (dependent on the hoop

membrane stiffness of the FRP jacket) so that the stress-strain curves are

monotonically ascending. The present chapter extends the work of Teng and Lam

(2004) in the following aspects:

(a) apart from the four models mentioned above, the model of Mirmiran and

Shahawy (1997a) together with three more recent models (Marques et al. 2004;

Binici 2005 and Teng et al. 2007a) are also included in the critical review and

assessment;

(b) a much more thorough assessment is presented for all these models by

considering different levels of confinement covering both the ascending and

descending types of stress-strain curves, with the emphasis being on the accuracy

41

of the key elements of such models, including the lateral-to-axial strain

relationship for FRP-confined concrete, as well as the stress-strain equation and

the peak axial stress point of the active-confinement base model;

(c) new results from recent tests conducted by the author are employed in the

assessment and some of these tests are for FRP-confined concrete with a

descending stress-strain curve (i.e. weakly-confined concrete) which is less well

understood and for concrete with very strong FRP confinement which has

received little attention in previous work.

(d) the lateral-to-axial strain relationship, which takes various forms and is

essential to this type of models (Teng and Lam 2004), is thoroughly examined;

and

(e) finally, a refined version of Teng et al.’s (2007a) model is presented to provide

more accurate predictions, particularly for weakly-confined concrete.

3.2 TEST DATABASE

3.2.1 General

A test database, containing the results of axial compression tests of 48

FRP-confined concrete cylinders (diameter 152D = mm and height

mm), is employed herein to evaluate the performance of existing analysis-oriented

stress-strain models. All these tests have recently been conducted at The Hong

Kong Polytechnic University, so the tests were conducted under standardized

conditions and all information required for evaluating stress-strain models can be

readily and accurately extracted.

305H =

Among these 48 tests, 25 of them have been published [i.e. specimens 01 to 13 in

Lam and Teng (2004); specimens 14 to 19 in Lam et al. (2006) and Specimens 20

to 25 in Teng et al. (2007b)] and were used in developing Teng et al.’s (2007a)

model, the other 23 tests are new tests that have never been published or used in

developing any of the existing stress-strain models. The new tests significantly

42

widened the range of confinement ratios from 0.08-0.46 for the 25 published tests

to 0.07-0.99 for the 48 tests considered in this chapter; some of the test

stress-strain curves feature a descending branch while others are rapidly ascending.

The confinement ratio 'l cof f refers to the ratio of the confinement pressure lf

at jacket rupture to the compressive strength of unconfined concrete 'cof , with

,2 frp h rupl

E tf

Dε

= (3.1)

where frpE and are the elastic modulus and the thickness of FRP jacket

respectively, and

t

,h rupε is the hoop rupture strain of FRP jacket. In the present

database, the maximum increase in concrete strength as a result of FRP

confinement is about 320%. Further details of this test database are available in

Table 3.1.

For ease of discussion, the terms “weakly-confined concrete”,

“moderately-confined concrete” and “heavily-confined concrete” are used herein.

Weakly-confined concrete refers to concrete whose stress-strain curves feature a

descending branch. Moderately-confined concrete and heavily-confined concrete

both refer to concrete whose stress-strain curves are of the bi-linear ascending

type. The latter two are differentiated using the 'cu co

'f f ratio, where 'cuf is the

axial stress at ultimate axial strain of FRP-confined concrete. When ' ' 2cu cof f < ,

the concrete is said to be moderately-confined, while cases with ' ' 2cu cof f ≥ are

said to be heavily-confined.

3.2.2 Specimens and Instrumentation

The preparation of all 48 specimens followed a standard procedure, which is

described below. All the specimens were cast in steel formworks and were cured

at ambient temperature in laboratory. Once the concrete was cured, the steel

formworks were removed and the concrete cylinders were sanded to remove the

attached dust for better bonding with the confining jackets. The FRP jackets were

43

all formed via the wet lay-up process. A layer of resin was first applied to the

surface of the concrete core and followed by the wrapping of the fiber sheets

which had already been saturated by the resin. The wrapping of the fiber sheets

was continuous with the finishing end overlapping the starting end by 150mm. It

should be noted that all the fiber sheets had hoop fibers only. In addition, two FRP

strips of 25mm in width were wrapped at the two ends to prevent possible

premature failure there. A plastic sheet was then used to squeeze out the air voids

as well as the redundant resin to ensure a compact bond between the concrete core

and the confining jacket and also a smooth surface of the FRP for the convenience

of bonding strain gauges. All the specimens were capped to ensure a smooth

loading surface before testing. For each batch of concrete, 2 or 3 control

specimens of the same size were also tested, from which the average values of the

compressive strength of unconfined concrete 'cof and the corresponding axial

strain coε were found.

For each control specimen, two longitudinal strain gauges, with a gauge length of

120 mm covering the mid-height region, were placed at 180° apart to measure the

axial strains. Another two strain gauges, with a gauge length of 60 mm, were

placed at 180° apart to measure the hoop strains. For each FRP-confined specimen,

either six or eight unidirectional strain gauges, with a gauge length of 20 mm,

were installed at mid-height to measure the hoop strains and another two strain

gauges of the same type were installed at mid-height to measure the axial strains.

In addition, axial strains were also measured by two linear variable displacement

transducers (LVDTs) at 180° apart and covering the mid-height region of 120 mm

for both unconfined and confined specimens. All axial strains reported in Table

3.1 are the average values of readings from the two LVDTs. The instrumentation

of control specimens as well as FRP-confined specimens are shown in Fig. 3.1.

Among the hoop strain gauges installed on FRP-confined specimens, five of them

equally spaced at 45 degrees were located outside the 150 mm wide overlapping

zone (SG1 to SG5 in Fig 3.1b), from which the average hoop strain was found.

The hoop strain readings within the overlapping zone are generally smaller than

those elsewhere as the overlapping zone has a larger jacket thickness. These

44

readings therefore reflect neither the actual strain capacity of the confining jacket

nor the actual dilation properties of the confined concrete, and should thus be

excluded when interpreting the behavior of FRP-confined concrete (Lam and Teng

2004). It should be noted that such important processing of the hoop strain

readings is not possible with existing tests reported by other researchers, for

which the precise number and locations of strain gauges for measuring hoop

strains are generally not reported.

Since the 48 tests were conducted within different research projects, there were

some differences in the loading methods employed. Specimens 01 to 13 were

tested with load control at a constant rate of 300kN/min, while all other specimens

were tested with displacement control, at a constant rate of either 0.18 mm/min or

0.6 mm/min; both rates are acceptable for such tests. All test data, including the

readings of the axial load, strain gauges and the LVDTs were collected with a data

logger and simultaneously saved in a computer. Fig. 3.2 shows the test setup of a

FRP-confined specimen before testing.

3.2.3 Test Results

All the FRP-confined specimens were found to fail by the sudden rupture of the

FRP jacket outside the overlapping zone (the typical failure mode is shown in Fig.

3.3). The key test results are given in Table 3.1. The compressive strength of

confined concrete 'ccf was obtained by dividing the maximum load by the

cross-sectional area of the specimen. Stress-strain curves of both the ascending

and the descending types were captured. When the stress-strain curve is of the

descending type, the axial stress at ultimate axial strain 'cuf is also reported in Table

3.1.

The stress-strain curves from all tests of the present database are shown in Fig. 3.4,

where the lateral strains lε are shown on the left and the axial strains cε are

shown on the right. Both the axial strain and the lateral strain are normalized by

the corresponding value of coε , while the axial stress cσ is normalized by the

45

corresponding value of 'cof . The following sign convention is adopted: in the

concrete, compressive stresses and strains are positive, but in the FRP, tensile

stresses and strains are positive. The predictions of the refined version of Teng et

al.’s (2007a) model are also provided in Fig. 3.4. The predicted curves end at a

hoop rupture strain averaged from either two or three physically identical

specimens. The refinement of Teng et al.’s (2007a) model is discussed later in

chapter.

The pair of specimens 20 and 21 as well as the pair of specimens 28 and 29

showed a descending branch in their stress-strain curves and the compressive

strengths of these specimens are only slightly higher than those of the unconfined

specimens. These specimens were all confined with a one-ply GFRP jacket, and

based on the experimental hoop rupture strains, the average confinement ratios of

the two pairs are 0.079 and 0.067 respectively. By contrast, all other

FRP-confined specimens shown in Fig. 3.4 exhibited the well-known bilinear

stress-strain curve of the ascending type. It should be noted that the pair of

specimens 42 and 43, which had a smaller confinement ratio than that of the pair

of specimens 20 and 21, also exhibited the ascending type stress-strain curve. This

is because the elastic modulus of the FRP jacket of the former pair is much larger

than that of the latter pair, indicating that the nature of the stress-strain curve

depends not only on the confinement ratio but also on the stiffness of the jacket.

3.3 EXISTING ANALYSIS-ORIENTED MODELS FOR FRP-CONFINED

CONCRETE

3.3.1 General Concept

The concept of establishing a passive-confinement stress-strain model from an

active-confinement base model through an incremental approach has previously

been employed for steel-confined concrete by Ahmad and Shah (1982) and Madas

and Elnashai (1992). The first documented attempt to extend this approach to

FRP-confined concrete was made by Mirmiran and Shahawy (1996). This model

follows the procedure proposed by Madas and Elnashai (1992). However, as some

46

of its parameters are not clearly defined (Teng and Lam 2004), this model was not

included in the assessment undertaken by Teng and Lam (2004) and is also not

included in the present comparisons. A later version of this model proposed by the

same authors (Mirmiran and Shahawy 1997a) does not specify the

active-confinement base model and was thus also excluded from the assessment

given in Teng and Lam (2004). In the present study, the model of Mander et al.

(1988) is assumed as the active-confinement model for use in this later version, as

this base model is employed in the earlier version of their model (Mirmiran and

Shahawy 1996). Following the work of Mirmiran and Shahawy (1996, 1997a), a

number of models of this kind have been proposed, including Spoelstra and Monti

(1999), Fam and Rizkalla (2001), Chun and Park (2002), Harries and Kharel

(2002), Marques et al. (2004), Binici (2005) and Teng et al. (2007a).

These models are all built on the assumption that the axial stress and the axial

strain of concrete confined with FRP at a given lateral strain are the same as those

of the same concrete actively confined with a constant confining pressure equal to

that supplied by the FRP jacket (Teng et al. 2007a). This assumption is equivalent

to assuming that the stress path of the confined concrete does not affect its

stress-strain behavior. As a result of this assumption, the stress-strain curve of

FRP-confined concrete can be obtained through the following procedure:

(1) for a given axial strain, find the corresponding lateral strain according to the

lateral-to-axial strain relationship;

(2) based on force equilibrium and radial displacement compatibility between the

concrete core and the FRP jacket, calculate the corresponding lateral confining

pressure provided by the FRP jacket;

(3) use the axial strain and the confining pressure obtained from steps (1) and (2)

in conjunction with an active-confinement base model to evaluate the

corresponding axial stress, leading to the identification of one point on the

stress-strain curve of FRP-confined concrete; and

(4) repeat the above steps to generate the entire stress-strain curve. Fig. 3.5

47

illustrates the concept of this incremental approach.

It is not difficult to realise that in the above procedure, the key elements that

determine the accuracy of the predictions are the active-confinement model and

the lateral-to-axial strain relationship. The performance of the active-confinement

model depends on: (a) the peak axial stress (i.e. failure surface) and the

corresponding axial strain; and (b) the stress-strain equation. The lateral-to-axial

strain relationship, which depicts the unique dilation property of FRP-confined

concrete, is either implicitly or explicitly given in existing models. An iterative

procedure is required in steps (1) and (2) to determine the correct lateral strain that

corresponds to the current axial strain.

In the remainder of this section, the existing models are reviewed in terms of the

three key aspects: the stress-strain equation and the peak axial stress point of the

active confinement model, and the lateral-to-axial strain relationship. The existing

models are also summarized in Table 3.2.

3.3.2 Peak Axial Stress Point

3.3.2.1 Peak axial stress

The peak axial stress on the stress-strain curve of actively-confined concrete is the

compressive strength of such concrete and the peak stress equation defines the

failure surface of such concrete. The models of Mirmiran and Shahawy (1997a),

Spoelstra and Monti (1999), Fam and Rizkalla (2001) and Chun and Park (2002)

directly employ the “five parameter” multiaxial failure surface given by Willam

and Warnke (1975) to define the peak axial stress:

'* '' '2.254 1 7.94 2 1.254l l

cc coco co

f ff fσ σ⎛ ⎞

= + − −⎜ ⎟⎜ ⎟⎝ ⎠

(3.2)

where '*ccf is the peak axial stress of concrete under a specific constant confining

pressure lσ .

48

Of the other four models, Harries and Kharel (2002) adopted the following

equation proposed by Mirmiran and Shahawy (1997b):

'* ' 0.5874.269cc co lf f σ= + (3.3)

Marques et al. (2004) adopted the following equation proposed by Razvi and

Saatcioglu (1999):

'* ' 0.836.7cc co lf f σ= + (3.4)

Binici (2005) employed the Leon-Pramono criterion (Pramono and Willam 1989),

which reduces to Eq. 3.5 if the tensile strength of unconfined concrete is taken to

be 0.1 times of its compressive strength.

'* '' '1 9.9 l l

cc coco co

f ff fσ σ⎛ ⎞

= + +⎜⎜⎝ ⎠

⎟⎟ (3.5)

Teng et al. (2007a) proposed the following linear function to define the peak axial

stress:

'* ' 3.5cc co lf f σ= + (3.6)

The above equations for the peak axial stress are compared in Fig. 3.6, showing

large differences between each other.

3.3.2.2 Axial strain at peak axial stress

Except the model of Marques et al. (2004), all existing models employ the

following equation initially proposed by Richart et al. (1928) to define the axial

strain at peak axial stress:

'*

*'1 5 1cc

cc coco

ff

ε ε⎡ ⎤⎛ ⎞

= + −⎢ ⎥⎜⎝ ⎠

⎟⎣ ⎦

(3.7)

49

where coε and *ccε are respectively the axial strains at '

cof and '*ccf . Marques

et al. (2004) used Eq. 3.7 for concrete with ' 40cof ≤ MPa but for concrete of

higher strength, Eq. 3.7 was modified with a factor introduced by Razvi and

Saatcioglu (1999) to account for the reduced effectiveness in the enhancement of

axial strain for high strength concrete. In Fig. 3.7, the predictions from all models

of the axial strain at peak axial stress are compared, where the differences stem

from the different expressions for the peak axial stress.

3.3.2.3 Stress-strain equation

All models except those of Harries and Kharel (2002) and Binici (2005) employ

the following equation originally proposed by Popovics (1973) and later adopted

by Mander et al. (1988) for steel-confined concrete:

( )( )

*

'* *1c ccc

qcc c cc

qf q

ε εσε ε

=− +

(3.8)

where the constant is defined by q

'* *c

c cc cc

EqE f ε

=−

(3.9)

where is the elastic modulus of concrete. cE

In the model of Harries and Kharel (2002), the stress-strain equation of

actively-confined concrete described by Eq. 3.8 is modified by a factor which was

introduced to control the slope of the descending branch. The model of Binici

(2005) employs three separate expressions to describe the full stress-strain curve.

The ascending branch is described using a linear expression followed by Eq. 3.8,

while the descending branch is described using an exponential expression.

50

3.3.2.4 Lateral-to-axial strain relationship

The lateral-to-axial strain relationship, not available in an active-confinement

model, provides the essential connection between the response of the concrete

core and the response of the FRP jacket, in a passive-confinement model for

FRP-confined concrete. This relationship has been established via different

approaches, and is either explicitly stated or implicitly given.

Explicit lateral-to-axial strain relationships are used in the models of Mirmiran

and Shahawy (1997a), Harries and Kharel (2002) and Teng et al. (2007a).

Mirmiran and Shahawy (1997a) used the tangent dilation ratio tµ (the absolute

value of the tangent slope of the lateral-to-axial strain curve of FRP-confined

concrete; i.e. d / dt l cµ ε ε= ) to link the lateral strain and the axial strain. A

fractional function was introduced by these authors to describe the tangent

dilation ratio based on their own test results of FRP-confined concrete. Harries

and Kharel (2002) instead used the secant dilation ratio sµ (the absolute value of

the secant slope of the lateral-to-axial strain curve of FRP-confined concrete; i.e.

/s l cµ ε ε= ), which is also based on their own test results of FRP-confined

concrete, to relate the axial strain to the lateral strain. A simplified tri-linear

equation was used to describe the variation of the secant dilation ratio. It should

be noted that Harries and Kharel (2002) used different equations to predict the

lateral strains of CFRP-confined and GFRP-confined concrete respectively. Based

on a careful interpretation of the dilation properties of confined and unconfined

concrete, Teng et al. (2007a) proposed the following lateral-to-axial strain

equation that is applicable to un-confined, actively confined and FRP-confined

concrete:

0.7

'0.85 1 8 1 0.75 exp 7c l l

co co co cofε σ εε ε

⎧ ⎫⎡ ⎤ ⎡⎛ ⎞ ⎛ ⎞ ⎛−⎪ ⎪= + + − −⎨ ⎬⎢ ⎥ ⎢⎜ ⎟ ⎜ ⎟ ⎜⎝ ⎠ ⎝ ⎠ ⎝⎣ ⎦ ⎣⎪ ⎪⎩ ⎭

lεε

⎤⎞−⎥⎟⎠⎦

(3.10)

where lσ is the confining pressure. In actively confined concrete, the concrete is

subjected to a constant confining pressure lσ throughout the loading process, but

51

in FRP-confined concrete, the passive lateral confining pressure depends on the

stiffness of the FRP jacket and increases continuously with the hoop strain of

FRP hε . For a given hoop/lateral strain, the confining pressure supplied by the

FRP jacket is

frp h frp ll

E t E tR Rε ε

σ = = − (3.11)

Implicit lateral-to-axial strain relationships were adopted by other researchers.

Spoelstra and Monti (1999) adopted the simple constitutive model proposed by

Pantazopoulou and Mills (1995), which describes the decrease of secant modulus

of concrete with an increasing area strain, to determine the lateral strain. Marques

et al. (2004) employed a similar constitutive model, but they argued that the one

used by Spoelstra and Monti (1999) does not ensure ( )' ,co cof ε corresponds to the

peak point on the stress-strain curve of unconfined concrete. A coefficient was

thus introduced to overcome this shortcoming. In the model of Fam and Rizkalla

(2001), an equation representing the variation of secant dilation ratio with

confining pressure was developed based on the results of Gardner (1969) from

triaxial compression tests of concrete to determine the lateral strain. In the model

of Chun and Park (2002), a cubic polynomial equation developed by Elwi and

Murray (1979) based on the results of Kupfer et al. (1969) from uniaxial

compression tests of concrete was used. In the model of Binici (2005), the secant

dilation ratio was set to be a constant in the elastic stage, beyond which it was

assumed to vary with the confining pressure.

3.4 ASSESSMENT OF EXISTING MODELS

3.4.1 Test Data

For the assessment of the accuracy of existing analysis-oriented stress-strain

models in predicting the lateral strain-axial strain curve and the stress-strain curve,

only comparisons with selected tests (specimens 28 and 29; 24 and 25; 34 and 35

in Table 3.1) are reported here for brevity. The observations made below are also

52

supported by comparisons conducted using the remainder of the database not

reported here.

The three sets of specimens correspond to confinement ratios of 0.067, 0.25, and

0.55 and 'cu co

'f f ratios of 0.88, 1.60, and 2.86 respectively to represent

weakly-confined, moderately-confined and heavily-confined concrete. The

comparisons focus on the dilation properties of FRP-confined concrete, although

the full-range axial stress-strain behavior is also examined. Following these

comparisons, the ultimate axial strain and the corresponding axial stress from

analysis-oriented models are also compared with the results of all 48 tests in the

test database.

3.4.2 Dilation Properties

It has been widely accepted that under axial compression, unconfined concrete

experiences volumetric compaction up to 90% of the peak stress. Thereafter the

concrete shows unstable volumetric dilation due to the rapidly increasing

lateral-to-axial strain ratio. However, this lateral dilation can be effectively

constrained by an FRP jacket (Lam and Teng 2003; Teng et al. 2007a). In

FRP-confined concrete, this lateral dilation results in a continuously increasing

lateral confining pressure provided by the FRP jacket. The dilation properties of

FRP-confined concrete are reflected by its lateral-to-axial strain relationship. A

more direct way to investigate the dilation properties of FRP-confined concrete is

to examine the tangent dilation ratio or the secant dilation ratio.

Figs 3.8 to 3.10 show the experimental variations of lateral strain, tangent dilation

ratio and secant dilation ratio as the axial strain increases and those from existing

analysis-oriented stress-strain models. These test data allow the following

observations to be made: (a) in the initial stage of deformation, the tangent

dilation ratio/secant dilation ratio is almost constant and is very close to the secant

dilation ratio of unconfined concrete; (b) afterwards, the tangent dilation

ratio/secant dilation ratio gradually increases, and the FRP jacket is increasingly

mobilized to confine the concrete; (c) a typical lateral-to-axial strain curve of

53

FRP-confined concrete features an inflection point corresponding to the maximum

value of the tangent dilation ratio.

The three sets of comparisons are for weakly-confined, moderately-confined, and

heavily-confined concrete specimens respectively (Figs 3.8 to 3.10). Each set of

comparisons includes four types of curves, being the lateral-to-axial strain curves,

the tangent dilation ratio curves, the secant dilation ratio curves and the

stress-strain curves. All curves terminate at the point when the average FRP hoop

rupture strain from the corresponding pair of tests is reached. In plotting the test

tangent dilation ratio, some of the data points were filtered out to remove

disturbances from local fluctuations due to data resolutions of closely spaced

readings.

From the comparisons, it can be seen that the existing models lead to rather

different predictions. The models of Fam and Rizkalla (2001) and Binici (2005)

do not predict an inflection point on the lateral-to-axial strain curve (i.e. the

predicted tangent dilation ratio/secant dilation ratio continuously increases with

the axial strain). For the other models, although they are capable of predicting the

inflection point, the ultimate axial strain and the maximum values of the tangent

dilation ratio and the secant dilation ratio are poorly predicted in most cases.

Mirmiran and Shahawy (1997a) proposed that the maximum value of the tangent

dilation ratio occurs when the axial strain reaches coε , however, the present test

results show that the tangent dilation ratio does not reach its maximum value at

such an early stage. Instead, this point occurs at an axial strain of

approximately 2 coε . In addition, the tangent dilation ratio predicted by Mirmiran

and Shahawy’s (1997a) model can be negative under heavy confinement. The

predictions of Spoelstra and Monti’s (1999) model and Marques et al.’s (2004)

model show some similarity in shape and they predict initial values of the tangent

dilation ratio and the secant dilation ratio which are much smaller than those

obtained from tests. The model of Harries and Kharel (2002) is excluded from Fig.

3.10, since the predicted tangent dilation ratio and secant dilation ratio are

unreasonably small. This shortcoming is due to their logarithmic equation for the

ultimate secant dilation ratio, which results in negative values under heavy

54

confinement. In addition, their definition of the ultimate secant dilation ratio is in

doubt, since it only considers the effect of jacket stiffness, but ignores the effects

of specimen size and compressive strength of unconfined concrete. Chun and

Park’s (2002) model suffers from the same drawback. These authors also used a

logarithmic equation, which may predict negative values, to determine the

ultimate secant dilation ratio. The model of Teng et al. (2007a) is seen to be the

most accurate one. This model provides the most accurate predictions of the

ultimate axial strains in all three situations. Apart from the weakly-confined

concrete specimens, the tangent dilation ratio and the secant dilation ratio are both

accurately predicted by this model.

It may be noted that although some of the models predict the ultimate axial strain

quite accurately in some cases, the shape of the lateral-to-axial strain curve is not

correctly captured. Ideally, both the ultimate point and the shape of the

lateral-to-axial strain curve should be accurately predicted. However, it is found

that given the same ultimate point, the stress-strain curve can be closely predicted

as long as the overall trend of the lateral-to-axial strain ratio can be reasonably but

not necessarily accurately predicted. Taking Teng et al.’s (2007a) model as an

example, Fig. 3.11 demonstrates that for both the ascending and descending types

of stress-strain curves, even if the lateral-to-axial stain curve is simply described

using a straight line, which is obviously incorrect but reasonably close to the

predicted lateral-to-axial strain curve, the predicted stress-strain curves are only

slightly affected. This indicates that local inaccuracy of the lateral-to-axial strain

curve only leads to small deviations in the predicted stress-strain curve. The

parameters used to generate Fig. 3.11 are summarized in Table 3.3.


The axial stress-axial strain curves are shown in Figs 3.8d, 3.9d and 3.10d. It can

be seen that for the weakly-confined and moderately-confined concrete specimens,

all examined models overestimate the ultimate axial strain and the corresponding

axial stress. However, for the heavily-confined concrete specimens, the

performance of these models improves. Among them, the model of Teng et al.

(2007a) is again seen to be the most accurate one. A significant deficiency of this

55

model is that it overestimates the axial stress at ultimate axial strain for

weakly-confined concrete and moderately-confined concrete, especially for the

former. In Fig. 3.8d, this model fails to predict the post-peak descending branch.


For brevity, only three sets of typical test results are compared with existing

models in the preceding sub-sections. These comparisons are mainly concerned

with the lateral-to-axial strain relationship, although the axial stress-strain curves

are also discussed. An alternative way to assess the predicted stress-strain

behavior is to compare the predicted ultimate axial strains and the corresponding

axial stresses with the test values, as shown in Figs 3.12 to 3.19, which allows a

much large number of tests to be included in the comparison. It should be noted

that in assessing the model of Harries and Kharel (2002), six specimens with the

largest confinement ratios are not included for the reason mentioned earlier.

On the whole, most models give poor predictions for the ultimate axial strain,

mainly because the accuracy of the predicted ultimate axial strains depends

heavily on the accuracy of the lateral-to-axial strain equation, which needs

improvement as shown earlier. The models of Mirmiran and Shahawy (1997a) and

Teng et al. (2007a) are the better models in predicting the ultimate axial strain. For

the axial stress at ultimate axial strain, the performance of all models except that

of Harries and Kharel (2002) becomes much better, with the models of Marques et

al. (2004), Binici (2005) and Teng et al. (2007a) giving more accurate predictions.

It is interesting to note that the models of Mirmiran and Shahawy (1997a),

Spoelstra and Monti (1999), Fam and Rizkalla (2001) and Chun and Park (2003)

show very similar performance in predicting the axial stress at ultimate axial

strain. A common feature of these four models is that they all employ Eq. 3.2 to

predict the peak axial stress of actively-confined concrete, which suggests that the

use of Eq. 3.2, which is widely accepted for steel-confined concrete, in an

active-confinement base model does not lead to an accurate passive-confinement

model for FRP-confined concrete. The model of Harries and Kharel (2002)

performs worst in predicting the axial stress at ultimate axial strain. This is also

mainly due to the inappropriate definition of the peak axial stress in the base

56

model. If a modified failure surface is used, this model will perform better (Teng

and Lam 2004). Among all these models, Teng et al.’s (2007a) model provides the

most accurate predictions for both the ultimate axial strain and its corresponding

axial stress.

3.5 REFINEMENT OF TENG ET AL.’S MODEL

3.5.1 General

From the comparisons presented in the above section, it is clear that Teng et al.’s

(2007a) model is the most accurate of all examined models. This model correctly

captures the unique dilation properties of FRP-confined concrete, which is central

to models of this kind, and shows much better performance over the other models

in predicting the ultimate condition. Nevertheless, this model still suffers from one

significant deficiency: it overestimates the axial stress at ultimate axial strain for

weakly-confined concrete and to a lesser degree for moderately-confined concrete

(Figs 3.8d and 3.9d). Teng et al. (2007a) noticed that their model overestimated

the responses of some weakly-confined specimens and suggested that any future

improvements to their model should be focused on weakly-confined concrete.

This problem was not resolved by Teng et al. (2007a), primarily due to the

insufficient information available on concrete with weak FRP confinement and the

scatter of the relevant test data available then. This problem becomes more

obvious when the model is compared with the more precise test data of the present

test database. A refinement of this model to eliminate this deficiency is presented

in the present section, so that stress-strain curves of the descending type can also

be accurately predicted by this model.

To identify a way of refining Teng et al.’s (2007a) model, its key elements need to

be screened for possible improvements. The stress-strain equation in the

active-confinement base model of Teng et al. (2007a) is also commonly employed

by other models for FRP-confined concrete and is not expected to be a source of

error. Fig. 3.22 provides some evidence for the accuracy of Eq. 3.8 (Popovics’s

equation). The lateral-to-axial strain relationship proposed by Teng et al. (2007a)

overestimates the axial strain at a given lateral strain for weakly-confined concrete,

57

although it is accurate for moderately-confined and heavily-confined concrete.

However, this overestimation of the axial strain for weakly-confined concrete is

not the cause of inaccuracy in predicting the descending branch. Indeed, if this

overestimation is corrected, the accuracy of the predicted stress-strain curve will

further degrade. Given the limited number of tests available on weakly-confined

concrete and the good overall performance of the lateral-to-axial strain equation

proposed by Teng et al. (2007a) (Figs 3.8 to 3.10), it is difficult to propose

improvements to or to justify any modifications of this equation. The definition of

the peak point of the stress-strain curve in the base model is therefore believed to

be the main source of error. In particular, for the axial strain at peak axial stress,

Eq. 3.7 which was initially proposed by Richart et al. (1928), is employed by all

models without any critical examination, although different equations (Eqs 3.2 to

3.6) have been proposed for the peak axial stress. As a result, the definition of the

peak axial stress and the corresponding axial strain are examined here to develop a

more satisfactory stress-strain model for FRP-confined concrete.

To this end, test results from four recent studies (Imran and Pantazopoulou 1996;

Ansari and Li 1998; Sfer et al. 2002 and Tan and Sun 2004) on actively-confined

concrete were collected and analysed. Since the stress-strain behavior of high

strength concrete under active confinement is known to differ from that of normal

strength concrete (Ansari and Li 1998), only test results of normal strength

concrete ( 'cof = 20 to 50 MPa) reported in the above four studies are included in

the analysis [the 51.8 MPa series of Tan and Sun (2004) is also included]. The

range of confinement ratios of these tests is from 0.04 to 0.91, which is very close

to that of FRP-confined concrete in the current database. These

active-confinement test results are given in Table 3.4.

3.5.2 Peak Axial Stress in the Base Model

The classical work on concrete under active confinement conducted by Richart et

al. (1928) led to the following equation for the peak axial stress:

58

'*

1' 1cc l

co co

f k 'f fσ

= + (3.12)

where is the confinement effectiveness coefficient and = 4.1. While a value of

4.1 for is commonly quoted, a wide range of other values has also been

proposed by different researchers based on their own test data on

actively-confined concrete (see Candappa et al. 2001). In Teng et al.’s (2007a)

model, a value of 3.5 is used for (see Eq. 3.6), which was deduced from test

results of FRP-confined concrete and is within the existing range of proposed

values for (Candappa et al. 2001).

1k

1k

1k

1k

As shown in Fig. 3.20, the dotted line, representing Eq. 3.6 for the peak axial

stress, agrees well with the test results. It is important to note that although Eq. 3.6

was deduced from test results of FRP-confined concrete, it does provide accurate

predictions for the test data of actively-confined normal strength concrete.

3.5.3 Axial Strain at Peak Axial Stress in the Base Model

Richart et al. (1928) also suggested that the effectiveness in the enhancement of

axial strain is around 5 times that in the enhancement of axial stress. Eq. 3.7 was

thus proposed for the axial strain at peak axial stress. Substitution of Eq. 3.12 into

Eq. 3.7 yields

*

1 '1 5cc l

co co

kf

ε σε

= + (3.13)

In the analysis-oriented stress-strain models assessed in the present chapter, Eq.

3.13 is accepted without any modification, except Marques et al. (2004), although

different equations were proposed for the peak axial stress. A more rational

approach to predict the degree of strain enhancement is to separate it from the

definition of the peak axial stress. That is, the relationship between the strain

enhancement ratio *cc coε ε and the confinement ratio '/l cofσ should be directly

59

established from active-confinement test data.

Based on the test data shown in Fig. 3.21, the following nonlinear equation is

proposed for the axial strain at peak axial stress:

1.2*

'1 17.5cc l

co cofε σε

⎛ ⎞= + ⎜ ⎟

⎝ ⎠ (3.14)

Fig. 3.21 shows that Eq. 3.14 provides accurate predictions of *ccε deduced from

the test results of Lam et al. (2006) for specimens confined with a 0.33 mm thick

CFRP jacket (specimens 17-19). In Fig. 3.21, the values of *ccε for specimens

17-19 were deduced from their axial stresses cσ and axial strains cε at

different confining pressures lσ using Eqs 3.6, 3.8 and 3.9. It can also be seen

that Eq. 3.14 provides reasonably close predictions of the test results of

actively-confined concrete (Imran and Pantazopoulou 1996; Ansari and Li 1998;

Sfer et al. 2002 and Tan and Sun 2004), given the wide scatter exhibited by these

test results.

Eq. 3.14 is thus proposed to replace its counterpart in Teng et al.’s (2007a)

original model which can be written as

*

'1 17.5cc l

co cofε σε

⎛ ⎞= + ⎜

⎝ ⎠⎟ (3.15)

Eq. 3.14 is compared with Eqs 3.2 to 3.6 in Fig. 3.7. It is interesting to note that

Eq. 3.14 predicts a trend that is opposite to that of the other equations. Teng et

al.’s (2007a) model with Eq. 3.15 replaced by Eq. 3.14 is referred to as the refined

model.

It should be noted that the incorporation of Eq. 3.14 into the base model shifts the

location of the peak point of the stress-strain curve and slightly modifies the

overall shape of the stress-strain curve. Fig. 3.22 shows the test stress-strain

60

curves of the first three tests by Ansari and Li (1998) in Table 3.4 as well as the

stress-strain curves predicted using the original base model where the peak point

is defined by Eqs 3.6 and 3.15 and using the modified base model where the peak

point is defined by Eqs 3.6 and 3.14. These tests were chosen for comparison as

the curves were clearly reported in the original paper and their peak stresses are

similar to the predictions of Eq. 3.6, which enables a more reliable and direct

comparison. It can be seen that the curves predicted using the modified base

model as well as the original base model are both in reasonably close agreement

with the test curves, considering the large scatter of test values of *ccε shown in

Fig. 3.21. This indicates that the modified definition of the peak point is at least as

valid as the original definition when judged on the basis of these test results.

Fig. 3.23 shows the predictions of the axial stress at ultimate axial strain of the

modified model versus the test data. It can be seen that the overestimation by Teng

et al.’s (2007a) original model of the axial stress at ultimate axial strain for

weakly-confined and moderately-confined concrete (Fig. 3.19b) has been

corrected while the prediction for heavily-confined concrete is only very slightly

affected (Fig. 3.23). The advantage of the refined model over the original model in

predicting the entire stress-strain curve is demonstrated for selected specimens

(Figs 3.4d and 3.4e). In these figures, the end of each curve is provided with a

symbol to indicate the group it belongs to for easy comparisons. The refined

model is seen to perform much better than the original model for weakly-confined

specimens. The difference between the original and refined models reduces as the

hoop membrane stiffness of the FRP jacket increases (Figs 3.4d and 3.4e).

Comparisons shown in Figs 3.4a to 3.4c and 3.4f to 3.4h between the refined

model and the test data show that overall, the refined model provide accurate

predictions.

In all comparisons with test data in this chapter, the test values of 'cof and

coε were used. The elastic modulus and the Poisson’s ratio of unconfined

concrete were either those specified in an individual model or taken to be

cE

'4730 cof (MPa) and 0.2 if they are not specified in the model. For Binici’s

61

(2005) model, the compressive fracture energy was found from

'8.8fc cofG = ( fcG in MPa/mm and 'cof in MPa) (Nakamura and Higai 2001)

and the characteristic length of the specimen in the loading direction was

taken to be the specimen height (i.e. 305 mm).

cl

Teng et al. (2007a) suggested that when test values of coε are not available, a value

of 0.0022 should be used with their model. For more accurate predictions, it is

proposed here that when the refined model is used to predict the behavior of

FRP-confined concrete, the following equation proposed by Popovics (1973)

should be used unless a test value is available:

'40.000937co cofε = ( 'cof in MPa) (3.16)

3.6 CONCLUSIONS

This chapter has presented a thorough assessment of the performance of eight

existing analysis-oriented stress-strain models for FRP-confined concrete which

employ an active-confinement model as the base model, leading to the

identification of Teng et al.’s (2007a) model as the most accurate through this

assessment. A refined version of Teng et al.’s (2007a) model has also been

proposed. The comparisons and discussions presented in this chapter allow the

following conclusions to be drawn:

1) The lateral-to-axial strain relationship, which reflects the unique dilation

properties of FRP-confined concrete, is central to models of this kind. A

successful model should accurately predict this relationship. Nevertheless,

provided the overall trend of this relationship is reasonably well described,

the axial stress-strain curve can be closely predicted, even if local

inaccuracies exist in the lateral-to-axial strain equation;

2) The definitions of the peak axial stress and the corresponding axial strain in

the active-confinement base model are also important to ensure the accuracy

62

of an analysis-oriented model for FRP-confined concrete;

3) Most of the eight models examined in this chapter correctly capture the

dilation properties of FRP-confined concrete, but provide poor predictions of

the ultimate axial strain. Their performance in predicting the axial stress at

ultimate axial strain is however much better. The accuracy of the predicted

ultimate axial strain depends mainly on the accuracy of the lateral-to-axial

strain equation, while that of the predicted axial stress at ultimate axial strain

depends mainly on the definitions of the peak axial stress and its

corresponding strain in the base model.

4) The model of Teng et al. (2007a) performs the best among the eight models

examined in this chapter. This model provides accurate predictions of both the

lateral-axial strain relationship and the ultimate condition, except that it

overestimates the axial stress at ultimate axial strain for weakly-confined and

to a lesser extent for moderately-confined concrete. As a result, its predictions

are likely to be inaccurate for stress-strain curves with a descending branch.

5) The performance of the model of Teng et al. (2007a) for weakly-confined

concrete can be significantly improved by replacing its equation for the strain

at peak axial stress with a nonlinear equation proposed in this chapter.

63

3.7 REFERENCES

Ahmad, S.H. and Shah, S.P. (1982). “Stress-strain curves of concrete confined by spiral reinforcement”, ACI Journal, 79(6), 484-490.

Ansari, F. and Li, Q.B. (1998). “High-strength concrete subjected to triaxial compression”, ACI Materials Journal, 95(6), 747-755.

Becque, J., Patnaik, A.K. and Rizkalla, S.H. (2003). “Analytical models for concrete confined with FRP tubes”, Journal of Composites for Construction, ASCE, 7(1), 31-38.


Candappa, D.C., Sanjayan, J.G. and Setunge, S. (2001). “Complete triaxial stress-strain curves of high-strength concrete”, Journal of Materials in Civil Engineering, ASCE, 13(3), 209-215.

Chun, S.S. and Park, H.C. (2002). “Load carrying capacity and ductility of RC columns confined by carbon fiber reinforced polymer.” Proceedings, 3rd International Conference on Composites in Infrastructure (CD-Rom), University of Arizona, San Francisco, USA.

Elwi, A.A. and Murray, D.W. (1979). “A 3D hypoelastic concrete constitutive relationship”, Journal of the Engineering Mechanics Division, ASCE, 105(4), 623-641.


Gardner, N.J. (1969). “Triaxial behavior of concrete”, ACI Journal, 66(2), 136-146.

Harmon, T.G., Ramakrishnan S. and Wang, E.H. (1998). “Confined concrete subjected to uniaxial monotonic loading”, Journal of Engineering Mechanics, ASCE, 124(12), 1303-1308.


Imran, I. and Pantazopoulou, S.J. (1996). “Experimental study of plain concrete under triaxial stress”, ACI Materials Journal, 93(6), 589-601.

Kupfer, H.B., Hilsdorf, H.K. and Rusch, H. (1969). “Behavior of concrete under biaxial stresses”, ACI Journal, 66(8), 656-666.

Lam, L. and Teng, J.G. (2003). “Design-oriented stress-strain model for FRP-confined concrete”, Construction and Building Materials, 17(6-7), 471-489.

64

http://wos01.isiknowledge.com/?SID=B5IOMeE2ehL9Cmn1HN@&Func=Abstract&doc=4/1

http://wos01.isiknowledge.com/?SID=B5IOMeE2ehL9Cmn1HN@&Func=Abstract&doc=4/1





Lam, L. and Teng, J.G. (2004). “Ultimate condition of fiber reinforced polymer-confined concrete”, Journal of Composites for Construction, ASCE, 8(6), 539-548.

Lam, L., Teng, J.G., Cheung, C.H. and Xiao, Y. (2006). “FRP-confined concrete under axial cyclic compression”, Cement and Concrete Composites, 28(10), 949-958.

Madas, P. and Elnashai, A.S. (1992). “A new passive confinement model for the analysis of concrete structures subjected to cyclic and transient dynamic loading”, Earthquake Engineering and Structural Dynamics, 21(5), 409-431.

Mander, J.B., Priestley, M.J.N. and Park, R. (1988). “Theoretical stress-strain model for confined concrete”, Journal of Structural Engineering, ASCE, 114(8), 1804-1826.

Marques, S.P.C. Marques, D.C.S.C., da Silva J.L. and Cavalcante, M.A.A. (2004). “Model for analysis of short columns of concrete confined by fiber-reinforced polymer”, Journal of Composites for Construction, ASCE, 8(4), 332-340.

Mirmiran, A. and Shahawy, M. (1996). “A new concrete-filled hollow FRP composite column”, Composites Part B-Engineering, 27(3-4), 263-268.

Mirmiran, A. and Shahawy, M. (1997a). “Dilation characteristics of confined concrete”, Mechanics of Cohesive-Frictional Materials, 2(3), 237-249.

Mirmiran, A. and Shahawy, M. (1997b). “Behavior of concrete columns confined by fiber composites”, Journal of Structural Engineering, ASCE, 123(5), 583-590.

Nakamura, H. and Higai, T. (2001). “Compressive fracture energy and fracture zone length of concrete.” Modeling of Inelastic Behavior of RC Structures Under Seismic Loads, Edited by P.B. Shing, T. Tanabe, ASCE, 471-487.

Popovics, S. (1973). “Numerical approach to the complete stress-strain relation for concrete”, Cement and Concrete Research, 3(5), 583-599.

Pantazopoulou, S.J. and Mills, R.H. (1995). “Microstructural aspects of the mechanical response of plain concrete”, ACI Materials Journal, 92(6), 605-616.

Pramono, E. and Willam K. (1989). “Fracture-energy based plasticity formulation of plain concrete”, Journal of Engineering Mechanics, ASCE, 115(6), 1183-1204.

Razvi, S. and Saatcioglu, M. (1999). “Confinement model for high-strength concrete”, Cement and Concrete Research, 125(3), 281-289.

Richart, F.E., Brandtzaeg, A. and Brown, R.L. (1928). A Study of the Failure of Concrete under Combined Compressive Stresses, Engineering Experiment Station, University of Illinois, Urbana, U.S.A.

65












Sfer, D., Carol, I., Gettu, R. and Etse, G. (2002). “Study of the behavior of concrete under traxial compression”, Journal of Engineering Mechanics, ASCE, 128(2), 156-163.

Spoelstra, M.R. and Monti, G. (1999). “FRP-confined concrete model”, Journal of Composites for Construction, ASCE, 3(3), 143-150.

Tan, K.H. and Sun, X. (2004). “Failure criteria of concrete under triaxial compression”, Proceedings, International Symposium on Confined Concrete (CD-Rom), 12-14 June, Changsha, China,

Teng, J.G. and Lam, L. (2004). “Behavior and modeling of fiber reinforced polymer-confined concrete”, Journal of Structural Engineering, ASCE, 130(11), 1713-1723.

Teng, J.G., Huang, Y.L. Lam. L and Ye L.P. (2007a). “Theoretical model for fiber reinforced polymer-confined concrete”, Journal of Composites for Construction, ASCE, 11(2).

Teng, J.G., Yu, T. Wong, Y.L. and Dong, S.L. (2007b). “Hybrid FRP-concrete-steel tubular columns: concept and behaviour”, Construction and Building Materials, 21(4), 846-854.

Willam, K.J. and Warnke, E.P. (1975). “Constitutive model for the triaxial behaviour of concrete”, Proceedings, International Association for Bridge and Structural Engineering, 19, 1-30.

66



Table 3.1 Test database of FRP-confined concrete cylinders

Source Specimen

D

(mm)H

(mm)'

cof (MPa)

coε (%)

Fiber type

t (mm)

frpE (GPa)

,h rupε (%)

'ccf ( '

cuf ) (MPa)

cuε (%)

1 152 305 35.9 0.203 Carbon 0.165 250.5 0.969 47.2 1.1062 152 305 35.9 0.203 Carbon 0.165 250.5 0.981 53.2 1.2923 152 305 35.9 0.203 Carbon 0.165 250.5 1.147 50.4 1.2734 152 305 35.9 0.203 Carbon 0.33 250.5 0.949 71.6 1.855 152 305 35.9 0.203 Carbon 0.33 250.5 0.988 68.7 1.6836 152 305 35.9 0.203 Carbon 0.33 250.5 1.001 69.9 1.9627 152 305 34.3 0.188 Carbon 0.495 250.5 0.799 82.6 2.0468 152 305 34.3 0.188 Carbon 0.495 250.5 0.884 90.4 2.4139 152 305 34.3 0.188 Carbon 0.495 250.5 0.968 97.3 2.51610 152 305 38.5 0.223 Glass 1.27 21.8 1.440 51.9 1.31511 152 305 38.5 0.223 Glass 1.27 21.8 1.890 58.3 1.45912 152 305 38.5 0.223 Glass 2.54 21.8 1.670 77.3 2.188

Lam and Teng(2004)

13 152 305 38.5 0.223 Glass 2.54 21.8 1.760 75.7 2.45714 152 305 41.1 0.256 Carbon 0.165 250 0.810 52.6 0.90015 152 305 41.1 0.256 Carbon 0.165 250 1.080 57.0 1.21016 152 305 41.1 0.256 Carbon 0.165 250 1.070 55.4 1.11017 152 305 38.9 0.250 Carbon 0.33 247 1.060 76.8 1.91018 152 305 38.9 0.250 Carbon 0.33 247 1.130 79.1 2.080

Lam et al. (2006)

19 152 305 38.9 0.250 Carbon 0.33 247 0.790 65.8 1.25020 152 305 39.6 0.263 Glass 0.17 80.1 1.869 41.5 (38.8) 0.82521 152 305 39.6 0.263 Glass 0.17 80.1 1.609 40.8 (37.2) 0.94222 152 305 39.6 0.263 Glass 0.34 80.1 2.040 54.6 2.13023 152 305 39.6 0.263 Glass 0.34 80.1 2.061 56.3 1.825

Teng et al. (2007b)

24 152 305 39.6 0.263 Glass 0.51 80.1 1.955 65.7 2.558

67

25 152 305 39.6 0.263 Glass 0.51 80.1 1.667 60.9 1.79226 152 305 33.1 0.309 Glass 0.17 80.1 2.080 42.4 1.30327 152 305 33.1 0.309 Glass 0.17 80.1 1.758 41.6 1.26828 152 305 45.9 0.243 Glass 0.17 80.1 1.523 48.4 (40.5) 0.81329 152 305 45.9 0.243 Glass 0.17 80.1 1.915 46.0 (40.5) 1.06330 152 305 45.9 0.243 Glass 0.34 80.1 1.639 52.8 1.20331 152 305 45.9 0.243 Glass 0.34 80.1 1.799 55.2 1.25432 152 305 45.9 0.243 Glass 0.51 80.1 1.594 64.6 1.55433 152 305 45.9 0.243 Glass 0.51 80.1 1.940 65.9 1.90434 152 305 38.0 0.217 Carbon 0.68 240.7 0.977 110.1 2.55135 152 305 38.0 0.217 Carbon 0.68 240.7 0.965 107.4 2.61336 152 305 38.0 0.217 Carbon 1.02 240.7 0.892 129.0 2.79437 152 305 38.0 0.217 Carbon 1.02 240.7 0.927 135.7 3.08238 152 305 38.0 0.217 Carbon 1.36 240.7 0.872 161.3 3.70039 152 305 38.0 0.217 Carbon 1.36 240.7 0.877 158.5 3.54440 152 305 37.7 0.275 Carbon 0.11 260 0.935 48.5 0.89541 152 305 37.7 0.275 Carbon 0.11 260 1.092 50.3 0.91442 152 305 44.2 0.260 Carbon 0.11 260 0.734 48.1 0.69143 152 305 44.2 0.260 Carbon 0.11 260 0.969 51.1 0.88844 152 305 44.2 0.260 Carbon 0.22 260 1.184 65.7 1.30445 152 305 44.2 0.260 Carbon 0.22 260 0.938 62.9 1.02546 152 305 47.6 0.279 Carbon 0.33 250.5 0.902 82.7 1.30447 152 305 47.6 0.279 Carbon 0.33 250.5 1.130 85.5 1.936

Present study

48 152 305 47.6 0.279 Carbon 0.33 250.5 1.064 85.5 1.821

68

Table 3.2 Summary of analysis-oriented models for FRP-confined concrete

Peak Point Model Stress-strain Equation Stress Strain

Lateral-to-Axial Strain Relationship

Mirmiran and Shahawy (1997a)

Popovics (1973) Eq. 3.2 Eq. 3.7

Explicit, Mirmiran and Shahawy

(1997a)

Spoelstra and Monti (1999)

Popovics (1973) Eq. 3.2 Eq. 3.7

Implicit, Pantazopoulou and Mills

(1995) Fam and Rizkalla (2001)

Popovics (1973) Eq. 3.2 Eq. 3.7 Implicit,

Fam and Rizkalla (2001)

Chun and Park (2002)

Popovics (1973) Eq. 3.2 Eq. 3.7 Implicit,

Elwi and Murray (1979)

Harries and Kharel (2002)

Modified from

Popovics (1973)

Eq. 3.3 Eq. 3.7 Explicit, Harries and Kharel (2002)

Marques et al. (2004)

Popovics (1973) Eq. 3.4

Modified from

Eq. 3.7

Implicit, Pantazopoulou and Mills

(1995)

Binici (2005)

Modified from

Popovics (1973)

Eq. 3.5 Eq. 3.7 Implicit, Binici (2005)

Teng et al. (2007a)

Popovics (1973) Eq. 3.6 Eq. 3.7 Explicit, Eq. 3.10

Teng et al. (2007a)

69

Table 3.3 Summary of parameters used for generating Fig. 3.11

Figure No. D

(mm) '

cof (MPa)

coε (%)

t (mm)

frpE (GPa)

,h rupε (%)

Fig. 3.11a,b 152 40 0.22 0.1 80 1.5 Fig. 3.11c,d 152 40 0.22 0.6 80 1.5

70

Table 3.4 Test results of concrete under active confinement

Source '

cof (MPa)

coε (%)

lf (MPa)

'l cof f '*

ccf (MPa)

*ccε

(%) 28.6 0.260 1.05 0.037 33.6 0.47028.6 0.260 2.1 0.073 36.4 0.67528.6 0.260 4.2 0.147 48.1 1.38528.6 0.260 8.4 0.294 65.2 2.37528.6 0.260 14.7 0.514 92.3 3.42528.6 0.260 21.0 0.734 114.5 4.46047.4 0.280 2.15 0.045 57.7 0.43047.4 0.280 4.3 0.091 67.3 0.69047.4 0.280 8.6 0.181 83.6 1.46047.4 0.280 17.2 0.363 118.1 2.53047.4 0.280 30.1 0.635 161.1 3.600

Imran and Pantazopoulou (1996)

47.4 0.28 43.0 0.907 204.7 4.73047.2 0.202 8.3 0.176 79.7 1.34947.2 0.202 16.5 0.351 109.7 1.56847.2 0.202 24.8 0.527 130.7 2.04947.2 0.202 33.1 0.702 144.2 2.420

Ansari and Li (1998)

47.2 0.202 41.4 0.878 166.9 2.95032.8 0.180 1.5 0.046 45.5 0.26032.8 0.180 4.5 0.137 55.3 0.41032.8 0.180 9.0 0.274 65.7 0.83038.8 0.210 1.5 0.039 47.8 0.34038.8 0.210 4.5 0.116 58.2 0.520

Sfer et al. (2002)

38.8 0.210 9.0 0.232 66.5 0.63027.2 0.182 1.875 0.069 36.2 0.30027.2 0.182 1.875 0.069 35.7 0.28927.2 0.182 7.5 0.276 50.1 0.43527.2 0.182 7.5 0.276 47.5 0.57327.2 0.182 15.0 0.551 72.1 0.74427.2 0.182 15.0 0.551 66.6 0.80251.8 0.238 1.875 0.036 64.8 0.32951.8 0.238 1.875 0.036 66 0.38651.8 0.238 7.5 0.145 86.6 0.45651.8 0.238 7.5 0.145 84.2 0.48951.8 0.238 12.5 0.241 99.3 0.492

Tan and Sun (2004)

51.8 0.238 12.5 0.241 103.3 0.662

71

LVDT2

LVDT1

SG2

SG3

SG1

SG4

(a) Unconfined specimens

LVDT2

LVDT1

SG5

SG4

SG3

SG2

SG1

SG6

45°

overlapping zone

SG8

SG7

(b) FRP-confined specimens

Fig 3.1 Instrumentation of specimens

72

Fig. 3.2 Test setup

Fig. 3.3 Typical failure mode

73

-10 -5 0 5 10 150

1

2

3

Normalized Strain

Nor

mal

ized

Axi

al S

tress

σc

/f ′co

Specimens 01, 02, 03fl /f′co=0.16



TestRefined Version ofTeng et al.′s (2007a)Model

(a) Specimens 01 to 09

-10 -5 0 5 10 150

0.5

1

1.5

2

2.5

Normalized Strain

Nor

mal

ized

Axi

al S

tress

σc

/f ′co

Specimens 10, 11fl /f′co=0.16



(b) Specimens 10 to 13

74

-5 0 5 100

0.5

1

1.5

2

2.5

Normalized Strain

Nor

mal

ized

Axi

al S

tress

σc

/f ′co




(c) Specimens 14 to 19

-10 -5 0 5 100

0.5

1

1.5

2

Normalized Strain

Nor

mal

ized

Axi

al S

tress

σc

/f ′co




TestOriginal Version of Teng et al.′s (2007a) ModelRefined Version of Teng et al.′s (2007a) Model

(d) Specimens 20 to 25

75

-10 -5 0 5 100

0.6

1.2

1.8

Normalized Strain

Nor

mal

ized

Axi

al S

tress

σc

/f ′co




TestOriginal Version of Teng et al.′s (2007a) ModelRefined Version of Teng et al.′s (2007a) Model

(e) Specimens 28 to 33

-8 -4 0 4 80

0.5

1

1.5

2

Normalized Strain

Nor

mal

ized

Axi

al S

tress

σc

/f ′co





(f) Specimens 34 to 39

76

-6 -4 -2 0 2 4 60

0.4

0.8

1.2

1.6

Normalized Strain

Nor

mal

ized

Axi

al S

tress

σc

/f ′co




(g) Specimens 42to 45

-8 -4 0 4 80

0.5

1

1.5

2

Normalized Strain

Nor

mal

ized

Axi

al S

tress

σc

/f ′co





(h) Specimens 26,27; 40,41; 46,47,48

Fig. 3.4 Experimental stress-strain curves of FRP-confined concrete

77

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Normalized Axial Strain εc /εco

Nor

mal

ized

Axi

al S

tress

σc

/f ′co

Active-Confinement ModelPassive-Confinement Model

Fig. 3.5 Generation of a stress-strain curve for FRP-confined concrete

78

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3

3.5

4

4.5

5

Confinement Ratio fl /f′co

Nor

mal

ized

Pea

k A

xial

Stre

ss f′ * cc

/f′ c

o

Eq. 3.2Eq. 3.3Eq. 3.4Eq. 3.5Eq. 3.6

Fig. 3.6 Comparison of predictions for the peak axial stress

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

20


Nor

mal

ized

Axi

al S

train

at P

eak

Axi

al S

tress

ε* cc

/ εco

Eq. 3.2 & Eq. 3.7Eq. 3.3 & Eq. 3.7Eq. 3.4 & Eq. 3.7Eq. 3.5 & Eq. 3.7Eq. 3.6 & Eq. 3.7Eq. 3.14

Fig. 3.7 Comparison of predictions for the axial strain at peak axial stress

79

0 0.005 0.01 0.015 0.02 0.025-0.02

-0.015

-0.01

-0.005

0

Axial Strain εc

Late

ral S

train

εl

Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Harries and Kharel (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 28, 29)

(a) Lateral-to-axial strain curves

0 0.005 0.01 0.015 0.02 0.0250

1

2

3

4

5

6

Axial Strain εc

Tan

gent

Dila

tion

Rat

io µ

t


(b) Tangent dilation ratio

80

0 0.005 0.01 0.015 0.02 0.0250

0.5

1

1.5

2

2.5

3

Axial Strain εc

Sec

ant D

ilatio

n R

atio

µs


(c) Secant dilation ratio

0 0.005 0.01 0.015 0.02 0.0250

10

20

30

40

50

60

70

Axial Strain εc

Axi

al S

tress

σc

(MP

a)


(d) Stress-strain curves

Fig. 3.8 Weakly-confined concrete

81

0 0.01 0.02 0.03 0.04 0.05-0.02

-0.015

-0.01

-0.005

0

Axial Strain εc

Late

ral S

train

εl



0 0.01 0.02 0.03 0.04 0.050

0.5

1

1.5

2

2.5

3

Axial Strain εc

Tan

gent

Dila

tion

Rat

io µ

t



82

0 0.01 0.02 0.03 0.04 0.050

0.4

0.8

1.2

1.6

Axial Strain εc

Sec

ant D

ilatio

n R

atio

µs



0 0.01 0.02 0.03 0.04 0.050

10

20

30

40

50

60

70

80

90

Axial Strain εc

Axi

al S

tress

σc

(MP

a)



Fig. 3.9 Moderately-confined concrete

83

0 0.01 0.02 0.03 0.04 0.05-0.01

-0.008

-0.006

-0.004

-0.002

0

Axial Strain εc

Late

ral S

train

εl

Mirmiran and Shahawy (1997a)Spoelstra and Monti (1999)Fam and Rizkalla (2001)Chun and Park (2002)Marques et al. (2004)Binici (2005)Teng et al. (2007a)Test (Specimens 34, 35)


0 0.01 0.02 0.03 0.04 0.050

0.3

0.6

0.9

1.2

Axial Strain εc

Tan

gent

Dila

tion

Rat

io µ

t



84

0 0.01 0.02 0.03 0.04 0.050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Axial Strain εc

Sec

ant D

ilatio

n R

atio

µs



0 0.01 0.02 0.03 0.04 0.050

20

40

60

80

100

120

Axial Strain εc

Axi

al S

tress

σc

(MP

a)



Fig. 3.10 Heavily-confined concrete

85

0 0.002 0.004 0.006 0.008 0.01-0.015

-0.012

-0.009

-0.006

-0.003

0

Axial Strain εc

Late

ral S

train

εl

Linear EquationTeng et al. (2007a)

(a) Descending type: lateral-to-axial strain curves

0 0.002 0.004 0.006 0.008 0.010

10

20

30

40

50

Axial Strain εc

Axi

al S

tress

σc

(MP

a)


(b) Descending type: stress-strain curves

86

0 0.004 0.008 0.012 0.016 0.02-0.015

-0.012

-0.009

-0.006

-0.003

0

Axial Strain εc

Late

ral S

train

εl


(c) Ascending type: lateral-to-axial strain curves

0 0.004 0.008 0.012 0.016 0.020

10

20

30

40

50

60

70

80

Axial Strain εc

Axi

al S

tress

σc

(MP

a)


(d) Ascending type: stress-strain curves

Fig. 3.11 Stress-strain curves predicted by Teng et al.’s (2007a) model: effect of

lateral-to-axial strain equation

87

0 5 10 15 200

5

10

15

20

Normalized Ultimate Axial Strain εcu /εco - Test

Nor

mal

ized

Ulti

mat

e A

xial

Stra

inε cu

/ εco

- P

redi

cted

Model ofMirmiran and Shahawy (1997a)

Test Results:Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP

(a) Ultimate axial strain

0 1 2 3 4 50

1

2

3

4

5

Normalized Axial Stress at Ultimate Axial Strainf′cu /f′co - Test

Nor

mal

ized

Axi

al S

tress

at U

ltim

ate

Axi

al S

train

f ′cu

/f′ co

- P

redi

cted

Model ofMirmiran and Shahawy (1997a)


(b) Axial stress at ultimate axial strain

Fig. 3.12 Performance of Mirmiran and Shahawy’s model in predicting the

ultimate condition

88

0 5 10 15 200

5

10

15

20


Nor

mal

ized

Ulti

mat

e A

xial

Stra

inε cu

/ εco

- P

redi

cted

Model ofSpoelstra and Monti (1999)



0 1 2 3 4 50

1

2

3

4

5


Nor

mal

ized

Axi

al S

tress

at U

ltim

ate

Axi

al S

train

f ′cu

/f′ co

- P

redi

cted

Model ofSpoelstra and Monti (1999)



Fig. 3.13 Performance of Spoelstra and Monti’s model in predicting the ultimate

condition

89

0 5 10 15 200

5

10

15

20


Nor

mal

ized

Ulti

mat

e A

xial

Stra

inε cu

/ εco

- P

redi

cted

Model ofFam and Rizkalla (2001)



0 1 2 3 4 50

1

2

3

4

5


Nor

mal

ized

Axi

al S

tress

at U

ltim

ate

Axi

al S

train

f ′cu

/f′ co

- P

redi

cted

Model ofFam and Rizkalla (2001)



Fig. 3.14 Performance of Fam and Rizkalla’s model in predicting the ultimate

condition

90

0 10 20 30 40 500

10

20

30

40

50


Nor

mal

ized

Ulti

mat

e A

xial

Stra

inε cu

/ εco

- P

redi

cted

Model ofChun and Park (2002)



0 1 2 3 4 50

1

2

3

4

5


Nor

mal

ized

Axi

al S

tress

at U

ltim

ate

Axi

al S

train

f ′cu

/f′ co

- P

redi

cted

Model ofChun and Park (2002)



Fig. 3.15 Performance of Chun and Park’s model in predicting the ultimate

condition

91

0 5 10 15 200

5

10

15

20


Nor

mal

ized

Ulti

mat

e A

xial

Stra

inε cu

/ εco

- P

redi

cted

Model ofHarries and Kharel (2002)



0 1 2 3 4 50

1

2

3

4

5


Nor

mal

ized

Axi

al S

tress

at U

ltim

ate

Axi

al S

train

f ′cu

/f′ co

- P

redi

cted

Model ofHarries and Kharel (2002)



Fig. 3.16 Performance of Harries and Kharel’s model in predicting the ultimate

condition

92

0 5 10 15 200

5

10

15

20


Nor

mal

ized

Ulti

mat

e A

xial

Stra

inε cu

/ εco

- P

redi

cted

Model ofMarques et al. (2004)



0 1 2 3 4 50

1

2

3

4

5


Nor

mal

ized

Axi

al S

tress

at U

ltim

ate

Axi

al S

train

f ′cu

/f′ co

- P

redi

cted

Model ofMarques et al. (2004)



Fig. 3.17 Performance of Marques et al.’s model in predicting the ultimate

condition

93

0 5 10 15 200

5

10

15

20


Nor

mal

ized

Ulti

mat

e A

xial

Stra

inε cu

/ εco

- P

redi

cted

Model ofBinici (2005)



0 1 2 3 4 50

1

2

3

4

5


Nor

mal

ized

Axi

al S

tress

at U

ltim

ate

Axi

al S

train

f ′cu

/f′ co

- P

redi

cted

Model ofBinici (2005)



Fig. 3.18 Performance of Binici’s model in predicting the ultimate condition

94

0 5 10 15 20 250

5

10

15

20

25


Nor

mal

ized

Ulti

mat

e A

xial

Stra

inε cu

/ εco

- P

redi

cted

Model ofTeng et al. (2007a)



0 1 2 3 4 50

1

2

3

4

5


Nor

mal

ized

Axi

al S

tress

at U

ltim

ate

Axi

al S

train

f ′cu

/f′ co

- P

redi

cted

Model ofTeng et al. (2007a)



Fig. 3.19 Performance of Teng et al.’s model in predicting the ultimate condition

95

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5


Nor

mal

ized

Pea

k A

xial

Stre

ss f′ * cc

/f′ c

o

↑Eq. 3.6

Imran and Pantazopoulou (1996)Ansari and Li (1998)Sfer et al. (2002)Tan and Sun (2004)

Fig. 3.20 Normalized peak axial stress versus confinement ratio

0 0.2 0.4 0.6 0.8 10

5

10

15

20


Nor

mal

ized

Axi

al S

train

at P

eak

Axi

al S

tress

ε* cc

/ εco

← Eq. 3.14

Imran and Pantazopoulou (1996)Ansari and Li (1998)Sfer et al. (2002)Tan and Sun (2004)Specimen 17Specimen 18Specimen 19

Fig. 3.21 Normalized axial strain at peak axial stress versus confinement ratio

96

0 0.01 0.02 0.03 0.04 0.050

20

40

60

80

100

120

140

Axial Strain εc

Axi

al S

tress

σc

(MP

a)

Test (Ansari and Li 1998)Original Base ModelModified Base Model

Fig. 3.22 Comparisons of original base model with modified base model

0 1 2 3 4 50

1

2

3

4

5


Nor

mal

ized

Axi

al S

tress

at U

ltim

ate

Axi

al S

train

f ′cu

/f′ co

- P

redi

cted

Refined Version ofTeng et al.′s (2007a) Model


Fig. 3.23 Performance of refined version of Teng et al.’s model

97

CHAPTER 4

DESIGN-ORIENTED STRESS-STRAIN MODELS FOR

FRP-CONFINED CONCRETE

4.1 INTRODUCTION

Chapter 3 has presented a comprehensive assessment of analysis-oriented models.

Although the complex nature of analysis-oriented models prevents them from

being directly employed for design use, they can be used to produce numerical

results for the development of design-oriented stress-strain models. The refined

version of Teng et al. (2007a)’s model presented in Chapter 3 is used in this

chapter for this purpose.

A great number of design-oriented models have been proposed (Fardis and Khalili

1982; Karbhari and Gao 1997; Samaan et al. 1998; Miyauchi et al. 1999; Saafi et

al. 1999; Toutanji 1999; Lillistone and Jolly 2000; Xiao and Wu 2000, 2003; Lam

and Teng 2003; Berthet et al. 2006; Harajli 2006; Saenz and Pantelides 2007; Wu

et al. 2007; Youssef et al. 2007). Design-oriented models generally comprise a

closed-form stress-strain equation and ultimate condition equations derived

directly from the interpretation of experimental results. The accuracy of design-

oriented models depends highly on the definition of the ultimate condition of

FRP-confined concrete. Existing design-oriented models have been assessed by a

number of studies (De Lorenzis and Tepfers 2003; Teng and Lam 2004; Bisby et

al. 2005). Among these models, the model proposed by Lam and Teng (2003)

appears to be advantageous over other models due to its simplicity and accuracy.

This model, with some modification, has been adopted by the design guidance for

the strengthening of concrete structures using FRP issued by the Concrete Society

98

(Concrete Society 2004) in the UK. More recently, this model has also been

adopted by ACI-440.2R (2008) with only very slight modifications. This model

adopts a simple form that naturally reduces to that for unconfined concrete when

no FRP is provided. Its simple form also caters for easy improvements to the

definition of the ultimate condition (the ultimate axial strain and compressive

strength) of FRP-confined concrete, which are the key to the accurate prediction

of stress-strain curves of FRP-confined concrete by this model.

Although Lam and Teng’s (2003) model was developed on the basis of a large test

database, a number of significant issues could not be readily resolved using the

test database available to them at that time. In particular, there was considerable

uncertainty with the hoop tensile rupture strain reached by the FRP jacket, which

has an important bearing on the definition of the ultimate condition. According to

a subsequent study by the same authors (Lam and Teng 2004), the distribution of

FRP hoop strain is highly non-uniform and the FRP hoop strains measured in the

overlapping zone of the FRP jacket are much lower than those measured

elsewhere. The lower FRP hoop strains in the overlapping zone reduce the

average hoop strain but do not result in lower confining pressure in this zone

because the FRP jacket is thicker there (Lam and Teng 2004). This observation

suggests that hoop strain readings within the overlapping zone have to be

excluded when interpreting the behavior of FRP-confined concrete, as these

readings reflect neither the actual strain capacity of the confining jacket nor the

actual dilation properties of the confined concrete. However, such important

processing of the hoop strain readings is not possible with test data collected by

Lam and Teng (2003) from the existing literature at that time, for which the

precise number and locations of strain gauges for measuring hoop strains are

generally not reported.

In addition to the uncertainty in the FRP hoop strains, the different testing

procedures (particularly the methods for axial strain measurement) adopted by

different researchers also have a bearing on the interpretation of the test data

covered by that test database. To address the deficiencies of that database and

hence those of Lam and Teng’s (2003) stress-strain model based on that database,

a large number of additional tests on FRP-confined concrete cylinders were

99

conducted under standardized testing conditions at the Hong Kong Polytechnic

University. The test database has been presented in Chapter 3 and it comprises test

data reported in Lam and Teng (2004), Lam et al. (2006) and Teng et al. (2007b)

plus test data from new tests conducted by the author (see Chapter 3). In parallel

with the experimental work, theoretical modeling work on FRP-confined concrete

was also carried out (see Chapter 3). With these new test results and the new

understandings from experimental and theoretical work, it then became feasible to

explore the refinement of Lam and Teng’s (2003) stress-strain model.

In this chapter, more accurate expressions for the ultimate axial strain and the

compressive strength of FRP-confined concrete are first proposed on a combined

experimental and analytical basis. Two modified versions of Lam and Teng’s

model based on these new expressions are next presented. The first version

involves only simple modification to the original model by updating the ultimate

condition equations. The second version differs from the first version in that it is

capable of predicting stress-strain curves with a descending branch when the

confinement is weak.

As in Chapter 3, the term “stress-strain” is generally used to refer to “axial stress-

axial strain”. The latter is used only when the axial stress-lateral strain response of

the concrete is also discussed. The sign convention adopted is the same as that of

Chapter 3: in the concrete, compressive stresses and strains are positive, but in the

FRP, tensile stresses and strains are positive.

4.2 TEST DATABASE

4.2.1 General

The test database used in this chapter is exactly the same as the one reported in

Chapter 3. A brief recap is given here. This database contains the results of 48

tests on concrete cylinders (152 mm × 305 mm) confined with varying amounts of

carbon FRP (CFRP) and Glass FRP (GFRP), with the compressive strength of

100

unconfined concrete ranging from 33.1 MPa to 47.6 MPa. The key features of

this test database are summarized as follows

'cof

1) the FRP jackets were all formed via the wet lay-up process and all had hoop

fibers only. For each batch of concrete, two or three control specimens of the

same size were also tested, from which the average values of the compressive

strength of unconfined concrete 'cof and the corresponding axial strain coε

were found;

2) the hoop strain hε of the FRP jacket was found as the average value of the

readings from five hoop strain gauges with a gauge length of 20 mm located

outside the overlapping zone (150 mm in length);

3) the axial strain of concrete cε was found as the average value of the readings

from two linear variable displacement transducers (LVDTs) at 180° apart and

covering the mid-height region of 120 mm. The lateral strain of concrete lε

was assumed to have the same magnitude as but the opposite sign to the

corresponding hε according to the sign convention adopted in this chapter;

and

4) the test database covers a wide range of FRP confinement levels. The most

heavily-confined specimen experienced an increase in the concrete strength of

about 320% while the most weakly-confined specimen exhibited a stress-

strain curve with a post-peak descending branch with negligible strength

enhancement.

For ease of discussion of the test results, some basic ratios are explained here,

namely, the confinement ratio 'l cof f , the confinement stiffness ratio Kρ and the

strain ratio ερ . The confinement ratio is commonly accepted to identify the

amount of FRP confinement with lf being the maximum confining pressure

provided by an FRP jacket. The confinement stiffness ratio represents the stiffness

ratio of the FRP jacket to the concrete core, and the strain ratio is a measure of the

101

strain capacity of the FRP jacket. It should be noted that the confinement ratio can

be taken as the product of the other two. The mathematical definitions of these

three ratios are given below

,' '

2 frp h ruplK

co co

E tff f D ε

ερ ρ= = (4.1a)

( )'

2 frpK

co co

E tf D

ρε

= (4.1b)

,h rup

coε

ερ

ε= (4.1c)

where frpE is the elastic modulus of FRP in the hoop direction, t is the thickness

of the FRP jacket, ,h rupε is the hoop strain of FRP at the rupture of the jacket due

to hoop tensile stresses, and is the diameter of the confined concrete cylinder. D


The stress-strain curves as well as the key results from all tests are given in

Chapter 3. Only the results of some typical tests are given here. Eight typical

stress-strain curves are shown in Fig. 4.1, where the lateral strains lε are shown

on the left and the axial strains cε are shown on the right. Both the axial strain and

the lateral strain are normalized by the corresponding value of coε , while the axial

stress cσ is normalized by the corresponding value of 'cof . In Fig. 4.1, both the

ascending type and descending type stress-strain curves are shown. All the

ascending type stress-strain curves exhibit the well-known bi-linear shape, and

both the compressive strength 'ccf and the ultimate axial strain cuε of confined

concrete are reached at the same point representing the rupture of the confining

jacket. Significant enhancement in both the strength and axial strain of concrete is

seen for this type of stress-strain curves. By contrast, 'ccf is reached before the

rupture of the jacket for the descending type stress-strain curves with little

strength enhancement. For this type of stress-strain curves, if the axial stress at

102

ultimate axial strain of confined concrete 'cuf falls below the compressive strength

of unconfined concrete 'cof , the concrete is referred to as insufficiently-confined

concrete in this chapter. On the other hand, concrete whose stress-strain curve

falls into the remainder of the descending type or the ascending type stress-strain

curves is referred to as sufficiently-confined concrete. The key information of

these eight specimens, together with some other specimens that are used for

discussion or comparison in this chapter can be found in Table 3.1. The naming of

the specimens follows that used in Chapter 3.


Using an analysis-oriented model for FRP-confined concrete (Spoelstra and Monti

1999), Lam and Teng (2003) demonstrated that the stiffness of the FRP jacket

affects both the ultimate axial strain and the compressive strength of FRP-

confined concrete. The present tests offer clear experimental evidence on the

effect of jacket stiffness, as illustrated in Fig. 4.2. Fig. 4.2a shows the

experimental stress-strain curves of specimens 31 and 45. The former is confined

with a GFRP jacket with 'cof = 45.9 MPa and a confinement ratio of 0.140, and the

latter is confined with a CFRP jacket with 'cof = 44.2 MPa and a confinement ratio

of 0.141. It can be seen that although the unconfined concrete strength and the

confinement ratio are both very similar for the two cases, the stress-strain curves

deviate from each other significantly because of the difference in the confinement

stiffness. The stress-strain curve of specimen 31 having a smaller confinement

stiffness ratio terminates at a lower axial stress but a larger axial strain. Similarly,

Fig. 4.2b shows the stress-strain curves of specimens 28 and 42. The former had '

cof = 45.9 MPa and a confinement ratio of 0.059 while the latter had 'cof = 44.2

MPa and a confinement ratio of 0.055. It is interesting to note that specimen 28

having a smaller confinement stiffness ratio exhibits a stress-strain curve of the

descending type while specimen 42 exhibits a stress-strain curve of the ascending

type. This observation suggests that the effect of confinement stiffness on the

ultimate condition of FRP-confined concrete should not be neglected when

developing an accurate design-oriented stress-strain model. It further suggests that

103

the confinement stiffness also plays a significant role in determining whether

sufficient confinement is achieved.

4.3 LAM AND TENG’S STRESS-STRAIN MODEL FOR FRP-

CONFINED CONCRETE

Lam and Teng’s design-oriented stress-strain model (Lam and Teng 2003) was

based on the following assumptions: (i) the stress-strain curve consists of a

parabolic first portion and a linear second portion, as given in Fig. 4.3; (ii) the

slope of the parabola at zero axial strain (the initial slope) is the same as the

elastic modulus of unconfined concrete; (iii) the nonlinear part of the first portion

is affected to some degree by the presence of an FRP jacket; (iv) the parabolic

first portion meets the linear second portion smoothly (i.e. there is no change in

slope between the two portions where they meet); (v) the linear second portion

terminates at a point where both the compressive strength and the ultimate axial

strain of confined concrete are reached; and (vi) the linear second portion

intercepts the axial stress axis at a stress level equal to the compressive strength of

unconfined concrete. The justifications for the above assumptions are given in

detail in Lam and Teng (2003) and are thus not discussed herein.

Based on these assumptions, Lam and Teng’s stress-strain model for FRP-

confined concrete is described by the following expressions:

( )22 2

'4c

c c cco

E EE

f cσ ε ε−

= − for 0 c tε ε≤ < (4.2a)

'2c co cf Eσ ε= + for t c cuε ε ε≤ ≤ (4.2b)

where cσ and cε are the axial stress and the axial strain, is the elastic modulus

of unconfined concrete, is the slope of the linear second portion. The parabolic

first portion meets the linear second portion with a smooth transition at

cE

2E

tε which is

given by

104

'

2

2 cot

c

fE E

ε =−

(4.3)

The slope of the linear second portion is given by 2E

' '

2cc co

cu

f fEε−

= (4.4)

This model allows the use of test values or values specified by design codes for

the elastic modulus of unconfined concrete . Lam and Teng (2003) proposed

the following equation to predict the ultimate axial strain

cE

cuε of confined concrete:

1.451.75 12cuK

coε

ε ρ ρε

= + (4.5)

Lam and Teng’s (2003) compressive strength equation takes the following form

'

' '1 3.3cc l

co co

f ff f

= + , if ' 0.07l

co

ff

≥ (4.6a)

'

' 1cc

co

ff

= , if ' 0.07l

co

ff

< (4.6b)

Eqs 4.6a and 4.6b are for sufficiently and insufficiently confined concrete

respectively. A minimum value of 'l cof f = 0.07 for sufficient confinement was

originally suggested by Spoelstra and Monti (1999) and was used in Lam and

Teng’s (2003) model with justification using test data available to them.

A comparison of Lam and Teng’s model with the test data of the present database

is shown in Fig. 4.4. In predicting the compressive strength and the ultimate axial

strain, a constant value of coε = 0.002 and experimental value of ,h rupε were used.

The elastic modulus of unconfined concrete was taken to be '4730c cE f= o (in

MPa). It can be seen from Fig. 4.4 that Eqs 4.5 and 4.6 overestimate the ultimate

105

axial strain of concrete at high levels of confinement and the compressive strength

of concrete at low levels of confinement. In addition, the effect of confinement

stiffness is only accounted for in the ultimate axial strain equation, but not in the

compressive strength equation. Refinement of these equations is therefore

necessary to provide more accurate predictions.

4.4 GENERALIZATION OF EQUATIONS

To take the effect of confinement stiffness into account, the expressions for the

ultimate axial strain and the compressive strength of FRP-confined concrete are

generalized here. In this regard, Eq. 4.5 can be cast into the following form:

( ) ( )cuK

co

C F fε ε ε εε ρ ρε

= + (4.7)

where ( )KFε ρ and ( )fε ερ are functions of these two ratios respectively, and Cε is

a constant.

Since the confinement ratio is the product of the confinement stiffness ratio and

the strain ratio, Eq. 4.6 can be cast into the following general form which is

similar to Eq. 4.7

'

' ( ) (ccK

co

f C F ff

)σ σ σ ερ ρ= + (4.8)

where ( )KFσ ρ and ( )fσ ερ are also functions of the confinement stiffness ratio

and the strain ratio respectively, and Cσ is a constant.

The above generalization allows the effect of confinement stiffness to be

explicitly accounted for in both the ultimate axial strain and the compressive

strength equations.

106

4.5 NEW EQUATIONS FOR THE ULTIMATE CONDITION

4.5.1 Ultimate Axial Strain

Based mainly on the interpretation of the test results in the test database, the

following improved equation for the ultimate axial strain of FRP-confined

concrete is proposed based on the best-fit results

0.8 1.451.75 6.5cuK

coε

ε ρ ρε

= + (4.9)

where the strain at the compressive strength of unconfined concrete coε was taken

to be 0.002 in determining the strain ratio ερ , since this value is commonly

accepted in existing design codes for RC structures. It is therefore suggested that

this value be used when Eq. 4.9 is used in a design specification. The first term on

the right side of Eq. 4.9 was taken to be 1.75 so that it predicts 0.0035cuε = when

no FRP confinement is provided and the coefficient and the exponents of the

second term were determined by the best-fit values of the test results. It should be

noted that 0.0035cuε = is also a commonly accepted value for the ultimate axial

strain of unconfined concrete [e.g. ENV 1992-1-1 (1992); BS 8110 (1997)] and

the constant 1.75 may be adjusted to meet the requirement of a specific design

code. Close agreement between the test results and the predictions of Eq. 4.9 is

seen in Fig. 4.5.

4.5.2 Compressive Strength

The compressive strength equation was refined on a combined experimental and

analytical basis. Fig. 4.1 has shown that both the axial stress-lateral strain curves

and the axial stress-axial strain curves exhibit a clearly bi-linear shape, with the

two portions smoothly connected by a transition zone near the compressive

strength of unconfined concrete. The shape of the second portion (the portion after

the transition zone) is very close to a straight line. It can also be seen that the

second portion of an axial stress-lateral strain curve is closer to a straight line than

107

that of an axial stress-axial strain curve. A careful study showed that the second

portion of the experimental axial stress-lateral strain curves from all tests

intercepted the axial stress axis at a stress level which is very close to the

corresponding value of compressive strength of unconfined concrete when this

portion was approximated using a fitted straight line. Such straight lines can be

expressed using the following equation

' 1c

co co

Kf

lσ εε

= + (4.10)

where constant Cσ is taken to be unity because of the reason given above. K is

the slope of the fitted straight line. It is obvious that the axial stress cσ reaches

'cuf when lε = ,h rupε− , and Eq. 4.10 becomes

'

,' 1 1h rupcu

co co

f Kf

K ε

ερ

ε= − = − (4.11)

Comparing Eq. 4.11 with Eq. 4.8 leads to 1Cσ = and ( )Fε ε ερ ρ= , while the

slope ( )KK Fσ ρ= remains undetermined.

To establish the function ( )KFσ ρ in Eq. 4.11, a parametric study was conducted

using the refined version of the analysis-oriented stress-strain model of Teng et al.

(2007a) for FRP-confined concrete (see Chapter 3). The parametric study covered

concrete cylinders of 152 mm in diameter confined with either CFRP or GFRP,

with 'cof ranging from 20 to 50 MPa and a wide range of jacket thickness to

represent different values of confinement stiffness. The material properties and

parameters examined are given in Table 4.1. In this parametric study, it was

assumed coε = 4 49.37 10 co'f−× ⋅ (Popovics 1973) as recommended in Chapter 3.

The parametric study consisted of three steps: 1) produce a family of axial stress-

lateral strain curves of a concrete cylinder confined with a certain type of FRP

jacket with varying jacket stiffness; 2) fit the second portion of these axial stress-

lateral strain curves using the best-fit straight lines with the point of interception

108

with the vertical axis ( 'c

cofσ ) fixed at unity and find the slopes of the fitted straight

lines; and 3) identify ( )KFσ ρ by finding the relationship between the slope and

ig. 4.6a demonstrates the first two steps for a concrete cylinder with

the confinement stiffness ratio.

F 'cof = 30

MPa confined with a CFRP jacket with varying thicknesses. Each stress-strain

curve in Fig. 4.6a corresponds to a particular value of the confinement stiffness

ratio. For each stress-strain curve with the portion starting from 0.5l coε ε= − was

fitted using a straight line as indicated by a dashed line in Fig. 4.6a. The slope of

the fitted line was then plotted against the confinement stiffness ratio as shown in

Fig. 4.6b, which is the last step of the process. A curve for GFRP-confined

concrete is also shown in Fig. 4.6b. This curve was generated using the numerical

results for the same concrete cylinder used in Fig. 4.6a but confined with a GFRP

jacket. It can be seen that the two curves almost overlap with each other and they

can be closely fitted using the following expression

(4.12)

q. 4.12 also provides accurate predictions for other values of

0.9( ) 3.2 0.06K KK Fσ ρ ρ= = − +

E 'cof studied. With

( )KFσ ρ determined, Eq. 4.11 becomes

( )'

cuf 0.9' 1 3.2 0.06K

cof ερ ρ= + − (4.13)

ig. 4.7 shows that Eq. 4.13 compares well with the test data of the test database. F

In predicting 'cuf for use in Fig. 4.7, experimental values of coε were used, aiming

to verify the validity of Eq. 4.13 experimentally although slightly different coε

values were used in the parametric study.

As the nonlinear relationship between ( )KFσ ρ and Kρ in Eq. 4.13 is slightly

inconvenient for design use, a simple linear equation is proposed as follows

109

( )'

' 1 3.5cuK

co

ff

0.01 ερ ρ= + − (4.14)

It should be noted that Eq. 4.14 predicts the axial stress at the ultimate axial strain,

but not the compressive strength

'ccf of FRP-confined concrete, although they are

nly different from each other when the stress-strain curve has a descending o

branch. Since Lam and Teng’s (2003) model neglects the small strength

enhancement in insufficiently confined concrete and employs a horizontal line to

represent any possible descending branch, Eq. 4.14 can readily be modified so that

it can be incorporated into Lam and Teng’s model (2003) for predicting 'ccf . It is

proposed that

( )'

1 3.5 0.01ccf' K

cof ερ ρ= + − , if 0.01 (4.15a) Kρ ≥

'

' 1cc

co

ff

= , if 0.01Kρ < (4.15b)

The performance of Eq. against

redicting

4.15 the test results is shown in Fig. 4.8. In

p 'ccf , experimental values of the FRP hoop rupture strain was used but

coε = 0.002 was used for the reason mentioned earlier. Eq. 4.15 defines a

minimum confinement stiffness ratio Kρ of 0.01 below which the FRP is assumed

to result in enhancement in the compressive strength of confined concrete.

4.6 MODIFICATION TO LAM AND TENG’S MODEL: VERSION (I)

The newly

no

defined ultimate axial strain and compressive strength equations for

RP-confined concrete as given by Eqs 4.9 and 4.15 can be directly incorporated

on

) of the modified Lam and Teng model. Fig. 4.9 shows a comparison between

F

into Lam and Teng’s (2003) model. The resulting model is referred to as Versi

(I

the predictions from the original model (Lam and Teng 2003) and Version (I) of

the modified model for three sets of specimens. Version (II) is presented in the

110

next section. The first two sets of specimens (specimens 22 and 23 in Fig. 4.9a

and specimens 24 and 25 in Fig. 4.9b) had a compressive strength of unconfined

concrete of 39.6 MPa and were confined with 2 and 3 plies of GFRP, respectively.

The confinement ratio was 0.186 for the 2-ply GFRP jacket and 0.278 for the 3-

ply GFRP jacket, while the confinement stiffness ratio was 0.018 for the former

and 0.027 for the latter. The last set of specimens (specimens 17, 18 and 19 in Fig

8c) had a compressive strength of unconfined concrete of 38.9 MPa and were

confined with 2 plies of CFRP, with a confinement ratio of 0.274 and a

confinement stiffness ratio of 0.055. The confinement stiffness ratios are based on

a constant value of coε = 0.002. This constant value of coε = 0.002 was also used

with experimental values of ,h rupε in determining other ratios and predicting the

stress-strain curves.

It can be seen that the original model gives close predictions for the specimens

with two plies of CFRP (Fig. 4.9c), but not for those with two and three plies of

GFRP (Figs 4.8b and 4.8c). Note that the confinement ratio for the 3-ply GFRP

cket is similar to that for the 2-ply CFRP jacket (0.278 versus 0.274), but the

eng model simply use a horizontal line to represent the post-peak descending

m

nd Teng’s model reduces naturally to the stress-strain model specified in

ja

confinement stiffness ratio for the former (0.027) is only half that for the latter

(0.055). The inaccuracy of the original model in this case is due to the omission of

the effect of confinement stiffness in the compressive strength equation (Eq. 4.5).

It is evident that the use of Eqs 4.9 and 4.15 to replace Eqs 4.5 and 4.6 improves

the performance of the model considerably for cases of low confinement stiffness.

4.7 MODIFICATION TO LAM AND TENG’S MODEL: VERSION (II)

Both the original Lam and Teng model and Version (I) of the modified Lam and

T

branch resulting from weak confinement. This simplification ensures that La

a

Eurocode 2 (ENV 1992-1-1 1992) for unconfined concrete when no FRP

confinement is provided. However, in applications where ductility is the major

concern, for example, in the seismic retrofit of RC columns, a small amount of

FRP confinement may suffice in terms of ductility improvement even though such

111

a small amount of confinement may not be sufficient for a significant

enhancement in the compressive strength. In such a situation, a more precise

definition of the post-peak stress-strain response of FRP-confined concrete is of

interest to structural engineers. In this section, an alternative modification to Lam

and Teng’s model to better cater for such cases is presented. The resulting stress-

strain model is referred to as Version (II) of the modified Lam and Teng model.

In order to cover both the ascedning and descending types of stress-strain curves

of FRP-confined concrete, the stress-strain curve of unconfined concrete for

design use is defined in a form similar to the well-known Hognestad’s (1951)

tress-strain curve. It is assumed that the stress-strain curve of unconfined s

concrete has a linear post-peak descending branch, which terminates at an axial

strain of 0.0035 after a 15% drop from the compressive strength of unconfined

concrete. Note that in Hognestad’s (1951) model the ultimate axial strain of

unconfined concrete is defined as 0.0038, instead of 0.0035. The latter is specified

in some design codes such as BS 8110 (1997) and ENV 1992-1-1 (1992).

For FRP-confined concrete, the parabolic first portion of Lam and Teng’s model

as given by Eq. 4.2a remains unchanged, while the linear second portion as given

by Eq. 4.2b is modified, leading to the following expressions:

( )22

c'

'2

( 0 )4

if 0.01

cc c t

co

c co c K

E EE

f

f E' '

t' ( )( ) if 0.01 c cuco cu

co c co Kcu co

f ff

ε ε ε

σ ε ρε ε ε

ε ε ρε ε

⎪⎪ < ≤⎨ −⎪ − − <⎪⎪ −⎩⎩

(4.16)

where

⎧ −− ≤ ≤⎪

⎪⎪= ⎧⎨ + ≥

2E , cuε and 'ccf are determined by Eqs 4.4, 4.9 and 4.15 respectively, and

'cuf is determined by Eq. 4.14 but is subjected to the following conditions:

0.85 , if 0; or 0.85 , if 0cf f fρ ρ≥ > = = (4.17)

' ' '

K Ku co co

112

Eq. 4.17 limits the axial stress at the ultimate strain of FRP-confined concrete 'cuf

to a minimum value of 0.85 'cof , because the concrete is deem

is stress level, no matter whether the FRP jacket has ruptured or not. Moreover,

ed to have failed at

th

it is suggested that if 'cuf is predicted by Eq. 4.14 to be equal to or smaller than

0.85 'cof , the concrete should be treated as unconfined. In such cases, the values

of 'cuf and cuε in Eq. 4.16 should be taken as 0.85 '

cof and 0.0035, respectively.

It sh ld be noted that Version (II) differs from Version (I) only when the

concrete is insufficiently confined ( 0.01K

ou

ρ < ) and Version (II) is illustrated in

ig. 4.10. The performance of Version (II) of the modified Lam and Teng model

onfined with only 1 ply of GFRP. The first pair (specimens

0 and 21) had a compressive strength of unconfined concrete of 39.6 MPa (Fig.

F

is shown in Fig. 4.11.

Fig. 4.11 is exclusively for two pairs of insufficiently confined specimens. All

these specimens were c

2

4.11a) while this value for the second pair (specimens 28 and 29) was 45.9 MPa

(Fig. 4.11b). The corresponding confinement ratio and confinement stiffness ratio

were 0.079 and 0.0090 for the former, and were 0.067 and 0.0078 for the latter,

respectively. Same as in Fig. 4.7, a constant value of 0.002coε = and

experimental values of ,h rupε were used in predicting the stress-strain curves.

Note that for the specimens covered by Fig. 4.11a, the confinement ratio is greater

than the minimum value of 0.07 for achieving sufficient confinement as suggested

y Lam and Teng (2003), but for the specimens covered by Fig. 4.11b, the

linear second portion for specimens of both Figs 4.10a and 4.10b, which do not

b

confinement ratio is smaller than this minimum value. In both cases, the

confinement stiffness ratios are smaller than the critical value of 0.01 as suggested

in the present study. It can be seen that the original model predicts a

monotonically increasing stress-strain curve for the specimens in Fig. 4.11a, but a

stress-strain curve with a horizontal second portion for the specimens in Fig.

4.11b, although in both cases the test curves are of the decreasing type. The

modified Lam and Teng model (I) predicts stress-strain curves with a horizontal

113

match the test curves well but provide reasonably close approximations. In

comparison, the predictions from the modified Lam and Teng model (II) are in

better agreement with the test stress-strain curves.

4.8 CONCLUSIONS

This paper has presented the results of a recent study aimed at the refinement of a

esign-oriented stress-strain model for FRP-confined concrete developed by Lam

weaknesses of this model have been examined through

omparisons with new test results obtained at The Hong Kong Polytechnic

th a small amount of FRP.

ent stiffness. Version (II) of the modified Lam and Teng model

performs better than Version (I) in predicting stress-strain curves with a

d

and Teng (2003). Some

c

University under standardized test conditions. New equations for predicting the

ultimate axial strain and the compressive strength of FRP-confined concrete have

been developed based on the interpretation of the new test results and results from

a parametric study using a refined version of Teng et al.’s (2007a) recent analysis-

oriented model. Two modified versions of Lam and Teng’s (2003) design-oriented

stress-strain model have been proposed. Version (I) of the modified Lam and

Teng model I) involves only a simple modification to the original model by

updating the ultimate axial strain and compressive strength. Version (II) of the

modified Lam and Teng model attempts to cater for the descending type stress-

strain curves, which are not covered by the original model. The following

conclusions can be drawn:

1) The original Lam and Teng model overestimates the ultimate strain of

concrete confined with a large amount of FRP and the compressive strength

of concrete confined wi

2) The new ultimate strain and compressive strength equations account for the

effect of confinement stiffness explicitly and provide close predictions of test

results.

3) Both modified versions of Lam and Teng’s model provide much closer

predictions of test stress-strain curves than the original model for cases of low

confinem

114

descending branch, although the latter also predicts such stress-strain curves

reasonably well.

115

4.9 REFERENCES


Berthet, J.F., Ferrier, E. and Hamelin, P. (2006). “Compressive behavior of concrete externally confined by composite jackets - Part B: modeling”, Construction and Building Materials, 20(5), 338-347.


BS 8110 (1997). Structural Use of Concrete, Part 1. Code of Practice for Design and Construction, British Standards Institution, London, UK.


Chun, S.S. and Park, H.C. (2002). “Load carrying capacity and ductility of RC columns confined by carbon fiber reinforced polymer.” Proceedings, 3rd International Conference on Composites in Infrastructure (CD-Rom), San Francisco.



Fardis, M.N. and Khalili, H. (1982). “FRP-encased concrete as a structural material”, Magazine of Concrete Research, 34(122), 191-202.

Harajli, M.H. (2006). “Axial stress-strain relationship for FRP confined circular and rectangular concrete columns”, Cement & Concrete Composites, 28(10), 938-948.


Hognestad, E. (1951). A Study of Combined Bending and Axial Load in Reinforced Concrete Members, Bulletin Series No. 399, Engineering Experiment Station, University of Illinois, Urbana, III.

Karbhari, V.M. and Gao, Y. (1997). “Composite jacketed concrete under uniaxial compression–verification of simple design equations”, Journal of Materials in Civil Engineering, ASCE, 9(4), 185-193.

116









Lam, L., and Teng, J.G. (2003). “Design-oriented stress-strain model for FRP-confined concrete”, Construction and Building Materials, 17(6-7), 471-489.

Lam, L., and Teng, J.G. (2004). “Ultimate condition of fiber reinforced polymer-confined concrete”, Journal of Composites for Construction, ASCE, 8(6), 539-548.

Lam, L., Teng, J.G., Cheng, C.H., and Xiao, Y. (2006). “FRP-confined concrete under axial cyclic compression”, Cement and Concrete Composites, 28(10), 949-958.

Lillistone, D. and Jolly, C.K. (2000). “An innovative form of reinforcement for concrete columns using advanced composites”, The Structural Engineer, 78(23/24), 20-28.

Marques, S.P.C., Marques, D.C.S.C., da Silva J.L. and Cavalcante, M.A.A. (2004). “Model for analysis of short columns of concrete confined by fiber-reinforced polymer”, Journal of Composites for Construction, ASCE, 8(4), 332-340.

Mirmiran, A. and Shahawy, M. (1997). “Dilation characteristics of confined concrete”, Mechanics of Cohesive-Frictional Materials, 2 (3), 237-249.

Miyauchi, K., Inoue, S., Kuroda, T. and Kobayashi, (1999). “Strengthening effects of concrete columns with carbon fiber sheet”, Transactions of the Japan Concrete Institute, 21, 143-150.


Popovics, S. (1973). “Numerical approach to the complete stress-strain relation for concrete”, Cement and Concrete Research, 3(5), 583-599.


Saafi, M., Toutanji, H.A. and Li, Z. (1999). “Behavior of concrete columns confined with fiber reinforced polymer tubes”, ACI Materials Journal, 96(4), 500-509.

Samaan, M., Mirmiran, A., and Shahawy, M. (1998). “Model of concrete confined by fiber composite.” Journal of Structural Engineering, ASCE, 124(9), 1025-1031.

Saenz, N. and Pantelides, C.P. (2007). “Strain-based confinement model for FRP-confined concrete”, Journal of Structural Engineering, ASCE, 133 (6), 825-833.

Spoelstra, M.R., and Monti, G. (1999). “FRP-confined concrete model”, Journal of Composites for Construction, ASCE, 3(3), 143-150.

117










Teng, J.G., and Lam, L. (2004). “Behavior and modeling of fiber reinforced polymer-confined concrete”, Journal of Structural Engineering, ASCE, 130(11), 1713-1723.

Teng, J.G., Huang, Y.L., Lam, L., and Ye, L.P. (2007a). “Theoretical model for fiber reinforced polymer-confined concrete”, Journal of Composites for Construction, ASCE, 11(2), 201-210.

Teng, J.G., Yu, T. Wong, Y.L. and Dong, S.L. (2007b). “Hybrid FRP-concrete-steel tubular columns: concept and behaviour”, Construction and Building Materials, 21(4), 846-854.

Toutanji, H.A. (1999). “Stress-strain characteristics of concrete columns externally confined with advanced fiber composite sheets”, ACI Materials Journal, 96(3), 397-404.

Wu, G., Wu, Z.S. and Lu, Z.T. (2007). “Design-oriented stress-strain model for concrete prisms confined with FRP composites”, Construction and Building Materials, 21(5), 1107-1121.

Xiao, Y. and Wu, H. (2000), “Compressive behavior of concrete confined by carbon fiber composite jackets”, Journal of Materials in Civil Engineering, ASCE, 12(2), 139-146.

Xiao, Y. and Wu, H. (2003), “Compressive behavior of concrete confined by various types of FRP composite jackets’, Journal of Reinforced Plastics and Composites, 22(13), 1187-1201.

Youssef, M.N., Feng, M.Q. and Mosallam, A.S. (2007). “Stress-strain model for concrete confined by FRP composites”, Composites Part B – Engineering, 38(5-6), 614-628.

118









Table 4.1 Parameters used in the parametric study

Concrete 'cof (MPa) 20 to 50 at an interval of 5

frpE (GPa) 230

ruph,ε 0.0075 CFRP t (mm) 0 to 1 at an interval of 0.1 frpE (GPa) 80

ruph,ε 0.015 GFRP t (mm) 0 to 1.5 at an interval of 0.1

119

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Normalized Axial Strain εc/εco

Nor

mal

ized

Axi

al S

tress

σc/

f ′ co

Specimens 20,21ρK=0.009, ρε=8.7




(a) Axial stress-axial strain curves

-10 -8 -6 -4 -2 00

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Normalized Lateral Strain ε l /εco

Nor

mal

ized

Axi

al S

tress

σc/

f ′ co





(b) Axial stress-lateral strain curves

Fig. 4.1 Typical stress-strain curves of FRP-confined concrete

120

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Nor

mal

ized

Axi

al S

tress

σc

/f ′co

Specimen 31Specimen 45

(a) Comparison between specimen 31 and specimen 45

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1


Nor

mal

ized

Axi

al S

tress

σc

/f ′co

Specimen 28Specimen 42

(b) Comparison between specimen 28 and specimen 42

Fig. 4.2 Effect of confinement stiffness on stress-strain behavior of FRP-confined concrete

121

Axial Strain εc

Axi

al S

tress

σc

Unconfined concrete(ENV 1992)FRP-confined concete(Lam and Teng)

f′cc

f′co

εco 0.0035εt εcu

Fig. 4.3 Illustration of Lam and Teng’s model

122

0 5 10 15 20 250

5

10

15

20

25

Normalized Ultimate Axial Strainεcu /εco - Test

Nor

mal

ized

Ulti

mat

e A

xial

Stra

inε cu

/ εco

- P

redi

cted

Lam and Teng (2004), CFRPLam and Teng (2004), GFRPLam et al. (2006), CFRPTeng et al. (2007b), GFRPPresent Study, GFRPPresent Study, CFRP


0 1 2 3 4 50

1

2

3

4

5

Normalized Compressive Strengthf′cc /f′co - Test

Nor

mal

ized

Com

pres

sive

Stre

ngth

f ′ cc

/f′ co

- P

redi

cted


(b) Compressive strength

Fig. 4.4 Performance of Lam and Teng’ s model against test results

123

0 5 10 15 20 250

5

10

15

20

25

Normalized Ultimate Axial Strainεcu /εco - Test

Nor

mal

ized

Ulti

mat

e A

xial

Stra

inε cu

/ εco

- P

redi

cted


Fig. 4.5 Performance of Eq. 4.9

124

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 00

0.5

1

1.5

2

2.5

3

3.5

4

Normalized Lateral Strain ε l /εco

Nor

mal

ized

Axi

al S

tress

σc /

f′ co

Refined Version ofTeng et al.′s ModelFitted Line

(a) First two steps

0 0.05 0.1 0.15 0.2 0.25-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Confinement Stiffness Ratio ρK

Slo

pe K

CFRP-confinedGFRP-confinedProposed Equation

(b) Last step

Fig. 4.6 Demonstration of the parametric study

125

0 1 2 3 4 50

1

2

3

4

5


Nor

mal

ized

Axi

al S

tress

at U

ltim

ate

Axi

al S

train

f ′cu

/f′ co

- P

redi

cted



0 1 2 3 4 50

1

2

3

4

5

Normalized Compressive Strengthf′cc /f′co - Test

Nor

mal

ized

Com

pres

sive

Stre

ngth

f ′ cc

/f′ co

- P

redi

cted



126

0 0.005 0.01 0.015 0.02 0.0250

10

20

30

40

50

60

70

Axial Strain εc

Axi

al S

tress

σc

(MP

a)

f′co = 39.6MPa

Efrp = 80100MPa

t = 0.34mmD = 152mmεh,rup = 0.0205

Lam and Teng (2003)Modified Model (I)Test (Specimens 22 and 23)

(a) Specimens 22 and 23

0 0.005 0.01 0.015 0.02 0.025 0.030

10

20

30

40

50

60

70

80

Axial Strain εc

Axi

al S

tress

σc

(MP

a)

f′co = 39.6MPa

Efrp = 80100MPa

t = 0.51mmD = 152mmεh,rup = 0.0181

Lam and Teng (2003)Modified Model (I)Test (Specimens 24 and 25)

(b) Specimens 24 and 25

127

0 0.005 0.01 0.015 0.02 0.0250

10

20

30

40

50

60

70

80

90

Axial Strain εc

Axi

al S

tress

σc

(MP

a)

f′co = 38.9MPa

Efrp = 247000MPa

t = 0.33mmD = 152mmεh,rup = 0.00994

Lam and Teng (2003)Modified Model (I)Test (Specimens 17 to 19)

(c) Specimens 17,18 and 19

Fig. 4.9 Performance of Version (I) of the modified Lam and Teng model

128

Axial Strain εc

Axi

al S

tress

σc

f′cc

f′cof′cu

0.85f′co

εco 0.0035 εcu εcu εcu

ρK>0.01

ρK=0.01

ρK<0.01

Unconfined

Fig. 4.10 Schematic of Version (II) of the modified Lam and Teng model

129

0 0.002 0.004 0.006 0.008 0.01 0.0120

5

10

15

20

25

30

35

40

45

50

Axial Strain εc

Axi

al S

tress

σc

(MP

a)

f′co = 39.6MPa

Efrp = 80100MPa

t = 0.17mmD = 152mmεh,rup = 0.0174

Lam and Teng (2003)Modified Model (I)Modified Model (II)Test (Specimens 20 and 21)

(a) Specimens 20 and 21

0 0.002 0.004 0.006 0.008 0.01 0.0120

5

10

15

20

25

30

35

40

45

50

Axial Strain εc

Axi

al S

tress

σc

(MP

a)

f′co = 45.9MPa

Efrp = 80100MPa

t = 0.17mmD = 152mmεh,rup = 0.0172

Lam and Teng (2003)Modified Model (I)Modified Model (II)Test (Specimens 28 and 29)

(b) Specimens 28 and 29

Fig. 4.11 Prediction of descending type of stress-strain curves

130

CHAPTER 5

DESIGN OF SHORT FRP-CONFINED RC COLUMNS

5.1 INTRODUCTION

The ultimate goal of a design-oriented stress-strain model for FRP-confined

concrete is to facilitate the design of FRP jackets for strengthening RC columns.

Relevant design provisions are now available in various design guidelines (fib

2001; ISIS 2001; ACI-440.2R 2002, 2008; JSCE 2002; CNR-DT 2004; Concrete

Society 2004) for strengthening RC structures. However, current provisions are

only applicable to short columns, where the slenderness effect is negligible.

Moreover, most of the design guidelines are only concerned with short columns

under concentric compression. Only Concrete Society (2004) and ACI-440.2R

(2008) have recommended a general procedure for the section analysis of short

FRP-confined RC columns so that the axial load-bending moment interaction

diagram can be constructed accordingly, but no corresponding design equations

are specified in both design guidelines. Although such a section analysis

procedure can fulfill the design needs, a much simpler method comprising a set of

explicit equations is still highly desirable. This chapter is concerned with the

development of such explicit equations for short FRP-confined RC columns. The

analysis and design of slender FRP-confined RC columns is dealt with in Chapters

7 to 9. It should be noted that the present study has been partially motivated by the

development of the Chinese Code for the Structural Use of FRP Composites in

Construction, which is formulated within the framework of the current Chinese

Code for Design of Concrete Structures (GB-50010 2002). Therefore, some of the

considerations in the present study follow the specifications given in GB-50010

(2002) and these considerations are highlighted where appropriate. The present

131

study is limited to FRP jackets that are continuous over a strengthened region of

the column and possess fibers oriented solely or predominantly in the hoop

direction. Partial safety factors are taken to be unity for convenience of

presentation.

5.2 SECTION ANALYSIS

5.2.1 The Strength of FRP-confined RC sections

Once the stress-strain model for FRP-confined concrete is defined, design of FRP-

confined RC sections can be carried out using the conventional section analysis. It

is necessary to note that the stress-strain models discussed in the preceding two

chapters are all for FRP-confined concrete under concentric compression, but in a

column under combined bending and axial compression, a strain gradient exists.

For conventional RC columns, the assumption that the stress-strain curve of

concrete in an eccentrically-loaded column is the same as that of concrete under

concentric compression is widely used. For FRP-confined RC columns, it has

been concluded in Chapter 2 that it is also reasonable to adopt the same

assumption. Indeed, this assumption has been adopted by previous researchers

(e.g. Saadatmanesh et al. 1994; Mirmiran et al. 2000; Monti et al. 2001; Yuan and

Mirmiran 2001; Cheng et al. 2002; Teng et al. 2002; Binici 2008; Yuan et al.

2008).

When the stress-strain curve of FRP-confined concrete from concentric

compression is directly used in a section analysis, the analysis procedure is similar

to that for conventional RC columns as described in numerous reinforced concrete

textbooks (e.g. Park and Paulay 1975; Kong and Evans 1987). The only difference

in the analysis procedure introduced by the presence of FRP confinement is the

use of a different concrete stress-strain relationship that considers the confinement

effect of the FRP. Numerical integration over the section can still be carried out

using the layer method in which the column section is divided into many small

horizontal layers as shown in Fig. 5.1. In the present study, the modified Lam and

Teng model (I) is used. A small adjustment needs to be made to this model so that

132

it reduces to the stress-strain model for unconfined concrete in GB-50010 (2002).

In GB-50010 (2002), normal strength concrete is assumed to have co 0.002ε =

an ultimate axial strain of 0.0033 and its stress-strain curve consists of

Hognestad’s parabola and a horizontal line. As a result, the original value of 1.75

for the first term on the right hand side of the ultimate axial strain equation (Eq.

4.9) is replaced by 1.65, so that the stress-strain model for FRP-confined concrete

can reduce to that for unconfined normal strength concrete adopted by GB-50010

(2002) when no FRP is provided. Complete composite action between concrete

and FRP is assumed. Compressive stresses are taken to be positive and the tensile

strength of concrete is ignored. Plane sections are assumed to remain plane. Only

columns with limited hoop steel reinforcement are considered so any confinement

effect from the hoop steel reinforcement is ignored. The longitudinal steel

reinforcement is assumed to have an elastic-perfectly plastic stress-strain curve.

The axial load N and the bending moment

and

M at any stage of loading carried by

the section with the reference axis going through the centre of the section are

found by integrating the stresses over the section:

1

( )c n

nR

c c c si c siR xi

N b dλ

σ λ σ σ= −

=

= + −∑∫ A

)

(5.1a)

(1

( )c n

nR

c c c c si c si siR xi

M b d A R dλ

σ λ λ σ σ= −

=

= + −∑∫ − (5.1b)

where R is the radius of the section, is the width of the section at a distance cb cλ

from the reference axis, nx is the depth of the neutral axis, siσ is the stress in the

th layer of longitudinal steel reinforcement, and i siA is the corresponding cross-

sectional area of the longitudinal steel reinforcement. The stress of concrete cσ in

the compression zone can be determined from Eq. 4.2. siσ can be calculated from

si sE siσ ε= if ysi

s

fE

ε < (5.2a)

133

sisi y

si

fεσε

= if ysi

s

fE

ε ≥ (5.2b)

Eq. 5.1 is applicable at any stage of loading. The ultimate limit state of the column

is reached when the strain at the extreme concrete compression fiber reaches the

ultimate strain of FRP-confined concrete, signifying crushing of concrete due to

FRP rupture. This ultimate strain is defined by Eq. 4.9 with the original value of

1.75 for the first term on the right hand side replaced by 1.65.

Section analysis was performed on a reference circular RC column (Fig. 5.2) with

a diameter D 600 mm. Altogether 12 steel bars of 25 mm in diameter are

distributed evenly around the section. The circle defined by the centers of these

bars has a diameter mm. The strengths of concrete and steel

reinforcement were taken to be common values as specified in GB-50010 (2002).

The concrete was assumed to be grade C30, representing a characteristic cube

strength of MPa and a corresponding compressive strength

MPa. The steel was assumed to be grade II with a

characteristic yield strength

=

500d =

30cuf =

' 0.67 20.1co cuf f= =

335yf = MPa and an elastic modulus of 200

GPa. Note that the strengths of unconfined concrete and steel reinforcement given

above are used throughout the present study, unless otherwise stated. The column

was either unconfined or wrapped with a three-ply or six-ply CFRP jacket, with a

nominal ply thickness of 0.165 mm. The elastic modulus and hoop rupture strain

were assumed to be 230 GPa and 0.0075 for the CFRP jacket.

sE =

The interaction curves produced by the section analysis are shown in Fig. 5.3.

These interaction curves are normalized by the axial load capacity (concentric

compression) and moment capacity

uoN

uoM (pure bending) of the reference column

when no FRP confinement is provided. It can be seen that the maximum benefit of

FRP confinement occurs when the section fails in pure compression, but

confinement is much less beneficial when the column fails in pure bending. The

sharp slope change in the interaction curves at high axial loads for the wrapped

134

columns (Fig. 5.3) occurs when the neutral axis begins to fall outside the column

section. As the neutral axis depth moves further away from the column section,

the parabolic portion of the confined concrete stress-strain curve gradually moves

outside the section, with stresses over the section being eventually governed

entirely by the linear second portion of the stress-strain curve.

5.2.2 Moment-Curvature Curves of FRP-confined RC Sections

For a given column section, there exists a unique moment-curvature curve under a

particular axial load . This moment-curvature curve can be readily constructed

using the approach described in the preceding section. For a given axial load ,

the corresponding moment-curvature curve can be generated by specifying a

series of suitable strain values for the extreme compression fiber of concrete

N

N

cfε

up to its ultimate value cuε . For each strain value, the curvature φ is varied until

the resultant axial force acting on the section, calculated from Eq. 5.1a, equals the

applied axial load. Once the neutral axis position has been determined, the

moment can be evaluated using Eq. 5.1b. Fig. 5.4 shows three typical moment

curvature curves for the reference column described in the preceding section with

a 6-ply CFRP jacket. These three curves are for three different levels of axial load,

which represent low, moderate and high axial loads respectively.

5.2.3 Comparison with Test Results

Only a very limited number of experimental studies have been conducted on FRP-

confined circular RC columns subjected to eccentric loading (Tao et al. 2004;

Hadi 2006; Fitzwilliam and Bisby 2006; Ranger and Bisby 2007). The columns

reported in these studies were subjected to the coupled effect of load eccentricity

and slenderness. In addition, these studies were more concerned with the overall

column behavior than the section behavior (e.g. load-deflections curves were

reported in these studies, but none of them reported the curvature distribution of

the columns). As a result, it is difficult to extract experimental data to verify the

section analysis procedure presented above. However, the test data reported in the

135

above studies are used in Chapter 7 to verify a theoretical model capable of

dealing with the slenderness effect in FRP-confined RC columns.

The only two studies that reported the moment-curvature data appear to be the

study by Sheikh and Yau (2002) and Harmon and Gould (2002). Sheikh and Yau

(2002) tested FRP-confined circular RC columns (356 mm in diameter) under

constant axial loading and reversed cyclic lateral loading. The moment-curvature

data measured in the plastic hinge region were reported for all the columns they

tested. The moment-curvature data have recently been used by Binici (2008) to

verify a similar section analysis procedure for FRP-confined RC columns. The

columns used for comparison herein had six evenly distributed longitudinal steel

reinforcing bars with a diameter of 25 mm and only had a very small amount of

transverse steel reinforcement. The geometric and material properties of these

columns are summarized in Table 5.1. The experimental and the theoretical

moment-curvature curves are compared in Fig. 5.5. It can be seen that the

theoretical curves are in good agreement with the experimental envelope curves

for columns ST-2NT and ST-3NT which were tested under a relatively high axial

load (about 50% of ). On the other hand, the ultimate curvatures of columns

ST-4NT and ST-5NT tested under a relatively low axial load (about 25% of )

are considerably underestimated. Binici (2008) made similar observations in his

comparison with the same test data. He argued that the discrepancy might be due

to some unexpected experimental factors, because column ST-5NT showed a

larger deformation capacity than column ST-4NT while it only had about half the

amount of FRP confinement provided to column ST-4NT.

uoN

uoN

Harmon and Gould (2002) conducted similar tests on FRP-confined RC columns

to those reported by Sheikh and Yau (2002). Harmon and Gould’s (2002) columns

were 180 mm in diameter and were either 600 mm or 1200 mm in length. All the

columns were reinforced with six evenly distributed longitudinal steel reinforcing

bars with a diameter of 12.7 mm and had no hoop steel reinforcement. The

concrete strength ranged from 41.1 MPa to 55.2 MPa while the yield strength of

steel was 604 MPa. All the columns were provided with a filament wound GFRP

jacket with the fibers oriented solely in the hoop direction. The thickness of the

136

GFRP jacket was varied to achieve different fiber-to-concrete volume ratios (the

volume of fiber to the volume of confined concrete) of 1%, 3%, and 5%. Some of

the columns were subjected to a constant axial load of 133 kN while others were

subjected to a higher axial load of 578 kN. The elastic modulus of the GFRP

jacket was 68950 MPa (Harmon 2008). The geometric and material properties of

these columns are summarized in Table 5.2. It should be noted that columns 0600-

5-133 and 0600-5-578 were excluded from the present comparisons, because no

curvature data were available due to the failure of the measurement method

initially employed in their study (Gould 1999). Gould and Harmon (2002)

reported that most of their columns failed by the fracture of the steel reinforcing

bars before the rupture of the FRP jacket. FRP rupture was only observed in

column 1200-1-133 which had a length of 1200 mm, a fiber-to-concrete volume

ratio of 1%, and was subjected to a constant axial load of 133 kN. The maximum

FRP hoop strain measured at the critical position of each column is not reported in

Gould (1999) and Gould and Harmon (2002); it is only reported in Gould and

Harmon (2002) that the maximum FRP hoop strain reading recorded in the entire

test series was about 2%. As a result, it is not possible to make precise

comparisons for these columns; an FRP hoop strain of 2% was assumed when

producing the theoretical moment-curvature curves for these columns. Fig. 5.6

shows the comparisons between the experimental and the theoretical results. It

should be noted that in each of Figs 5.6a to 5.6c, a pair of experimental curves are

shown for a pair of columns with the same configuration except their lengths.

Obviously, the theoretical moment-curvature curves are identical for such a pair

of columns. Two theoretical curves corresponding to the lower bound and the

upper bound of concrete strength (41.1 Mpa and 55.2 MPa respectively) are

shown in Fig. 5.6. It can be seen that within this range of concrete strength, the

theoretical curves are only slightly affected by the concrete strength. It should also

be noted that was assumed when producing the theoretical curves as the

exact thickness of the concrete cover is not reported in Gould and Harmon (2002);

the thickness of the concrete cover only has a small effect on the theoretical

moment-curvature curve.

0.8d = D

Fig. 5.6a shows the comparison for columns 0600-1-133 and 1200-1-133. The

theoretical curves are close to the experimental curves. It should be noted that

137

column 1200-1-133 failed by FRP rupture. As a result, the comparison for these

two columns should be more reliable than that for the other columns. It should

also be noted that these two column were subjected to a very small axial load (133

kN), which is only about 7% of . The section analysis procedure presented

above is still reasonably accurate for this case. Fig. 5.6b shows that the theoretical

ultimate curvature is considerably smaller than the experimental value. This might

be due to some unexpected experimental factors, because the columns of Fig. 5.6b

only differed from those of Fig. 5.6a in that the former were subjected to a higher

axial load (578kN). Therefore, the ultimate curvatures of the former are expected

to be smaller than those of the latter, but the excremental ultimate curvatures

shown in Fig. 5.6b contradict this expectation. Besides, a significant discrepancy

also exists between the two experimental curves in Fig. 5.6b although these two

curves are expected to be very similar. Fig 5.7c shows that the theoretical curves

are reasonably close to the experimental curves. By contrast, the theoretical curves

in Figs 5.7d and 5.7e terminate at a much larger ultimate curvature than the

experimental curves. This may be due to the premature occurrence of steel

fracture before the FRP jacket was fully utilized.

uoN

Based on the above comparisons and given the fact that this assumption has been

adopted by many previous researchers, the present section analysis procedure is

deemed to be reasonable. However, more tests are still needed to fully clarify the

possible effect of load eccentricity on the stress-strain behavior of FRP-confined

concrete.

5.3 EQUIVALENT STRESS BLOCK

In the design of RC members, the stress profile of concrete in compression is

generally simplified using an equivalent stress block, over which stresses are

uniformly distributed. This equivalent stress block can be described by two factors,

the magnitude of the stresses over and the depth of this equivalent stress block,

which can be determined from the criterion that the equivalent stress block must

resist the same axial force and bending moment as the original stress profile.

Because of the FRP confinement, existing values of stress block factors adopted

138

for the design of conventional RC members are no longer suitable. This section is

therefore concerned with the development of appropriate stress block factors for

FRP-confined concrete in circular columns. In the present study, the mean stress

factor 1α is defined as the ratio of the uniform stress over the stress block to the

compressive strength of FRP-confined concrete and the block depth factor 1β is

defined as the ratio of the depth of the stress block to that of the neutral axis.

The stress distributions over the compression zone are examined for different

neutral axis positions to find the equivalent stress block factors. The thickness of

the confining CFRP jacket was taken as a variable. The maximum thickness is

such that it leads to a strength enhancement ratio ' ' 2cc cof f = . Fig. 5.7 shows the

variations of the stress block factors against the strength enhancement ratio. It can

be seen that the mean stress factor decreases as the strength enhancement ratio

increases and the block depth factor varies only slightly against the strength

enhancement ratio. It can also be seen that for circular sections, both stress block

factors also depend slightly on the neutral axial depth nx , as the width of

horizontal layers in a circular section varies with its depth. For simplicity, it is

suggested that

1 0.9β = (5.3)

Once 1β is fixed, 1α can then be recalculated according to the criterion that the

equivalent stress block must resist the same axial force as the original stress

profile. The recalculated values of 1α are shown in Fig. 5.8. The following simple

linear equation is suggested

' '

1 1.17 0.2 cc cof fα = − (5.4)

It should be noted that the numerical results shown here are for concrete confined

with a CFRP jacket having a rupture strain of 0.0075. However, Eqs 5.3 and 5.4

are also applicable to concrete confined with other types of FRP jackets. The

139

overall performance of Eqs 5.3 and 5.4 are examined through a comprehensive

parametric study later in the chapter.

5.4 DESIGN EQUATIONS

Using the stress block factors proposed above, design equations based on a

simplified section analysis method are presented herein, following a similar

approach adopted by GB-50010 (2002). This method is only applicable to

columns that have six or more evenly distributed longitudinal steel reinforcing

bars.

As shown in Fig. 5.9, the steel reinforcing bars are smeared into an equivalent

steel cylinder of the same total cross-sectional area and with longitudinal strength

only. 02πθ , 2πθ , 12πθ , and '22πθ are respectively the central angles

corresponding to the depths of the neutral axis, the equivalent stress block, the

yielded compressive steel reinforcement and the yielded tensile steel

reinforcement. 0θ , 1θ , θ and '2θ can be calculated from Eq. 5.5 and it is obvious

that 2θ = 1- '2θ .

( )0

ar cos 1 1 /n sR Rξθ

π− +⎡ ⎤⎣= ⎦ (5.5a)

( )1ar cos 1 1 /n sR Rβ ξθ

π− +⎡ ⎤⎣= ⎦ (5.5b)

( ) ( )1

ar cos / 1 1 /s n sR R Rβ ξθ

π− − +⎡ ⎤⎣ ⎦=

R (5.5c)

( ) ( )'2

ar cos / 1 1 /s nR R R Rβ ξθ

π− + +⎡ ⎤⎣ ⎦= s (5.5d)

where sR is the radius of the imaginary steel cylinder, 0/n nx hξ = is the ratio of

the neutral axis depth nx to the effective height of the section (0h sR R= + ), and

β is the ratio of the yield strain of steel reinforcement to the strain of the extreme

compressive concrete fiber as given by

140

y

s cu

fE

βε

= (5.6)

The axial load and bending moment carried by concrete can then be calculated

from

'1

sin(2 )12c ccN f A πθα θπθ

⎛= −⎜⎝ ⎠

⎞⎟ (5.7a)

3'

12 sin (3c ccM f AR )πθα

π= (5.7b)

where A is the gross area of the cross section and 2Rπ= . The axial load and

bending moment carried by steel reinforcement can be calculated from

( ) ( )1 2s y s c y s tN f A k f A kθ θ= + − + (5.7c)

1sin( ) sin( )c ts y s s y s s

m mM f A R f A R 2πθπ π

πθ+ += + (5.7d)

where sA is the total cross-sectional area of longitudinal steel bars, and

( ) ( )( )

0 1 0 11 / / sin( ) sin( )1 /

n s sc

n s

R R R Rk

R Rξ π θ θ πθ πθ

πβξ+ − − + −⎡ ⎤⎣ ⎦=

+ (5.8a)

( ) ( )

( )

' '2 0 2 01 / / sin( ) sin(

1 /n s s

tn s

R R R Rk

R R)ξ π θ θ πθ πθ

πβξ

+ − − + −⎡ ⎤⎣ ⎦= −+

(5.8b)

( ) [ ] ( )

( )

0 1 0 10 1

sin(2 ) sin(2 )1 / / sin( ) sin( )2 4

1 /

n s s

cn s

R R R Rm

R R

π θ θ πθ πξ πθ πθ

πβξ

− −+ − − + +⎡ ⎤⎣ ⎦

=+

θ

(5.8c)

( ) ( )

( )

' '2 0' 2 0

2 0sin(2 ) sin(2 )1 / / sin( ) sin( )

2 41 /

n s s

tn s

R R R Rm

R R

π θ θ πθ πξ πθ πθ

πβξ

− −⎡ ⎤+ − − + +⎡ ⎤⎣ ⎦ ⎣ ⎦=

+

θ

(5.8d)

141

Obviously, Eqs 5.7 and 5.8 are too complex for design use. For simplicity, the

following approximate expressions are proposed:

10 1.25 0.125 1c ckθ θ θ≤ = + = − ≤ (5.9a)

20 1.125 1.5 1t tkθ θ≤ = + = − ≤θ (5.9b)

Using Eq. 5.9, the terms 1 ckθ + and 2 tkθ + in Eq. 5.7c can be reasonably well

approximated by cθ and tθ respectively (Fig. 5.10a). In the mean time, the terms

1sin( ) cmπθ + and 2sin( ) tmπθ + in Eq. 5.7d can be approximated by sin( )cπθ and

sin( )tπθ respectively (Fig. 5.10b). It should be noted that Eq. 5.9 was derived

with an assumed value of / sR R =1.16, which was used in developing similar

expressions for GB-50010 (2002). Nevertheless, Eq. 5.9 is still sufficiently

accurate for columns with other values of / sR R , as proven by the parametric

study presented in the next section.

Based on the above discussions, the following expressions are obtained for the

ultimate axial load capacity and bending moment capacity uN uM of an FRP-

confined circular column:

( )'1

sin(2 )12u cc c t yN f A fπθθα θ θπθ

⎡ ⎤= − + −⎢ ⎥⎣ ⎦sA (5.10a)

3

'1

sin( ) sin( )2 sin ( )3

cu cc y sM f AR f A R tπθ ππθα

π+

= +θ

π (5.10b)

5.5 PERFORMANCE OF PROPOSED DESIGN EQUATIONS

In this section, numerical results from a comprehensive parametric study are

presented to assess the performance of Eq. 5.10. In this parametric study, the

column was assumed to have a diameter of 600 mm and to have 12 evenly

distributed longitudinal steel reinforcing bars. The concrete was assumed to be

grade C30 with MPa and the steel was assumed to be grade II with ' 20.1cof =

142

335yf = MPa. With the above parameters fixed, the strength of any given section

can be determined if the values of the following parameters are also specified: 1)

the eccentricity e ; 2) the thickness of the concrete cover, which can be

represented by either d D or sR R . The former ratio is used throughout the

present study and is referred to as the depth ratio; 3) the steel reinforcement ratio

sρ ; 4) the amount and the properties of the confining FRP jacket. The strength

enhancement ratio ' 'cc cof f and the strain ratio ,h rup coε ε were chosen to describe

the effect of FRP confinement. The values of the variables considered in the

parametric study are summarized in Table 5.3.

These parameters lead to 1620 cases of combination. The justifications for the

ranges of the parameters considered are as follows. The lower bound of the

normalized eccentricity is close to the minimum eccentricity , which is

generally specified by existing design codes for RC structures

[ mm in GB-50010 (2002) and

mine

min / 30 20≥e D= min 0.05 20e D= ≤ mm in BS-

8110 (1997)]. On the other hand, MacGregor et al. (1970) stated that the vast

majority of RC columns have a normalized eccentricity less than 0.84 based on

the results of a survey of more than 20,000 columns. The range of sρ studied is

comparable to the allowable range specified by existing design codes for RC

structures [1% to 8% in ACI-318 (2005) and 0.5% to 5% in GB-50010 (2002)].

The depth ratio actually reflects the thickness of the concrete cover. The same

range of the depth ratio (0.7 to 0.9) has been used by Hellesland (2005) for similar

analysis of RC columns and this range is also similar to that given in the ACI

design handbook [ACI 340R-97 (1997)] which provides design charts for circular

RC sections possessing a depth ratio ranging from 0.6 to 0.9. The strength

enhancement ratio goes up to 2, which is believed to cover all the cases of

practical applications. The three values chosen for the strain ratio represent

respectively the typical rupture strains of high modulus CFRP, CFRP and GFRP

jackets when they are used to confine a circular column. However, it should be

noted that the design value of the rupture strain of a given type of FRP composite

is much smaller than the actual material rupture strain of the same type of FRP

composite (i.e. the rupture strain obtained from tensile coupon tests). The reasons

143

for the much smaller design value are: 1) when an FRP jacket is used to confine a

concrete cylinder, its average hoop rupture strain is much smaller than the

material rupture strain due to the non-uniform deformation of concrete and other

factors (see Chapter 2); 2) in limit state design, the characteristic value of the

strain is divided by a partial safety factor to arrive at the design value to ensure a

desired level of safety reserve; and 3) to account for long-term degradation, an

additional reduction factor also needs to be applied to the FRP materials. As a

result, when the values of the FRP rupture strain used in the parametric study are

interpreted as the design values, the maximum value studied (1.5%) actually

corresponds to a material rupture strain of over 3% for a GFRP material.

Therefore, the range covered by the three strain ratios is believed to cover most

commercially available FRP materials.

The axial load capacity of all cases was calculated using both the section analysis

based on Eq. 5.1 and the proposed design equations (Eq. 5.10). The numerical

results from both approaches for all cases are compared in Fig. 5.11, in which the

axial load capacity is normalized by . It can be seen that the majority of cases

fall within the bounds.

uoN

5%±

Fig. 5.12 presents the interaction curves for three selected cases. The column in

Fig. 5.12 is confined with either a high modulus CFRP, a CFRP or a GFRP jacket.

All the information needed to produce these curves is given in Fig. 5.12. It can be

seen that the proposed design equations give results of excellent accuracy except

at high axial load levels. It should be noted that the interaction curves produced by

the design equations terminate at a high axial load level when the neutral axis

starts to fall outside the cross section. This is because the stress block factors are

based on the assumption that the neutral axis stays within the cross section, which

means that the use of Eq. 5.10 leads to errors when the neutral axis falls outside

the cross section. Nevertheless, this limitation of the design equations is

insignificant as such cases are not normally encountered in design due to the need

to include at least the minimum eccentricity for all columns.

144

5.6 CONCLUSIONS

This chapter has been concerned with the development of design equations for

short FRP-confined circular RC columns. Section analysis of such columns

employing Lam and Teng’s design-oriented stress-strain model was first

discussed. Design equations based on a simplified section analysis were next

presented. In this simplified analysis, the contribution of the confined concrete to

the load capacity of the section is approximated by transforming the stress profile

of concrete into an equivalent stress block and the contribution of the longitudinal

steel reinforcing bars to the load capacity of the section is approximated by

smearing the bars into an equivalent steel ring. The proposed design equations are

simple in form and provide an accurate approximation of the results from the

accurate section analysis.

145

5.7 REFERENCES

ACI 340R-97 (1997). ACI Design Handbook: Design of Structural Reinforced Concrete Elements in Accordance with the Strength Design Method of ACI 318-95, ACI Committee 340, American Concrete Institute.



ACI-318 (2005). Building Code Requirements for Structural Concrete and Commentary, ACI Committee 318, American Concrete Institute, Farmington Hills, Michigan, USA..

Binici, B. (2008). “Design of FRPs in circular bridge column retrofits for ductility enhancement”, Engineering Structures, 30(3), 766-776.



CNR-DT 200 (2004), Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Existing Structures, Advisory Committee on Technical Recommendations For Construction, National Research Council, Rome, Italy.



Fitzwilliam, J. and Bisby, L.A. (2006). “Slenderness effects on circular FRP-wrapped reinforced concrete columns”, Proceedings, 3rd International Conference on FRP Composites in Civil Engineering, December 13-15, Miami, Florida, USA, 499-502.


Gould, N.C. (1999). A Mechanistic Model and Design Procedure for Composite-confined Concrete Columns, Ph.D. thesis, Washington University, St. Louis.

146

Gould, N.C. and Harmon, T.G. (2002). “Confined concrete columns subjected to axial load, cyclic shear, and cyclic flexure – part II: experimental program”, ACI Structural Journal, 99(1), 42-50.


Harmon, T.G. (2008). Private communication.



Kong, F.K and Evans, R.H. (1987). Reinforced and Prestressed Concrete, Third Edition. Chapman & Hall, London, UK.

MacGregor, J.G., Breen, J.E. and Pfrang E.O. (1970). “Design of slender concrete columns”, ACI Journal, 67(1), 6-28.

Mirmiran, A., Naguib, W. and Shahawy, M. (2000). “Principle and analysis of concrete filled composite tubes”, Journal of Advanced Materials, 32(4), 16-23.


Park, R. and Paulay, T. (1975) Reinforced Concrete Structures, John Wiley & Sons, N.Y., USA.



Sheikh, S.A. and Yau, G. (2002). “Seismic behavior of concrete columns confined with steel and fiber-reinforced polymers”, ACI Structural Journal, 99(1), 72-80.

Tao, Z., Teng, J.G., Han, L.H. and Lam, L. (2004). “Experimental behaviour of FRP-confined slender RC columns under eccentric loading”, Proceedings, Second International Conference on Advanced Polymer Composites for Structural Applications in Construction, University of Surrey, Guildford, UK, 203-212.

147

Teng, J.G., Chen, J.F., Smith, S.T. and Lam, L. (2002) FRP-Strengthened RC Structures, John Wiley and Sons, Inc., UK.

Yuan, W. and Mirmiran, A. (2001). “Buckling analysis of concrete-filled FRP tubes”, International Journal of Structural Stability and Dynamics, 1(3):367-383.

Yuan, X.F., Xia, S.H., Lam, L. and Smith, S.T. (2008). “Analysis and behaviour of FRP-confined short concrete columns subjected to eccentric loading”, Journal of Zhejiang University Science A, 9(1), 38-49.

148

Table 5.1 Summary of Sheikh and Yau’s (2002) tests

Specimen D (mm)

H (mm)

d (mm)

frpE (GPa)

t(mm)

,h rupε (%)

yf(Mpa)

'cof

(Mpa) N

(kN) ST-2NT 356 1470 271 21 2.5 2.00 490 40.4 2570 ST-3NT 356 1470 271 75.5 1.0 1.3 490 40.4 2570 ST-4NT 356 1470 271 151 0.5 1.3 490 44.8 1380 ST-5NT 356 1470 271 21 1.25 2.00 490 40.8 1290

Table 5.2 Summary of Gould and Harmon’s (2002) tests

Specimen D (mm)

H (mm)

frpE (GPa)

t(mm)

yf(Mpa)

N(kN)

0600-1-133 180 600 68.95 0.45 604 133 0600-1-578 180 600 68.95 0.45 604 578 0600-3-578 180 600 68.95 1.35 604 578 1200-1-133 180 600 68.95 0.45 604 133 1200-1-578 180 600 68.95 0.45 604 578 1200-3-578 180 600 68.95 1.35 604 578 1200-5-133 180 600 68.95 2.25 604 133 1200-5-578 180 600 68.95 2.25 604 578

Table 5.3 Values of parameters used in the parametric study

Parameter Values e D 0.05,0.1,0.15,0.2,0.25,0.3,0.4,0.6,0.8

sρ 1%, 2%, 3%, 4%, 5% d D 0.7, 0.8, 0.9 ' '

cc cof f 1.25, 1.5, 1.75, 2

,h rup coε ε 1, 3.75, 7.5

149

xn

cu

R

cc'f

d

si

bc

sidsi

D

c

c

Fig. 5.1 Strains and stresses over an FRP-confined circular column section

Dd

Fig. 5.2 Cross section of the reference RC column

150

0 0.5 1 1.5 20

0.5

1

1.5

2

Normalized Moment Capacity Mu/Muo

Nor

mal

ized

Loa

d C

apac

ity N

u/N

uo

Unconfined3-ply CFRP6-ply CFRP

Fig. 5.3 Axial load-bending moment interaction curves for the reference column

0 0.01 0.02 0.03 0.04 0.050

100

200

300

400

500

600

700

800

Curvature φ (1/m)

Mom

ent M

(kN ⋅

m)

D = 600mmd = 500mmf′co = 20.1MPafy = 335MPaρs = 2.08%Efrp = 230GPa

t = 0.495mmεh,rup = 0.0075

1500 kN4500 kN7500 kN

Fig. 5.4 Bending moment-curvature curves for the reference column

151

-100 -50 0 50 100-300

-200

-100

0

100

200

300

Curvature φ (1/km)

Mom

ent

M (k

N ⋅m

)

ST-2NT

(a) Specimen ST-2NT

-100 -50 0 50 100-300

-200

-100

0

100

200

300

Curvature φ (1/km)

Mom

ent

M (k

N ⋅m

)

ST-3NT

(b) Specimen ST-3NT

152

-150 -100 -50 0 50 100 150-300

-200

-100

0

100

200

300

Curvature φ (1/km)

Mom

ent

M (k

N ⋅m

)

ST-4NT

(c) Specimen ST-4NT

-200 -150 -100 -50 0 50 100 150 200-300

-200

-100

0

100

200

300

Curvature φ (1/km)

Mom

ent

M (k

N ⋅m

)

ST-5NT

(d) Specimen ST-5NT

Fig. 5.5 Comparison with Sheikh and Yau’s (2002) tests

153

-400 -300 -200 -100 0 100 200 300 400-60

-40

-20

0

20

40

60

Curvature φ (1/km)

Mom

ent

M (k

N ⋅m

)

Test (0600-1-133)Test (1200-1-133)Analysis (f′co

=41.1 MPa)

Analysis (f′co=55.2 MPa)

(a) Specimens 0600-1-133 and 1200-1-133

-800 -600 -400 -200 0 200 400 600 800-80

-60

-40

-20

0

20

40

60

80

Curvature φ (1/km)

Mom

ent

M (k

N ⋅m

)


=41.1 MPa)


(b) Specimens 0600-1-578 and 1200-1-578

154

-600 -400 -200 0 200 400 600-100

-80

-60

-40

-20

0

20

40

60

80

100

Curvature φ (1/km)

Mom

ent

M (k

N ⋅m

)


=41.1 MPa)


(c) Specimens 0600-3-578 and 1200-3-578

-1500 -1000 -500 0 500 1000 1500-60

-40

-20

0

20

40

60

Curvature φ (1/km)

Mom

ent

M (k

N ⋅m

)

Test (1200-5-133)Analysis (f′co

=41.1 MPa)


(d) Specimen 1200-5-133

155

-800 -600 -400 -200 0 200 400 600 800-80

-60

-40

-20

0

20

40

60

80

Curvature φ (1/km)

Mom

ent

M (k

N ⋅m

)

Test (1200-5-578)Analysis (f′co

=41.1 MPa)


(e) Specimen 1200-5-578

Fig. 5.6 Comparison with Gould and Harmon’s (2002) tests

156

1 1.2 1.4 1.6 1.8 20.8

0.85

0.9

0.95

1

Strength Enhancement Ratio f′cc / f′co

Mea

n S

tress

Fac

tor α

1

xn = 0.1R

xn = 1R

xn = 1.5R

xn = 2R

(a) Mean stress factor

1 1.2 1.4 1.6 1.8 20.82

0.84

0.86

0.88

0.9

0.92

Strength Enhancement Ratio f′cc / f′co

Blo

ck D

epth

Fac

tor

β 1

xn = 0.1R

xn = 1R

xn = 1.5R

xn = 2R

(b) Block depth factor

Fig. 5.7 Stress block factors for CFRP-confined concrete in circular sections

157

1 1.2 1.4 1.6 1.8 20.75

0.8

0.85

0.9

0.95

1

1.05

←α1 = 1.17-0.2f′cc/f′co

Strength Enhancement Ratio f′cc/f′co

Mea

n S

tress

Fac

tor α

1

xn = 0.1R

xn = 1R

xn = 1.5R

xn = 2R

Fig. 5.8 Mean stress factor for FRP-confined concrete in circular sections

for 1β =0.9

1xnxn

cu

0

R

Rs

yf

yf cc

'

1

2

2

y

y

'f f 'cc

1

Fig. 5.9 Schematic of the simplified section analysis approach

158

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θ

θ 1+k

c, θ

2+k t

θ1+kc

θ2+kt

θc=1.25θ-0.125

θt=1.125-1.5θ

(a) Approximation of 1 ckθ + and 2 tkθ + using cθ and tθ

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

θ

sin(

πθ1)

+mc,

sin

( πθ 2

)+m

t

sin(πθ1)+mc

sin(πθ2)+mt

sin(πθc)

sin(πθt)

(b) Approximation of 1sin( ) cmπθ + and 2sin( ) tmπθ + using sin( )cπθ and sin( )tπθ

Fig. 5.10 Simplifications for strength contributions from steel reinforcement

159

0 0.5 1 1.50

0.5

1

1.5

Normalized Axial Load CapacityNu/Nuo - Section Analysis

Nor

mal

ized

Axi

al L

oad

Cap

acity

Nu/N

uo -

Des

ign

Equ

atio

ns

5%

-5%

Fig. 5.11 Performance of the proposed design equations

160

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4


Nor

mal

ized

Loa

d C

apac

ity N

u/N

uo

D=600mmd=500mmEfrp=700GPaεh,rup=0.002ρs=1%f′cc/f′co=1.35

Section AnalysisDesign Equations

(a) High modulus CFRP jacket

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4


Nor

mal

ized

Loa

d C

apac

ity N

u/N

uo

D=600mmd=500mmEfrp=230GPaεh,rup=0.0075ρs=2%f′cc/f′co=1.5


(b) CFRP jacket

161

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Nor

mal

ized

Loa

d C

apac

ity N

u/N

uo

D=600mmd=450mmEfrp=80GPaεh,rup=0.015ρs=1.5%f′cc/f′co=1.75


(c) GFRP jacket

Fig. 5.12 Comparisons between accurate and approximate analyses

162

CHAPTER 6

ANALYSIS OF ELASTIC COLUMNS

6.1 INTROCUCTION

Before developing a design procedure for slender FRP-confined RC columns, it is

desirable to understand the rationale behind the current design approach for

slender RC columns. It is well-known that this approach comprises a first-order

analysis of the frame to determine the first-order moments plus a second-order

analysis of an individual column isolated from the frame with idealized end

restraints representing the adjacent restraining members in the frame to determine

the second-order moments. The second-order moments are approximated by an

amplification of the first-order moments in order for the design to be related to the

section strength. The isolated column with end restraints (referred to as the

restrained column hereafter) is then replaced by an equivalent hinged column

through appropriate treatment of the end restraints of the restrained column.

This approach originated from the analysis of elastic columns. Therefore, the

rationale on which this approach is based is best explained through an analysis of

elastic columns. As a preparation for the analysis of slender FRP-confined RC

columns, this chapter examines all the components forming the current design

approach for slender RC columns through an analysis of non-sway elastic

columns with elastic end restraints. The exact solution for restrained columns is

derived first, followed by a careful investigation into the behavior of such

columns. In addition, new design equations for elastic columns are proposed

based on thorough interpretations of the current design approach.

163

6.2 THE PROLBLEM OF COLUMN DESIGN

As can be seen in later sections of this chapter, the design of a restrained column

can be finally transformed into the design of an equivalent hinged column

subjected to uniform first-order moments. It is advisable to use this equivalent

column to address the problem of column design. The first-order moment is

always enlarged by the interaction between the axial load and the deflection of the

column. For such a column, the maximum moment always occurs at the

mid-height of the column. If the first-order moment and the column properties are

known, the designer needs to determine how much axial load the column can

resist. The section axial load-bending moment diagram offers the answer to this

question. When the combination of the axial load and moment at the critical

section exceeds the interaction curve, it indicates that the column will fail by the

exhaustion of the material strength. It is clear that the key is to determine the

maximum moment of this column and to compare the combination of the axial

load and the maximum moment to the section axial load-bending moment

interaction curve. For elastic columns, the interaction curve features a straight line,

as shown in Fig. 6.1. This straight line can be mathematically described by Eq. 6.1

max 1uo uo

MNN M

+ = (6.1)

where is the maximum axial load the section can take in the absence of

bending moments while

uoN

uoM is the maximum bending moment the section can

take in the absence of the axial load.

The maximum moment of a column can be found using the numerical integration

method developed by Newmark (1943). For elastic columns, an exact expression

for the maximum moment exists and can be derived by solving the governing

differential equations, as presented in the following section.

6.3 EXACT SOLUTIONS

164

6.3.1 General

In this section, the exact solution for restrained columns is derived first and

hinged columns are then treated as a special case of restrained columns with zero

end restraints.

6.3.2 Exact Solution for Restrained Columns

Consider a restrained column as shown in Fig. 6.2a. The length of the column is l

nd the end restraints are represented by two elastic springs with rotational

stiffnesses of

a

1R and 2R respectively. The column is subjected to two external

end moments 1eM and 2eM plus an axial load . For an elastic column, the

final status of the column is the same regardless of whether the loads have been

applied in a proportional manner or not. For ease of discussion in later sections, it

is assumed throughout this chapter that the axial load is applied after the end

moments. The deflections due to the end moments and those due to the axial load

are solved independently and the final deflections of the column are taken as the

sum of the deflections from the two sources.

N

6.3.2.1 Deflection caused by the first-order moments

In Fig. 6.2b, an initially straight column is bent into a deflected shape as the

end moments

v

1M and 2M are applied. A pair of shear forces arises if the

end moments are not equal. It should be noted that the external end moments

V

1eM

and 2eM are shared by the column and the end restraints; 1M and 2M are the

moments transmitted to the ends of the column while the rest is resisted by the

end restraints. 1M and 2M can be considered as the moments at the ends of a

column obtained from a conventional first-order frame analysis and they are used

as input in the design of this column when it is isolated from the frame. In this

chapter, 2M is always assigned a non-negative value; 1M can have a negative

value but its absolute value is always smaller than or equal to that of 2M . The

latter is referred to as the larger first-order end moment or simply the larger end

165

moment for brevity hereafter and the ratio of 1M to 2M is referred to as the

first-order end moment ratio in this chapter.

For the case depicted by Fig. 6.2b, the moment at any given height of the column

can be written as

'' 2 12

M MM EIv M xl−

= − = − (6.2)

where E is the elastic modulus, I is the second moment of area and is any

given height along the column and

x

0x = represents the end where 2M is

applied.

From Eq 6.2 and the boundary conditions that no deflection occurs at either end of

the column, the deflection shape of the column can be easily found to be

(3 22 1 21 2

1 26 2 6

M M M lv x x MEI l

−⎡ ⎤= − + +⎢ ⎥⎣ ⎦)M x (6.3)

6.3.2.2 Deflection caused by the axial load

In Fig. 6.2c, an axial load is applied on the same column where N 1M and

2M are already present. An additional deflection is then introduced by the

application of this axial load. It should be noted that once the axial load is applied,

the end restraints begin to bear additional end moments triggered by the end

rotations as a result of the additional deflection. The end moments are thus no

longer

w

1M and 2M , but are replaced by the end moments defined by the

following equations where the moments from the end restraints are subtracted

from the original end moments

'

2 2 2 (0)M M R w= − (6.4a)

'1 1 1 (l )M M R w= − (6.4b)

166

The pair of shear forces is thus changed to

2 1M MVl−

= (6.5)

As illustrated in Fig. 6.2d, the moment at any given height of the column can then

be written as

( ) ( ''2 )M M V x N v w EI v w= − + + = − + (6.6)

After some rearrangement, Eq. 6.7 can be obtained

( )'' 3 22 1 21 2

' '2 (0) 1 ( )'

2 (0)

26 2 6

l

M M MN lEIw Nw x x M M xEI l

R w R wR w x

l

−⎡ ⎤+ = − − + +⎢ ⎥⎣ ⎦⎡ ⎤−

+ −⎢ ⎥⎢ ⎥⎣ ⎦

(6.7)

If Eq. 6.7 is differentiated four times, it becomes

6 2 4 0w k w+ = (6.8a)

NkEI

= (6.8b)

The general solution to Eq. 6.8 is

3 2

0 0 0 0 0sin cosw a kx b kx c x d x e x f= + + + + 0+ (6.9)

By differentiating Eq. 6.9 twice and comparing it with Eq. 6.7, it can be easily

found that

20 6

1M McEIl−

= − (6.10)

167

and

20 2

MdEI

= (6.11)

From the boundary condition

(0) 0w = (6.12)

it can be found that

0 0f b= − (6.13)

The other three coefficients can be found using the following three boundary

conditions

( ) 0lw = (6.14)

' ''2 (0) (0) 0R w EIw− = (6.15)

' ''1 ( ) ( ) 0l lR w EIw+ = (6.16)

Eqs 6.14 to 6.16 can be expanded and finally written in the following matrix form

0

2 2

1 1

2 21 2

2

21 21 1

sin cos 1

cos sin sin cos

26

2

kl kl l aR k N R b

R k kl N kl R k kl N kl R e

M M k lNMM MM R k l

N

−⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥− − −⎣ ⎦

+⎡ ⎤−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+− −⎢ ⎥⎣ ⎦

0

1 0

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(6.17)

The expressions for , and are lengthy, but they can be arranged in a

neat and unified form as

0a 0b 0e

168

01 02 030

01 02 03

cos sincos sin

a a kl a klad d kl d kl

+ +=

+ + (6.18)

01 02 030

01 02 03

cos sincos sin

b b kl b klbd d kl d kl

+ +=

+ + (6.19)

01 02 030

01 02 03

cos sincos sin

e e kl e kled d kl d kl

+ +=

+ + (6.20)

where

( ) ( )21 2 1 1 2 11 2 1 1 2

01

26 2

kl M M R kl M M R RNlM M R M Rak k N

+ ++= + + + 2 (6.21a)

( ) ( )21 2 2 1 2 12 2 1 1 2

02

26 2

kl M M R kl M M R RNlM M R M Rak k N

+ ++= − − − − 2 (6.21b)

( )2 21 2 1

03 2 1

26

k l M M R Ra lM R

N+

= − − 2 (6.21c)

( )2 21 2 1

01 1 2

26

k l M M R Rb lM R

N+

= − − 2 (6.22a)

02 03b a= (6.22b)

03 02b a= − (6.22c)

( ) (2

01 1 2 1 2 2 1 1 22k le M M R R M R MN

= − + + − )R (6.23a)

( ) ( ) ( )(2 2 2

02 1 2 1 2 2 1 1 2 1 2 1 222 6k l k le M M R R M R M R M M RN

= + − − + + + )R (6.23b)

( ) ( ) ( )2 2

03 1 2 1 2 1 1 2 1 222 6

N kl kl ke M M M M R M M N Rk N

⎛ ⎞= − − − + − + −⎜

⎝ ⎠R ⎟

R

(6.23c)

01 1 22d R= (6.24a)

( )02 1 2 1 22d R R Nl R R= − − + (6.24b)

( ) 21 2

03 1 2

N R R N ld

k+ +

= klR R− (6.24c)

6.3.2.3 Final deflections

The final deflected shape of the column can be taken as the sum of and v w

169

( )1 20 0 0

2sin cos

6M M l

0f v w a kx b kx e x fEI

+⎡ ⎤= + = + + + +⎢

⎣ ⎦⎥ (6.25)

The moment distribution along the column can then be described by

( )'' 20 0sin cosM EIf EIk a kx b kx= − = + (6.26)

The maximum moment occurs at a location x which is defined by

(30 0cos sin 0dM EIk a k x b k x

dx= − ) = (6.27)

It can be derived from Eq. 6.27 that

0

0

tan ak xb

= (6.28)

By substituting Eq. 6.28 into Eq. 6.26 , the maximum moment can be found as

( )2 2max 0 0 0tan sin cos 2M EIk b k x k x k x N a b= + = + (6.29a)

It should be noted that the solution of Eq. 6.27 may result in a value of x smaller

than zero or larger than , which implies that the maximum moment defined by

Eq. 6.29a may occur outside the column height. A careful examination of the issue

revealed that this special case occurs when and only when , . For

this case, the maximum moment within the column height is either the moment at

or the moment at . For the former case, the maximum moment can be

found from Eq. 6.27 to be

l

0 0a < 0 0b >

0x = x l=

max 0M Nb= (6.29b)

while for the latter case, the maximum moment is

170

max 0 0sin cosM N a kl b kl= + (6.29c)

The absolute value is taken in Eq. 6.29c because the latter case only occurs when

the first-order end moment ratio approaches -1 and the restraint at the end where

the larger first-order end moment acts is much stiffer than that at the opposite end.

The maximum moment has a negative value according to the current sign

convention in such a case.

For all the other cases, the maximum moment is defined by Eq. 6.29a and it

always occurs somewhere between the two ends.

6.3.3 Exact Solution for Hinged Columns

The exact solution for hinged columns can be found in various textbooks (e.g.

Galambos 1968; Chen and Atsuta 1976). Here hinged columns are treated as a

special case of restrained columns with 1 2 0R R= = . With

substituted into the derivation presented in the preceding sub-section, the final

deflection of a hinged column can be written as

1 2 0R R= =

1 1

2 2 2

cos 1sin cos 1

sin

M MklM M Mf kx kxN kl l

⎛ ⎞− −⎜ ⎟⎜ ⎟= + +⎜ ⎟⎜ ⎟⎝ ⎠

x − (6.30)

The moment at any given height of the column is

1

'' 22

cossin cos

sin

M klMM EIf M kx kx

kl

⎛ ⎞−⎜ ⎟⎜ ⎟= − = +⎜ ⎟⎜ ⎟⎝ ⎠

(6.31)

The maximum moment occurs at a height of x which is defined by

171

1

22

coscos sin 0

sin

M klMdM M k k x k x

dx kl

⎛ ⎞−⎜ ⎟⎜ ⎟= −⎜ ⎟⎜ ⎟⎝ ⎠

= (6.32)

By rearranging Eq. 6.32, the following equation can be obtained

1

2

costan

sin

M klMk x

kl

−= (6.33)

By substituting Eq. 6.33 into Eq. 6.31 and after some rearrangement, the

maximum moment can be written as

2

1 1

2 2max 2

1 2 cos

sin

cr

cr

M M NM M N

M MN

N

π

π

⎛ ⎞⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝=

⎛ ⎞⎜ ⎟⎝ ⎠

⎠ (6.34a)

where is the Euler load of the column and crN2

2

EIl

π= for a hinged column.

Similar to the situation examined in the previous section for restrained columns,

the maximum moment defined by Eq. 6.34a is only applicable to cases where the

maximum moment does occur within the column height. However, in the absence

of end restraints, x l> is not possible. It can be found from Eq. 6.33 that 0x ≥

is ensured when 1

2

coscr

M NM N

π⎛ ⎞

≥ ⎜⎜⎝ ⎠

⎟⎟ . By contrast, when 1

2

coscr

M NM N

π⎛ ⎞

< ⎜ ⎟⎜ ⎟⎝ ⎠

,

the maximum moment defined by Eq. 6.34a occurs outside the column ( 0x < ).

For such cases, the maximum moment within the column height occurs at ,

thus

0x =

max 2M M= (6.34b)

172

6.4 BEHAVIOR OF RESTRAINED COLUMNS

The behavior of hinged columns is easy to understand; the maximum moment of a

hinged column is either enlarged or remains unaffected as a result of the

interaction between the axial load and the deflection. To be specific, the maximum

moment of a hinged column subjected to uniform first-order moments is always

enlarged by the second-order moments, but the maximum moment of a hinged

column subjected to oblique first-order moments may remain unaffected by the

second-order moments. By contrast, the behavior of restrained columns is more

complicated because the end restraints produce additional moments which do not

exist in hinged columns. It is thus necessary to understand how the end restraints

affect the behavior of restrained columns before investigating the design of such

columns.

A good way to investigate the behavior of a column is to monitor the process of

how its moment distribution changes as the axial load increases. This process is

illustrated in Fig. 6.3, where a series of moment diagrams corresponding to

different loading stages is shown. These moment diagrams were produced using

the exact solution presented in the previous section. In each sub-figure, the

first-order moment diagram is indicated by the shaded area (labeled 0). The area

defined by a dashed line is the moment diagram for a given level of axial loading.

The dashed line denoted by 2 is for the final loading level when the axial load

capacity is reached and the other dashed line dentoed by 1 is for an intermediate

loading level.

It can be seen from Fig. 6.3 that irrespective of the first-order end moment ratio,

the larger end moment 2M decreases as the axial load increases; the direction of

2M may even be reversed if its magnitude is not large enough and the end

restraints are strong enough. The rules of the development of 1M is however

dependent on the first-order end moment ratio, which is discussed in detail below.

173

When 1

2

1 0MM

≥ ≥ 1, M changes in the same manner as 2M . It continuously

decreases as the axial load increases and it may also change direction. When

1

2

0 MM

> ≥ −0.9 , the magnitude of 1M continuously increases as the axial load

increases. It should be noted that the lower limit for 1

2

MM

depends on the

magnitude of the end restraints but it is always close to -1. When 1

2

0.9 1MM

− > ≥ − ,

the magnitude of 1M initially decreases as the axial load increases but then

increases as the axial load further increases.

An important observation is that unlike the hinged column, the maximum moment

of a restrained column can reduce as the axial load increases, as indicated by the

curve denoted by 1 in Fig. 6.3b. This implies that the second-order effect can

enhance the axial load capacity of a restrained column in certain cases. This

phenomenon arises from the existence of the end restraints and is explained in

detail below. The second-order moments can be decomposed into two parts. The

first part is the moments caused by the interaction between the axial load and the

deflection, which can be mathematically described as the product of the axial load

and the total deflection of the column. The second part is the moments produced

by the end rotations. This part of moments can be mathematically described as the

product of the end rotation associated with the additional deflection caused by the

axial load and the stiffness of the end restraints. The increments of end rotations

are always opposite in direction to the increments of end moments induced by

these rotations. As a result, these two parts of moments may combine to yield such

an overall effect that the maximum second-order moment always occurs away

from the column end where 2M is applied and it is always of the same sign as

2M while the second-order moments in the neighborhood of the same column

end are of the opposite sign to 2M . In other words, the maximum moment of a

restrained column subjected to uniform first-order moments is always enlarged by

the second-order moments, but the maximum moment of a restrained column

subjected to oblique first-order moments may however be reduced by the

174

second-order moments.

6.5 DESIGN OF RESTRATNED COLUMNS

6.5.1 General

The current design approach for restrained columns in the existing design codes

(e.g. ENV-1992-1-1 1992; BS-8110 1997; ACI-318 2005) is based on the

philosophy that a restrained column can be replaced by an equivalent hinged

column subjected to symmetrical bending. The latter is referred to as the standard

hinged column in this chapter. The pursuit of the standard hinged column can be

done in two steps. The fist step is to find a hinged column with two end moments

which are equal to the first-order end moments of the restrained column. The key

issue in this step is to determine the length of the hinged column which ensures

that it has a maximum moment equal to that of the restrained column and thus the

same axial load capacity. It is obvious that the hinged column obtained in this step

is subjected to oblique first-order moments unless the column is symmetrically

bent. The second step is to transform the hinged column obtained in the previous

step into the standard hinged column. The latter is of an equal length to the former

and the two also need to have equal maximum moments and axial load capacities.

In other words, an equivalent uniform first-order moment distribution is sought in

step two to replace the oblique first-order moment distribution of the hinged

column obtained in step one. The key issue in this step is to find the relationship

between the two moment distributions. Step two is examined first since it is more

straightforward than step one and it can be independently considered as the

solution to the design of hinged columns.

6.5.2 From a Hinged column to a Standard Hinged Column

Theoretically speaking, the design of hinged columns can be dealt with in an exact

manner provided the maximum moment along the column is found from Eq. 6.34.

However, much simpler design equations are available and they can be found in

many textbooks (e.g. Galambos 1968; Chen and Atsuta 1976). Fig. 6.4 reveals the

philosophy behind the design of hinged columns as explained in the preceding

175

paragraph. In Fig. 6.4, the column on the left hand side is subjected to oblique

first-order moments while the one on the right hand side is subjected to uniform

first-order moments eqM with such a magnitude that the maximum moments in

the two columns are equal. The equivalent moment eqM can be expressed as a

portion of 2M , as 2eq mM C M= , where is the equivalent moment factor and

it can be determined from the derivation given below.

mC

Substituting 1 2 eqM M M= = into Eq. 6.34a, the maximum moment of the

standard hinged column is given by

max 1

cos2

eqeq

cr

MM M

NN

ϕπ

= =⎛ ⎞⎜ ⎟⎝ ⎠

(6.35)

where 1ϕ is the moment amplification factor of the standard hinged column. By

comparing Eq. 6.35 and Eq. 6.34, it can be found that

2

1 1

2 2

1 2 cos

2sin2

crm

cr

M M NM M N

CN

N

π

π

⎛ ⎞⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠=

⎛ ⎞⎜ ⎟⎝ ⎠

, if 1

2

coscr

M NM N

π⎛ ⎞

≥ ⎜⎜⎝ ⎠

⎟⎟ (6.36a)

cos2m

cr

NCN

π⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠, if 1

2

coscr

M NM

π⎛ ⎞

< ⎜⎜⎝ ⎠N ⎟⎟ (6.36b)

Now it is very clear that the maximum moment of any hinged column subjected to

two end moments and an axial load can be written as

max 1 2 2mM C M Mϕ ϕ= = (6.37)

where reflects the effect of moment gradient, mC 1ϕ reflects the moment

amplification of the corresponding standard hinged column and ϕ is the moment

176

amplification factor of the column considered. Conventional design methods are

based on the simplification of and mC 1ϕ .

Fig. 6.5 shows that 1ϕ can be closely approximated by the following expression

proposed by Lai and MacGregor (1983)

1

1 0.25

1cr

cr

NN

NN

ϕ+

=−

(6.38)

In existing design codes, the term 0.25cr

NN

is omitted and 1ϕ is approximated

using the following expression

11

1cr

NN

ϕ =−

(6.39)

The exact expression and the two approximate expressions are compared in Fig.

6.5, which clearly shows that Eq. 6.39 is un-conservative for the full range of the

axial load level while Eq. 6.38 provides a much closer prediction. It is believed

that the term 0.25cr

NN

is omitted in existing design codes as the error so induced

becomes much less significant in the much more complicated design procedure

for columns involving material nonlinearity. The un-conservativeness introduced

by the omission of 0.25cr

NN

can be counterbalanced by other appropriate

considerations in dealing with factors such as material nonlinearity and end

restraints. The design equations in existing design codes for RC structures do not

aim to provide exact answers, but reasonably accurate yet conservative answers in

a reasonably simple manner. However, when material nonlinearity is not involved,

the design problem becomes much more straightforward and design equations

with higher accuracy may be sought. As a result, for elastic columns, the term

177

0.25cr

NN

may be retained.

Eq. 6.36 indicates that is a function of two ratios, that is mC 1

2

MM

and cr

NN

.

The effects of these two ratios on are illustrated in Fig. 6.6. In Fig. 6.6, the

family of thin solid curves is for

mC

cr

NN

increasing from 0 to 1 at an interval of 0.1.

Each of these curves shows how the values of vary with mC 1

2

MM

. Each of them

consists of two portions separated by a small circle which corresponds to

1

2

coscr

M NM N

π⎛ ⎞

= ⎜⎜⎝ ⎠

⎟⎟ . The first portion is a horizontal line and is mathematically

described by Eq. 6.36b while the second portion features an ascending shape and

is mathematically described by Eq. 6.36a. All these curves converge to the point

where when . The shaded area represents cases where the

maximum moment of the column does occur within the column height and is

always larger than

1mC = crN N=

2M while the remaining area represents cases where 2M is

the maximum moment of the column. This indicates that only the values located

in the shaded area need to be reasonably approximated because when the

maximum moment calculated using an approximate expression for is smaller

than

mC

2M , the maximum moment is always taken to be 2M . Many approximate

expressions have been proposed for and a brief discussion and comparison

of these expressions can be found in Austin (1961). Among all these expressions,

the following equation proposed by Austin (1961) is widely accepted because of

its simplicity and accuracy. The values of predicted using Eq. 6.40 is shown

by the dashed line in Fig. 6.6.

mC

mC

1

2

0.6 0.4 0.4mMCM

= + ≥ (6.40)

Eq. 6.40 neglects the effect of the axial load and the value of has a lower mC

178

limit of 0.4 because Eq. 6.40 was originally derived for elastic lateral-torsinal

buckling and this lower limit might be removed for in-plane bending (Chen and

Lui 1991). Lai and MacGregor (1983) however suggested that this lower limit be

retained considering the uncertainties of the behavior of columns with a 1

2

MM

ratio of -0.5 to -1 (MacGregor et al. 1970). It can be seen from Fig. 6.6 that the

dashed line passes through the shaded area, which indicates Eq. 6.40 is

un-conservative for cases falling into the shaded area above the dashed line. This

un-conservativeness is however small as can be seen in Fig. 6.7, where the exact

values of max

2

MM

and the approximate results predicted from Eqs 6.38 and 6.40

are compared. The approximate predictions shown in Fig. 6.7 are generally

conservative but close to the exact results except for 1

2

1MM

= − . When 1

2

1MM

= − ,

the exact curve is a horizontal line since the exact maximum moment is always

2M irrespective of the axial load level. The approximate curve for this case

however overlaps with that for 1

2

0.5MM

= − because 0.4mC = when 1

2

0.5MM

< − .

If this limitation is removed, Eqs 6.38 and 6.40 provide less conservative

predictions for cases where 1

2

0.5MM

< − .

In summary, the proposed approximate equation for the maximum moment of a

hinged column is

max 2 2

1 0.25

1cr

m

cr

NNM C MN

N

+=

−M≥ (6.41)

where Eq. 6.40 is employed to determine . mC

6.5.3 From a Restrained Column to a Hinged Column

179

The final uncertainty in the design of restrained columns is how to determine the

length of the standard hinged column. To sustain the deflected shape of a column,

the deflections, curvatures and moments of this column must maintain certain

relationships as required by the governing differential equation given in the

previous section; these relationships form the basis of the well-known numerical

integration method (e.g. Newmark 1943; Craston 1972). An important implication

of these relationships is that a restrained column can be replaced by an equivalent

hinged column with a length of over which the moment distribution is

exactly the same as that of the restrained column over the same length. It should

be noted that the choice of the equivalent hinged column is not unique since the

equivalent hinged column can be any portion of the restrained column provided

that the end moments of the equivalent hinged column are the same as the

corresponding moments in the restrained column. In practical design, the

first-order end moments

eql

1M and 2M of the real restrained column are always

known so that it is advisable to choose such an equivalent hinged column that it is

subjected to end moments equal to 1M and 2M . The determination of this

equivalent hinged column is illustrated in Fig. 6.8. The length of the equivalent

hinged column is equal to the distance between the two intersection points, where

the moment of the restrained column is equal to the first-order moment of the

adjacent column end. It should be noted that does not exist when the

maximum moment of the restrained column is smaller than the larger first-order

end moment. In such a case, the maximum moment can be taken to be the larger

first-order end moment for a conservative design. Once is known, the

maximum moment in the equivalent hinged column can be readily found from Eq.

6.41 with the use of

eql

eql

2

2creq

EINl

π= .

Now it is clear that the key to a satisfactory design is how to determine . The

exact expression for can be derived from the exact analysis presented in the

previous section, but it is obviously too complicated for design use. The current

design approach adopted in existing design codes (e.g. ACI 2005, EC2) employs

the effective length of the real restrained column as an approximation of .

eql

eql

effl eql

180

The use of the effective length to estimate the equivalent length was originally

proposed by Bijlaard et al. (1953) for restrained elastic columns with equal end

restraints and equal first-order end moments. Bijlaard et al. (1954) extended this

approach to more general cases so that restrained elastic columns with unequal

end restraints and unequal first-order end moments can also be dealt with. The

effective length of a column is the length between the points of inflection of the

buckled shape of a column subjected to pure axial compression. The effective

length can be exactly determined by solving Eq. 6.17 with

imposed. For a non-trivial solution, the determinant of the first matrix on the left

hand side must vanish, which leads to the following eqaution

1 2 0M M= =

( )2

1 2 1 2

2 tan22 1 4 1

tanG G G G

π ππ µ

π πµµ µ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎛ ⎞⎜ ⎟ ⎜ ⎟ 0µ+ + − + − =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(6.42)

where µ is the effective length factor and effll

= . The effective length factor as a

closed-form solution cannot be achieved; it has to be found using a numerical

approach. The current design approach in existing design codes (e.g. ACI-318

2005, ENV-1992-1-1 1992) employs an alternative graphical solution which was

originally proposed by Julian and Lawrence (1959). The value of µ can be

readily found from the alignment charts developed by Julian and Lawrence (1959)

provided that the magnitudes of the end restraints are known. A detailed

derivation of the effective length can be found in and Kavanagh (1960). In Eq.

6.42, and are the column-to-beam stiffness ratios at the two ends. It

should be noted that the conditions

1G 2G

11

2EIRG l

= and 22

2EIRG l

= are used when

deriving Eq. 6.42. These two conditions result from the assumptions made in

deriving Eq. 6.42. Eq. 6.42 is developed for an idealized frame of infinite storeys

and bays with all the columns in the top storey axially loaded. All the columns in

this frame reach their buckling load simultaneously as a result of the idealization.

In real frames, the situation is different but a discussion of the differences between

the idealized frame and a real frame is beyond the scope of this chapter. Interested

181

readers may refer to Johnston (1976) in which attempts were made to take some

of the particular conditions into account. For most columns in real frames, Eq.

6.42 does provide a useful approximation.

From the definitions of the effective length and the equivalent length, it is obvious

that the equivalent length is equal to the effective length when a column subjected

to pure axial compression reaches its buckling load. In the presence of the

first-order end moments, these two lengths become different. Lai and MacGregor

(1983) performed a study to investigate how the equivalent length varies with a

number of parameters. A similar but more comprehensive parametric study was

carried out using the exact solution given in the previous section to see if the use

of the effective length as an approximation of the equivalent length is adequate for

design use.

Following Lai and MacGregor (1983), a careful study revealed that the equivalent

length varies with the following factors: the first-order end moment ratio, the axial

load and the magnitude of the end restraints. Four parameters, namely, 1

2

MM

, 1

2

RR

,

µ and cr

NN

, which completely reflect the effects of the above factors were

varied in the parametric study. From the definitions of the above four parameters,

it is obvious that 1

2

MM

varies from -1 to 1, 1

2

RR

can have any value between zero

and infinity, µ ranges from 0.5 for fixed-fixed columns to 1 for hinged-hinged

columns, and the value of the normalized axial load cr

NN

is in the range of 0 to 1

(here is the buckling load of a restrained column subjected to pure axial

compression and

crN

2

2 2

EIl

πµ

= ).

Figs 6.9 to 6.11 shows the results of the parametric study. In these figures, the

vertical axis is the ratio of the equivalent length to the effective length while the

horizontal axis is the normalized axial load. Figs 6.9 to 6.11 shows respectively

182

the effect of one of the first three parameters as the last parameter varies. It should

be pointed out that the equivalent length is known for the two extreme cases of

loading: when , the equivalent length is equal to the effective length;

when , the equivalent length is equal to the real length of the restrained

column, but it can change dramatically when the axial load approaches zero.

Considering the fact that

crN N=

0N =

0N = is just an idealized situation and for a neater

illustration, the axial load was assigned a small value to represent when

producing Figs 6.9 to 6.11.

0N =

Fig. 6.9 shows the effect of 1

2

MM

. In all the three sub-figures, the effective length

factor is fixed at 0.75, but the end restraint ratio varies. Fig. 6.9a is for a

symmetrically restrained case. It can be seen that when the column is subjected to

uniform first-order moments, the equivalent length is always larger than the

effective length, which means that the effective length approach is

un-conservative. When the column is subjected to oblique first-order moments,

the equivalent length can either be larger or smaller than the effective length and it

should be noted that there is a range of axial load in which the equivalent length

does not exist. This range corresponds to cases where max 2M M< , as discussed in

previous sections and the size of this range depends on the first-order end moment

ratio as well as the magnitude of the end restraints. Fig. 6.9b is for and

. The three curves in the middle converge to a certain value when the axial

load approaches zero. The converged value indicates that when no axial load is

applied, the equivalent length is exactly the same as the real height of the

restrained column. The curve for

1R = ∞

2 0R =

1 2M M= should also converge to this value,

but it remains at a lower position for the reason stated earlier. The curve for

1 2M M= − suddenly drops at a certain axial load level. This is because in this

case the magnitude of 1M is initially reduced by the axial load, as illustrated in

Fig. 6.3c. The equivalent length thus does not exist when the axial load is in the

lower range. Fig. 6.9c is for 1 0R = and 2R = ∞ . The curves for 1 2M M= and

1 2M M= − are the same as those in Fig. 6.9b, because the interchange of the end

restraints does not affect the behavior of the column when the first-order end

183

moments are either symmetrically or anti-symmetrically distributed. The other

three curves however feature a reverse trend compared with those in Fig. 6.9b.

Fig. 6.10 illustrates the effect of the effective length factor. In both sub-figures,

the two end restraints are equally stiff. Fig. 6.10a is for 1 2M M= . It clearly

shows that at a given axial load level, the effective length approach becomes more

un-conservative for stiffer end restraints. A similar observation can be made about

Fig. 6.10b, where only one end of the column is subjected to a first-order end

moment.

Fig. 6.11 shows the effect of the end restraint ratio. Fig. 6.11a is for 1 2M M= .

The lowest curve in Fig. 6.11a is for the symmetrically restrained case and the

other two curves overlap for the reason given earlier. Fig. 6.11b is for cases where

only one end of the column is subjected to a first-order end moment.

Based on the results of the parametric study, it can be concluded that the

equivalent length in most cases is different from the effective length and the

effective length approach can be very un-conservative for certain cases. A possible

solution to overcome the shortcomings of the effective length approach is to

develop a more accurate alternative approximation for the equivalent length.

However, the present parametric study indicates that the equivalent length varies

with a number of equally important parameters. A close approximation must thus

take all these parameters into account, which makes the task difficult, particularly

if a relatively simple approach is sought. An alternative option is to develop a

conservative but simple approximation of the equivalent length but with some

sacrifice in the accuracy. Such a conservative approach can be achieved by

developing an approximation for the effective length that includes an appropriate

degree of overestimation. One such approximation has been proposed by Lai and

MacGregor (1983). Lai and MacGregor (1983) suggested the effective length

factor be taken as the smaller of

( )1 20.7 0.05 1G Gµ = + + ≤ (6.43a)

0.85 0.05 1sGµ = + ≤ (6.43b)

184

where sG is the smaller of and . Eq. 6.43 was originally proposed by

Cranston (1972) based on the numerical results of an elastic analysis similar to the

one presented in this chapter; his original intention was however to provide a

rough estimation of the effective length of restrained RC columns for a safe

design. Eq. 6.43 has been adopted in various design codes (e.g. ACI 2005, BS

1997, EC2). The final approach suggested by Lai and MacGregor (1983) for the

design of a single restrained column was to use Eq. 6.43 for the determination of

in Eq. 6.41, but Lai and MacGregor (1983) did not verify their proposed

approach. A study of the approach proposed by Lai and MacGregor (1983)

showed that their approach could be unnecessarily conservative in some cases,

particularly when the end restraints are strong. A further problem is that in some

other cases, their approach is extremely un-conservative, which must be avoided

in design. Fig. 6.12 gives some evidence of the shortcomings of their approach

and the causes are discussed in detail in the next section.

1G 2G

crN

6.5.4 Proposed Equations

Although the values of the equivalent length can vary with a number of

parameters, they generally fall in the neighborhood of the value of the effective

length. The direct use of the effective length has the advantages of simplicity and

familiarity to engineers. The associated un-conservativeness may be

counterbalanced through other considerations in the design approach. The

following equation is therefore proposed for the design of an isolated column:

( )max 2 2

1 1.25

1cr

m

cr

NNM C N

N

µ+ −=

−M M≥ (6.44)

where is defined by Eq. 6.40 and mC2

2 2crEINl

πµ

= with µ being found from

Eq. 6.42. Eq. 6.44 differs from Eq. 6.41 in that the coefficient 0.25 in the

numerator of Eq. 6.41 is replaced by 1.25 u− in Eq. 6.44. Eq. 6.44 reduces to Eq.

185

6.41 for hinged columns. It can be seen from later comparisons that this

adjustment can effectively eliminates the un-conservativeness associated with the

direct use of the effective length for the majority of cases.

In Figs 6.12 to 6.15, the exact values of the maximum moment are compared with

the values predicted using Eq. 6.44. Figs 6.12 to 6.15 are for four different values

of the first-order end moment ratio ( 1

2

1,0.5,0, 0.5MM

= − ). A comparison for

1

2

1MM

= − is not shown because the predictions of Eq. 6.44 are very conservative

for this case as is assigned a constant value of 0.4. Only selected numerical

results are shown here due to space limitation, and the attention is focused on the

upper bound and the lower bound of the

mC

max

2

MM

ratio. The upper bound and the

lower bound curves are the envelope curves of all curves corresponding to

different magnitudes of the end restraints, but they can be very closely

approximated by the curves for the following four extreme cases: 1) ,

2) , 3) and 4)

1 2 0R R= =

1 2R R= = ∞ 1 20,R R= = ∞ 01 2,R R= ∞ = . For clarity,

comparisons for the first two cases and those for the next two cases are shown in

two separate sub-figures.

Fig. 6.12 is for cases where 1 2M M= . The exact results are shown as the solid

lines while the approximate results from Eq. 6.44 are shown as the dashed lines.

The results from Lai and MacGregor’s (1983) approach are also shown in Fig.

6.12, but not in Figs 6.13 to 6.15 because the performance of their approach is

similar in all cases. Fig. 6.12a shows that when both ends of a column are hinged,

the predictions of Eq. 6.44 and those of Lai and MacGregor’s (1983) approach are

exactly the same and they are both very close to the exact results. When both ends

of a column are fixed, Eq. 6.44 becomes slightly un-conservative while Lai and

MacGregor’s (1983) approach is very un-conservative at lower axial loads and

becomes very conservative at high axial loads. The un-conservativeness of Lai

and MacGregor’s (1983) approach for lower axial loads arises from the

overestimation of the effective length by Eq. 6.43. The minimum value of the

186

effective length factor predicted by Eq. 6.43 is 0.7 while the exact value is 0.5. As

a result of the overestimation of the effective length, the value of in Eq. 6.41

is underestimated by Eq. 6.43 so that the term

crN

1cr

NN

− in Eq. 6.41 can become

negative. When this happens, the maximum moment according to Eq. 6.41 is 2M ,

as indicated by the horizontal line in Fig. 6.12a. Fig. 6.12b shows the comparison

for hinged-fixed columns. Since the results shown in Fig. 6.12b are for columns

with uniform first-order moments, the two solid curves representing the exact

solution overlap with each other. Both approximate approaches neglect the effect

of the end restraint ratio so that both of them predict only a single curve for cases

(3) and (4) in Figs 6.12b to 6.15b. For the cases shown in Fig. 6.12b, Eq. 6.44

provides accurate predictions while the performance of Lai and MacGregor’s

(1983) approach is similar to that revealed by Fig. 6.12a.

Similar observations can be made about Figs 6.13 to 6.15. For case (2), Eq. 6.44 is

always conservative while for case (1), Eq. 6.44 is generally conservative except

when the predicted maximum moment is similar to 2M . Similarly, For case (3)

Eq. 6.44 is always conservative while for case (4), Eq. 6.44 is generally

conservative except when the predicted maximum moment is similar to 2M .For

other cases not shown here, Eq. 6.44 provides consistently conservative

predictions for the majority of them and slightly un-conservative predictions for

the rest of them. In summary, Eq. 6.44 possesses a simple form which makes it

easy to use in design and provides reasonably accurate predictions which are

acceptable for design use.

6.6 CONCLUSIONS

This chapter has been concerned with the analysis of elastic columns with elastic

end restraints as a preparation for the analysis of slender FRP-confined RC

columns presented in Chapters 7 to 9. The exact solution for the lateral deflection

of such columns induced by combined bending and axial compression was

derived first. The rationale behind the current column design approach was next

explained. Finally, approximate design equations for elastic columns were

187

presented, which represent an improvement to existing approaches.

It was shown that the current design approach transforms a restrained column into

an equivalent hinged column with equal end eccentricities. The length of the

equivalent hinged column, referred to as the equivalent length in this chapter, is

generally different from the effective length of the restrained column. However, it

was shown that the equivalent length does not exist in certain cases. In order to

develop a unified design approach, the effective length can be used as a

reasonable estimate of the equivalent length and the errors so introduced may be

counterbalanced by other appropriate considerations in dealing with factors such

as the end restraints and the equivalent uniform moment factor. That is why this

design approach is commonly known as the “effective length approach”. It was

also shown that even for elastic columns the effective length approach is

approximate in nature because it was developed from idealized frame conditions.

The effective length approach is popularly used in the design of inelastic columns,

such as RC columns. With the presence of material nonlinearity, the problem of

column design becomes even more complicated and simple closed-form design

equations with great accuracy are thus not possible. Existing design equations for

RC columns are generally based on the rationale explained in this chapter and

they aim to provide reasonable but conservative predictions through appropriate

treatment of the key elements of the effective length approach: 1) in determining

the effective length of a restrained RC column, some codes (e.g. ACI-318 2005)

use simple charts developed from Eq. 6.42 to relate the effective length to the

column-to-beam stiffness ratio, while others (e.g. BS-8110 1997) allow the

effective length to be roughly estimated according to the end conditions of a

column; 2) in determining the equivalent uniform moment factor, Eq. 6.40 is

generally adopted because of its simplicity. As a result, it must be borne in mind

that although the current design approach has a rational basis, it is approximate in

nature.

188

6.7 REFERENCES

ACI 318-05 (2005). Building Code Requirements for Structural Concrete and Commentary, ACI Committee 318, American Concrete Institute.

Austin, W.J. (1961).”Strength and design of metal beam-columns”, Journal of the Structrual Division, ASCE, 87(ST4), 1-32.

Bijlaard, P.P., Fisher, G.P. and Winter, G. (1953). “Strength of columns elastically restrained and eccentrically loaded”, Proceedings, ASCE, Separate No. 292.

Bijlaard, P.P. (1954). “Buckling of columns with equal and unequal end eccentricites and equal and unequal end restraints”, Procceedings, Second Natl. Congress of Applied Mechanics, Ann Arbor, Mich.



Chen, W.F. and Atsuta, T. (1976). Theroy of Beam-Columns, New York, McGraw-Hill.

Chen, W.F. and Liu, E.M. (1991). Stability Design of Steel Frames, Boca Raton, CRC Press.


Galambos, T.V. (1968). Structrual Members and Frames, Englewood Cliffs, N.J., Prentice-Hall.

Johnston, B.G. (1976). Guide to Stability Design Criteria for Metal Structures, New York, Wiley.

Julian, O.G. and Lawrence, L.S. (1959). “Notes on J and L Nomographs for Determination of Effective Lengths, unpublished report.

Kavanagh, T.C. (1960). “Effective length of framed columns”, Proceedings, ASCE, 86(ST2), 81-101.

Lai, S.M.A and MacGregor, J. G. (1983). “Geometric nonlinearities in nonsway frames”, Journal of Structural Engineering, ASCE, 109(12), 2770-2785.


Newmark, N.M. (1943). “Numerical procedure for coputing deflection s, moments, and buckling loads”, Transactions, ASCE, 108, 1161-1234.

189

Winter, G. (1954). “Compression members in trusses and frames”, The Philosophy of Column Design, Proceedings, 4th Technical Session, Column Research Council, Lehigh University.

190

Moment M

Axi

al L

oad

N

Nuo

Muo

Fig. 6.1 Axial load-bending moment diagram of an elastic column

l

N

N

Me1

Me2

M1

M2

V

V

v

M1

v w

N

N

M2

V

V

N

N

v wf

M

f

x

M2

V

V

(a) Restrained column

(b) Column subjected to end moments only

(c) Application of axial load

(d) Moment at a given height of the column

Fig. 6.2 Forces and corresponding deflections of a restrained column

191

0 1 2

0

21

1

2

0

(a) 1 21 0M M≥ ≥ (b) 1 20 0M M> ≥ − .9 (c) 1 20.9 1M M− > ≥ −

Fig. 6.3 Moment distributions of restrained columns with different first-order end

moment ratios

M1

M2

Mmax

Mmax

Meq

Meq

Fig. 6.4 Transformation of a hinged column into a standard hinged column

192

0 0.2 0.4 0.6 0.8 11

2

3

4

5

Normalized Axial Load N/Ncr

Nor

mal

ized

Max

imum

Mom

ent

Mm

ax/M

2

Exact SolutionEquation 6.38Equation 6.39

Fig. 6.5 Moment amplification factor of the standard hinged column

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

End Moment Ratio M1/M2

Equ

ival

ent U

nifo

rm M

omen

t Fac

tor

Cm

N/Ncr = 0 to 1

Eq. 6.40

Fig. 6.6 Equivalent moment factor

193

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5


Nor

mal

ized

Max

imum

Mom

ent

Mm

ax/M

2

Exact SolutionEquation 6.41

M1/M2 = 1

M1/M2 = 0.5

M1/M2 = 0

M1/M2 = -1

M1/M2 = -0.5

Fig. 6.7 Performance of the proposed equation for the design of hinged columns

M2

M1M1

M2

leq

Fig. 6.8 Transformation of a restrained column into an equivalent hinged column

194

0 0.2 0.4 0.6 0.8 10.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

N/Ncr

µ equ

/µ

µ=0.75

R1/R2=1

M1/M2 = 1M1/M2 = 0.5M1/M2 = 0M1/M2 = -0.5M1/M2 = -1

(a)

0 0.2 0.4 0.6 0.8 11

1.05

1.1

1.15

1.2

1.25

1.3

1.35

N/Ncr

µ equ

/µ

µ=0.75R1/R2=Inf

M1/M2 = 1M1/M2 = 0.5M1/M2 = 0M1/M2 = -0.5M1/M2 = -1

(b)

195

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

1.15

N/Ncr

µ equ

/µ

µ=0.75R1/R2=0

M1/M2 = 1M1/M2 = 0.5M1/M2 = 0M1/M2 = -0.5M1/M2 = -1

(c)

Fig. 6.9 Effect of first-order end moment ratio on equivalent length

196

0 0.2 0.4 0.6 0.8 11

1.05

1.1

1.15

N/Ncr

µ equ

/µ

M1/M2=1

R1/R2=1

µ = 0.6µ = 0.75µ = 0.9

(a)

0 0.2 0.4 0.6 0.8 10.95

1

1.05

1.1

N/Ncr

µ equ

/µ

M1/M2=0

R1/R2=1

µ = 0.6µ = 0.75µ = 0.9

(b)

Fig. 6.10 Effect of effective length factor on equivalent length

197

0 0.2 0.4 0.6 0.8 11

1.01

1.02

1.03

1.04

1.05

1.06

1.07

N/Ncr

µ equ

/µ

M1/M2=1µ=0.75

R1/R2 = 0R1/R2 = 1R1/R2 = Inf

(a)

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

N/Ncr

µ equ

/µ

M1/M2=0

µ=0.75

R1/R2 = 0R1/R2 = 1R1/R2 = Inf

(b)

Fig. 6.11 Effect of end restraint ratio on equivalent length

198

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5


Nor

mal

ized

Max

imum

Mom

ent

Mm

ax/M

2

M1/M2=1

both

end

s hi

nged

both ends f

ixed

Exact SolutionEquation 6.44Lai and MacGregor (1983)

(a) Cases 1 and 2

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5


Nor

mal

ized

Max

imum

Mom

ent

Mm

ax/M

2

M1/M2=1

one e

nd fix

ed, o

ne e

nd h

inged

Exact SolutionEquation 6.44Lai and MacGregor (1983)

(b) Cases 3 and 4

Fig. 6.12 Comparisons of exact results with approximate results for 1 2 1M M =

199

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5


Nor

mal

ized

Max

imum

Mom

ent

Mm

ax/M

2

M1/M2=0.5M1/M2=0.5

both

end

s hi

nged

both ends fixed


(a) Cases 1 and 2

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5


Nor

mal

ized

Max

imum

Mom

ent

Mm

ax/M

2

M1/M2=0.5

end

1 hi

nged

, en

d 2

fixed

end 2 hinged

end 1 fixed,


(b) Cases 3 and 4

Fig. 6.13 Comparisons of exact results with approximate results for 1 2 0.5M M =

200

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5


Nor

mal

ized

Max

imum

Mom

ent

Mm

ax/M

2

M1/M2=0

both

end

s hi

nged

both

end

s fix

ed


(a) Cases 1 and 2

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5


Nor

mal

ized

Max

imum

Mom

ent

Mm

ax/M

2

M1/M2=0

end

2 fix

ed

end

1 hi

nged

,

end 1 fixed, end 2 hinged


(b) Cases 3 and 4

Fig. 6.14 Comparisons of exact results with approximate results for 1 2 0M M =

201

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5


Nor

mal

ized

Max

imum

Mom

ent

Mm

ax/M

2

M1/M2=-0.5

both

end

s hi

nged

both

end

sfix

ed


(a) Cases 1 and 2

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5


Nor

mal

ized

Max

imum

Mom

ent

Mm

ax/M

2

M1/M2=-0.5

end

2 fix

ed

end

1 hi

nged

,

end 1 fixed, end

2 hin

ged


(b) Cases 3 and 4

Fig. 6.15 Comparisons of exact results with approximate results for

1 2 0.5M M =−

202

CHAPTER 7

THEORETICAL MODEL FOR

SLENDER FRP-CONFINED RC COLUMNS

7.1 INTRODUCTION

Chapter 5 has presented a set of design equations for FRP-confined RC columns

that are short enough for the slenderness effect to be ignored. Chapter 6 has shown

how the slenderness of an elastic column can undermine its strength through the

interaction between the axial load it sustains and its lateral deflection. The lateral

deflection of an elastic column can be exactly evaluated, but this is not possible

for an FRP-confined RC column due to the nonlinearity of its constituent

materials. As a result, a rational theoretical model based on certain numerical

procedures is required for the analysis of slender FRP-confined RC columns.

Although various theoretical models have been proposed to analyze slender

columns made of different materials (Chen and Atsuta 1976), to the best

knowledge of the author, none of them has been used to model slender FRP-

confined RC columns. A recent study by Yuan and Mirmiran (2001) on the

modeling of slender concrete-filled FRP tubes appears to be the only existing

analytical study addressing the slenderness effect in concrete columns receiving

FRP confinement. In their model, the total eccentricity of a column (i.e. load

eccentricity plus lateral deflection) is assumed to be a portion of a cosine wave

depending on the load eccentricity at either column end. However, as their

columns did not have internal reinforcing steel bars and the FRP tubes had

significant axial stiffness, their model and the results obtained from their study are

not directly applicable to slender FRP-confined RC columns.

203

Against this background, this chapter presents a theoretical model for slender

FRP-confined RC columns. The proposed model is more sophisticated than the

model of Yuan and Mirmiran (2001) in that no assumption is made about the

lateral deflected shape of the column; it is instead sought in an iterative manner.

The theoretical model is described in detail first and its verification using

experimental results of both RC columns and FRP-confined RC columns is next

discussed. It should be noted that only a very limited number of tests on slender

FRP-confined RC columns have been reported in the open literature and all those

available to the author are used herein to verify the proposed model.

7.2 THEORETICAL MODEL

7.2.1 General

The method of analysis used herein is the well-known numerical integration

method. This method was originally proposed by Newmark (1943) and has been

widely adopted in the analysis of RC columns (e.g. Pfrang and Siess 1961;

Cranston 1972), steel columns (e.g. Shen and Lu 1983) and composite columns

(e.g. Choo et al. 2006; Tikka and Mirza 2006). However, to the best knowledge of

the author, this method has not been used in the analysis of FRP-confined RC

columns. In the present study, this method is applied to pin-ended FRP-confined

RC columns subjected to eccentric loads at both ends with the end eccentricities

being and , respectively (Fig. 7.1). It should be noted that is taken to be

the one with a larger absolute value and is always assigned a non-negative value.

This indicates that has a negative value when the column is bent in double

curvature. The column with a length l is equally divided into a desirable number

of segments with a length of

1e 2e 2e

1e

l∆ . The column section at each grid point is divided

into a desirable number of horizontal layers. The lateral displacement at each grid

point at a particular loading stage can be sought in an iterative manner by making

use of the axial load-moment-curvature ( N M φ− − ) relationship of the column

section and the numerical integration function of the column. The full-range axial

load-lateral deflection curve of the column (referred to as the load-deflection

204

curve for brevity hereafter) can then be traced in an incremental manner using

either a force-control or deflection-control technique. For the ascending branch of

the load-deflection curve, it is more convenient to use the force-control technique

(increasing the axial load by small steps). In the case of a stability failure, a

descending branch of the load-deflection curve exists and the deflection-control

technique (increasing the deflection of a particular grid point by small steps)

should be used to trace the descending branch. In summary, the load-moment-

curvature relationship and the numerical integration function are the two key

elements of the numerical integration method and they are discussed in detail in

the following sub-sections, where the procedure for generating the full-range load-

deflection curve is also described.

7.2.2 Construction of Axial Load-Moment-Curvature Curves

The procedures for the construction of the moment-curvature curve under a given

axial load have been presented in detail in Chapter 5, so they are not repeated

herein. The stress profile of confined concrete in compression is described using

the modified Lam and Teng Model (I) with the value of 1.75 for the first term on

the right hand side of Eq. 4.9 replaced by 1.65; the tensile strength of concrete is

ignored. The longitudinal steel reinforcement is assumed to have an elastic-

perfectly plastic stress-strain response and any confinement effect from the hoop

steel reinforcement is ignored. In the present analysis, every section is divided

into 50 horizontal layers and the solution is considered successful once the

difference between the resultant axial force and the applied axial load is within

. Each moment-curvature curve produced in the present analysis consists of

201 points, which ensures sufficient numerical accuracy.

610 N−

7.2.3 Numerical Integration for the Column Deflection

Since the lateral deflections of the column are very small compared to the length

of the column, the curvature can be taken to be the second order derivative of the

lateral deflection of the column. Using the central difference equation, the

relationship between the lateral deflections and the curvatures can be expressed as

205

( )( 1) ( ) ( 1)

( )2

2i i ii

f f f

lφ+ −− +

= −∆

(7.1)

where and ( )if ( )iφ are the lateral displacement and curvature at the grid point

respectively and . i is the index of the grid point and is the

number of segments that the column is equally divided into (Fig. 7.1). is

used in the present analysis. Eq. 7.1 can be rewritten as

thi

2,3, 1i m= − 1m−

31m =

( )2( 1) ( ) ( 1) ( )2i i i if f f lφ+ −= − − ∆ (7.2)

Eq. 7.2 is the numerical integration function used to find the lateral deflection of

the column. The implementation of this equation is described in the following

section.

7.2.4 Generation of the Ascending Branch of the Load-Deflection Curve

The axial load is increased by small increments to generate the ascending branch

of the load-deflection curve. For a given axial load , the first step is to construct

the corresponding moment-curvature curve using the procedure described above.

Under this axial load, the first order moment at each grid point can be easily

calculated

N

1,( ) ( )i iM N e= ⋅ (7.3)

where and are the first order moment and the initial eccentricity at the

grid point respectively. Note that the initial eccentricity follows a linear

distribution with and

1,( )iM ( )ie

thi

(1) 2e e= ( ) 1me e= . If the lateral deflection of the column is

known, the second order moment can be expressed as

2,( ) ( )i iM N f= ⋅ (7.4)

206

and the total moment can then be expressed as

( )( ) 1,( ) 2,( ) ( ) ( )i i i i iM M M N e f= + = ⋅ + (7.5)

Now assume a value for (2)f . The assumed value can be any reasonably small

value or simply zero. The moment at this grid point (2)M can then be evaluated

using Eq. 7.5 and the curvature at this grid point (2)φ can be retrieved from the

moment-curvature curve corresponding to the present axial load. With (2)M and

(2)φ known and note that (1) 0f = , the lateral displacement of the third grid point

(3)f can be evaluated using Eq. 7.2. It is evident that the lateral deflection of the

column can be traced from grid point to grid point by repeating the above

procedure. Once the calculations eventually reach the other end of the column, its

lateral displacement needs to be examined to see if it satisfies . If this

condition is satisfied, then the correct lateral deflection of the column is found.

Otherwise, the assumed value of

( ) 0mf =

(2)f needs to be adjusted until is

satisfied. It is suggested that the new value of

( ) 0mf =

(2)f be taken as the previous value

minus ( )

1mf

m −. In the present analysis, the solution is considered to be acceptable if

the calculated ( )mf has an absolute value smaller than 410− mm.

The above procedure determines the lateral deflection of the column under a

particular axial load. The ascending branch of the load-deflection curve can be

obtained by calculating the deflection of the column for a series of successively

increasing loads. In the present analysis, an initial load increment (i.e. load step

size) of is used, where is the axial load capacity of the column section

under consideration when subjected to concentric compression as given by

10.1 uN 1uN

'

1u cc c yN f A f= + sA (7.6)

207

where and cA sA are the total cross-sectional areas of concrete and longitudinal

steel bars, respectively. After several load increments, the applied load eventually

exceeds the maximum load that the column can sustain. This situation arises when

the moment at any grid point calculated by Eq. 7.5 exceeds the maximum moment

on the moment-curvature curve under the present axial load. When this occurs, the

calculation process needs to restart from the last load level and a smaller load

increment of is used for the subsequent load steps. After some steps, a

similar treatment needs to be adopted with the load step size reduced by a factor

of 10. The same process is repeated until the load step size is eventually reduced

to . The axial load capacity of the column is then taken to be the

maximum load for which a convergent solution of the lateral deflection can be

found. The corresponding value of

10.01 uN

6110 uN−

uN

(2)f is recorded as a reference displacement

value reff for use in the generation of the descending branch of the load-

deflection curve, as discussed in the following sub-section.

7.2.5 Generation of the Descending Branch of the Load-Deflection Curve

If the column is controlled by material failure, its load-deflection curve has no

descending branch. However, if the column is slender enough to trigger stability

failure, a descending branch of the load-deflection curve exists. In this case, the

displacement-control technique should be used to trace the descending branch.

The numerical procedure is similar to that described in the previous sub-section

with the only difference being that the aim is to find the correct axial load under a

prescribed value of (2)f .

A displacement increment of 0.1 reff is initially used (i.e. the prescribed value of

(2)f in the first step is 1.1 reff ). The initial assumed value of the corresponding

axial load can be taken as . The corresponding deflected configuration can

then be calculated using the numerical procedure described in the previous sub-

section. It should be noted that the calculated

uN

( )mf always has a negative value,

because the actual axial load must be smaller than . The assumed axial load is uN

208

thus successively reduced at steps of 0 until the calculated .01 uN ( )mf has a

positive value. The correct axial load can then be determined using the bisection

method by setting the last two assumed values of axial load to be the upper bound

and lower bound respectively in the bisection method. The final axial load derived

from the bisection method is the solution for the current step and it is used as the

initial value in the next step. After several increments of (2)f , the analysis

eventually fails to find a convergent solution. This situation arises when the

moment at any grid point calculated by Eq. 7.5 exceeds the maximum moment on

the moment-curvature curve under the present axial load. When this occurs, the

calculation needs to restart from the prescribed value of (2)f and with a smaller

displacement increment of 0.01 reff . The entire process is repeated and the

analysis stops when the increment is reduced to 610 reff− .

It should be noted that the accuracy of the present analysis is affected by a number

of factors (i.e. the number of segments the column is divided into, the number of

horizontal layers the cross section is divided into, the number of points the

moment-curvature curve consists of, and the tolerances adopted in the analysis). A

convergence study showed that all these factors have been very well looked after

in the present study (i.e. any refinement to these factors will not have any

significant effect on the numerical results). A computer program was developed to

implement the numerical procedure described above using Matlab 7.1.

7.3 VERIFICATION OF THE THEORETICAL MODEL

7.3.1 Comparison with Cranston’s Numerical Results

The validity of the present model is first verified using the numerical results of

Cranston’s (1972) theoretical model for RC columns, which was based on a

similar numerical integration procedure. Cranston’s (1972) model used the same

stress-strain relationships for both concrete and steel as those adopted by the

present study except that the concrete was assumed to have an ultimate axial strain

of 0.0035. This value was also used in the present model for this set of

209

comparisons. Table 7.1 lists the properties of the columns analyzed by Cranston

(1972). These columns were all pin-ended and were eccentrically loaded at only

one end. The cross-sectional shape was rectangular rather than circular and it is

illustrated in Fig. 7.2. The height and the width of the cross section are denoted by

and respectively and the eccentricity is in the height direction. The steel

reinforcement ratio is denoted by

h b

sρ and the distance between the steel

reinforcement is denoted by . 'h cuf is the characteristic cube strength of concrete

and yf is the characteristic yield strength of steel reinforcement. Cranston (1972)

normalized the load-deflection curves (Fig. 7.3) using appropriate reference

values. The lateral deflection at mid-height of the column midf was normalized by

the section height while the axial load was normalized by the design value of the

section axial load capacity under concentric compression , which was

determined from the design values of material strengths (the numbers bracketed in

Table 7.1; the partial safety factors for concrete and steel are 1.5 and 1.15

respectively). However, the characteristic material strengths were used in the

numerical analysis. The predictions of both models are shown in Fig. 7.3. It can

be clearly seen that these predictions are in excellent agreement for both the

material failure and stability failure cases.

uoN

7.3.2 Comparison with Experimental Results

Predictions from the present theoretical model are also compared with exisiting

experimental results of both RC columns and FRP-confined RC columns in this

sub-section. First, Kim and Yang’s (1995) tests on RC columns are used for

comparison. Details of Kim and Yang’s tests are listed in Table 7.2. These

specimens were square in shape and had a wide range of concrete strength. All

these specimens were bent in symmetrical single curvature ( ). and

are the test and theoretical axial load capacities of a column, respectively.

In Kim and Yang’s (1995) tests, two physically identical columns were prepared

for each configuration. Close agreement between the present predictions and the

test results can be seen in the last column of Table 7.2. In addition, the full-range

load-deflection curves were also reported by Kim and Yang (1995), and those of

1 2e e= ,u testN

,u theoN

210

the normal strength series are compared with predictions of the present model in

Fig. 7.4. No further comparisons for RC columns only are discussed herein

because the present method of analysis has long been well-accepted for RC

columns (e.g. Pfrang and Siess 1961; Cranston 1972).

Only a very limited number of experimental studies have been carried out on

FRP-confined circular RC columns. These studies include Tao et al. (2004), Hadi

(2006), Fitzwilliam and Bisby (2006) and Ranger and Bisby (2007). Hadi (2006)

tested five small-scale (150 mm in diameter) circular normal strength concrete

columns wrapped with CFRP and subjected to axial loading with the same end

eccentricity of 42.5 mm. One reference column which received no FRP wrapping

was also tested. Four of the five FRP-confined columns were not provided with

internal steel reinforcement so they failed by the cracking of concrete on the

tension face of these columns. Unfortunately, the hoop strains on the FRP jacket

were not reported which makes it difficult for these test results to be used to verify

the proposed theoretical model.

The two test series of Fitzwilliam and Bisby (2006) and Ranger and Bisby (2007)

were conducted by the same research group and the test configurations of the two

series are similar. Therefore, these two test series are discussed together, although

they had different test objectives. Fitzwilliam and Bisby (2006) varied the column

height but fixed the load eccentricity while Ranger and Bisby (2007) varied the

load eccentricity but fixed the column height. In Ranger and Bisby’s (2007) test

series, all the columns were 152mm in diameter and 600 mm in height, and were

connected to a steel system at both column ends to create the pinned end condition

and the load eccentricity. All the columns were reinforced with four 6.4 mm

diameter steel bars longitudinally and 6.4 mm diameter steel ties spaced at 100

mm transversely with a 25 mm concrete cover to the longitudinal reinforcement.

A total of six load eccentricities were considered: 0, 5, 10, 20, 30, or 40 mm. For

each load eccentricity, a column confined with a single ply of CFRP as well as an

unconfined reference column was tested. The FRP jacket included a 100 mm

overlapping zone with its centerline being at the same circumferential position as

the centerline of the less compressed face of the column. The concrete had a

cylinder strength of 33.2 MPa, the steel reinforcement had an yield strength of 710

211

MPa, and the FRP had an elastic modulus of 90 GPa (based on a nominal

thickness of 0.381 mm) and a rupture strain of 1.12% obtained from tensile

coupon tests. A summary of Ranger and Bisby’s (2007) tests is given in Table 7.3.

In Fitzwilliam and Bisby’s (2006) test series, the column height varied from 300

mm to 1200 mm at an interval of 300 mm and all the columns were tested with a

fixed load eccentricity of 20 mm. The steel reinforcement and FRP used in the

study were the same as those used in Ranger and Bisby (2007). The concrete

cylinder strength averaged from three batches of cylinder tests conducted during

the period of column tests was 35.5 MPa. The other test parameters were similar

to those of Ranger and Bisby’s (2007) tests. It should be noted that some columns

received longitudinal FRP wrapping before hoop FRP wrapping, but these tests

have been excluded from the present comparison. Fitzwilliam and Bisby’s (2006)

tests are summarized in Table 7.4. In both test series, the lateral deflection of the

column was monitored at three different vertical locations with one being located

at the mid-height of the column.

Ancillary tests on standard concrete cylinders under concentric compression were

also conducted in both test series. The concrete cylinders were made from the

same concrete and confined with the same type and amount of FRP as the

columns. The predictions of Lam and Teng’s stress-strain model are first checked

against these cylinder tests before the proposed theoretical model is verified using

the column tests. These cylinder tests are summarized in Table 7.5. The concrete

cylinders in Ranger and Bisby (2007) were only confined with a 1-ply CFRP

jacket while some of the cylinders in Fitzwilliam and Bisby (2006) were confined

with a 2-ply CFRP jacket as some of the columns tested by Fitzwilliam and Bisby

(2006) were also confined with a 2-ply CFRP jacket. The compressive strength of

unconfined concrete varied slightly in Fitzwilliam and Bisby’s (2006) tests

because the concrete cylinders were tested at different ages during column testing.

It should be noted that in Ranger and Bisby (2007), the FRP hoop rupture strain

found from their cylinder tests was reported to be 0.62%, a much smaller value

than that found from Fitzwilliam and Bisby’s (2006) cylinder tests (1.17%),

although the test configurations were almost the same in the two test series. It was

later confirmed by Bisby (2008) that this small value arose from an editorial error

212

and the correct value is 1.15%. The compressive strength 'ccf and the ultimate

axial strain cuε predicted by Lam and Teng’s stress-strain model [modified

Version (I)] are listed in the last two columns of Table 7.5. It can be seen that the

predicted values of the compressive strength are reasonably close to the

experimental values, however, the predicted values of the ultimate axial strain are

much larger than the experimental values, particular for the specimens confined

with a 1-ply FRP jacket. To minimize the errors that might arise from this

discrepancy in modeling the column behavior, the experimental values of 'ccf and

cuε were directly incorporated in Lam and Teng’s stress-strain model [modified

Version (I)] in predicting the behavior of test columns. MPa and ' 44.2ccf =

0.86%cuε = were used in modeling the columns tested by Ranger and Bisby

(2007). For the columns in Fitzwilliam and Bisby (2006), MPa and ' 40.7ccf =

0.788%cuε = were used for the 1-ply jacket while and ' 60.1ccf = 1.443%cuε =

were used for the 2-ply jacket. The predicted values of 'ccf and cuε were also used

in predicting the behavior of test columns as a reference.

The predicted axial load capacities of all columns are compared with the

experimental values in Tables 7.3 and 7.4. It can be noted that for all the

unconfined columns in Ranger and Bisby’s (2007) study, their axial load

capacities are overestimated by some 20% while this overestimation is much

smaller for Fitzwilliam and Bisby’s (2006) tests. The relatively large

overestimation observed for Ranger and Bisby’s (2007) tests may be due to: 1)

additional eccentricities due to geometric/material imperfections and inaccurate

alignment of load; and 2) possible spalling of the concrete cover in unconfined

columns during testing which reduces the effective cross-sectional area. In

particular, it can be shown that the theoretical results are sensitive to an additional

eccentricity, especially when the nominal load eccentricity is small. For example,

if a 10 mm additional eccentricity is assumed, then the predicted axial load

capacity of column U-0 becomes 525 kN, which is much closer to the

experimental value. However, in the comparisons for these two test series, no

additional eccentricity was used except for columns U-0 and C-0 (subjected to

nominally concentric compression) where a small eccentricity of 1 mm was used.

213

For a column subjected to concentric compression, a small load eccentricity (or

other forms of imperfection) needs to be introduced into the theoretical model as

otherwise no lateral deflections can be predicted by the theoretical model. For

FRP-confined columns, in most cases, their axial load capacity is underestimated

by about 5% to 15%. The only exceptions occurred in columns C-30 and C-40

which had relatively large load eccentricities. Their axial load capacities were

overestimated by nearly 20%. However, it is difficult to accept this overestimation

as clear evidence that the load eccentricity has a detrimental effect on confinement

effectiveness because: 1) no such trend can be found for the range of smaller load

eccentricities (0, 5, 10, 15, 20 mm); 2) the theoretical predictions are sensitive to

additional eccentricity whose exact value is unknown; and 3) scatter may exist in

the test results.

The theoretical and experimental full-range load-deflection curves of FRP-

confined columns tested by Ranger and Bisby (2007) and Fitzwilliam and Bisby

(2006) are compared in Fig. 7.6 and Fig. 7.7 respectively. Two theoretical curves

are shown for each column; they were produced using the experimental and

predicted 'ccf and cuε values respectively. All the theoretical curves terminate

when the extreme compression fiber reaches the ultimate axial strain of confined

concrete. It should be noted that column C-0 in Ranger and Bisby’s (2007) test

series and the pair of physically identical columns 300C-1-0A and 300C-1-0B in

Fitzwilliam and Bisby’s (2006) test series are excluded from the present

comparisons, because only very small lateral deflection of these columns were

recorded. This is due to the fact that the former had a relatively small height-to-

diameter ratio of four and was tested under nominally concentric compression

while the latter only had a length-to-diameter ratio of two, so the slenderness

effect in these columns was not significant enough to induce large lateral

deflections. It can be seen in Figs 7.6 and 7.7 that for the same column, the

theoretical curve produced using the 'ccf and cuε values predicted by Lam and

Teng’s stress-strain model [modified Versi (I)] terminates at a larger

deformation value, which is closer to test results, than the other theoretical curve.

This is because the prediction of the deformability capacity of these columns

largely depends on the value used for the ultimate axial strain of confined concrete,

on

214

which is overestimated by Lam and Teng’ stress-strain model as indicated by the

cylinder tests results (see Table 7.5). This observation also implies that

eccentricity might have an effect on the stress-strain behavior of FRP-confined

concrete and this possible effect needs to be fully clarified in the future. It can also

be seen in Figs 7.6 and 7.7 that there exist considerable discrepancies between the

theoretical and experimental stiffnesses of these columns. This may be due to

inaccurate displacement measurements at the initial loading stage, because the

lateral deflections might be too small to be precisely measured (it is worth noting

that even some negative lateral displacements at the mid-height of column C-5

were recorded). It should also be noted that the lateral displacement of column C-

10 was not accurate (see Fig. 7.5b), as confirmed by Bisby (2008). Fig. 7.5b

shows that column C-10 possesses a much larger deformation capacity than

column C-20. This contradicts engineering intuition, because column C-20 should

have a larger deformation capacity than column C-10, given the fact that the

former was loaded with a larger initial load eccentricity and both columns failed at

very similar hoop rupture strains of the FRP jacket (1.07% for column C-10 and

1.15% for column C-20).

Tests on slender FRP-confined circular RC columns performed by Tao et al.

(2004) have also been simulated using the present theoretical model. A total of 16

columns were tested and the properties of these columns are listed in Table 7.6.

All the columns were 150mm in diameter and reinforced with four 12 mm

longitudinal steel bars and 6 mm steel hoops spaced at 100 mm. The columns had

a 21 mm concrete cover to the longitudinal steel reinforcement. The C1 series had

a length-to-diameter ratio of 8.4 while this ratio for the C2 series was 20.4. Each

series included four different load eccentricities (0, 50, 100 and 150 mm) and for

each eccentricity, one unconfined column and one FRP-confined column were

tested. As the load eccentricities adopted were relatively large, all the columns

were cast with corbel ends and capped with a steel plate with V-shaped grooves to

achieve the required load eccentricities and a pinned end condition. A 150 mm

overlapping zone was adopted in forming the FRP jacket and the position of the

overlapping zone is similar to that adopted by Ranger and Bisby (2007). Apart

from the column tests, a series of ancillary cylinder compression tests were also

conducted to determine the material properties. The average cylinder compressive

215

strength of unconfined concrete was found from six standard concrete cylinders to

be 48.2 MPa. The longitudinal steel reinforcement had a yield strength of 388.7

MPa. The CFRP had an elastic modulus of 255 GPa based on a nominal thickness

of 0.17 mm per ply and a rupture strain of 1.67% based on tensile coupon tests. In

addition, three FRP-confined concrete cylinders were tested under concentric

compression. These cylinders were confined with the same type and amount (2-

plies of CFRP) of FRP as the columns. The stress-strain curves predicted using

Lam and Teng’s stress-strain model [modified Version (I)] are compared with the

experimental curves in Fig. 7.7. It should be noted that only two experimental

curves are shown in Fig. 7.7 because the third specimen experienced an

unexpected experimental error during testing. The hoop rupture strain averaged

from the two cylinders was 1.32% and this value was used when generating the

predicted stress-strain curve. It can be seen that the predicted and experimental

curves are in close agreement so Lam and Teng’s model was directly used in the

present theoretical model for the modeling of this series of column tests.

The predicted axial load capacities of all columns are listed in Table 7.6. Again, a

small eccentricity of 1 mm was used when analysing columns tested under

nominally concentric compression (i.e., columns C1-1U, C1-1R, C2-1U, and C2-

1R). It is surprising to note that for all the unconfined columns, the predicted axial

load capacity is considerably larger than the experimental value, particularly for

those columns subjected to nominally concentric loading. By contrast, the

predictions for FRP-confined columns are much more reasonable but the same

trend can still be observed. This might be due to the same reasons as given earlier:

additional eccentricities from geometric/material imperfections, inaccurate

alignment of load, and the possible spalling of the concrete cover in unconfined

columns. In an internal report (Yu et al. 2004), it was suggested that an additional

eccentricity of 15 mm be used for unconfined columns and an additional

eccentricity of 7.5 mm be used for FRP-confined columns when modeling Tao et

al.’s (2004) column tests. When this suggestion is adopted, the predicted values

(bracketed in Table 7.6) become much closer to the excremental values. With the

inclusion of the additional eccentricity in the theoretical model, the average

, ,u theo u testN N ratio for the six FRP-confined columns with a non-zero nominal

216

load eccentr

nother point worth noting is that the theoretical model only predicts a marginal

he theoretical and experimental full-range load-deflection curves of FRP-

curve of column C2-3R is shown in Fig. 7.8c using a dotted line. However, the

icity decreases from 1.11 to 1.01. However, the same ratio for

unconfined columns is still 1.19, indicating significant overestimation.

A

increase in the axial load capacity due to FRP wrapping, which is particularly true

for series C2. This observation indicates that the effectiveness of FRP

confinement decreases as columns become more slender. The same trend can also

be observed in the predictions for Fitzwilliam and Bisby’s (2006) tests. This is

because when columns become more slender, they are more susceptible to

stability failure. When a column is short enough for its load capacity to be

controlled by material failure, failure of the column is predicted when the extreme

compression fiber of the critical section reaches the ultimate axial strain of

confined concrete. By contrast, when the column is slender enough, stability

failure controls and the column reaches its axial load capacity when the extreme

compression fiber has not yet reached its ultimate axial strain.

T

confined columns are also compared (Fig. 7.8). An additional eccentricity of 7.5

mm was added to the nominal load eccentricity when producing the theoretical

curves shown in Fig. 7.8. For specimens C1-1R and C2-1R, two theoretical curves

are shown. The upper one is for an additional eccentricity of 7.5 mm while the

lower one is for an additional eccentricity of 15 mm. These two specimens were

tested under nominally concentric compression, so their behavior is more sensitive

to the additional eccentricity. It can be seen that the use of a 15 mm additional

eccentricity produced closer predictions, but the curves for a 7.5 mm additional

eccentricity are also reasonably close to the experimental curves. It can be seen in

Figs 7.8c and 7.8d that the deformation capacities of columns of series C2 are

significantly overestimated, this is because these columns were very slender and

no FRP rupture was observed even at very large lateral deflection when the tests

stopped. Taking column C2-3R as an example, the ultimate hoop strain recorded

at the compression face of the column was only 0.226%. If this value is used in

Lam and Teng’s stress-strain model, the theoretical deformation capacity of the

column becomes much closer to the experimental value. The new theoretical

217

new theoretical curve almost overlaps with the old theoretical curve so a small

triangle is included to mark the ending point of the new theoretical curve.

7.4 CONCLUSIONS

This chapter has been concerned with the development and verification of a

eoretical model for slender FRP-confined RC columns. The theoretical model

successfully used to model

slender steel columns, RC columns and steel-concrete composite columns.

2) curate in predicting the axial

load capacity of slender FRP-confined RC columns. However, the proposed

3) ith the external FRP strengthening of

RC columns for the enhancement of their axial load capacity. For this purpose,

th

incorporates Version (I) of the modified Lam and Teng model at the section

behavior level and finds the lateral deflection of a column through numerical

integration at the column behavior level. The theoretical model was verified

against a similar model for RC columns through comparison of numerical results

from both models. Besides, the theoretical model was also verified using

experimental results of both RC columns and FRP-confined RC columns, with the

latter being emphasized. The comparisons and discussions presented in this

chapter allow the following conclusions to be drawn:

1) The numerical integration method has long been

However, to the best knowledge of the author, the work presented in this

chapter is the first attempt to extend the numerical integration method to the

analysis of slender FRP-confined RC columns.

The proposed theoretical model is reasonably ac

model performs worse in predicting the deformation capacity of slender FRP-

confined RC columns. One possible reason is that the eccentricity has certain

effect on the ultimate axial strain of FRP-confined concrete, which is the key

to the prediction of deformation capacity of FRP-confined RC columns. More

research is needed to clarify this issue.

The present thesis is only concerned w

the proposed theoretical model is satisfactory as demonstrated by the

comparisons presented in this chapter. However, when RC columns are

218

strengthened with FRP wraps primarily for the enhancement of their

deformation capacity, such as in the case of seismic retrofit of RC columns,

more research is needed to clarify whether the present numerical scheme is

sufficiently accurate for such cases.

219

7.5 REFERENCES

Bisby, L.A. (2008). Private communication.

Chen, W.F. and Atsuta, T. (1976). Theory of Beam-Columns, McGraw-Hill, New York.

Choo, C.C., Harik, I.E. and Gesund, H. (2006). “Strength of rectangular concrete columns reinforced with fiber-reinforced polymer bars”, ACI Structural Journal, 103(3), 452-459.


Fitzwilliam, J. and Bisby, L.A. (2006). “Slenderness effects on circular FRP-wrapped reinforced concrete columns”, Proceedings, 3rd International conference on FRP Composites in Civil Engineering, December 13-15, Miami, Florida, USA, 499-502.


Kim, J.K. and Yang, J.K. (1995). “Buckling behaviour of slender high-strength concrete columns”, Engineering Structures, 17(1), 39-51.

Newmark, N.M. (1943). “Numerical rocedure for computing deflections, moments, and buckling loads”, ASCE Transactions, 108, 1161-1234.

Pfrang, E.O. and Siess, C.P. (1961). “Analytical study of the behavior of long restrained reinforced concrete columns subjected to eccentric loads”, Structural research series No. 214, University of Illinois, Urbana, Illinos.


Shen, Z.Y. and Lu, L.W. (1983). “Analysis of initially crooked, end restrained steel columns”, Journal of Constructional steel research, 3(1), 10-18.

Tao, Z., Teng, J.G., Han, L.H. and Lam, L. (2004). “Experimental behaviour of FRP-confined slender RC columns under eccentric loading”, Proceedings, 2nd International Conference on Advanced Polymer Composites for Structural Applications in Construction, University of Surrey, Guildford, UK, 203-212.

Tikka, T.M. and Mirza, S.A. (2006). “Nonlinear equation for flexural stiffness of slender composite columns in major axis bending”, Journal of Structural Engineering, ASCE, 132(3), 387-399.

220

Yuan, W., and Mirmiran, A. (2001). “Buckling analysis of concrete-filled FRP tubes”, International Journal of Structural Stability and Dynamics, 1(3), 367-383.

Yu, Q., Tao, Z., Gao, X., Yang, Y.F., Han, L.H and Zhuang, J.P. (2004). 大轴压比下 FRP 约束混凝土柱抗震性能研究 , Fuzhou University, China (in Chinese).

221

Table 7.1 Properties of columns in Fig. 7.3

Specimen cuf (MPa)

yf (MPa)

sρ (%)

'h h

1 2e e 2e h l h

Column 1 0.5 15 Column 2 0.5 25 Column 3 0.1 40 Column 4

31 (13.8)

414 (360) 6 0.7 0

0.5 40

222

Table 7.2 Summary of Kim and Yang’s (1995) tests

Specimen b

(mm)h

(mm)

'cof

(MPa) yf

(MPa)sρ

(%)l h 'h h 1 2e e 2e h ,u testN

(kN) ,u theoN

(kN) ,

,

u theo

u test

NN

60L2-1 63.7 1.0360L2-2 18 65.7 65.8 1.00 100L2-1 38.2 0.96100L2-2

80 80 25.5 387 1.9830

0.625 1 0.3

35.0 36.6 1.05 60M2-1 102.8 1.0860M2-2 18 113.5 111.1 0.98 100M2-1 45.2 1.22100M2-2

80 80 63.5 387 1.9830

0.625 1 0.3

47.6 55.1 1.16 60H2-1 122.1 1.1060H2-2 18 123.7 134.7 1.09 100H2-1 54.3 1.17100H2-2

80 80 86.2 387 1.9830

0.625 1 0.3

54.9 63.3 1.15

223

Table 7.3 Summary of Ranger and Bisby’s (2007) tests

Specimen D

(mm)l

(mm) 1

2

ee

2e (mm)

ConcreteCover (mm)

frpE (GPa)

t (mm)

'cof

(MPa)yf

(MPa) sρ

(%),u testN

(kN),u theoN

(kN) ,

,

u theo

u test

NN

U-0 90 0 497 641 1.29C-0 152 600 1 0 25 90 0.381 33.2 710 0.71 873 786 0.90U-5 90 0 459 584 1.27C-5 152 600 1 5 25 90 0.381 33.2 710 0.71 770 725 0.94

U-10 90 0 447 525 1.17C-10 152 600 1 10 25 90 0.381 33.2 710 0.71 664 655 0.99U-20 90 0 351 420 1.20C-20 152 600 1 20 25 90 0.381 33.2 710 0.71 579 518 0.89U-30 90 0 253 322 1.27C-30 152 600 1 30 25 90 0.381 33.2 710 0.71 337 402 1.19U-40 90 0 179 242 1.35C-40 152 600 1 40 25 90 0.381 33.2 710 0.71 264 305 1.16

224

Table 7.4 Summary of Fitzwilliam and Bisby’s (2006) tests

Specimen D

(mm) l

(mm) 1

2

ee

2e (mm)

Concrete Cover (mm)

frpE (GPa)

t (mm)

'cof

(MPa)yf

(MPa)sρ

(%),u testN

(kN)

,u theoN

(kN)

,

,

u theo

u test

NN

300U-A 471300U-B 152 300 1 20 25 90 0 35.5 710 0.71 462 458 0.98

300C-1-0-A 675300C-1-0-B 152 300 1 20 25 90 0.381 35.5 710 0.71 679 531 0.78

300C-2-0-B 152 300 1 20 25 90 0.762 35.5 710 0.71 911 684 0.75600U-A 152 600 20 25 90 0 35.5 710 0.71 428 448 1.05

600C-1-0-A 152 600 1 20 25 90 0.381 35.5 710 0.71 563 505 0.90900U-A 152 900 1 20 25 90 0 35.5 710 0.71 398 432 1.09

900C-1-0-A 152 900 1 20 25 90 0.381 35.5 710 0.71 549 468 0.851200U-A 3891200U-B 152 1200 1 20 25 90 0 35.5 710 0.71 411 411 1.03

1200C-1-0-A 4511200C-1-0-B 152 1200 1 20 25 90 0.381 35.5 710 0.71 481 433 0.93

1200C-2-0-A 152 1200 1 20 25 90 0.762 35.5 710 0.71 539 466 0.86

225

Table 7.5 Cylinder tests in Fitzwilliam and Bisby (2006) and Ranger and Bisby (2007)

Source '

cof (MPa)

frpE (GPa)

t (mm)

'ccf

(MPa) cuε

(%) ,h rupε

(%)

'ccf -predicted

(MPa) cuε -predicted

(%) 41 0.804 1.140

39.7 0.768 1.16534.6

90 0.38141.4 0.793 1.192

46.0 1.24

35.8 90 0.762 59.8 0.165 1.111 63.9 1.7658 0.112 1.175

Fitzwilliam and Bisby

(2006) 36.4 90 0.762 62.5 0.156 1.151 65.6 1.83

Ranger and Bisby (2007)

33.2 90 0.381 44.2 0.860 1.15 44.7 1.25

226

Table 7.6 Summary of Tao et al.’s (2004) tests

Specimen D

(mm)l

(mm) 1

2

ee

2e (mm)

ConcreteCover (mm)

'cof

(MPa)yf

(MPa)sρ

(%)frpE

(GPa) t

(mm),u testN

(kN)

,u theoN (kN)

,

,

u theo

u test

NN

C1-1U 455 942(658) 2.07(1.45)C1-1R 150 1260 1 0 21 48.2 388.7 2.56 255 0.34 765 1018(871) 1.33(1.14)C1-2U 149 273(198) 1.83(1.33)C1-2R 150 1260 1 50 21 48.2 388.7 2.56 255 0.34 248 288(243) 1.16(0.98)C1-3U 88 119(101) 1.35(1.15)C1-3R 150 1260 1 100 21 48.2 388.7 2.56 255 0.34 124 131(119) 1.06(0.96)C1-4U 54.5 75.9(68.6) 1.39(1.26)C1-4R 150 1260 1 150 21 48.2 388.7 2.56 255 0.34 77 79.5(75.0) 1.03(0.97)C2-1U 276 668(392) 2.42(1.42)C2-1R 150 3060 1 0 21 48.2 388.7 2.56 255 0.34 386 700(543) 1.81(1.41)C2-2U 108 146(114) 1.35(1.06)C2-2R 150 3060 1 50 21 48.2 388.7 2.56 255 0.34 126 149(131) 1.18(1.04)C2-3U 62.5 76.1(67.4) 1.22(1.08)C2-3R 150 3060 1 100 21 48.2 388.7 2.56 255 0.34 71.5 77.6(72.9) 1.08(1.02)C2-4U 39 54.0(49.7) 1.83(1.27)C2-4R 150 3060 1 150 21 48.2 388.7 2.56 255 0.34 47.5 55.1(52.8) 1.16(1.11)

227

Dl

N

2e

2

l

m-2m-1

N e1

m

d

3

1

Fig. 7.1 Schematic of the theoretical model

228

' hh

b

Fig. 7.2 Cross section of the columns used in Fig. 7.3

229

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

Normalized Lateral Displacement fmid / h

Nor

mal

ized

Axi

al L

oad

N/N

uo

Cranston (1972)Present study

(a) Columns 1 and 2

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Normalized Lateral Displacement fmid / h

Nor

mal

ized

Axi

al L

oad

N/N

uo

Cranston (1972)Present study

(b) Columns 3 and 4

Fig. 7.3 Comparisons with Cranston’s theoretical model

230

0 10 20 30 40 50

0

10

20

30

40

50

60

70

Lateral Displacement fmid (mm)

Axi

al L

oad

N (k

N)

60L2-2 (Test)60L2-2 (Predicted)100L2-1 (Test)100L2-1 (Predicted)

Fig. 7.4 Comparisons with Kim and Yang’s tests on RC columns

231

-1 0 1 2 3 4 5 60

100

200

300

400

500

600

700

800


Axi

al L

oad

N (k

N)

C-5 (Test)C-5 (Predicted)C-5 (Predicted, Lam and Teng)

(a) Column C-5

0 2 4 6 8 10 120

100

200

300

400

500

600

700


Axi

al L

oad

N (k

N)

C-10 (Test)C-10 (Predicted)C-10 (Predicted, Lam and Teng)C-20 (Test)C-20 (Predicted)C-20 (Predicted, Lam and Teng)

(b) Columns C-10 and C-20

232

0 5 10 15 200

50

100

150

200

250

300

350

400

450


Axi

al L

oad

N (k

N)

C-30 (Test)C-30 (Predicted)C-30 (Predicted, Lam and Teng)C-40 (Test)C-40 (Predicted)C-40 (Predicted, Lam and Teng)

(c) Columns C-30 and C-40

Fig. 7.5 Comparison with Ranger and Bisby’s tests

233

0 2 4 6 8 100

100

200

300

400

500

600


Axi

al L

oad

N (k

N)

600C-1-0-A (Test)600C-1-0-A (Predicted)600C-1-0-A (Predicted, Lam and Teng)900C-1-0-A (Test)900C-1-0-A (Predicted)900C-1-0-A (Predicted, Lam and Teng)

(a) Columns 600C-1-0-A and 900C-1-0-A

0 10 20 30 40 500

100

200

300

400

500

600


Axi

al L

oad

N (k

N)

1200C-1-0-A (Test)1200C-1-0-B (Test)1200C-1-0 (Predicted)1200C-1-0 (Predicted, Lam and Teng)1200C-2-0-A (Test)1200C-2-0-A (Predicted)1200C-2-0-A (Predicted, Lam and Teng)

(b) Columns 1200C-1-0-A, 1200C-1-0-B and 1200C-2-0-A

Fig. 7.6 Comparison with Fitzwilliam and Bisby’s tests

234

0 0.005 0.01 0.015 0.02 0.0250

10

20

30

40

50

60

70

80

90

100

Axial Strain εc

Axi

al S

tress

σc

(MP

a)

D = 150mmf′co = 48.2MPa

Efrp = 255GPa

t = 0.34mmεh,rup = 0.0132

Test (2 Specimens)Lam and Teng′s Model

Fig. 7.7 Comparison with Tao et al.’s cylinder tests

235

0 10 20 30 40 50 600

100

200

300

400

500

600

700

800

900


Axi

al L

oad

N (k

N)

C1-1R (Test)C1-1R (Predicted)C1-2R (Test)C1-2R (Predicted)

e2 = 15 mm

e2 = 7.5 mm

(a) Columns C1-1R and C1-2R

0 10 20 30 40 50 600

20

40

60

80

100

120

140


Axi

al L

oad

N (k

N)


(b) Columns C1-3R and C1-4R

236

0 50 100 150 2000

100

200

300

400

500

600


Axi

al L

oad

N (k

N)


e2 = 7.5 mm

e2 = 15 mm

(c) Columns C2-1R and C2-2R

0 50 100 150 200 2500

10

20

30

40

50

60

70

80


Axi

al L

oad

N (k

N)

C2-3R (Test)C2-3R (Predicted, old)C2-3R (Predicted, new)C2-4R (Test)C2-4R (Predicted)

(d) Columns C2-3R and C2-4R

Fig. 7.8 Comparisons with Tao et al.’s tests on FRP-confined RC columns

237

CHAPTER 8

SLENDERNESS LIMIT FOR

SHORT FRP-CONFINED RC COLUMNS

8.1 INTRODUCTION

Design equations for short FRP-confined RC columns have been presented in

Chapter 5. However, for the application of these equations, a clear definition of

short columns is needed. In most design codes for RC structures (e.g. ENV-1992-

1-1 1992; BS-8110 1997; ACI-318 2005), a simple check is generally required to

determine whether an RC column is short or slender before applying

corresponding design equations. If the column is classified as a short column, the

design procedure is more straightforward compared to that for slender columns as

the slenderness effect can be ingored. Therefore, a proper definition of the

slenderness value that separates short columns from slender columns is important

in design. In this chapter, this particular slenderness value is referred to as the

slenderness limit for short columns, or simply the slenderness limit for brevity. It

should be noted that although a number of design guidelines for FRP-strengthened

RC structures (fib 2001; ISIS 2001; ACI-440.2R 2002, 2008; JSCE 2002; CNR-

DT200 2004; Concrete Society 2004) have been developed, such a slenderness

limit is only available in ISIS (2001), which is only intended for columns with no

significant bending (i.e. concentric compression or slightly eccentric compression).

This is mainly due to the fact that only a very limited number of studies have

investigated the behavior of slender concrete columns confined with FRP. These

studies include Mirmiran et al. (2001a) and Yuan and Mirmiran (2001) on slender

concrete-filled FRP tubes and Tao et al. (2004), Fitzwilliam and Bisby (2006) and

Ranger and Bisby (2007) on slender FRP-confined RC columns. It has been

238

realized in these studies that when FRP confinement exists, concrete columns tend

to experience more severe slenderness effects. Consequently, existing slenderness

limit expressions for RC columns [Excellent summaries can be found in

Hellesland (2005) and Mari and Hellesland (2005)] are not directly applicable to

concrete columns confined with FRP. Mirmiran et al. (2001a) tested

concentrically-loaded slender concrete-filled FRP tubes and developed a

theoretical model for such columns. Both their experimental and theoretical

results support the above conclusion. Yuan and Mirmiran (2001) further

developed a theoretical model for slender concrete-filled FRP tubes subjected to

eccentric compression (this model has been briefly described in Chapter 7). Their

analysis also led to the same result More recently, Tao et al. (2004), Fitzwilliam

and Bisby (2006) and Ranger and Bisby (2007) carried out tests on FRP-confined

RC columns. These studies found that FRP-confined RC columns experienced a

larger loss in the axial load capacity compared to corresponding RC columns. The

findings of these studies indicate that with FRP confinement, an RC column

which is originally classified as a short column may need to be re-classified as a

slender column, and a proper slenderness limit expression thus needs to be

developed for design use.

Yuan and Mirmiran (2001) recommended that for columns subjected to equal end

eccentricities, the current slenderness limit of 22 for RC columns as specified in

ACI-318 (2005) be reduced to 11 for the slenderness limit for concrete-filled FRP

tubes. Although the slenderness limit value proposed by Yuan and Mirmiran

(2001) is simple and conservative, it should be noted that it only corresponds to a

effective length-to-diameter ratio of 2.75. As a result, most columns would be

classified as slender columns according to their expression. In fact, it is shown in

later sections of this chapter that the magnitudes of the end eccentricities have a

significant effect on the slenderness limit; a more accurate slenderness limit

expression can be developed so that a much wider range of columns can be

classified as short columns.

This chapter is concerned with the development of such a slenderness limit

expression for FRP-confined circular RC columns (RC columns are treated as a

special case where no FRP confinement is provided) in braced frames. It should

239

be noted that existing design codes for RC structures generally specify different

slenderness limit expressions for columns in braced frames and unbraced frames

respectively; the latter are assigned with a smaller slenderness limit (Hellesland

2005). This chapter is only concerned with columns in braced frames. To this end,

a comprehensive parametric study was performed using the theoretical model

developed in Chapter 7 to examine the effects of various parameters on the

slenderness limit. Based on the results of this parametric study, a simple

slenderness limit expression for design use is proposed. This expression considers

all the governing parameters and can reduce to a form that is identical or similar to

the slenderness limit expressions for conventional RC columns in current design

codes. The present study has been partially motivated by the development of the

Chinese Code for the Application of FRP Composites in Construction, which is

formulated within the general framework of the current Chinese Code for Design

of Concrete Structures (GB-50010 2002). Therefore, some of the considerations

herein follow the specifications given in GB-50010 (2002) and they are

highlighted where appropriate.

8.2 DEFINITION OF SLENDERNESS LIMIT

The concept of the slenderness limit can be illustrated by examining the behavior

of a column as its slenderness varies. For simplicity, consider a hinged RC

column under an increasing axial load N bent in symmetrical single curvature

( ). In such a case, the critical section is located at the mid-height of the

column. Fig. 8.1 shows the axial load-bending moment loading paths of the

critical section of the column for three different values of column slenderness; the

load eccentricity is fixed. In the absence of the slenderness effect (i.e. the height

of the columns is very small), the loading path follows the straight line all the

way up until it intersects the cross-section interaction curve at point

1e e= 2

OA

A , which

marks the occurrence of material failure. In such a case, the critical section is only

subjected to the first-order moment 2N e⋅ and the axial load capacity of the

column is defined by point A . Once the column has a certain height, the lateral

displacement of the mid-height section midf induces a second-order moment

and thus causes the loading path to deviate from OA . If the column is not midN f⋅

240

too slender, the loading path is still of an ascending shape when it intersects with

the cross-section interaction curve at point B , which indicates that the column is

still controlled by material failure and the axial load capacity is defined by point

B . By contrast, if the column is very slender, the loading path may exhibit a

descending branch (path ODE ), which indicates that the column is no longer

controlled by material failure, and instead, stability failure occurs prior to material

failure. The axial load capacity of such a column is defined by the peak point D

of the loading path. It is obvious that the larger the column slenderness, the larger

the loss in the axial load capacity due to the second-order moment.

The slenderness limit for short RC columns is commonly defined to ensure that

the slenderness effect leads to only a small amplification of the first-order moment

(i.e. the second-order moment only constitutes a small fraction of the first-order

moment) at the critical section or a small reduction (commonly 5% or 10%) of the

axial load capacity. In this chapter, the criterion of a 5% reduction in the axial

load capacity is adopted to define the slenderness limit for short columns. The

slenderness ratio is defined as

efflr

(8.1) λ =

where is the effective length of a column and r is the radius of gyration

and for circular columns, where is the diameter of the cross section. It

should be noted that the numerical study presented in this chapter is limited to

hinged columns whose effective length is equal to their physical length

effl

/ 4D= D

For restrained columns, it has been concluded in Chapter 6 that the effective

length can be determined following the approximate approach in existing codes

(e.g. ACI-318 2005; ENV-1992-1-1 1992) where simple charts developed from

Eq. 6.42 are provided to relate the effective length to the column-to-beam stiffness

ratio. No further research has been undertaken in the present study on the effective

length of FRP-confined RC columns. It is suggested that the methods given in

241

existing design codes for RC structures be used to estimate the effective length of

FRP-confined RC columns.

.

8.3 PARAMETRIC STUDY

8.3.1 Parameters Considered

A comprehensive parametric study was carried out in order to investigate the

effects of various factors on the slenderness limit for short FRP–confined RC

columns. The parametric study was carried out on the same reference column (Fig.

8.2) as used in Chapter 5. More specifically, the reference RC column had a

diameter of 600 mm and was longitudinally reinforced with 12 evenly distributed

steel bars. The strengths of concrete and steel reinforcement were taken to be

common values as specified in GB-50010 (2002). The concrete was assumed to be

grade C30, representing a characteristic cube strength of MPa and a

corresponding cylinder strength MPa. The steel was assumed

to be grade II with a characteristic yield strength

30cuf =

' '0.67 20.1co cuf f= =

335yf = MPa and an elastic

modulus of 200 GPa. It is evident that all the parameters that influence the

structural behavior of a slender column have some effect on the slenderness limit.

Based on previous studies on slender RC columns (Pfrang and Siess 1961;

MacGregor et al. 1970), the main parameters are identified to be

: the eccentricity ratio

sE =

1 2e e , the normalized eccentricity 2e D , the steel

reinforcement ratio sρ , and the depth ratio d D . When FRP confinement is taken

into account, two additional parameters should be considered, namely, the

stiffness and the strain capacity of the FRP jacket. To keep the slenderness limit

equation as simple as possible, the strength enhancement ratio ' 'cc cof f and the

strain ratio ερ were chosen to reflect the effect of FRP confinement based on a

careful consideration. The values of all the parameters considered in the

parametric study are summarized in Table 8.1.

242

The combinations of these parameters led to about 3000 cases. It should be noted

that most parameters studied herein are the same as those used in the parametric

study presented in Chapter 5. The end eccentricity ratio is the only new parameter.

Theoretically speaking, the end eccentricity ratio ranges from -1 to 1. The use of

instead of is based on computational considerations. Under the

unobtainable idealization of

0.99− 1−

1 2 1e e = − , the lateral deflection of the column is

exactly anti-symmetrical. However, with a slightest disturbance introduced as

always is the case in reality, the column tends to behave in a significantly different

way due to the phenomenon which is commonly known as “unwrap” (Pfrang and

Siess 1961). For the strength enhancement ratio, ' ' 1cc cof f = represents the case of

unconfined concrete, but not cases of insufficiently confined concrete.

The results of the parametric study are presented in two parts. The first part deals

with RC columns while the second part considers the effect of FRP confinement.

In all the figures presenting the results of the parametric study, the vertical axis is

always the slenderness limit while the horizontal axis is one of the six parameters

considered. A family of curves is shown in each figure, showing the variation of

the slenderness limit with another parameter. The values of the other parameters

used to generate these numerical results are also given in each figure.

8.3.2 Results for RC Columns

First, the effect of the end eccentricity ratio is investigated. The numerical results

are shown in Fig. 8.3. Each curve shows the variation of the slenderness limit with

the end eccentricity ratio for a particular normalized eccentricity. It can be clearly

seen that for a given normalized eccentricity, the slenderness limit decreases

almost linearly as the end eccentricity ratio increases. This can be easily

understood, since a column bent in single curvature with always

experiences the largest slenderness effect. It can also be noted that as the

normalized eccentricity increases, the slenderness limit generally increases rapidly,

which is more clearly reflected in Fig. 8.4 and is further discussed later. It should

be noted that these numerical results are for

1e e= 2

0.8d D = and 3%sρ = since the

243

results for other values of these two parameters are similar and are not presented

herein for brevity. The effects of these two parameters are further discussed later.

As stated above, the slenderness limit generally increases with the normalized

eccentricity. The general trend of the curves in Fig. 8.4 can be explained as

follows. It is obvious that the larger the normalized eccentricity, the lower the

axial load level. When approaches infinity, the column approaches the state of

pure bending and the slenderness limit approaches infinity. By contrast, when

, since no lateral deflection would occur at this ideal condition, Fig. 8.4

shows that the end eccentricity ratio hardly has any effect when the end

eccentricity is very small. In Fig. 8.4, the different curves all converge to the same

point obtained using a very small normalized eccentricity of

2e

2 0e =

2 0.001e D = . It

should be noted that this small value of normalized eccentricity was only used in

Fig. 8.4 for illustrative purposes but was not used in the rest of the parametric

study.

Fig. 8.5 shows the effect of the depth ratio. Fig. 8.5a is for a relatively high axial

load level ( 2 0.05e D = ) while Fig. 8.5b is for a relatively low axial load level

( 2 0.8e D = ). It can be seen that for both cases the slenderness limit slightly

increases with the depth ratio. This is because an increase in the depth ratio

increases the lever arm of the steel reinforcement and thus enlarges the load

contribution from the steel reinforcement, which helps the column to resist the

slenderness effect. However, this parameter has a much smaller effect than the

first two parameters discussed above.

Lastly, the effect of the steel reinforcement ratio is investigated. The numerical

results in Fig. 8.6 also cover both a high axial load level and a low axial load level.

Increasing the amount of steel reinforcement increases the slenderness limit

although not very effective. The reason is similar to that given in the preceding

paragraph.

This set of numerical results indicates that the end eccentricity ratio and the

normalized eccentricity are the two primary parameters for RC columns and they

244

should both be taken into account when developing a design expression for the

slenderness limit. The depth ratio and steel reinforcement ratio have negligible

effects on the slenderness limit and they may thus be ignored in the design

expression.

8.3.3 Results for FRP-confined RC Columns

All the results for FRP-confined RC columns presented herein are for

0.7d D = and 1%sρ = since these two parameters are expected to have only

minor effects on the slenderness limit and the numerical results obtained using

these values are expected to be conservative for columns with a higher depth ratio

and a larger steel reinforcement ratio.

The effect of the strength enhancement ratio is investigated first. As stated earlier, ' ' 1cc cof f = represents the case of no FRP confinement. Fig. 8.7 is for 3.75ερ =

which represents the strain capacity of common CFRPs. Fig. 8.7 includes a set of

sub-figures, corresponding to different axial load levels. The effect of the strength

enhancement ratio is substantial. It can be seen that among all the curves shown in

Fig. 8.7, the uppermost curve passing through small circles in Fig. 8.7a has the

sharpest slope. This indicates that the smaller the end eccentricity ratio and the

higher the axial load level, the more significant the effect of the strength

enhancement ratio.

As discussed in Chapter 4, for a given strength enhancement ratio, an FRP jacket

with a larger strain capacity always yields a larger ultimate axial strain for FRP-

confined concrete. As a result, the strength enhancement ratio alone is not

sufficient to reflect the effect of FRP confinement. Fig. 8.8 shows the effect of the

strain capacity of FRP materials. Figs 8.8a and 8.8b show the numerical results for

a relatively high axial load level ( 2 0.05e D = ) and two different strength

enhancement ratios respectively. It is clear that larger strain ratios result in smaller

slenderness limits. This is because for a given strength enhancement ratio, the

second portion of stress-strain curve of FRP-confined concrete defined by a larger

strain ratio is always flatter, which leads to a more severe slenderness effect. It

245

can also be easily understood that the strain ratio has a more significant effect for

a larger amount of confinement. The above observations also hold for Figs 8.8c

and 8.8d, where a relatively low axial load level ( 2 0.8e D = ) is considered.

However, as the axial load level decreases, the strain ratio has a smaller effect.

The above discussions indicate that both the strength enhancement ratio and the

strain ratio have a significant effect on the slenderness limit. Although FRP

confinement enhances the load capacity of an RC column, it also amplifies the

slenderness effect. This is because FRP confinement can substantially increase the

axial lad capacity of a section without significantly enhancing its flexural rigidity.

The effect of the strain ratio varies mainly with the axial load level. At low axial

load levels, its effect may be ignored.

8.4 SLENDERNESS LIMIT EXPRESSIONS FOR DESIGN USE

Based on the results and discussions of the parametric study, the six parameters

examined can be classified into three categories in terms of their significance. The

primary parameters include the strength enhancement ratio, the end eccentricity

ratio, and the normalized eccentricity; these three parameters must be considered

in a slenderness limit expression for use in design. The minor parameters include

the depth ratio and the steel reinforcement ratio; these two parameters may be

ignored in the design expression. The strain ratio has a moderate effect and lies

between the above two categories. In this section, a design expression is first

proposed for RC columns. This expression is then extended to FRP-confined RC

columns by introducing additional terms that reflect the effect of FRP

confinement.

8.4.1 Slenderness Limit for RC Columns

Based on a careful interpretation of the numerical results for RC columns, the

following simple equation is proposed for the slenderness limit for short RC

columns

246

2 1

2

60 (1 ) 20crite eD e

λ = − + (8.2)

Eq. 8.2 accounts for the two primary parameters for RC columns and can reduce

to a form that is similar to the design equations given in existing design codes

where fewer parameters are considered. GB-50010 (2002) specifies the following

expression for the slenderness limit for short RC columns

20critλ = (8.3)

This expression is based on the most conservative condition of 1 2 1e e = and

ignores the effect of load eccentricity. It is easy to see that with these conditions,

Eq. 8.2 reduces to Eq. 8.3. In addition, the current slenderness limit expression in

ACI-318 (2005) is

1

2

34 12critMM

λ = − (8.4)

where 1M and 2M are the first-order moments at the two ends respectively. If Eq.

8.4 is written in terms of the end eccentricity ratio, it becomes

1

2

34 12critee

λ = − (8.5)

Eq. 8.5 considers the end eccentricity ratio but is developed with a fixed load

eccentricity of 2 0.2e D = (Mirmiran et al. 2001b). When 2 0.2e D = , Eq. 8.2

reduces to 1

2

32 12critee

λ = − , which is similar to but slightly more conservative

than Eq. 8.5. The author is aware that the ACI expression was developed using the

moment magnifier method and it was initially based on the same criterion adopted

in this chapter (5% axial load reduction), as documented in MacGregor et al.

(1970). However, in a revisited paper by MacGregor et al. (1993), a new criterion

was adopted (5% first-order moment amplification) but this equation remained

unchanged. Fig. 8.9 shows the performance of Eq. 8.2 and Eq. 8.4. The

247

predictions of both expressions for relatively low values of slenderness limit are

similar. However, the ACI expression is very conservative at high values of

slenderness limit, which occur at low axial load levels, particularly when the

column is bent in double curvature. Both expressions produce a few slightly un-

conservative results at low values of slenderness limit.

8.4.2 Slenderness Limit for FRP-confined RC Columns

Fig. 8.7 shows that the slenderness limit for an FRP-confined RC column can be

roughly approximated by dividing the slenderness limit for the corresponding RC

column without FRP confinement by the strength enhancement ratio (shown as

dashed lines in Fig. 8.7). Thus, the following equation is proposed for the

slenderness limit for short FRP-confined RC columns

2 1

2'

'

60 (1 ) 20

critcc

co

e eD e

ff

λ− +

= (8.6)

This expression has a clear physical meaning: the numerator defines the

slenderness limit for short RC columns without FRP confinement, while the

denominator accounts for the effect of FRP confinement. It should be noted that

the effect of the strain ratio is ignored in this equation. Numerical results for the

slenderness limit corresponding to the 5% axial load reduction criterion are shown

in Fig. 8.10a. These results were generated using the most critical combination of

the depth ratio and the steel reinforcement ratio ( 0.7d D = and 1%sρ = ) since

RC columns with a higher depth ratio and a larger steel reinforcement ratio are

less affected by the slenderness effect. It can be seen that Eq. 8.6 is un-

conservative for a few cases at very low slenderness limit values. Even when the

10% axial load reduction criterion is adopted, Eq. 8.6 still yields un-conservative

results at very low slenderness limit values (Fig. 8.10b). A careful study of the

numerical results showed that all the un-conservative predictions are for columns

bent in symmetrical single curvature ( 1 2 1e e = ). However, in braced frames most

columns are bent in double curvature, which means that most columns have a

248

negative end eccentricity ratio. If the results for 1 2 1e e = are removed from Fig.

8.10a, very few un-conservative predictions rema even for these cases the

predictions are only slightly un-conservative (Fig. 8.10c).

in and

he parametric study has shown that the strain ratio has some effect on the T

slenderness limit in some cases. To account for this effect, the following equation

is proposed:

2 1

2'

,'

60 (1 ) 20

(1 0.06 )crit

h rupcc

co co

e eD e

ff

λεε

− +=

+ (8.7)

umerical results for the slenderness limit based on the 5% axial load reduction N

criterion are shown in Fig. 8.11a. These results are also for 1%sρ = and

0.7d D = . It can be seen from Fig. 8.11a that Eq. 8.7 is conservative for all cases

en the slenderness limit is very small. However, if a 10% loss of axial

load capacity is acceptable, Eq. 8.7 provides a lower bound prediction for all cases,

as shown in Fig. 8.11b. A 10% reduction in the axial load capacity has been

adopted as the criterion for permitted second effects in the existing literature (e.g.

CEB-FIP 1993).

except wh

summary, Eq. 8.6 possesses a simpler form, but may be un-conservative for

.5 CONCLUSIONS

his chapter has been concerned with the development of a slenderness limit

In

low values of slenderness limit. Although Eq. 8.7 takes a slightly more

complicated form, it provides a lower bound prediction for all the cases studied in

this chapter. Eq 8.7 is recommended for design use.

8

T

expression for FRP-confined RC columns for use in design. A comprehensive

parametric study was carried out to investigate the effects of various parameters

on the slenderness limit using the theoretical model developed in Chapter 7.

249

Based on the results of the parametric study, expressions for the slenderness limit

of FRP-confined RC columns in braced frames were proposed. The results and

discussions presented in this chapter allow the following conclusions to be drawn:

1) Although FRP confinement can significantly increase the axial load capacity

) The six parameters examined in the parametric study have different effects on

) The proposed slenderness limit expression has a clear physical meaning: the

of RC columns, it also introduces a greater slenderness effect. This is because

FRP confinement can substantially increase the axial load capacity of an RC

section but affects little the flexural rigidity of the section.

2

the slenderness limit for FRP-confined RC columns. The strength

enhancement ratio, the end eccentricity ratio and the normalized eccentricity

were identified as the primary parameters while the depth ratio and steel

reinforcement ratio were identified as the minor parameters. The strain ratio

was shown to have an effect between the above two categories.

3

numerator defines the slenderness limit for short RC columns without FRP

confinement, while the denominator accounts for the effect of FRP

confinement. The numerator of the proposed expression is slightly more

complex than the slenderness limit expressions for RC columns given in some

of the current design codes for RC structures. This is because an RC column

originally classified as a short column may need to be considered as a slender

column when it is confined with FRP. Therefore, the proposed expression

aims at greater accuracy at the sacrifice of a certain degree of simplicity so

that a much wider range of FRP-confined RC columns can be classified as

short columns to avoid the extra complications involved in the design of

slender columns. It is worth pointing out that existing slenderness limit

expressions for RC columns of different forms can be readily upgraded for

use in the design of FRP-confined RC columns by incorporating the

denominator of the proposed expression.

250

8.6 REFERENCES

ACI-440.2R (2002). Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures, ACI Committee 440, American Concrete Institute.

ACI-318 (2005). Building Code Requirements for Structural Concrete and Commentary, ACI Committee 318, American Concrete Institute..


CEB-FIP (1993). Model Code 1990, CEB-Bulletin No. 213/214, Comité Euro-International du Beton.






Hellesland, J. (2005). “Nonslender column limits for braced and unbraced reinforced concrete members”, ACI Structural Journal, 102(1), 12-21.

Mari, A.R. and Hellesland, J. (2005). “Lower slenderness limits for rectangular reinforced concrete columns”, Journal of Structural Engineering, ASCE, 131(1), 85-95.


MacGregor, J.G. (1993). “Design of slender concrete columns-revisited”, ACI Structural Journal, 90(3), 302-309.

Mirmiran, A., Shahawy, M. and Beitleman, T. (2001a). “Slenderness limit for hybrid FRP-concrete columns”, Journal of Composites for Construction, 5(1), 26-34.

251

Mirmiran, A., Yuan, W. and Chen, X. (2001b). “Design for slenderness in concrete columns internally reinforced with fiber-reinforced polymer bars”, ACI Structural Journal, 98(1), 116-125.

Pfrang, E.O. and Siess, C.P. (1961). “Analytical study of the behavior of long restrained reinforced concrete columns subjected to eccentric loads”, Structural research series No. 214, University of Illinois, Urbana, Illinos.


Tao, Z., Teng, J.G., Han, L.H. and Lam, L. (2004). “Experimental behaviour of FRP-confined slender RC columns under eccentric loading”, Proceedings, 2nd International Conference on Advanced Polymer Composites for Structural Applications in Construction, University of Surrey, Guildford, UK, 203-212.


252

Table 8.1 Values of parameters used in the parametric study

Parameter Values 1 2e e 1, 0.5, 0, -0.5, -0.99 e D 0.05, 0.1 ,0.2, 0.4, 0.8

sρ 1%, 3%, 5% d D 0.7, 0.8, 0.9 ' '

cc cof f 1, 1.25, 1.5, 1.75, 2

,h rup coε ε 1, 3.75, 7.5

253

Moment M

Axi

al L

oad

N

O

AB

CD

E

e2 = Const

e2

1

N⋅e2

N⋅fmid

Peak Point

Fig. 8.1 Behavior of slender columns

Dl

N

2e

N e1

d

Fig. 8.2 Definition of the reference column

254

-1 -0.5 0 0.5 120

40

60

80

100

120

140

160

180d/D=0.8

ρs=3%

End Eccentricity Ratio e1/e2

Sle

nder

ness

Lim

it λ

crit

e2/D = 0.05e2/D = 0.1e2/D = 0.2e2/D = 0.4e2/D = 0.8

Fig. 8.3 Effect of end eccentricity ratio

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.820

40

60

80

100

120

140

160

180d/D=0.8

ρs=3%

Normalized Eccentricity e2/D

Sle

nder

ness

Lim

it λ

crit

e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

Fig. 8.4 Effect of eccentricity

255

0.7 0.75 0.8 0.85 0.90

20

40

60

80

e2/D=0.05 ρs=1%

Depth Ratio d/D

Sle

nder

ness

Lim

it λ

crit

e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(a) 2 0.05e D =

0.7 0.75 0.8 0.85 0.90

20

40

60

80

100

e2/D=0.8 ρs=1%

Depth Ratio d/D

Sle

nder

ness

Lim

it λ

crit

e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(b) 2 0.8e D =

Fig.8.5 Effect of depth ratio

256

0.01 0.02 0.03 0.04 0.050

20

40

60

80

100

e2/D=0.05 d/D=0.9

Steel Reinforcement Ratio ρs

Sle

nder

ness

Lim

it λ

crit

e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(a) 2 0.05e D =

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

20

40

60

80

100

120

140

160

180

e2/D=0.8 d/D=0.9

Steel Reinforcement Ratio ρs

Sle

nder

ness

Lim

it λ

crit

e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(b) 2 0.8e D =

Fig. 8.6 Effect of steel reinforcement ratio

257

1 1.25 1.5 1.75 20

10

20

30

40

50

60

70

80


Sle

nder

ness

Lim

it λ

crit

εh,rup/εco=3.75 e2/D=0.05

d/D=0.7 ρs=0.01

e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(a) 2 0.05e D =

1 1.25 1.5 1.75 20

10

20

30

40

50

60

70

80

90


Sle

nder

ness

Lim

it λ

crit


d/D=0.7 ρs=1%

e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(b) 2 0.1e D =

258

1 1.25 1.5 1.75 20

20

40

60

80

100


Sle

nder

ness

Lim

it λ

crit


d/D=0.7 ρs=0.01e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(c) 2 0.2e D =

1 1.25 1.5 1.75 20

20

40

60

80

100

120


Sle

nder

ness

Lim

it λ

crit


d/D=0.7 ρs=0.01e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(d) 2 0.4e D =

259

1 1.25 1.5 1.75 20

20

40

60

80

100

120

140

160


Sle

nder

ness

Lim

it λ

crit


d/D=0.7 ρs=0.01e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(e) 2 0.8e D =

Fig. 8.7 Effect of strength enhancement ratio

260

0 2 4 6 80

10

20

30

40

50

60

Strain Ratio εh,rup/εco

Sle

nder

ness

Lim

it λ

crit

f′cc/f′co=1.25 e2/D=0.05d/D=0.7 ρs=0.01

e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(a) ' '

2 0.05, 1.25cc coe D f f= =

0 2 4 6 80

10

20

30

40


Sle

nder

ness

Lim

it λ

crit

f′cc/f′co=2 e2/D=0.05

d/D=0.7 ρs=0.01e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(b) ' '

2 0.05, 2cc coe D f f= =

261

0 2 4 6 80

20

40

60

80

100

120

140

160

180


Sle

nder

ness

Lim

it λ

crit

f′cc/f′co=1.25 e2/D=0.8

d/D=0.7 ρs=0.01

e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(c) ' '

2 0.8, 1.25cc coe D f f= =

0 2 4 6 80

20

40

60

80

100

120

140


Sle

nder

ness

Lim

it λ

crit

f′cc/f′co=2 e2/D=0.8

d/D=0.7 ρs=0.01e1/e2 = -0.99e1/e2 = -0.5e1/e2 = 0e1/e2 = 0.5e1/e2 = 1

(d) ' '

2 0.8, 2cc coe D f f= =

Fig. 8.8 Effect of strain ratio

262

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180

Slenderness Limit λcrit - Accurate Analysis

Sle

nder

ness

Lim

it λcr

it - D

esig

n E

quat

ion

5% Axial Load Reduction

(a) Proposed expression

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180


Sle

nder

ness

Lim

it λcr

it - D

esig

n E

quat

ion


(b) ACI expression

Fig. 8.9 Performance of the proposed expression and the ACI expression for

RC columns

263

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180


Sle

nder

ness

Lim

it λcr

it - D

esig

n E

quat

ion


(a) Slenderness limits based on a 5% axial load reduction

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180


Sle

nder

ness

Lim

it λcr

it - D

esig

n E

quat

ion


(b) Slenderness limits based on a 10% axial load reduction

264

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180


Sle

nder

ness

Lim

it λcr

it - D

esig

n E

quat

ion


(c) Slenderness limits based on a 5% axial load reduction

(results for 1e e2= excluded)

Fig. 8.10 Performance of Eq. 8.6 for FRP-confined RC columns

265

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180


Sle

nder

ness

Lim

it λcr

it - D

esig

n E

quat

ion


(a) Slenderness limits based on a 5% axial load reduction

0 20 40 60 80 100 120 140 160 1800

20

40

60

80

100

120

140

160

180


Sle

nder

ness

Lim

it λcr

it - D

esig

n E

quat

ion


(b) Slenderness limits based on a 10% axial load reduction

Fig. 8.11 Performance of Eq. 8.7 for FRP-confined RC columns

266

CHAPTER 9

DESIGN OF SLENDER

FRP-CONFINED RC COLUMNS

9.1 INTRODUCTION

Design equations for short FRP-confined RC columns have been presented in

Chapter 5 and expression for the slenderness limit that differentiates short

columns from slender columns has been given in Chapter 8. The information

given in Chapters 5 and 8 provides a complete procedure for the design of short

FRP-confined RC columns. This chapter deals with the design of slender FRP-

confined RC columns. It has been pointed out in Chapter 2 that no existing design

guidelines include a design procedure for slender FRP-confined RC columns. This

has mainly been due to the fact that only a limited number of studies have

investigated the behavior of such columns (Mirmiran et al. 2001; Yuan and

Mirmiran 2001; Tao et al. 2004; Fitzwilliam and Bisby 2006; Ranger and Bisby

2007). These studies have been reviewed in Chapters 2, 7 and 8, so the same

information is not repeated herein.

In practice, the majority of columns are restrained at both ends and are

eccentrically loaded, with the eccentricities at the two ends often being different.

Nevertheless, it has been shown in Chapter 6 that such a restrained column with

different end eccentricities can be transformed into an equivalent hinged column

where the eccentricities at the two ends are the same through the effective length

approach. Such an equivalent column is referred to as the standard hinged column.

This chapter is therefore limited to the analysis and design of slender FRP-

267

confined circular RC columns in the form of the standard hinged column. The

effective length of a retrained FRP-confined RC column may be estimated using

the methods given in existing design codes for RC structures, as suggested in

Chapter 8. The equivalent uniform moment factor may be evaluated using Eq.

6.40, as done in most existing design codes for the design of RC columns.

This chapter first describes a simple theoretical model for slender FRP-confined

RC columns. This simple model is exclusively for the analysis of standard hinged

columns. Despite its simplicity and higher computational efficiency, the model

leads to accurate predictions. A careful study is then performed to determine the

maximum allowable amount of FRP confinement and the slenderness limit of RC

columns beyond which the use of FRP for strengthening is not recommended.

Finally, design equations are proposed and their performance is assessed through

comprehensive comparisons with the numerical results from the simple theoretical

model. Once again, some of the considerations herein follow the specifications

given in GB-50010 (2002) and they are highlighted where appropriate.

9.2 SIMPLE THEORETICAL MODEL

9.2.1 General

A rigorous theoretical model for the analysis of FRP-confined RC columns has

been presented in Chapter 7. This model allows the eccentricities at the two

column ends to be unequal and finds the lateral deflection of the column at a

particular load level iteratively. It is obvious that this rigorous theoretical model

can be used as an analytical tool for the development of design equations for

slender FRP-confined RC columns. Nevertheless, as only columns with equal end

eccentricities are considered in this chapter, a much simpler model which still

provides accurate predictions was developed and used to produce numerical

results for the development of design equations.

The simple theoretical model differs from the rigorous model in that the deflected

shape of a column is now assumed to be a half-sine wave and equilibrium is only

checked at the critical section (the section at the mid-height of the column) where

268

the maximum lateral deflection of the column takes place.

As a result, the numerical procedure presented in Chapter 7 can be significantly

simplified. The present method of analysis has been widely adopted in similar

studies of hinged columns with equal end eccentricities (e.g. Bazant et al. 1991)

and has proven to be very successful. It is worth noting that the design equations

for slender RC columns specified in existing design codes (e.g. ENV-1992-1-1

1992; GB-50010 2002) are based on the present method of analysis, which is

another reason for the use of this simple model in the development of design

equations for slender FRP-confined RC columns.

9.2.2 Method of Analysis

The assumption that the deflected shape of the columns under consideration can

be closely approximated using a half sine wave can be mathematically expressed

as

sinmidf flπ⎛= − ⎜⎝ ⎠

x ⎞⎟ (9.1)

where midf is the lateral displacement at the critical section and is the distance

from the origin (see Fig. 9.1). Differentiating Eq. 9.1 twice gives

x

2

2 sinmidf xl lπ πφ ⎛= ⎜

⎝ ⎠⎞⎟ (9.2)

so midf is related to the curvature at the critical section midφ through

2

2mid midlf φπ

= (9.3)

The moment acting on the critical section can then be written as

269

( )2

2mid mid midlM N e f N e φπ

⎛ ⎞= + = +⎜

⎝ ⎠⎟ (9.4)

This moment must be balanced by the stresses on the critical section. These

stresses can be determined using the conventional section analysis described in

Chapter 5. For the present purpose, it is more convenient to seek the values of

and for a given value of midM N midφ . That is, for a given value of midφ , assume a

strain value for the extreme compression fiber of concrete to evaluate the resultant

axial force and moment of the critical section and check to see if they satisfy Eq.

9.4. Once Eq. 9.4 is satisfied, the solution for the present value of midφ is found.

Otherwise, adjust the assumed strain value until Eq. 9.4 is satisfied. Once the

solution is found, a point on the full-range load-deflection curve of the column is

identified and the entire curve can be generated by finding successive solutions of

and for increasing values of midM N midφ . The analysis stops when the extreme

compression fiber of concrete reaches its ultimate axial strain. In the present study,

the critical section was divided into 50 horizontal layers and the solution was

considered successful when the difference between the two sides of Eq. 9.4 is

within . It should be noted that equilibrium is only guaranteed at the

critical section within the present theoretical model. A computer program was

written using Matlab 7.1 to fulfill the above numerical procedure.

610 midM−

9.2.3 Accuracy of the Simple Theoretical Model

The present simple model has been well accepted for the analysis of RC columns

(e.g. Bazant et al. 1991). To understand the accuracy of the simple model for

FRP-confined circular RC columns, the predictions of the simple model are

compared herein with the predictions of the rigorous model for the columns tested

by Tao et al. (2004). These tests were used to verify the rigorous model presented

in Chapter 7. The properties of these columns are given in Table 7.6. The

experimental load-deflection curves are compared with the predicted curves of

both the simple model and the rigorous model in Fig. 9.2. All theoretical curves

were obtained with a 7.5 mm additional eccentricity, as was done in Chapter 7.

All theoretical curves terminate when the extreme compression fiber of the critical

270

section reaches the ultimate axial strain of confined concrete based on the hoop

rupture strain of 1.32% as found from the Tao et al.’s (2004) cylinder tests.

A comparison of the theoretical curves reveals the following difference in all

cases: the simple model predicts a slightly quicker ascending branch, a slightly

higher peak load (by about 1% to 2%), and a slightly quicker descending branches

than the rigorous model. Furthermore, the simple model predicts a much longer

descending branch which terminates at a much larger lateral deflection despite

that the same hoop rupture strain was used in both models. This phenomenon is

more pronounced for the C2 series (Figs 9.2c and 9.2d) which has a larger

slenderness ratio than the C1 series (Figs 9.2a and 9.2b). It should be noted that

the theoretical curves for the C2 series terminate at some 700 mm, but only their

early portions are shown in Figs 9.2c and 9.2d. This phenomenon arises from the

half-sine wave assumption and is explained below in detail by taking column C2-

3R as an example. Fig. 9.3a compares the lateral deflection distributions over the

column corresponding to two key deformation states, namely, the peak axial load

and the ultimate lateral deflection, predicted by both models. Figs 9.3b and 9.3c

respectively compare the distributions of the curvature and the moment for the

same two deformation states. The predictions of the simple model at the peak

axial load are shown as thick solid lines while the corresponding predictions from

the rigorous model are shown as thin solid lines; those at the ultimate lateral

deflection are shown as dotted lines. It can be seen that at the peak axial load, the

distributions of lateral deflection, moment and curvature predicted by the two

models are all in close agreement, which indicates that the half-sine wave

provides a good approximation to the deflected shape of the column. This

agreement explains why the two models lead to closely similar predictions for the

ascending branch. It is worth pointing out that the half-sine wave assumption

implies that the curvature is zero at either column end, which is not realistic unless

the initial load eccentricity is zero. After the peak axial load is achieved, a plastic

hinge gradually forms at the mid-height region of the column, as predicted by the

rigorous model (Fig. 9.3b). Therefore, the distributions of the curvature and the

lateral deflection predicted by the two models are no longer similar; the localized

plastic hinge can never be captured using the half-sine wave assumption. As a

result, the simple model predicts a much longer descending branch. Despite this

271

inaccuracy, the peak axial loads predicted by the simple model are in close

agreement with those predicted by the rigorous model. Both sets of theoretical

peak axial loads are close to the experimental results. Therefore, the simple model

is sufficiently accurate as a basis for the development of design equations for the

axial load capacities of slender FRP-confined RC columns.

9.3 LIMITS ON THE USE OF FRP

In the design of FRP-confined RC columns, it is important to ensure that the FRP

material is used in a safe and economical manner. Existing tests on standard FRP-

confined concrete cylinders showed that the concrete strength can be increased by

over three times provided the FRP confinement is strong enough (see Chapter 3).

The failure of strongly-confined specimens can be explosive and is undesirable in

practice. As columns become more slender, the effectiveness of FRP confinement

is reduced (see Chapter 7) and the use of FRP may eventually become

uneconomical. Another concern with FRP-confined RC columns is that the lateral

deflections may exceed an acceptable limit in design. It is thus advisable to

impose certain limits on the use of FRP to ensure that the FRP is used in a safe

and economical manner. To this end, a parametric study was conducted using the

simple theoretical model presented in the preceding section to examine how the

effectiveness of FRP confinement is affected by various parameters.

The studywas carried out on the same reference column that has been used in

Chapter 8 except that the reference column used herein is subjected to equal end

eccentricities , as shown in Fig. 9.1. The parameters studied herein are similar to

those studied in Chapter 8. The end eccentricity ratio is no more a variable but is

fixed at unity. A new and very important parameter is the slenderness ratio

e

λ .

The values used for these parameters are summarized in Table 9.1. These values

are similar to those used in Chapters 5 and 8.

The combinations of these parameters led to about 8,000 cases. The slenderness

ratio goes up to 50 as a preliminary study showed that beyond this slenderness the

confining effect of FRP is very limited. The remaining values used for all

parameters are the same as those used in Chapter 5.

272

Figs 9.4 to 9.9 show the results of the parametric study. In Figs 9.4 to 9.9, the

vertical axis is the axial load capacity enhancement ratio ,u u refN N , where

is the axial load capacity of an RC column before it is confined with FRP. The

horizontal axis is one of the six parameters considered. A series of curves is

provided in each of these figures, showing the variation of another parameter, and

the values of the other four parameters used to generate these numerical results are

also given in each figure.

,u refN

Fig. 9.4 shows the effect of the strength enhancement ratio. This figure is for a

CFRP-confined column with a small amount of steel reinforcement and an

intermediate end eccentricity. It is obvious that the axial load capacity of this

column increases as the amount of FRP confinement increases, but the

effectiveness of FRP confinement drops rapidly as the column becomes slender.

The strain ratio also appears to be an important parameter. It can be seen in Fig.

9.5 that with the same level of concrete strength enhancement, the enhancement of

the axial load capacity is smaller when the confining jacket has a larger strain

capacity. This can be easily understood as an FRP jacket with a larger strain

capacity results in larger lateral deflections. Figs 9.6 and 9.7 respectively show

the effects of the end eccentricity and the slenderness of the column. The

confinement effect is most pronounced in columns with a small slenderness

subjected to a small end eccentricity. It should be noted that the increase in the

axial load capacity may be very limited (in some cases less than 5%) even under

very heavy confinement when the column is slender and is subjected to a large

end eccentricity. In such cases, the use of FRP may not be economical. Compared

with the above four parameters, the depth ratio and the steel reinforcement ratio

are less important. The effects of these two parameters on the axial load capacity

enhancement ratio may be neglected in practice, as suggested by Figs 9.8 and 9.9.

An examination of the numerical results showed that under very heavy

confinement that results in , the lateral deflections at failure exceed the

commonly accepted limit of

' 2cc cof f= '

50l in design (Cranston 1972) in some cases.

273

Therefore, it is suggested that the maximum allowable amount of FRP

confinement be limited to

'

' 1.75cc

co

ff

≤ (9.5)

The present numerical results indicate that with the above limit, the maximum

possible increase in the axial load capacity of an RC column is approximately

50%. For cases where the slenderness and the end eccentricity of the column are

within a reasonable range (e.g. 20λ ≤ and 0.3e D≤ for columns confined with

an FRP jacket possessing a strain capacity of 0.75%), this limit generally ensures

that increases in the axial load capacity of more than 15% can be realized.

Another important conclusion that can be drawn from the parametric study is that

the confinement effect may become very limited when a column is very slender. It

is thus advisable to impose a limit on the slenderness beyond which the use of

FRP should not be recommended. It is suggested herein that a minimum increase

in the axial load capacity of 10% should be achieved as a result of FRP

confinement. The following approximate equation is proposed based on this

criterion

max 50 3 ελ ρ= − (9.6)

where maxλ is the upper bound of the slenderness ratio of a column whose axial

load capacity still benefit significantly from the provision of an FRP jacket. Eq.

9.6 was derived for a reference column with a steel reinforcement ratio of 1%, a

section depth ratio 0.8d D = , and an end eccentricity ratio 0.2e D = . The

concrete strength of the reference column was increased by 75% as a result of

FRP confinement. It should be noted that when the eccentricity ratio is larger than

0.2, Eq. 9.6 may lead to an increase in the axial load capacity smaller than 10%.

Some of the values listed in Table 9.1 do not satisfy the conditions set by Eqs 9.5

and 9.6; these values were thus removed to form a reduced set a parametric study

274

cases for use in later sections of this chapter. The values of the parameters in the

reduced set are listed in Table 9.2. It should be noted that the values of maxλ

defined by Eq. 9.6 for the three different strain ratios considered are 47, 38.75,

and 27.5 respectively. Values of 50, 40, and 30 were however used instead for

simplicity.

9.4 DESIGN METHOD

9.4.1 Review of Current Design Methods for RC Columns

The current design approach adopted in various design codes for RC columns (e.g.

ENV-1992-1-1 1992; BS-8110 1997; GB-50010 2002; ACI-318 2005)

approximate the second-order moments by an amplification of the first-order

moment so that the failure load can be related to the strength of the critical section.

In other words, the current design approach transforms the design of a slender

column into the design of a section with an equivalent eccentricity, which consists

of the initial end eccentricity of the slender column and a an additional

eccentricity equal to the nominal lateral displacement of the critical section nomf .

The concept of this additional eccentricity is illustrated in Fig. 9.10. Fig. 9.10 is

similar to Fig. 8.1 and it shows three loading paths, OA , and , for a

standard hinged column with three different values of column slenderness but a

fixed initial end eccentricity. Graphically, the design approach seeks the straight

loading paths OB and shown as dashed lines in Fig. 9.10 to replace the

original curved loading paths OB and shown as solid lines for material

failure and stability failure respectively. It is obvious that for material failure,

OB ODE

OC

ODE

nomf

is the real lateral displacement of the critical section at failure. However, for

stability failure, nomf is a fictitious lateral displacement. This point must be borne

in mind and is further discussed in a later section. It is now clear that the key

element of the current design approach is to find nomf .

There are two main approaches in the current design codes, namely, the moment

magnifier method and the nominal curvature method. The moment magnifier

275

method has been adopted by ACI-318 (2005) and many of its previous versions,

among others. This approach originated from the elastic analysis of columns,

where the lateral deflections can be exactly determined provided the section

flexural stiffness is known. When this approach is used for the design of RC

columns, the key is to find the equivalent section flexural stiffness that accounts

for the effect of the nonlinearities in the constituent materials. By contrast, the

nominal curvature method was originally proposed by Aas-Jakobsen and Aas-

Jakobsen (1968) has been adopted by ENV-1992-1-1 (1992), BS-8110 (1997),

and GB-50010 (2002), among others. This approach relates the lateral deflection

to the curvature through the relationship defined by Eq. 9.3. It is obvious that the

key to this approach is to determine the nominal curvature nomφ corresponding to

nomf . This chapter follows the framework of the nominal curvature method to

develop a design procedure that is consistent with GB-50010 (2002). Therefore,

the moment magnifier method is not further pursued.

9.4.2 Nominal Curvature

The nominal curvature and the nominal lateral displacement can be related by the

following equation

2

2nom nomlf φπ

= (9.7)

As explained earlier, the nominal curvature sought in the nominal curvature

method for material failure is the real curvature of the critical section at failure

failφ . However, the nominal curvature for stability failure needs some further

explanation, which can be achieved by making use of the moment-curvature

diagram shown in Fig. 9.11. Fig. 9.11 shows the moment-curvature curve of the

critical section when the column reaches its axial load capacity . This curve

shows how the internal moment varies with the curvature of the critical section

under this particular axial load. The curvature at the end of this curve,

uN

secφ , is the

maximum curvature that the critical section can sustain under this particular axial

load. The inclined straight line represents how the external moment varies as the

276

curvature at the critical section increases. This inclined line can be mathematically

described by Eq. 9.4. This line has a slope of 2 2uN el π and intersects the vertical

axis at a value of . At the point where the inclined line meets the moment-

curvature curve, the external moment is equilibrated by the internal moment. It is

obvious that for material failure (Fig. 9.11a), the inclined line must intersect the

moment-curvature curve at the end of the moment-curvature curve

(

uN e

secnom failφ φ φ= = ). However, when stability failure occurs, the above three

curvatures have different values, as illustrated in Fig. 9.11b. When stability failure

occurs, the inclined line must meet the moment-curvature curve at the point of

failure for moment equilibrium. Therefore, the inclined line must be a tangent to

the moment-curvature curve at the point of failure, where the curvature is failφ .

The nominal curvature can be found as the curvature at the intersection point of

the inclined line and the horizontal line since this point corresponds to the point

on the section interaction curve at . Obviously, the nominal curvature always

has a value larger than

uN

failφ but smaller than secφ .

The failure modes of an RC section can be classified into three categories: 1)

balanced failure; 2) compression failure and 3) tension failure. At balanced failure,

the extreme compression fiber of concrete reaches the ultimate compressive strain

of concrete when the most highly-tensioned longitudinal steel bar(s) on the

opposite side of the section reaches its tensile yield strain. The particular axial

load corresponding to balanced failure is denoted by . When compression

failure occurs, the concrete reaches its ultimate compressive strain before the steel

reinforcement yields. Axial loads corresponding to compression failure are always

larger than . Tension failure is the opposite to compression failure and axial

loads corresponding to tension failure are always smaller than . According to

the definitions given above, the curvature at balanced failure can be easily found

to be

balN

balN

balN

2 cu ybal D d

ε εφ

+=

+ (9.8a)

277

balφ is used as the basis to evaluate nomφ in the nominal curvature method. For

material failure, a factor 1ξ is used to reflect the effect of axial load when the axial

load is other than . For stability failure, an additional factor balN 2ξ needs to be

introduced to explain the difference between nomφ and secφ . In summary, nomφ can

be related to balφ using the following equation

1 2nom balφ ξ ξ φ= (9.8b)

It can be concluded from the above discussions that , balN 1ξ and 2ξ are the

essential elements in the nominal curvature method and they are discussed in

detail in the following sub-sections.

9.4.3 Axial Load at Balanced Failure

GB-50010 (2002) specifies the following equation as an estimate of the axial load

at balanced failure for RC sections

'0.5bal coN f= A (9.9)

A careful study showed that Eq. 9.9 is no longer reasonable when used to predict

the axial load at balanced failure of FRP-confined RC sections. Fig. 9.12 shows a

series of interaction curves for an RC section before and after being confined with

various amounts of FRP. Balanced failures at various confinement levels are

indicated by different markers in Fig. 9.12. It is interesting to note that these

markers gradually deviate from the maximum moment point on the interaction

curve as the confinement level increases. The same observation has also been

reported by Cheng et al. (2002). The axial load at balanced failure of an FRP-

confined RC section depends on the following four parameters sρ , d D , ' 'cc cof f

and ερ . To develop an approximate expression for , a simple parametric

study was conducted using the reduced set of parameters given in Table 9.2.

balN

278

Based on the numerical results, the following simple equation is proposed for

FRP-confined RC sections

'0.8bal ccN f= A (9.10)

Fig. 9.13 compares the predictions of Eq. 9.10 with the exact numerical results for

FRP-confined RC sections. It can be seen that Eq. 9.10 only provides a rough

estimation but it will be seen that it is sufficiently accurate for design use. The

predictions of Eq. 9.9 for RC sections are also shown in Fig. 9.13.

9.4.4 Factors 1ξ and 2ξ

GB-50010 (2002) specifies the following equation for 1ξ

1 1bal

u

NN

ξ = ≤ (9.11)

while BS-8110 (1997) and ENV-1992-1-1 (1992) employ the following equation

1 1uo u

uo bal

N NN N

ξ −= ≤

− (9.12)

where is the axial load capacity of an RC section when it is concentrically

compressed. Fig. 9.14 compares the predictions of Eqs 9.11 and 9.12 with the

exact results from section analysis. Fig 9.14a is for an RC section while Fig. 9.14b

is for an FRP-confined RC section with

uoN

' ' 1.5cc cof f = . It should be noted that the

value of used in the comparison was the exact value to eliminate the

discrepancy introduced by the different approximate equations employed in these

codes. The axial loads and curvatures in Fig. 9.14 were normalized using and

balN

balN

balφ respectively. The small circle in Fig. 9.14 represents balanced failure while

the small square represents the case where the section is subjected to the minimum

eccentricity. The definition of the minimum eccentricity in existing design codes

279

for RC structures has been discussed in Chapter 5. The minimum eccentricity was

taken as min 0.05e D= for simplicity in the present study. It can be seen that the

curvature decreases as the axial load increases and in the case of concentric

compression the curvature is zero. Although Eq. 9.12 satisfies the boundary

condition at concentric compression, the overall performance of Eq. 9.11 is better

since the small range beyond the small square is excluded by the minimum

eccentricity and thus does not need to be considered in design. When the FRP

confinement is provided, the range corresponding to compression failure becomes

smaller. It is interesting to note that although the exact results indicate that the

curvatures at tension failure are always larger value balφ , it is limited to balφ in all

the codes mentioned above. This limit on 1ξ is not explained in Aas-Jakobsen and

Aas-Jakobsen (1968) in which the nominal curvature method was originally

proposed. To the best knowledge of the author, this issue has not been clearly

explained so far. An attempt is made below to explain this issue.

When assessing the role played by 1ξ in the nominal curvature method, it is

advisable to combine 1ξ with 2ξ to see the overall effect of these two factors. GB-

50010 (2002) specifies the following equation for 2ξ

2 1.15 0.01 1lD

ξ = − ≤ (9.13)

2ξ is limited to 1 because when a column is relatively short it fails in the mode of

material failure. It has already been explained that for material failure, only 1ξ

needs to be considered and 2ξ always remains unity (Fig. 9.11a). Only when the

column is slender enough to cause stability failure does 2ξ need to be considered.

In such a case, it always has a value smaller than 1, as clearly illustrated in Fig.

9.11b.

A careful study revealed that if the exact values of 1ξ are used, the development

of an expression for 2ξ becomes difficult. This is because in such cases the value

280

of 2ξ depends strongly on the end eccentricity besides the slenderness of the

column. However, if 1ξ is limited to unity, the dependence of 2ξ on the end

eccentricity is much less so that 2ξ may be taken as a function of the slenderness

only. It should be noted that although limiting 1ξ to 1 has the advantage of

simplicity, it can make the nominal curvature method un-conservative in

predicting the axial load capacity of columns subjected to material failure and

with an axial load capacity smaller than . Such a case can happen in a short

column with a large end eccentricity. Such a column has a curvature larger than

balN

balφ but its curvature is forced to be balφ in the above approach, which gives rise to

the un-conservativeness. Nevertheless, as the second order effects in such

columns are limited, the un-conservativeness will be seen to be within a

reasonable range. It should be noted that the above explanation is completely

based on the theoretical results but has not been verified using any test data.

As a result of the above discussions, it is suggested that for FRP-confined RC

columns, the form of Eq. 9.11 be retained but be estimated using Eq. 9.10,

leading to Eq. 9.14a

balN

'

10.8 1bal cc

u u

N f AN N

ξ = = ≤ (9.14a)

Based on the values of 1ξ given by Eq. 9.14a, Eq. 9.14b is proposed for 2ξ for use

in the design of slender FRP-confined circular RC columns

2 (1.15 0.06 ) (0.01 0.012 ) 1lDεξ ρ ρ= + − + ≤ε (9.14b)

It should be noted that Eq. 9.14b ignores the effects of a number of factors, such

as the end eccentricity for simplicity in design. It is shown in a later section of this

chapter that Eq. 9.14b is sufficiently accurate for design use. Eq. 9.14b also

reduces to Eq. 9.13 when no FRP confinement is provided.

281

9.4.5 Proposed Design Equations

With , balN 1ξ and 2ξ determined, the full set of design equations is given below.

( )'1

sin 212u cc c tN f A fπθθα θ θπθ

⎛ ⎞= − + −⎜ ⎟⎝ ⎠

y sA (9.15a)

2 3'

1 2 12

sin sin2 sin3

cu bal cc y s

lN e f AR f A R tπθ ππθξ ξ φ απ π

⎛ ⎞ ++ = +⎜ ⎟

⎝ ⎠

θπ

(9.15b)

' '1 1.17 0.2 cc cof fα = − (9.15c)

0 1.25 0.125 1cθ θ≤ = − ≤ (9.15d)

0 1.125 1.5 1tθ θ≤ = − ≤ (9.15e)

On the right hand side of Eqs 9.15a and 9.15b are the approximate expressions for

the section interaction curve that have been presented in Chapter 5.

When the axial load and the associated initial end eccentricity are known, the

design of the FRP jacket should follow the steps listed below

1) Select the type of FRP and check to see if the slenderness of the column

satisfies Eq. 9.6;

2) Assume a jacket thickness and calculate and 'ccf cuε ;

3) Determine the value of θ through a trial-and-error process until Eqs 9.15a

and 9.15b are both satisfied;

4) Check to see if the axial load capacity calculated from Eq. 9.15a is larger than

the applied axial load;

5) If step 4) is satisfied, the FRP confinement assumed in step 2) is strong

enough to resist the applied axial load; otherwise, increase the jacket

thickness and go through steps 2) to 4) again until step 4) is satisfied;

282

6) If step 4) still cannot be satisfied when the confinement has already been

increased to a very high level that exceeds the limit given in Eq. 9.5, it

indicates that the use of FRP in this case is either inefficient or uneconomical

and other means of strengthening should be used instead (e.g. increase the

cross-sectional area).

It should be noted that in the above procedure, it is assumed that all the geometric

and material properties of the original RC section are known, as is generally the

case in the retrofit of existing RC columns.

Fig. 9.15 compares a series of interaction curves predicted using the proposed

design approach with those produced using the simple theoretical model. The

interaction curves are for a series of CFRP-confined RC columns of the same

section but with a range of slenderness ratios from 10λ = to 40λ = at an

interaval of 10. It should be noted that 40λ = is slighter larger than the maximum

allowable slenderness ratio ( 38.75λ = ) defined by Eq. 9.6. The interaction curves

are cut off by a straight line that represents the minimum eccentricity. The column

was assumed to have a 600 mm diameter. The compressive strength of concrete

and the yield strength of the steel reinforcement were assumed to be 20.1 MPa and

335 MPa respectively. The other parameters used to produce the interaction

curves in Fig. 9.15 are all given in Fig. 9.15. The approximate curve for 10λ =

from the design equation is in excellent agreement with the exact curve as the

second order effect is very limited at this slenderness value. The approximate

curve for 20λ = is un-conservative for a certain range of end eccentricities. This

overestimation mainly stems from the fact that at this slenderness material failure

is still the predominant failure mode and this range of end eccentricities

corresponds to tensile failure. As a result, the exact nominal curvature found using

the simple theoretical model must be larger than balφ but it is forced to be equal to

balφ in the proposed design approach. However, the overestimation is not

significant and is comparable to that found in the current design approach adopted

by GB-50010 (2002) for RC columns.

283

Fig. 9.16 shows the overall performance of the proposed design approach. The

axial load capacities calculated using the simple theoretical model and the

proposed design approach are normalized by . Fig. 9.16 includes the

numerical results of all FRP-confined columns considered in the reduced set of

parametric cases listed in Table 9.2. It can be seen that the majority of cases fall

within the error lines. The maximum overestimation found among the

cases studied is 12.3%.

uoN

10%±

BS-8110 (1997) specifies that for simplicity 1 2 1ξ ξ= = can be used in the design

of RC columns with some sacrifice of accuracy; this simplification is always

conservative. It should be noted that the simplification of 1 1ξ = makes the design

procedure much more straightforward because otherwise 1ξ has to be evaluated

through a trial-and-error process ( 1ξ is a function of as defined in Eqs 9.12,

9.13 and 9.14a). If

uN

1 2 1ξ ξ= = is adopted in the proposed design approach, the

results for the same cases shown in Fig. 9.16 are shown in Fig. 9.17. It can be seen

that with this simplification, the predictions become more conservative but the

majority of cases still fall within the 10%± error lines. The maximum

underestimation found among the cases studied is 18.2%.

When the proposed equations are applied to design FRP jackets for existing RC

columns, it is possible that the axial load capacity for the strengthened column

calculated using the proposed design equations is smaller than that of the original

RC column calculated using the design equations given in GB-50010 (2002). This

situation can occur when the slenderness of the column approaches maxλ because

the increase in the axial load capacity is then very limited and the design

equations in GB-50010 (2002) are slightly un-conservative. Therefore, the

designer must be careful in the design of such columns. The use of more advanced

methods such as the theoretical model presented in this chapter is recommended

for such cases.

Finally, the proposed design equations are used to predict the axial load capacities

of FRP-confined RC columns reported in existing studies (Tao et al. 2004;

284

Fitzwilliam and Bisby 2006; Ranger and Bisby 2007). These column tests have

been described in detail in Chapter 7. The FRP hoop rupture strains used in the

design equations when making predictions for these column tests in these three

studies are 1.32%, 1.16%, and 1.15% respectively, as found from the

corresponding ancillary cylinder tests. It should be noted that some of the columns

do not satisfy the conditions set by Eqs 9.5 and 9.6. The columns of Tao et al.

(2004) and those confined with a 2-ply CFRP jacket in Fitzwilliam and Bisby

(2006) and Ranger and Bisby (2007) have a ' 'cc cof f ratio slightly larger than 1.75.

Besides, the columns of Tao et al. (2004) had a slenderness ratio of 33.6 for series

C1 and 81.6 for series C2 which exceed the maximum allowable slenderness ratio

(30.2) defined by Eq. 9.6. In the present set of comparisons, series C2 in Tao et al.

(2004) was excluded while the remaining columns reported in these three studies

were retained. An additional eccentricity was considered when using the design

equations to predict the axial load capacities, as required by GB-50010 (2002).

GB-50010 (2002) specifies that the additional eccentricity should be taken as the

larger of 20 mm and 1/30 of the diameter for circular columns. Obviously, this

provision is for realistically-sized columns; the use of a 20 mm additional

eccentricity is thus unreasonable for the small-scale columns under consideration.

As a result, an additional eccentricity of 7.5 mm was used for all the cases under

consideration. The predicted axial load capacities using the proposed design

equations and the corresponding experimental values are compared in Fig. 9.18. It

can be seen in Fig. 9.18 that the predictions of the proposed design equations are

reasonably close to the experimental values and are conservative in most cases.

Some slight overestimation can be observed for column C1-1R of Tao et al. (2004)

and columns C-30 and C-40 of Ranger and Bisby (2007). These observations are

consistent with those found in the predictions by the rigorous theoretical model

for the same columns (see Chapter 7).

9.5 CONCLUSIONS

This chapter has been concerned with the development of design equations for

slender FRP-confined circular RC columns. To this end, a simpler theoretical

285

model than the one presented in Chapter 7 was developed for modeling slender

FRP-confined RC columns. This model is exclusively for hinged columns with

equal end eccentricities and assumes the deflected shape of such columns to have

the shape of a half-sine wave. Subsequently, two practical limits on the use of

FRP for the strengthening of RC columns were proposed to ensure a safe and

economical strengthening scheme based on the numerical results produced by the

simple theoretical model. Finally, design equations were developed using the

nominal curvature method. The design equations provide close agreement with the

theoretical results. The results and discussions presented in this chapter allow the

following conclusions to be drawn:

1) The load-deflection curves predicted using the simple theoretical model

presented in this chapter and those predicted using the rigorous theoretical

model presented in Chapter 7 have a very similar ascending branch and a very

similar peak axial load. The simple theoretical model however predicts a

much longer descending branch of the load-deflection curves than the

rigorous model since the half-sine wave assumption fails to capture the

formation of the plastic hinge at the mid-height region of a column. Despite

this deficiency, the simple theoretical model is sufficiently accurate as a basis

for the development of design equations for the axial load capacities of

slender FRP-confined RC columns.

2) The simple theoretical model revealed the same phenomenon as has been

observed in existing tests: the effectiveness of FRP confinement decreases as

the columns become more slender. The theoretical model also indicated that

strong confinement may result in excessive lateral deflection that is not

acceptable in design. As a result, practical limits need to be imposed on the

level of confinement and the column slenderness to ensure a safe and

economical strengthening scheme.

3) The nominal curvature method was fully discussed, including some original

insight into a subtle issue in this method which has not been properly

explained to the best knowledge of the author.

286

4) The proposed design equations have a simple form that is familiar to

engineers and provide predictions that are in close agreement with the

theoretical results. A simplified version of these design equations with a small

sacrifice in accuracy was also presented, which is easier for design use and

leads to more conservative predictions than the original version.

5) The design equations were shown to provide reasonable and generally

conservative predictions for FRP-confined RC columns reported in existing

studies.

287

9.6 REFERENCES

Aas-Jakobsen, A. and Aas-Jakobsen, K. (1968). “Buckling of slender columns”, Bulletin d’ Information, Comite Europeen du Beton, No. 69, 201-270.

ACI-318 (2005). Building Code Requirements for Structural Concrete and Commentary, ACI Committee 318, American Concrete Institue.

Bazant, Z.P., Cedolin, L. and Tabbara, M.R. (1991). “New method of analysis for slender columns”, ACI Structural Journal, 88(4), 391-401.









Mirmiran, A., Shahawy, M. and Beitleman, T. (2001). “Slenderness limit for hybrid FRP-concrete columns”, Journal of Composites for Construction, 5(1), 26-34.


Tao, Z., Teng, J.G., Han, L.H. and Lam, L. (2004). “Experimental behaviour of FRP-confined slender RC columns under eccentric loading”, Proceedings, 2nd

288

International Conference on Advanced Polymer Composites for Structural Applications in Construction, University of Surrey, Guildford, UK, 203-212.


289

Table 9.1 Entire set of parametric study cases

Parameter Values λ 10, 20, 30, 40, 50

e D 0.05,0.1,0.15,0.2,0.25,0.3,0.4,0.6,0.8

sρ 1%, 2%, 3%, 4%, 5% d D 0.7, 0.8, 0.9 ' '

cc cof f 1.25, 1.5, 1.75, 2

,h rup coε ε 1, 3.75, 7.5

Table 9.2 Reduced set of parametric study cases

Parameter Values 10, 20, 30, 40, 50 for , 1h rup coε ε = 10, 20, 30, 40 for , 3.75h rup coε ε = λ

10, 20, 30 for , 7.5h rup coε ε = e D 0.05,0.1,0.15,0.2,0.25,0.3,0.4,0.6,0.8

sρ 1%, 2%, 3%, 4%, 5% d D 0.7, 0.8, 0.9 ' '

cc cof f 1.25, 1.5, 1.75

290

Ddl

N

N

e

e

f

x

fmid

Fig. 9.1 Schematic of the simple theoretical model

291

0 20 40 60 80 1000

100

200

300

400

500

600

700

800

900


Axi

al L

oad

N (k

N)

C1-1R (Test)C1-1R (Rigorous Model)C1-1R (Simple Model)C1-2R (Test)C1-2R (Rigorous Model)C1-2R (Simple Model)

(a) Columns C1-1R and C1-2R

0 20 40 60 80 100 1200

20

40

60

80

100

120

140


Axi

al L

oad

N (k

N)


(b) Columns C1-3R and C1-4R

292

0 50 100 150 2000

100

200

300

400

500

600


Axi

al L

oad

N (k

N)


(c) Columns C2-1R and C2-2R

0 50 100 150 200 2500

10

20

30

40

50

60

70

80


Axi

al L

oad

N (k

N)


(d) Columns C2-3R and C2-4R

Fig. 9.2 Comparison with Tao’s (2004) tests on FRP-confined circular RC

columns

293

0 100 200 300 400 500 600 700 8000

500

1000

1500

2000

2500

3000

Lateral Deflection (mm)

Col

umn

Hei

ght

(mm

)

Peak axial load

Ultimate lateral deflection

(a) Distribution of lateral deflection

0 1 2 3 4 5 6 7 8

x 10-4

0

500

1000

1500

2000

2500

3000

Curvature (1/mm)

Col

umn

Hei

ght

(mm

)


Peak axial load

(b) Distribution of curvature

294

0 2 4 6 8 10 12 14

x 106

0

500

1000

1500

2000

2500

3000

Moment (N⋅mm)

Col

umn

Hei

ght

(mm

)


Peak axial load

(c) Distribution of moment

Fig. 9.3 Illustration of differences between the two theoretical models

295

1 1.2 1.4 1.6 1.8 21

1.1

1.2

1.3

1.4

1.5

1.6

d/D=0.8ρs=1%

e/D=0.2εh,rup/εco=3.75


Axi

al L

oad

Cap

acity

Enh

ance

men

t Rat

ioN

u/Nu,

ref

λ = 10λ = 20λ = 30λ = 40λ = 50

Fig. 9.4 Effect of strength enhancement ratio

0 2 4 6 81

1.1

1.2

1.3

1.4

1.5

1.6d/D=0.8ρs=1%

e/D=0.05f′cc/f′co=1.75


Axi

al L

oad

Cap

acity

Enh

ance

men

t Rat

ioN

u/Nu,

ref

λ = 10λ = 20λ = 30λ = 40λ = 50

Fig. 9.5 Effect of strain ratio

296

0 0.2 0.4 0.6 0.81

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

d/D=0.8

ρs=1%

λ=20

εh,rup/εco=3.75

Normalized Eccentricity e/D

Axi

al L

oad

Cap

acity

Enh

ance

men

t Rat

ioN

u/Nu,

ref

f′cc/f′co=1.25

f′cc/f′co=1.5

f′cc/f′co=1.75

f′cc/f′co=2

Fig. 9.6 Effect of end eccentricity

10 20 30 40 501

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

d/D=0.8

ρs=1%

f′cc/f′co=1.75

εh,rup/εco=3.75

Slenderness Ratio λ

Axi

al L

oad

Cap

acity

Enh

ance

men

t Rat

ioN

u/Nu,

ref

e/D=0.1e/D=0.2e/D=0.3e/D=0.4e/D=0.6

Fig. 9.7 Effect of slenderness ratio

297

0.01 0.02 0.03 0.04 0.050.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

d/D=0.8

λ=20e/D=0.2εh,rup/εco=3.75

Steel Reinforcement Ratio λ

Axi

al L

oad

Cap

acity

Enh

ance

men

t Rat

ioN

u/Nu,

ref

f′cc/f′co=1.25

f′cc/f′co=1.5

f′cc/f′co=1.75

f′cc/f′co=2

Fig. 9.8 Effect of steel reinforcement ratio

0.7 0.75 0.8 0.85 0.91

1.05

1.1

1.15

1.2

ρs=1%

λ=30

e/D=0.2

εh,rup/εco=3.75

Depth Ratio d/D

Axi

al L

oad

Cap

acity

Enh

ance

men

t Rat

ioN

u/Nu,

ref

f′cc/f′co=1.25

f′cc/f′co=1.5

f′cc/f′co=1.75

f′cc/f′co=2

Fig. 9.9 Effect of depth ratio

298

Moment M

Axi

al L

oad

N

O

AB

CD

E

e = Const

e

1

NBe

NBfnom

NCe NCfnom

Fig. 9.10 Concept of the fictitious lateral displacement

299

Curvature φ

Mom

ent

Mφnom = φfail = φsec = ξ1φbal

(a) Material failure

Curvature φ

Mom

ent

M φsec =ξ1φbalφnom = ξ2φsec

φfail

N = Const

(b) Stability failure

Fig. 9.11 Determination of the nominal curvature

300

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Normalized Moment Mu / Muo

Nor

mal

ized

Axi

al L

oad

Nu

/ Nuo

Unconfinedf′cc/f′co=1.25

f′cc/f′co=1.5

f′cc/f′co=1.75

Fig. 9.12 Axial loads at balanced failure

0 2000 4000 6000 8000 100000

2000

4000

6000

8000

10000

Axial Load at Balanced FailureNbal - Analysis (kN)

Axi

al L

oad

at B

alan

ced

Failu

reN

bal -

App

roxi

mat

ion

(kN

)

Unconfinedf′cc/f′co=1.25

f′cc/f′co=1.5

f′cc/f′co=1.75

Fig. 9.13 Performance of Eqs 9.9 and 9.10

301

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

Normalized Axial Load Nu/Nbal

Nor

mal

ized

Cur

vatu

re φ u

/ φba

l

RC Section Section AnalysisGB EquationBS Equation

(a) RC section

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.5

1

1.5

2

2.5

3

3.5

4

Normalized Axial Load Nu/Nbal

Nor

mal

ized

Cur

vatu

re φ u

/ φba

l

FRP-confined RC Sectionf′cc/f′co=1.5

Section AnalysisGB EquationBS Equation

(b) FRP-confined RC section

Fig. 9.14 Factor 1ξ

302

0 100 200 300 400 500 6000

1000

2000

3000

4000

5000

6000

7000

8000

9000d/D=0.8ρs=1%f′cc/f′co=1.5εh,rup/εco=3.75

Moment M (kN⋅m)

Axi

al L

oad

N (k

N)

Accurate AnalysisDesign Equations

λ=10

λ=20

λ=30

λ=40e=0.05D

Fig. 9.15 Typical interaction curves

303

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

Normalized Axial Load CapacityNu/Nuo - Accurate Analysis

Nor

mal

ized

Axi

al L

oad

Cap

acity

Nu/N

uo -

Des

ign

Equ

atio

ns

10%

-10%

Fig. 9.16 Performance of the proposed design equations

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

Normalized Axial Load CapacityNu/Nuo - Accurate Analysis

Nor

mal

ized

Axi

al L

oad

Cap

acity

Nu/N

uo -

Des

ign

Equ

atio

ns

10%

-10%

Fig. 9.17 Performance of the proposed design equations with 121 == ξξ

304

0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

1000

Axial Load Capacity Nu (kN) - Test

Axi

al L

oad

Cap

acity

Nu

(kN

)- D

esig

n E

quat

ions

Ranger and Bisby (2007)Fitzwilliam and Bisby (2006)Tao et al. (2004)

C40C30

C1-1R

Fig. 9.18 Predictions of the proposed design equations against test results

305

CHAPTER 10

CONCLUSIONS

10.1 INTRODUCTION

This thesis has presented a systematic study which aims to develop a rational

design procedure for FRP-confined RC columns. Such a design procedure is not

available in current design guidelines for FRP-strengthened RC structures and is

thus urgently needed for the confident and effective use of FRP wraps to enhance

the axial load capacity of RC columns.

A series of axial compression tests on FRP-confined concrete cylinders was first

presented in the thesis to gain a good understanding of the stress-strain behavior

of FRP-confined concrete, which is fundamental and essential to the analysis and

design of FRP-confined RC columns. Stress-strain models for FRP-confined

concrete of different levels of sophistication and for different purposes were next

developed as a prerequisite for the analysis of FRP-confined RC columns.

Subsequently, a simple but accurate stress-strain model for FRP-confined concrete

was incorporated in a conventional section analysis procedure to develop design

equations for short FRP-confined RC columns with a negligible slenderness effect.

Finally, two theoretical models of different levels of sophistication were

developed to deal with the slenderness effect in slender FRP-confined RC

columns. The rigorous theoretical model was used to develop an expression to

classify columns into short columns and slender columns while the simple

theoretical model was used to develop design equations for slender columns. The

proposed design procedure includes a set of design equations for short columns, a

simple expression to separate short columns from slender columns, and a set of

306

design equations for slender columns.

It should be noted that the present study is limited to circular columns. Therefore,

the conclusions given in this chapter is for circular columns while further research

needs for rectangular columns are highlighted towards the end of this chapter.

10.2 BEHAVIOR OF FRP-CONFINED CONCRETE

A large number of FRP-confined concrete cylinders were tested under

standardized conditions and the test results were reported in Chapters 3 and 4.

These tests confirmed the following observations which have also been made by

previous researchers:

1) The confining action of FRP jackets is passive in nature. The lateral dilation

of concrete results in a continuously increasing lateral confining pressure

provided by the FRP jacket which in turn reduces the rate of the lateral dilation.

2) The hoop rupture strains of FRP measured in compression tests on

FRP-confined concrete are smaller than those obtained from material tensile

tests. Previous researchers have attributed this phenomenon to a number of

factors including the non-uniform deformation of concrete, the effect of jacket

curvature, local misalignment or waviness of fibers, as well as residual strains,

multi-axial stress states, and the existence of an overlapping zone in the jacket.

3) With a sufficient amount of confinement, the stress-strain curves of

FRP-confined concrete feature an ascending bi-linear shape with both the

compressive strength and the ultimate axial strain of concrete significantly

enhanced; the stress-strain curves may exhibit a descending branch with little

strength enhancement when the confinement is weak.

4) The ultimate condition (the compressive strength and the ultimate axial strain)

of FRP-confined concrete depends on both the stiffness and the strain

capacity of the confining jacket. The ultimate condition of concrete subjected

to the same confining pressure at the rupture of the confining jacket can be

307

considerably different if the jacket stiffness is different.

10.3 MODELING OF FRP-CONFINED CONCRETE

Analysis-oriented stress-strain models and design-oriented models for

FRP-confined concrete were discussed in Chapters 3 and 4 respectively.

Analysis-oriented models generate the stress-strain curves in an

incremental-iterative manner by accounting for the interaction between the FRP

jacket and the concrete core while design oriented models are in closed-form and

are suitable for design use. The following conclusions can be drawn based on a

comprehensive assessment of analysis-oriented models which employs an active

confinement model as the base model:

1) The lateral-to-axial strain relationship, which reflects the unique dilation

properties of FRP-confined concrete, is central to models of this kind. A

successful model should accurately predict this relationship. Nevertheless,

provided the overall trend of this relationship is reasonably well described, the

axial stress-strain curve can be closely predicted, even if local inaccuracies

exist in the lateral-to-axial strain equation.

2) The definitions of the peak axial stress and the corresponding axial strain in the

active-confinement base model are also important to ensure the accuracy of an

analysis-oriented model for FRP-confined concrete.

3) The analysis-oriented model proposed in Chapter 3 represents an

improvement to the best-performance model identified by the assessment,

particularly for weakly-confined concrete.

The original Lam and Teng’s design-oriented stress-strain model was refined in

Chapter 4 on a combined experimental and theoretical basis. A simple stress-strain

equation and an accurate definition of the ultimate condition of FRP-confined

concrete are central to a successful model of this type. The proposed ultimate

condition equations represent an improvement to existing ones in that they relate

the ultimate condition to both the stiffness and the strain capacity of the confining

308

jacket rather than solely to the ultimate confining pressure. The model proposed in

Chapter 4 also naturally reduces to a stress-strain model for unconfined concrete

adopted in existing design codes for RC structures.

10.4 ANALYSIS AND DESIGN OF FRP-CONFINED RC COLUMNS

The proposed design procedure for FRP-confined RC columns follows the

conventional procedure for RC columns: a column needs to be classified to be a

short column or a slender column before applying the corresponding set of design

equations. The slenderness effect is ignored in short columns while it must be

accounted for in slender columns.

The design of short columns is simply a matter of constructing the section axial

load-bending moment interaction diagram. Section analysis incorporating the

refined Lam and Teng’s stress-strain model proposed in Chapter 4 was carried out

in Chapter 5. The section analysis served as a basis to develop design equations: the

contribution of the confined concrete to the load capacity of the section was

approximated by transforming the stress profile of concrete into an equivalent

stress block; the contribution of the longitudinal steel reinforcing bars to the load

capacity of the section was approximated by smearing the bars into an equivalent

steel ring. The proposed design equations are in a simple form that is familiar to

civil engineers and their performance was shown to be very good by a

comprehensive parametric study.

A rigorous theoretical model was developed in Chapter 7 to deal with the

slenderness effect in slender FRP-confined RC columns. This model was shown to

provide reasonably close predictions for existing tests. It was also shown that

FRP-confined RC columns are subjected to a more profound slenderness effect than

their RC counterparts because FRP confinement can substantially increase the axial

load capacity of an RC section but affects little the flexural rigidity of the section.

As a result, a short column may become a slender column when it is confined with

FRP. A simple expression was developed in Chapter 8 to define short

FRP-confined RC columns based on a comprehensive parametric study using the

rigorous theoretical model. This expression provides lower-bound predictions and

309

can reduce to an expression that is identical or similar to the slenderness limit

expressions for RC columns given in current design codes for RC structures when

no FRP confinement is provided.

To develop design equations for slender FRP-confined RC columns, the rationale

behind the current design approach (the effective length approach) for slender

columns was first explained through analysis of elastic columns with elastic end

restraints in Chapter 6. A simple theoretical model was then developed in Chapter

9 to analyze standard hinged FRP-confined RC columns. This analysis showed

that some practical limits need to be imposed on the use of FRP to ensure an

effective and safe strengthening scheme: 1) the strength enhancement of FRP

should be limited as strong confinement might lead to excessive lateral deflections;

and 2) the slenderness of columns should be limited as the confinement

effectiveness decreases as the column becomes more slender. Under these two

constraints, approximate design equations were developed using the nominal

curvature method. The key elements in the nominal curvature method were

carefully examined to accommodate necessary adjustments needed to reflect the

effect of FRP confinement. The proposed design equations were shown to be in

close agreement with the theoretical results and provide reasonably accurate yet

generally conservative predictions for existing tests.

10.5 FURTHER RESEARCH

This thesis has primarily been concerned with FRP-confined circular RC columns,

but the framework presented in this thesis can be readily extended to

FRP-confined rectangular RC columns when an accurate stress-strain model for

FRP-confined concrete in rectangular sections becomes available. The common

issues for circular columns and rectangular columns as well as some particular

issues for rectangular columns needing further research are highlighted in this

section.

The behavior of FRP-confined concrete in circular columns under concentric

compression has now been well understood, but much less is known about the

behavior of FRP-confined concrete in rectangular columns under concentric

310

compression due to the non-uniform nature of the confinement in such columns.

On the experimental side, more advanced measuring techniques are desirable to

capture the non-uniform stress distribution in such columns to gain a thorough

understanding of the confining mechanism. On the theoretical side, finite element

analysis incorporating a sophisticated constitutive model for concrete represents a

powerful tool to simulate the complex stress state in such concrete.

The theoretical work presented in this thesis has been based on the assumption

that stress-strain models derived from concentric compression tests are directly

applicable in the analysis of columns subject to combined bending and axial

compression. It was noted in Chapter 2 that this assumption had been adopted by

the majority of researchers in the analysis of circular columns. This assumption

was further examined in Chapter 5 using existing test results. Based on the limited

test results, it was deemed reasonable to adopt this assumption for research on

slender column behavior with an emphasis on the load-carrying capacity rather

than the ultimate deformation capacity. Chapter 9 further showed that the design

equations derived on the basis of this assumption are reasonably accurate yet

generally conservative, as demonstrated by existing experimental evidence.

However, more research is needed to clarify the possible effect of load

eccentricity on the stress-strain behavior of FRP-confined concrete in RC columns.

It should be noted that even if further research indicates that the effect of load

eccentricity should be accounted for when defining the ultimate condition (the

compressive strength and the ultimate axial strain) of FRP-confined concrete, the

proposed design equations in the present study will still be valid except that the

ultimate condition of the stress-strain curve needs to be refined. This is because in

the modified Lam and Teng stress-strain model, the effects of the stiffness and the

strain capacity of the confining jacket on the ultimate condition of FRP-confined

concrete in circular columns are already separated from each other. As a result,

when the ultimate condition of FRP-confined concrete is affected by the existence

of eccentricity, an equivalent confinement stiffness ratio and an equivalent strain

ratio which are a function of the eccentricity can be defined. These two equivalent

ratios can then be used in the proposed design procedure without any modification

of the design procedure. On the other hand, the effect of load eccentricity on the

stress-strain behavior of FRP-confined concrete in rectangular columns is much

311

vaguer due to: 1) the stress-strain behavior of FRP-confined concrete in

rectangular columns under concentric compression is not clear yet; 2) the

stress-strain behavior of FRP-confined concrete in rectangular columns under

eccentric compression may be different when bent about the major axis and the

minor axis; and 3) only very limited relevant tests have been conducted.

Another important issue that needs to be clarified is the possible effect of size on

the stress-strain behavior of FRP-confined concrete. It was noted in Chapter 2 that

the behavior of large-scale circular columns can be reasonably extrapolated from

the behavior of small-scale circular columns. However, the size effect is much

more uncertain in rectangular columns due mainly to the very limited test data and

the large scatter of test data. With the possible effects of size and eccentricity on

the behavior of FRP-confined concrete in rectangular columns clarified, a reliable

stress-strain model can be developed and incorporated in the theoretical models

presented in this thesis for the analysis of slender columns to develop

corresponding design equations.

Finally, more tests on both circular and rectangular columns with a broad range of

slenderness are needed to verify the design equations proposed in this thesis for

circular columns and the design equations to be developed for rectangular

columns following the framework proposed in this thesis.

In summary, the effects of size and eccentricity on the behavior of FRP-confined

concrete and the effect of slenderness on the behavior of FRP-confined RC

columns are the main issues that need much further research.

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frp-confined rc columns: analysis, · 2020. 6. 29. · a proper design procedure for frp-confined...

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