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The Generic Simulation of Reinforced Concrete

Beams with Prestressing and External

Reinforcement

Daniel Knight

B.E Civil & Structural Engineering (Hons)

B. Finance

Thesis submitted for the degree of Doctor of Philosophy

The School of Civil, Environmental and Mining Engineering

The University of Adelaide

Australia

-January 2014-

-iii-

ABSTRACT

This thesis presents a series of journal papers in which a new segmental moment-rotation

(M/Ө) approach is developed for simulating the instantaneous and sustained loading

behaviour of reinforced concrete (RC) beams with prestressing and external reinforcement.

The M/Ө approach is formed on the fundamental Euler-Bernoulli postulation that plane

sections remain plane, but not necessarily on the Euler-Bernoulli corollary of a linear strain

profile. Further adaption of the well-established mechanics of partial-interaction (PI) theory

introduces a fundamental baseline concept in which residual strains due to time-effects,

thermal gradients and prestressing are accounted for in simulating the formation and gradual

widening of cracks and the associated effects of tension-stiffening allowing for bond-slip.

The effects of concrete softening are incorporated into the M/Ө approach through a size

dependent concrete stress-strain relationship based on the mechanics of shear-friction theory

which simulates the behaviour of a member once a concrete softening wedge forms. The

approach is shown to be able to quantify segmental equivalent flexural rigidities for both

instantaneous and time-dependent behaviour, thus removing the reliance on empiricism in

quantifying the effects of concrete cracking and softening.

In defining the segmental equivalent flexural rigidities of RC beams with both post-tensioned

and pre-tensioned reinforcement it is shown how the approach is used to quantify the load-

deflection behaviour of the entire member through the application of conventional analysis

techniques. The established M/Ө approach is then generically applied to RC beams with both

prestressed fibre reinforced polymer (FRP) and steel reinforcement in quantifying the beams

instantaneous and sustained loading behaviour through being able to accommodate any

conventional method of quantifying the time-dependent parameters. Thus the broad

application of the M/Ө approach provides a novel method of simulating, through mechanics,

the full-range of behaviour of a prestressed beam, that is from prestress application through

serviceability loading and to collapse. Moreover, the reliance on empiricisms, as typically

relied upon in standard analysis methods, are removed with the only empirical components

required being in defining the material properties.

Having established the M/Ө approach for the instantaneous and sustained loading of

conventional prestressed beams, the approach is extended to simulate the behaviour of RC

beams with unbonded post-tensioned FRP and steel tendons. Through understanding the

individual segmental behaviour, a global approach is introduced in which the behaviour of

the unbonded reinforcement can be quantified from the deformation based analysis. The

approach is then further extended to incorporate the analysis of RC beams with mechanical-

fastened (MF) FRP allowing for the PI behaviour at the fasteners. This extension forms the

basis of a generic technique which can subsequently be used in the design of MF systems,

with and without prestress, and therefore provide the foundation in developing design

guidelines.

The universal application of the developed residual strain PI M/Ө approach provides a novel

technique in simulating what is observed in practice for RC beams with prestressing and

external reinforcement. The approach is a useful extension to the current analysis techniques

in which the reliance on defining empiricisms through vast experimental testing procedures is

removed.

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TABLE OF CONTENT

ABSTRACT ............................................................................................................................. iii

STATEMENT OF ORIGINALITY .................................................................................... vii

LIST OF PUBLICATIONS ................................................................................................... ix

ACKNOWLEDGEMENTS ................................................................................................... xi

INTRODUCTION.................................................................................................................... 1

CHAPTER 1 ............................................................................................................................. 4

Background .............................................................................................................................. 4

List of manuscripts .................................................................................................................. 4

Flexural Rigidity of Reinforced Concrete Members Using a Deformation Based Analysis .. 7

Incorporating Residual Strains in the Flexural Rigidity of RC members ............................ 23

CHAPTER 2 .......................................................................................................................... 50

Background ............................................................................................................................ 50

List of manuscripts ................................................................................................................ 50

Short-term partial-interaction behaviour of RC beams with prestressed FRP and Steel....... 53

The time-dependent behaviour of RC beams with prestressed FRP and steel ...................... 74

CHAPTER 3 ........................................................................................................................... 95

Background ............................................................................................................................ 95

List of manuscripts ................................................................................................................ 95

Simulating RC beams with unbonded FRP and steel prestressing tendons .......................... 98

RC Beams with Mechanically Attached FRP Strips ........................................................... 116

CHAPTER 4 ......................................................................................................................... 134

Concluding Remarks ........................................................................................................... 134

Suggested future research .................................................................................................... 135

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STATEMENT OF ORIGINALITY

This work contains no material which has been accepted for the award of any other degree or

diploma in any university or any tertiary institution to Daniel Knight and, to the best of my

knowledge and belief, contains no material previously published or written by another

person, except where due reference has been made in the text.

I give consent to this copy of my thesis when deposited in the University Library, being made

available for load and photocopying, subject to the provisions of the Copyright Act 1968.

The author acknowledges that copyright of published works contained within this thesis (as

listed below) resides with the copyright holder(s) of those works.

I also give permission for the digital version of my thesis to be made available on the web,

via the University’s digital research repository, the Library catalogue, the Australasian

Digital Theses Program (ADTP) and also through web search engines, unless permission has

been granted by the University to restrict access for a period of time.

…………………………………………………… ………..……………

Daniel Knight Date

-viii-

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LIST OF PUBLICATIONS

Oehlers, DJ., Visintin, P,. Zhang, T., Chen, Y and Knight, D. Flexural Rigidity of Reinforced

Concrete Members Using a Deformation Based Analysis. Concrete in Australia 2012, 38(4)

50-56

Knight, D., Visintin, P., Oehlers, D.J and Jumaat., M.Z. Incorporating Residual Strains in the

Flexural Rigidity of RC members. Advances in Structural Engineering. DOI; 10.1260/1369-

4332.16.10.1701. Nov 12, 2013b

Knight, D., Visintin, P., Oehlers, D.J., and Mohamed Ali, M.S. Short-term partial-interaction

behaviour of RC beams with prestressed FRP and Steel. Journal of Composites for

Construction, 10.1061/(ASCE)CC.1943-5614.0000408 (Jun. 26, 2013a).

Knight, D., Visintin, P., Oehlers, D.J., and Mohamed Ali, M.S. The time-dependent

behaviour of RC beams with prestressed FRP and steel. Submitted to Engineering Structures

Knight, D., Visintin, P., Oehlers, D.J., and Mohamed Ali, M.S. Simulating RC beams with

unbonded FRP and steel prestressing tendons. Accepted to Composites B.

DOI;10.1016/j.compositesb.2013.12.039

Knight, D., Visintin, P., Oehlers, D.J., and Mohamed Ali, M.S. Simulation of RC Beams with

Mechanically Attached FRP Strips. Submitted to Composite Structures

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ACKNOWLEDGEMENTS

My PhD work has been a challenging and rewarding experience which I have shared with

many people. Their endless support and encouragement has been invaluable in many ways.

My sincerest thanks go to Emeritus Prof. Deric Oehlers for his guidance, patience and

wisdom. He has been an integral part in my learning and completion of this work and has

provided me with lifelong skills.

I would like to thank Dr Mohamed Ali for his ideas, guidance and mentoring throughout my

studies as well as Dr Phillip Visintin for the countless hours he made himself available to my

research.

Finally, thanks go to my family and wonderful girlfriend for their unwavering support and

endless motivation and to my close friends who were always there for support and a Friday

afternoon beer.

- 1 -

INTRODUCTION

The mechanics governing the instantaneous loading behaviour of prestressed concrete (PC)

flexural members is generally depicted in terms of flexural rigidity (EI). The standard

practice is to quantify an effective flexural rigidity (EIeff) through a strain based full-

interaction (FI) moment-curvature (M/χ) analysis, which applies at a two-dimensional section

of the member. Being strain based the approach cannot directly quantify the slip between the

reinforcements and adjacent concrete, that is when a crack intercepts a reinforcement layer,

and is therefore unable to simulate the associated effects of tension-stiffening. Furthermore,

the time-dependent strains due to creep and shrinkage of a PC member have a large impact on

the overall performance of the member, namely due to an increased crack width and

deflections due to prestress loss. In order to accommodate such time-effects the M/χ approach

is extended to quantify the sustained loading behaviour by varying material properties over

time. Typically derived empirical expressions are used to determine EIeff which is used to

allow for tension-stiffening in the vicinity of the cracked regions. While this approach is able

to provide reasonable estimates of EI for members with bonded steel and prestressing

reinforcement tested within the bounds of the experimental tests for which EIeff was

calibrated, when extended beyond this range a poor correlation between predicted and

observed results exists. Moreover, this approach becomes particularly problematic in

simulating beams with fibre reinforced polymer (FRP) reinforcement, where due to a low

reinforcement ratio the effects of tension-stiffening are commonly overestimated.

Furthermore, the absence of bond between the prestressing reinforcement and adjacent

concrete in reinforced concrete (RC) beams with unbonded prestressed reinforcement violates

the condition of strain compatibility. This is because the developed strains in unbonded

reinforcement are dependent on the total member deformation. Thus, in order to account for

unbonded reinforcements, empirically derived bond-reduction factors are typically adapted to

a conventional FI M/χ approach in order to define the developed stress in the unbonded

reinforcement along the member’s length. Thus being highly reliant on empirical components

and on firstly quantifying the member behaviour, the M/χ approach is unable to be

generically applied to any member and reinforcement type. A similar problem arises in

simulating the behaviour of RC members with mechanically-fastened (MF) FRP

reinforcement, in which further difficulty in accounting for the partial-interaction (PI)

behaviour at the fastener emerges. Furthermore, due to the relatively new interest and

complexity of analysis of RC beams with MF-FRP systems, no generic design guideline

currently exists.

This thesis presents a series of journal papers in which a new segmental moment-rotation

(M/Ө) approach is developed for simulating the instantaneous and sustained loading of

reinforced concrete (RC) beams with bonded and unbonded prestressed and external

reinforcement. The M/Ө approach is formulated in which a segment of a PC beam is

subjected to a constant moment in order to quantify the sectional flexural rigidity (EI) and its

variation with moment. The approach incorporates residual strains as a novel extension to the

- 2 -

established mechanics of PI theory to simulate both the instantaneous and time-dependent

behaviour of RC beams with prestressed reinforcement. The influence of concrete softening

on the overload behaviour of the member is incorporated through the application of a size-

dependant concrete stress-strain relationship, which generically simulates the mechanics of

shear-friction theory. The residual strain PI M/Ө approach is shown to be able to quantify the

equivalent flexural rigidity (EIequ) of a PC beam without the reliance on empiricisms, apart

from those required in defining material-properties. Being able to determine segmental

deformations along the length of a member with unbonded prestressed reinforcement, the

approach is shown to be able to directly quantify the unbonded reinforcement strain and thus

remove the reliance on empirical bond-reduction factors. The analysis of mechanically-

fastened FRP systems is a further novel extension in which case it is shown how the M/Ө

approach is able to accommodate both the unbonded MF-FRP as well as the PI behaviour at

the mechanical-fastener through a developed numerical member analysis. It is shown how the

generic nature of the numerical analysis and mechanics based segmental M/Ө approach

provide a foundation in determining design guidelines for RC beams with MF-FRP systems.

The manuscripts contained in this thesis are published, accepted or submitted to

internationally recognised journals. Each of the three chapters encompassing these

manuscripts contain: an introduction explaining the aim of the chapter and how the research

fits into the overall objective; a list of manuscripts contained within the chapter; and finally

the presentation of each manuscript.

Chapter 1 consists of two manuscripts. The first introduces the fundamental mechanisms of

the developed segmental deformation approach for the generic analysis of reinforced concrete

(RC) beams incorporating: residual strain partial-interaction (PI) theory to directly simulate

the effects of tension-stiffening as the internal bonded reinforcement pulls from the crack

face and a size-dependent stress-strain model to simulate the concrete wedge formation

associated with concrete softening. The second manuscript describes, in detail, the

instantaneous and sustained loading behaviour of RC beams whereby baseline approach is

introduced in order to accommodate residual strains due to time-effects, thermal gradients

and the presence of prestressed reinforcement.

Chapter 2 consists of two manuscripts which show how the presented residual strain

segmental PI M/Ө approach, introduced in Chapter 1, is capable of deriving the cross-

sectional behaviour of prestressed concrete (PC) beams, such as effective flexural rigidities.

The first manuscript shows how the instantaneous loading of PC beams can be generically

simulated through the segmental approach, allowing for concrete cracking and crack

widening and the associated effects of tension-stiffening. The second paper focuses on the

serviceability behaviour of PC beams under sustained loading. It is shown how concrete

shrinkage, creep and tendon relaxation is successfully incorporated in to the approach in

quantifying the variation in flexural rigidity of a beam.

- 3 -

Chapter 3 further extends the approach to being able to quantify the flexural behaviour of RC

beams with unbonded steel and FRP tendons. The first manuscript in this chapter outlines

how the cross-sectional analysis, presented in the preceding chapters, is used in a developed

numerical member analysis in order to define member deformations and thus determine the

behaviour of the unbonded reinforcement. The second manuscript in this chapter furthers the

approach to simulating the behaviour of MF-FRP systems providing a novel analysis

technique which may be suitable for providing generic design guidelines.

Chapter 4 of this thesis consists of the concluding remarks of this research as well as

suggestions for future research. The widespread applications of the developed residual strain

PI M/Ө approach for prestressed concrete beams provides a novel technique in simulating

what is actually observed in practice. The generic approach can be seen as a useful design

tool as an extension to current analysis techniques, in which the reliance on extensive

experimental testing, required to define empirical components, is reduced.

- 4 -

CHAPTER 1

Background

Chapter 1 presents the first manuscript ‘Flexural rigidity of reinforced concrete members

using a deformation based analysis’ which provides a background to the existing research in

the area and highlights the overall need for this research. Following this is the introduction of

the fundamental mechanisms which form the basis of this research for the remainder of the

thesis. This manuscript outlines the fundamentals of the deformation based moment-rotation

(M/Ө) approach which can simulate the mechanisms of tension-stiffening, wedge softening

and shear failure. Being mechanics based, it is shown how the approach reduces the reliance

on vast experimental testing and hence can be seen as a useful extension to the current

moment-curvature (M/χ) analysis.

The second manuscript in this chapter, ‘Incorporating residual strains in the flexural rigidity

of RC members’ provides a detailed outline of the segmental M/Ө approach. Initially a

segmental approach without tension-stiffening is presented, followed by a segmental partial-

interaction (PI) moment-rotation (M/Ө) approach which is shown to be able to simulate what

is actually observed in practice through the adaption of a numerical tension-stiffening

procedure which provides allowances for members with prestress and subjected to both

instantaneous and sustained loading. A baseline concept is introduced which enables the

approach to incorporate time-effects, thermal gradients and prestressing. It is shown how the

mechanics of the approach makes it suitable for generic application to any reinforcement,

bond and concrete type.

List of Manuscripts

Oehlers, DJ., Visintin, P,. Zhang, T., Chen, Y and Knight, D. Flexural Rigidity of Reinforced

Concrete Members Using a Deformation Based Analysis. Concrete in Australia 2012, 38(4)

50-56

Knight, D., Visintin, P., Oehlers, D.J and Jumaat., M.Z. Incorporating Residual Strains in the

Flexural Rigidity of RC members. Advances in Structural Engineering.DOI; 10.1260/1369-

4332.16.10.1701. Nov 12, 2013b)

- 5 -

Statement of Authorship

Flexural Rigidity of Reinforced Concrete Members Using a Deformation Based

Analysis. Concrete in Australia 2012, 38(4) 50-56.

Oehlers, DJ

Compiled manuscript and supervised research

I hereby certify that the statement of contribution is accurate and I give permission for the

inclusion of the paper in this thesis

Signed…………………………………………………………………………..Date…………

Visintin, P

Supervised and contributed to research



Signed………………………………………………………………………….Date…………

Zhang, T

Contributed to research



Signed………………………………………………………………………….Date…………

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Chen, Y




Signed………………………………………………………………………….Date…………

Knight, D (Candidate)




Signed………………………………………………………………………….Date…………

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a1172507

Text Box

A Oehlers, D.J., Visintin, P., Zhang, T., Chen, Y. & Knight, D. (2012) Flexural rigidity of reinforced concrete members using a deformation based analysis. Concrete Australia, v. 384), pp. 50-56

a1172507

Text Box

A NOTE: This publication is included on pages 7-20 in the print copy of the thesis held in the University of Adelaide Library.

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Incorporating Residual Strains in the Flexural Rigidity of RC members. Advances in

Structural Engineering. DOI; 10.1260/1369-4332.16.10.1701. Nov 12, 2013a


Performed analyses and developed model



Signed…………………………………………………………………………..Date…………

Visintin, P




Signed………………………………………………………………………….Date…………

Oehlers, DJ




Signed………………………………………………………………………….Date…………

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Jumaat, M.Z

Assisted in manuscript evaluation



- 23 -

a1172507

Text Box

A Knight, D., Visintin, P., Oehlers, D.J. & Jumaat, M.Z. (2013) Incorporating residual strains in the flexural rigidity of RC members with varying degrees of prestress and cracking. Advances in Structural Engineering, v. 16(10), pp. 1701-1718

a1172507

Text Box

NOTE: This publication is included on pages 23-49 in the print copy of the thesis held in the University of Adelaide Library. It is also available online to authorised users at: http://doi.org/10.1260/1369-4332.16.10.1701

http://doi.org/10.1260/1369-4332.16.10.1701

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CHAPTER 2

Background

Chapter 1 introduced a residual strain segmental M/Ө approach that can be generically

applied to any type of reinforced concrete (RC) beam with prestressed reinforcement.

Chapter 2 furthers this approach, firstly with the manuscript ‘Short-term partial-interaction

behaviour of RC beams with prestressed FRP’, which applies the developed M/Ө approach to

pre-tensioned and post-tensioned FRP and steel RC beams under instantaneous loading. It is

shown in this manuscript how the approach can be seen as an extension to conventional

analysis techniques through being able to determine the variation in effective flexural-rigidity

of a prestressed concrete beam, which can subsequently be incorporated into standard

analysis techniques for determining member deflection.

The second manuscript in this chapter, ‘The time-dependent behaviour of RC beams with

prestressed FRP and steel’, provides the fundamental analysis procedure for simulating the

behaviour of prestressed members subjected to a sustained load. The approach is shown to be

able to be generically applied to any reinforcement and concrete type and conveniently adopt

any conventional method for determining time-effects. The variation in flexural-rigidity at

predetermined time intervals is shown to produce accurate predictions of member behaviour;

moreover the approach is shown to be a novel technique in calibrating empirical code factors

without the reliance on conducting numerous experimental tests.

List of Manuscripts

Knight, D., Visintin, P., Oehlers, D.J., and Mohamed Ali, M.S. Short-term partial-interaction

behaviour of RC beams with prestressed FRP and Steel. Journal of Composites for

Construction, DOI;10.1061/(ASCE)CC.1943-5614.0000408 (Jun. 26, 2013a).

Knight, D., Visintin, P., Oehlers, D.J., and Mohamed Ali, M.S. The time-dependent

behaviour of RC beams with prestressed FRP and steel. Submitted to Engineering Structures

- 51 -


Short-term partial-interaction behaviour of RC beams with prestressed FRP and Steel.

Journal of Composites for Construction, DOI; 10.1061/(ASCE)CC.1943-5614.0000408 (Jun.

26, 2013a).


Performed all analyses, developed model and theory.



Signed…………………………………………………………………………..Date…………

Visintin, P




Signed………………………………………………………………………….Date…………

Oehlers, DJ




Signed………………………………………………………………………….Date…………

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Mohamed Ali, M.S




Signed………………………………………………………………………….Date…………

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a1172507

Text Box

NOTE: This publication is included on pages 53-71 in the print copy of the thesis held in the University of Adelaide Library. It is also available online to authorised users at: http://dx.doi.org/10.1061/(ASCE)CC.1943-5614.0000408

a1172507

Text Box

A Knight, D., Visintin, P., Oehlers, D.J. & Mohamed Ali, M.S. (2013) Short-term partial-interaction behaviour of RC beams with prestressed FRP and steel. Journal of Composites for Construction, v. 18(1), pp. 1-9

http://dx.doi.org/10.1061/(ASCE)CC.1943-5614.0000408

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The time-dependent behaviour of RC beams with prestressed FRP and steel. Submitted

to Engineering Structures





Signed…………………………………………………………………………..Date…………

Visintin, P




Signed………………………………………………………………………….Date…………

Oehlers, DJ




Signed………………………………………………………………………….Date…………

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Mohamed Ali, M.S




Signed………………………………………………………………………….Date…………

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The time-dependent behaviour of RC beams with prestressed FRP and

steel

Daniel Knight, Phillip Visintin, Deric J Oehlers and Mohamed Ali M.S

Abstract

A prestressed beam, when subjected to a sustained load has both an instantaneous and time

dependent response. In typical reinforced concrete (RC) beams, time dependent behaviour

may lead to serviceability failures in structural beams where deflections are excessive. In the

case of prestressed concrete (PC) beam however, understanding the time dependent

behaviour is crucial as time-effects under serviceability loading can result in a critical loss of

prestress causing failure. Conventional techniques to simulate the behaviour of PC beams are

reliant on a moment-curvature (M/χ) analysis to quantify the flexural rigidity (EI) of a beam,

the approach being mechanically correct prior to cracking as only the material stress-strain

relationships are empirically derived. Post-cracking, however, the M/χ approach has to be

semi-empirical and requires correction factors as it cannot generically simulate the effects of

tension-stiffening for different reinforcement types. This paper presents a displacement based

M/Ө approach for determining the behaviour of PC beams by applying the mechanics of

partial interaction (PI) theory which directly simulates the formation and widening of cracks

as the reinforcement pulls from the crack face, thus directly allowing for tension stiffening.

The PI M/Ө approach can quantify the equivalent flexural rigidities (EIequ) associated with

tension-stiffening which can be used in standard analysis techniques to quantify beam

behaviour. The approach is shown to accommodate time-effects, namely concrete creep,

shrinkage and reinforcement relaxation and can therefore be seen as a useful extension to

current analysis techniques, showing good correlation to experimental tests of beams under

sustained loading, without the reliance on empiricisms.

Keywords; prestressed concrete; tension-stiffening; flexural rigidity; bond-slip; FRP;

partial-interaction; post-tensioned; pre-tensioned; creep; shrinkage

Introduction

The time-dependent strains due to creep and shrinkage of a prestressed concrete (PC) beam

have a large impact on the overall performance of the beam, namely due to increased crack

widths and deflections due to a loss of initial prestressing force which can lead to ultimate

failure under serviceability loading. It is standard practice to depict the long-term loading

behaviour of PC beams in terms of effective flexural rigidities (EIeff) which are quantified

through the use of a strain based moment-curvature (M/χ) analysis (Bazant and Panula 1980,

Banson 1977, Bishcoff 2005, Branson and Trost 1982, CEB 1992, 2010, Gilbert and

Mickleborough 1990, Kawakami and Ghalim 1996, Thompson and Park 1980, Nawy 2010,

Warner et al 1998) which applies at a discrete section of the beam while the concrete material

properties are varied over time. Being strain based, the M/χ approach cannot directly simulate

slip between the reinforcements and adjacent concrete, which occurs at a crack face. While

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this approach can provide reasonable estimates of EIeff for steel beams reinforced to within

the bounds of the experimental tests from which EIeff was calibrated, when extended beyond

this range, a poor correlation between predicted and observed deflections are observed. This

poses a significant problem for beams reinforced with FRP, where, due to the high strength of

the tendons, reinforcement ratios are low resulting in tension-stiffening being commonly

overestimated. Despite extensive research to the tension-stiffening behaviour of FRP

reinforced concrete (CEB 1992, 2010, Gilbert and Ranzi 2011) the empirically derived

equations are still only applicable to within the bounds of the test data from which they are

derived.

To overcome this problem, the segmental M/Ө approach developed for beams reinforced

with either steel or FRP by Visintin et al (2012, 2013a, 2013b) and Oehlers (2005, 2011,

2012) and extended for application to short-term loading of prestressed concrete beams by

Knight et al (2013b), is further extended here to accommodate the time-dependent behaviour

of prestressed concrete beams. Recent work by the authors (Oehlers et al 2011) introduced

concept of a M/Ө approach to quantify the behaviour of a prestressed concrete beam under

short-term loading, while the fundamental concept of accommodating residual strains through

a base-line approach was introduced by Knight et al (2013a). This paper further develops

these concepts in order to quantify the flexural behaviour of both pre-tensioned and post-

tensioned beams over sustained loading periods. It is first shown how the segmental

behaviour at either pre-tensioned or post-tensioned application may accommodate creep and

shrinkage, consequently resulting in the prestress loss being quantified. The behaviour of an

uncracked segment under an applied load is then presented, demonstrating how the residual

strain M/Ө approach can be used to derive the equivalent flexural rigidity (EIequ) of a PC

beam. The approach then extends to the cracked segment behaviour in which the mechanics

of partial-interaction (PI) theory (Visintin et al 2013b, Oehlers et al 2005, 2011, 2012, 2013,

Knight et al 2013a, Gupta and Maestrini 1990, Haskett et al 2008, 2009a, 2009b, Foster et al

2010, Muhamed et al 2012, Mohamed Ali et al 2012) is further developed to simulate tension

stiffening, while accommodating residual strains.

It is therefore shown how the reliance on empiricisms is removed in defining the flexural-

rigidity (EI) of a PC beam, apart from that associated in defining material properties. The

residual strain PI M/Ө approach can be applied to a segment of a PC beam to derive the

equivalent flexural rigidity (EIequ) allowing for tension stiffening and the effects of prestress,

creep, shrinkage and reinforcement relaxation. The results of the M/Ө analysis can then be

converted to an equivalent M/χ relationship and consequently equivalent flexural rigidity

(EIequ) which can be used in a standard analysis to quantify beam behaviour. Comparisons are

made to prestressed steel and FRP beams as tested by Zou (2003) showing good correlation

over the sustained loading periods. The approach is shown to be capable of being generically

applied to accommodate a wide variety of reinforcement types, that is, steel, FRP and AFRP,

with varying surface treatments. A parametric study investigates time-dependent effects on

the tension-stiffening prism behaviour in order to quantify prestress loss over time and crack

spacing’s.

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Moment-Rotation analysis of a Segment

The time-dependent partial-interaction PI moment rotation M/Ө behaviour of either a pre-

tensioned or post-tensioned beam at prestress application is firstly considered followed by the

behaviour of a segment under an applied moment. A cracked segment analysis is then

outlined in which a residual strain tension-stiffening analysis is introduced in order to

accommodate both the presence of prestressing and time-effects.

Prestressing

Post-tension application

The PI M/Ө analysis for a segment of length 2Ldef extracted from a post-tensioned beam is

illustrated in Figure 1(b) with a cross-section X-X in Figure 1(a). For the case of the post-

tensioned beam, a tendon is fed through a horizontal duct at eccentricity e at time t after the

concrete is poured. Prior to any externally applied load, the concrete and reinforcements in

Figure 1(b) are of length 2Ldef; sections A-A will be referred to as the base lines for both the

concrete and reinforcement which is defined as any movement of the segment ends relative to

these positions or base lines will induce strains in the materials. Due to symmetry, we need

only consider one half of the segment of length Ldef such that deformations at either end are

equal, so that F-F can be taken as a stationary datum such that any deformation relative to

line F-F will result in the same strains or effective strains. A prestress strain in the tendon εpr

equal to post-tensioning strain εt-po resulting in a stress σt-po and consequently a force Ppr is

applied to the tendon which is at eccentricity e.Ppr can be taken as an external force that must

be resisted by the remaining reinforced concrete section, resulting in deformation B-B. Now

consider the case where due to a shrinkage strain εsh a deformation occurs in the concrete of

magnitude εshLdef, that is from A-A to C-C, such that C-C now becomes the baseline for

deformation in the concrete and A-A remains the baseline for deformations in the

reinforcements

Figure 1. M/Ө analysis with shrinkage for post-tensioned segment

It is now a matter of finding the deformation profile B-B at rotation Өpr-t in Figure 1(b) when

the resultant force in the reinforced concrete section FRC, that is the resultant of Frt, Fcc and Frb

in Figure 1(e), are equal to and in line with the prestressing force Ppr, at eccentricity e as

shown in Figure 1(e). An iterative procedure is employed whereby Өpr-t is fixed and a top

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fibre displacement δcc-po is guessed, thereby fixing the position of line B-B in Figure 1(b). As

the section remains uncracked the deformations over the cross-section can be divided by the

segment length Ldef to give the resulting strain profile in Figure 1(c). It is necessary to note

that two strain profiles now exist, one for the concrete and one for the reinforcements. It has

previously been explained that any deformation relative to F-F results in a strain such that the

movement from C-C to B-B divided by the length Ldef gives the strain profile E-E for the

concrete and similarly the deformation from A-A to B-B divided by Ldef gives the strain

profile G-G for the reinforcements where ten represents tension and comp represents

compression. It can also be seen that the strain profiles E-E and G-G are parallel and located

εsh apart, whilst the strain in the post-tensioned tendon εt-po in Figure 1(c) remains at the

applied prestress strain εpr. The strains in Figure 1(c) are real strains that are accommodated

by the material, hence, through material-stress strain relationships the corresponding stress

profile in Figure 1(d) can be determined and hence the forces within the section shown in the

force profile F in Figure 1(e), can be determined. If the algebraic sum of these forces is not

equal to Ppr then δcc-po may be adjusted accordingly, thereby shifting the location of the

neutral axis until equilibrium of forces is achieved. If at longitudinal equilibrium the resultant

force FRC is not in line with the prestressing tendon force Ppr then the rotation Өpr-t may be

adjusted and the analysis repeated until it does so in order to determine the deformation B-B.

Hence, the analysis provides the initial rotation Өpr-t of the PC segment corresponding to an

internal prestress moment Mpr equal to Ppre accounting for shrinkage, while the displacement

at the level of the post-tensioned tendon δt-po becomes an initial displacement that is carried

forward into the second step of analysis after the duct has been grouted.

Pre-tension application

Now consider the pre-tensioned segment in Figure 2(b) which has a cross-section X-X as

shown in Figure 2(a). Baselines for deformation are unchanged from Figure 1, that is C-C

represents the concrete after a shrinkage strain εsh and A-A represents the reinforcement,

while F-F remains a stationary datum such that any relative deformation will induce strains in

the materials. The eccentric prestress force due to initial prestress strain εpr results in

deformation B-B in Figure 2(b), however, in this instance as the tendon is fully bonded to the

surrounding hardened concrete when the prestressing force is transferred, a full-interaction

(FI) condition exists. Hence, the initial tendon strain εpr reduces by δt-pre/Ldef to εt-pre,

corresponding to a prestress force Ppr-pre which is now less that Ppr in Figure 1. This loss is

most commonly referred to as elastic shortening (Gilbert and Mickelborough 1996, Nawy

2010, Thompson and Park 1980, Glodowski and Lorenzetti 1972). The general analysis

procedure is now identical to Figure 1 such that deformation B-B must be determined for a

given rotation Өpr-t to satisfy both rotational and longitudinal equilibrium, however, now the

resultant force must now equal the reduced prestress force Ppr-pre, such that only a single

solution is possible and therefore δt-pre becomes the originating tendon deformation for further

analysis stages of a pre-tensioned section.

- 78 -

Figure 2. M/Ө analysis with shrinkage for a pre-tensioned segment

The outlined analyses in Figures 1 and 2 provide the initial rotation of either the post-

tensioned or pre-tensioned segment, due to both the effects of prestress, shrinkage, creep and

tendon relaxation, that is Өpr-t at point O in Figure 3(a) corresponding to a prestressing

moment Mpr equal to Ppre or Ppr-pree, respectively. The rotation Өpr-t due to the combined

effects can be converted to an equivalent curvature χpr-t, by dividing by the deformed length

Ldef, which is uniform along the length of a simply support prestressed beam. Furthermore, as

the section is uncracked, the curvature χpr-t is the same as that which could be determined

using a conventional FI analysis.

Figure 3. M/Ө, M/χ and M/EI variation

Applied loads and prior to cracking

Consider the segment in Figure 1(b) after the post-tensioned force has been applied, the

tendon is now grouted so that these deformations are the starting position prior to the

application of the transverse applied loads; this is shown in Figure 4(b). Firstly consider the

case whereby the applied loads are sustained for a period of time t such that the

corresponding applied moment M is not significant enough to cause concrete cracking, where

M is an applied moment measured from the origin O in Figure 3(a). The procedure begins in

Figure 5 by setting a rotation Өm such that the combination of shrinkage and applied moment

- 79 -

causes a total rotation Ө in Figure 4(b) equalling Өm-Өpr-t, corresponding to a guessed

deformation δcc-m and resulting in a total deformation shift from B-B to D-D.

Prior to the application of an applied external moment M, line B-B represents the segment

deformation and corresponds to the initial tendon deformation δt-po, occurring at a rotation

Өpr-t as shown in Figure 1(b), as determined in the first part of the analysis. As in Figures 1

and 2, the strain in the reinforcement and concrete are given by strain profiles E-E and G-G,

respectively, now shown in Figure 4(c), however the strain in the prestressing tendon is now

given by the post-tensioning strain εt-po prior to grouting plus the extension of the tendon due

to a change in deformation at tendon level from B-B to D-D, that is εt-po plus Δδt/Ldef.

For the pre-tensioned case, the tendon deformation δt-pre and corresponding strain εt-pre would

be carried over from the analysis in Figure 2(b), and therefore, the tendon strain would

become εt-pre plus Δδt/Ldef, that is δt-po in Figure 4(b) becomes δt-pre Furthermore, the effect of

concrete creep can be included in the analysis by adjusting the concrete modulus Ec for creep

when deriving the stress profile as in Figure 4(d) from the strain profile in Figure 4(c) and

similarly tendon relaxation can be accommodated by a change in tendon modulus Et.

Figure 4. Uncracked segment behaviour with shrinkage

- 80 -

Figure 5. Moment-Rotation procedure for segment

It is now matter of applying an iterative procedure as outlined in Figure 5 in order to

determine the location of deformation profile D-D such that axial force equilibrium occurs, at

which point the moment M may be determined corresponding to a rotation Ө. This analysis

defines a single point on the M/Ө relationship denoted as point A in Figure 3(a) and can be

repeated for increasing rotations to quantify the M/Ө relationship at time t, that is from O-B

until either: the maximum tensile strain exceeds the concrete rupture strain εct at which point

the section has cracked and the crack tip reaches a level of reinforcement; or the maximum

compressive strain exceeds the concrete peak strain capacity εpk, which is the strain at which

concrete softening occurs. However, in this paper concrete softening is not considered.

The abscissa in Figure 3(a) for line O-B can now be divided by the deformation length Ldef to

derive the moment-curvature M/χ relationship in Figure 3(b) where the secant stiffnesses are

the equivalent flexural rigidities EIequ, taken about the origin O, as illustrated in Figure 3(c).

Prior to the formation of cracks, the cross-section remains in full-interaction such that the

uncracked flexural rigidities EIuncr obtained from a traditional FI M/χ approach with time-

dependent strains, will be the same as those derived from the M/Ө approach as the reliance

on empiricisms is removed. It is necessary to remember that the M/EIequ relationship in

Figure 3(c) represents an applied moment, with the origin denoted by point O in Figure 3(a).

The inclusion of time-effects compared to an instantaneously loaded beam can be seen by the

reduced beam stiffness between O-B in Figure 3.

- 81 -

The M/Ө analysis in Figure 4, the results of which are in Figure 3(a), can be used to predict

the moment at the onset of cracking Mcr, that is when the tensile strain in the concrete in

Figure 4(c) exceeds the concrete tensile strain capacity εct. The formation of this crack is now

referred to as the initial crack; the analysis may be applied beyond this initial crack formation

that is until the crack tip intercepts a layer of reinforcement. Once the crack tip intercepts a

reinforcement layer, a slip between the reinforcement and adjacent concrete exists in order to

allow for separation of the crack faces, that is crack widening; this is referred to as tension-

stiffening.

Cracked segment analysis

Now let us consider the left hand side of the beam segment in Figure 6(b) in which the crack

tip is above the level of both the tensile reinforcement and prestressing tendon. The general

analysis procedure is identical to that procedure outlined in Figure 5 for the uncracked

segment in Figure 4, however now the force in the reinforcements crossing a crack face are

dependent on the reinforcement slip Δt and Δrb in Figure 6(b) which can be simulated using

the following numerical residual strain tension-stiffening analysis.

Figure 6. Cracked segment M/Ө analysis

Tension-stiffening with residual strains

Firstly consider the effect of residual strains on the tension stiffening numerical procedure

outlined by Visintin et al (2012, 2013a, 2013b) and Knight et al (2013a, 2013b) for an

element extracted from a tension-stiffening prism shown in Figure 6(a). The residual strain

partial-interaction load-slip behaviour allowing for both the influence of shrinkage and

prestress can be determined through an extension of the numerical technique as outlined by

Knight et al (2013b). In this adaption, it is first necessary to establish boundary conditions to

determine the effects of residual strains in the full-interaction (FI) region in Figure 7(a) as

this is the starting point for the behaviour of the partial-interaction (PI) region.

- 82 -

Figure 7. Slip-strain with shrinkage and prestress

Prior to any shrinkage of the concrete or applied prestressing strain, both the tendon and

concrete are equal in length as denoted by line T-T in Figure 7(a) with a total element length

Le. Line T-T is, therefore, the base line for deformation in the concrete and tendon when no

shrinkage is present, that is the reinforcement and concrete stresses are both zero. A

shrinkage strain εsh is present however, which if the concrete was unrestrained and free to

move, would contract by εshLe to C-C. The concrete baseline C-C is now the position of the

concrete when stress is zero. It is, therefore, a question of determining the position of the

element end A-A where εt-shLe is the contraction of the tendon within the element from its

baseline T-T that results in a reinforcement stress, and εc-shLe is the extension of the concrete

from its baseline C-C within the element which causes a concrete stress, as outlined by

Knight et al (2013b). Therefore, from equilibrium and compatibility, the residual strains in

the reinforcement εt-sh can be determined from Equation 1

cctt

ccshsht

AEAE

AEεε

(1)

and consequently the concrete strain εc-sh is the sum of εc-sh and εt-sh which equates to the total

shrinkage strain εsh.

Where in Equation 1 Et and Ec, At and Ac are the modulus and cross-sectional areas of the

tendon and concrete respectively, and hence the strains in the element due to shrinkage alone

are known.

Now consider a post-tensioned tendon strain εt-po equal to the applied prestressing strain εpr

resulting in tendon force Ppr which reacts against the concrete element with a force in Figure

7(b). For the linear elastic case, equilibrium is achieved when

cc

ttpot

pocAE

AEεε

(2)

- 83 -

The FI pre-tensioned behaviour is illustrated in Figure 7(c). The end C-C represents the

position of the concrete prior to the anchors being released. On release of the anchors, the

contraction of the tendon is resisted by the concrete element such that the change in strain

εc-pre is the same in both. Therefore the equilibrium position must be found whereby the force

in both the tendon and concrete are the same and from the linear elastic condition is given by

ttcc

ttpr

precAEAE

AEεε

(3)

The residual strains determined in Figures 7(a) and 7(b) applies to the post-tensioned element

prior to interface slip. Now consider the partial-interaction (PI) post-tensioned case in Figure

7(d) where at release the net strain in the tendon causes a net expansion from A-A by an

amount (εt-sh-εt-po)Le to T-T and any additional tensile force Pt-n that induces strain εt-n would

cause a further extension by εt-nLe to Tn-Tn. Similarly for the concrete prism, on release of the

net strain a contraction of (εc-sh-εc-po)Le from A-A to C-C occurs and any applied tensile force

Pc-n resulting in strain εc-n causes an expansion of εc-nLe to Cn-Cn with a total increase of slip

within the element being the distance between Tn-Tn and Cn-Cn given by

encntpocpotshcshtn )]Lε(ε)ε(ε)ε[(εδΔ (4)

which can be simplified to

encnteresn )Lε(εLΔδΔ (5)

where Δ’res is the residual slip-strain due to shrinkage and post-tensioning strains

It can also be seen in Equation 4 that the increase in slip within an element is due to the

applied strains εt-n-εc-n plus the additional components due to post-tensioning εt-po+εc-po and

shrinkage εt-sh+εc-sh. Furthermore, as with the M/Ө analysis, concrete creep may be

accommodated by changing the modulus Ec and similarly, tendon relaxation included by

changing Et.

The PI behaviour of a pre-tensioned element is illustrated in Figure 7(e) and is the same as in

Figure 7(c) such that Equation 4 applies however the subscript “po” is replaced by “pre”.

Moreover, for the analysis of untensioned reinforcement it is only necessary to include the

effects of shrinkage. It has been shown how residual strains are incorporated into a prism

segment such that it is now possible to outline the partial-interaction tension-stiffening

numerical analysis in quantifying the load-slip behaviour of either the prestressing tendon or

untensioned reinforcement, accounting for time-effects.

- 84 -

Partial-Interaction Segmental Analysis

It is common practice to simulate tension-stiffening both experimentally (Morza and Houde

1979, Somayaji and Shah 1981, Jiang et al 1984, Rizkalla and Hwang 1984, Lee and Kim

2008, Wu and Gilbert 2008) and theoretically (Oehlers et al 2011, Gupta and Maestrini 1990,

Haskett et al 2008, 2009a, 2009b, Muhamed et al 2012, Mohamed Ali et al 2012, Wu et al

1991, Marti et al 1998, Teng et al 2006, Choi and Cheung 1996) by using concentrically

loaded reinforced concrete prisms. The mechanism governing the partial-interaction tension-

stiffening analysis can now be illustrated by considering the same tendon embedded in a

concentrically loaded prism of length Ld shown in Figure 8(b). The prism with cross-section

in Figure 8(a) is comprised of n individual elements of length Le as in Figure 6.

Figure 8. Tension-stiffening analysis without residual strains

Firstly consider element 1 in Figure 8(b). It is now it is a matter of determining the force in

the tendon Pt1 corresponding to an imposed slip Δ1 at the crack face which represents the

accumulated slips in each of the n elements within the prism, and is shown as Δt in Figure

6(b). It is also noted that the left hand side of element 1 represents the crack face such that the

force in the concrete equals zero. The bond stress τ between the reinforcement and

surrounding concrete in Figure 8(b) varies over the contact surface area LeLper and is

dependent on the interface slip Δ1 between the reinforcement and concrete, such that the bond

force B1 in element 1 can be determined. The relationship between bond-stress and interface

slip (τ-Δ) are typically determined empirically (CEB 1992, 2010, Wu and Gilbert 2008, Teng

et al 2006, Eligehausen et al 1982m Seracino et al 2007a, 2007b). The analysis can therefore

be generically applied as it can accommodate varying reinforcement surface treatments of

any material type with the only empiricism necessary being the material stress-strain

relationship and bond-slip characteristic. The force in the tendon over Le of element 1 in

Figure 8(b) varies from Pt1 to Pt1 – B1 and, therefore, the force in the concrete increases from

zero at the crack face to B1. The forces on either side of element 1 can be used to determine

the mean forces within the element and, therefore, from conventional material-stress strain

relationships it is possible to derive the mean strain in the tendon εt1 and that in the concrete

εc1 due to an applied load. The total slip-strain (Δ’1) over element 1 is the difference between

the tendon and concrete strain (εt1-εc1) combined with slip-strain (Δ’res) due to shrinkage and

prestress, and thus the change in slip within the element 1 is δΔ1, as in Equation 5. Therefore,

- 85 -

as the slip within element 1 is Δδ1 then the slip in element 2 (Δ2) is equal to Δ1 minus Δδ1.

The analysis may now be repeated over subsequent elements in Figure 8(b) in order to

determine the variation along the length of the prism between the slip Δ and slip strain Δ’,

whereby the initial guess for Pt1 is adjusted until the desired boundary conditions are

achieved (Visintin et al 2012, 2013a, 2013b, Knight et al 2013b).

The numerical partial-interaction analysis outlined in Figure 8(b) may now be used to

determine the primary crack spacing Scr-p in Figure 8(a) by applying the known boundary

condition (Visintin et al 2012, 2013a, 2013b, Knight 2013b). Cracking occurs at a point of

full-interaction when the stress σc, which increases along the prism length due to developed

bond B in the concrete exceeds the tensile cracking stress fct. Once a crack has formed at Scr-p

a prism of length equal to the crack spacing as illustrated in Figure 8(b) must be considered in

which case the mechanics of the approach changes. Through applying partial-interaction (PI)

theory it is now possible to determine the tension-stiffening behaviour of the prism in Figure

8(b) as it is known that the reinforcement is pulled from each end with equal force and,

therefore, by symmetry the slip of the reinforcement at the mid-point of the prism is zero.

Similarly, when the stress in the concrete σc exceeds the tensile rupture stress of the concrete,

a secondary crack forms, in which case the prism length now reduces to Scr-p/4 as in Figure

8(c) which can then be used to both determine tension stiffening between cracks spaced at Scr-

p but also the onset of tertiary cracks should they occur.

Figure 9. Full-interaction RC prism boundary conditions

As outlined by Knight et al (2013a, 2013b), it is necessary to account for the prism size when

considering both un-tensioned reinforcement and a prestressing tendon within the same

prism. A uniform stress distribution in Figure 9(b) is assumed and the resulting force at any

cross-section being in line with Pt such that if this were not the case bending would occur and

thus reinforcement types are treated severalty and the outcomes combined in order to achieve

- 86 -

a result. Furthermore it is important to note that consideration needs to be taken in

quantifying the location of primary and secondary cracks as in Figure 9 as the stress in the

concrete prism σc is now a resultant of the combined bond stress B along the prism length Ld

from both the prestressing tendon and untensioned reinforcement (Knight et al 2013b).

Having defined the tension-stiffening behaviour using PI theory, the M/Ө analysis for the

cracked segment in Figure 6(b) can continue using the same general analysis procedure as

outlined in Figure 5. However, now the reinforcement slip Δrb and prestressing tendon slip Δt

are found geometrically from Figure 6(b) and can be used to determine the corresponding

load using the tension stiffening analysis with residual strains illustrated in Figures 7, 8 and 9.

The corresponding forces within the segment can be determined as in Figure 6(e) and the

maximum concrete displacement δcc-m varied in order to satisfy longitudinal equilibrium at

which point the moment M can be determined.

Hence, the analysis in Figure 6(b) can be applied for increasing rotations in order to quantify

the M/Ө for B-C in Figure 3(a), which can be effectively converted to a M/χ relationship in

Figure 3(b) by dividing the rotation Ө by the deformed length Ldef and, consequently, the

resulting M/EIequ can be quantified as illustrated in Figure 3(c) where EIcr is the equivalent

cracked flexural rigidity and M represents an applied moment. The M/χ and M/EIequ

relationships determined from the M/Ө analysis are mechanics based and will differ from a

standard analysis technique which is reliant on empirical components (Bishcoff 2005, CEB

1992, 2010, AS3600 2009) allowing for time effects. The change in the M/Ө, M/ χ and

M/EIequ accounting for time-effects compared to an instantaneous beam response is illustrated

in Figure 3 for a time interval t.

Beam Analysis

Having determined the M/EIequ relationship of the prestressed beam subjected to sustained

loading, as well as the equivalent uniform curvature χpr-t due to prestress and time-effects, it is

possible to determine the beam deflection. Firstly, consider the simply supported beam

illustrated in Figure 10(a). Prior to the application of an external traverse load, there exists a

uniform curvature χpr-t, as illustrated in Figure 10(d) which for a post-tensioned application is

determined from Figure 1(b) or pre-tensioned from Figure 2(b). An external traverse load P is

applied in Figure 10(a) resulting in the moment M distribution illustrated in Figure 10(b) and

hence from the M/EIequ relationship in Figure 3(c) it is possible to quantify the flexural-

rigidity EI of the prestressed beam along the length as in Figure 10(c). Knowing the EIequ in

Figure 10(c) and the moments in Figure 10(b) the variation in curvature along the beam

length can be determined and these are added to the residual uniform curvature χpr-t to give

the total curvature profile in Figure 10(d) which can be integrated twice to give beam

deflection.

- 87 -

Figure 10. Beam deflection numerical procedure

Application to Test Results

The outlined segmental moment-rotation approach presented in this paper may now be used

to predict the time-dependent behaviour of both reinforced concrete beams with steel and

FRP prestressing tendon under sustained loading as tested by Zou (2003).

Material Properties

The outlined M/Ө analysis is not dependent on any specific material property that is it can

accommodate any material property. The following material properties were used in the

analysis of these beams. However these could be substituted by alternative values if they

were thought to be more accurate.

For the elastic modulus of concrete at any desired point in time Ec(t, t0), an effective modulus

method outlined by AS3600 (2009) is used where for a given time t the effective modulus of

the concrete is given by

)tφ(t,1

)(tE)t(t,E

0

0c0c

(6)

where t0 is the time at first loading in days and φ represents the creep coefficient at a time t

for concrete first loaded at t0.

The design shrinkage strain εsh outlined by AS3600 (2009) is employed whereby the

following two components are required

(t)ε(t)ε(t)ε cs.ecs.dsh (7)

where the final subscripts d and e refer to drying and exothermic values respectively and are

derived from AS3600 (2009) based on the concrete compressive strength fc in MPa and time t

in days

)(1.0e10).50f(0.06(t)ε 0.1t6

ccs.e

(8)

csd.b41cs.d εkk(t)ε (9)

- 88 -

)0.15t(t

)t(αk

h

0.8

11 0.8 where

d)(bbdth

, h0.005te

1 1.20.8α

(10)

where 6. 10800 bcsd

and k4=0.65 for interior environments.

Prior to cracking, a linear stress strain relationship for tensile concrete is assumed. The

concrete stress σ in compression and up to the peak stress is based on the following parabolic

distribution by Hognestad et al (1955).

2

2

pk

c

pk

ccf

(11)

where εc is the concrete strain to cause a stress and fc in MPa is the concrete strength

corresponding to εpk being the peak concrete strain, defined empirically by Tasdemir et al

(2008) as

610).10539.29067.0( ccpk ff (12)

For reinforcements, the bond-stress slip model outlined by CEB (1992) is adopted in order to

accommodate FRP tendons, where for monotonic loading the bond stress between the

reinforcement and adjacent concrete is quantified as a function of the relative slip Δ

according to Equations 13-14

0.4

1maxFRP Δ

Δττ

for 10 (13)

-τ-ττ

1

1maxmaxFRP

for max1 (14)

where 6.01 , f 5.2max MPa

While for the steel strand, the bond stress-slip model from CEB (1992) is adopted in order to

accommodate the reduced bond stress experienced with uncoated/smooth prestressing

tendons. For monotonic loading, the bond stress τsteel between standard reinforcement and the

surrounding concrete can be calculated as a function of the relative displacement Δ as follow

p

0.4

1maxsteel n

ΔΔττ

for 10 (15)

pmaxsteel nττ for 21 (16)

p

23

2fmaxmaxsteel n

ΔΔ

ΔΔ)τ(τττ

for 32 (17)

pfsteel nττ for 3 (18)

- 89 -

where max4.0 f is in MPa for confined concrete, the surface type bond reduction factor as

specified by (CEB 1992) is ƞp =0.2 for smooth strands, ƞp =0.4 for spirally bound strands, ƞp

=0.6 for ribbed strands and ƞp =1.0 for untensioned ribbed reinforcing bars

Comparisons to Test Results

The presented M/Ө analysis is compared to test results of a series of 4 beams tested by Zou

(2003) in Figures 11(a-d). These beams are simply supported over a span of 6000mm with

width 150mm and depth 300mm with no traverse load for time period 0-56 days and loaded

at 2 points 2500mm from each support for an extended time period. Beams labelled ‘CFRP’

are pretensioned with 2 No. 8mm CFRP tendons while the ‘Steel’ beam is prestressed with 2

No. 9.3 mm steel strands, all with 65mm of cover. For all beams, the prestressing force is

transferred at a concrete age of 9 days when the concrete strength fc is 37MPa for CFRP-

1/Steel-1, 55MPa for CFRP-3 and 66 MPa for CFRP-2 with fc increasing to 52MPa, 74MPa

and 77Mpa at the age of 28 days, for the respective beams. Specimens CFRP-1/Steel-1 were

loaded with two point loads at the age of 56 days equating to slightly less than the cracking

moment, followed by an additional 3kN point loads at age 256 days to induce cracking,

whereas specimens CFRP-2 remained uncracked under the initial and sustained 3kN load

combined with the time-dependent effects. A 2-point applied load equating to 1.2 times the

cracking moment was applied to CFRP-3 at the age of 56 and sustained for 300 days.

Steel-1and CFRP-1 beams are identical except for the prestressing tendon material, that is

they have the same prestressing force as well as identical concrete material properties. Due to

the lower relaxation and modulus of elasticity of the CFRP tendons, it is observed that a

lower prestress loss is present in the initial self-weight loading stages, that is from 9-56 days

in Figures 11(a) and (c), resulting in a greater upwards deflection of beam CFRP-1 compared

to that of beam Steel-1. Beam CFRP-2 in Figure 11(b) remains uncracked under the sustained

traverse load over the time period 56-286 days, while beam CFRP-3 in Figure 11(d) cracks

under the initial traverse load applied at 56 days which is sustained for a period of 300 days.

As the long-term material properties are unpublished by Zou (2003), it is necessary to

quantify them through beam behaviour in the pre-loaded and uncracked stage of beam

loading that is during the period from 9 to 56 days; hence, omitting errors associated in

quantifying material behaviour using AS3600 (2009), which can have considerable variation

between predicted and actual material properties. This was simply done by adjusting the

material properties to get good correlation at the preloaded uncracked stage within the first 56

days. It can be seen in Figure 11(a) for the prestressed beam CFRP-1 that by quantifying the

long-term material behaviour in the uncracked region, a good correlation with experimental

results is achieved once the beam is cracked. The good correlation can also be seen for Steel-

1, CFRP-2 and CFRP-3 beams throughout loading, whether the beam is cracked or uncracked

over the sustained loading intervals.

The quantified material properties from the uncracked analysis, now used in the AS3600

(2009) code approach provides estimates of beam behaviour over time showing good

deflection predictions in the uncracked region. Once the beam is cracked, the approach

tended to overestimate beam stiffness resulting in lower beam deflections for FRP beams.

- 90 -

Correction factors (Zou 2003, AS3600 2009) are typically introduced in order to resolve the

deflection errors associated with the over-estimation of stiffnesses experienced in the code

approach, particularly for FRP reinforced beams. It can be seen for the prestress steel beam

Steel-1 in Figure 11(c), through quantifying the material properties in the uncracked stages,

the PI M/Ө approach provides a good prediction in the cracked regions, while the code

approach tends to only slightly overestimate beam deflection throughout the cracked time

intervals.

Figure 11. Time-Deflection experimental response and predictions for prestress CFRP and

Steel beams

Zou (2003) reports a primary crack spacing for beam CFRP-1 between 300-400mm while the

predicted crack spacing was Scr-p=345mm, while the crack spacing for beam CFRP-3 was

reported to be between 350-500mm compared to the predicted spacing of Scr-p=395mm,

showing good correlation with test results. Furthermore, it can be seen that the presented PI

M/Ө approach is capable of quantifying the equivalent flexural rigidity EIequ of a prestressed

concrete beam subjected to sustained loading and accounting for shrinkage, prestress, creep

and relaxation for both uncracked and cracked PC segments. Figure 12 illustrates the change

in M/EIequ for beam CFRP-1 for various sustained loading time intervals, which can be used

in a standard beam analysis to quantify beam behaviour. It can be seen that for a lower

sustained time interval, the cracked EIequ remains greater than for the extended time periods.

Furthermore, the instantaneous loss of moment due to crack instability at initial crack

formation is greater for longer time periods as a greater loss of prestress occurs due to loss of

flexural stiffness.

0 100 200 300 400 500 600-10

0

10

20

30

40CFRP-1

0 50 100 150 200 250 300-5

0

5

10CFRP-2

0 100 200 300 400 500 600-10

0

10

20

30

40Steel-1

Time (Days)

Defl

ecti

on

(m

m)

0 100 200 300 400-20

0

20

40

60

80CFRP-3

Predicted

Experiemental

Code

(a)

(d)

(b)

(c)

- 91 -

Figure 12. M/EIequ for beam CFRP-1

It can therefore be seen that by quantifying the long-term material properties, the PI M/Ө

approach provides good estimates of beam behaviour through quantifying beam equivalent

flexural rigidities EIequ for sustained loading. The EIequ can be used in a standard beam

analysis to quantify beam deflection, particularly in the cracked regions and without the need

for empiricisms or correction factors typically relied upon in code approaches for prestressed

design.

Conclusions

This paper presents a mechanics based residual strain moment-rotation approach that allows

for the slip between the reinforcements and concrete and for time effects. It has been shown

how it can be used to predict the long-term flexural behaviour of prestressed concrete beams.

This time-dependent M/Ө approach is capable of quantifying the rotation at all stages of

sustained loading; that is at the prestress application, through serviceability loading and can

accommodate the formation of cracks due to applied loads and time-effects. The PI M/Ө

approach quantifies equivalent flexural rigidities that can be used in standard analysis

techniques to derived beam deflections. The code approach is shown to rely upon correction

factors and empirical factors, namely in the cracked regions, to accommodate different

reinforcement types such as FRP with varying surface treatments. The presented PI M/Ө

analysis predicts well the time-dependent behaviour though a mechanics based approach

without the reliance on empiricisms and correction factors. Being mechanics based, the M/Ө

approach may be applied generically to allow for any type of prestressed concrete beam with

varying reinforcement and prestressing types and hence is ideally suited for FRP

reinforcement where the modulus and bond properties can vary widely.

Acknowledgements

The authors would like to acknowledge the support of both the Australian Research Council

ARC Discovery Project DP0985828 ‘A unified reinforced concrete model for flexure and

shear’.

0 2 4 6 8 10 12

x 1012

0

5

10

15

20

25

30

EIequ

(kNm2)

Ap

pli

ed

Mo

men

t (k

Nm

)

t = 9 days

t= 350

t = 150

- 92 -

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- 95 -

CHAPTER 3

Background

Having established the M/Ө approach for simulating the instantaneous and time-dependent

behaviour of conventional prestressed members, the fifth manuscript, ‘Simulating RC beams

with unbonded FRP and steel prestressing tendons’, furthers the approach to being able

simulate the behaviour of RC beams with unbonded reinforcement. The quantified

deformations of individual segments introduced in the preceding chapters are now

incorporated into a global analysis in order to determine the developed stress in the unbonded

reinforcement and thus the load deflection response of the beam. Being able to directly

quantify the behaviour of each segment enables the approach to remove the typical reliance

on empirical bond-reduction factors in quantifying the developed stress in unbonded

reinforcement.

Further adaption of the approach is presented in the sixth manuscript of this thesis,

‘Simulation of RC beams with mechanically fastened FRP strips’, which introduces a

numerical analysis for RC beams with mechanically-fastened (MF) FRP allowing for both the

PI behaviour at the fastener and at the bonded reinforcement, as well as the effects of

concrete softening. This analysis forms the basis of a generic technique which can

subsequently be used in the design of MF systems, with and without prestress, and therefore

provide the foundation for developing design guidelines for such beams.

List of Manuscripts

Knight, D., Visintin, P., Oehlers, D.J., and Mohamed Ali, M.S. Simulating RC beams with

unbonded FRP and steel prestressing tendons. Accepted to Composites Part B.

DOI;10.1016/j.compositesb.2013.12.039

Knight, D., Visintin, P., Oehlers, D.J., and Mohamed Ali, M.S. Simulation of RC beams with

mechanically fastened FRP strips. Submitted to Composite Structures

- 96 -


Simulating RC beams with unbonded FRP and steel prestressing tendons. Accepted to

Composites Part B. DOI; 10.1016/j.compositesb.2013.12.039





Signed…………………………………………………………………………..Date…………

Visintin, P




Signed………………………………………………………………………….Date…………

Oehlers, DJ




Signed………………………………………………………………………….Date…………

- 97 -

Mohamed Ali, M.S




Signed………………………………………………………………………….Date…………

- 98 -

Simulating RC beams with unbonded FRP and steel prestressing tendons

Daniel Knight, Phillip Visintin, Deric J Oehlers and Mohamed Ali, M.S

Abstract

A partial interaction based analysis to simulate the behaviour of RC beams with prestressed

unbonded tendons is proposed. Unlike bonded reinforcement, the strain developed in

unbonded reinforcing tendons under bending is uniform along the length of the member and

is thus member dependant. Conventional analysis techniques incorporate correction factors

and empirical components in defining the strain developed in both the unbonded and bonded

reinforcement. Being semi-empirical, the post-cracking analysis cannot directly simulate the

effects of tension-stiffening on the untensioned bonded reinforcement. Accordingly, this

paper presents a segmental moment-rotation approach for simulating the behaviour of RC

beams with unbonded prestressed reinforcement, such that the mechanics of the approach

removes the reliance on empiricisms in defining the reinforcement and unbonded tendon

behaviour. Validated against experimental results, the approach is shown to accommodate

concrete creep, shrinkage and reinforcement relaxation, thus enabling prestressing losses to

be quantified.

Keywords; prestressed concrete, FRP, partial-interaction, tension-stiffening, unbonded,

creep, shrinkage, numerical analysis.

Introduction

The use of unbonded prestressed reinforcement in conventional reinforced concrete (RC)

construction is attracting increased attention, as compared to traditional bonded prestressed

(Dolan and Swanson 2002) construction as the process is simplified by removing the need for

tendon grouting (Lou et al 2013, Barberi et al 2006, He and Liu 2010).

The absence of bond between the prestressed reinforcement and the surrounding concrete

violates the condition of strain compatibility, which is the basis of traditional full-interaction

moment-curvature (M/χ) analysis techniques. This is because the strains in the unbonded

reinforcement are dependent on the total deformation of the tendon along the total length of

the member. Hence, the behaviour of an RC member with unbonded reinforcement is

dependent on both: the deformation of the member; as well as the cross section.

Conventional analysis techniques resolve the stress increment fps in the unbonded

prestressing steel using a bond-reduction method (Naaman and Alkhairi 1991, Naaman et al

2002, Ghallab and Beeby 2008, ACI 318). Using this approach an empirically derived

correction factor is applied to the M/χ analysis in an attempt to reduce the cross-section to an

equivalent cross-section with bonded reinforcements. Being empirically defined and reliant

on firstly quantifying the behaviour of an equivalent bonded member, this approach is unable

to be generically applied to any member type and any loading scenario and notable

- 99 -

differences between the various empirical derivations exists (Harajli and Kanjim 1992).

Moreover, the reliance on empiricisms to define the mechanics of member behaviour means

the bond-reduction approach cannot simulate the behaviour observed in practice, that is, it

cannot simulate the mechanisms of crack formation and widening and the mechanism of

concrete softening.

In this paper, a new approach for the analysis of members with unbonded prestressed

reinforcement is presented. The proposed approach is an extension of the segmental moment-

rotation (M/Ө) approach which has been developed for the analysis of both conventional RC

beams and columns under instantaneous and long term loading (Oehlers et al 2012, 2013,

Visintin et al 2012a, 2012b, 2012c, 2013) and extended for the analysis of members with

bonded prestressed reinforcement under instantaneous and sustained loading (Knight et al

2013a, Knight at el 2013b).

The segmental M/Ө approach uses the mechanics of partial interaction (PI) theory (Visintin

et al 2012a) and (Gupta and Maestrini 1990, Oehlers et al 2005, Haskett et al 2008, 2009a,

2009b, Foster et al 2010, Oehlers et al 2011, Muhamed et al 2012, Mohamed Ali et al 2012)

to directly simulate the slip of bonded reinforcement relative to the concrete encasing it and,

hence, describes crack formation, crack widening and tension stiffening, including that

which is influenced by the residual strains associated with concrete creep, shrinkage and pre-

stressed reinforcement (Visintin et al 2013, Knight et al 2013a, 2013b). Furthermore, the

approach also uses the mechanics of shear friction theory (Visintin et al 2012a, 2013, Haskett

et al 2010, 2011) to describe a size dependent stress strain relationship (Chen et al 2013)

which can, through mechanics, simulate the formation of wedges associated with concrete

softening (Oehlers et al 2012). Hence, using the segmental M/Ө approach, the mechanics of

partial interaction and shear friction directly simulate the mechanisms of concrete cracking

and softening such that the only empiricisms required for analysis are those associated with

defining material properties.

In this paper, having described the segmental M/Ө approach, it will be shown how it can be

applied at a member level to determine the deflection of an RC member with unbonded

prestressed reinforcement, including the deflections associated with concrete creep and

shrinkage. The approach is then validated against a series of tests conducted by Harajli and

Kanjim (1992), Tao et al. (1985) and Saafi and Toutanjli (1998) and a parametric study is

carried out to show how the approach can be used to directly quantify the strain in the

unbonded reinforcement at all loading stages. Hence, it is shown how the M/θ approach can

be used to determine strain reduction coefficients which can be used in existing analysis

techniques without the need for experiments.

Moment-rotation of segment

Consider a segment of length 2Ldef in Figure 1(b), where 2Ldef corresponds to the crack

spacing, to which a constant prestressing moment Mpr-0 is applied. The segment is extracted

from a reinforced concrete RC member with unbonded prestressed reinforcement and is of a

cross section as in Figure 1(a). In order to accommodate the deformations due to both

concrete shrinkage and the application of prestress, let us first establish the total deformation

- 100 -

which results in the development of stress within the segment (Visintin et al 2013). That is, if

unrestrained, shrinkage will cause a deformation of the concrete in Figure 1(b) from A-A to

C-C by εshLdef without inducing a stress. Hence the deformation profile C-C may be used as a

baseline whereby any movement of the concrete from C-C to B-B relative to datum F-F

results in a strain to cause a stress in the concrete. Similarly, any movement of the

reinforcement from the initial position A-A relative to F-F will result in a strain to cause a

stress in the reinforcement. Let us now consider how the deformation of the segment can be

used to determine the segmental M/θ relationship.

Figure 1. Moment rotation analysis of a segment at prestress application

Prestress application

The application of prestress in a member with unbonded reinforcement is identical to that for

a prestressed member with bonded tendons (Knight et al 2013a, 2013b) which are initially

tensioned prior to grouting and is summarised in the flow chart in Figure 2.

Figure 2. M/Ө procedure at application of pre-stress

- 101 -

A tendon prestressing force Fpr-0 is applied to the segment in Figure 1(a) at an eccentricity e

causing the ends of the segment A-A in Figure 1(b) to both contract and rotate to B-B; it is,

therefore, a matter of determining the location of profile B-B such that both force and

rotational equilibrium are obtained. For analysis as described in Figure 2 using Figure 1, a

rotation of the segment end θpr-0 is assumed and a deformation of the concrete due to the

combination of concrete shrinkage and prestress application at the extreme compression face

δtop-0 is guessed, thereby, defining the deformation profile B-B in Figure 1(b). The strain

profile E-E which results in a stress in the concrete is, therefore, given by

(1)

where δ is the deformation between A-A and B-B at a specific level.

The strain profile G-G in Figure 1(c) resulting in a stress in the reinforcement, due to the

deformation form A-A to B-B, is given by

(2)

Having defined a strain profile for both the concrete and the reinforcement in Figure 1(c), the

internal stresses and forces as in Figures 1(d) and (e) are also known through the application

of standard material stress strain relationships. It is then a matter of iterating as in the flow

diagram in Figure 2 until both force and rotational equilibrium are achieved, that is the

resultant force in the section FRC in Figure 1(e) is equal to and in line with the prestressing

force Fpr-0 as shown. This gives the rotation Өpr-0 for a specific prestressing moment Mpr-0 as

plotted in Figure 3(a). An equivalent curvature χpr-0 in Figure 3(b) can also be obtained

knowing χ = Ө/Ldef. Finally, the relationship between the constant member moment Mpr-0 and

the contraction of the RC segment at the level of the tendon δext-0 is known such that δext-0/Ldef

in Figure 3(c) is an effective strain εlv-td in the segment at tendon level and which is uniform

along the member length.

Figure 3. (a) M/Ө (b)M/χ and (c) M/εlv-td relationships

shε

defL

δ

cε

defL

δ

rε

- 102 -

Moment Application

Let us now determine the behaviour of the segment upon application of an applied moment

M1 as in Figure 4(a).

Figure 4. cracked segment analysis

The analysis technique upon the application of a moment is summarised in the flow chart in

Figure 5 where the analysis is carried out for a given rotation θ at the segment end in Figure

4(a). As the force in the tendon Fpr depends on the deformation of the whole member and not

just that of a segment, Fpr is unknown at this stage of the analysis. A family of curves with

varying Fpr is therefore generated here for use in the ensuing member analysis.

Figure 5. Moment-rotation procedure for segment

- 103 -

For a given tendon force Fpr-1 in Figure 4, a change in rotation Δθ1 from the initial rotation θpr-

0 is set and the applied moment M1 to cause this total rotation θ is sought. A total deformation

δtop-1 due to time effects, prestress and M1 is set at the top fibre, thereby, defining the

deformation profile D-D.

The concrete strain profile E-E in Figure 4(b) can then be determined using Eq. 1 and using a

size dependent stress-strain relationship (Chen et al 2013) to determine the stress profile in

Figure 4(c), the force profile in Figure 4(d) is determined. The use of a size dependent stress

strain relationship, derived using shear friction theory where the length of the prism from

which the concrete material properties should be extracted corresponds to 2Ldef in Figure 4,

allows for concrete softening through the formation of a wedge as in Figure 4(a) without the

need for an empirical hinge length (Oehlers et al 2012, Chen et al 2013)

For bonded reinforcement located in the uncracked region, the stress developed can be

determined from the strain profile determined using Eq. 2 and, hence, the internal forces can

be obtained from material stress-strain relationships. For reinforcement located in the cracked

region, the load developed is a function of the slip of the reinforcement Δreinf in Figure 4

relative to the concrete in which it is encased; a behaviour which is commonly referred to as

tension stiffening.

The uncracked analysis in Figure 1 can be used to determine when the segment first cracks

and this will be referred to as the initial crack. Furthermore, each of the bars in Figure 1(a) is

encased by concrete and this will be referred to as a tension stiffening prism. Consider a

tension stiffening prism as in Figure 6(d) with an initial crack on the left hand side. The

reinforcement load Freinf in Figure 6(d) causes a slip δ between the reinforcement and the

adjacent concrete as in Figure 6(c). The accumulation of the interface slip along the length of

the bonded interface results in a total slip at the crack face of Δreinf. To quantify this slip and

the forces it induces requires the mechanics of partial-interaction theory that incorporates the

bond-slip (δ -δ) relationship. There are numerous closed form (Muhamed et al 2012,

Mohamed Ali et al 2012) or numerical solutions (Visintin et al 2012a, 2012b, 2012c, Knight

et al 2013, Gupta and Maestrini 1990, Haskett et al 2008, 2009a, 2009b, Wu et al 1991, Choi

and Cheung 1996, Marti et al 1998, Morza and Houde 1979, Lee and Kim 2008) readily

available that can quantify the variation in interface slip δ or interface slip-strain dδ/dx along

the member and the only difference in the solutions are the required boundary conditions.

Moreover, the effects of concrete creep and shrinkage are accommodated through a residual

strain approach as suggested by Visintin et al (2013) and Knight et al (2013b).

- 104 -

Figure 6. Partial-interaction tension-stiffening procedure

Let us first consider the behaviour at the initial crack shown in Figure 6(d) (Oehlers et al

2011, 2012, Visintin et al 2012a). Partial-interaction theory, which requires the bond-slip

properties (τ-δ) can be used to: quantify the variation in the slip-strain dδ/dx in Figure 6(a) in

which dδ/dx is the difference in strain across the bond interface, that is εr-εc in Figure 6(d);

the variation in the bond shear stress τ along the prism as in Figure 6(b); and the variation in

slip δ as in Figure 6(c).

For a long length of reinforcing bar, the boundary condition required for the single crack

analysis is that at some point located Lprim from the crack face, both dδ/dx and δ tend to zero.

The partial-interaction analysis in Figure 6(d) can predict the minimum position at which the

next crack could occur and hence provide the segment length Ldef in Figure 4(a) where the

primary crack spacing Lprim = 2Ldef, should the concrete stresses be large enough. Once

primary cracks occur, the partial-interaction analysis is that of a symmetrically loaded prism

in Figure 6(e) of length Lprim where by symmetry the slip at Lprim/2 is zero which is the new

boundary condition. The analysis of this prism can be used to predict when cracking could

occur at the mid-length Lprim/2 and should these secondary cracks occur the analysis is that

shown in Figure 6(f). Hence the relationship between Freinf and reinf required in the analysis

in Figure 4 when cracking occurs can be derived from the partial-interaction analyses

depicted in Figure 6.

- 105 -

Having now defined all the forces in Figure 4(d), whether the segment is cracked or not, δtop

can be adjusted until equilibrium of internal forces is obtained that is the algebraic sum of

forces in Figure 4(d) is zero after which moments can be taken to determine the applied

moment M1.

The above analysis provides the remaining component of M/ϴ in Figure 3(a), that is after the

initial prestress, and the corresponding M/χ and M/δext relationship for a given tendon pre-

stressing force Fpr. This can be used to determine the relationship for a family of pre-stressing

forces for use in the following member analyses.

Member analysis

The segmental M/Ө relationships extracted from a segment can now be applied at a member

level to determine member deflection. Consider the beam in Figure 7 where the un-bonded

pre-stressed reinforcement is anchored at the ends such that it can be assumed that no slip of

the anchorages occurs.

Figure 7. (a) Member analysis (b) curvature distribution

To determine the load deflection behaviour of the member an iterative approach as outlined

in Figure 8 is applied.

Figure 8. Unbonded member load-deflection response procedure

- 106 -

Having defined a M/Ө and corresponding M/χ and M/εlv-td relationship as shown in Figure 3

for a given applied moment Mapplied, it is now a matter of determining the corresponding

unbonded tendon force in Figure 7(a).

Using the M/εlv-td relationship in Figure 3(c), the strain variation over the member length is

known, hence, integration of this strain gives the total extension of the tendon. Through

knowing the tendon material properties the calculated tendon force Fpr-calc can then be

determined as Fpr-calc = fn(δlv-td). If the calculated tendon force is not equal to the initially

applied tendon force Fpr the moment distribution must be adjusted until it does so.

Having obtained the correct moment distribution, the distribution of curvature along the

member can be determined using the M/χ relationship shown in Figure 3(b) and the

deflection determined by integration using standard analysis techniques. The analysis can

then be repeated for an increasing applied moment Mapplied in order to produce the full

member load-deflection relationship.

It should be noted that any frictional resistance between the unbonded reinforcement and

adjacent concrete is deemed to be negligible as suggested by Alkhairi and Naaman (1993) for

straight tendons. Furthermore, we may also consider the partial-interaction behaviour of the

end anchorages, commonly referred to as wedge seating (Warner et al 1998, Nawy 2010,

Gilbert and Mickleborough 2002) which results in a loss of prestressing force due to

unbonded reinforcement shortening over length Lmember in Figure 7(a). Displacement due to

wedge seating is typically defined by the manufacturer and is specific to the anchorage type.

The general analysis procedure applies, as previously outlined, with the addition of a change

in tendon length due to an anchorage movement, such that Fpr-calc is recalculated based on

further iterations.

Comparison to test results

Having defined the behaviour of an RC member with unbonded prestressing reinforcement

using the segmental M/ϴ approach, it is now applied to simulate the load deflection response

of tests carried out by Harajli and Kanj (1992), Tao and Du (1985) and Saafi and Toutanji

(1998).

Beam FRP-1 tested by Saafi and Toutanji (1998) and illustrated in Figure 9(a) is of length

2500mm with cross-section dimensions 300x150mm prestressed with 4 unbonded AFRP

tendons with cross-sectional area 30mm2, ultimate strength 1330MPa and an elastic modulus

of 50GPa. The beam with a concrete strength of 41MPa is loaded under 4 point bending with

the applied loads located 700mm from the end supports. It was reported that the ultimate load

capacity of member is governed by the rupture of a single tendon, allowing for further

developments in load carrying capabilities due to additional stress increments in the 3

remaining un-bonded tendons. It can be seen that despite slight discrepancies between

predicted and experimental behaviour in the post-yield region, the general predicted trend

from the presented approach is very good.

- 107 -

Beams Steel-1 and Steel-2 in Figure 9(b) and (c) were tested by Tao and Du (1985) under 2

point loads and had a length of 4200mm with cross-sectional dimensions 280x160mm. Beam

Steel-1 is constructed from concrete of strength of 30.6-MPa, prestressed with a single

unbonded steel tendon with cross-sectional areas 58.8mm2 and reinforced with bonded steel

reinforcement of 157mm2. Beam Steel-2 is constructed form concrete of strength 33.1MPa, is

prestressed with a total area of tendon of 156.8mm2 and with bonded steel reinforcement of

total cross-sectional area of 804mm2.

It is reported that the beam Steel-1 in Figure 9(b) had a low combined reinforcement ratio

resulting in the internal bonded reinforcement yielding followed by ultimate failure due to

concrete crushing; represented by the descending region in the load-deflection response. The

model predicts well the ultimate load carrying capacity of beam Steel-1 in Figure 9(b) with

only slight discrepancies between predicted and experimental results. Similarly, the predicted

behaviour of beam Steel-2 correlates well to the experimental data through the entire loading

range.

Finally it can be seen that the presented approach provides a reasonable prediction of beam

Steel-3 as tested by Harajli and Kanj (1992). Beam Steel-3 is tested under 2 point loads and

has cross-sectional dimensions 127x228mm with length 3048mm and a concrete strength of

44MPa. The beam is reinforced with 2 No. 6mm reinforcing bars with a yield strength of

275MPa while the prestressed reinforcement consists of a single 5mm steel tendon with an

effective prestress of 1606MPa. Figure 9(d) shows that the predicted model tends to under

estimate the behaviour of the member throughout loading; however the general trend of the

load-deflection relationship is very good.

Figure 9. Comparisons to unbonded steel and FRP members

0 5 10 15 20 25 30 3510

20

30

40

50FRP-1

Deflection (mm)

Lo

ad

(k

N)

0 20 40 60 80 100 1200

10

20

30

40

Deflection (mm)

Lo

ad

(k

N)

Steel-1

0 10 20 30 400

50

100

150

Deflection (mm)

Lo

ad

(k

N)

Steel-2

Experimental

Predicted

Experimental

Predicted

Experimental

Predicted

0 10 20 30 400

2

4

6

8

Deflection (mm)

Lo

ad

(k

N)

Steel-3

data1

Experimental

Predicted

(b)

(c) (d)

(a)

- 108 -

Unbonded tendon stress

In order to determine the capacity of a member with un-bonded pre-stressed tendons it is

common practice to only define the tendon stress increment (fps,u) at the nominal flexural

strength of the unbonded prestressed member through the application of empirically derived

equations (Namman and Alkhairi 1991, ACI-318 2008, Harajli and Kanjim 1992, Tao and Du

1985, BSI-8110 1985). While there is no doubt that the empirically derived equations provide

a reasonable estimate of fps,u when applied within the bounds of the data set from which they

were extracted, as shown in Table 1, when applied to a more generalised data set correlations

are poor and the range of predictions implies that the full range of physical behaviour

occurring in practice is not captured in all empirical equations. While the reliance on first

defining these empirical equations to account for tendon type, beam dimensions, loading

types and member materials mean that they cannot be generically applied to any member type

(Harajli and Kanjim 1992).

Table 1: Predicted and Experimental fps,u for Steel-1 and Steel-3

Method tal)(Experimen

)(Predicted

ups,

ups,

f

f

Beam Steel-1 Beam Steel-3

ACI 318 (2008) 1.31 0.83

BS8110 (1985) 1.06 0.91

Harajli and Kanj (1992) 1.23 0.81

Tao and Du (1985) 0.86 1.01

M/Ө Approach 0.92 1.04

Furthermore, understanding the tendon stress increment (Δfps) at varying loading stages is

important for the serviceability design of unbonded prestressed members, particular in

quantifying time-effects. Conventionally, in order to quantify Δfps throughout loading and

prior to ultimate, the unbonded section is reduced to a bonded analysis through the use of a

bond-reduction coefficient such that strain compatibility applies. This bond-reduction

coefficient is determined in the uncracked state and subsequently adjusted based on the

flexural rigidity (EIeff) of the member in a cracked state, thus being reliant on firstly

quantifying Ieff, which is typically based on the well-known equation derived by Branson and

Trost (1982) which itself is empirical. Despite Branson’s equation being generally accepted

as providing a reasonable estimate of the effective flexural rigidity for members which are

reinforced to levels within the bounds of the experimental tests from which it was calibrated,

when used outside of the test bounds discrepancies between observed and predicted

deflections result, particularly for members with low levels of reinforcement (Gilbert and

Ranzi 1995).

Now consider Figure 10(a,b) which show a comparison of measured tendon stress increments

(Δfps) for beam Steel-1 as tested by Tao and Du (1985) and beam Steel-3 as tested by Harajli

and Kanj (1992) and the tendon stress increment which is directly determined through the

M/θ approach. It can be seen how, despite some deviations, the predicted results trend well to

the measured stress values throughout all loading stages. The final point of the predicted

stress increments in Figure 10(a,b) represent the tendon stress (fps,u) at ultimate and can

- 109 -

therefore be compared to theoretical approaches as presented in Table 1, whereby fps,u is

comprised of the initial stress (fpre ) due to prestressing plus the final stress increment (Δfps).

The approach presented in this paper may therefore be applied to the calibration empirical

factors for use in the codified bond-reduction approach without the reliance on extensive

experimental testing.

Furthermore, being able to quantify the tendon stress increment (Δfps) at varying loading

stages is useful in determining the long-term loading behaviour of the unbonded prestressed

beam, notably in quantifying the time-effects associated tendon relaxation.

Figure 10. Comparisons of unbonded tendon stress increment Δfps

Time-Effects

We can now consider time-dependent effects on the behaviour of unbonded prestressed

member by accounting for concrete creep, shrinkage and tendon relaxation. Consider member

with cross-sectional dimensions 250mmx150mm of length 3500mm prestressed with 2

unbonded CFRP tendons with an ultimate strength of 1590MPa combined with 4 bonded

steel reinforcing bars with a yield strength of 450MPa. It can be seen that the contribution of

creep in Figure 11(a) tends to increase with the applied load, whereas for shrinkage the

increase in deflection remains relatively constant over the loading as shown in Figure 11(b).

Figure 11(c) represents the load-deflection response for the member with a combination of

creep, shrinkage and tendon relaxation. A loss of prestressing force results in a lower ultimate

load carrying capacity, while the deflection throughout loading increases at every load

increment. Hence, the approach is able to quantify the prestress loss at specific time

increments which can be used to determine if re-stressing of the tendons or additional tendons

is required to maintain flexural strength or deflection control.

0 100 200 3000

1

2

3

4

5

6

7

8

9

Tendon Stress Increment (fps

) MPa

Ap

pli

ed L

oad

(k

N)

Steel-3

0 200 400 6000

5

10

15

20

25

30

35

40

45

Tendon Stress Increment (fps

) MPa

Ap

pli

ed L

oad

(k

N)

Steel-1

Experimental

Predicted

Experimental

Predicted

(a) (b)

- 110 -

Figure 11. Influence of creep, shrinkage and relaxation on member defection

Summary

A mechanics based partial-interaction (PI) moment-rotation (M/Ө) approach developed to

simulate the behaviour of reinforced (RC) beams with unbonded prestressing FRP and steel is

presented in this paper. It is shown that the approach can allow for both the instantaneous and

sustained loading of RC beams with unbonded prestressed tendons through the application of

a residual strain PI analysis to quantify local segmental behaviour, which is then used in a

global approach in quantifying entire member behaviour. Importantly, the approach

eliminates the need for empiricisms and additional bond-reduction coefficients that are

typically relied upon in the current analysis techniques, this should greatly reduce the cost of

developing new types of FRP by reducing the amount of experimental testing required.

Moreover the approach is generic in that it can be applied to the analysis of members with

either prestressed steel or FRP reinforcement. The presented approach is compared to

published experimental results showing good correlation. It is also suggested that the

approach can be a useful tool in practice for determining the developed unbonded

reinforcement stress at any loading stage, thus enabling for the time-dependent effects on

member behaviour to be quantified.

Acknowledgements

The authors would like to acknowledge the support of the Australian Research Council ARC

Discovery Project DP0985828 ‘A unified reinforced concrete model for flexure and shear’.

0 50 100 1500

5

10

15

20

25L

oad

(k

N)

0 50 100 1500

5

10

15

20

25

Deflection (mm)

= 0

= 1

=2

0 50 100 1500

5

10

15

20

25

=0, sh

=0, Rt=0%

=1, sh

=300, Rt=1%

=2, sh

=600, Rt=2%

sh = 0

sh =300

sh = 600

(a) (b) (c)

- 111 -

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- 114 -


Simulation of RC Beams with Mechanically Fastened FRP strips. Submitted to

Composite Structures





Signed…………………………………………………………………………..Date…………

Visintin, P




Signed………………………………………………………………………….Date…………

Oehlers, DJ




Signed………………………………………………………………………….Date…………

- 115 -

Mohamed Ali, M.S




Signed………………………………………………………………………….Date…………

- 116 -

Simulation of RC Beams with Mechanically Fastened FRP Strips

Daniel Knight, Phillip Visintin, Deric J Oehlers and Mohamed Ali, M.S

Abstract

A now common approach to the strengthening of reinforced concrete beams is by attaching

fibre reinforced polymer (FRP) strips to the beams’ soffit. Strips can be either externally

bonded (EB) with epoxy resin, a process which is labour intensive and requires careful

preparation in order to achieve adequate bonding, or attached by mechanical fasteners (MF),

a process which does not require any surface preparation and typically results in comparable

strength to epoxy bonding. Predicting the response of an RC beam with an MF-FRP strip is

complex as a full member analysis is required to determine the strains developed in the FRP

strip between the fasteners. Moreover, as there is no direct interaction between the concrete

and strip, strain compatibility does not apply and thus a conventional full-interaction (FI)

approach becomes reliant on an empirically derived member behaviour. This paper presents a

member dependent analysis in which a segmental M/Ө approach is used to determine beam

deformations and forces in beams with prestressed and unprestressed mechanically attached

FRP strips. Being mechanics based the approach incorporates: residual strain partial-

interaction (PI) theory to directly simulate the effects of tension-stiffening as the internal

bonded reinforcement pulls from the crack face; a size-dependent stress-strain model to

simulate the concrete wedge formation associated with concrete softening; and allows for

both the influence of concrete creep and shrinkage. It is shown that the approach can be used

in the design of flexural members in quantifying the required number of mechanical fasteners

and fastener spacing for strengthening comparable to or exceeding that of conventional EB

systems, and thus be used to develop design guidelines for MF-FRP systems.

Introduction

An alternative to epoxy bonding (Sika 1999, Master Builders 1998) is the use of mechanical

fastened (MF) systems, whereby the fibre reinforced polymer (FRP) strip is attached to the

beams substrate with mechanical fasteners at predefined intervals (Lamanna et al 2004,

Banks and Arora 2007, Napoli et al 2009, Matrin and Lamanna 2008, Nardone et al 2011).

The MF-FRP technique has been shown to be advantageous compared to conventional

externally bonded (EB) FRP as it allows for rapid installation with minimal surface

preparation, while maintaining suitable beam ductility (Banks and Arora 2007). However, the

absence of an adhesive bond between the MF-FRP strip and concrete substrate violates the

conditions of strain compatibility which is the basis of a conventional cross-sectional

moment-curvature (M/χ) analysis. Furthermore the strain developed in the MF-FRP strip is

dependent on the deformation of the beam not only between fasteners but also over the whole

length of the member.

- 117 -

Further complexity in the analysis arises as experimental tests on RC beams strengthened

with MF-FRP systems show that a significant slip can occur at the fastener-strip and fastener-

concrete interface due to developed fastener bearing stresses and as such the effect of partial-

interaction at the fastener cannot be neglected (Napoli et al 2009). Numerous analytical and

experimental investigations on the bearing stress-slip relationship at the fastener have been

undertaken (Lamanna et al 2004, Banks and Arora 2007, Martinelli et al 2012, Elsayed et al

2009) in which the total slip at the fastener is defined in terms of the fastener shear-bearing

stress. A typical design approach (Nardone et al 2011) for RC beams with MF-FRP strips

combines the fastener bearing stress-slip relationship in a conventional full-interaction (FI)

M/χ analysis, while the effects of tension stiffening of the internal reinforcement in the

cracked regions are ignored. In order to allow for cracking, such an approach adopts an

empirically derived flexural rigidity (EIemp) to accommodate the effects of tension stiffening

(Branson 1977, CEB 1992) which tends to restrict the application to the bounds of the

experimental tests form which the empiricisms were derived. Furthermore, the formation of

concrete compression wedges result in the wedge sliding relative to the adjacent concrete

(Haskett et al 2010, 2011) and as such cannot be directly simulated by the M/χ analysis as it

is a mechanism. The M/χ approach must therefore resort to the use of empirical softening

compression stress-strain relationships which tend to be size-dependent as shown by Chen et

al (2013).

The reliance on empiricisms to define the mechanics of the members behaviour means that

the FI M/χ approach is unable to directly simulate what is actually observed in practice,

particularly the mechanisms of crack formation and widening and that associated with

concrete softening. This can become particularly problematic as the stress developed in the

MF-FRP strip between subsequent fasteners is a function of member deformations. Other

approaches incorporate finite element analyses in order to quantify the developed strain in the

MF-FRP strip and consequently the member behaviour (Napoli et al 2009, Lee et al 2009,

Martinelli et al 2014), accounting for the effects of partial-interaction in both the internal and

MF-FRP reinforcements. Such approaches tend to provide good predictions of beam

behaviour (Martinelli et al 2014), however despite the vast interest in this new strengthening

technique there remains no international guideline in dealing with MF-FRP systems at this

time.

In this paper a member analysis is first introduced in order to outline the process required in

determining the overall behaviour of an RC beam with an MF-FRP strip. It is shown how a

generic iterative procedure can be applied to quantify the force developed in each of the

unbonded MF-FRP strips, that is between fasteners, along the length of a member for an

externally applied moment. In undertaking the member analysis, a segmental partial-

interaction (PI) moment-rotation (M/Ө) approach is required in order to define beam

segmental deformations between subsequent mechanical fasteners. This approach is an

extension of the M/Ө approach used to simulate the instantaneous and sustained loading of

typical RC beams (Oehlers et al 2011, 2012, 2013, Visintin et al 2012a, 2013a, 2013b, Knight

et al 2013a, 2013b) and prestressed RC beams, with bonded and unbonded reinforcement

(Knight et al 2013a, 2013b, 2013c). The approach applies the mechanics of partial-interaction

- 118 -

(PI) theory (Oehlers et al 2005, Haskett et al 2008, 2009a, 2009b, Gupta and Maestrini 1990,

Muhamed et al 2012, Mohamed Ali M.S et al 2012) to directly simulate the slip of the

internal bonded reinforcement as it pulls from the adjacent concrete, thus allowing for crack

formation, widening and the associated effects of tension stiffening and the associated effects

of concrete creep and shrinkage.

It is shown how the reliance on empiricisms in defining the MF-FRP strip strain in the

conventional M/χ analysis (Nardone et al 2011) can be removed as the mechanics associated

with the M/Ө approach is able to directly simulate beam deformations between subsequent

fasteners. In defining the FRP strip strain, the fastener bearing stress-slip is accounted for

using the results of direct bearing stress-slip relationships as derived by Elsayed et al (2009).

Moreover, concrete wedge formation associated with concrete softening (Oehlers et al 2005,

2011) is simulated through a size dependent stress-strain relationship (Chen et al 2013) based

on the mechanics of shear friction theory (Oehlers et al 2013, Visintin et al 2013b, Haskett et

al 2009a). Being able to simulate the partial-interaction mechanisms in the segmental M/Ө

approach shows it be a useful extension to the conventional M/χ technique, in which the

reliance on empiricisms to define the mechanisms associated with tension stiffening, concrete

softening and fastener slip are removed.

Having established the complete beam analysis procedure, the approach is then validated

against experimental beams tested by Martin and Lamanna (2008) and Ebead (2011),

showing good correlation to test results for the prediction of the full-member responses and

developed fastener loads. It is then shown how this novel approach can be used as a design

tool to outline the effect of fastener spacing and prestress and the subsequent changes in the

flexural behaviour of the beam and thus be seen as a step towards developing generic rules

for the design of such beams.

Analysis of MF-FRP RC Member

The force developed in the unbonded FRP strip between mechanical-fasteners is dependent

on the deformation of the beam within this region, deeming it necessary to perform an entire

beam analysis in order to quantify the overall behaviour. It is firstly shown how the FRP strip

forces are derived from the beam analysis and subsequently how the individual segmental

behaviour, quantified from a M/Ө approach, is used to determine beam deformations within

individual fastened lengths.

Consider the MF-FRP strengthened RC beam in Figure 1(b) with a cross-section as in Figure

1(a). In this particular instance, the FRP strip is attached to the beam with n+1 fasteners,

resulting in n fastened lengths (L) that are symmetrical about the middle of the beam. It is

initially assumed that the fasteners at a particular location have a stiffness denoted by (k), as

quantified using a typical bearing stress-slip model (Elsayed et al 2009), and that the number

of fasteners at a location may vary. In this analysis, it will be assumed that the beam is

symmetrical about its mid-span and also symmetrically loaded about its mid-span so that the

slip between the FRP strip and the adjacent beam surface at mid-span is zero through

symmetry as shown in Figure 1(b).

- 119 -

The beam analysis begins by fixing the FRP strip force PFRP-1 over the initially unstressed

length L1 in Figure 1(b). It is now a question of finding the applied moment Mapp in Figure

1(b) that causes or induces the strip force PFRP-1 over length L1. As the strip force PFRP-1 over

length L1 is known, the strip strain ԐFRP-1 over L1 is known. Subsequently the extended length

of the strip (LFRP-1) is given by L1+ԐFRP-1L1 which is shown in Figure 1(c). The magnitude of

the moment distribution Mapp in Figure 1(b) is now guessed over the length of the member,

such that the moment acting over the fixed fastened length L1 in Figure 1(b) is now known.

The extended length of the bottom or soffit of the RC beam, Lb-1 in Figure 1(c), that is the

beam adjacent to the FRP strip, due to the moment distribution Mapp is dependent on the

deformations of the beam segments within this region (the derivation allowing for creep,

shrinkage and cracking is explained in detail later). Thus the slip (s2) at fastener 2 is equal to

the difference in lengths Lb-1-LFRP-1 as shown in Figure 1(c). Therefore the developed load in

fastener 2 referred to as PF-2 is equal to k.s2. Thus over the adjacent length L2 in Figure 1(b)

the developed FRP strip force (PFRP-2) is equal to PFRP-1 - PF-2.

The above shooting method of analysis is repeated along the length of the beam to a known

boundary condition. In this example for lengths L2 to Ln where at the end of Ln the force in

the fastener is equal to the force in the FRP strip, that is PF-(n+1) equals PFRP-n. If over the

length Ln the boundary condition is not achieved, that is the force FFRP-3 does not equal the

fastener load PF-(n+1), then the magnitude of Mapp is varied until this condition is satisfied.

Once this boundary condition is reached the curvature profile in Figure 1(d) is known along

the beam length, as determined from the results of a segmental analysis used to derive the

deformation of the soffit such as Lb-1 and which is introduced in the following section. The

beam deflections may be evaluated from the variation in curvature using standard analysis

methods.

Figure 1. Member analysis

- 120 -

Understanding the segmental deformations within the fastened lengths is paramount in

determining the extended length at the soffit such as Lb-1 in Figure 1(c) required for the beam

analysis and in determining the curvature profile along the beam. Having established how the

developed force in the MF-FRP strip can be determined between subsequent fastened lengths

along the beam and its dependence on the deformation of the RC beam itself, it is now shown

how beam deformations may be quantified through a segmental partial interaction M/Ө

approach.

Moment-Rotation of a Segment

It has been shown using Figure 1 that the slip in the fasteners depends on the longitudinal

deformation of the FRP strip and on the longitudinal deformation of the RC beam at the soffit

that is adjacent to the FRP strip. The longitudinal deformation of the strip is relatively easy to

quantify as it can be obtained directly from the FRP strip material properties and the axial

force in the strip. The longitudinal deformation at the RC soffit is incredibly difficult to

quantify as it depends on: the distribution and width of primary and secondary flexural cracks

along the member; short term and time dependent concrete material properties such as that

due to creep, shrinkage and thermal gradients; the bond-slip between the internal

reinforcement and adjacent concrete; and concrete softening in compression regions. These

variables can be allowed for using empirically based approaches such as the use of

empirically derived flexural rigidities (Branson 1977, CEB 1992). However the moment-

rotation approach has been used in this research as it eliminates much of the empiricism

through the use of mechanics and in particular the use of mechanics to simulate the partial-

interaction behaviours of both tension-stiffening and concrete softening in compression

regions associated with the formation of hinges (Oehlers et al 2005, 2011, 2012, 2013,

Visintin et al 2012, 2013a, 2013b, Knight et al 2013a, 2014b, 2013c, Haskett et al 2009a).

The moment-rotation approach derives the properties of an RC beam through the mechanics

of a small segment of the member of length (2Ldef) (Oehlers et al 2011, Visintin et al 2012,

Haskett et al 2009a, 2009b) that is subjected constant stress resultants such as the moment M

and axial load P as illustrated in Figure 2. An Euler-Bernoulli linear deformation A-A at

rotation is imposed. Prior to tensile cracking of the concrete or compressive softening of the

concrete, this deformation such as A-B over length Ldef induces a real linear strain profile so

that the moment/curvature (M/) relationship can be derived through standard commonly

used techniques that is the corollary of the Euler-Bernoulli theorem can be applied.

Moreover, the presence of a shrinkage strain Ԑsh may be included through a baseline

movement by ԐshLdef as in Figure 2 and explained in detail by Visintin et al (2013a) and

Knight et al (2013b).

- 121 -

Figure 2. Segmental analysis

When flexural cracking occurs, then the crack widens through slip between the internal

reinforcement and the adjacent concrete such as st in Figure 2(b), the linear strain profile

within the material no longer exists, that is the corollary of the Euler-Bernoulli theorem no

longer directly applies. This deformation can be quantified using the partial-interaction

analyses of prisms with the reinforcement embedded as in Figure 2(a) as illustrated in Figure

3 and which depends on the bond-slip between the reinforcement and concrete. The analysis

depicted in Figure 3(a) can be used to quantify the primary crack spacing spr as illustrated in

Figure 3(a) and the force in the reinforcement to induce primary cracks; that in Figure 3(b)

depicts the relationship between the reinforcement force P and st for a length of prism 2Ldef

that is equal to the crack spacing spr as required in the analysis depicted Figure 2(b) and

which also gives the reinforcement force to induce secondary cracks; and that in Figure 3(c)

the tension stiffening behaviour when there are secondary cracks and their spacing (Visintin

et al 2012, Gupta et al 1990, Haskett et al 2008, 2009a, 2009b, Muhamed et al 2012,

Mohamed Ali M.S et al 2012, Jiang et al 1984, Choi and Cheung 1996, Lee and Kim 2008).

Furthermore the effects of creep, shrinkage and thermal gradients on the tension-stiffening

behaviour may included through a residual strain concept introduced by Visintin et al (2013a)

and Knight et al (2013b).

- 122 -

Figure 3. Tension stiffening analysis

When concrete compressive softening occurs through the formation of a wedge as in Figure

2(b), then there is additional deformation due to the slip of the wedge, such as sc. This

additional deformation in compression can be accommodated through the use of shear

friction theory (Haskett et al 2009a, Oehlers et al 2011, Mohamed Ali M.S et al 2012), or

through the use of size dependent stress-strain relationships based on shear friction theory

(Chen et al 2013).

Having established the basis of the segmental analysis in Figure 2, consider a segment of an

MF-FRP beam in Figure 4(b) for the cross-section in Figure 4(a); due to symmetry of a

segment about line D-D as in Figure 2, we only need to consider length Ldef in Figure 4(b).

As outlined in the preceding section, the force (PFRP) developed in the unbonded MF-FRP

strip is dependent on the beam deformations within the fastened region and thus cannot be

determined explicitly from the segmental analysis. Thus for analytical purposes it is a matter

of initially setting a force (PFRP) and solving for a family of forces and using the results for

the appropriate force from the beam analysis. An applied moment M in Figure 4 results in the

segmental deformation from A-B in Figure 4(b) and the linear strain profile C-C in Figure

4(c) in the uncracked and unsoftened region. Using conventional material stress-strain

properties, the stress distribution in these regions is derived as in Figure 4(d) and thus the

segmental forces in Figure 4(e) are known. In the cracked region, the reinforcement force

(Prb) is determined from the tension-stiffening analysis in Figure 3, while in the compression

region the force in the concrete wedge is determined from the size-dependent stress-strain

relationship (Chen et al 2013). The associated effects of concrete creep and shrinkage may

be included in this analysis using the established procedure outlined by Visintin et al (2013a)

and Knight et al (2013b). For the set strip force PFRP it is therefore a question of determining

the segmental moment M and corresponding rotation Ө that results in force equilibrium, thus

giving the segment displacement (δFRP) at the level of the FRP strip.

- 123 -

Figure 4. Analysis of an MF-FRP RC segment

The results of the numerical analysis give the M/Ө response allowing for the PI mechanisms

in the tension and compression regions. This rotation Ө divided by Ldef gives the variation in

the equivalent curvature (χ), thus the M/χ response as in Figure 5(a) is known, while the

displacement in the segment at the level of the MF-FRP strip (δFRP) divided by Ldef gives the

substrate equivalent strain Ԑsub such that the M/Ԑsub relationship in Figure 5(b) can be

determined. Moreover, integration of Ԑsub over a fastened length gives the required extended

length and thus length (Lb) due to beam deformations such as Lb-1 in Figure 1(c). A family of

M/χ and M/Ԑsub relationships are determined from this numerical procedure and for an

increasing strip force (PFRP) and are subsequently used in the member analysis outlined in

Figure 1. It is also shown in Figure 5(a) how the time dependent changes in the M/χ and

M/Ԑsub responses are accommodated in the segmental analysis, such that a shift in the abscissa

occurs.

Figure 5. (a) M/χ (b) M/εsub with and without time effects

- 124 -

Application to Test results

Having established the beam analysis procedure that uses segmental deformations that are

quantified through the M/Ө approach, the procedure is now validated against published

experimental results of beams tested by Martin and Lamanna (2008) and Ebead (2011).

Martin and Lamanna (2008) tested RC beams designated 6L, 10L and 12L. They were of

length of 3657mm, with a concrete strength 48MPa and cross-sectional dimensions

305x305mm and were strengthened with 2No. 25mm reinforcing bars in tension and 2No.

10mm reinforcing bars in compression. Beams 6L, 10L and 12L had varying quantities of

mechanical fasteners (MF) which provided fastened lengths of 152mm, 254mm and 305mm,

respectively, of the FRP strip which had an ultimate strength of 805MPa and modulus of

57GPa. Figures 6(a-c) provide a comparison between the experimental results and numerical

simulations. It can be seen that the general trend of the responses tends to be very good, with

slight discrepancies, particularly in the post-yield behaviour of the specimen. Post-yielding

of the internal steel reinforcement, the MF-FRP strip becomes the primary strengthening

mechanism of the RC beam such that the developed strains in the strip become highly reliant

on firstly quantifying the fastener slip; due to a general unavailability of test data, it can be

difficult to accurately predict the bearing failure behaviour at the fastener (Martin and

Lamanna 2008)

Beams designated M-P-D10 and M-F-10 beams were tested by Ebead (2011) and were of

length 2400mm with cross-sectional dimensions 150x250mm with 2No. 25mm internal steel

reinforcing bars and a concrete of strength 37MPa and 41MPa, respectively. Both beams

were strengthened with an MF-FRP strip with cross-sectional dimensions 3.2x102mm and an

ultimate strength of 1000MPa. The FRP strip for the strengthening of Beam M-P-D10 was of

length 1350mm and was symmetrical about the centreline of the beam and fastened by a total

of 15 fasteners, whereas, the FRP strip applied to Beam M-F-D10 spanned the beam full

length and was fastened at 23 locations. The predicted behaviour of both beams in Figure

6(d-e) tends to overestimate the experimental behaviour which may once again be attributed

to the bearing stress-slip characteristic used. However the shape is correct.

- 125 -

Figure 6. Comparisons to load-deflection test result

Figure 7(a-b) shows how the numerical approach is capable of quantifying the bearing load

on the fasteners at pre-defined applied transverse loads. Although there are some

discrepancies between the predicted and tested fastener bearing loads, the general trend along

half the beam span remains good. It can be seen how the member dependent M/Ө approach

can be a useful tool in the design of the mechanical fasteners through being able to quantify

the developed fastener bearing stresses.

0 10 20 30 400

100

200

300(a) Beam 6L

0 20 40 600

100

200

300(b) Beam 10L

0 20 40 60 800

100

200

300

Ap

pli

ed L

oad

(k

N)

(c) Beam 12L

0 20 40 600

20

40

60(d) Beam M-P-D10-1

0 10 20 30 40 500

50

100

Mid-Span Deflection (mm)

(e) M-F-D10-1

Predicted

Experimental

Predicted

Experimental

Predicted

Experimental

Predicted

Experimental

Experimental

Predicted

- 126 -

Figure 7. Comparisons to connector load

The variation in strip strains with lateral load from a numerical analysis of Beam MF-FRP is

shown in Figure 8. Also shown in Figure 8 is the IC debonding strain (Seracino et al 2007) if

the FRP strip had been adhesively bonded. It can be seen that the use of fasteners have

allowed the IC debonding strain to be exceeded which is a further benefit of this procedure

using mechanical fasteners.

Figure 8. Developed MF-FRP strip strain development for Beam 6L

0 200 400 600 800 1000 12000

2

4

6

8

10

12

14Beam FRP 6L

Fastener Distance from Support (mm)

Lo

ad o

n C

on

nec

tor

(kN

)

25kNm Predicted

50kNm Predicted

80kNm Predicted

25kNm Experimental

50kNm Experimental

80kNm Experimental

0 200 400 600 800 1000 12000

2

4

6

8

10Beam FRP 12L

-1500 -1000 -500 0 500 1000 15000

0.5

1

1.5

2

2.5

3

3.5x 10

-3

Distance Along Fastened Length, mm

FR

P S

trip

Str

ain

, M

F-F

RP

2.5kNm

25kNm

50kNm

75kNm

100kNm

117kNm

IC Debonding Strain

Legend

- 127 -

M/Ө approach for member design

Fastener Spacing

It has been shown how the numerical approach can be used to quantify the MF-FRP strip

strain at pre-defined transverse loading intervals. It is now shown how the numerical

approach can be a useful tool in determining the required spacing of the mechanical fasteners

in order to utilise the material properties of the FRP strip. A beam with geometric dimensions

as tested by Martin and Lamanna (2008) is considered, however the linear spacing of the

fasteners and thus the number of fasteners is varied in order to provide an understanding of

the effects on the MF-FRP strain ( ) as shown in Figure 9.

Figure 9(a) 6 Mechanical-Fasteners (b) 3 Mechanical-Fasteners per half span

It can be seen that a closer linear spacing due to a greater number of fasteners results in lower

developed strains in the FRP strip along the fastened length. This is apparent in Figure 9(a)

which illustrates the developed strip strains for 100mm fastener spacing’s in comparison to

Figure 8(b) which is an identical beam and loading scenario, however, the FRP strip is

attached with 3 fasteners per half span at a spacing of 500mm. Furthermore, it can be seen

from Figure 9(a) that the developed strip strain ( ) prior to beam failure does exceed

the intermediate crack debonding strain ( ), however the external moment at which this

occurs is less than that of the strengthened member in Figure 9(b). A reduced number of

fasteners and thus a greater fastener spacing results in greater fastener slip, this is particularly

apparent in the post-yielding stage of the internal steel reinforcement as the MF-FRP strip

becomes the primary tensile reinforcement. Hence, a larger number of fasteners results in the

division of the strip into smaller intervals resulting in the strains not fully developing and the

FRP strip strength being under-utilized, a similar conclusion as suggested by Martin and

Lamana (2008). Furthermore, by considering the load-deflection responses in Figure 10 for

beams with a varying linear spacing of mechanical fasteners, it can be seen that reducing the

fastener spacing results in a less ductile response.

FRPMF

-1500 -1000 -500 0 500 1000 15000

1

2

3

4

5x 10

-3 (b) 3 Fasteners at 500mm Spacing

15kNm

30kNm

60kNm

80kNm

90kNm

108kNm

PIC

Debonding

-1500 -1000 -500 0 500 1000 15000

0.5

1

1.5

2

2.5

3

3.5

4x 10

-3 (a) 6 Fasteners at 250mm Spacing

Fastened Distance from Mid-span (mm)

FR

P S

trip

Str

ain

, M

F-F

RP

7kNm

15kNm

30kNm

45kNm

60kNm

80kNm

104kNm

PIC

Debonding

Legend (b)

Legend (a)

FRPMF

FRPIC

- 128 -

Figure 10. Load-Deflection response of MF-FRP beams with/without prestress and with

varying fastener spacing’s

Post-tensioned MF-FRP Strips

It has been shown how the presented numerical approach can be used to quantify the

behaviour of an RC beam strengthened with an MF-FRP strip. It is also shown how the

efficiency of the strengthening procedure, when compared to an equivalent externally bonded

(EB) beam, is dependent on the mechanical fastening spacing. It is now shown how the

approach may accommodate prestressing (Knight et al 2013a, 2013b, 2013c) of the MF-FRP

strips in simulating beam behaviour, thus providing the basis for further research and

extending the application of the current MF-FRP technique.

Figure 11 illustrates the developed MF-FRP strip strain when an initial post-tensioning strain,

equivalent to the IC debonding strain in this case, is applied to the strip prior to mechanical

fastening. As outlined by Knight et al (2013a), the strain in conventionally bonded

prestressed reinforcement prior bonding, or fastening in this instance, remains uniform along

the beam length, thus no bearing slip occurs at the intermediate fasteners. However, this is

not the case at the end-fasteners as the bearing force will be equivalent to the initial

prestressing force, therefore, conventional prestressed FRP end-anchorage is required (Yang

et al 2009). Upon applying an external moment shown as a uniformly distributed load in

Figure 11, a further strain develops in the FRP strip between subsequent fasteners, as in

Figure 11, where it is shown that for a comparable moment in Figure 9(a), a greater strip

strain results. Furthermore, the load-deflection response of the beam can be seen in Figure 10,

whereby a much stiffer flexural response results from the addition of the post-tensioning

force. It is therefore suggested that the MF technique may prove to be beneficial for post-

tensioned FRP strip applications, however further research and experimental testing is

recommended.

0 10 20 30 40 50 60 70 800

50

100

150

200

250

300

350

Mid-Span Deflection (mm)

Ap

pli

ed L

oad

(k

N)

150mm Spacing

250mm Spacing

300mm Spacing

500mm Spacing

250mm Spacing Post-Tensioned

- 129 -

Figure 11. Average post-tensioned MF-FRP strip strain

Summary

Attaching fibre-reinforced polymer (FRP) strips to the soffit of a concrete beam with

mechanical fasteners (MF) has many advantages over conventional externally bonded (EB)

systems. The current analytical methods typically rely on full-interaction (FI) moment-

curvature (M/χ) analyses which are reliant on empirical components in order to quantify: the

load-slip behaviour as the reinforcement pulls from a crack face; and the mechanisms

associated with concrete softening. Accordingly, this paper outlines how the mechanics of a

partial-interaction moment-rotation (M/Ө) approach can be used to quantify the segmental

deformations of a MF-FRP RC beam without the reliance on empiricisms. The segmental

beam deformations are subsequently used in a full beam analysis in order to determine beam

behaviour and deflections under transverse loads. Model validation is carried out in the form

of comparisons to experimental load-deflection responses and resultant fastener loads along

the beam length. Furthermore, the M/Ө approach provides an accurate way of simulating the

developed load on individual fasteners and resultant slip, which is useful in the design of MF

systems. The result of this paper is a generic mechanics based analysis technique which can

be applied to any reinforced or prestressed concrete beam with any number of mechanical

fasteners, while being able to accommodate any bearing stress-slip characteristic at the

fastener.

Acknowledgements

The authors would like to acknowledge the support of the Australian Research Council ARC

Discovery Project DP0985828 ‘A unified reinforced concrete model for flexure and shear’.

-1500 -1000 -500 0 500 1000 15002

2.5

3

3.5

4

4.5

5

5.5

6

6.5x 10

-3

Fastener Distance from Mid-Span, mm

FR

P S

trip

Str

ain

, M

F-F

RP

15kNm

30kNm

60kNm

110kNm

IC Debonding Strain

- 130 -

Notation

A-A segment baseline prior to deformation

B-B deformation profile

C-C strain profile

D-D datum line

EB externally bonded

EI flexural rigidity

FI full-interaction

FRP fibre reinforced polymer

IC intermediate crack

L fastened length

Lb beam substrate extension due to Mapp

Ldef deformation length

LFRP extended length due to strip force

M moment

Mapp applied moment to member

MF mechanical-fastener

n number of fastened lengths

P force

Pc-mat P in compression concrete

Pc-ten P in tension concrete

Pc-wedge P in concrete softening wedge

PF P at fastener

PFRP P in MF-FRP strip

PI partial-interaction

Prb P bottom level reinforcement

Prt P in top level reinforcement

RC reinforced concrete

s slip

sc concrete wedge slip

spr primary crack spacing

ssec secondary crack spacing

st reinforcement slip

δ displacement; displacement profile

δFRP δ of MF-FRP within in concrete segment

ε strain; strain profile

εIC-FRP intermediate cracking debonding strain

εMF-FRP ε developed in MF-FRP

εrb ε in bottom reinforcement

εsh shrinkage strain

εsub strain in concrete substrate

σ stress; stress profile

χ curvature

Ө rotation

- 131 -

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- 134 -

Chapter 4

Concluding Remarks

This thesis has introduced a mechanics based segmental moment-rotation (M/Ө) analysis for

the simulation of both the instantaneous and sustained loading of reinforced concrete (RC)

beams with prestressing tendons or external reinforcement. Unlike conventional analysis

techniques, which assumes a linear strain profile and relies on empiricisms to simulate

flexural behaviour, the M/Ө approach uses the well-established mechanics of partial-

interaction (PI) and shear-friction theory. Being mechanics based, the M/Ө approach is able

to simulate what is actually observed in practice, that is the formation and widening of cracks

as the bonded reinforcement pulls from the adjacent concrete as well as the formation of

concrete softening wedges. A residual strain concept is introduced in this research in order to

account for the effects of prestress, shrinkage, creep, thermal gradients and reinforcement

relaxation, thus allowing for the simulation of the sustained loading of prestressed RC beams.

The mechanics of the M/Ө approach allows it to be generically applied to any cross-section,

with any concrete property and any reinforcement type, being either bonded and with any

bond characteristic, or unbonded. Having established the beam deformation analysis, the

approach extends to incorporate the instantaneous and sustained loading of RC beams with

unbonded fibre reinforced polymer (FRP) and steel prestressing tendons. Through being able

to quantify deformations along the length of a beam, the stress developed in unbonded

reinforcement can be determined without the reliance on empiricisms, therefore showing how

the approach can be useful in the design of such flexural members.

Further application of the approach enables the simulation of RC beams with mechanically

fastened (MF) FRP strips. A generic member numerical analysis outlines how the developed

M/Ө approach is used to determined segmental deformations within fastened lengths, thus

allowing for the partial-interaction behaviour in the cracked regions and at the fastener to be

quantified, as well as allowing for the shear-friction mechanism associated with concrete

softening.

The broad application of the presented M/Ө approach shows how it represents a generic

mechanics based solution for RC beams with prestressing and external reinforcement. The

M/Ө approach is shown to be able to accurately simulate the instantaneous and sustained

loading behaviour from prestress application, through serviceability loading and to collapse

of a wide variety of beams without the reliance on empiricisms, as in current approaches.

Thus the approach presented in this research may be seen as a useful extension to the

traditional analysis techniques and a fundamental step in developing generic design

procedures for RC beams with prestressing and external reinforcement.

- 135 -

Suggested Future Research

The final manuscript presented in Chapter 3 of this thesis outlines a generic numerical

analysis procedure for RC beams with mechanically fastened (MF) FRP strips. In this

manuscript it is shown how the presented M/Ө approach is able to incorporate the effects of

post-tensioning of the MF-FRP strip prior to fastening, and as such simulates the change in

flexural behaviour compared to a conventional MF-FRP RC beam, despite the general lack of

such experimental testing. Having established the analysis procedure in this research, further

research combined with relevant experiential testing schemes would reveal the effectiveness

and practicality of this technique for prestressed applications by removing the laborious and

sometimes ineffective procedures associated with adhesive bonding, particularly for existing

structures.

Moreover, the M/Ө approach presented in this research has been shown to be able to simulate

the instantaneous and sustained loading behaviour of RC beams with prestressed and external

reinforcement. In this research the primary application for the approach is reinforcement that

is straight along the length of the member, however in conventional prestressed RC design

parabolic or draped, bonded and unbonded, prestressing tendons are typically used due to the

effectiveness in being able to balance applied loads. In order to accommodate such

reinforcement the fundamental segmental analysis procedures presented throughout Chapters

1-3 of this thesis remains, however numerous segments are now required in order to account

for the changing eccentricity of the prestressing reinforcement along the members length.

The general analysis procedure would be as follows. Consider the member in Figure 1(a)

where the maximum tendon eccentricity (emax) occurs at half the member length, that is

Lmember/2 and such that the beam is also symmetrical about this point. Initially the member is

divided into (n) segments of length Lseg which are small enough that the tendon may be

assumed to be straight, rather than parabolic, over this length as in Figure 1(b). For a distance

(x) along the length of the beam the eccentricity of the tendon can be determined from

Equation 1.

2

max)( 4membermember

xL

x

L

xee (1)

Thus using Equation 1 the change in eccentricity along Lseg in Figure 1(b) may be determined

by finding e(1,2) –e(2,3), such that the subscript ‘(1, 2)’ refers to the eccentricity at the location

between segment 1 and segment 2. Once the change in change in eccentricity, or slope, of the

prestressing tendon over length Lseg is known the resultant force in the tendon may be

resolved from simple geometry, such that the segmental analysis presented in Chapters 2 and

3 may be repeated for (n) segments. Having defined the M/Ө relationship for each segment of

the member in Figure 1(a) the moment-curvature (M/χ) relationship and variation in flexural

rigidity (EI) along the member length for varying moments, thus M/EI is known, such that

the behaviour of the beam can be derived from standard integration techniques.

- 136 -

Figure 1. (a) Parabolic member analysis (b) Parabolic segment analysis

The broad applications of the presented approach in this research lends itself to a vast range

of applications, such that further extension to other RC beam types with prestressing and

external reinforcement may lead to the further calibration of current design approaches,

without the reliance on extensive experimental testing.

the generic simulation of reinforced concrete beams with ... · reinforcement ratio the effects of...

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