A Generic Mechanics Approach for Predicting

Shear Strength of Reinforced Concrete Beams

by

Tao Zhang

Thesis submitted for the degree of Doctor

of Philosophy at The University of

Adelaide

(The School of Civil, Environmental and

Mining Engineering)

Australia

- December 2014 -

i

TABLE OF CONTENTS

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

STATEMENT OF ORIGINALITY............................................................................... v

LIST OF PUBLICATIONS ......................................................................................... vii

ACKNOWLEDGEMENTS .......................................................................................... ix

INTRODUCTION & GENERAL OVERVIEW .......................................................... 1

CHAPTER 1 .................................................................................................................... 3

Background ....................................................................................................................... 3

List of Manuscripts ............................................................................................................ 3

Flexural rigidity of reinforced concrete members using a deformation based analysis .... 6

Presliding shear failure in prestressed RC beams. I: Partial-Interaction mechanism ...... 20

Presliding shear failure in prestressed RC beams. II: Behavior ...................................... 43

CHAPTER 2 .................................................................................................................. 61

Background ..................................................................................................................... 61

List of Manuscripts .......................................................................................................... 61

Shear strength of steel RC beams and slabs without stirrups .......................................... 63

Shear strength of FRP RC beams and one-way slabs without stirrups ........................... 81

CHAPTER 3 ................................................................................................................ 117

Background ................................................................................................................... 117

List of Manuscripts ........................................................................................................ 117

Partial-interaction tension-stiffening properties for numerical simulations .................. 119

Shear strength of RC beams with steel stirrups ............................................................. 136

CHAPTER 4 ................................................................................................................ 173

Background ................................................................................................................... 173

List of Manuscripts ........................................................................................................ 173

Shear strength of RC beams subjected to axial load ..................................................... 175

CHAPTER 5 ................................................................................................................ 211

Background ................................................................................................................... 211

List of Manuscripts ........................................................................................................ 211

Shear sliding tests on concrete blocks under low confinement ..................................... 212

Concrete shear-friction material properties: application to shear capacity of RC beams

of all sizes ...................................................................................................................... 242

CHAPTER 6 ................................................................................................................ 263

Concluding Remarks ..................................................................................................... 263

ii

iii

ABSTRACT

This thesis includes a series of journal articles in which a mechanics based segmental

approach is developed for simulating shear behaviour of reinforced concrete (RC)

beams. Using the well-established theories of partial interaction and shear friction, the

generic mechanics approach simulates the formation and widening of diagonal cracks

and shear sliding failure for RC beams. Being mechanics based, the proposed approach

can be generally applied to various kinds of structures, that is any cross section, with

any type of concrete and reinforcement and with any bond properties. Moreover, no

component of the proposed approach relies on empiricism to account for the mechanics

of shear failure, and the approach can accommodate any material characteristics which

with time can be refined and revisited to improve the accuracy of shear strength

simulation.

In developing the mechanics of the segmental approach for prestressed RC beams, it is

shown how the approach is applied to analyse shear behaviour and simulate shear

failure of prestressed beams. Parametric studies are conducted to explain the effect of

prestress on shear behaviour. For verification, the proposed approach is applied to 102

specimens and the analytical and experimental results are in good agreement.

The generic nature of the mechanics approach is shown by its application to steel and

fibre-reinforced polymer (FRP) reinforced beams and one-way slabs without stirrups.

From the mechanics of the segmental approach, closed form solutions are derived for

shear design and validated by comparisons with test results and code predictions of 626

steel and 209 FRP reinforced specimens.

Having developed closed form solutions for beams without stirrup, the approach is

extended to incorporate shear reinforcement. Significantly, the partial interaction

analyses of longitudinal and transverse reinforcements are directly linked. Furthermore,

simple solutions are derived through mechanics for tension stiffening and can be

applied for shear and flexure analysis in the segmental approach. The numerical and

closed form solutions are applied to 194 specimens and validated with good correlation

of predicted and measured results.

The generic mechanics approach is further extended to accommodate the effect of axial

load on shear strength. The proposed approaches are applied to 61 specimens and

simulation results show good agreement with test data.

A series of push-off tests are conducted to investigate the shear friction parameters for

initially uncracked concrete under low levels of confinement. In addition, it is shown

that the concrete shear friction properties can be extracted from simple confined

cylinder tests and then applied in the segmental approach to predict shear sliding

capacity. Thus this research highlights the potential to reduce the significant cost of

empiricisms in terms of time and money when developing innovative RC products and

generic design guidelines.

The broad application of the mechanics based segmental approach presents a general

solution to simulate shear strength of RC beams. Thus the generic mechanics approach

is a good extension of traditional shear analysis techniques as it obviates the necessity

of empiricisms through huge amount of testings to determine shear strength of RC

members.

iv

v

STATEMENT OF ORIGINALITY

I certify that this work contains no material which has been accepted for the award of

any other degree or diploma in my name, in any university or other tertiary institution

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. In

addition, I certify that no part of this work will, in the future, be used in a submission in

my name, for any other degree or diploma in any university or other tertiary institution

without the prior approval of the University of Adelaide and where applicable, any

partner institution responsible for the joint-award of this degree.

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

made available for loan and photocopying, subject to the provisions of the Copyright

Act 1968.

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

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 Search and also

through web search engines, unless permission has been granted by the University to

restrict access for a period of time.

________________________________________ ______________

Tao Zhang Date

vi

vii

LIST OF PUBLICATIONS

Journal Papers

Oehlers, D. J., Visintin, P., Zhang, T., Chen, Y., and Knight, D. (2012). “Flexural

rigidity of reinforced concrete members using a deformation based analysis.” Concrete

in Australia, 38(4), 50-56.

Zhang, T., Visintin, P., Oehlers, D. J. and Griffith, M. C. (2014). “Presliding shear

failure in prestressed RC beams. I: Partial-Interaction mechanism.” ASCE Journal of

Structural Engineering, 10.1061/(ASCE)ST.1943-541X.0000988, 04014069.

Zhang, T., Visintin, P., Oehlers, D. J. and Griffith, M. C. (2014). “Presliding shear

failure in prestressed RC beams. II: Behavior.” ASCE Journal of Structural Engineering,

10.1061/(ASCE)ST.1943-541X.0000984, 04014070.

Zhang, T., Visintin, P. and Oehlers, D. J. (2014). “Shear strength of steel RC beams and

slabs without stirrups.” Submitted to ICE Structures and Buildings.

Zhang, T., Oehlers, D. J. and Visintin, P. (2014). “Shear strength of FRP RC beams and

one-way slabs without stirrups.” ASCE Journal of Composites for Construction,

10.1061/(ASCE)CC.1943-5614.0000469, 04014007.

Zhang, T., Visintin, P. and Oehlers, D. J. (2014). “Partial-interaction tension-stiffening

properties for numerical simulations.” Submitted to Structures.

Zhang, T., Visintin, P. and Oehlers, D. J. (2014). “Shear strength of RC beams with

steel stirrups.” Submitted to ASCE Journal of Structural Engineering.

Zhang, T., Visintin, P. and Oehlers, D. J. (2014). “Shear strength of RC beams

subjected to axial load.” Submitted to ASCE Journal of Structural Engineering.

Chen, Y., Zhang, T., Visintin, P., and Oehlers, D. J. (2014). “Concrete shear-friction

material properties: application to shear capacity of RC beams of all sizes”, Submitted

to Advances in Structural Engineering.

Conference Paper

Zhang, T., Visintin, P., and Griffith, M. C. (2014). “A Unified Solution for Shear

Design of FRP Reinforced Concrete Structures.” Proc., The 7th International

Conference on FRP Composites in Civil Engineering, CICE, Vancouver, Canada.

viii

ix

ACKNOWLEDGEMENTS

My sincere gratitude goes first to Emeritus Prof. Deric Oehlers, who guided me through

the ups and downs of the PhD study and led me into the wonderful palace of research.

His excellent supervision, invaluable advice, continuous support and encouragement,

were an enormous help to me in the completion of this work.

I would also like to thank Prof. Michael Griffith for his great guidance and patience

throughout my studies, and Dr. Phillip Visintin for his prompt and precious suggestions

and ideas whenever there was a problem in the research.

I am very thankful to laboratory staff, Mr. Ian Ogier, Mr. Jon Ayoub, Mr. Steven

Huskinson, and Mr. Ian Cates, for their assistance with the experimental work. Special

thanks are given to Mr. Ian Ogier for his support and cooperation, and impressive

craftsmanship in the test configuration.

I also thank my colleague Yongjian Chen for his help in the experiment, and his

collaboration and discussion in the research.

The financial support from the China Scholarship Council (CSC) and the University of

Adelaide (UoA) are greatly appreciated.

Finally, to my parents and my partner, I am eternally grateful for their love, support,

motivation, encouragement.

x

1

INTRODUCTION & GENERAL OVERVIEW

Due to the complex mechanism of shear failure in reinforced concrete (RC) members,

most current approaches and code guidelines for predicting shear strength are empirical

or semi-empirical. Thus they do not physically explain the shear failure mechanism seen

in practice and should only be used within the bounds of the testing regimes from which

they were derived. Therefore, a generic mechanics approach, which can cope with any

type of concrete and reinforcement, has been developed in this thesis to explain and

simulate the shear behaviour of RC members. Furthermore, the proposed approach is

simplified for design purposes and generic closed form solutions are derived for

quantifying the shear strength of RC members.

This thesis is a collection of manuscripts that are either in preparation, submitted,

accepted or published in internationally recognised journals, where the titles of Chapters

reflect the overall research outcomes. Each chapter takes the following format: an

introduction explaining the key theory and results of the chapter, a list of all the

manuscripts presented in the chapter, after which the presentation of each manuscript.

In Chapter 1, a deformation based segmental approach is proposed for analysis of

tension stiffening, wedge softening and shear failure; a mechanics solution is described

to quantify the shear strength of RC member. Based on the theories of partial interaction

and shear friction, a generic mechanics based segmental approach is developed to

simulate the shear behaviour and failure of prestressed concrete beams both with and

without stirrups. This numerical approach is applied to predict the shear behaviour of

102 published test specimens and validated by comparisons of experimental strength

and code prediction.

In Chapter 2, to show the generic nature of the model, the mechanics approach is

applied to steel or fiber-reinforced polymer (FRP) reinforced beams and one-way slabs

without stirrups. From the mechanics of the approach, generic closed form solutions

suitable for use in design office are derived and applied to predict the strength of 626

steel reinforced specimens and 209 FRP specimens and validated by comparison with

experimental strength. The predictions of proposed design equations are also compared

with that of code approaches.

In Chapter 3, closed form solutions from the mechanics approach are extended to

incorporate steel stirrups. Importantly, the partial interaction analysis of transverse

reinforcements is directly linked with that of longitudinal reinforcements. Moreover, a

semi-mechanical model for tension-stiffening is developed, and simple equations are

derived for the simulation of partial interaction mechanism and applied in the closed

form solutions for shear strength. The numerical and closed form solutions of the

mechanics approach are used to simulate 194 published test specimens, and good

correlations between the predicted and measured strengths are achieved.

In Chapter 4, the generic segmental approach is further extended to account for the

influence of axial load on shear strength of RC members. The generic closed form

solutions are derived and applied to predict the shear strength of RC members with axial

load. The proposed solutions are validated with good agreement between the predicted

and measured strengths of test specimens.

In Chapter 5, a series of push-off test are carried out to study the shear friction

parameters for initially uncracked concrete under low levels of confinement. It is also

shown how the concrete shear friction properties required for the segmental analysis can

2

be easily extracted from simple actively confined cylinder tests and then applied to

analyse the shear sliding behaviour of initially uncracked concrete and predict the shear

strength of RC beams.

3

CHAPTER 1

Background

This chapter describes the generic mechanics based segmental approach, which is based

on the theories of partial interaction and shear friction and can simulate presliding shear

failure in prestressed reinforced concrete (RC) beams.

In the first publication ‘Flexural rigidity of reinforced concrete members using a

deformation based analysis’ a background to the existing research in the area is

provided and the overall need for this research is highlighted. The introduction of the

fundamental mechanisms is given. 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.

In the second publication “Presliding shear failure in prestressed RC beams. I: Partial-

Interaction mechanism”, the development of the mechanics based segmental approach is

presented. This forms the basis of this research for the remainder of the thesis. For

prestressed concrete beams with and without stirrups the interaction between flexure

and shear has been investigated at three stages: upon application of prestressing force;

precracking analysis; postcracking analysis. Importantly, the contributions of the

concrete and the stirrups to shear strength are treated and analysed together rather than

separately.

In the third publication “Presliding shear failure in prestressed RC beams. II: Behavior”,

using generic material properties, the mechanics based segmental approach is applied to

study the shear behaviour of prestressed beams with and without transverse

reinforcement. Following this is the investigation of the effect of prestress on shear

behaviour and the effectiveness of transverse reinforcement in prestressed beams. The

approach is verified by comparison of the predicted and measured shear strength of 102

published test specimens failing in shear.

List of Manuscripts

Oehlers, D. J., Visintin, P., Zhang, T., Chen, Y., and Knight, D. (2012). “Flexural

rigidity of reinforced concrete members using a deformation based analysis.” Concrete

in Australia, 38(4), 50-56.

Zhang, T., Visintin, P., Oehlers, D. J. and Griffith, M. C. (2014). “Presliding shear

failure in prestressed RC beams. I: Partial-Interaction mechanism.” ASCE Journal of

Structural Engineering, 10.1061/(ASCE)ST.1943-541X.0000988, 04014069.

Zhang, T., Visintin, P., Oehlers, D. J. and Griffith, M. C. (2014). “Presliding shear

failure in prestressed RC beams. II: Behavior.” ASCE Journal of Structural Engineering,

10.1061/(ASCE)ST.1943-541X.0000984, 04014070.

4

5

Knight, D.

Contributed to research

I hereby certify that the statement of contribution is accurate and I give permission for the inclusion of the paper in this thesis

6

Flexural rigidity of reinforced concrete members using a deformation based analysis

Oehlers, D. J., Visintin, P., Zhang, T., Chen, Y., and Knight, D.

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 in Australia, vol. 38 (4), pp. 50-56

NOTE:

This publication is included on pages 6-18 in the print copy of the thesis held in the University of Adelaide Library.

19

20

Presliding shear failure in prestressed RC beams. I: Partial-Interaction

mechanism

Zhang, T., Visintin, P., Oehlers, D.J., Griffith, M.C.

Abstract

Despite significant experimental, numerical and analytical research, the shear behaviour

of reinforced concrete members remains one of the least well understood mechanisms in

reinforced concrete. Due to the complexity of shear behaviour, empirical or semi-

empirical analysis approaches have typically been developed and these are widely

employed in codes of practice. As with all empirical models, they should only be

applied within the bounds of the tests from which they were derived which restricts the

wide application of innovative materials as expensive testing must be performed to

adjust existing empirical formulae or develop empirical formulae specific to the new

materials. There is, therefore, a strong need to develop a generic, mechanics based

model to describe shear failure which is the subject of this paper. The model is based on

the mechanics of partial interaction: that is slip between reinforcement and adjacent

concrete which allows for crack formation and widening and is commonly referred to as

tension-stiffening; and slip across sliding planes in concrete associated with shear

failure which is referred to as shear-friction.

Keywords: Shear capacity; Prestressed concrete; Shear friction; Partial interaction.

Introduction

There are numerous widely ranging approaches used in the study of shear failure of

reinforced concrete members (Bažant and Sun 1987; Bažant and Kazemi 1991; Bentz et

al. 2006; Choi et al. 2007; Collins et al. 1996, 2008a, 2008b; Hoang and Jensen 2010;

Hoang and Nielsen 1998; Hsu et al. 1987; Jensen and Hoang 2009; Kani 1964; Mattock

and Hawkins 1972; Nielsen et al. 1978; Park et al. 2006; Reineck et al. 2003; Reineck

1991; Vecchio and Collins 1986, 1988; Yu and Bažant 2011; Zararis and Papadakis

2001; Zhang 1997), because of the incredible complexity of the problem, particularly in

the case where prestressed reinforcement is present (Avendaño and Bayrak 2011;

Bažant and Cao 1986; Bennett and Debaiky 1974; Bennett and Mlingwa 1980; Bruce

1962; Cladera and Mari 2006; Elzanaty et al. 1986; Hanson 1964; Hernandez 1958; Hsu

et al. 2010; Kang et al. 1989; Kar 1969; Kaufman and Ramirez 1988; Kordina et al.

1989; Laskar et al. 2010; MacGregor 1960; MacGregor et al. 1965; Maruyama and

Rizkalla 1988; Mcmullen and Woodhead 1973; Ojha 1967; Olesen et al. 1967; Park et

al. 2013; Robertson and Durrani 1987; Saqan and Frosch 2009; Shahawy and Cai 1999;

Sheikh et al. 1968; Sozen et al. 1959; Teng et al. 1998; Wolf and Frosch 2007; Zwoyer

1953). Several different approaches including shear friction theory (Birkeland and

Birkeland 1966; Mattock and Hawkins 1972; Walraven et al. 1987), plastic theory

(Nielsen et al. 1978; Zhang 1997; Hoang and Nielsen 1998), strut-and-tie modelling

(Hwang and Lee 2002; Mörsch 1909; Park and Kuchma 2007; Ritter 1899), and the

modified compression field theory (Vecchio and Collins 1986, 1988; Bentz et al. 2007),

have been proposed to quantify the shear resistance of RC members. While these

approaches have been applied with varying degrees of success, they do not in general

allow for the direct interaction between shear and flexural loading nor for partial-

interaction between reinforcement and adjacent concrete which is the essence of the

model that is described in this paper.

21

For prestressed RC members confined with stirrups, codes and design guidelines

[American Concrete Institute (ACI) Committee 318 2008; Standards Australia 2009;

European Committee for Standardization (CEN) 1991; Joint ACI-ASCE Committee 445

1998] split the shear capacity of a member, Vn into two components which are usually

treated separately for ease of calculation: the shear strength attributed to the concrete Vc;

and the shear strength attributed to the shear reinforcement Vs.

Although commonly applied, the ACI shear strength equations (ACI Committee 318

2008) are empirical and several shortcomings in their formulation have been widely

recognised. For instance, the ACI code approach does not account for the variation of

the longitudinal reinforcement ratio, which is known to be significant in controlling the

shear strength of prestressed concrete beams (Elzanaty et al. 1986; Saqan and Frosch

2009; Tompos and Frosch 2002). Moreover, as the concrete and stirrup contributions to

the shear strength are treated separately and independently, it is implied that the

confinement of stirrups has no effect on the shear strength provided by the concrete,

which is again not consistent with the fundamental principles observed in practice

(Bažant and Sun 1987; Russo and Puleri 1997; Tompos and Frosch 2002; Cladera and

Mari 2006). Finally, it is known that shear strength is not a linear function of the shear

reinforcement ratio (Yu and Bažant 2011), although most design codes assume that it is.

Together, these factors, which exist as a result of our tendency as researches to modify

existing empirical equations, in this case to extend previous empirical formulations,

rather than develop mechanics based solutions, mean that code approaches are typically

overly conservative when applied to prestressed concrete members (Bennett and

Debaiky 1974; Elzanaty et al. 1986; Kaufman and Ramirez 1988). All of these

shortcomings highlight the incredible complexity of shear failure.

In this paper, an alternative mechanics based segmental approach is developed to

address the fundamental deficiencies identified in the existing empirical approaches and

to describe the shear behaviour and failure of prestressed concrete beams with and

without transverse reinforcement. It is shown that the flexural forces affect the shear

capacity along potential shear failure planes and, hence, the interaction between flexure

and shear has been quantified. This mechanics based segmental approach has been

applied to analyse published test specimens in a companion paper (Zhang et al. 2013)

and the analytical results are shown to have good correlation with experimental results,

which validates the proposed model; the significance of the parameters that affect the

shear behaviour of the prestressed members are investigated using the proposed model

and the analysis can explain the influence of important variables on shear behaviour.

The Shear Resistance Mechanism

The pretensioned or posttensioned prestressed concrete beam, herein referred to as

prestressed concrete beam, is subjected to transverse load Va in Fig. 1. In Fig. 1(b), only

half of the specimen is shown as the other half is symmetrically loaded. Let flexural

cracks form at a discrete spacing of Scr as shown. Potential diagonal cracks such as A-

A*, A-C, A-D or even vertically along A-E can form. In practice, the diagonal cracks

initiate from the vertical flexural cracks and then extend to the top in a non-linear way

as A-A* and A-C in Fig. 1(b). Theoretically, these diagonal cracks can be considered as

linearly inclined cracks as A-D in Fig. 1(b). To better follow the major trend line of the

diagonal crack in practice, the upper tip of the diagonal cracks are assumed to be at

point A located a distance Scr/2 from the plane section B-B*; as Scr/2 is much smaller

than the shear span a, this assumption has a very minor influence as explained later.

Hence these diagonal cracks extend from the compression region at the upper point A

and to the tension region at the position of a flexural crack as shown. The angle at

22

which inclined cracks form is, therefore, limited to a discrete sequence which is a

function of the tensile crack spacing (Muhamad et al. 2011, 2012; Visintin et al. 2012a).

Fig. 1. Prestressed concrete beam with shear loading

(a) cross section (b) typical diagonal crack patterns

According to Zhang (1997), shear failure of a section without transverse reinforcement

will occur along a plane at a given angle of failure when the applied transverse load Va is both sufficient to cause the formation of a crack along the plane as well as shear

sliding along the cracked plane. It is, therefore, possible to analyse each potential

sliding plane within the discrete sequence identified in Fig. 1 to determine the load to

cause inclined cracking Vcr and the load to cause sliding Vsl and, hence, to identify the

angle of the critical diagonal crack CDC at which sliding failure will occur.

In the case of a member without transverse reinforcement as in Fig. 2(b), it is known

(Zhang 1997) that the load to cause inclined cracking Vcr will always increase with a

reduction in the angle as shown in Fig. 2(a) and, furthermore, the load to cause sliding Vsl will always decrease as shown. If it were possible to have a diagonal crack at any

angle in Fig. 2(b), then failure would occur at cap-min at a shear load Vcap-min that is at the intercept of Vsl and Vcr in Fig. 2(a). However, cracks occur at discrete positions

(Muhamad et al. 2011, 2012; Visintin et al. 2012a) and as such there are only a finite

number of possible positions of diagonal cracks. Take for example the potential

diagonal crack A-C in Fig. 2(b). The shear load to cause cracking Vcr-2 is greater than

the shear load to cause sliding Vsl-2, as in Fig. 2(a), so that shear failure occurs

immediately after a diagonal crack forms at Vcr-2; this could be described as unstable or

catastrophic shear failure. In contrast, the diagonal crack A-D forms at Vcr-1 and the

shear load can be gradually increased to Vsl-1 before failure occurs such that this is a

stable shear failure. As in this example Vcr-2 is less than Vsl-1, the shear capacity Vcap is

governed by the diagonal A-C and is equal to Vcr-2 which is slightly greater than Vcap-min.

This example helps to illustrate the mechanics of the random nature of the shear

capacity as it is governed by the mechanics of the random nature of flexural cracking

(Muhamad et al. 2012). In general for a member without stirrups, one or two diagonal

crack will form and sliding along one of these planes will cause shear failure.

Scr/2a

n /2 (n=1,3,5...)Scr

ScrA*

Pre-stressed tendon

b

d

B*

AB

β

Va

C D

(a) (b)

E

23

Fig. 2. Shear failure of RC beams without transverse reinforcement

A member with transverse reinforcement is shown in Fig. 3(b). Let us first consider the

case of a member with no prestress. According to Lucas (2011), the sliding capacity Vsl

need not be monotonically decreasing with the reduction of as shown in Fig. 3(a).

This is because as reduces more stirrups are engaged across the sliding plane as illustrated in Fig. 3(b) where no stirrups cross the potential sliding plane A-E and at

least four stirrups cross the sliding plane A-A*. These stirrups increase the shear

capacity directly by resisting the applied shear Va and indirectly by providing

confinement across the cracked plane as will be explained later. Sliding failure Vcap-min,

therefore, corresponds to the local minimum of Vsl. The cracking capacity Vcr need not

be considered as it is generally smaller than Vsl (Jensen et al. 2010; Jensen and Hoang

2009) and hence an intercept of Vcr and Vsl is unlikely.

Scr/2

A* B*

A Bβ

Va

βCDC

90

Vcap

V

Vcr Vsl

Ma

C D

Vcap-min

βcap-min

E

Vcr-2

Vcr-1

Vsl-1

Vsl-2

(b)

(a)

Scrn /2 (n=1,3,5...)Scr

2Δrt

24

Fig. 3. Shear failure of RC beams with transverse reinforcement

For beams with prestressed longitudinal reinforcement and transverse reinforcement,

the shear to cause cracking Vcr-pre in Fig. 3(a) can be significantly greater than Vcr due to

the compressive stress in the concrete due to the prestress. However, the sliding

capacity Vsl-pre may change but not necessarily be always greater than the cracking load

Vcr-pre, unlike the case for non-prestressed beams with stirrups, in which Vsl will always

be greater than Vcr. This is because the presence of a prestress restrains crack openings

so that the stirrups may not be sufficiently engaged to cause a large increase in sliding

capacity seen in conventionally reinforced beams as decreases. Consequently, with high levels of prestress the failure capacity Vcap-pre occurs at the intercept of Vcr-pre and

Vsl-pre as in a section without stirrups.

Having established that in order to determine the shear capacity Vcap it is necessary to

determine the load to cause inclined cracking Vcr-pre and the load to cause sliding Vsl-pre,

let us consider how each component may be determined individually for a given

potential sliding plane which forms at angle .

Shear to Cause a Diagonal Crack (Vcr-pre)

Consider the free body to the right of the diagonal crack in Fig. 1 now shown in Fig. 4

from which the cracking load Vcr-pre needs to be determined. As it is generally

recognised that transverse reinforcement has little or no effect on the inclined cracking

load (Elzanaty et al. 1986; Hoang and Jensen 2010; Hoang and Nielsen 1998; Jensen

and Hoang 2009), that is the load to cause the inclined crack, it will not be considered

here.

Scr/2

B*

A B

β

Va

90

Vcap-min

V

Vcr

Vsl-pre

Ma

β

Vcap-pre

A* C D E

Vcr-pre

Vsl

(b)

(a)

Scr

n /2 (n=1,3,5...)Scr

25

Fig. 4. Shear load to cause cracking

The shear load to cause a diagonal crack Vcr-pre can be determined by taking moments

about point A (Zhang 1997; Zhang et al. 2012) so that,

2/ sin / 2

1/ 2

tef pr pt

cr pre

cr

f b d P d cV

a S

where Ppr is the prestressing force applied in the tendon, a is the shear span, d is the

beam depth, cpt is the concrete cover for the tendon and ftef is the effective tensile

strength of the concrete (Zhang 1997).

Sliding Capacity along a Diagonal Plane (Vsl-pre)

A segment of a beam, with the cross section in Fig. 5(a), is shown in Fig. 5(b) with the

forces external to the free body required for equilibrium; from the loads applied to the

beam, there is a combination of shear Va and moment Ma as can be seen in Fig. 5(b). To

resist the vertical load Va, besides the force in the stirrup Pstp, an additional shear force S

must be acting along the inclined plane (Lucas et al. 2011) and thus the vertical

equilibrium can be satisfied as follows

sin 2a stpV S P

Fig. 5. Shear capacity along the inclined plane

(a) cross section (b) shear analysis for the segment For the given applied stress resultants Va and Ma, all the forces acting on the segmental

free body as in Fig. 5(b), such as the concrete tensile force Pcont, forces developed in the

Scr/2

A* B*

A B

Va

β

Ma

Ppr

n /2 (n=1,3,5...)Scr

ftef

a

cpt

d

Scr/2

A* B*

A B

Va

Ma

Ppt

n /2 (n=1,3,5...)Scr

dNA

cpt

Compression region

d

b

crt

Prt

Pstp

Pcont

PrcPconc

β

S

(b)(a)

26

compression reinforcement Prc, in the tension reinforcement Prt, in the prestressed

tendon Ppt and in the stirrups Pstp, can be determined using a segmental analysis which

will be described later. This segmental analysis also gives the depth of the compression

zone dNA.

The sliding capacity Vsl in Fig. 2 is only applicable after a diagonal crack has formed at

Vcr. Hence we are dealing with a cracked section, that is a section with a compressive

region over the depth dNA in Fig. 5 and a tension region below this. Shear friction theory

shows that shear can only be resisted across a potential cracked sliding plane if there are

compressive forces normal to this plane (Mohamed Ali et al. 2008; Lucas 2011). Hence,

prior to sliding, the tension region cannot resist shear and, consequently, the only region

that resists shear along the sliding plane is the compression region. The sliding capacity

Vsl, that is the shear to cause sliding which is also the shear capacity prior to sliding,

depends on the shear-friction capacity of the concrete along the potential sliding plane

in the compression region which is shown in Fig. 6. It is essential to emphasise that the

shear force S and the concrete compressive force Pconc are still the same as in Fig. 5.

The compressive force Pconc can be resolved into a normal force Pconcsin to the plane

and a shear force Pconccos along the plane. Hence the diagonal sliding plane has to

resist both S and Pconccos and the resistance to sliding of this plane is enhanced by the

normal force Pconcsin.

Fig. 6. Shear forces along the inclined plane

Having established the shear forces which must be resisted along sliding plane A-A* in

Fig. 6, let us now determine the capacity to resist sliding of the compression region of

the sliding plane which has a surface area of bdNA/sin. The shear friction capacity of the initially uncracked plane Zcap is given by

3sin

NAcap N

dZ b

where τN is the shear friction strength which is assumed to be uniform along the

diagonal plane in the compression region. Furthermore, according to the well-

established shear friction theory (Birkeland and Birkeland 1966; Haskett et al. 2010;

Haskett et al. 2011; Mattock and Hawkins 1972; Walraven et al. 1987), it is typically a

function of the normal stress σN acting directly on the plane which is also assumed to be

uniformly distributed over the compression zone of the concrete, that is, Pconcsin divided by the diagonal interface area in the compression region.

2sin

4concNNA

bP

d

A*

APconcsinββ

S

Pconccosβ

S

Pconcsinβ

Potential shear-friction sliding plane

Pconc≡( )

27

Consequently, the maximum value of the shear force S along the sliding plane Smax that

the shear plane can resist is

max cos 5cap concS Z P

Importantly, it is worth noting that part of the shear strength of the diagonal sliding

plane Zcap has to be used to resist the shear component of the concrete compressive

force acting along the inclined plane Pconccos in order to maintain equilibrium; the rest of Zcap is available to resist the shear force S along the plane. The omission of this shear

component of the compressive force in previous research (Nielsen et al. 1978; Lucas et

al. 2011) may explain the tendency for the direct application of shear-friction theory to

overestimate shear strength.

Substituting Smax in Eq. (5) for S in Eq. (2) gives the maximum shear load that can be

resisted, namely the sliding capacity Va-sl corresponding to Va as follows:

sin cos sin 6a sl cap conc stpV Z P P

where for simplicity and conservatism, the dowel action of the longitudinal

reinforcement is not considered here.

If Va-sl is not less than Va, sliding failure does not occur and thus the beam can undertake

further loading otherwise sliding failure does occur. It is now a question of quantifying

both Pconc and Pstp in Eq. (6) using the following segmental analysis approach.

The Segmental Analysis Approach

Having developed the shear failure criteria for prestressed concrete members, it is now

necessary to develop a methodology to solve Eq. (6), that is to determine: the magnitude

and location of the internal actions for a given applied load Va and the potential sliding

plane forming at an angle of . To do this the segmental approach proposed in Visintin et al. (2012a, 2012b) for the flexural analysis of RC members and extended to the

flexural analysis of prestressed beams in Knight et al. (2013a, 2013b) is adapted here

for application along an inclined plane. This segmental analysis approach is applied

here as it provides a mechanics based procedure for the analysis of reinforced concrete

using the mechanics of partial interaction (Haskett et al. 2008; Muhamad et al. 2011,

2012; Oehlers et al. 2011a, 2011b) to simulate the slip of the reinforcement relative to

the concrete in which it is encased. Hence, the only empiricisms required for the

analysis are related to defining material properties and not in describing any form of the

mechanics of member behaviour. The approach is developed here in three stages: (1)

upon application of prestress, (2) precracking analysis and (3) postcracking analysis. It

should be noted here that the analysis procedure presented is only applied until the

commencement of concrete softening, however, this is not considered to be a limitation

as the presliding capacity derived in this work will generally occur prior to concrete

softening.

Application of Prestressing Force

For a prestressed concrete member, the first step of the analysis is to consider the

behaviour at the application of prestress, that is, prior to any external loads being

applied. At this stage, the prestressing cable is not bonded within the duct in Fig. 7(a) so

28

there is no direct interaction through bond between the prestressing tendon and the

concrete section; the prestressing force Ppr, which induces a strain in the tendon pr, can be considered as an external force which must be resisted. It is, therefore, a question of

determining the deformation of the segment from A-A* to C-C

* at rotation θpr at which

the resultant force PRC in the reinforced concrete section (that is the sum of the forces in

the concrete Pconc, in the compression reinforcement Prc and in the ordinary tension

reinforcement Prt) is equal to and in line with the prestressing force Ppr as in Fig. 7(e).

Fig. 7. Segment for analysis on application of prestress

This analysis can be performed iteratively by setting an initial guess for the rotation due

to prestress θpr in Fig. 7(a) and guessing a deformation of the concrete at the top fiber

Δtop, thereby, defining the deformation profile C-C as in Fig. 7(b) with the assumption

of a plane section remaining plane. Having defined the deformation profile, the

distribution of strain is simply the deformation from A-A* to C-C

* divided by the length

over which it must be accommodated Ldef which varies along the depth as shown in Fig.

7(a). The deformation length Ldef throughout the depth of the section can be determined

through simple geometry as the length of the segment varies from the half crack spacing

Scr/2 at the top fibre to a multiple of the half crack spacing at the bottom fibre. As the

deformation length varies over the depth of the section, the strain profile is non-linear as

in Fig. 7(c); to determine the stress profile in Fig. 7(d) from the application of standard

material stress-strain relationships, it may be convenient to use a layered approach.

From the stress distribution, the distribution of internal forces in Fig. 7(e) is known and

Δtop must be updated until force equilibrium is achieved. At the point of force

equilibrium, if the resultant PRC does not align with Ppr and is not of the same

magnitude the initial guess for θpr and Δtop are incorrect and must be updated.

Following the application of prestressing, the prestressing duct is grouted and hence full

interaction between the tendon and the RC section is established so that the force in the

prestressing tendon is now considered as an internal force. The above analysis in Fig. 7

provides the initial rotation θpr and deformation at the level of the prestressing tendon

Δpr-res. At this deformation Δpr-res, the strain in the tendon is pr and any deformation of the tendon away from Δpr-res will result in a change in the force in the tendon.

Application of Transverse Loads prior to Cracking

Scr/2

A* B*

A B

Ppr

n /2 (n=1,3,5...)Scr

cptcrt

Prt

Prc

Pconc

βC

C*

θpr

Ldef

Δtop Δtop

Δpr-res

ε Δ= /Ldef

(a) segment (b) deformation (c) strain (d) stress (e) force

A C

A* C*

Δ

Pre-stressing cable

PRC

29

Let us now consider the application of an external load which causes a shear Va and

corresponding moment Ma at B-B in Fig. 8(a). For analysis, a rotation θ is imposed on

the section causing a deformation from A-A to D-D. As the prestressing duct is now

grouted, the force in prestressed tendon Ppt now acts as part of the reinforced concrete

section rather than as an external force as in the previous analysis.

Fig. 8. Segment for precracking analysis

Ignoring the influence of stirrups prior to cracking, from horizontal equilibrium of

forces in Fig. 8

cos 7pt rc conc bf rtP P P S P

where Pconc-bf is the concrete compressive force before flexural cracking occurs, and Prt

is in compression according to the deformation profile shown in Fig. 8(b).

Similarly, from vertical equilibrium

sin 8aV S

Combining Eqs. (7) and (8) to eliminate the shear force S along the sliding plane and

rearranging in terms of the applied shear Va yields

tan 9a pt rt conc bf rcV P P P P

Taking moments about the top point A in Fig. 8(a), rotational equilibrium is achieved

when

/ 2 10a a cr pt pt conc bf conc bf rc rc rt rtM V S P d c P d P c P d c

where dconc-bf is the distance from the centroid of the concrete compressive force to the

top layer before flexural cracking, and crt the concrete cover of ordinary reinforcement.

Hence it is now a question of determining the deformation from A-A* to D-D

* in Fig.

8(a) such that both force and rotational equilibrium are achieved. To do this, a rotation θ

Va

Ma

Scr/2

A* B*

A B

Ppt

n /2 (n=1,3,5...)Scr

Prt

Prc

Pconc-bf

βC

C*

θ

Ldef

Δtop

Δpr-res

ε Δ= /Ldef

(a) segment (b) deformation (c) strain (d) stress (e) force

D

D*

Prc

Pconc-bf

Ppt

Prt

A D C

Δtop

A* D* C*

Δ

S ScosβSsinβ

ScosβSsinβ

Δpt-bf

dconc-bf

30

is first set and the deformation at the top fibre Δtop is guessed. As was the case in the

analysis prior to the application of external loads this defines the deformation profile D-

D*.

For the concrete and initially un-tensioned reinforcement, the strains in Fig. 8(c) can be

determined directly from the deformation profile A-A* to D-D

* in Fig. 8(b) by dividing

the deformation Δ by the deformation length Ldef. For the prestressed tendon, the

application of the prestressing force Ppr in Fig. 7 causes a deformation C-C*, where, at

the level of the tendon the deformation of the segment is pr-res and the strain in the

tendon is pr. The deformation lines C-C* and A-A

* in Fig. 8 are referred to as base lines

for the prestressed tendon and the concrete respectively and any deformation from the

base line causes stress. The change in deformation at the level of the tendon shown as

(pr-res-pt-bf) from C-C* to D-D

* in Fig. 8(b) will further tension the tendon and

therefore cause the strain in the tendon to increase from pr. Hence the strain in the

tendon pt will be pr plus (pr-res-pt-bf)/Ldef where Ldef is the length of the segment at the level of the tendon.

From the strain profile in Fig. 8(c), the stresses in Fig. 8(d) and hence forces in Fig. 8(e)

are known and Δtop can be adjusted until force equilibrium in Eq. (9) is achieved. Having

satisfied force equilibrium, if rotational equilibrium in Eq. (10) is not obtained the initial

rotation θ must be updated and the analysis repeated until it does. From the known

magnitudes and location of the internal forces Eq. (6) can be solved in order to

determine the corresponding shear sliding capacity Va-sl.

The above analysis can then be repeated for increasing magnitudes of the applied load

Va to produce the relationship between the applied stress resultants Va and Ma and the

corresponding shear sliding capacity Va-sl in Fig. 9. The above analysis applies prior to

cracking and hence can be used to derive the points from O to A after which flexural

cracking takes place and the crack tip crosses the longitudinal reinforcement. From this

point onwards, with the moment Mfl-cr causing flexural cracking at a shear load of Vfl-cr,

the forces developed in all the reinforcements crossing a crack must be determined from

partial interaction theory.

Fig. 9. Shear sliding capacity for a diagonal crack at angle of β

It is interesting to note that rearranging Eq. (10) gives

11

/ 2

pt pt conc bf conc bf rc rc rt rt

a

cr

P d c P d P c P d cV

a S

Vfl-cr

shear sliding capacityVsl-pre

Mfl-cr

V

O

A

applied loadV

a

B

0 M

cracking point

31

where the shear span a = Ma/Va.

Substituting Eq. (11) into Eq. (8), we can derive

12/ 2 sin

pt pt conc bf conc bf rc rc rt rt

cr

P d c P d P c P d cS

a S

From Eq. (12), it can be seen that the shear force S along the inclined plane is increased

by the tensile force in the tendon while decreased by both the compressive forces in the

concrete and the longitudinal reinforcement.

Further substituting Eq. (12) into Eq. (7) gives

13

pt pt rc rc rt rt

conc bf

conc bf

P e d c P e c P e d cP

e d

where e=(a-Scr/2)tan.

Eq. (13) shows that Pconc-bf, on which the shear capacity of the uncracked region Zcap

depends, as described previously in Eqs. (3) and (4), is also increased by the tensile

force in the tendon while decreased by the compressive forces in the longitudinal

reinforcements.

Fig. 10. Segment for postcracking analysis

Application of Transverse Loads after Cracking

Let us now consider the segment in Fig. 10 which is subjected to applied loads Va and

Ma which cause the formation of a flexural crack that crosses the longitudinal

reinforcements.

From horizontal equilibrium of forces

Pstpi Va

Ma

Scr/2

A* B*

A B

Ppt

n /2 (n=1,3,5...)Scr

Prt

Prc

Pconc-af

β

θLdef

Δtop

nΔpt

ε Δ= /Ldef

(a) segment (b)deformation(c)strain(d)stress(e)force

PrcPconc-af

Ppt

Prt

A

Δtop

nΔrt

Δstpi

E

E*

NAPcont

Pcont

Δ

S

ΣPstpi

ScosβSsinβ

dconc-afdcont

dstpi

E

A*E*

32

cos 14pt rt cont conc af rcP P P P P S

where Pconc-af is the concrete compressive force after flexural cracking, and Pcont is the

tensile force in the concrete.

From vertical equilibrium

1

sin 15n

a stpi

i

V S P

where Pstpi is the force developed in the i-th stirrup, and n is the number of the stirrups

crossing the diagonal crack.

Combining Eqs. (14) and (15) to eliminate the shear force along the sliding plane S

yields

1

tan 16n

a pt rt conc af rc stpi

i

V P P P P P

Taking moments about the upper point A, rotational equilibrium is obtained when

1

172

na cr

a stpi stpi pt pt rt rt cont cont conc af conc af rc rc

i

V SM P d P d c P d c P d P d P c

where dstpi is the horizontal distance from the position of the i-th stirrup to the point A,

dconc-af is the distance from the centroid of the concrete compressive force to the top

layer after flexural cracking, and dcont the distance from the concrete tensile force to the

top layer.

Rearranging Eq. (17) gives

1 18

/ 2

n

stpi stpi pt pt rt rt cont cont conc af conc af rc rc

ia

cr

P d P d c P d c P d P d P c

Va S

Substituting Eq. (18) into Eq. (15) gives

1

219

/ 2 sin

ncr

pt pt rt rt cont cont conc af conc af rc rc stpi stpi

i

cr

SP d c P d c P d P d P c P a d

Sa S

Comparing Eq. (19) with Eq. (12), it is shown that the shear force S along the diagonal

crack is further reduced by the forces in the stirrups.

Combining Eqs. (19) and (14) gives

33

1

220

ncr

pt pt rt rt cont cont rc rc stpi stpi

iconc af

conc af

SP e d c P e d c P e d P e c P a d

Pe d

where e=(a-Scr/2)tan.

For the case without stirrups, Pconc-af can be obtained by simply ignoring the stirrup

component in Eq. (20). Comparing Eq. (20) with Eq. (13), it can be seen that the

concrete compressive force Pconc-af after flexural cracking increases with the increase of

forces in stirrups. This is as expected owing to the direct resistance of stirrups to the

vertical load Va, thereby correspondingly decreasing the shear force S along the inclined

plane, and increasing the concrete compressive force. It is worth noting that an increase

of Pconc-af tends to increase Zcap, the shear capacity of the uncracked region. This shows

the indirect but beneficial effect of stirrups on the shear sliding capacity of the diagonal

plane.

In the case of a cracked member, the same general analysis procedure is followed, that

is, a rotation θ is imposed and the deformation profile adjusted until force equilibrium

given by Eq. (16) is satisfied. Having satisfied force equilibrium, if rotational

equilibrium in Eq. (17) is not obtained θ is adjusted until it does.

When determining the internal forces in Fig. 10(e), for the uncracked concrete and the

reinforcement contained within the uncracked region, the strains, stresses and forces in

Figs. 10(c), (d) and (e) respectively are determined as in the uncracked analysis. For any

reinforcement crossing a tensile crack, the load developed in the reinforcement is a

function of the slip of the bar relative to the concrete in which it is embedded, that is Δpt,

Δrt and Δstp in Fig. 10(a) and must be determined using partial interaction theory

(Haskett et al. 2008; Muhamad et al. 2011, 2012; Oehlers et al. 2011a, 2011b) as

explained in the following section.

In Fig. 10(a), when considering the total slip for the tensile reinforcement to

accommodate multiple cracking, the slips at each crack position shall be accumulated.

For example, if Δrt is half the crack width that is the slip at one crack face at the level of

un-tensioned reinforcement as in Fig. 2(b), for diagonal crack A-A*, the total slip of the

reinforcement is Δrt plus the sum of the diagonal crack widths which are located

between A-A* and B-B

*, that is A-C, A-D, and A-E in this case; so the final slip for

mild reinforcement is 7Δrt, and the nΔrt in Fig. 10(a) is 7Δrt. Meanwhile, it is equally

important to note that the force in the steel will still be Prt, the one to accommodate Δrt

in the partial interaction analysis, rather than the one to accommodate nΔrt. The same

principle applies to prestressed tendon as well.

Tension Stiffening Analysis

Consider the reinforcing bar located concentrically in a concrete prism of area Ac which

has been extracted from the segment as in Fig. 11(a). If a slip Δcf at the crack face (Δ1) is

imposed as shown (which could correspond to Δrt or Δpt in Fig. 10), it is a question of

determining the force in the reinforcing bar Pr when this slip occurs. Generic closed

form solutions (Mohamed Ali et al. 2006; Mohamed Ali et al. 2012; Muhamad et al.

2011, 2012) have been developed for this using partial interaction theory. Alternatively,

an iterative solution (Haskett et al. 2008; Oehlers et al. 2011a, 2011b) can be used to do

this by first guessing or estimating the force Pr to induce Δcf and then using known

boundary conditions to determine the correct value for Pr.

34

To determine the crack spacing Scr, the boundary condition (Muhamad et al. 2011, 2012)

in Fig. 11(a) can be used; that is at point B along the prism Lpri=Scr from the crack face,

both slip and slip-strain Δ' tend to zero and the strain in the concrete εc reaches the

concrete tensile cracking strain εct.

Fig. 11. Boundary conditions for partial interaction analysis

(a) longitudinal reinforcement (b) transverse reinforcement

To allow for tension stiffening between flexural cracks as required for the analysis in

Fig. 2(b), let us take cracks at point E and D for example. After primary cracks form

with a spacing of Scr, the force in the concrete Pc becomes zero at both crack faces, and

since the reinforcement is tensioned at both crack faces, by symmetry, the boundary

condition changes into Δ=0 at the position of half the prism length, that is Scr/2

(Muhamad et al. 2011, 2012) at point C in Fig. 11(a).

For a prestressed tendon, at each segment due to the applied prestress (Knight et al.

2013a, 2013b) there is a residual strain in the reinforcement εr-pr = Ppr/(ErAr) where Er

and Ar are respectively the elastic modulus and area of the reinforcement and the

residual strain in the concrete is εc-pr = εr-pr(ErAr/EcAc)where Ec is the elastic modulus of

the concrete . The addition of the residual strains results in a change to the boundary

condition applied to determine the crack spacing. For a prestressed prism, a crack forms

at the location where Δ tends to zero and Δ' tends to εr-pr+εc-pr and εc tends to the rupture

strain εct at point B as in Fig. 11(b). No change in the tension stiffening boundary

condition occurs and only the incorporation of the residual strains due to prestress at an

elemental level (Zhang et al. 2012) need to be considered.

To determine the load developed in the transverse reinforcement in Fig. 10(a) the

boundary conditions in Fig. 11(b) can be used. For a given total crack opening 2Δstp, the

reinforcement is considered to be fully anchored at the ends of each stirrup leg. As

shown in Fig. 11(b), 2Δstp is then considered to comprise of the individual slip from

each side of the crack face Δstp1 and Δstp2. For analysis, for a fixed 2Δstp, the slip Δstp1 is

guessed and the corresponding force Pstp1 from Δstp1 and Pstp2 arising from Δstp2=2Δstp-

Δstp1 is determined. If Pstp1 does not equal Pstp2, it is known that the initial guess of Δstp1 is incorrect and it is adjusted until it is. In this way a relationship between the total slip

2Δstp and Pstp as required for the analysis in Fig. 10(a) is known.

Having defined the force developed in the un-tensioned Prt and prestressed Ppt

reinforcement as a function of the slips Δrt, Δpt as well as the load developed in the

stirrups Pstp as a function of the crack width 2Δstp, the analysis in Fig. 10 can be carried

on iterating θ and Δtop until both force and rotational equilibrium are achieved. This

analysis will provide the magnitudes of all the internal forces Pconc-af, Pcont, Prc, Prt, Ppt

and Pstp as well as the depth to the neutral axis allowing the presliding capacity for the

cracked section, between A and B in Fig. 9 to be determined.

Δcf

Pr

Scr

crack face

2ΔstpΔstp1

Δstp2

Pstp1

Pstp2

Δ=0

Δ=0(a)

(b)

LstpP’ r

Pc

Pc

S /2cr

ABC

35

Conclusion

This paper has presented the development and application of a new mechanics based

segmental approach to predict the presliding shear capacity of prestressed RC beams

with and without stirrups. The segmental approach is based on the mechanics of partial

interaction and shear friction and builds upon the presliding shear friction failure criteria

and segmental analysis technique for flexure previously developed by the authors.

Importantly the segmental approach addresses the commonly identified shortcomings of

the existing empirical approaches, that is: it allows for the variation of longitudinal

reinforcement ratio; it does not fix or assume the angle at which the diagonal failure

plane will occur; it does not treat the contribution of the stirrups and the concrete to the

shear capacity separately, conversely it shows the importance of treating them together;

and quantifies the interaction between flexure and shear. This partial-interaction

segmental approach has been applied in a companion paper (Zhang et al. 2013) to

analyse published prestressed concrete members and has been validated by comparing

the numerical results with test strength; using the proposed model the major variables

influencing the shear strength of prestressed members are studied and from the analysis

the effect of these parameters on shear behaviour can be explained.

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’. The first author also thanks the China Scholarship Council for financial

support.

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43

Presliding shear failure in prestressed RC beams. II: Behavior

Zhang, T., Visintin, P., Oehlers, D.J., Griffith, M.C.

Abstract

In a previous companion paper, based on the theories of partial interaction and shear

friction, a mechanics based segmental approach, which can cope with any cross section

and material property, was developed to simulate the shear behaviour and failure of

prestressed concrete beams with and without stirrups; the included equations and

mechanisms are purely mechanics based and independent of empirical material

properties. In this paper, this numerical approach has been applied to describe the shear

behaviour of prestressed RC members. The effect of prestress on the shear behaviour is

explained, and the results of parametric studies on stirrup effectiveness are also shown.

102 published test beams with and without stirrups have been analysed by the proposed

model and analytical results show good agreement with the experimental data; the

average predicted results for the beams without stirrups being 96% of the test results

and that for the beams with stirrups being 91% with coefficients of variation of 0.10 and

0.08 respectively. The equations provided by ACI 318-08 have been used to calculate

the shear strength of the same test specimens, and the ACI shear provisions are shown

to be quite conservative with an average of 67% and 74% and coefficients of variation

of 0.29 and 0.18 for beams with and without stirrups. The influence of the random

nature of cracks on the shear strength is also investigated.

Keywords: Prestressed beams; Shear behaviour; Shear failure mechanism; Shear

capacity

Introduction

Current approaches for deriving the shear strength of prestressed RC members (ACI

Committee 318 2008; Eurocode 1991; ASCE-ACI Committee 445 1998; Standards

Australia 2009) are predominantly empirical relying on calibration through a large

amount of specimen testing. Being empirically based, they can only be used within the

bounds of the range of tests from which they were extracted. Furthermore, being

empirically based, they do not simulate shear failure and, hence, provide little

understanding of the shear failure mechanism. An alternative mechanics based approach,

which simulates the shear failure mechanism of prestressed beams using partial-

interaction theory and allows directly for the interaction between flexure and shear, has

been described in a companion paper (Zhang et al. 2013b).

In this paper, the fundamental concepts behind this mechanics based approach are first

explained. The mechanics based approach is then used to study the behaviour of both

prestressed beams without transverse reinforcement and those with transverse

reinforcement. This is then followed by a study of the effect of prestress on the shear

behaviour and the effectiveness of stirrups in prestressed beams. Having used the

mechanics based model to describe the shear behaviour of prestressed beams and in

particular the parameters that affect their strength, the model is then compared with 102

published tests half of which have stirrups and half without. It is shown that the

mechanics based approach has good correlation with test results and, as would be

expected, code approaches tend to be unduly conservative. This would suggest that in

the long term, but not in this paper, that this mechanics based approach could be the

44

basis for developing closed form solutions for shear strength and for refining existing

design equations.

Partial-Interaction Shear Failure Model

The partial-interaction shear failure model is fully described in a companion paper

(Zhang et al. 2013b). It is recognised (Zhang 1997; Hoang and Nielsen 1998; Jensen

and Hoang 2009; Hoang and Jensen 2010) that shear failure along a critical diagonal

plane depends on both the shear load to cause the diagonal crack Vcr-pre and the shear

sliding resistance along the diagonal crack Vsl-pre; whichever is larger controls the shear

failure. For a prestressed member, after the cracking load Vcr-pre and sliding strength Vsl-

pre are determined for diagonal planes with various angles as in Fig. 1, the final shear capacity Vcap-pre (Zhang et al. 2013b) can be considered as either the intercept of Vsl-pre

and Vcr-pre, or the local minimum of Vsl-pre if it is larger than the corresponding Vcr-pre.

Consider a diagonal crack of an angle of as in Fig. 2(b), the cracking capacity Vcr-pre can be determined (Zhang 1997; Zhang et al. 2013b) using Eq. (1).

2/ sin / 2

1/ 2

tef pr pt

cr pre

cr

f b d P d cV

a S

where Ppr is the prestressing force applied in the tendon, a is the shear span, b is the

width of beam cross section, d is the overall depth of the section, cpt is the concrete

cover for the prestressed tendon, Scr is the primary crack spacing and ftef is the effective

tensile strength of the concrete (Zhang 1997).

Fig. 1. Shear failure of prestressed RC members

Now let us determine the sliding strength Vsl-pre for the diagonal crack as in Fig. 2(b). At

a combination of applied shear Va and moment Ma, all the internal forces in the segment

analysed can be obtained by using the proposed approach (Zhang et al. 2013b). In this

approach, the potential sliding plane A-A* is displaced to C-C

*. It is a question of

finding the correct rotation θ and neutral axis dNA to satisfy both the force and rotational

equilibrium. Before flexural cracking, full interaction can be applied to solve all the

forces, since there is no slip between the reinforcement and concrete, that is the analysis

is strain based. For example at the level of the reinforcement where the segment length

is Lrt as shown, the displacement at the level of the reinforcement rt induces a strain

rt/Lrt and consequently a stress and consequently a force in the reinforcement Prt that can be used in deriving equilibrium. However after flexural cracking, slip occurs

between the longitudinal and transverse reinforcement and the adjacent concrete (such

as rt of the reinforcement, pt of the prestressed tendon and stp of a stirrup) as in Fig. 2(b) and this has to be accommodated by the use of the partial interaction model as

described in the companion paper (Zhang et al. 2013b). In this case as an example, rt is

β

V

Vsl-pre

Vcap-pre

Vcr-pre

45

now the slip at the crack face which induces a force Prt that depends on the bond slip

characteristics that can only be determined using partial-interaction theory.

Fig. 2. Shear behaviour along a diagonal plane:

(a) cross seciton; (b) segment for shear analysis

With all internal forces known, the corresponding sliding shear capacity Va-sl along A-

A*, at a specific value of in Fig. 2(b), can now be determined through shear friction

theory as explained in the companion paper (Zhang et al. 2013b). This is illustrated in

Fig. 3. The sliding capacity depends on the normal force to the sliding plane Pconcsin and can be determined from shear-friction material properties. This sliding capacity has

to resist the shear force Pconccos such that the remainder is available to resist the shear force S which itself is affected by the presence of stirrups as derived from the analysis

in Fig. 2(b). The sliding capacity can be resolved vertically to get Va-sl. If Va does not

exceed Va-sl, then sliding does not occur and the specimen can undergo further loading.

This process can be repeated for different combinations of Va and Ma at that specific

value