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Evaluation of Shear Strength of High

Strength Concrete Beams

Submitted by

Attaullah Shah

Department of Civil Engineering

University of Engineering & Technology

Taxila-Pakistan

June 2009

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Evaluation of Shear Strength of High

Strength Concrete Beams

Submitted by

Attaullah Shah(Registration No.01/UET/PhD/CE-02)

This thesis is submitted in partial fulfillment of therequirements for the PhD Civil Engineering

PhD SupervisorProf Dr. Saeed Ahmad

Department of Civil Engineering

University of Engineering & Technology

Taxila-Pakistan

June 2009

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Abstract

In this thesis, the shear properties of High Strength Reinforced Concrete (HSRC)

beams have been investigated on the basis of available research data and

experimental work at Structural Laboratories of University of Engineering andTechnology Taxila-Pakistan. The shear capacity of High Strength Reinforced

Concrete (HSRC) beams is relatively less investigated in the contemporary

research, as most of the research data available is based on the results from

normal strength reinforced concrete with compressive strength of 40MPa or less.

There is a general consensus amongst the researchers in the field of Structural

Engineering and Concrete Technology that the shear strength of HSRC beams,

unlike the Normal Strength Reinforced Concrete (NSRC) does not increase, in

the same proportion as the increase in the compressive strength of concrete, due

to brittle behaviour of the High Strength Concrete. Hence the current empirical

equations proposed by most of the building and bridges codes for shear strength

of HSRC beams are less conservative as compared to the Normal Strength

Reinforced Concrete (NSRC) beams. This major observation by the researchers

is the main focus of this research.

An extensive literature review of the shear properties of Normal Strength

Reinforced Concrete (NSRC) beams and High Strength Reinforced Concrete

(HSRC) beams was undertaken. Additionally the shear strength of disturbed

region (D-Region) was also studied. In disturbed region the ordinary beams

theory based on Bernoullis theorem is not applicable. In the literature review of

disturbed regions special emphasis was laid over Strut and Tie Model (STM),

which is an emerging analysis and design tool in the current research in

reinforced concrete.

The literature review was followed by the experimental work, which comprised of

70 high strength reinforced concrete beams and 9 two ways high strength

concrete cobles. Beams were cast in two sets of 35 beams each, one set without

web reinforcement and other with web reinforcement. For each set of 35 beams

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five values of longitudinal reinforcement and seven values of shear span to depth

ratio were selected to mainly study the behaviour of slender beams, where

typical shear failure can be anticipated. These beams were tested under

monotonic load at the mid span to examine the contribution of various

parameters like longitudinal steel, shear span to depth ratio, and web

reinforcement, on the shear capacity of HSRC beams. It has been observed that

the shear strength of beams has been increased with the increase in longitudinal

steel and shear reinforcement but it has reduced with the increase in the shear

span to depth ratio. The beams with low longitudinal steel ratio and no web

reinforcement failed mainly due to shear flexure cracks. However the beams with

longitudinal steel ratio of 1% and more failed mainly due to beam action in shear

tension failure. The beams with small shear span to depth ratio and large valuesof longitudinal steel ratio however failed due to shear compression failure.

The shear failure of HSC beams with large values of longitudinal steel and shear

span to depth ratio was however more sudden and brittle, giving no sufficient

warning before failure, which has been observed as serious phenomena in the

shear failure of HSC beams.

The addition of web reinforcement increased the shear strength of all beamstested. The failure mode was also affected. The obvious contribution of the

minimum web reinforcement was avoiding the sudden failure of the HSC beams.

These test results were also compared with the equations of some international

building and bridges codes and methods for shear strength of HSRC beams. It

has been noticed that these equations do not provide equal level of safety in the

shear design of HSRC beams. Some of the codes are over conservative, while

few others are less conservative for the shear design of HSRC beams.

Comparison of the observed shear strength of tested HSRC beams with the

results of the codes equations used, reveal that most of these equations are less

conservative for shear design of HSRC beams at lower values of longitudinal

steel for both cases of beams with and without web reinforcement, particularly for

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longitudinal steel ratio less than1%. Hence additional care may be required for

shear design of HSRC beams at large values of shear span to depth ratios.

To analyze the behaviour of typical disturbed region in concrete structures, the

basic rationale of Strut and Tie Model (STM) was used for the analysis and

design of two way corbels. These corbels were tested under monotonic loads

applied at the overhanging portion of the corbels. The actual shear capacities of

these corbels were compared with the theoretical shear capacities of the corbels

worked out with the STM. The actual and theoretical values of the shear were

falling close to each other. Their comparison reveals that STM can be further

tested as more simple and reliable tool for analysis and design of disturbed

region (D-Region) in concrete structures, through more experimental research.

Further research work on shear properties of HSRC beams with higher values of

compressive strength of concrete in the beam region and more experimental

research on the disturbed region including pile caps, deep beams, dapped ended

beams and corbels has been recommended at Engineering University-Taxila

Pakistan.

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Acknowledgement

The higher study has been both my ambition and dream since my graduation but

the job and family commitments always impeded to realize it. The historic

decision of Higher Education Commission (HEC)-Pakistan, to strengthen theUniversities in Pakistan and taking initiatives for promoting research, ushered a

new era of innovation and higher education in Universities and institutes of higher

learning. I was offered PhD admission both from UET Peshawar and UET Taxila

at the same time but I preferred the later as it is closely located to my place of

job.

PhD studies at UET Taxila-Pakistan, had been an enterprising experience of my

life which transformed me from a predominantly Servicing officer into an

academician with more thirst for learning, innovation and interaction with

scholarly people. My PhD supervisor Prof Dr. Saeed Ahmad actively involved me

in the research work of post graduate students, their examination and viva voce

exams right from the beginning and provided me an opportunity to learn more

about the latest trends and developments in the Civil Engineering, besides my

core area of research. In these endeavors I had been able to work on many

projects with him which mainly included, High Range Water Reducers,

(Superplasticizers), Self Compacting Concrete, Very Early Strength (VES)

Concrete, High Strength Concrete (HSC), Retrofitting and Rehabilitation of the

damaged structures etc. These efforts on the part of my supervisor enabled me

to bridge the knowledge gap and tackle the PhD studies more seriously and

rigorously. I must appreciate his patience and straightforwardness as I have

always found him a sincere and upright person. He had been very kind

throughout the research work and provided me, his guidance at all stages of mystudies.

Interaction with the staff at UET Taxila turned a pleasant opportunity. While

working with the Laboratory staff, academicians and other administrative staff at

different times, I have received their due support and kindness. I remember

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taking lunch with the Concrete and Structure Laboratories staff during casting

and testing of beams. I always felt as part of the family of employees at UET

Taxila, and received due regards from all of them. The staff of Labs worked with

me tirelessly in the afternoon and I must appreciate their kindness and support.

I received due support from the Chairman Civil Engineering Prof. Dr M.A.Kamal

and Ex-Chairman Prof. Dr A.R Ghumman in discussing my problems regarding

the funding of faculty research project and other such matters.

The staff of Directorate of Advanced Studies Research and Technology

Development had always been very kind and cooperative in forwarding my

requests for grants to the competent authority, which enabled me to get two

grants of Rs 200,000 each for faculty research with my supervisor.

I was always duly encouraged by Prof Dr. Muhammad Ilyas UET Lahore and

Dr. Tariq Mehmood Zaib, Pakistan Atomic Energy Commission (PAEC), during

my PhD studies and editing of the thesis. Their support and positive attitudes

always provided a hope to complete my work. In the days of despair they always

encouraged me.

At last but not the least I feel highly indebted to Prof Dr. Habibullah Jamal Ex-

Vice Chancellor and incumbent Vice Chancellor UET Taxila Prof. Dr. M. Akram

Javed for their support and guidance.

Today when I am writing the closing chapter of my PhD thesis, I feel proud and

highly grateful to Almighty Allah, that in my efforts to broaden my vision and

knowledge, I was fortunate to meet with very friendly people and as a result I,

feel part of UET Taxila today. In my endeavors my parents my family and my

personal staff, always supported me. My children kept missing me while I was

working at my office in writing this thesis and conducting experimental works.

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I pray to the almighty Allah that this work may pave ways for further innovation &

research and this nation and the Engineering professionals may benefit from the

findings-Amen.

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List of Figures

Figure No. Description Page

Figure 2.1Cracks appeared when vertical load is applied at the mid span of a beam(Jose,2000)

25

Figure 2.2Distribution of bending and shear stresses across the section of a beamelement and stress state in element A2 and corresponding Mohrscircle(Jose,2000)

27

Figure 2.3 Types of cracks expected in the reinforced concrete beams (Jose, 2000). 29

Figure 2.4Forces acting in a beam element within the shear span and internal arches ina RC beam (Russo et al., 2004).

30

Figure 2.5Shear in beam with no transverse reinforcement. (Stratford and Burgoyne,2003)

32

Figure 2.6 Comparison of theoretical and test results of shear failure of beams(Kani.1964)

33

Figure 2.7Parallel chord truss model. The struts are intercepted by the stirrups atspacing of d (Ritter, 1989).

35

Figure 2.8 Shear strength of RC beams with shear reinforcement (ACI-ASCE,1998) 36

Figure 2.9 Size-effect law (Baant et al.1986). 38

Figure 2.10 Kanis Tooth Model (Kani,1964). 44

Figure 2.11 Compression Field Theories (Mitchell and Collins,1974) 48

Figure 2.12 Description of Modified compression Field Theory (Vecchio and Collins,1986) 52

Figure 2.13Values of and for RC members with at least minimum shearreinforcement.

57

Figure 2.14Values of and for RC members with less than minimum shearreinforcement (Vecchio and Collins1986).

59

Figure 2.15 Transmission of forces across the crack. ( Bentz. et al,2006) 61

Figure 2.16 Variable truss Model of RC beams ( Mitchell, 1986) 68

Figure 2.17 Shear Friction Hypothesis of Birkeland and Birkeland (1966) 69

Figure 2.18Comparison of CSA and ACI amounts of minimum shear reinforcement (Yoonet al, 1996).

79

Figure 3.1 World Trade Centre (USA) 88

Figure 3.2The world Highest Tower Burj Dubai,UAE (2651 feet) (162 floors, scheduledconstruction, 2008)

88

Figure 3.3Variation of compressive stress-strain curves with increasing compressivestrength.( Adapted from Collins and Mitchell, (1997).

108

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Figure No. Description Page

Figure 4.1Example of B& D-Regions in a Common Building Structure (Schlaich et al

,1987)122

Figure 4.2Example of B&D-Regions in a Common Bridge Structure. (Schlaich. et al.1987).

122

Figure 4.3 Some Typical Strut and Tie models as proposed by ACI 318-06(ACI-ASCE,1996)

123

Figure 4.4 Classifications of Nodes (ACI- 318-06) 128

Figure 4.5 Proposed STM for Deep beams under applied external load 130

Figure 4.6 Proposed STM for one way corbel under applied external load. 130

Figure 4.7 Proposed STM for two way corbel under applied external load. 130

Figure 4.8 Proposed STM for dapped beam end under applied external load 131

Figure 4.9 Proposed STM for pile cap under applied external load. 131

Figure 5.1 Flowchart for use of the NCHRP simplified design method ( NHRP, 2006). 149

Figure 6.1 Details of beams used in the testing. 155

Figure 6.2 Details of loading arrangement for the testing of RC beams. 157

Figure 6.3 Details of roller supports and deflection gauges used for the beams. 157

Figure 6.4 Wet sand filled around the beams for curing. 160

Figure 6.5Failure of beams without web reinforcement due to diagonal tension shearfailure mode of the beam.

165

Figure 6.6

Failure of beams without web reinforcement due to diagonal tension shear

failure mode of the beam. The failure angles have been reduced with theincrease in longitudinal steel.

167

Figure 6.7 Flexural shear failure of beams without web reinforcement having a/d>5. 168

Figure 6.8

Typical shear failures of beams without web reinforcement. The failure is morebrittle and sudden amongst all. The crack causing failure of the beam was notnoticed in the beginning and beams failed very suddenly due to tension shearfailure.

169

Figure 6.9Effect of longitudinal Steel ratio on the shear strength of concrete beamswithout stirrups for same value of a/d.

171

Figure 6.10

Effect of longitudinal Steel ratio on the shear strength of concrete beams with

web reinforcement for same value of a/d. 171

Figure 6.11Effect of shear span to depth ratio on the shear strength of concrete beamswithout stirrups for same value of longitudinal steel ratio.

173

Figure 6.12Effect of shear span to depth ratio on the shear strength of concrete beamswithout stirrups for same value of longitudinal steel ratio.

173

Figure 6.13Beam shear failure or diagonal tension shear failure in beams with webreinforcement.

174

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Figure No. Description Page

Figure 6.14 Load deflection curves for beams without web reinforcement and =0.0073 178

Figure 6.15 load deflection curves for beams without web reinforcement and =0.02 179

Figure 7.1 Geometry of the proposed two way corbel and proposed STM. 181

Figure 7.2 Reinforcement Form work used for the two way corbels. 181

Figure 7.3 Loading arrangement for HSC two way corbels. 183

Figure 7.4 Details of embedment strain gauge 184

Figure 7.5 Strain Data Logging system used. 184

Figure 7.6 Member Forces in strut and Tie model for two way corbel. 185

Figure 7.7 Details of reinforcement, formwork and embedment gauges. 186

Figure 7.8 Typical shear failures of the two ways HSC corbels. 188

Figure 9.1 Plot of the proposed model generated by the software. 214

Figure 9.2Comparison of actual values of shear stress with the predicted values byproposed regression model and other models for HSC beams without webreinforcement.

221

Figure 9.3Comparison of actual shear stress of beams having stirrups with the proposedregression model and other models.

222

Figure A-1 Geometry of Two way corbel. 249

Figure A-2 Geometry of assumed Strut and Tie Model ( STM) 250

Figure A-3 Member Force in strut and Tie model for two way corbel. 252

Figure A-4

Reinforcement details of two way corbel designed for 80 Kips (355KN) load by

STM. 254

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List of Tables

Table No. Description Page

Table 2.1Comparison of experimental results with the full MCFT, simplified MCFT and

ACI equation for shear strength of RC beams.( Bentz et al, 2006)

64

Table 2.2Comparison of the shear strength of RC beams proposed by Zararis , ACI

And EC-2 ( Zararis P.D,2003)73

Table 3.1Definition of HPC as per SHRP (Zia et al, 1993)

90

Table 3.2Volume of coarse aggregate per unit of volume of concrete. (ACI-211.1)

98

Table 3.3Upper limits of specified compressive strength of concrete for HSC andStandard test specimen. (Paultre and Mitchell (2003). 103

Table 3.4

Comparison of values of load factors, strength reduction factors and material

strength reduction factor proposed by various codes (Paultre and Mitchell,2003). 104

Table 3.5

Comparison of values of modulus of elasticity modulus of rapture and minflexure reinforcement proposed by various codes (Paultre and Mitchell(2003).

105

Table5.1Summary of Major Code Expressions for the Concrete Contribution to ShearResistance.

143

Table 5.2Summary of Research Results conducted at various Universities.

144

Table 5.3Comparison of test values and Codes values based on shear data base(NCHRP; 2006)

145

Table 6.1Mix Proportioning/ Designing of High Strength Concrete.

154

Table 6.2 Details of reinforcing bars used in the beams 154

Table 6.3Reinforcement details of beams.

156

Table 6. 4Shear span to depth ratio and corresponding span of seven beams in eachset of longitudinal reinforcement.

156

Table6.5Details of Series-I beams without web reinforcement ( 35 Nos)

159

Table 6.6 Details of Series-II beams with web reinforcement ( 35 Nos) 159

Table 6.7 Total applied failure load at the beams without web reinforcement 161

Table 6.8 Total applied failure load at the beams with web reinforcement 162

Table 6.9Shear Strength and failure angles of 35 HSC beams, without webreinforcement 163

Table 6.10 Shear Failure mode of 35 beams with web reinforcement 163

Table 6.11Shear Strength ,failure angles and failure modes of 35 HSC beams, withweb reinforcement. 164

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Table No. Description Page

Table 6.12Effect of the longitudinal steel on the shear strength of beams for constanta/d values. 170

Table 6.13Shear strength, failure mode and failure angles for 35 HSRC beams withweb reinforcement.

175

Table 6.14Increase in the shear strength due to addition of web reinforcement in HSRCbeams.

177

Table 7.1 Mix Proportioning/ Designing of High Strength Concrete Double Corbels 182

Table 7.2Details of technical parameters and member forces in assumed STM

186

Table 7.3 Comparison of theoretical and actual failure loads of HSC double corbels 187

Table 8.1Comparison of the shear strength of beams without web reinforcement withthe provisions of the ACI 318-08

190

Table 8.2Comparison of the shear capacity of beams with web reinforcement with theprovisions of the ACI 318-08 191

Table 8.3Comparison of increase in shear strength due to stirrups and ACI-318provision for stirrups contribution

192

Table 8.4Comparison of the shear Strength of beams without web reinforcement withthe provisions of the Canadian Standards (Simplified Method) 194

Table.8.5Comparison of the shear Strength of beams with web reinforcement with theprovisions of the Canadian Standards (Simplified Method) 195

Table 8.6Comparison of the shear Strength of beams without web reinforcement withthe provisions of MCFT( LRFD Method) 197

Table 8.7

Comparison of the shear Strength of beams with web reinforcement with the

provisions of MCFT ( LRFD Method) 198

Table 8.8Comparison of the shear Strength of beams without web reinforcement withthe provisions of EC-02

200

Table 8.9Comparison of the shear Strength of beams with web reinforcement with theprovisions of EC-02 201

Table 8.10Comparison of the shear Strength of beams without web reinforcement withequation proposed in new theory of Zararis,P.D. 203

Table 8.11Comparison of the shear Strength of beams with web reinforcement withequation proposed in new theory of Zararis,P.D. 204

Table 8.12Comparison of Vtest/VCode for ACI, CSA, MCFT, EC-02 and New Equation forbeams without web reinforcement. 206

Table 8.13Comparison of Vtest/VCode for ACI, CSA, MCFT, EC-02 and New Equation forbeams with web reinforcement. 207

Table 8.14:Summary of means of the ratios of observed values and different codeValues for shear strength of beams without web reinforcement

208

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Table No.Description Page

Table 8.15Summary of means of the ratios of observed values and different codeValues for shear Strength of beams with web reinforcement. 208

Table 9.1 Comparison of actual and predicated values of shear stress of High Strengthconcrete beams without web reinforcement for three proposed models. 216

Table 9.2Comparison of actual and predicted values of shear stress of high strengthconcrete beams with web reinforcement.

218

Table 9.3Comparison of proposed model ACI equation and model proposed byG.Russo et al. (2004) for beams without web reinforcement..

220

Table 9.4Comparison of actual shear stress of beams having no stirrups with theproposed model and other models of ACI, Bazant and Russo

224

Table 9.5Comparison oftest/pred by the proposed model and other models for beamswithout shear reinforcement ( 35 Nos). ( For constant steel ratio and variablea/d)

228

Table 9.6 Comparison oftest/pred by the proposed model and other models for beamswith shear reinforcement ( 35 Nos) ( For constant steel ratio and variablea/d)

229

Table 9.7Comparison oftest/pred by the proposed model and other models for beamswithout shear reinforcement ( 35 Nos). ( For constant a/d and variable steelratio)

230

Table 9.8Comparison oftest/pred by the proposed model and other models for beamswith shear reinforcement ( 35 Nos) ( For constant a/d and variable steelratio)

231

Table A-1

Forces in Truss of double corbel after analysis.251

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Chapter Description Page

4Shear design of disturbed region (D-region) in reinforced

concrete.121

4.1 The basic concept of Beam and Disturbed region 121

4.2 Basic design principles for shear design of disturbed region 124

4.3 Using Strut and Tie Model for the shear design of Structuralcomponents.

124

4.4 Choosing the Strut and Tie Model (STM). 126

4.5 Procedure for shear deign of disturbed region with STM. 129

4.6 Some latest research on the shear design of disturbed region

with STM.

131

5Provisions of international building codes for the shear design ofNormal & High Strength Concrete.

136

5.1British Standards (BS-8110) 136

5.2European Code EC2-2003. 137

5.3ACI Code 318-06 (American Concrete Institute) 138

5.4 Canadian Standards for design of Concrete structures. CSA A-23.3-94. 140

5.5AASHTO LRFD (Load Reduction Factor Design) Bridge Design

Specifications -1996.141

5.6 Empirical methods for beams without shear reinforcement. 142

5.7Results of High Strength concrete beams at differentUniversities, in near past.

143

5.8Evaluation of shear design methods of different building codesbased on test data base by National Cooperative HighwayProgram ( NCHRP).

145

5.9Variations in the provisions of international building code forshear capacity of beams.

150

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Chapter Description Page

6

Experimental Program

Experimental programme and discussion of test results of HSC

beams ( B-Region).

153

6.1 Introduction to experimental programme. 153

6.2 Test Specimen. 154

6.3 Test set up 157

6.4 Experimental results. 161

6.5 Discussion of results. 165

7

Experimental Programme on disturbed Region ( D-region) inconcrete and observations.

180

7.1Experimental Programme for testing of disturbed region in

concrete(D-region).185

7.2Design of the two way corbel by Strut and Tie Model

( STM)187

7.3 Test results and discussion of two way corbel testing. 187

8

Comparison of the observed values with the provisions of

International building and bridges codes.189

8.1 ACI Code 318-08 (American Concrete Institute) 189

8.2 Canadian Standards for design of Concrete structures. (CSAA23.3-94).

193

8.3AASHTOs LRFD DESIGN SPECIFICATION ( 1994).(Modified Compression Field theory-MCFT).

196

8.4Comparison of observed values with the provisions of

Eurocode-02200

8.5 New Theory Proposed by Prodromos D.Zararis (2003) 203

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Chapter Description Page

9

Statistical Model for the prediction of shear strength of High

Strength Concrete beams.211

9.1 Regression model and its application in Civil Engineering. 211

9.2 Regression Model for beams with web reinforcement 213

9.3 Regression Models for shear strength of beams with webreinforcement.

217

9.4 Comparison of the proposed models with ACI-318 Codeand other models:

219

9.5 Discussion on the proposed regression models 232

10

Conclusions and Recommendations. 234

10.1 Conclusions 234

10.2 Conclusions on the work in disturbed region 237

10.3 Recommendations for future work 238

References 239

Appendix A Design of Two way corbel using STM 249

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Chapter No1.

Introduction.

The strength of concrete is one of the most important properties of this versatile

construction material. High Strength Concrete has been widely used in the

construction industry for last few decades. The development of new water

reducing admixtures and the mineral admixtures is making it possible to achieve

more reliable high strength concretes in the recent years. High Performance

Concrete (HPC) is referred to the specialized series of concretes designed to

provide several benefits in the construction of concrete structures. High Strength

Concrete therefore belongs to the High Performance Concrete series, due to its

peculiar properties. The use of High Strength Concrete is likely to increasefurther in 21st century with the construction of more high-rise buildings, long span

pre-stressed bridges, and pre-cast elements in concrete structures.

Concrete unlike steel is relatively non-homogenous material; hence its different

structural properties are likely to change with increase in compressive strength.

The high strength concrete is comparatively a brittle material as the sound matrix

of aggregates and cement paste provides a smoother shear failure plane, which

leads to its abrupt failure. Consequently the shear strength of High Strength

Concrete does not increase in the same way, as its compressive strength. The

availability of limited experimental work on the high strength concrete makes it

difficult to safely predict the shear capacity of high strength reinforced concrete

members.

The shear capacity of reinforced concrete members is presently evaluated on the

basis of empirical equations proposed by different International Building Codeswith certain modifications in the equations for normal strength concrete. As most

of these equations have been derived on the basis of experimental data of

concrete with compressive strength of 6000 psi (40 MPa) or less, therefore their

application to higher values of compressive strength always raise questions in

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the minds of researchers. To further rationalize and generalize, these empirical

equations for shear design of high strength reinforced concrete members,

extensive research is required. This research is therefore an effort in this

direction.

1.1 Problem Statement

To better understand the behaviour of High Strength Reinforced Concrete beams

in shear.

1.2 Aim and Objectives of Research

The main aim of the research is to improve the understanding about the

behaviour of high strength reinforced concrete members in shear and to developsome more rational procedure for the shear design of the High Strength Concrete

members, based on the literature review and experimental work. The relative

objectives of research are further explained as follows;

- To evaluate the shear strength of High Strength Reinforced Concrete

(HSRC) beams with and without web reinforcement.

- To study the effect of various variables on the shear strength of the high

strength concrete beams.

- To compare the provisions and procedures in different International

Building and Bridges Codes and latest developments for the shear design

of high strength concrete beams.

- To discuss the latest trends in the shear design of non-linear and

disturbed regions in the high strength concrete structures, where ordinary

beams theory cannot be applied.

1.3 Scope of the research study

The scope of the research study is as follows;

- Shear Behaviour of High Strength Reinforced Concrete (HSRC) beams

having compressive strength of 52 MPa (8200psi) has been studied.

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- Slender beams with shear span to depth ratio a/d from 3 to 6 have been

selected for research and the results obtained can be generalized for only

this range of beams.

- Five levels of longitudinal steel ratio have been selected, starting from

minimum longitudinal steel ratio of 200/fy to 2% level. Hence the results

mainly cover this range of longitudinal steel ratio from 0.33% to 2%.

- The proposed regression model to predict the shear strength of HSRC

beams is based on the observations of 70 beams tested. Hence its

generalization would require further research.

- For comparison of the observed shear strength of HSRC beams with the

provisions of five building and bridges codes have been selected i.e. ACI-

318, Canadian Code, Euro code (EC-02), AASHTO LRFD bridge designspecification based on Modified Compression Field Theory ( MCFT).

- For the study the shear strength of disturbed region, the basic Strut and

Tie Model (STM), was applied to High Strength Concrete corbels.

1.4 Methodology/Programme

To study the effect of various parameters on the shear strength of HSRC beams,

the following research methodology was adopted;

The experimental work was divided into two regions namely beam region (B-

region) and disturbed region (D-region). For beam region, the following

methodology was adopted.

i. To study the shear behaviour of HSRC beams, 70 beams of size 9inx12in

(23cmx30 cm) were selected in two sets of 35 beams each, such that in

first set no web reinforcement was provided, whereas in second set of 35

beams, web reinforcement corresponding to minimum shear

reinforcement given by ACI-318-08 was provided.

ii. Five levels of longitudinal steel ratio (0.33%, 0.73%, 1%, 1.5% and 2%)

was selected to study the effect of longitudinal steel ratio on the shear

strength of HSRC beams.

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iii. To study the effect of shear span to depth ratio seven values of a/d were

selected as 3, 3.5, 4, 4.5, 5, 5.5 and 6 to mainly cover the behaviour of

slender HSRC beams in shear.

The beams were tested under monotonic loads and the observations were

recorded in terms of cracking pattern, failure mode, and ultimate failure capacity,

deflections of beams at mid span and critical sections at distance d from the

face of supports.

The shear strength of the beams was determined at the failure point and the

observed values were compared mutually and with the provisions of selected

Building and Bridges Codes. The effect of various parameters on the shear

strength of HSRC beams was studied on the basis of observations from thetesting.

An attempt was made to develop regression equation to predict the shear

strength of beams based on the sample date of tested beams; however its

generalization would require extensive experimental work.

To study the shear behaviour of RC structures in disturbed region, where the

shear span to depth ratio is less than 3.0, focus was laid on the Strut and Tie

Model (STM) and nine high strength concrete corbels designed on the basis of

STM for an assumed external load were tested. The actual and theoretical shear

failure loads were compared to check the suitability of STM for analysis and

design of disturbed region in concrete.

1.5 Layout of the thesis

The thesis has been divided into ten chapters. Next to the introduction, in

Chapter 2, shear strength of reinforced concrete and various factors affecting

shear strength of concrete have been discussed. Some latest approaches like

Modified Compression Field Theory (MCFT), Simplified Compression Field,

theory and truss approaches have been discussed in quite details. At the end of

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the chapter, two design examples on MCFT and one example on use of

specialized software Response-2000 based on MCFT, have been added.

Additionally some latest review work by using MCFT and simplified MCFT has

been included in the Chapter 2.

In Chapter 3, various properties of high performance concrete and high strength

concrete have been discussed with special emphasis over the selection of

material, admixtures, mix proportioning, transportation, placement and structural

properties of high strength concrete. Codes provision for measuring the

compressive strength, flexural strength, modulus of elasticity and other structural

properties of HSC in European code (EC-02 and CEB-MC-90), Canadian code

(CSA A23.3-94), American Concrete Institute (ACI-318-02) and New Zealandcode (NZS 3101-95) have been discussed. In literature review of shear strength

of high strength concrete, current state of the research in shear strength of high

strength reinforced concrete beams has been elucidated, which forms basis for

onwards study of the problem. Some latest approaches to address the problem

of shear in high strength concrete have also been discussed in the chapter.

In Chapter 4, shear strength of disturbed regions (D-region) in concrete

structures has been discussed, in the light of latest research. The literature

review on the shear design of disturbed region has revealed that shears design

of disturbed region with new tools like Strut and Tie Model (STM), is as an

emerging area in the shear design of high strength concrete members. However,

there are many challenges in application of STM for the design of concrete

structures. The growing use of new concept of Strut and Tie Modeling of

disturbed region in concrete structures necessitated, to dedicate some

experimental work to this emerging concept for design of concrete structures.

In Chapter 5, provisions of some important International Building and Bridges

Codes for Normal and High Strength Concrete beams have been discussed and

references to the relevant clauses of respective Building Codes has been given.

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In Chapter 6, the experimental program for beam region has been given. In the

beams region seventy beams of high strength concrete in two sets of 35 beams

with web reinforcement and 35 beams without web reinforcement have been

tested with reference to the effect of different parameters on the shear strength of

high strength concrete beams. Each set of beams is comprised of five values of

longitudinal steel ratio and seven values of shear span to depth ratio. This is

followed by the observations and test results and discussion thereon.

In Chapter 7, experimental work on disturbed region has been explained with

special reference to high strength concrete corbels. The testing setup and other

instruments used for measuring the shear strength of the corbels have been

given. The test results have been discussed in term of the suitability of STM forshear design of two way corbels.

In Chapter 8, the actual values of the shear strength of HSC beams have been

compared with the values worked out with the equations proposed by some

international building and bridges codes.

In Chapter 9, efforts have been made to develop some statistical regression

model for predicting the shear strength of HSRC beams on the basis of

experimental results and these have been compared with some other models.

The validity and generalization of the proposed model is however limited due to

insufficient date. However graduate research to propose some more rational

models, which can best fit the available shear database of high strength concrete

beams incorporating more parameters, can be undertaken in the next phase of

research by other graduate students. This preliminary effort can pave way for the

same.

In chapter 10, conclusions and recommendations for future research have been

proposed and at the end references are given.

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The shear stress in a homogenous elastic beam is given as

Ib

VQ (2.1)

Where , V= Shear force at section under consideration.

Q = Static moment about the neutral axis of that portion of cross section lying

between a line through point in question parallel to neutral axis and nearest face

of the beam.

I= Moment of Inertia of the cross section about neutral axis.

b = Width of the beam at a given point.

The small infinitesimal elements A1 & A2 of the rectangular beam in Figure 2.2are shown with the tensile normal stress ftand shear stress across the plane a1-

a1 and a2-a2at distance y from the neutral axis.

The internal stresses acting on elements A1 & A2 are also shown in Figure 2.2.

Using Mohrs circle, the principal stresses for element A2 in the tensile zone

below the neutral axis can be found as

)2

(2

2

(max) tt

t

fff _______________Principal tension (2.2)

)2

(2

2

(max) tt

c

fff _______________________Principal compression (2.3)

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Figure 2.2: Distribution of bending and shear stresses across the section of a beam

element and stress state in element A2 and corresponding Mohrs circle(Jose,2000)

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considered relatively less brittle than the diagonal tension failure due to the

stress redistribution. Yet it is, in fact, a brittle type of failure with limited warning,

and such as design should be avoided completely. This failure is often called as

compression failure or web shear failure.

(a)

Figure 2.3 Types of cracks expected in the reinforced concrete beams (Jose, 2000).

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2.3 Shear strength of normal strength reinforced concrete beams

The research on shear strength of concrete has shown that reinforced concrete

beams without transverse reinforcement can resist the shear and flexure by

means of beam and arch actions, also sometimes called concrete mechanisms

(Russo et al, 2002). These forcesacting on the beam element in its shear span

are shown in Figure 2.4. It was assumed that the resultant of the aggregates

interlocking at thecrack interface can be replaced by Va as shown in the Figure

2.4, whose direction passes through the point of application of the internal

compression force C. The shear contribution due to dowel Vd is negligible at the

rotation equilibrium. The resultant bending moment is given by

Mc = Vc.x = T.jd .. (2.4)

Where Vc is the shear force due to concrete resisting contribution, T is tensileforce in the longitudinal reinforcement and x is the distance between the support

and the point where crack has been appeared.

The sheer force is the derivative of the bending moment Vc= dMc/dx

Vc = jd dx

d T + T.dx

djd ............................... (2.5)

Forces acting in a beam element within the shear span

b. Internal arches in RC beams.

Figure 2.4 Forces acting in a beam element within the shear span and internal arches in

a RC beam (Kani, 1964., Russo et al., 2004).

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The first term in equation 2.5, is the resistance to shear as contribution of the

beam action, whereas the second part is called arch action.

In beam action, the lever arm is constant and the tensile force in the steel bars is

supposed to vary. The beam action is related to the crack pattern in the shear

span, in which the tensile zone is generally divided into blocks or teeth.

Beam action describes shear transfer by changes in the magnitude of the

compression-zone concrete and flexural reinforcement actions, with a constant

lever-arm, requiring load-transfer between the two forces. In a cracked beam,

load-transfer from the flexural reinforcement to the compression-zone occurs

through the teeth of concrete between cracks, requiring bond between the

concrete and reinforcement. Bending and failure of this concrete is studied bytooth models.

The second part of the equation shows the shear resisting contribution due to

arch action, which is characterized by the internal variation of the lever arm jd

with the T constant. The arch mechanism transfers the vertical loads to the

supports through the arch route.

Arch action occurs in the un-cracked part of concrete near the end of a beam,

where load is carried from the compression-zone to the support by a

compressive strut. The vertical component of this strut transfers shear to the

support, while the constant horizontal component is reacted by the tensile

flexural reinforcement. Both beam action and arch action can act in the same

region (Stratford and Burgoyne,2003). Thus shear transfer in the beam can take

place by one of the two mechanisms i.e. variation in the magnitude of internal

actions and variation in the lever arm between the actions. The details are shown

in Figure 2.5. Before cracking of the beams, the shear is resisted by the beam by

all the elements of the beams shown in the paths I, II and III ( Figure 2.4).

However after the cracks, only the un-cracked part of the beams is resisting the

shear by transferring it to the supports.

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Figure 2.6 Comparison of theoretical and test results of shear failure of beams (Kani.1964)

The joint committee ASCE-ACI-426in 1973 and later in 1998 reported the following

five mechanisms for resisting the shear in reinforced concrete sections (NTRB,

2005).

i. Shear in the un-cracked concrete zone

In cracked concrete member, the un-cracked compression zone offers some

resistance to the shear but for slender beams with no axial force, this part is very

negligible due to small depth of compression zone.

ii. Residual tensile stresses

When concrete is cracked and loaded in uni-axial tension, it can transmit tensile

stresses until crack widths reach 0.06 mm to 0.16 mm, which adds to the shear

capacity of the concrete. When the crack opening is small, the resistanceprovided by residual tensile stresses is significant. However in a large member,

the contribution of crack tip tensile stresses to shear resistance is less significant

due to the large crack widths that occur before failure in suchmembers.

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Figure 2.8 Shear strength of RC beams with shear reinforcement (ACI-ASCE,1998)

Experimental studies (Talbot, 1909), reported that the shear capacity of beams was

greater than predicted by the truss model and the idea of concrete contribution was

developed.

Kani (1969) provided a quite different explanation for the role of web reinforcement

in resisting the shear, called as Rational Theory. With the help of actual tests

results, he explained that the purpose of web reinforcement is to provide reactions to

the internal arching which supports the compression zone of the beams and not to

carry the shear force or any part of it. Hence no direct relationship can be expected

between the magnitude of shear force and requirement of web reinforcement. This

was certainly in sharp contrast with the conventional shear theory based on truss

model. He himself declared his proposed rational theory not reconciling with the

conventional shear theory.

Chana (1987), reported that the failure mechanisms of RC beams with transverse

reinforcement is different than the beams without shear reinforcement. Hence Vs and

Vc mutually influence each other and simply adding the two terms may not give valid

results.

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Kani (1967), pointed out, a strong size effect of RC beams in shear without

transverse reinforcement. A reduction of 40% in relative strength was observed in

the size range of 150 to 1200 mm.

Kani (1967) presented the concept of valley of diagonal shear failure for the RC

beams without web reinforcement. After testing 133 beams to study the effect of

concrete strength, longitudinal steel ratio and shear span to depth ratio a/d, he

came up with the following significant results;

i. The shear strength of RC beams does not depend on the compressive

strength of concrete for the range studied ( 2500< fc < 5000 psi)

ii. The amount of longitudinal steel reinforcement has significant effect on the

relative beam strength i.e. Mu/Mfl, where Mu is the moment corresponding tothe diagonal cracking of the beam and M fl is the flexural moment capacity of

the beams for given longitudinal steel.

iii. The relative beam strength is much more suitable indicator rather than the

ultimate shearc, which depend on the a/d ratio and longitudinal steel ratio.

According to Kani (1967), the web reinforcement is required to increase the Mu to

the level of Mfl, so that diagonal cracking is avoided before flexural failure of RC

beams. Hence the shear design of beams with web reinforcement is an attempt to fill

the gap between Mu and Mfl.

Kani(1967),further elaborated the effect of beam depth on the shear strength of RC

beams and showed with the help of actual tests results that increasing the beam

depth leads to considerable reduction in the relative beam strength.

The shear strength of concrete has inverse relation with the depth of the beam.Shioya et al. (1989), has experimentally showed that the shear strength of 3000 mm

deep beam was merely one third of the shear strength of 600 mm for beams without

shear reinforcement.

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The size effect is marked for beams without transverse reinforcement. The test data

has shown that the size effect plays its role in case of beams without transverse

reinforcement. Collins et al. (1996) have demonstrated that the size effect

disappears when beams without stirrups contain well distributed longitudinal

reinforcement to restrict the propagation of shear diagonal cracks.

2.4.2 Shear span to effective depth a/d or moment to shear ratio and support

conditions

ASCE- ACI Committee 326 (1998) has showed the shear capacity as function of

shear to moment ratio. The basic equation for the shear strength of RC concrete

beams proposed by ACI-318-98, makes the shear span to depth ratio as one of the

basic parameters for calculating the shear capacity of RC section.

When the shear span to depth ratio becomes less than 2.5, the shear capacity of the

RC becomes larger than that of slender beams as the shear is directly transferred to

supports through compression struts. However the supports condition strongly

influences the formation of compression strut. Compressive strut is more likely to

form when beam is loaded from upper face and supports to the bottom face (Adebar

1994).

Kotsovos.M.D ( 1984) studied the effect of web reinforcement for the RC beams

having a/d ratio between 1 and 2.5 with the help of non linear finite element analysis

and observed that placement of web reinforcement in the middle third rather than in

the shear span results in improved ductility and load carrying capacity of RC beams.

In one of the latest studies by Kotsovos and Pavlovic (2004), they used finite

element analysis to study the size effect in beams with smaller shear span to depth

ratio less than 2 and compared the results of theoretical model with the actual

experiment. They concluded that the shear and flexural capacity of beams with

shear span to depth ratio less than 2, is independent of the size of members and the

size effect vanishes for such beams.

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The shear span to depth ratio a/d has accounted for by most of the building and

bridges codes in the world.

2.4.3 Axial force

The axial tensile force tends to decrease the shear strength of concrete members

whereas the axial compression increases the shear capacity. However members

with no shear reinforcement subjected to large axial forces may fail in brittle manner,

not giving sufficient warning. The ACI building Code approach for concrete members

subjected to axial compression has been reported as un-conservative by Gupta and

Collins (1993).

2.4.4 Crushing strength of the beam webSome codes limit the crushing strength of concrete to 0.20 fc in case of vertical

stirrups and 0.25fc in case of 450 stirrups. ACI limits for the cracks control is given

as

v= 8 fc(psi) or v= 0.70 f c(MPa) (2.10)

2.4.5 Yielding of stirrups

The yielding of stirrups is also an important failure mode when the beam is subjectedto flexure and shear stress.

The contribution of longitudinal steel also called dowel action was assumed to be

independent of the shear reinforcement initially, but the later work of Chana (1987)

and Sarsam et al.. (1992), proved that this was an incorrect assumption as the

stirrups keep the longitudinal steel bars in place and prevent shear crack from

opening.

The shear capacity of RC beams is mostly determined on the basis of semi-

empirical or statistically derived equations. The shear capacity of the beams without

shear reinforcement Vc is simply added to the stirrups contribution Vs, which is

determined on the basis of parallel truss model with constant 450 inclinations.

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CFT uses four conditions for the analysis of a section:

1. Equilibrium of the section is considered under external shear force, respective

components of the concrete diagonal compression force, vertical stirrups and

longitudinal steel

2. Strain compatibility of the cracked concrete

3. Stress strain relationship of reinforcement

4. Stress stains relationship of cracked concrete in compression

The shear stress in cracked section due to applied external load causes tensile

stresses fsx in the longitudinal reinforcement and fsy in the transverse reinforcementbesides compressive forces f2 in the cracked concrete, which is inclined at to the

longitudinal axis. Due to these stresses, the longitudinal steel is elongated by x and

transverse reinforcement by y whereas cracked concrete is compressed by 2.

On the basis of experimental results Collins (1978), suggested that the following

relationship for the compressive stress max2f required to fail the concrete in

compression.

'

'

max2/21

6.3

cm

cff

(2.11)

Where 'cf = 28 days cylindrical compressive strength of concrete

m = diameter of the strain circle (1+ 2 ) and

'

c = strain of the concrete at which the cylinder stress reaches maximum value of'

cf

For values of 2f less than max2f , the strain is given as

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due to uni-axial compression of concrete outside the shear span may lead to

collapse of the RC structure. Hence the shear brittle failure can be avoided by

identifying the possible location of such failure on one hand and ensuring the ductile

failure of concrete by increasing its strength to the required level on the other hand.

Kotsovos M.D (1988) while further elaborating his concept of Compressive Force

Path observed that the shear resistance associated with the region along the

compressive forces is transmitted to the supports and not by the beam below the

neutral axis. This leads to substantial increase in the concrete strength due to tri-

axial action. He further advised that the relevant provisions of the building codes

may be revised on the basis of Compressive Force Path, as the existing

procedures are not helpful in avoiding brittle failure of RC structures. This fact has

been verified by Collins et al (2008), in one their latest work on the shear design

procedures. Kotsovos and Bobrowski (1993) later developed a detailed design

method for flexure and shear of RC beams based on the Compressive Force Path

concept. The proposed new design method can be applied to any structural skeleton

according to them. The brittle failure of the structures can be avoided, while

developing the model on the basis of actual behaviour of RC structures, obtained

from experimental studies of such structures. The critical section for flexure and

shear can be identified with the Compressive Force Path Method and the requisite

reinforcement to avoid brittle failure of RC structures can be provided at these critical

sections, while providing nominal reinforcement in the rest of the structure.

Designing structures by this method would certainly bring economy and reliability,

but extensive experimental research will be required for substituting the existing

flexural and shear theory of beams with the Compressive Force Path Method

proposed by them.

2.6.1.1 Modified Compression Filed Theory (MCFT)

Vecchio and Collins (1986)further developed the CFT into Modified Compression

Field Theory (MCFT) that accounts for the influence of tensile stresses on the post

cracking shear behaviour of concrete. The basic theory has been described in

Figure 2.12

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The average principal tensile stress after cracking as suggested by Collins and

Mitchell (1996) is given as

11 5001 cr

ff

(psi) (2.19)

Where'4 ccr ff

The conditions at the crack are also required to be checked for equilibrium as

In X-direction 0)cos()cos)sin( crcrcicrxsxcr AAAf

In y-direction 0)sin()sin)cos( crcrcicrvsxcr AAAf

Where crA is the crack plane, ci is the interface shear stress at the crack

From the above equation we can deduce

cotcot cixsxcrf (2.20)

tantan civsxcrf (2.21)

Form these equations; it is apparent that as ci at a crack increases, the stress in the

longitudinal reinforcement increases but the stress in the transverse reinforcement

decreases. On the basis of work by Walraven(1981) and Bhide and Collins (1986),

the following limitation was imposed the shear stress at the crack by Vecchio and

Collins(1989).

63.0

2430.0

16.2 '

a

w

fcci ( psi and in) (2.22)

16

2430.0

18.0 '

a

w

fcci (MPa, mm) (2.23)

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MCFT is an improvement of CFT, as it can predict the shear strength of those

members without shear reinforcement.

The design procedure for shear design of RC member by MCFT assumes that the

shear stress in the web is equal to the shear force divided by the effective shear

area bwdvand that the shear steel yields at failure under equilibrium. The following

steps are involved;

Vn = Vc+ Vs + Vp (2.29)

Vc = Shear Strength provided by the cracked concrete.

Vs = Shear strength provided tensile stress in stirrups

Vp = Vertical component of applied Pre-stressed tendons.

pyvvwn VfAdbfV )cot(cot1 (2.30)

pyvvwcn VfAdbfV )cot(' (2.31)

= Concrete tensile stress factor indicating the ability of diagonally cracked concrete

to resist shear. dv 0.9 d = the minimum web depth.

The shear stress resisted by the web of beam is function of the longitudinal strain

and decreases with its increase. The highest value of longitudinal strain is

approximated to the strain in the tension chord and is given by

pspss

popsuuvu

x AEAE

fAVNdM

cot5.05.0/< 0.002 (2.32)

Where fp0 = stress in the tendons when the surrounding concrete is at zero stress

and is taken as 1.1 times the effective stress in the pre-stressing steel after all

losses.

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Asp = Area of the pre-stressed longitudinal reinforcement.

As = Area of the non-pre-stressed longitudinal reinforcement.

Nu = Ultimate applied load which is taken as positive when the tensile force is

resulted and negative when compression.

Mu = Ultimate moment at the section.

For RC members containing at least the minimum shear reinforcement, the values of

and can be determined from theFigure 2.13, given on next page.

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Figure 2.13 values of and for RC members with at least minimum shear reinforcement.

(Vecchio and Collins1986).

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Figure 2.14 values of and for RC members with less than minimum shear reinforcement

(Vecchio and Collins1986).

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The following additional considerations and precautions are required while shear

designing of RC members by MCFT.

The first section to be checked for shear is at distance 0.5 dv Cot form the

face of support, which is approximately equal to d.

The required amount of shear reinforcement at other locations can be

checked at 10th points of the span.

To avoid the failure due to yielding of the longitudinal reinforcement, the

following equation must be satisfied

psspys fAfA dvMu +

VpVsVuNu 50.05.0 Cot] (2.34)

Value of in radians.

The reinforcement provided at the supports must be detailed such that the

tension force can be safely resisted which is given as ;

T=

ps

u VVV

50.0

Cot but T 0.50

p

u VV

Cot (2.35)

The longitudinal reinforcement must be extended by a distance d beyond

the point where it is no longer required to resist the flexure.

2.6.1.2 Simplified Compression Filed Theory (SCFT)

The solution of shear strength problem with the MCFT involves determination of two

important parameters and . However by hand solution of such problem is difficult

as it involves a tedious process. Computer software like Response-2000 can be

used to determine the load deformations response of the reinforced concrete

membrane elements.

Bentz et al. (2006),proposed a simplified MCFT for quick and convenient calculation

of the shear strength of RC beams. This method according to authors provided

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Table 2.1 Comparison of experimental results with the full MCFT, simplified MCFT and ACI

equation for shear strength of RC beams.( Bentz et al, 2006)

Beam 'cf

MPa

x

%

yxf

MP

a

xs

mm

'/ cyz ff Axial

load

/xf

'exp/ cf

predicted /exp

Full

MCFT

Simp

MCFT

ACI

ecchio and Collins ( 1982) ; ag =6 mm

PV1 34.5 1.79 483 51 0.235 0 0.23 0.93 0.96 1.37

PV2 23.5 0.18 428 51 0.033 0 0.049 1.47 1.41 0.48

PV3 26.6 0.48 662 51 0.120 0 0.115 0.95 0.96 0.63

PV4 26.6 1.03 242 51 0.096 0 0.109 1.12 1.13 0.68

PV5 28.3 0.74 621 102 0.163 0 0.150 0.91 0.92 0.80

PV6 29.8 1.79 266 51 0.159 0 0.153 0.95 0.95 0.84

PV10 14.5 1.79 276 51 0.190 0 0.27 1.06 1.10 1.05

PV11 15.6 1.79 235 51 0.197 0 0.23 0.98 0.98 0.90PV12 16.0 1.79 469 51 0.075 0 0.196 1.09 1.19 1.24

PV16 21.7 0.74 255 51 0.087 0 0.099 1.12 1.12 0.62

PV18 19.5 1.79 431 51 0.067 0 0.156 1.08 1.08 1.10

PV19 19.0 1.79 458 51 0.112 0 0.21 0.95 1.06 1.10

PV20 19.6 1.79 460 51 0.134 0 0.22 0.93 1.00 1.04

PV21 19.5 1.79 458 51 0.201 0 0.26 0.91 1.03 1.14

PV22 19.6 1.79 458 51 0.327 0 0.31 0.98 1.24 1.38

PV26 21.3 1.79 456 51 0.219 0 0.25 0.88 1.02 1.18

PV27 20.5 1.79 442 51 0.385 0 0.31 0.96 1.24 1.41

PV30 19.1 1.79 437 51 0.249 0 0.27 0.88 1.07 1.18

Bhide and Collins ag = 9mm

PB11 25.9 1.09 433 90 0 0 0.049 1.02 1.03 0.75

PB12 23.1 1.09 433 90 0 0 0.066 1.28 1.30 0.96

PB4 16.4 1.09 423 90 0 1.00 0.071 1.25 1.35 1.40

PB6 17.7 1.09 425 90 0 1.00 0.065 1.28 1.30 1.33

PB7 20.2 1.09 425 90 0 1.90 0.043 0.97 1.05 1.34

PB8 20.4 1.09 425 90 0 3.00 0.039 0.99 1.08 1.74

PB10 24.0 1.09 433 90 0 5.94 0.023 0.92 0.99 2.10

PB13 23.4 1.09 414 90 0 0 0.201* 1.04 1.06 1.06

PB24 20.4 1.10 407 90 0 0 0.236* 1.08 1.10 1.10

PB15 38.4 2.02 485 45 0 0 0.051 1.02 1.16 0.95

PB16 41.7 2.02 502. 45 0 1.96 0.035 0.98 1.13 1.61

PB14 41.1 2.02 489 45 0 3.01 0.037 1.13 1.34 2.32

PB18 25.3 2.20 402 45 0 0 0.067 1.06 1.13 1.02

PB19 20.0 2.20 411 45 0 1.01 0.064 0.98 1.09 1.40

PB20 21.7 2.20 424 45 0 2.04 0.065 1.16 1.33 2.25

PB28 22.7 2.20 424 45 0 1.98 0.067 1.23 1.40 2.32

PB21 21.8 2.20 402 45 0 3.08 0.065 1.26 1.46 3.09

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Table 2.1 ContdBeam '

cf

MPa

x

%

yxf

MPa

xs

mm

'/ cyz ff Axial load

/xf

'exp/ cf

predicted /exp

Full

MCFT

Simp

MCFT

ACI

PB22 17.6 2.20 433 45 0 6.09 0.059 1.13 1.38 4.62

PB25 20.6 2.20 414 45 0 4.05 0.485* 1.10 1.10 1.10

PB29 41.6 2.02 496 45 0 2.02 0.036 1.02 1.15 1.69

PB30 40.04 2.02 496 45 0 2.96 0.037 1.10 1.27 2.29

PB31 43.4 2.02 496 45 0 5.78 0.026 0.97 1.18 3.13

amaguchi et alag =20mm

S21 19.0 4.28 378 150 0.849 0 0.34 0.89 1.37 1.50

S31 30.2 4.28 378 150 0.535 0 0.28 0.80 1.10 1.52

S32 30.8 3.38 381 150 0.481 0 0.28 0.87 1.14 1.58

S33 31.4 2.58 392 150 0.323 0 0.26 0.86 1.04 1.46

S34 34.6 1.91 418 150 0.230 0 0.21 0.91 0.92 1.25

S35 34.6 1.33 370 150 0.142 0 0.163 1.15 1.15 0.97

S41 38.7 4.28 409 150 0.452 0 0.31 0.95 1.23 1.91

S42 38.7 4.28 409 150 0.452 0 0.33 1.02 1.32 2.06

S43 41.0 4.28 409 150 0.427 0 0.29 0.91 1.16 1.86

S44 41.0 4.28 409 150 0.427 0 0.30 0.94 1.19 1.91

S61 60.7 4.28 409 150 0.288 0 0.25 0.90 1.01 1.98

S62 60.7 4.28 409 150 0.288 0 0.26 0.91 1.03 2.01

S81 79.7 4.28 4.9 150 0.220 0 0.20 0.92 0.92 1.82

S82 79.7 4.28 409 150 0.220 0 0.20 0.92 0.93 1.83

Andre ag =9mm, KP ag =20mm

TPI 22.1 2.04 450 45 0.208 0 0.26 0.92 1.02 1.21TPIA 25.6 2.04 450 45 0.179 0 0.22 0.89 0.90 1.14

KPI 25.2 2.04 430 89 0.174 0 0.22 0.89 0.90 1.12

TP2 23.1 2.04 450 45 0.199 3.00 0.114 1.01 1.02 0.72

KP2 24.3 2.04 430 89 0.180 3.00 0.106 1.03 1.06 0.68

TP3 20.8 2.04 450 45 0 3.00 0.061 1.27 1.34 2.75

KP3 21.0 2.04 430 89 0 3.00 0.054 1.15 1.22 2.47

TP4 23.2 2.04 450 45 0.396 0 0.35 1.09 1.39 1.68

TP4A 24.9 2.04 450 45 0.369 0 0.35 1.14 1.41 1.77

KP4 23.0 2.04 430 89 0.381 0 0.30 0.94 1.20 1.44

TP5 20.9 2.04 450 45 0 0 0.093 1.49 1.42 1.28

KP5 20.9 2.04 430 89 0 0 0.063 1.01 0.98 0.87

Krischner and Khalifa ag = 10 mm

SEI 42.5 2.92 492 72 0.110 0 0.159 0.90 0.94 1.04

SE5 25.9 4.50 492 72 0.855 0 0.31 0.89 1.26 1.60

SE6 40.0 2.92 492 72 0.040 0 0.094 0.95 0.99 1.02

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Figure 2.17Shear Friction Hypothesis of Birkeland and Birkeland (1966)

The shear friction was adopted by ACI-318 code in 1973 and the value of has

been reduced as against suggested by Birkeland and Birkeland (1966) . The

Canadian Code has recently introduced modified friction formula.

2.6.4 Strut and Tie Model (STM)

It is essentially an equilibrium model where the designer specifies at least one load

path and ensures that no part of this path has been overstressed. The term truss is

used for Disturbed or D-region and term B is used for Beam or B-region, although

both the terms designate an assemblage of pin jointed, Uni-axially stressed

compression or tension members. In B-region the beam behaviour is expected .i.e.

plane section remains plane and uniform compression field can be found inresponse to shearing load. In design of D-region, complex load paths emulate from

the concentrated load, which converge towards support or flow onwards and hence

arch action is exhibited.

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wwccn fff 65.050.0 )(02.0)(15.0 For normal strength concrtete beams (2.51)

wwccn fff 65.050.0 )(02.0)(12.0 For high strength Concrete beams (2.52)

Chi et al(2007) proposed a unified theoretical model for the shear strength of beams

with and without web reinforcement and observed that the proposed strength model

can address the slender beams in a better way.

Somo and Hong (2006) analysed the modelig error of the shear prediction models

proposed by ACI, CSA, MCFT, Shear firction method and Zusttys equation for data

base of 1146 beams and reported that the Zusttys equation has given the bestmodel amongst the models studied. However for beams with strirrups, MCFT

provides most accurate results.

Tompos and Frosch(2002) studied the effect of various parameters like beam size,

longitudinal steel abd stirrups and reported that the current shear design provisons

of ACI are based on database of the beams sizes, not commonly used sizes in

actual practice. They further reported that for longitudinal steel of 1% or low, the

shear strength of beams has been reduced for all sizes of beam.

Bokhari.I and Ahamd.S (2008) analyzed the data of shear strength of 122 HSRC

beams and reported that the shear provisons are conservative for a/d less than 2.5.

Shear Strenghtening of RC and pre-stressed beams with Carbon Fibre Reinforced

Polymers ( CRFP) has been increasisgnly used in the recent years. Whiteland and

Ibell ( 2005) worked on the fibre reinforced RC beams and gave some guidlines fordeveloping the basic design methods.

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Table 2.2 Contd'

cf

MPa

b

cm

d

cm

da /

ReinforcementExp

uV

kN

ACI EC-2Theory of

Zararis.P

%

'

%

v yvvfMPa

uV

kN

ACuVV /

uV

kN

ECuVV /

uV

kN

theoryuVV /

12.8 15.2 27.2 3.36 1.46 0.34 0.21 0.58 70.0 50.7 1.381 49.0 1.428 68.3 1.025

31.3 15.2 27.2 3.36 1.46 0.34 0.21 0.58 84.5 64.0 1.320 72.6 1.164 82.8 1.020

30.3 15.2 27.2 3.36 1.46 0.34 0.43 1.15 119.8 87.0 1.377 92.7 1.292 112.4 1.066

42.5 15.2 27.2 3.36 1.46 0.34 0.210 0.58 89.9 70.2 1.281 84.2 1.068 86.1 1.044

48.1 15.2 27.2 3.60 4.16 0.37 0.43 1.15 160.0 101.5 1.576 119.0 1.344 149.4 1.071

29.5 15.2 27.2 4.50 1.46 0.34 0.21 0.58 79.6 62.2 1.280 70.7 1.126 84.5 0.942

30.9 15.2 27.2 5.05 4.16 2.61 0.21 0.58 98.6 66.5 1.482 78.4 1.258 96.7 1.020

30.8 15.2 27.2 3.60 4.16 2.61 0.21 0.58 111.9 68.8 1.626 78.2 1.430 92.6 1.208

31.6 15.2 27.2 3.60 4.16 2.61 0.84 2.25 191.9 138.3 1.387 141.3 1.358 190.1 1.009

Swamy and Andriopoulos 29.4 7.6 9.5 3.00 1.97 0.22 0.16 0.44 15.6 10.2 1.522 13.7 1.142 14.7 1.061

29.4 7.6 9.5 3.00 1.97 0.22 0.38 0.79 18.1 12.8 1.417 15.9 1.138 17.9 1.005

29.4 7.6 9.5 3.00 1.97 0.22 0.43 1.09 20.5 14.9 1.372 17.9 1.146 20.6 0.995

28.7 7.6 9.5 4.00 1.97 0.22 0.06 0.17 13.6 8.0 1.695 11.7 1.162 12.2 1.114

28.3 7.6 13.2 3.00 3.95 0.16 0.12 0.31 25.4 13.9 1.828 17.2 1.480 22.2 1.144

25.9 7.6 13.2 3.00 3.95 0.16 0.34 0.61 27.8 16.5 1.682 19.0 1.460 25.4 1.094

25.9 7.6 13.2 3.00 3.95 0.16 0.60 1.33 28.9 23.8 1.216 25.5 1.132 34.1 0.848

28.3 7.6 13.2 4.00 3.95 0.16 0.12 0.31 20.0 13.3 1.500 17.2 1.166 22.3 0.897

25.9 7.6 13.2 4.00 3.95 0.16 0.34 0.61 25.6 16.0 1.600 19.0 1.344 26.3 0.973

28.3 7.6 13.2 5.00 3.95 0.16 0.12 0.31 18.9 13.0 1.454 17.2 1.100 22.4 0.844

Mphonde and Frantz

22.1 15.2 29.8 3.60 3.36 0.31 0.12 0.35 76.3 56.6 1.348 63.0 1.210 82.9 0.92039.9 15.2 29.8 3.60 3.36 0.31 0.12 0.35 93.9 68.4 1.373 86.5 1.085 98.0 0.958

59.8 15.2 29.8 3.60 3.36 0.31 0.12 0.35 97.9 78.6 1.245 109.0 0.898 108.0 0.906

83.0 15.2 29.8 3.60 3.36 0.31 0.12 0.35 111.4 82.9 1.344 132.1 0.843 117.0 0.952

27.9 15.2 29.8 3.60 3.36 0.31 0.26 0.70 95.4 76.7 1.244 85.5 1.116 110.9 0.860

47.1 15.2 29.8 3.60 3.36 0.31 0.26 0.70 120.5 88.2 1.366 109.3 1.102 124.0 0.952

68.6 15.2 29.8 3.60 3.36 0.31 0.26 0.70 151.2 98.5 1.535 132.3 1.143 134.0 1.128

82.0 15.2 29.8 3.60 3.36 0.31 0.26 0.70 115.8 98.7 1.173 145.4 0.796 138.8 0.834

28.7 15.2 29.8 3.60 3.36 0.31 0.38 1.03 138.0 92.7 1.500 99.7 1.394 133.1 1.044

46.6 15.2 29.8 3.60 3.36 0.31 0.38 1.03 133.4 102.9 1.297 121.9 1.095 145.3 0.918

69.6 15.2 29.8 3.60 3.36 0.31 0.38 1.03 161.6 113.9 1.419 146.4 1.103 155.9 1.036

82.8 15.2 29.8 3.60 3.36 0.31 0.38 1.03 150.0 119.3 1.257 159.3 0.941 160.7 0.933

Elzanatly, Nilson, and State 62.8 17.8 26.6 4.0 3.30 0.13 0.17 0.65 149.1 97.5 1.530 132.5 1.125 139.8 1.066

40.0 17.8 26.6 4.0 2.50 0.13 0.17 0.65 111.3 83.7 1.329 105.3 1.057 117.0 0.951

20.7 17.8 26.6 4.0 2.50 0.13 0.17 0.65 78.2 70.3 1.113 77.7 1.006 94.8 0.825

Johnson and Ramirez

36.4 30.4 53.8 3.10 2.49 0.79 0.14 0.69 338.8 293.0 1.156 301.9 1.122 346.2 0.979

36.4 30.4 53.8 3.10 2.49 0.79 0.07 0.35 222.1 237.4 0.935 251.8 0.882 274.2 0.810

72.4 30.4 53.8 3.10 2.49 0.79 0.07 0.35 263.0 295.5 0.890 368.2 0.714 324.3 0.811

72.4 30.4 53.8 3.10 2.49 0.79 0.07 0.35 316.0 295.5 1.069 368.2 0.858 324.3 0.975

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Table 2.2 contd'

cf

MPa

b

cm

d

cm

da /

ReinforcementExp

uV

kN

ACI EC-2Theory of

Zararis.P

%

'

%

v yvvf

MPa

uV

kN

ACuVV /

uV

kN

ECuVV /

uV

kN

theoryuVV /

oon, Cook, and Mitchell

36.0 37.5 65.5 3.28 2.80 0.06 ___ ___ 249.0 271.4 0.917 281.0 0.886 300.0 0.830

36.0 37.5 65.5 3.28 2.80 0.06 0.08 0.35 457.0 357.3 1.279 358.3 1.275 413.5 1.105

36.0 37.5 65.5 3.28 2.80 0.06 0.08 0.35 263.0 357.3 1.016 358.3 1.013 413.5 0.878

36.0 37.5 65.5 3.28 2.80 0.06 0.12 0.50 483.0 394.2 1.225 391.5 1.234 462.1 1.045

6.70 37.5 65.5 3.28 2.80 0.06 ___ ___ 296.0 357.1 0.829 425.4 0.696 362.6 0.816

6.70 37.5 65.5 3.28 2.80 0.06 0.08 0.35 405.0 443.1 0.914 502.7 0.805 476.0 0.851

6.70 37.5 65.5 3.28 2.80 0.06 0.12 0.50 552.0 479.9 1.150 535.9 1.030 524.7 1.052

6.70 37.5 65.5 3.28 2.80 0.06 0.16 0.70 689.0 529.0 1.302 580.1 1.188 589.5 1.168

87.0 37.5 65.5 3.28 2.80 0.06 ___ ___ 327.0 362.0 0.903 506.3 0.646 389.6 0.839

87.0 37.5 65.5 3.28 2.80 0.06 0.08 0.35 483.0 448.0 1.078 583.7 0.827 503.0 0.960

87.0 37.5 65.5 3.28 2.80 0.06 0.14 0.60 598.0 509.4 1.174 638.9 0.936 584.1 1.023

87.0 37.5 65.5 3.28 2.80 0.06 0.23 1.00 721.0 647.6 1.113 727.3 0.991 713.8 1.010

Kong and Rangan

60.4 25.0 29.2 2.50 2.80 0.31 0.16 0.89 228.3 169.6 1.346 212.8 1.073 213.7 1.068

60.4 25.0 29.2 2.50 2.80 0.31 0.16 0.89 208.3 169.6 1.228 212.8 0.979 213.7 0.975

68.9 25.0 29.2 2.50 2.80 0.31 0.16 0.89 253.3 175.8 1.441 226.9 1.116 219.6 1.153

68.9 25.0 29.2 2.50 2.80 0.31 0.16 0.89 219.4 175.8 1.248 226.9 0.967 219.6 0.999

64.0 25.0 29.7 2.49 1.66 0.31 0.10 0.64 209.2 151.0 1.385 194.2 1.077 171.2 1.222

64.0 25.0 29.7 2.49 1.66 0.31 0.10 0.64 178.0 151.0 1.179 194.2 0.916 171.2 1.039

64.0 25.0 29.3 2.49 2.80 0.31 0.10 0.64 228.6 154.1 1.483 230.0 1.126 195.7 1.168

64.0 25.0 29.3 2.49 2.80 0.31 0.10 0.64 174.9 154.1 1.135 203.0 0.861 195.7 0.894

83.0 25.0 34.6 2.40 2.85 0.26 0.16 0.89 243.4 220.5 1.104 286.0 0.851 266.7 0.913

83.0 25.0 29.2 2.50 2.80 0.31 0.16 0.89 258.1 185.2 1.393 249.2 1.035 228.3 1.130

84.9 25.0 29.2 3.01 2.80 0.31 0.16 0.89 241.7 184.1 1.313 252.1 0.958 233.2 1.036

84.9 25.0 29.2 2.74 2.80 0.31 0.16 0.89 259.9 185.2 1.403 252.1 1.031 231.6 1.122

84.9 25.0 29.2 2.50 2.80 0.31 0.16 0.89 243.8 186.5 1.307 252.1 0.967 229.8 1.061

65.4 25.0 29.3 2.73 2.80 0.31 0.10 0.64 178.4 153.9 1.159 205.4 0.868 198.0 0.902

65.4 25.0 29.3 2.73 2.80 0.31 0.10 0.64 214.4 153.9 1.393 205.4 1.044 198.0 1.083

71.0 25.0 29.4 3.30 4.47 1.23 0.10 0.60 217.2 158.7 1.368 212.5 1.022 219.8 0.988

71.0 25.0 29.4 3.30 4.47 1.23 0.13 0.72 205.4 167.5 1.226 220.4 0.932 231.5 0.887

71.0 25.0 29.4 3.30 4.47 1.23 0.16 0.89 246.5 180.0 1.369 231.6 1.064 248.6 0.992

71.0 25.0 29.4 3.30 4.47 1.23 0.20 1.12 273.6 196.9 1.389 246.9 1.108 270.3 1.012

71.0 25.0 29.4 3.30 4.47 1.23 0.22 1.27 304.4 208.0 1.464 256.8 1.185 285.8 1.065

71.0 25.0 29.4 3.30 4.47 1.23 0.26 1.49 310.6 224.1 1.386 271.3 1.145 306.8 1.012

Zararis and Papadakis

24.9 14.0 23.5 3.60 1.37 0.30 ___ ___ 32.3 28.4 1.138 35.1 0.919 34.7 0.931

22.4 14.0 23.5 3.60 1.37 0.30 0.09 0.24 40.2 34.9 1.152 39.8 1.010 45.6 0.882

23.9 14.0 23.5 3.60 1.37 0.30 0.14 0.37 49.7 40.0 1.242 45.1 1.101 52.4 0.949

22.5 14.0 23.5 3.60 1.37 0.30 0.19 0.50 59.2 43.5 1.359 47.6 1.243 57.4 1.040

23.0 14.0 23.5 3.60 1.37 0.30 0.28 0.73 63.5 51.4 1.235 54.9 1.156 68.3 0.930

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Table 2.2 Cotd'

cf

MPa

b

cm

d

cm

da /

ReinforcementExp

uV

kN

ACI EC-2Theory of

Zararis.P

%

'

%

v yvvf

MPa

uV

kN

ACuVV /

uV

kN

ECuVV /

uV

kN

theoryuVV /

22.4 14.0 23.5 3.60 1.37 0.30 0.06 0.16 36.2 32.3 1.120 37.5 0.966 41.5 0.872

23.9 14.0 23.5 3.60 1.37 0.30 0.09 0.23 43.7 35.4 1.233 41.0 1.066 45.7 0.956

20.8 14.0 23.5 3.60 1.37 0.30 0.12 0.31 44.7 36.3 1.230 40.3 1.108 47.6 0.939

21.6 14.0 23.5 3.60 0.68 0.30 0.27 0.73 56.2 49.5 1.135 48.5 1.158 60.3 0.982

21.3 14.0 23.5 3.60 0.68 0.30 0.17 0.46 47.2 40.5 1.166 40.3 1.172 47.9 0.985

Karayiannis and Chalioris

26.0 20.0 26.0 2.77 1.47 0.59 ___ ___ 60.2 55.6 1.083 57.4 1.049 57.9 1.039

26.0 20.0 26.0 2.77 1.47 0.59 0.08 0.21 64.0 66.6 0.961 67.2 0.952 71.0 0.901

26.0 20.0 26.0 2.77 1.47 0.59 0.12 0.32 89.0 72.3 1.231 72.4 1.229 77.7 1.145

26.0 20.0 26.0 2.77 1.47 0.59 0.16 0.43 89.2 78.0 1.143 77.5 1.151 84.3 1.058

26.0 20.0 26.0 2.77 1.47 0.59 0.25 0.64 93.0 88.9 1.046 87.3 1.064 97.4 0.955

26.0 20.0 26.0 3.46 1.96 0.59 ___ ___ 71.6 56.0 1.279 63.7 1.124 62.7 1.141

26.0 20.0 26.0 3.46 1.96 0.59 0.04 0.11 71.2 61.7 1.154 68.8 1.035 70.7 1.007

26.0 20.0 26.0 3.46 1.96 0.59 0.07 0.17 71.2 64.8 1.099 71.6 0.994 74.5 0.953

26.0 20.0 26.0 3.46 1.96 0.59 0.09 0.23 76.7 67.9 1.129 74.5 1.030 78.8 0.973

26.0 20.0 26.0 3.46 1.96 0.59 0.13 0.34 84.8 73.6 1.152 79.6 1.065 86.8 0.977

Collins and Kuchma

71.0 29.5 92.0 2.50 1.03 1.03 0.16 0.80 516.0 602.0 0.857 589.1 0.875 486.0 1.061

75.0 29.5 92.0 2.50 1.36 1.36 0.16 0.80 583.0 616.7 0.945 637.3 0.914 514.7 1.132

74.0 16.9 45.9 2.72 1.03 1.03 0.13 0.65 139.0 158.5 0.877 177.4 0.783 148.0 0.939

74.0 16.9 45.9 2.72 1.16 1.16 0.13 0.65 152.0 159.1 0.955 181.6 0.836 152.8 0.995

Angelakos, Bentz, and Collins

32.0 30.0 92.5 2.92 0.50 0.14 0.08 0.40 263.0 370.2 0.710 305.4 0.861 278.1 0.946

21.0 30.0 92.5 2.92 1.01 0.14 0.08 0.40 282.0 330.7 0.852 277.9 1.014 303.9 0.928

38.0 30.0 92.5 2.92 1.01 0.14 0.08 0.40 277.0 401.0 0.690 364.0 0.761 330.2 0.839

65.0 30.0 92.5 2.92 1.01 0.14 0.08 0.40 452.0 485.3 0.931 477.7 0.946 370.0 1.221

80.0 30.0 92.5 2.92 1.01 0.14 0.08 0.40 395.0 496.2 0.796 533.8 0.740 378.3 1.044

47.0 30.0 92.5 2.92 0.76 0.14 0.08 0.40 342.0 427.4 0.800 385.3 0.887 325.4 1.051

Mean of 174 test beams 1.252 1.092 1.004

CoV ( %) 16.78 18.26 10.23

Rengina and Appleton (1997) studied the behaviour of shear strengthened beams

with jacketing and shotcrete and showed that shotcrete and mortar jackets provide

simple and efficient shear strenghthening techniques.

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Kotsovos.M.D (2007) emphasized the fact that the basic assumptions of the current

design approaches of ACI-318 and EC-02 for flexure and shear are not compatible

with the actual strcutural beavior of RC members. There is a need to revise the

current RC design meethods for shear and flexure on the basis of actual behaviour

of RC beams to make it more compatible.

2.7 Minimum Amount of Shear Reinforcement

The purpose of minimum shear reinforcement is to prevent brittle shear failures and

to provide adequate control of shear cracks at service load levels. Both the

Canadian Standards CSA Standard (CSA A23.3-84), and ACI Code required a

minimum area of shear reinforcement equal to 0.35bw

s/fy,

such that the stirrups areassumed to carry 50 psi minimum shear stress. This value is independent of the

concrete strength. As the concrete compressive and tensile strengths increase, the

cracking shear also increases. This increase in cracking shear requires an increase

in minimum shear reinforcement such that a brittle shear failure does not occur upon

cracking. The 1994 CSA Standard (CSA A23.3-94) makes the minimum amount of

shear reinforcement a function of not only fy, but also fc to account for the higher

cracking shear as the specified concrete strength is increased. Where shear

reinforcement is required, the minimum area of shear reinforcement shall be such

that:

cv fA 06.0y

w

f

sb (2.52)

Figure 2.18 gives comparison of the CSA 1994 and ACI-1999 amounts of minimum

shear reinforcement. The CSA requirements provide a more gradual increase in the

required amount of minimum shear reinforcement as the concrete strength

increases.

Tests carried out by Yoon et al (1996), on large beams with concrete strengths

varying from 36 MPa to 87 MPa indicated that the amount of minimum shear

reinforcement prescribed by the 1994 CSA Standard provides adequate control of

diagonal cracks at service load levels and provide reasonable levels of ductility.

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Figure 2.18 Comparison of CSA and ACI amounts of minimum shear reinforcement

(Yoon et al, 1996)

2.8 Future of research on shear design of RC members.

Shear is one of the most researched properties of RC members in last 6 decades.

Regan (1993), classified research on shear into three broad groups;

i. The first of kind of research relates to shear sensitive areas like shear in fire,

shear connections between members, shear in high strength concrete and

punching shear. This group of research aims at filling the knowledge gap in

the above areas.

ii. The second group relates to understand the behaviour of basic material at

fundamental level. In this group of research, topics like role of aggregateinterlocking in shear , Size effect on shear and other basic concepts of

fracture mechanics related to shear are investigated. This group of research

is related to more basic and fundamental topics in shear strength of RC

members.

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effect, and concrete strength in a reliable way, hence it can considered

a suitable subsitutue of the traditional ACI equation. However the

complexity in application of MCFT for the design of RC members

would need further simplifiction.

vi. An attempt to use the Simplified Modified Compression Field theory

based equations, would reduce the complexity to some extent and it

seems more advisable that the modified MCFT is used instead of

traditional ACI equation, which would ensure ductile failure of RC

structures and at the same time would also satisfy the basic ACI

equation.

To sum up the liteature review on the shear design of normal strength RC beams,we

can infer that research on shear design of RC members will continue to be an area

of interest for many young resereachers to come and the riddle of shear failure will

continue to be the focus of future research.

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Chapter Appendix 2.1

Solved Example with Modified Compression Field Theory.

Case 1. RC beams with shear reinforcement

Applied factored shear force Vu= 200 kN

Web width of the beam bw= 300 mm

Total depth of beam= 450 mm

f'c = 55 MPa

Shear span a = 1800

Mu = 200x1800 kN-mm

Longitudinal steel = 3-700mm2 +2-300mm2

Solution.

03.0554509.0300

000,200''

cvw

u

c

u

fdb

V

f

v

cot1085.11064.1

)30027003(000,200

cot000,2005.0)405/1800000,200(cot5.05.0/

43

ss

uuvux

AE

VNdM

From Figure 2.11, the value of for 05.0'

c

u

f

vand x between 1.5x 10

-3 and 2 x 10-3

=42o which gives x = 1.84 x 10-3 and = 0.15

)270@10(270

0,200/40511.12751404053005515.0)cot('

mmmmmms

ss

dfAdbfV

vyv

vwcn

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84

Taking case 2 from actual beams tested in the experimental program.

Beam Bs1.5,5

Applied factored shear force Vu= 1.6(67.3) +1.2 ( 2.22)= 110.34 kN

Web width of the beam bw= 225 mm

Total depth of beam h = 300 mm

f'c = 52 MPa

Clear span = 2790 mm

Shear span a = 1395 mm

Mu = 110344x1395 kN-mm

Longitudinal steel ratio= = 0.015

Yield stress of longitudinal steel fyl= 414MPa

Yield stress of transverse steel fyv = 275 MPaSolution.

0308.0523009.0225

110344''

cvw

u

c

u

fdb

V

f

v

cot1002.310128.3

3009.022501.0000,200

cot1103445.0)3009.0/139510344(cot5.05.0/

43

5

1

ss

uuvux

AE

VNdM

The value is more than the admissible values of 0.002, hence we may take the

Maximum value of x =0.002. From Figure 2.10, the value of for 05.0' c

u

f

v

=43 which gives

x = 0.002 and

= 0.14

mms

ss

dfAdbfV

vyvvwcn

201

82755/27007.1275652702255214.0)cot('

Provided 7mm @150mm. O.K

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Solution of the problem with program Response-2000

Step 1. Define section properties.

Concrete cylinder strength =52 MPa

Yield strength of the longitudinal steel = 463 MPa

Yield strength of transverse steel = 275 MPa

Pre-stress steel type = None.

Width of the beam section= 225 mm

Height of the section= 300 mm

Top steel =2#10

Bottom steel= 3#20

Stirrups type= Closed loop

Stirrup area per leg = 32 mm2

Step 2. Loads

Shear load = 110.34 kN

Moment= 110344x1395 kN-mm

Step 3. Full member properties

Length subjected to shear; Shear Span = 1395 mm

Constant shear analysis

Supports on bottom

Solution:

The various graphs given by the software are shown on the next page.