evaluation of shear strength of high strength concrete...
TRANSCRIPT
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 the requirements for the PhD Civil Engineering
PhD Supervisor
Prof 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 and
Technology 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 Bernoulli’s 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 values
of 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 beams
tested. 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 the
Universities 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 my
studies.
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 2.1 Cracks appeared when vertical load is applied at the mid span of a beam (Jose,2000)
25
Figure 2.2 Distribution of bending and shear stresses across the section of a beam element and stress state in element A2 and corresponding Mohr’s circle(Jose,2000)
27
Figure 2.3 Types of cracks expected in the reinforced concrete beams (Jose, 2000). 29
Figure 2.4 Forces acting in a beam element within the shear span and internal arches in a RC beam (Russo et al., 2004).
30
Figure 2.5 Shear 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 2.7 Parallel chord truss model. The struts are intercepted by the stirrups at spacing 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 (Bažant et al. 1986). 38
Figure 2.10 Kani’s 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 2.13 Values of β and θ for RC members with at least minimum shear reinforcement.
57
Figure 2.14 Values of β and θ for RC members with less than minimum shear reinforcement (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 2.18 Comparison of CSA and ACI amounts of minimum shear reinforcement (Yoon et al, 1996).
79
Figure 3.1 World Trade Centre (USA) 88
Figure 3.2 The world Highest Tower Burj Dubai,UAE (2651 feet) (162 floors, scheduled construction, 2008)
88
Figure 3.3 Variation of compressive stress-strain curves with increasing compressive strength.( Adapted from Collins and Mitchell, (1997).
108
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Figure No. Description Page
Figure 4.1 Example of B& D-Regions in a Common Building Structure (Schlaich et al
,1987) 122
Figure 4.2 Example 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 6.5
Failure of beams without web reinforcement due to diagonal tension shear failure 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 the increase 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 more brittle and sudden amongst all. The crack causing failure of the beam was not noticed in the beginning and beams failed very suddenly due to tension shear failure.
169
Figure 6.9 Effect of longitudinal Steel ratio on the shear strength of concrete beams without 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 6.11 Effect of shear span to depth ratio on the shear strength of concrete beams without stirrups for same value of longitudinal steel ratio.
173
Figure 6.12 Effect of shear span to depth ratio on the shear strength of concrete beams without stirrups for same value of longitudinal steel ratio.
173
Figure 6.13 Beam shear failure or diagonal tension shear failure in beams with web reinforcement.
174
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Figure No. Description Page
Figure 6.14 Load deflection curves for beams without web reinforcement and ρ=0.0073 178
Figure 6.15 load deflection curves for beams without web reinforcement and ρ=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 9.2 Comparison of actual values of shear stress with the predicted values by proposed regression model and other models for HSC beams without web reinforcement.
221
Figure 9.3 Comparison of actual shear stress of beams having stirrups with the proposed regression 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 2.1 Comparison of experimental results with the full MCFT, simplified MCFT and ACI equation for shear strength of RC beams.( Bentz et al, 2006)
64
Table 2.2 Comparison of the shear strength of RC beams proposed by Zararis , ACI And EC-2 ( Zararis P.D,2003)
73
Table 3.1 Definition of HPC as per SHRP (Zia et al, 1993)
90
Table 3.2 Volume of coarse aggregate per unit of volume of concrete. (ACI-211.1)
98
Table 3.3 Upper limits of specified compressive strength of concrete for HSC and Standard 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 min flexure reinforcement proposed by various codes (Paultre and Mitchell (2003).
105
Table5.1 Summary of Major Code Expressions for the Concrete Contribution to Shear Resistance.
143
Table 5.2 Summary of Research Results conducted at various Universities.
144
Table 5.3 Comparison of test values and Codes values based on shear data base (NCHRP; 2006)
145
Table 6.1 Mix Proportioning/ Designing of High Strength Concrete.
154
Table 6.2 Details of reinforcing bars used in the beams 154
Table 6.3 Reinforcement details of beams.
156
Table 6. 4 Shear span to depth ratio and corresponding span of seven beams in each set of longitudinal reinforcement.
156
Table6.5 Details 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 6.9 Shear Strength and failure angles of 35 HSC beams, without web reinforcement
163
Table 6.10 Shear Failure mode of 35 beams with web reinforcement 163
Table 6.11 Shear Strength ,failure angles and failure modes of 35 HSC beams, with web reinforcement.
164
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Table No. Description Page
Table 6.12 Effect of the longitudinal steel on the shear strength of beams for constant a/d values.
170
Table 6.13 Shear strength, failure mode and failure angles for 35 HSRC beams with web reinforcement.
175
Table 6.14 Increase in the shear strength due to addition of web reinforcement in HSRC beams.
177
Table 7.1 Mix Proportioning/ Designing of High Strength Concrete Double Corbels 182
Table 7.2 Details 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 8.1 Comparison of the shear strength of beams without web reinforcement with the provisions of the ACI 318-08
190
Table 8.2 Comparison of the shear capacity of beams with web reinforcement with the provisions of the ACI 318-08
191
Table 8.3 Comparison of increase in shear strength due to stirrups and ACI-318 provision for stirrups contribution
192
Table 8.4 Comparison of the shear Strength of beams without web reinforcement with the provisions of the Canadian Standards (Simplified Method)
194
Table.8.5 Comparison of the shear Strength of beams with web reinforcement with the provisions of the Canadian Standards (Simplified Method)
195
Table 8.6 Comparison of the shear Strength of beams without web reinforcement with the 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 8.8 Comparison of the shear Strength of beams without web reinforcement with the provisions of EC-02
200
Table 8.9 Comparison of the shear Strength of beams with web reinforcement with the provisions of EC-02
201
Table 8.10 Comparison of the shear Strength of beams without web reinforcement with equation proposed in new theory of Zararis,P.D.
203
Table 8.11 Comparison of the shear Strength of beams with web reinforcement with equation proposed in new theory of Zararis,P.D.
204
Table 8.12 Comparison of Vtest/VCode for ACI, CSA, MCFT, EC-02 and New Equation for beams without web reinforcement.
206
Table 8.13 Comparison of Vtest/VCode for ACI, CSA, MCFT, EC-02 and New Equation for beams with web reinforcement.
207
Table 8.14: Summary of means of the ratios of observed values and different code Values for shear strength of beams without web reinforcement
208
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Table No. Description
Page
Table 8.15
Summary of means of the ratios of observed values and different code Values for shear Strength of beams with web reinforcement.
208
Table 9.1
Comparison of actual and predicated values of shear stress of High Strength concrete beams without web reinforcement for three proposed models.
216
Table 9.2
Comparison of actual and predicted values of shear stress of high strength concrete beams with web reinforcement.
218
Table 9.3
Comparison of proposed model ACI equation and model proposed by G.Russo et al. (2004) for beams without web reinforcement..
220
Table 9.4
Comparison of actual shear stress of beams having no stirrups with the proposed model and other models of ACI, Bazant and Russo
224
Table 9.5
Comparison of test/pred by the proposed model and other models for beams without shear reinforcement ( 35 Nos). ( For constant steel ratio and variable a/d)
228
Table 9.6
Comparison of test/pred by the proposed model and other models for beams with shear reinforcement ( 35 Nos) ( For constant steel ratio and variable a/d)
229
Table 9.7
Comparison of test/pred by the proposed model and other models for beams without shear reinforcement ( 35 Nos). ( For constant a/d and variable steel ratio)
230
Table 9.8
Comparison of test/pred by the proposed model and other models for beams with shear reinforcement ( 35 Nos) ( For constant a/d and variable steel ratio)
231
Table A-1
Forces in Truss of double corbel after analysis.
251
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Table of Contents.
Chapter Description Page
1. Introduction 19
1.1 Problem Statement 20
1.2 Aim and Objectives of Research 20
1.3 Scope of the research study 20
1.4 Methodology/Programme 21
1.5 Layout of the thesis 22
Literature Review.
2. Shear strength of RC beams. 25
2.1 Introduction to shear strength of beams 25
2.2 Mode of Failure of concrete beams in shear. 28
2.3 Shear strength of Normal Strength Reinforced Concrete (NSRC) beams.
23
2.4 Factors affecting shear strength of RC beams 37
2.5 Historical development of shear design of reinforced concrete beams
43
2.6 Recent approaches in the Shear Design of reinforced Concrete beams.
47
2.7 Minimum Amount of Shear Reinforcement. 78
2.8 Future of research on shear design of RC members. 79
Chapter Appendix 2.1 Solved Example with MCFT 83
3 Shear strength of high Performance reinforced concrete beams 88
3.1 High Performance Concrete. ( HPC) 88
3.2 High Strength Concrete. 91
3.3 Codes Provisions for High Strength Concrete. 103
3.4 Mechanical properties of high strength concrete 106
3.5 Stress strain behaviour and shear strength of HSC 107
Summary 119
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Chapter Description Page
4 Shear 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 Structural components.
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
5
Provisions of international building codes for the shear design of Normal & High Strength Concrete.
136
5.1
British Standards (BS-8110) 136
5.2
European Code EC2-2003. 137
5.3
ACI Code 318-06 (American Concrete Institute) 138
5.4 Canadian Standards for design of Concrete structures. CSA A- 23.3-94.
140
5.5 AASHTO LRFD (Load Reduction Factor Design) Bridge Design
Specifications -1996. 141
5.6 Empirical methods for beams without shear reinforcement. 142
5.7 Results of High Strength concrete beams at different Universities, in near past.
143
5.8 Evaluation of shear design methods of different building codes based on test data base by National Cooperative Highway Program ( NCHRP).
145
5.9 Variations in the provisions of international building code for shear 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) in
concrete and observations. 180
7.1 Experimental Programme for testing of disturbed region in
concrete (D-region). 185
7.2 Design 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. (CSA A23.3-94).
193
8.3 AASHTO’s LRFD DESIGN SPECIFICATION ( 1994). (Modified Compression Field theory-MCFT).
196
8.4 Comparison of observed values with the provisions of
Eurocode-02 200
8.5 New Theory Proposed by Prodromos D.Zararis (2003) 203
18
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 web reinforcement.
217
9.4 Comparison of the proposed models with ACI-318 Code and 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
19
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 increase
further 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 Codes
with 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
20
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 develop
some 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.
21
- 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 design
specification 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.
22
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 the
testing.
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
23
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 Zealand
code (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.
24
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 for
shear 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.
25
Chapter No. 2
Shear strength of reinforced concrete beams
Chapter Introduction: This chapter explains the basic concepts of the shear strength of concrete beams and their application to the shear strength of Reinforced Concrete (RC) beams. The factors affecting the shear strength of RC beams are also discussed in quite details. The historical perspective and recent approaches in the area of shear strength of concrete have been elucidated and some modern research findings have been discussed in more details at the end of the chapter.
2.1 Introduction to shear strength of RC beams
The shear stress acts parallel or tangential to the section of a material. When a
simple beam is subjected to bending, the fibers above the neutral axis are in
compression and the fibers located below the neutral axis are in tension. A
concrete beam with longitudinal steel when subjected to external loads will
develop diagonal tensile stresses which will tend to produce cracks. These
cracks are vertical at the centre of the span and will become inclined as they
reach the support of the beam as shown in Figure 2.1. The stress that causes the
inclined cracks in the beam is called diagonal tension stresse (Jose M.A, 2002).
Figure 2.1 Cracks appeared when vertical load is applied at the mid span of a beam
(Jose.M.A, 2000)
26
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 2.2
are shown with the tensile normal stress ft and shear stress ν across the plane a1-
a1 and a2-a2 at distance y from the neutral axis.
The internal stresses acting on elements A1 & A2 are also shown in Figure 2.2.
Using Mohr’s 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)
27
Figure 2.2: Distribution of bending and shear stresses across the section of a beam
element and stress state in element A2 and corresponding Mohr’s circle(Jose,2000)
28
2.2 Mode of failure of concrete beams in shear.
Various failure modes in RC beams are shown in Figure 2.3(a). In the region of
flexural failure, cracks are mainly vertical in the middle third of the beam span
and perpendicular to the lines of principal stress. These cracks result from a very
small shear stress v and a dominant flexural stress f which results in an almost
horizontal principal stress ft(max).
Diagonal tension failure happens, if the strength of the beam in diagonal tension
is lower than its strength in flexure. The shear span-to-depth ratio is of
intermediate magnitude for diagonal failure, varying between 2.5 and 5.5 for the
case of concentrated loading. Such beams can be considered of intermediate
slenderness. Cracking starts with the development of a few fine vertical flexural
cracks at mid span, followed by the destruction of the bond between the
reinforcing steel and the surrounding concrete at the support. Thereafter, without
ample warning of impending failure, two or three diagonal cracks develop at
about 1½d to 2d distance from the face of the support in the case of reinforced
concrete beams, and usually at about a quarter of the span in the case of pre-
stressed concrete beams. As they stabilize, one of the diagonal cracks widens
into a principal diagonal tension crack and extends to the top compression fibers
of the beam, as seen in Figure 2.3 (b) (Jose,2000).
In beams having shear span to depth ratio less than 2.5, a few fine flexural
cracks start to develop at mid span and stop propagating as destruction of the
bond occurs between the longitudinal bars and the surrounding concrete at the
support region. Thereafter, an inclined crack steeper than in the diagonal tension
case suddenly develops and proceeds to propagate toward the neutral axis. The
rate of its progress is reduced with the crushing of the concrete in the top
compression fibers and a redistribution of stresses within the top region occurs.
Sudden failure takes place as the principal inclined crack dynamically joins the
crushed concrete zone, as illustrated in Figure 2.3(c). This type of failure can be
29
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).
30
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 forces acting 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 the crack 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 tensile
force 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).
31
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 by
tooth 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.
32
Figure 2.5 Shear in beam with no transverse reinforcement.
(Stratford and Burgoyne, 2003)
In one of the earliest research on shear failure, at University of Toronto Canada,
Kani (1964) defined the regions of beam action and arch actions for resisting the
shear in RC beams, for the first time. It was pointed out by him that initially the
shear is resisted by the teeth of cracked concrete, but after destruction of the
resistance by teeth of the cracked beam, a quite different mechanism through
tied arches in the compression zone occurs. On the basis of actual test results,
Kani (1964), reported that in the region of low values of shear span to depth ratio
(a/d), the shear capacity of the structure is determined by the strength of
remaining arch, whereas in the region with medium value of a/d, the capacity of
teeth of cracked concrete determines the shear capacity of the beams. He also
proposed an expression for the boundary point separating the two regions. In
Figure 2.6, the boundary for shear failure of the beams tested in Toronto has
been given, which shows that up to a/d of 2.5, shear failure due to arch action is
dominant whereas in the region with a/d more than 2.5 and up to 5.75 or 6 beam
action due to concrete teeth ( beam action) is dominant and the shear capacity
due to arch action is very small.
33
Figure 2.6 Comparison of theoretical and test results of shear failure of beams (Kani.1964)
The joint committee ASCE-ACI-426 in 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 resistance
provided 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 such members.
34
iii. Interface shear transfer
The contribution of interface shear transfer to shear strength is a function of the
crack width and aggregate size. Thus, the magnitude decreases as the crack
width increases and as the aggregate size decreases. Consequently, this
component is also called “aggregate interlock” denoted by Va. However, it is now
considered more appropriate to use the terminology “interface shear transfer” or
"friction".
iv. Dowel action
When a crack forms across longitudinal bars, the dowelling action Vd , of the
longitudinal bars provides a resisting shear force, which depends on the amount
of concrete cover beneath the longitudinal bars and the degree to which vertical
displacements of those bars at the inclined crack are restrained by transverse
reinforcement.
v. Shear reinforcement
This forms the main part of the shear capacity of the beams with web
reinforcement and is typically modeled with 45 degree truss model.
The ASCE-ACI Committee 426 has reported the following equation for the concrete
shear strength incorporating the longitudinal reinforcement.
Vc = 10080.0
12
cf ≤ 0.192
12
cf [MPa] (2.6)
For beams with transverse reinforcement, the basic model to explain the mechanism
for carrying the shear was proposed by Ritter (1899). The load was assumed to flow
down the concrete diagonal struts and then lifted to the compression chord by
transverse tension ties on its way to support as shown in Figure 2.7 below.
35
Figure 2.7 Parallel chord truss model. The struts are intercepted by the stirrups at spacing of
d (Ritter, 1989)
Traditionally one truss with stirrups at the longitudinal spacing “d” was assumed but
in fact Ritter showed that there was continuous diagonal compression carried up and
over cracks by a band of stirrups as shown in Fig 2.6
For 45 degree truss model, the capacity provided by the shear reinforcement is
equal to the capacity of an individual stirrup multiplied by the number of stirrups over
the length d, which is approximately equal to “d/s”. The shear carried by the stirrups
is given as;
Vs =
s
dfA yv (2.7)
The shear strength of RC beams with transverse reinforcement is traditionally
determined by summing the individual contribution of concrete and steel as shown in
Figure 2.7 (ACI-ASCE,1998) .i.e.
Vn = Vc+Vs (2.8)
36
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.
37
2.4 Factors affecting shear strength of concrete beams
One of the major reasons for limited understanding of the shear behaviour and
diagonal failure of the RC beams is greater number of parameters involved in the
problem. Kani (1967), identified the following parameters affecting the diagonal
cracking of RC beams.
i. Grade of steel (tensile strength of longitudinal steel).
ii. Compressive strength of Concrete.
iii. Cross section and shape of beams ( web width, depth etc.)
iv. Shear arm or shear span.
v. Types, arrangements, quantity and location of web reinforcement
vi. Types of loadings.
vii. Types of beams supports (Simply supported or continuous).
viii. Pre-stress forces and its point of application etc.
In addition to the basic five echanisms for shear transfer discussed in section 2.3,
the effect of the other significant parameters on the shear strength of RC beams is
explained as follows;
2.4.1 Depth of member or size effect
Size effect refers to the fact that shear strength is not constant for a given
compressive strength of concrete but varies with the size of the beam, both its depth
and length. The phenomena become more obvious for lightly reinforced RC beams.
Current design methods for shear in RC members are based almost entirely on the
results of tests specimen having maximum size of 300 mm. Hence the assumption
of constant shear strength by most of the equations like ACI318-08, simple
equation, is contradicting the actual behaviour of the RC beams.
Dimensional analysis shows that the structural size effect for geometrically similar
specimens or structures is governed by the simple relation given by Bazant et al
(1984).
38
Bazant et al (1984), used the following reduction factor to account for the size effect.
o
tN
d
d
Bf
1
'
(2.9)
N = P/bd = nominal stress at failure
P = maximum load (that is, failure load);
b = thickness;
d = characteristic dimension of the specimen or structure;
'tf = direct tensile strength; and
B, do = empirical constants, do being a certain multiple of the maximum size of in
homogeneities in the material.
The size-effect law has been used by Bažant and Sun (1987); Bažant and Sener
(1988); and Bažant, Sener, and Pratt (1988) to predict the size effects for shear,
torsion, and bond pullout testing of concrete. The law has been shown in Figure 2.9
Figure 2.9 Size-effect law (Bažant et al. 1986).
39
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 to
the diagonal cracking of the beam and Mfl 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 shear νc, 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.
40
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.
41
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 web
Some 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 fc′ (MPa) (2.10)
2.4.5 Yielding of stirrups
The yielding of stirrups is also an important failure mode when the beam is subjected
to 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.
42
The Artificial Neural Network study of the slender beams by Chabib et al.(2006), has
shown that the assumption of superimposing the individual theoretical contribution of
concrete and steel in resisting the shear, as practiced by most of the building codes
is not justified. It was reported that the effect of shear reinforcement was more, at
lower shear reinforcement ratio than the relatively higher ratio. They further
explained that the shear strength of beams with moderate shear reinforcement is 75-
80% higher than the corresponding values calculated by ACI equation.
2.4.6 Failure of tension chord.
The tension in the longitudinal reinforcement is function of moment at distance “d”
effective depth from the nearest location of maximum moment. The ACI-318
therefore requires that the longitudinal reinforcement must be extended at least
distance “d” from the point where it is no more required.
2.4.7 Failure of stirrups anchorage
At the ultimate loads, the stress in the stirrups approaches the yield strength at every
point, where the inclined crack is intercepted by a stirrup. The upper end of crack
may be close to the compression zone hence the part of the stirrup above the crack
may fail due to slippage or failure of anchorage and hence the stirrups must be
closed looped or anchored by hook or T-heads.
2.4.8 Serviceability failure due to excessive crack width at service load
The ACI limits the maximum shear to 8√fc′ (psi) or 0.70√fc′ (MPa) to control the
crack under the Service loads.
2.4.9 Loading conditions
The shear strength of RC beams also depends on the condition of the loads applied.
M.D Brown et al. (2004) reported on the basis of 1200 tests data of beams that the
shear strength of beams subjected to uniformly distributed loads is more than the
beams subjected to concentrated loads. The current codes provisions are safe for
such types of loading. However they have reported that the provisions of ACI318-05
43
are not safe for the beams subjected to concentrated loads between 2d and 6d from
the face of support, which are usually called slender beams.
2.5 Historical development of shear design of reinforced concrete beams
The American Concrete Institute (ACI) Standard Specification No.23 (1920) allowed
the shear stress of 0.025 fc′ subject to a maximum of 0.41 MPa (60 psi) for the
members without shear reinforcement. However the value of shear stress was
increased to 0.03 fc′ where the mechanical anchorage with 1800 hook was provided
for the longitudinal reinforcement.
The distinction between members with and without mechanical anchorage was
removed in 1951 when the shear stress of members without shear reinforced was
revised as 0.03 fc′ and for members with shear reinforcement as 0.12 fc′. The ACI
318-51 was based on 450 truss analogy .i.e. the beam was idealized as parallel
chord with compression diagonal inclined at 450 to the longitudinal reinforcement.
For RC members without web reinforcement, various models for shear transfer were
considered since early 1960’s, which are broadly classified into the following three
major groups; ( ASCE-ACI, 1998)
i. Mechanical or Physical models for structural behaviour and failure.
ii. Fracture mechanics approaches
iii. Non Linear Finite elements analysis
In these models, “Kani tooth model (1964)” has a pioneering role in explaining the
shear flexure cracking of RC beams. With the help of his proposed Kani’s Tooth
Model (1964), he tried to explain the development of secondary diagonal cracks due
to bending of concrete teeth between two adjacent flexural cracks. The concrete
between two flexural cracks resemble the teeth of a comb as shown in Figure 2.10.
These concrete teeth act as cantilevers attached to the compression zone and
loaded horizontally by the shear from the bonded longitudinal reinforcement. The
shear failure of RC beams takes place when one of these cantilevers fails in flexure.
This is sometimes known as famous “Kani’s hypothesis of shear failure”.
44
Figure 2.10 Kani’s Tooth Model (Kani, 1964)
Fenwick and Paulay (1968) and Taylor (1974), further evaluated the Kani’s tooth
model. They explained that the teeth of cracked concrete restrict the freely bending
of beams due to resistance from crack friction and dowel action of longitudinal steel.
Many researchers later worked on the tooth model to study the flexure shear
cracking mechanism for RC slender beams without web reinforcement (Macgroger
and Walter, 1967; Hamadi and Regan, 1980; Chana, 1987).
The fracture mechanics approaches are based on the fact that when the diagonal
cracks develop in RC beams, there is a peak tensile stress at the tip of the crack,
which reduces along the crack, also called as softening of cracked zone. In case
where the failure of the RC beams is caused by single critical diagonal crack, the
application of fracture mechanics can develop more reliable results than the
empirical methods. The application of fracture mechanics involves numerical
modeling of the complex tensile stress crack displacement relationship and empirical
relationships are developed in terms of fracture mechanics parameters, having little
explanation of the structural behaviour (ACI-ASCE, 1998)
In the non linear finite element analysis, simple Strut and Tie Model is widely used
for the members like deep beams and other non prismatic members. Due to specific
geometry of the structures, significant re-distribution of the stresses after cracking
requires that sufficient steel is provided in all the direction for the ductile failure of the
RC structure. Application of Strut and Tie Model for more slender beams without
transverse reinforcement may lead to unsafe solution. In such cases the diagonal
crushing strength of concrete is required to be reduced (Collins and Mitchell, 1986).
45
Kotsovos (1986) analyzed the shear behaviour of RC beams with web reinforcement
and having shear span to depth ratio greater than 2.5 under two point loads. He
compared the test results with the Finite Element Analysis (FEA) results of the same
beams and made the following significant conclusions:
i. The predicted behaviour of the beams by FEA is incompatible with
the actual shear behaviour at critical section of RC beams with
various arrangements of stirrups
ii. Shear behaviour is associated with the development of tensile stress
within the compression zone and particularly in the region of
compression zone between sections at load and sections twice the
beam depth.
iii. The stirrups resist the tensile stresses within the compression zone
rather than transforming the beam into truss as widely considered.
This negates the concept of truss model most commonly used for the
design of beams with web reinforcement.
iv. The tensile forces develop, when the destruction of bond between
steel and concrete takes place when the bond stresses are critical.
After the bond failure, the transfer of stresses from concrete to steel
is prevented. This is significant observation, which negates the
famous Kani’s hypothesis as due to failure of bond between steel and
concrete teeth; the cantilever action ceases to act.
The findings of Kotsovos (1986), led to a new era of research in the shear analysis
of RC beams, where the tensile forces developed in the compression zone after
formation of the diagonal cracks became the major focus of subsequent researches.
The famous Modified Compression Field (MCFT) theory of Vecchio and Collins
(1986) has been based on the above findings of Kotsovos (1986).
46
The latest approaches are however based on the varying truss angle within certain
limits suggested on the basis of theory of plasticity, referred to as “Standard Truss
Model”, with no concrete contribution. Here the shear strength of concrete is
assumed to be mainly due to aggregate interlocking, and dowel action of concrete.
The modification of varying truss angle and concrete contribution was used in the
“Modified Truss Model”
Schaliac et al. (1987), introduced the concepts of D (disturbed) and B (Beam)
regions. The distribution of strain is non-linear in the D- region and linear in the B-
region. Mitchell and Collins (1974) abandoned the concept of linear elasticity and
introduced the concept of “Compression Field Theory” (CFT), for members
subjected to shear and flexure.
Vecchio and Collins (1986), presented the “Modified Compression theory” (MCFT)
which provided a more realistic assessment for wide range of shear reinforcement
and also for the cases with no shear reinforcement.
At the same time the general theory of shear was also developed on the basis of
constitutive laws of friction by determining the strain and deformation in the web.
According to this method, the discrete formation of cracks, the crack spacing, the
width of crack must be determined and equilibrium checked along the crack to
evaluate the crack slip mechanism.
Bentz et al.(2006), introduced the concept of simplified compression field theory for
the shear design of concrete beams. The method provides a simplified version of
MCFT, where the calculation of full load deformation analysis is not needed. The
details of these methods are given in the subsequent sections.
47
2.6 Recent approaches in the shear design of reinforced concrete beams
More rational approaches for the shear design of concrete have been evolved in the
last 25 years. The more recent approaches in the shear design of concrete beams
are
i. Compression Field Theory
ii. Truss approaches with concrete contributions
iii. Shear friction theory
iv. Strut and Tie Model.
v. Some latest research work on the shear design of reinforced concrete
beams.
2.6.1 Compression field approaches
In this approach the tensile stresses along the cracked concrete is also taken into
consideration, which was neglected in the earlier approaches. The shear stress
applied to the cracked concrete section causes tensile stresses fsx in the longitudinal
reinforcement, fsy in reinforcement, and compressive stress f2 in the cracked
concrete inclined at angle θ to the longitudinal reinforcement. The value of θ is
determined by considering the deformation of the transverse reinforcement, the
longitudinal reinforcement and diagonally stressed concrete.
The truss models with diagonals were attended by Kupfer (1964) and Baumann
(1972). They presented the approaches for determining the angle θ, assuming that
the cracked concrete and reinforcement were linearly elastic. Methods for
determining the value of θ for full range of loads was developed by Collins and
Mitchell (1974) on the basis of Wagner procedure and the approach was called
“Compression Field theory (CFT)”
The basic assumptions of the CFT are shown in the Figure 2.11, which idealizes
cracked concrete as material with coinciding principal stresses and strain axes,
which are free to adopt their direction as per applied loads.
48
Figure 2.11 Compression Field Theories (Mitchell and Collins,1974)
49
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 reinforcement
besides compressive forces f2 in the cracked concrete, which is inclined at θ to the
longitudinal axis. Due to these stresses, the longitudinal steel is elongated by εx and
transverse reinforcement by εy whereas cracked concrete is compressed by ε2.
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 (ε1+ ε2 ) 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
50
''2
2ccf
f
(2.12)
Hence it was shown that the diagonally compressed concrete fails at relatively lower
compressive stress as the stress is transmitted through relatively wide crack.
Typically the diagonal cracks are wider than the flexural cracks. When the
longitudinal steel and transverse steel is same in magnitude, then θ is equal to 45
degrees. However practically, the transverse steel is often less than longitudinal
steel and the θ is less than 45o, then significant shear stresses are transmitted
across the cracks. The magnitude of the shear transferred across the crack depends
on the crack width, which is further related to the tensile straining of concrete. The
principal tensile strain is given by the following equation.
221 cot)( xx (2.13)
For shear stress less than the value causing the first yielding of the reinforcement, a
simple expression is given for as follows;
)1
1/()1
1(tan 4
vx nn (2.14)
Where x is longitudinal steel ratio and v transverse steel ratio.
n = modular ration=Es/Ec and '
'
c
cc
fE
(2.15)
Kotsovos M.D (1983), presented the concept of “Compressive Force Path” and
explained that there is no single cause of diagonal shear failure in RC members
Various mechanisms for the shear failure are dependent on the effect of shear force
on the compressive force path. Hence the earlier theories based on unique
mechanism of truss models may not lead to realistic design as the actual behavior of
the structures is not considered. He further argued that even if such procedures are
effective in preventing the diagonal failure in the shear span, localized brittle failure
51
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
52
Figure 2.12 Description of Modified compression Field Theory (Vecchio and Collins1986)
The equilibrium conditions when applied to the Figure 2.12, we get
1cot ff sxx (2.16)
1tan ff syv (2.17)
12 )cot(tan ff (2.18)
53
The average principal tensile stress after cracking as suggested by Collins and
Mitchell (1996) is given as
1
15001
crff (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
f cci ( psi and in) (2.22)
16
2430.0
18.0 '
a
w
f cci (MPa, mm) (2.23)
54
a; stands for aggregates width.
w ; Width of crack
The crack width can be obtained as average crack width
msw 1
mymxm ss
s
cossin
/1 (2.24)
)(25.010
2 1x
bxxxmx
dk
scs
(2.25)
)(25.010
2 1v
byyymy
dk
scs
(2.26)
xc , yc distance between midsection and longitudinal and transverse reinforcement,
respectively, xs ys spacing of longitudinal and transverse reinforcement,
respectively
1k : Coefficient for bond characteristics of bars (0.4 for deformed bars, 0.8 for plain bars)
yx bb , Bar diameter of longitudinal and transverse reinforcement, respectively.
Under high loads, the average strain in the stirrups exceeds the yield strain and we
get the following equation
tan
63.0
2430.0
16.2tan
'
1
a
wf
f cci (psi and in ) (2.27)
tan
16
2430.0
18.0tan
'
1
a
wf
f cci (MPa, mm) (2.28)
This equation limits the principal tensile stress in cracked concrete, so that possible
failure of the aggregates interlock mechanism is taken into account in the MCFT,
55
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 bwdv and 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)
β = 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
popsuuvux 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.
56
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
β and θ can be determined from the Figure 2.13, given on next page.
57
Figure 2.13 values of β and θ for RC members with at least minimum shear reinforcement.
(Vecchio and Collins1986).
58
For the design of RC members without shear reinforcement or shear steel less than
the minimum shear, the diagonal cracks are widely spaced as compared to beams
with shear steel due to reduced inclination of θ. For the conditions, when the value of
θ becomes 90o, the spacing is denoted by Sx. The maximum value of Sx is 2000 mm
and the maximum aggregates sizes a are taken as 19mm. For RC members with no
or less than minimum shear steel, values of θ and β for depends on the longitudinal
strain parameter, which in turn depends on the distance between of the longitudinal
steel in the vertical axis.
For RC beams having aggregates sizes other than 19 mm and less than minimum
shear steel, the equivalent spacing Sex can be determined as
Sex= Sx 16
35
a (2.33)
The values of θ and β for members having less than the minimum shear steel, is
determined from Figure 2.14
59
Figure 2.14 values of β and θ for RC members with less than minimum shear reinforcement
(Vecchio and Collins1986).
60
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 θ 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 ≥
dv
Mu
+
VpVs
VuNu50.05.0
Cot θ ] (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 θ but T ≥ 0.50
p
u VV
Cot θ (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 β and θ. 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
61
excellent prediction of shear strength of RC concrete beams with only 13%
coefficient of variation.
According to the basic assumptions of MCFT, at yield point, the strain in transverse
reinforcement has to be more than 0.002 and the strain in the concrete along the
crack has also to be about 0.002. Hence it was shown by Bentz et al (2006) that the
maximum shear stress will be 0.28f’c, whereas for very low value of εx , it was
deduced that the failure shear stress is 0.32f’c . However as conservative estimate a
value of 0.25f’c was selected for estimating the shear stress of RC beams before
yielding of the transverse reinforcements. Based on these assumptions, the value of
β proposed was given as;
15001
cot33.0
(2.36)
The value of β must also satisfy the equation given as
)16(4231.0
18.0
gaw (2.37)
The crack width w is determined by the crack spacing s and principal tensile
strain 1 as shown in Figure 2.15 and ga represents the maximum coarse aggregate
size.
Figure 2.15 Transmission of forces across the crack.( Bentz et al, 2006)
62
For elements with no transverse reinforcement, Eq. 2.37 can be expressed as
sin/686.031.0
18.0
1xes (2.38)
For high strength concrete particularly where 'cf > 70 MPa (10,000 psi), the concrete
matrix is strong enough and the shear cracks break through the aggregates and
ga is taken as zero. For maximum post cracking shear capacity of members without
transverse reinforcement will occur when Eq. 2.37 and Eq. 2.38 will give the same
values. This requirement would lead to the following equation for
1
1
5001
sin/258.10568tan
xes
(2.39)
The longitudinal strain x is related to the principal strain 1 as
1 = 5001(15000
cot)cot1(
1
42
x (2. 40)
The value of and concrete shear strength for members with no transverse
reinforcement decreases, with the increase in the crack spacing xes . That is why that
long RC beams, with no transverse reinforcement fail at lower shear stress, than
smaller beams, as the crack spacing increase in the large beams. This is also
referred to as size effect in shear.
Thus for members without transverse reinforcement, the value of depends on
the longitudinal strain x and crack spacing parameter xes . Bentz et al, called these
two effects as “strain effect factor” and “Size effect factor” respectively.
They further proposed, a simple and conservative equation for , which combines
x and xes as follows
63
exx s
1000
1300.
15001
4.0
(2.41)
The simplified MCFT uses the following equation for determination of
deg752500
88.0)(7000deg29( xex
sree (2.42)
Hence the simplified MCFT, determines the value of shear strength of RC members
when several iterations are made to reach at the converged values of and
Bentz et al (2006) used the experimental results of pure shear tests of 112 beams
data to compare the experimental shear values and shear values determined by full
MCFT and simplified MCFT as well as ACI-318. The comparison has been given in
Table 2.1
The coefficient of variation (CoV) for full MCFT has been worked out as 12.2% and
for simplified MCFT as 13%. For ACI, equation the CoV is 46.7%. The comparison
shows that the simplified MCFT gives results very close to the full MCFT, hence
simplified MCFT provides a relatively quicker method for the design of RC member
failing in shear. Table 2.1 shows summary of the comparison of experimental results
with the full MCFT, simplified MCFT and ACI equation for shear strength of RC
beams. Simplified MCFT, when compared with the detailed MCFT results have
given almost similar results.
64
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
Vecchio 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 0.90
PV12 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
65
Table 2.1 Cont’d Beam '
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
Yamaguchi et al ag =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 1.21
TPIA 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
66
Table 2.1 Cont’d Beam '
cf
MPa
x
%
yxf
MPa
xs
mm
'/ cyz ff
Axial load
/xf
'exp/ cf
predicted /exp
Full
MCFT
Simp
MCFT
ACI
Porsaz and Beidermann; ag = 10 mm
SE11 70.8 2.93 478 34 0.063 0 0.093 0.83 0.90 0.91
SE12 75.9 2.94 450 72 0.060 0 0.098 0.96 1.01 0.99
SE13 80.5 6.39 509 54 0.115 0 0.149 0.82 0.86 1.34
SE14 60.4 4.48 509 72 0.378 0 0.30 1.03 1.19 2.32
Marti and Mayboom; ag = 13 mm
PPI 27 1.95 480 108 0.116 0 0.183 0.98 1.02 1.02
PP2 28.1 1.59 563 108 0.111 -0.38 0.196 1.06 1.08 0.95
PP3 27.7 1.24 684 108 0.113 -0.80 0.199 1.03 1.02 0.86
Vecchio et al ; ag = 10 mm
PA1 49.9 1.65 606 45 0.086 0 0.126 0.94 1.03 0.95
PA2 43 1.66 606 45 0.100 0 0.145 0.94 1.02 0.96
PHS1 72.2 3.25 606 44 0 0 0.037 1.07 1.08 0.97
PHS2 66.1 3.25 606 44 0.033 0 0.093 1.13 1.25 1.27
PHS3 58.4 3.25 606 44 0.074 0 0.140 0.99 1.13 1.20
PHS8 55.9 3.25 606 44 0.115 0 0.193 1.02 1.15 1.45
PC1 25.1 1.65 500 50 0.163 0 0.197 0.84 0.87 0.99
Pang and Hsu ag = 19 mm
A2 41.3 1.19 463 189 0.134 0 0.136 1.01 1.01 0.87
A3 41.6 1.79 447 189 0.192 0 0.190 0.98 0.99 1.23
A4 42.5 2.98 470 189 0.330 0 0.28 0.97 1.11 1.82
B1 45.2 1.19 463 189 0.056 0 0.092 1.01 1.08 0.87
B2 44.1 1.79 447 189 0.126 0 0.146 0.96 0.96 0.97
B3 44.9 1.79 447 189 0.057 0 0.102 0.94 1.05 0.96
B4 44.8 2.99 470 189 0.057 0 0.119 0.92 1.10 1.12
B5 44.8 2.98 470 189 0.129 0 0.177 0.89 0.96 1.16
B6 42.8 2.98 470 189 0.194 0 0.23 0.95 0.96 1.53
Zhang and Hsu ag = 19 mm
VA1 95.1 1.19 445 94 0.056 0 0.068 1.04 1.20 0.75
VA2 98.2 2.39 409 94 0.100 0 0.103 1.03 1.03 1.02
VA3 94.6 3.59 455 94 0.173 0 0.163 0.94 0.94 1.59
VA4 103.1 5.24 470 94 0.239 0 0.22 1.00 0.91 2.21
VB1 98.2 2.39 409 94 0.054 0 0.080 1.01 1.07 0.91
VB2 97.6 3.59 455 94 0.054 0 0.097 0.95 1.13 1.10
VB3 102.3 5.98 445 94 0.052 0 0.099 0.90 1.08 1.17
VB4 96.9 1.79 455 189 0.027 0 0.052 0.97 1.12 0.85
Average 1.01 1.11 1.40
CoV 12.2 13.0 46.7
67
2.6.2 Truss approaches with concrete contributions
In the traditional approaches for the shear design of concrete beams, it is assumed
that compression struts are formed parallel to cracks and no stresses are transferred
across the cracks, hence the concrete contribution due to transfer of stresses
across the cracks is usually neglected, but this often leads to conservative results.
The more recent approaches also take into account the following two contributions;
Tensile stresses that exists transverse to the crack.
Shear stress that is transferred along the inclined crack by aggregates
interlocking.
The truss model was, however later on modified by Ramirez and Breen (1991) and
the nominal shear strength of concrete was given as
Vn = Vc + Vs (2.43)
The Vc suggested by Rameriz and Breen ( 1991) is given as;
Vn = dbwcr 32
1 (2.44)
Here cr = Shear stress resulting in the first diagonal tension cracking in the conc.
In some latest research work, the original truss method has been changed into the
variable angle truss model. This model accounts for the fact that the concrete struts
are generally not inclined at 45°, but may instead be in a range from about 25° to
65°. The new proposed model has been shown in Figure 2.16 (Mitchel,1986)
68
Figure 2.16 Variable truss Model of RC beams. (Mitchel,1986)
2.6.3. Shear friction approach
Shear friction approach was first introduced by Birkeland and Birkeland (1966), to
deal with transfer of forces across the joints in pre-cast concrete construction as
shown in Figure 2.17. When concrete is subjected to shear and compression forces,
cracks are formed and the roughness of the crack will create separation δ between
the two halves. The reinforcement is provided across the interface, provides an
external clamping force T.
The roughness may be visualized as series of saw toothed frictionless fine saw
toothed ramp having a slope of tanθ. The separation is sufficient to yield
reinforcement across the crack. This nominal shear resistance of concrete is given
as
Vn = μ Asfy (2.45)
Where μ = 1.7 for monolithic concrete
μ = 1.4 for artificial roughened concrete and 0.80 to 1.0 for ordinary.
69
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 μ 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 in
response 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.
70
Strut and Tie Model (STM) is one of the most rational and relatively simple design
approaches for non flexural members. The STM has been used in Europe for many
years in Europe and has been adopted by Canadian Code in 1984, AASHTO LRFD
bridge design in 1994 (James et al. 2003), It was incorporated in ACI building Code
381-02 as appendix A and later was recommended as an optional design procedure
for disturbed region in ACI 318-06 building code.
The STM is based on lower bond theory of plasticity assuming that steel and
concrete are frequently plastic and efficiency factors are applied to uni-axial strength
of concrete to account for concrete softening. The STM design is not unique as it
depends on the shape of non flexural structure, material, design perception and
understanding of the structure. However the method has opened a great venue for
research in the design of disturbed regions.
The joint ACI-ASCE Committee 445(1998) report on Shear and Torsion has given
detailed commentary on the STM. The Strut and Tie Model has been used for the
design of disturbed regions like Deep beams, Dapped ended beam, corbels,
brackets, pile caps, opening in slabs and non prismatic structural members. Further
details of the STM have been given in chapter No. 5, under Shear Design of
Disturbed Region.
2.6.5: Some latest research on shear design of reinforced concrete beams.
Zararis P.D (2003), has reported a new concept for the design of shear
reinforcement, in which the shear strength of beam without shear reinforcement is
expressed as follows;
bdfd
cd
d
aV ctcr )()(2.02.1
(2.46)
Where dd
a)(2.02.1 ≥ 0.65 ( d in m )
f ct = 0.30 fc′ 2/3
c; depth of compression zone which is determined by the quadratic equation;
71
0)/(
600.600
cc f
dd
d
c
fd
c (2.47)
For beams with shear reinforcement, the steel contribution is added which is
expressed as
Vs = ( 0.50 + 0.25 d
a) ρ vf yv b d
The shear of RC beams in complete form is as follows:
bdfd
af
d
cd
d
ayvvct )25.05.0()..2.02.1 . (2.48)
Zararis (2003) coampred experiemntal results of various researchers with the
theoretical shear strength of RC beams worked out with the equtaions proposed by
him and reported the least Coefficient of Variation ( CoV) for 174 beams data. The
deatils of comparison has been given in Table 2.2 Zararis (2003) reported that the
coefficient of variation proposed by his new theory was the least one, when
compared with ACI and EC-02 equations.
Arsalan G (2007) developed the following equation for predicting the diagonal
cracking shear stress of beams without stuirrups.
65.050.0 )(02.0)(15.0 cccr ff For Normal Strength Concrete (2.49)
65.0)(12.0 ccr f For High Strength Concrete (2.50)
By comparing the test values and predicted values from the propsoed model, it was
deduced that the proposed equation gave same results as the ACI simplified
equation.
Guray A (2008), further proposed the following expressions for the shear strength of
RC beams with stirrups;
72
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 Zustty’s equation for data
base of 1146 beams and reported that the Zustty’s equation has given the best
model 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 for
developing the basic design methods.
73
Table 2.2 Comparison of the shear strength of RC beams proposed by Zararis , ACI and
EC-2 ( Zararis P.D,2003)
'cf
MPa
b
cm
d
cm
da /
Reinforcement
Exp
uV
kN
ACI EC-2 Theory of
Zararis.P
%
'
%
v yvvf
MPa
uV
kN
ACIu VV /
uV
kN
ECu VV /
uV
kN
theoryu VV /
Leonhardt and Walther ( 1962)
30.4 19 27 2.78 2.47 0.33 0.41 1.52 170.5 131 1.301 139.9 1.218 168.6 1.011
28.2 19 27 2.78 2.47 0.33 0.42 1.63 186.0 135 1.378 141.6 1.314 173.7 1.070
30.4 19 27 2.78 2.47 0.33 0.59 1.54 187.5 132 1.420 140.8 1.331 169.9 1.103
30.4 19 27 2.78 2.47 0.33 0.58 1.61 189.0 135.6 1.394 144.0 1.312 174.1 1.085
Bresler and Scorelies (1963)
24.1 30.7 46.6 3.92 1.80 0.18 0.10 0.33 233.2 170.7 1.366 179.1 1.302 210.9 1.105
24.3 30.5 46.4 4.93 2.28 0.18 0.10 0.33 244.8 172.4 1.420 183.5 1.334 216.1 1.132
24.8 23.1 46.1 3.95 2.43 0.24 0.15 0.48 222.5 147.1 1.512 154.2 1.443 185.3 1.200
23.2 22.9 46.6 4.91 2.43 0.24 0.15 0.48 200.2 142.5 1.405 149.4 1.340 191.0 1.048
29.6 15.5 46.4 3.95 1.80 0.36 0.20 0.65 156.1 114.9 1.358 120.8 1.292 143.8 1.085
23.8 15.2 46.4 4.93 3.66 0.37 0.20 0.66 161.5 110.5 1.462 111.4 1.450 156.5 1.031
Bresler and Scorelies (1964)
25.1 30.5 46.0 3.98 1.69 0.18 0.10 0.35 168.4 171.7 0.981 179.2 0.940 206.5 0.816
23.6 22.9 45.7 4.01 2.28 0.24 0.15 0.51 172.7 144.8 1.192 151.4 1.141 188.4 0.917
24.4 15.5 45.8 4.00 1.67 0.35 0.20 0.69 118.6 110.1 1.077 111.0 1.068 137.1 0.865
26.3 30.5 45.7 4.01 1.71 0.18 0.10 0.35 214.6 173.2 1.239 183.3 1.171 211.3 1.015
23.2 22.9 45.9 3.99 2.26 0.23 0.15 0.51 203.9 144.6 1.409 150.6 1.354 192.0 1.061
26.7 15.2 46.0 3.96 1.69 0.36 0.20 0.70 143.3 111.8 1.282 114.1 1.255 141.3 1.014
25.2 30.5 46.2 3.95 1.76 0.18 0.10 0.35 219.8 173.1 1.269 182.0 1.208 212.8 1.032
26.5 23.1 46.0 3.97 2.34 0.24 0.15 0.51 201.9 152.3 1.325 161.8 1.247 198.2 1.018
24.9 15.5 46.0 3.97 1.75 0.35 0.20 0.69 142.6 111.4 1.280 113.4 1.257 142.3 1.002
26.3 30.5 46.1 3.96 1.77 0.18 0.10 0.35 241.8 175.2 1.380 186.1 1.300 241.8 1.124
26.3 30.5 46.0 3.97 1.77 0.18 0.10 0.35 207.6 175.0 1.186 186.0 1.116 207.6 0.968
Bahl
26.8 24 30 3 1.26 0.22 0.15 0.66 130.0 112.4 1.157 117.7 1.104 138.1 0.941
25.1 24 60 3 1.26 0.11 0.15 0.66 252.5 220.7 1.144 196.0 1.288 246.7 1.023
26.3 24 90 3 1.26 0.07 0.15 0.66 372.5 335.2 1.111 299.1 1.245 332.6 1.119
25.4 24 120 3 1.26 0.05 0.15 0.66 468.0 442.9 1.057 393.7 1.189 440.6 1.062
Placas and Regan
26.7 15.2 27.2 3.36 1.46 0.34 0.21 0.58 79.6 61.2 1.300 67.5 1.179 78.7 1.011
29.6 15.2 27.2 3.36 1.46 0.34 0.43 1.15 104.5 86.6 1.207 91.9 1.137 111.9 0.934
29.6 15.2 27.2 3.36 0.98 0.34 0.21 0.58 75.5 62.0 1.218 65..5 1.153 73.3 1.030
26.2 15.2 27.2 3.36 0.98 0.34 0.21 0.58 89.5 61.9 1.445 71.9 1.245 83.6 1.070
33.9 15.2 27.2 3.60 4.16 0.37 0.21 0.58 117.3 70.6 1.661 82.0 1.431 105.7 1.109
32.3 15.2 27.2 3.60 4.16 0.37 0.43 1.15 160.0 93.2 1.717 101.2 1.580 137.2 1.166
29.0 15.2 27.2 3.36 1.46 0.34 0.14 0.38 89.5 54.4 1.645 62.6 1.430 69.0 1.277
29.9 15.2 27.2 3.60 4.16 1.49 0.43 1.15 149.6 91.8 1.629 98.4 1.521 130.0 1.150
31.6 15.2 27.2 3.60 4.16 2.96 0.43 1.15 149.6 92.8 1.611 100.4 1.490 125.6 1.191
74
Table 2.2 Cont’d '
cf
MPa
b
cm
d
cm
da /
Reinforcement
Exp
uV
kN
ACI EC-2 Theory of
Zararis.P
%
'
%
v yvvfMPa
uV
kN
ACu VV /
uV
kN
ECu VV /
uV
kN
theoryu VV /
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 0.920
39.9 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
75
Table 2.2 Cont’d '
cf
MPa
b
cm
d
cm
da / Reinforcement
Exp
uV
kN
ACI EC-2 Theory of
Zararis.P
%
'
%
v yvvf
MPa
uV
kN
ACu VV /
uV
kN
ECu VV /
uV
kN
theoryu VV /
55.8 30.4 53.8 3.10 2.49 0.79 0.14 0.69 382.9 330.6 1.158 367.8 1.041 377.1 1.015
51.3 30.4 53.8 3.10 2.49 0.79 0.07 0.35 280.9 267.0 1.052 303.2 0.926 298.7 0.941
51.3 30.4 53.8 3.10 2.49 0.79 0.07 0.35 258.3 267.0 0.967 303.2 0.852 298.7 0.865
29.2 40.6 34.5 2.65 2.32 1.02 0.39 2.14 460.1 441.7 1.042 444.9 1.034 526.6 0.874
32.2 40.6 34.5 2.65 2.31 1.02 0.39 2.14 549.1 447.8 1.226 456.7 1.202 533.9 1.028
32.4 40.6 34.5 2.65 2.31 1.02 0.39 2.14 504.6 448.2 1.126 457.4 1.103 533.9 0.945
33.8 40.6 34.5 2.65 2.31 1.02 0.39 2.14 584.7 450.9 1.297 462.8 1.263 538.0 1.086
Roller and Russell
120.2 35.6 55.9 2.50 1.65 ___ 0.07 0.28 298.0 340.7 0.874 542.8 0.549 363.5 0.820
120.2 35.6 55.9 2.50 3.03 ___ 0.43 1.94 100.0 689.7 1.595 877.2 1.254 863.6 1.273
120.2 35.6 55.9 2.50 4.55 ___ 0.88 4.02 658.7 124.3 1.475 249.7 1.327 402.6 1.182
120.2 35.6 55.9 2.50 6.06 ___ 1.25 5.73 944.3 485.1 1.309 556.0 1.249 844.4 1.054
120.2 35.6 55.9 2.50 6.97 ___ 1.75 8.03 239.5 955.2 1.145 968.0 1.138 403.0 0.932
72.4 45.7 76.2 3.00 1.73 ___ 0.08 0.36 665.5 619.1 1.075 713.5 0.932 581.1 1.145
72.4 45.7 76.2 3.00 1.88 ___ 0.16 0.71 788.0 744.0 1.059 842.4 0.935 745.7 1.056
125.4 45.7 76.2 3.00 1.88 ___ 0.08 0.36 482.9 622.1 0.776 1006 0.480 658.5 0.734
125.4 45.7 76.2 3.00 2.35 ___ 0.16 0.71 749.6 753.2 0.995 1138 0.658 860.8 0.871
125.4 45.7 76.2 3.00 2.89 ___ 0.23 1.04 172.4 878.8 1.334 241.8 0.944 052.3 1.114
Sarzam and Al - Musawi
40.4 18.0 23.5 4.00 2.23 0.37 0.09 0.76 114.7 79.2 1.449 100.3 1.143 111.7 1.026
75.3 18.0 23.5 4.00 2.23 0.37 0.09 0.76 122.6 94.9 1.292 137.0 0.894 124.6 0.984
75.7 18.0 23.5 4.00 2.82 0.37 0.09 0.76 138.3 96.1 1.439 137.4 1.006 132.4 1.044
70.1 18.0 23.5 4.00 3.51 0.37 0.09 0.76 147.2 95.1 1.547 132.0 1.115 137.5 1.069
Xie et al
37.7 12.7 21.6 3.00 2.07 ___ ___ ___ 36.7 30.1 1.219 44.8 0.819 37.5 0.978
40.7 12.7 20.3 3.00 3.20 1.00 0.49 1.58 87.1 71.7 1.215 81.4 1.069 94.3 0.924
98.9 12.7 21.6 3.00 2.07 ___ ___ ___ 45.7 39.5 1.157 85.3 0.536 49.3 0.927
98.3 12.7 19.8 3.00 4.54 1.03 0.51 1.65 102.4 81.1 1.262 116.1 0.882 114.2 0.897
89.8 12.7 19.8 3.00 4.54 1.03 0.65 2.10 108.3 92.4 1.172 121.7 0.889 122.6 0.883
103.2 12.7 19.8 3.00 4.54 1.03 0.78 2.53 122.6 103.2 1.187 138.7 0.884 138.0 0.888
McGormley, Creary, and Ramirez
42.2 20.3 41.9 3.27 3.03 1.20 0.34 1.45 271.5 225.2 1.205 238.9 1.136 287.3 0.945
43.2 20.3 41.9 3.27 3.03 1.20 0.34 1.45 298.1 226.0 1.319 240.9 1.237 287.8 1.035
45.5 20.3 41.9 3.27 3.03 1.20 0.34 1.45 275.9 228.4 1.208 245.5 1.123 289.1 0.954
44.4 20.3 41.9 3.27 3.03 1.20 0.34 1.45 307.0 227.2 1.351 243.3 1.262 289.0 1.062
35.3 20.3 41.9 3.27 3.03 1.20 0.34 1.45 315.9 217.4 1.453 224.6 1.406 278.8 1.133
48.3 20.3 41.9 3.27 3.03 1.20 0.34 1.45 311.5 231.1 1.347 251.0 1.241 293.3 1.062
50.0 20.3 41.9 3.27 3.03 1.20 0.34 1.45 333.7 232.8 1.433 254.3 1.312 295.7 1.128
50.5 20.3 41.9 3.27 3.03 1.20 0.34 1.45 320.4 233.3 1.373 255.2 1.255 295.8 1.083
53.4 20.3 41.9 3.27 3.03 1.20 0.34 1.45 289.2 236.0 1.225 260.7 1.109 298.3 0.970
55.1 20.3 41.9 3.27 3.03 1.20 0.34 1.45 311.5 237.6 1.311 263.8 1.180 299.6 1.039
56.7 20.3 41.9 3.27 3.03 1.20 0.34 1.45 267.0 239.0 1.117 266.8 1.001 300.2 0.890
56.1 20.3 41.9 3.27 3.03 1.20 0.34 1.45 267.0 238.5 1.119 265.7 1.005 299.9 0.890
76
Table 2.2 cont’d '
cf
MPa
b
cm
d
cm
da /
Reinforcement
Exp
uV
kN
ACI EC-2 Theory of
Zararis.P
%
'
%
v yvvf
MPa
uV
kN
ACu VV /
uV
kN
ECu VV /
uV
kN
theoryu VV /
Yoon, 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
77
Table 2.2 Cot’d '
cf
MPa
b
cm
d
cm
da /
Reinforcement
Exp
uV
kN
ACI EC-2 Theory of
Zararis.P
%
'
%
v yvvf
MPa
uV
kN
ACu VV /
uV
kN
ECu VV /
uV
kN
theoryu VV /
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.
78
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 0.35bws/fy, such that the stirrups are
assumed 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 f’c 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 06.0
y
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.
79
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 aggregate
interlocking 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.
80
iii. The third group is engaged in translating the research results into a more
meaningful tool for the building codes in the form of methods and rules for the
shear analysis and design of RC members.
There is a general feeling in the minds of many researchers, that enough research
has been carried out on this topic and there seems no more room for further
research in this field. Regan (1993) tried to answer this basic question, where
research on shear is waste of time or service to humanity? After reviewing the
research of last 4-5 decades, Regan (1993), highlighted the significance of the
research on the shear of RC beams in the following ways;
i. The research on shear for 40 years has enabled the structural engineers to
design the RC members without web reinforcement, pre-stressed beams and
flat slab buildings more accurately.
ii. The research on shear has been focused on making the design provisions of
building codes more rational and comprehensive. Considerable achievements
have been made in this direction. In these endeavors many misconceptions
and doubts were also created, which were clarified in later works.
iii. Most of the proposed models developed in the meanwhile were based on the
existing data but these models could poorly predict the behaviour of actual
beams, mainly due to the fact that important variables were not considered in
the models at times.
iv. More experimental tests and researches are required for significant
improvement in the shear design concept for its further rationalization
involving parametric studies.
Despite of the fact that research on shear strength of RC beams, has been
condcuted for more than six decades,but even then the riddle of shear failure
initiated by Kani(1964) is still unexplained. The exact beahvior of RC concrete in
shear is still an active areas in contemporary research.
81
Mitchell et al. (2008) in a long term project, reviwed the results of 1849 tests on the
shear strength of RC beams to judge the adequacy and safety provided by the shear
equations used in North America. The findnigs of the research provide the latest
state of researech on the shear strength the Some of the important findings and
conclusions of the research of Mitchell et al (2008) are given as follows:
i. The traditional appraoch to design the shear reinformcent for the
region where the external shear is exceeding the concrete shear
capacity dbfV wcc'2
may lead to un-conservative results and the
chances of brittle failure may enhance. Hence there is a need to
revised and rationalize the shear design equation of ACI and
particularly the simplified shear design equation.
ii. The new load factors introduced in ACI-318-02, have led to increased
flexural stresses in felxural reinforcement at service loads, which have
furrther reduced the safety agianst shear failure.
iii. The design engineers must understand that the shear strength of RC
beams is also affected by member depth, crack roughness and strain
in longitudinal reinforcement, in addition to concrete strength.
iv. The recent research data shows that for RC members without web
reinforcement, the influence of strain in longitudinal steel is more
pronounced. High strength in the longitudinal steel and wider crack
widths may decrease the shear strengths of RC members.
v. In high strength concrete with small aggreagtes sizes, the cracks
surafces are reltively somoother and can lead to reduction in the shear
capacity of RC members. The equations based on the Modfied
Compression Field Thoery ( MCFT) accounts for the strain effect, size
82
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.
83
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 θ for 05.0'
c
u
f
v and εx between 1.5x 10-3 and 2 x 10-3
θ =42o which gives x = 1.84 x 10-3 and β = 0.15
)270@10(270
000,200/40511.12751404053005515.0)cot('
mmmmmms
ss
dfAdbfV vyv
vwcn
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=ρ = 0.015
Yield stress of longitudinal steel fyl = 414MPa
Yield stress of transverse steel fyv = 275 MPa
Solution.
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 θ for 05.0'
c
u
f
v
θ =43˚ which gives
x = 0.002 and
β = 0.14
mms
ss
dfAdbfV
vyvvwcn
201
82755/27007.1275652702255214.0)cot('
Provided 7mm @150mm. O.K
85
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.
86
Shear design
Attaullah Shah 2009/3/4
All dimensions in millimetresClear cover to transverse reinforcement = 40 mm
Inertia (mm4) x 106
Area (mm2) x 103
yt (mm)
yb (mm)
St (mm3) x 103
Sb (mm3) x 103
69.0
517.5
150
150
3450.0
3450.0
75.0
569.8
155
145
3682.5
3922.9
Gross Conc. Trans (n=6.48)
Geometric Properties
Crack Spacing
Loading (N,M,V + dN,dM,dV)
2 x dist + 0.1 db /
0.0 , 0.0 , 0.0 + 0.0 , 1.0 , 0.0
230
300
2 - 10
Av = 32 mm2 per leg
@ 150 mm
3 - 20
Concrete
c' = 2.28 mm/m
fc' = 52.0 MPa
a = 19 mmft = 2.19 MPa (auto)
Rebar
s = 100.0 mm/m
fu = 695 MPa
Trans, fy= 275
Long, fy= 463
1. General Properties displayed by the software
Cross Section Longitudinal Strain
-3.02 12.78
top
bot
Transverse Straintop
bot
Crack Diagram
3.08
2.05
1.81
1.02
0.13
Shear Straintop
bot
Shear Stresstop
bot
Principal Compressive Stress
-52.0
top
bot
Shear on Crack
3.99
top
bot
Principal Tensile Stress
2.19
top
bot
Control : M-Phi
93.3
99.4
Control : M-ex
-2.2 5.7
99.4
87
2. Details of cracking.
Cross Section Longitudinal Strain
-3.02 12.78
top
bot
Principal Tensile Strain
12.8
top
bot
Crack Diagram
3.08
2.05
1.81
1.02
0.13
Crack Widths
3.34
top
bot
Average Angle
90.0
top
bot
Long. Crack Spacing
300.0
top
bot
Transverse Crack Spacingtop
bot
Diagonal Spacing
300.0
top
bot
Control : M-Phi
93.3
99.4
Control : M-ex
-2.2 5.7
99.4
3. Reinforcement details.
Cross Section Longitudinal Strain
-3.02 12.78
top
bot
Transverse Straintop
bot
Long. Reinforcement Stress
-56.1 476.8
top
bot
Long. Reinf Stress at Crack
498.0
top
bot
Long. Average Bondtop
bot
Stirrup Stresstop
bot
Stirrup Stress at Cracktop
bot
Transverse Average Bondtop
bot
Control : M-Phi
93.3
99.4
Control : M-ex
-2.2 5.7
99.4
88
Chapter No. 3
Shear strength of high performance reinforced concrete
beams
Chapter Introduction: This chapter mainly addresses the issues in the shear of High strength concrete. The chapter starts with the definition of HSC, its historical development and extensive use across the world. Then the structural properties of High Strength concrete have been discussed and lastly the shear strength of high strength concrete beams has been explained on the basis of experimental research and empirical relationship developed by different researchers in last two decades.
Figure 3.1: World Trade Centre (USA) Figure 3.2:The world Highest Tower Burj Dubai,UAE (2651 feet) (162 floors, scheduled construction, 2008)
3.1 High Performance Concrete (HPC)
The term High Performance Concrete (HPC) is used to describe concretes that are
made with carefully selected high quality ingredients, optimized mixture designs, and
which are batched, mixed, placed, consolidated and cured to the highest industry
standards. Typically, HPC will have a water-binder material ratio (w/b) of 0.4 or less.
Achievement of these low w/b concretes often depends on the effective use of
admixtures to achieve high workability, another common characteristic of HPC
mixes.
89
Forster [1994] defined HPC as "a concrete made with appropriate materials
combined according to a selected mix design and properly mixed, transported,
placed, consolidated, and cured so that the resulting concrete will give excellent
performance in the structure in which it will be exposed, and with the loads to which
it will be subjected for its design life."
American Concrete Institute has defined HPC as;
“Concrete meeting special combinations of performance and uniformity requirements
that cannot always be achieved routinely using conventional constituents and normal
mixing, placing and curing practices” (ACI-318.116R,2006).
The requirements may involve enhancements of characteristics such as placement
and compaction without segregation, long-term mechanical properties, early-age
strength, volume stability, or service life in severe environments. Concretes
possessing many of these characteristics often achieve higher strength. Therefore
HPC is often of high strength, but high strength concrete may not necessarily be of
High-Performance Concrete at times.
National Concrete Bridge Council (USA) has defined HPC as; “…concrete that
attains mechanical properties, durability or constructability properties exceeding
those of normal concrete.”
Zia et al ( 1993), while working on the States Highways Research Project-USA
( SHRP) C-205, defined various types of HPC, as given in Table 3.1
90
Table 3.1 Definition of HPC as per SHRP (Zia et al, 1993)
Category of HPC
Minimum Compressive
Strength
Maximum Water/
Cement Ratio
Minimum FrostDurability
Factor
1.Very early strength VES
Option A(with Type III cement)
2,000 psi (14 MPa)in 6 hours
0.40 80%
Option B(with PBC-XT cement)
2,500 psi (17.5 MPa)
in 4 hours
0.29 80%
2. High early strength (HES) (with Type III cement)
5,000 psi (17.5 MPa)
in 24 hours
0.35 80%
3. Very high strength (VHS) (with Type I cement)
10,000 psi (70 MPa)
in 28 hours
0.35 80%
More recently the term Ultra High Strength Concrete (UHSC) is also used in the
literature, which refers to the HPC having the compressive strength of concrete in
excess of 100 MPa.
The production of HPC has been possible with development of new material besides
the conventional cement and aggregates. These may include mineral and chemical
admixtures.
The use of fly ash and Silica fume, (also called condensed silica fume or micro
silica), continues to be popular element of high performance concrete, and
especially high strength concrete. Mehta (1994), Aitcin (1993), Goldman and Bentur
[1993], Bharatkumar et al (2005) and O.Kiyali (2005) examined the effects of silica
fume on mechanical behaviour. These reports confirm findings that a silica fume
tends to improve both mechanical properties and durability.
91
A number of issues with frost resistance of concrete containing silica fume have
been investigated, including the need for any entrained air when working with very
low W/CM ratio concretes. According to ACI 318-95 [1995], the quantity of silica
fume in concrete, exposed to deicing salts is limited to no more than 10 percent.
The use of cementitious systems with very high quantities of fly ash has also been
investigated. Malhotra [1990] reports that performance in rapid freezing and thawing
of concrete with high volumes of class F fly ash was adequate but that the concrete
with very high quantities of fly ash performed poorly in deicer scaling tests. Naik et
al. (1994) found that although concrete made with high volumes of class C fly ash
passed ASTM C-944 for abrasion resistance, better abrasion resistance was
obtained for concrete without the high fly ash content. The small and spherical fly
ash particles filled the voids or airspaces and increased the density. The smaller
particle size of fly ash with a higher surface area and glassy phase content also
improved the pozzolanic reaction and (Isaia et al,2003). Therefore, the CFA ( Class
C-Fly ash) made the blended cement paste more homogeneous and denser as well
as having a higher pozzolanic reaction than the one containing the original fly ash,
and this resulted in an increase in the compressive strength(Mehta,2003).
The mix proportioning of HPC has been attempted in various projects. Field trials of
High Early Strength (HES), Very Early Strength (VES) and Very High Strength
(VHS) concretes in SHRP C-205 and C-206 indicated that existing proportioning
methods remain valid, with minor modifications, for these mixes.
3.2 High strength concrete:
The definition of High Strength Concrete (HSC), has been changing with time due to
advancement in the concrete and material technology. At times the compressive
strength of 40 MPa was considered as high strength, however with improved mixed
design, ultra high range water reducers (Superplasticisers), and mineral admixtures,
concretes with compressive strength above 100 MPa are easily obtained in the field.
ACI-318 committee revealed that in the 1960’s, 52 MPa (7500 psi) concrete was
92
considered high-strength concrete and in the 1970’s, 62 MPa (9000psi) concrete
was considered as HSC. The committee also recognized that the definition of the
high-strength concrete varies on a geographical basis. In regions where 62 MPa
(9000 psi) concrete is already being produced commercially, high-strength concrete
might be in the range of 83 to 103 MPa (12,000 to 15,000psi). However in
developing countries like Pakistan, achieving 60 MPa concrete is still underway.
The High Strength Concrete has been successfully used in the construction of pre-
stressed bridges in the world such as Braker Lane Bridge, built in Austin, Texas,
USA in 1990, with concrete strengths ranging from 75.8 MPa to 96.5 MPa at 28
days. The Red River Cable-Stayed Bridge Guangxi, China ( 65 MPa) , Normandy
Bridge, France (60 MPa) and Portneuf Bridge Quebec, Canada (60 MPa) [ Bickley
and Mitechlles,2001]
Due to substantial increase in the strength of concrete, the term ultra High strength
Concrete is also used. In Europe, high-performance steel fiber reinforced
cementitious composite referred to as Reactive Powder Concrete (RPC) was
developed. Steam curing at 90οC and densest packing design enabled to produce
pre-cast concrete having high-performance and ultra high-strength of around 200
MPa. The actual applications of RPC have been done by around 35 projects in the
world (Toru Kawai, 2005). The High Strength Concrete has been used for the
construction of pre-cast and high rise building across the world.
Until recently, the world's tallest buildings were in the United States, but in 1993, the
tall building construction boom shifted to Asia with the erection of the 1207 ft (368 m)
Central Plaza office tower in Hong Kong. Two major high-rises in Asia are the 1371
ft (418 m) Jin Mao Tower in Shanghai, China, and the 1378 ft (420 m), Petronas twin
towers in Kuala Lumpur, Malaysia. These monumental towers use composite
structural systems, combining vertical components such as cores, columns, and
shear walls of concrete that have strengths of up to 80 MPa (11,600 psi) with
structural steel horizontal members to resist lateral and vertical forces [PCA,2005].
93
The two tallest concrete buildings in the United States were completed in Chicago in
1989. Both the 969 ft (295 m), 311 South Wacker Building and the 920 ft (276 m),
Two Prudential Plaza Buildings took advantage of 83 MPa (12,000 psi) high-strength
concrete in the fabrication of cast-in-place, steel-reinforced columns and walls at the
buildings' lower levels to support the total dead and live loads of the structures. The
middle and upper levels of the buildings, where total accumulated forces are lower,
were constructed with concrete in strengths ranging from 27.6 MPa (4000 psi) to 69
MPa (10,000 psi) [PCA,2005].
The use of HSC at local level has also been increased during the last one decade
due to construction of pre-stressed bridges, girders and other infrastructure projects
in Pakistan. In multi-span bridges constructed in Pakistan Motorway project and high
rise buildings and tower being constructed, extensively use the pre-stressed
concrete technology in Pakistan. Hence HSC of compressive strength 60Mpa and
above is used in these projects. In the following sections, important considerations
in the development of high strength concrete have been discussed, which are mainly
based on the ACI-State of the Art Report on the high Strength Concrete by ACI
Committee, 363-97
3.2.1 Selection of materials for high strength concrete.
3.2.1.1 Cement: The selection of appropriate type of cement is critical for the HSC,
the ACI committee 363[2005], has given the following recommendations on testing
and selection of the cement.
Initially, silo test certificates should be obtained from potential suppliers for the
previous 6 to 12 months. If the tri-calcium silicate content varies by more than 4
percent, the ignition loss by more than 0.5 percent, or the fineness by more than 375
cm2/g (Blaine), then problems in maintaining a uniform high strength may result.
Sulfate (SO4) levels should be maintained at optimum with variations limited to
±0 .20 percent.
94
3.2.1.2 Admixtures in high strength concrete.
3.2.1.2.1 Mineral admixtures;
In the use of mineral admixtures like fly ash, Silica fumes, Slag cement (Ground
granulated Blast furnace Slag), the following guiding principles may be kept in mind;
i. Specifications for fly ash are covered in ASTM C 618-08a(2008). Methods for
sampling and testing are found in ASTM C 311-07(2007), which shall be
followed for uniformity of the supply.
ii. To ensure uniform textures and properties of fly ash supply, appropriate
testing of shipments and increased frequency of sampling shall be followed.
iii. For silica fumes. Thus, it is necessary to quickly cover the surfaces of freshly
placed silica-fume concrete to prevent rapid water evaporation.
iv. Specifications for ground granulated blast furnace slag are given in ASTM C-
989-09(2009).
v. The evaluation of mineral admixtures with the laboratory tests is an important
step in the selection of mineral admixtures.
3.2.1.2.2 Chemical admixtures;
Chemical admixtures are widely used in the development of high-strength concrete.
The super-plasticizers (SP), refer to high range water reducing admixtures by ASTM
C494-05(2005). There are four major groups of super-plasticizers.
i. Sulfonated Naphthalene Formaldehyde Condense (SNF)
ii. Sulfonated Melamine Formaldehyde Condense (SMF)
iii. Modified Liognosulfate ( MLS)
iv. Other types including polyarcylatyes, polystyrene sulfonates and
polymers etc.
These high water reducing agents change the properties of fresh and hardened
concrete in the following ways (Ahmad et al. 2004)
i. Reduction in the interfacial tension
ii. Multilayered absorption of organic molecules
95
iii. Protective adherent Sheath layer of water molecules
iv. Release of water trapped amongst the cement particles.
v. Retarding effect of cement hydration
vi. Change in the morphology of hydrated cement.
Various types of chemical admixtures are discussed as follows;
i. Air entraining admixtures (ASTM C-260). In concrete subjected to freezing
and thawing during initial stages, air entrainment helps in decreased water
cement ratio and improved air-void ratio. The Air entraining agents reduce the
compressive strength of concrete and its use is recommended only where the
durability of concrete is main concern.
ii. Retarding agents (ASTM C-494 Type B&D).
A retarder can control the rate of hardening in the forms to eliminate cold joints
and provide more flexibility in placement schedules.
The dosage of retarders depends on the temperature during the setting time of
concrete. Initially if the temperature is too much, high dosage of retarders is
recommended, however, it may be reduced if the temperature declines.
iii. Normal range water reducers (ASTM C-494 Type A)
These generally increase the strength of concrete without affecting the rate of
hardening. The increased dosage of normal water reducers may reduce the
strength of concrete but may improve the hardening of concrete.
iv. High range water reducers (ASTM C-494 Type F&G)
In high-strength concrete, HRWR may serve the purpose of increasing strength
at the slump or increasing slump. The method of addition should distribute the
admixture throughout the concrete. Adequate mixing is critical to uniform
performance.
96
v. Accelerators (ASTM C 494, Types C and E)-
In HSC, accelerators are used selectively, where removal of the form work is
required at an early stage from columns and walls. These mostly reduce the
compressive strength of concrete in the long run.
3.2.1.3 Aggregates
For HSC fine aggregates in round shape are preferred for its particle shape and
smooth textures. Fine aggregates with fineness modulus of 2.5 may give very sticky
concrete, which may be difficult in placement, whereas for FM, of 3.0 the workability
may improve substantially. The amounts passing sieves No 50 and 100 may be
kept low but still within the requirements of ASTM C-33. The sand gradation may not
have significant effect on the early stage but at later ages, it may become an
important parameter in the strength of HSC. Hence FM between 2.5 to 3 is preferred
for HSC.
In case of coarse aggregates, the maximum sizes of ½ in or ¾ in are preferred. The
crushed stones have given better results as compared with the rounded aggregates.
For HSC, coarse aggregate should be clean, cubical, angular, 100 percent crushed
aggregate with a minimum of flat and elongated particles.
The water absorption capacity of aggregates plays an important role in the strength
development of the HSC. If aggregates are capable of absorbing a moderate
amount of water, they can act as tiny curing-water reservoirs distributed throughout
the concrete, thereby providing the added curing water which is beneficial to these
low water-cement ratio pastes in HSC. The high strength of aggregates is also an
important in selection of coarse aggregates for HSC.
97
3.2.1.4 Water
Usually, water for concrete is specified to be of potable quality, available from
municipality lines or local tube wells.
3.2.2 Mix proportioning of high strength concrete
Mix proportioning of high strength concrete is more important as involves the
selection of appropriate admixtures besides other basic ingredients in the normal
concrete. For achieving the requisite strength, testing of mix designed concrete is
frequently carried out and concrete is accepted if the following conditions are
fulfilled;
a) The average of all sets of three consecutive strength test results shall equal or
exceed the required fc'.
b) No individual strength test (average of two cylinders) shall fall below fc' by more
than 500 psi (3.4 MPa). However, some designers have specified higher or lower
over design strengths than called for in ACI 318 regardless of established
performance.
The age of concrete specimen is critical to the strength of concrete. In HSC,
substantial strength gain has been observed at later ages beyond 28 days and
mostly at 56 and 90 days. Hence it is recommended to employ accelerated tests for
prediction of later age strength of concrete as per ACI- publication SP-56.
The water cement ratio mostly referred as water binder ratio in HSC, is very
important for the high strength of concrete. The increase in strength is achieved by
increasing the quantity of cementitious material and hence additional water is
required for its hydration. For HSC not containing water reducers may require a
slump of 2-4 in depending on the placement conditions and forms. The use of high
range water reducers may reduce the water requirement and a range of w/c ratio
from 0.27 to 0.50 is recommended for such kinds of HSC.
98
The cement contents have also direct effect on the compressive strength of
concrete. Most commonly a range of 392 to 557 kg/m3 has been used for developing
high strength concrete.
To optimize the mix proportioning of various material in the HSC, rigorous testing of
various ingredients of HSC are very important. The type and brand of cement is also
an important consideration in the selection of material. The quantity of cement
beyond certain desirable level may lead to workability problems and reduction in the
compressive strength of concrete. The maximum desirable quantity of cement for
HSC may also depend on the temperature and retarding admixtures and ice may be
required to control the temperature of the additional cement used in HSC.
The proportioning of aggregates has direct bearing on the strength of HSC. Low fine
aggregate contents have resulted in a reduction in paste requirements and normally
have been more economical. Such proportions also have made it possible to
produce higher strengths for a given amount of cementitious materials. However, if
the proportion of sand is too low, serious problems in workability become apparent,
due to less fluidity and flow-ability of concrete. The optimum amount of coarse
aggregates for HSC will mainly depend on the properties of sand. The following
Table 3.2 illustrates the mix proportioning of aggregates as per ACI-211.1
Table 3.2 -Volume of coarse aggregate per unit of volume of concrete. (ACI-211.1)
Max sizes of
aggregates (in)
Volume of dry rodded coarse aggregates for different fine moduli of sand
2.40 2.60 2.80 3.00
3/8 0.50 0.48 0.76 0.44
½ 0.59 0.57 0.55 0.53
¾ 0.66 0.64 0.62 0.60
1 0.71 0.69 0.67 0.65
1 ½ 0.75 0.73 0.71 0.69
2 0.78 0.76 0.74 0.72
3 0.82 0.80 0.78 0.76
6 0.87 0.85 0.83 0.81
99
3.2.3 Mixing, transportation and curing of high strength concrete
The control and storing of material for HSC is almost the same as normal strength
concrete. The temperature of all the constituents must be kept as low as possible
before batching and the mixing and batching facility must closer to the site.
ACI 304-05 recommends that cements and pozzolans be weighed with automatic
equipment. To maintain the proper water-cement ratios necessary to secure high
strength concrete, accurate moisture determination in the fine aggregate is essential.
In hot weather use of ice may be recommended to control the temperature rather to
use the cold water.
Canadian Standards Association’s Preliminary Standard A 266.5-M 1981, tests have
shown that high-range water-reducing admixtures are most effective and produce
the most consistent results when added at the end of the mixing cycle after all other
ingredients have been introduced and thoroughly mixed. If there is evidence of
improper mixing and non uniform slump during discharge, procedures used to
charge truck and central mixers should be modified to insure uniformity of mixing as
required by ASTM C 94-05.
Due to the relatively low water content and high cement content and the usual
absence of large coarse aggregate, the efficient mixing of high-strength concrete is
more difficult than conventional concrete. Hence it is more important in case of HSC
to check the mixer performance. ACI 304-05 recommends usual specifications, such
as 1 min for 1 cu-yd (0.75 cu-m) plus 1/4 min for each additional cu-yd of capacity,
are used as satisfactory guides for establishing mixing time.
In case of ready mix concrete, retarding admixtures are used to prolong the time the
concrete will respond to vibration after it has been placed in the forms. Withholding
some of the mixing water until the truck arrives at the job site is sometimes
desirable. Then after adding the remaining required water, an additional 30
100
revolutions at mixing speed are used to incorporate the additional water into the
mixture adequately as recommended by ACI 304-05.
In case of transportation of truck mixed concrete, if the haulage is more and there
are chances of concrete hardening before placement, dry proportioned material are
mixed in the truck and transported to the site and water is added at the site and
material is mixed. However if there is some free water in the aggregates, it may
cause the hydration of the cement or part thereof. High-strength concrete is likely to
have a high cement content and small maximum size aggregate which can facilitate
concrete pumping. In the field, the pump should be located as near to the placing
areas as practicable. Direct communication is essential between the pump operator
and the concrete placing crew. Continuous pumping is desirable because if the
pump is stopped, movement of the concrete in the line may be difficult or impossible
to start again.
The delivery of concrete to the job site must be scheduled so it will be placed
promptly on arrival, particularly the first batch. Equipment for placing the concrete
must have adequate capacity to perform its functions efficiently so there will be no
delays at distance. Sufficient vibrating machinery and manpower must be available
for the mixing of concrete. Vibration almost to the point of excess may be required
for high-strength concrete to achieve its full potential.
For placement of HSC in a framed structure, all columns, floors and beams must be
placed from the same grade of high strength concrete, wherever possible. In case
the use of two different concretes is inevitable in column and floor construction, it is
important that placement of high-strength concrete in column and adjoining areas
must be carried out before the floor concrete.
For all types of concretes, curing is an essential element of quality production of
concrete. HSC must be water cured at the early age due to very high heat of
hydration in the initial stage. Additional water for curing may be required for HSC
101
with low w/c ratio such as 0.29. It has been proposed that moist curing of HSC may
be continued for 28 days. The commonly used curing methods for HSC may include
immersion or pounding, sprinkling. Use of Burlap, cotton mats, rugs, and other
coverings of absorbent materials will hold water on the surface, whether horizontal
or vertical. Liquid membrane-forming curing compounds retain the original moisture
in the concrete but do not provide additional moisture.
For quality control of HSC, many researchers have recommended that the
specification for compressive strength should be modified from the typical 28-day
criterion to either 56 or 90 days. This extension of test age would then allow, for
example, the use of 7000 psi (48 MPa) concrete at 56 days in lieu of 6000 psi (41
MPa) at 28 days for design purposes. In actual field, since HSC is used in high rise
building, hence full load application of the service load is not possible at the lower
floors and more time can be provided for gaining additional strength. A close check
of the field results and maintenance of records in the form of control charts or other
means are necessary to maintain the desired control. Early-age control of concrete
strength such as the accelerated curing and testing of compression test specimens
according to ASTM C 684-99(2003) is often used, especially where later-age (56 or
90 days) strength tests are the final acceptance criterion.
For the test samples of HSC, ASTM standards specify a cylindrical specimen 6 in.
(152 mm) in diameter and 12 in. (305 mm) long. This size specimen has evolved
over a period of time, apparently from practical considerations. Designers generally
assume 6 x 12 in. (152 x 305 mm) regardless of the specimen size, as the
compressive stress is transferred through the loading platen-specimen interface, a
complex, tri-axial distribution of stresses in the specimen end may develop which
can radically alter the specimen failure mode and affect results.
102
3.2.4 Structural properties of high strength concrete
The shear properties and stress strain behaviour of HSC, is separately discussed in
next chapter. Other properties of HSC may include modulus of elasticity, modulus of
rapture, poison’s ratio.
The modulus of elasticity of HSC may range from 4.5 to 6.5 x l06 psi (31 to 45 GPa).
The following relation has been reported between the compressive strength of
concrete and modulus of elasticity (Carrasquillo et al. 1981)
Ec = 40,000 'cf + 1.0 x l06 psi for 3000 psi < fc,’ < 12,000 psi (3.1)
The Poisson’s ratio of high strength concrete tends to decrease with the increase of
water cement ratio. Based on the available information, Poisson’s ratio of High-
strength concrete in the elastic range seems comparable to the expected range of
values for lower-strength concretes.
The modulus of rupture of concrete may range as
fr' = 11.7 'cf psi for 3000 psi < '
cf < 12,000 psi ( 3.2)
The tensile splitting stress of HSC is given as 7.4 'cf psi for 3000 psi < fc,’<12,000 psi.
To the extent that is known, the fatigue strength of high-strength concrete is the
same as that for concretes of lower strengths. The measured values of the unit
weight of high strength concrete are slightly higher than lower-strength concrete
made with the same materials.
The Freeze thaw resistance of HSC has been observed to increase as compared to
normal concrete due to greatly reduced freezable water contents and the increased
tensile strength of high-strength concrete.
103
The shrinkage of high strength concrete containing high-range water reducers was
observed as less than for lower-strength concrete. The total creep of high strength
concrete has been observed same as the normal strength concrete.
3.2.5 Structural Designing considerations in design of HSC beams
For strain in HSC, The constant value of strain at extreme concrete compression
fiber of 0.003 prescribed by ACI 318-05 is seen to represent satisfactorily the
experimental results for high-strength as well as lower-strength concrete, although it
is not as conservative for high strength concrete.
3.3 Codes Provisions for High Strength Concrete
Paultre and Mitchell (2003), provided a detailed commentary on the HSC provisions
of four building codes used in Europe, Canada, USA and New Zealand, i.e. CEB-
MC-90, EC-02, CSA A23.3-94, ACI-318-02 and NZS 3101-95 respectively. The
important considerations of the codes are summarized as follows;
3.3.1 Concrete compressive strength
The upper limits on the compressive strength of concrete proposed by these codes
are given in Table3.3
Table 3.3 Upper limits of specified compressive strength of concrete for HSC and Standard test specimen. (Paultre and Mitchell (2003). Country /Region Code Year Max specified Concrete
compressive strength
MPa
Standard test
specimen.
(mm)
Europe CEB-FIP MC-90 1993 80 Cyl.150x300
Cube.200x300
Europe EC-02 2002 90 Cyl.150x300
Cube.200x200
Canada CSA A23.3-94 1994 80 Cyl.150x300
Cube.100x200
USA ACI318-02 2002 No limit Cyl. 152x1304
Cube. 152x304
New Zealand NZS 3101-95 1995 100 Cyl. 152x304
Cube.152x304
104
3.3.2 Load and Resistance factors
The load factors, strength reduction factor and material reduction factors have been
shown in Table 3.4. Model Code MC-90 introduced the high strength reduction factor
as hsc =
5001.1
1
ckf
………………………………………………( 3.3)
ckf will be defined below.
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).
3.3.3 Specified and Characteristic Strengths
ACI and NZ uses 28days compressive strength'
cf , which is less than the average
strength '
crf defined as follows;
'crf =
'cf +2.33s-3.45 ( MPa) for MPaf c 35'
………………………………………( 3.4)
'crf = 0.9
'cf +2.33s ( MPa) for MPaf c 35'
………………………………………. (3.5)
Where” s” is standard deviation of the sample data.
Code
Load Factor Strength Reduction factors Material Strength reduction factor
Dead
Live Flexure
Flexure
and Axial
load
Shear Concrete Steel
ACI318-02
1.2 1.6 0.65-0.9 0.65-0.9 0.75 - -
CSA A23.3-94 1.25 1.5 - - - 0.6 0.85
EC-02/ MC-90 1.35 1.5 - - - 1/1.15 1/1.15
NZS 3101-95 1.2 1.6 0.85 0.65-0.85 0.75 - -
105
European Code, on the other hand use several values of strength based on the
characteristic strength ckf , which is related to mean compressive strength cmf as
follows; ffcmf ck where f is 1.64s and is usually taken as 8MPa.
3.3.3.1 Modulus of Elasticity, concrete tensile strength and minimum flexural
reinforcement.
Table 3.5 gives the values proposed by these codes for modulus of Elasticity and
minimum reinforcement for flexure.
Table 3.5 Comparison of values of modulus of elasticity modulus of rupture and min flexure reinforcement proposed by various codes (Paultre and Mitchell (2003).
Code Modulus of Elasticity Modulus of rapture
Min flexural reinforcement
ACI318-02
'5.1043.0 ccc fwE MPa
For Normal density
concrete
'4700 cc fE MPa
Modulus of rapture
'6.0 cffr
dbf
dbf
fA w
yw
y
cs
4.1
4
'
min
MPa and mm
EC-02/
MC-90
3.0)10
(22000 cmmc
fE
Uses direct tensile
strength ctmf
3/230.0 ckctm ff MPa
for MPafck 50
)10
1ln(12.2 cmctm
ff MPa
for MPafck 50
dbdbf
fctmA tt
yks 0013.026.0min wher
e tb is mean width of the
concrete zone in tension.
CSA
A23.3-94
5.1' )2300
)(69003200(wc
fE cc
hb
f
fA w
y
cs
'
min 2.0
Slightly more than ACI
NZS 3101-
95
'8.0 cffr dbf
dbf
fA w
yw
y
cs
4.1
4
'
min
106
Due to wide variation in the Codes approaches to assess various structural
properties of HSC, Paultre and Mitchell (2003) have recommended that further
experimental and analytical studies may be initiated to better understand the
behaviour of HSC and to arrive at more rational and internationally acceptable
building code for HSC.
3.4 Mechanical properties of high strength concrete.
The various mechanical properties of HSC, as given in the “ACI Committee 363-
Roeprt on the State of the Art on High Strength Concrete” are discussed as follows;
The compressive strength of concrete has direct effect on the properties of concrete
both in fresh and hardened forms. In fresh concrete properties such as viscosity,
flow ability, workability depends on the compressive strength of concrete, as the
later depends on cement content. The high strength concrete normally involves high
cement content and low water binder ratio, leading to low workability despite of
increase in the compressive strength of concrete. In hardened form many structural
properties like flexural strength, shear strength, creep and shrinkage also depend on
the compressive strength of concrete.
The modulus of elasticity of concrete is frequently expressed as the function of its
compressive strength and is proportional to its square root value. Other properties of
concrete like modulus of rapture, workability, bond strength etc are also directly
related to the compressive strength of concrete.
The tensile splitting strength is usually measured as indirect tensile strength also
called as “Cylinder splitting strength”. The indirect tensile strength is usually 10% of
the compressive strength of concrete for normal strength concrete, but at higher
strength of concrete, it may be reduced to 5%. Further indirect tensile strength was
found as 70% of the flexural strength at 28 days. The test data on the fatigue
strength of HSC is very limited; however it has been shown that the fatigue stress of
HSC is almost the same NSC.
107
The unit weight of HSC is slightly greater than the lower strength concrete made with
same material. The thermal properties of HSC fall within the range prescribed for the
NSC. The strength gain of HSC is relatively more than NSC at early ages mainly due
to 1) an increase in the internal curing temperature in the concrete cylinders due to a
higher heat of hydration and (2) shorter distance between hydrated particles in high-
strength concrete due to low water-cement ratio.
The freeze and thaw resistance of HSC has been observed as greater than NSC
due to reduced water cement ratio in concrete.
The shrinkage of HSC has been observed as same as NSC in most of the cases,
however the shrinkage of HSC having high range water reducers, has been reduced
for HSC.
The creep and shrinkage properties of High Strength Concrete were studied at
Magnel Laboratory (Taerwe, 1995); it was found that with the increase of concrete
strength, the final value of creep is overestimated for both normal and high strength
concrete. The creep rate proposed by the CEB-FIP Model Code 1990 can better
represent the creep development of high strength concrete than normal strength
concrete. Due to high heat of hydration owning to relatively more cement content,
high strength concrete elements might show some early age thermal cracking and
hence proper curing is required for HSC.
3.5 Stress strain behaviour and shear strength of HSC.
The stress strain behaviour of high strength concrete significantly varies from that of
the normal strength concrete. The stress strain relationship for various values of
compressive strengths as reported by Collins and Mitchells (1996) given in Figure
3.3 (a) , shows that the downwards sloping branch of the curve gets steeper with the
increase in the compressive strength of concrete, which means that HSC is relatively
brittle material.
108
Ahmad et al. (1995) worked on the shear critical HSRC beams and showed that the
post peak branch of the mid span load deflection curve in the shear critical high
strength concrete beams is relatively steeper than that of the NSRC beams as
shown in Figure 3.3(b).
(Adapted from Collins and Mitchell, (1997).
NNN3 (41MPa) and NNH3(104Mpa)
( Adopted from Ahmad et al,1995)
Figure 3.3 Variation of compressive stress-strain curves with increasing compressive strength
109
The main variations in these curves are illustrated as follows;
1. The stress strain curve is getting more linear with the increase of compressive
strength of concrete.
2. The strain at maximum stress for HSC is relatively more as compared to
NSC.
3. The descending part of the curve after the peak is steeper in case of HSC.
The brittle behavior of HSC can be explained as follows; (ACI report 363R-27,1997)
1. The difference in rigidity between cement paste and aggregates leads to
concentration of stresses at the contact zones of the two ingredient of HSC
and at certain overall stress level, a distributed micro crack pattern forms at
the contact points.
2. When the overall stress level further increases, a substantial part of the
increased energy is used in developing a clearer crack pattern. The stress
strain curve at this stage tends to deviate from linear elastic line.
3. With further increase of stresses, the micro crack pattern will provide an
efficient re-distribution of the stress and a tough and brittle failure is obtained.
As the compressive strength of concrete increases, the difference in the strength of
aggregates and cements matrix decreases and the HSC behaves like a
homogenous material and the stress-strain curve becomes more linear as compared
to NSC. This relatively uniform re-distribution of stresses leads to sudden failure of
HSC.
The shear of high strength concrete RC structure is comprised of two main parts
i. Vc - The nominal concrete contribution includes, in an undefined way, the
contributions of the still un-cracked concrete at the head of a hypothetical
diagonal crack, the resistance provided by aggregate interlock along the
110
diagonal crack face, and the dowel resistance provided by the main
reinforcing steel.
ii. Vs- The shear resistance provided by the stirrups.
As already explained, the failure of HSC in uni-axial compression is sudden as
compared to NSC as the failure surface is relatively smooth along the crack plane. In
bi-axial compression, the diagonal compression from the load point and support is
combined with the diagonal tension in the perpendicular direction. In high strength
concrete, the diagonal tension cracks are expected to have smooth plane, due to
very high strength of concrete matrix. This leads to crushing of aggregates as shown
in Figure 3.4, and as result the aggregate interlocking is not playing significant role in
resisting the shear. There is a general consensus amongst the researchers, that the
aggregates interlocking decreases, when the compressive strength of RC concrete
increases, due to peculiar failure of the HSC (Cladera and Mari,2005). The decrease
of aggregates interlocking may lead to reduction in the shear strength of HSC.
Figure 3.4 Crack in high-strength concrete. The crack goes through the aggregates (Cladera and Mari,2005)
The research of Ahmed et al. (1986) have shown that the current design methods
are not conservative for HSC particularly for larger shear to depth ratio (Slender
beams a/d >3.0) and relatively low longitudinal steel ratio.
Kaufman and Ramirez (1988), demonstrated the benefits of using HSC in pre-
stressed beams, where the strength of the diagonal truss members is increased due
111
to increase in the compressive strength of concrete in case of HSC. This in turn
leads to increased efficiency of web reinforcement, as more stirrups are mobilized.
Further research is however required to identify the minimum web reinforcement for
a particular level of HSC, to avoid brittle and sudden failure.
The research data on the shear strength of high strength concrete beams is limited
particularly for the compressive strength of 70 MPa and more. Following four
challenges are pointed by Duthinh and Carino (1996), while dealing with the
problem of shear design of high strength concrete.
1. The current provision and empirical equations used for the shear design by
various codes are mostly based on the research carried with concrete of 40
MPa or less. Again these equations used in various design codes for shear
strength of concrete, are at times complex and difficult to understand. Hence
there is a need to further simplify these equations for better understanding
and easy application by the designers.
2. The minimum shear reinforcement for HSC beams needs to be rationalized to
avoid brittle failure of the beams and adequate control of the shear cracks.
3. The relatively little role of the aggregate interlocking in HSC due to stronger
matrix, the shear friction of HSC can be expected 30-35% less than the NSC.
4. The compression capacity of the cracked web is reduced due to transverse
tension, which is sometimes referred to as “Softening of concrete”, which
depends on the concrete strength.
The test data shows that the aggregate interlocking decreases with the increase of
compressive strength of concrete. The data from tests by Ahmad and Khaloo (1991)
showed that the provisions of ACI-318 for shear design may be un-conservative for
112
high strength concrete beams for higher shear span to depth ratios and relatively low
steel.
Duthin and Carino (1996) further pointed out that most of the current shear design
techniques either do not acknowledge the loss in the aggregate interlock mechanism
in high strength concrete or simply do not account for the influence of adding shear
reinforcement to other shear transfer mechanisms. Johnson and Ramirez (1989),
reported that for a constant low shear reinforcement, the overall reserve shear
strength after diagonal cracking diminishes with increase in the compressive
strength of concrete.
S. Sarkar, et al (1999) conducted an interesting research to study the contribution of
the compression zone concrete υcz, aggregate interlocking υa and dowel action of
the longitudinal steel υd , to the shear capacity for high strength reinforced concrete
( HSRC) beams without transverse reinforcement. The research was carried out on
beams with compressive strength ranging from 40Mpa to 110Mpa. The following
inferences were made;
1. The role of aggregate interlocking mechanism at higher concrete strengths is
slightly enhanced. In addition, this mechanism had a predominant influence
on the ultimate load carried by the beam. In other words, the contribution of
this mechanism to the total shear strength carried by the beam was around
42% for higher concrete strength beams with compressive strength of 110
Mpa as compared to 34% for NSC of 40 Mpa. However this increase in the
shear contribution due to aggregates interlocking is much less than the
increase in the compressive strength of concrete.
2. The contribution of the compression zone concrete remained fairly constant at
higher compressive strength with very little increase from 13% to 17% with
the increase of compressive strength from 40Mpa to 110Mpa.
113
3. The contribution of dowel action remained the main part in the absence of the
aggregate interlocking but it decreased from 53% to 43% with the increase of
concrete strength from 40 Mpa to 110 MPa.
The earlier research by Kumar.D (1992) reported the following share of compression
zone, dowel action and aggregate interlocking in the shear strength of beams of
NSC, without web reinforcement;
Shear in compression zone Vc 20 to 40% Shear from dowel action Vd 15 to 25% Shear from aggregate interlock Va 35 to 50% Sherwood et al. (2006) has reported that at least 60% of the vertical shear is carried
by aggregate interlock at flexural cracks, with the remaining proportion being carried
in the compression zone and through dowel forces. Hence the observations by S.
Sarkar, et al (1999), need further experimental validation.
The contribution of dowel action to shear resistance is a function of the amount of
concrete cover beneath the longitudinal bars and the degree to which vertical
displacements of those bars at the inclined crack are restrained by transverse
reinforcement. Typically, little dowel action can be provided by reinforcement that is
near the tension face of a member without transverse reinforcement because that
action is then limited by the tensile strength of concrete.
Thus the aggregates interlocking appear to be a major share in resisting the vertical
shear of RC beams without web reinforcement. Experimental studies reveal that
aggregate interlock is directly related to some concrete properties such as the
tensile strength, the maximum aggregate size and the shape of the aggregate (river
or crushed aggregate). For this reason, the effect of aggregate interlock on the shear
capacity of R.C. members becomes less effective when shear cracks widen.
114
Consequently, the dowel action plays a major role in preventing shear failure of
beams without web reinforcement.
S. Sarkar, et al (1999) used the Zsutty’s equation (1968) Eq-5.28, to develop an
expression for the shear capacity of HSC beams without transverse reinforcement
for the concrete strength range of 40Mpa to 110Mpa on the basis of regression
analysis. They classified the data into two categories. (i), beams having a/d≤ 2 and
(ii) beams having a/d >2.0. The general form of equation for shear stress of beams
without web reinforcement is given as;
ncn adf /.. …………………………………. (3.6)
Where
; is polynomial regression constant,
cf ; is the specified compressive strength of concrete,
; is the longitudinal steel ratio,
d ; is effective depth of beam,
a ; is shear span and
n ; The polynomial exponent.
The following two equations were developed on the basis of existing data base of
shear stress of beams without web reinforcement,
For beams having a/d≤2 66.0/..13.4 adf cn ……………… (3.7)
For beams having a/d>2 55.0/..05.3 adfcn ………….. (3.8)
Bazant and Kim (1984) proposed a very reliable expression for computing the shear
strength of RC beams, without transverse reinforcement which is given as ;
2/56/52/13/1 ).(9.206.83.0
d
af cuc ………………….. ( 3.9)
115
Where
)
251/1
ad
d
; is a function taking into account the size effect of aggregates.
ad ; Max aggregates sizes
On the basis of the above equation, Russo,G .et.al (2004), proposed the following
expression for shear strength of HSC concrete beams without transverse
reinforcement.
33.296.0
38.091.02/146.0 )/(2.0.97.0 dafff
lyccuc …………. (3.10)
lyf ; yield stress of longitudinal steel
They further proposed the following expression for the shear strength of HSRC
beams with transverse reinforcement using the above expression.
33.296.0
38.091.02/146.0 )/(2.0.97.0 dafff
lyccuc + yvvb fI 75.1 (3.11)
v ; Steel ratio of web reinforcement.
yvf ; Yield stress of web reinforcement.
The factor Ib is given by the equation:
33.296.038.0
91.02/146.0
2/146.0
)/(2.0.97.0
.97.0
dafff
fI
lycc
c
b
………….. (3.12)
To check whether the shear failure is due to beam action or arch action, the author
further proposed a critical value as
116
(a/d)c = 05.0
41.019.0
57.0c
ly
f
f
………………………………………………..(3.13)
Hence bI = 0.57 which means that for
i). a/d < (a/d)c bI <0.57, arch action prevails
ii). a/d > (a/d)c bI >0.57,beam action prevails
The proposed expression gave least coefficient of variation when compared with the
provisions of ACI-318, Euro code and CEB/FIB model on the basis of data of 116
beams already tested.
Ko et al (2001) researched the plastic rotation capacity of reinforced high strength
concrete beams in the range of 60-80 MPa and proposed the following new equation
for ultimate compressive strain of extreme fibers as the theoretical estimate of
003.0cu underestimates the test results.
)(00054.0)(
144.1003.0
,
2,
ccu f …………………… (3.14)
Mohiuddin A. Khan et al. (2000) applied the concept of Fracturing Truss Model
( FTM ) rather than MCFT and Strut and Tie Model ( STM), to HSC concrete beams
and compared the test results with the theoretical results. They observed that the
assumption of FTM is more consistent with actual beam failure as compared to
MCFT. They also examined the provisions of ACI-318 and recommended to include
an alternate Fracturing truss model (FTM) in the future codes. They also observed
that the concretes having different tensile stresses have significant effect on the
shear capacity of beam, concrete stresses and steel strain. Hence biaxial tests must
be conducted rather than split cylinder test for determining the exact concrete tensile
stresses.
Cladera and Mari (2004), developed an Artificial Neural Network (ANN) to predict the
shear strength of RC beams, using a large database of experimental results and
made the following important conclusions:
117
1. The influence of the amount of web reinforcement on the shear strength RC
beams is not linearly proportional to the amount of web reinforcement. i.e. the
shear strength due to increase in shear reinforcement is not increasing in the
same ratio. The effectiveness of stirrups decreases with their increase within the
range of transverse steel ratio of 0.33% to 3.57%. The more the stirrups the less
effective they are.
2. Due to increase in size at low shear reinforcement, the shear strength has been
reduced by 25% when the size of beam has been increased from 250 to 750mm.
3. The influence of compressive strength of concrete also changes with the amount
of web reinforcement.
4. AASHTO LRFD design equation gives relatively good results as compared with
the ACI-318 and Eurocode-2
They further proposed a simplified shear design methods and compared the tests
results with the model.
Hamad and Najar (2001) studied the role of transverse reinforcement in confining the
tension lap in high strength concrete beams. They reported that the flexure failure in
the HSRC beams with no web reinforcement is more abrupt, whereas in beams with
web reinforcement, the failure was relatively ductile. The flexure cracks widths have
been reduced in the later case and this leads to ductile failure of the beams.
Shehta et al. (2003) developed theoretical models for the minimum flexural, shear
and torsional for RC beams made with different compressive strengths of concrete.
They reported that due to little test results available, there is great difference in the
minimum values proposed by different Codes and hence more experimental
research has been recommended by them.
Cladera and Mari (2005) worked on the HSRC beams failing in shear and reported a
very brittle failure of the HSRC beams without shear reinforcement. The failure was
observed as more sudden with further increase in the strength of concrete. However,
the failure shear strength of beams was observed to increase with the increase in
118
the compressive strength for such beams. They also proposed an expression for
minimum web reinforcement of HSRC beams to avoid brittle failure of the beams,
which is given as follows;
Mpaf
sbfA
y
wmctw 5.7
,min,
(3.15)
Here mctf , stands for tensile strength of HSC, which is given as;
3,
2
30.0 cmct ff MPa if fc < 60MPa fc is specified compressive strength of
concrete
2, 58.0 cmct ff MPa if if fc > 60MPa (3.16)
They also concluded that the limitation of 2% longitudinal steel for HSRC beams with
web reinforcement is also not justified.
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Summary The definition of HSC has been changing from time to time and region to region.
Preparation of HSC requires special selection of material like aggregates, cement,
admixtures and high range water reducers. HSC also requires special
considerations in mixing, transportation, placing and curing.
The compressive stress strain curves of HSC shows that it is a relatively brittle
material as compared to NSC. The smooth plane of cracks in HSC reduces the
aggregates interlocking which leads to reduction of share of aggregates interlocking
in shear strength of concrete as compared to NSC.
Four challenges are faced in the shear design HSC members in the contemporary
research. Firstly; the available research data base is limited to research carried out
with the RC beams with compressive strength of 40MPa or less, secondly there no
consensus on the minimum shear reinforcement to avoid brittle failure of HSC
reinforced concrete beams, thirdly the decrease in the aggregate interlocking in
resisting the shear with the increase of compressive strength of concrete and finally
decrease in the compression strength of cracked web due to transverse tension,
also called softening of concrete and its measurement.
The literature review of work on the shear strength of HSRC beams thus identifies
the following research areas;
1. The relationship between aggregate interlocking share and compressive
strength of concrete in shear strength of HSRC beams. A reduction factor
may be explored to correlate the aggregates interlocking share of HSC and
compressive strength of concrete.
2. The level of minimum web reinforcement for HSRC beams, to avoid sudden
and brittle failure of beams may be identified.
120
3. Parametric study for the shear strength of HSRC beams, incorporating
important parameters affecting the shear strength of HSRC beams is
required, supported by the experimental work to develop more rational
equations to determine shear strength of HSRC beams.
121
Chapter No. 4
Shear Strength prediction of disturbed region (D-region) in
reinforced concrete.
Chapter Introduction: This chapter explains the basic concept of the beam (B) region and Disturbed (D) region. The need for special attention to the design of disturbed region has been discussed. The basic philosophy of Strut and Tie Model (STM) for the design of disturbed region has been given in quite details. It is followed by the explanation of basic design principles and steps for the design of D-region. At the end some STM have been given for various types of disturbed regions in concrete structures.
4.1 The basic concept of beam and disturbed region:
Structures are sometimes classified as either B- (Beam or Bernoulli) Regions or D-
(Disturbed or Discontinuity) Regions, for selection of appropriate design procedure.
B-Regions are parts of a structure in which Bernoulli's hypothesis of straight-line
strain profiles applies. D-Regions, on the other hand, are parts of a structure with a
complex variation in strain. D-Regions include portions near abrupt changes in
geometry (geometrical discontinuities) or concentrated forces (statical
discontinuities), such as deep beams, corbel, pile caps, dapped ended beams,
brackets etc.
Figure 4.1 and Figure 4.2 show examples of the division between B-Regions and D-
Regions in building and bridge structures, respectively. In these figures, the un-
shaded area with a notation B indicates B-Region, and the shaded area with a
notation D is used to indicate D-Region. The notations h1, h2, h3 ... are used to
denote the depth of structural members. The notations b1 and b2 denote the flange
width of structural members.
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Figure 4.1: Example of B & D-Regions in a Common Building Structure (Schlaich et al
,1987)
Figure 4.2: Example of B& D-Regions in a Common Bridge Structure. (Schlaich. et al. 1987)
123
Figure 4.3 Typical D regions shown as shaded areas, adapted from Schlaich et al. (1987).
124
4.2 Basic design principles for shear design of disturbed region
Most design practices for B-Regions are based on conventional beam theory or
flexural theory, while the design for shear is based on the well-known parallel chord
truss analogy. In contrast, the most familiar types of D-Regions, such as deep
beams, corbels, beam-column joints, and pile caps, are currently still designed by
empirical approaches or by using common detailing practices. For most other types
of D-Regions, code provisions provide little guidance to designers. Presently the
following four approaches are used for the design of disturbed region in RC
structures;
1. ACI equation and detailing methods
2. Truss analogy
3. AASHTO-LRFD Standards
4. Strut and Tie Model (STM).
The Strut-and-Tie Method (STM) is an emerging methodology for the design of all
types of D-Regions in structural concrete. It is worth noting that although the STM is
equally applicable to both B- and D-Region problems; it is not practical to apply the
method to B-Region problems. The conventional beam theory for flexure and parallel
chord truss analogy for shear are recommended for those designs. The idea of the
strut-and-tie method came from the truss analogy method introduced independently
by Ritter in the early 1900 (Riltter, 1899) for shear design of B-Regions.
This method employs the so-called truss model as its design basis. The model was
used to idealize the flow of force in a cracked concrete beam. In parallel with the
increasing availability of experimental results and the development of limit analysis
in plasticity theory, the truss analogy method has been validated and improved
considerably in the form of full member or sectional design procedures. The truss
model has also been used as the design basis for torsion.
125
4.3 Use of Strut and Tie Model (STM) as a design tool for structural components.
Strut-and-Tie (STM) is a unified approach that considers all load effects (M, N, V,
and T for moment, axial force, shear force and torsion, respectively) simultaneously.
The Strut-and-Tie model (STM) approach evolved as one of the most useful design
methods for shear critical structures and for other disturbed regions in concrete
structures. The model provides a rational approach by representing a complex
structural member with an appropriate simplified truss model. There is no single,
unique STM for most design situations encountered. There are, however, some
techniques and rules which help the designer to develop an appropriate model.
However the selection of appropriate truss model is an uphill task. The STM more
accurately predicts the shear strength of the beams whereas a/d is less than 2.5. For
slender beams having a/d>2.5, a sectional model approach, that also includes Vc
caused by tensile stresses in the concrete is more appropriate.
The basic assumptions for application of STM to disturbed region are (Fu, 2005)
i. STM is a strength design method and the serviceability should also be
checked .i.e. stresses in any part of the structure must not exceed the
allowable stresses.
ii. Equilibrium of internal and external forces must be maintained
iii. Tension in concrete is neglected and usually the concrete struts are assumed
to take the compressive force and the steel the tension.
iv. Forces in struts and ties are uni-axial. Planar and uni-axial analysis is done
due to specific geometry of the structures.
v. External forces apply at nodes, like trusses.
vi. Pre-stressing is treated as a load, applied at the nodes of the truss.
vii. Detailing for adequate anchorage is provided, to ensure proper anchorage of
the steel bars.
4.4 Steps involved in the design of D-region using STM.
The joint committee report of ACI and ASCE on Shear and Torsion ( ACI-ASCE-
445,99) has given the following steps for the design of disturbed region in RC
structures
126
4.4.1 Choosing a STM for the structure (D-region)
The first step in STM is to visualize the flow of forces with compressive struts
representing the flow of concentrated compressive stresses in the concrete and
tension ties representing the steel. The struts and ties are essential for the
equilibrium. Schlaich et al (1987) have suggested that the strut and tie model may be
selected on the basis of elastic analysis, so that the angles of compression
diagonals are + 15 degree of the angle of resultant. The compressive struts are
bulging between the load points and supports, causing a transverse tension
idealized by tension ties. However for simplification, the compressive struts are
idealized as straight line members following the centerline of the compressive struts
as shown in Figure 4.4. Hence most of the codes require a minimum additional
reinforcement in the struts and ties to control cracking.
Figure 4.4 Crack control reinforcement required with assumed straight-line compressive struts, proposed by Schlaich et al. (1987) ( ACI-ASCE-445R-99).
4.4.2 Checking compressive stresses in struts
In the shear design by STM, it is necessary to check that crushing of compressive
struts do not occur. Bregmeister et al. (1991) suggested the stress limit of for the
unconfined bearing plate node, the factored bearing strength as;
c
bc
fA
A
ffc 5.0))(
25.15.0( ( 4.1)
127
Ramirez (1990) proposed the stress limits listed in Table 4.1 Table 4.1 Effective stress level in the concrete struts (Ramirez, 1990)
Effective Stress level
Concrete Struts Proposed by
0.80fc’ Undisturbed and uni-axial state of compressive stress that may exist for prismatic strut
Schlaich et al. (1987)
0.60fc’ Tensile strains and/or reinforcement perpendicular to the axis of the strut may cause
cracking parallel to the strut with normal crack width Schlaich et al. (1987)
0.51fc’
For skew cracks with extraordinary crack width. Skew cracks would be expected if modeling of the struts departed significantly from the theory of elasticity’s flow of internal forces
Schlaich et al. (1987)
0.34fc’ Moderately confined diagonal struts going directly from point load to support with shear
span-to-depth ratio less than 2.0 Schlaich et al. (1987)
0.75fc’ Struts forming arch mechanism Alshegeir and
Ramirez (1990)
0.50fc’ Arch members in pre-stressed beams and fan compression members Alshegeir and
Ramirez (1990)
0.95fc’ Undisturbed and highly stressed compression struts Alshegeir and
Ramirez (1990)
v2fc’ Uncracked uniaxially stressed struts or fields Alshegeir and
Ramirez (1990)
v2(0.80) fc’ Struts cracked longitudinally in bulging compression fields with transverse
reinforcement Alshegeir and Ramirez (1990)
v2(0.65) fc’ Struts cracked longitudinally in bulging compression fields without transverse
reinforcement Alshegeir and Ramirez (1990)
v2(0.60) fc’ Struts in cracked zone with transverse tensions from transverse reinforcement MacGregor (1997)
v2(0.30) fc’ Severely cracked webs of slender beams with q = 30 degrees MacGregor (1997)
v2(0.30) fc’ Severely cracked webs of slender beams with q = 45 degrees MacGregor (1997)
Note: v2 = 0.5 + 1.25/Öf ¢c in MPa after Bergmeister et al. (1991).
Jisra et al.( 1991) recommended an effective concrete strut stress level of 0.8 fc’
4.4.3 Design of nodal zones
The types of nodal zones and dimensions of struts have been shown in the following
Figure 4.5. These nodal zones are defined as
CCT: Nodal zone bounded by compression Struts and one tension ties
CCC: Nodal zone bounded by compression struts only.
CTT: Nodal zone bounded by compression struts and tension ties in two or more
directions.
TTT: Nodal zone bounded by Tension Ties only.
128
Figure 4.5 Classifications of Nodes (Ref: ACI- 318-06)
The Canadian standard (“Design” 1984) limits the concrete stress in the nodal zones
to the following values:
- 0.85φfc’ in node regions bounded by compressive struts and bearing areas (CCC
nodes);
- 0.75f φfc’ in node regions anchoring a tension tie in only one direction (CCT
nodes); and
- 0. 60φfc’ in node regions anchoring tension ties in more than one direction (CTT
nodes), where f is the capacity- reduction factor for bearing.
Bergmeister et al. (1991) proposed several equations of effective concrete strength
for various kinds of nodes, including reinforcement, unconfined nodes with bearing
plates, and tri-axially confined nodes.
129
4.4.4 Design of tension ties
The area of reinforcement for tension ties are determined as
usepspsyst NffAfA ))(( (4.2)
Where Ast = area of reinforcing bars; Aps = area of pre-stressed reinforcement; fy = yield strength of reinforcing bars; fps = stress in pre-stressed reinforcement at ultimate; φ= capacity-reduction factor for axial tension (0.9); and fse = stress in pre-stressing steel after all losses.
4.4.5 Anchorage of tension ties
The anchorage of tension ties must be sufficient to develop the required stress in
reinforcement. The anchorage zone must be spread over large area so that there is
no crushing of the nodal zone and the embedment of reinforcement must be
sufficient.
Some the basic models for different disturbed regions are as given in Figure 4.6 to
Figure 4.9
130
a. Column with double corbel
b. Column with single corbel
c. Beams with double ledged supports
d. Beam with single ledge support.
Figure 4.6 Strut-and-tie model idealizations for brackets, ledges, and corbels (Cook and Mitchell 1988).
131
Figure 4.6 Proposed STM for Deep beams under applied external load (Schlaich et al.1987)
Figure 4.7 Proposed STM for one way corbel under applied external load (Fu,2005).
132
Figure 4.8 Proposed STM for two way corbel under applied external load (FU,2005).
Figure 4.9 Proposed STM for dapped beam end under applied external load (Fu,2005).
133
Figure 4.10 Proposed STM for pile cap under applied external load (Fu,2005).
4.5 Some latest research on the shear design of disturbed region with STM:
Shyh-Jiann Hwang and Hung-Jen Lee (2002) developed a simple procedure on the
basis softened STM for predicting the shear strength of discontinuity region failing in
diagonal compression. A simplified equation was proposed which incorporates the
shear resisting mechanism of softened STM. The experimental values of 449
disturbed regions like deep beams, corbels, squat walls and beam column joints
were compared with the proposed model and the coefficient of variation was
observed as only 6%.
Foster and Adnan (2002) introduced an efficiency factor for softening of concrete
and other strength reduction effects such as that of transverse tension strains. They
analyzed three basic models such as (i) Models based on the concrete strength (ii)
Multi-parameter models and (iii) Model based loosely on Modified Compression
Field Theory ( MCFT). In this study, it was observed that models based on concrete
strength did not correlate well with the experimental data of non flexural members.
The multi-parameter model of Batchelor et al.(1984) and Chen also did not correlate
well with the experimental data. The model based on strut angle (Shear span to
134
depth ratio) gave the best prediction of the efficiency factor to calculate the capacity
of concrete struts. They further reported that the influence of boundary effects is of
considerable importance in the determination of the mode of failure of non flexural
members.
Wight (2001) explained the key features of ACI-318-02, building Code for the use of
Strut and Tie Model by the designer, however later the methods was incorporated in
ACI318-06, as an alternate design method for some structural components.
Hamed and M.Salem (2004) introduced the concept of micro truss model for the
design and checking the non linear response of concrete structures. The micro truss
model is formulated with the simple stiffness method, where careful non linear
algorithm is applied. The micro truss model is generalized form of STM which can be
developed without much experience. It was observed in this research that the
reinforcement worked out on the basis of Micro Truss model was less than the
general STM model. The Schalaich Model of STM has been based on equilibrium of
STM and hence it is lower bound solution. Therefore over estimation of
reinforcement may exist. Again contribution of the non cracked concrete (tension
stiffening) and contribution of cracked concrete (tension softening) is neglected by
Schalaich, as reported by the researchers.
Tan et al. (2001), applied the STM for the design of pre-stressed deep beams by
proposing a simple and direct model and evaluated the results of 39 deep beams
with the proposed model. They reported that the proposed STM for pre-stressed
deep beams has given consistent and accurate prediction of small and large pre-
stressed beams for different geometrical properties, various pre-stressing force and
different web reinforcement.
Tan (2004) worked on the use of STM for the design of non prismatic members
including beams with recesses and geometric discontinuities. He reported that STM
provides a simple and straightforward solution to otherwise a complicated problem.
135
Tjen et al. (2002) developed a software as Computer Aided Strut and Tie ( CAST)
for generating the STM for a particular problem, which involves all aspects of the
problem like definition of D-region, selection of STM, truss analysis, members
definitions, and design summary. This tools is being used and improved with the
feedback from users and researchers.
Vollum and Tay ( 2001), tried to estimate the effect of node dimensions on the shear
strength of short span beam as predicted under STM by testing 12 short span
beams of a/d ratio as 1.6. They also studied the influence of the concrete
compressive strength on the shear strength of beams given by the relevant
provisions of EC2, and Collins et al (MC90). From their research, they made the
following important inferences;
- STM has overestimated the influence of node dimensions on the shear strength
of RC short beams.
- STM has failed to predict the observed failure modes of the beams.
- STM could not predict the influence of concrete strength on the shear strength of
the RC short beams, if the provisions of EC2 and MC90 are used.
- MC90 has been observed as more conservative for shear design of RC short
beams.
- They have concluded that the current empirical design equations proposed by
BS8110 and EC2 are more practical for design of short span RC beams than
simple STM. The same is also true for the shear design of Beam Column
connections/. However the authors have suggested that STM can better assist in
detailing of reinforcement in the RC beams.
According to ACI 318-06, it shall be permitted to design structural concrete members
or D-regions in such members, by modeling the member or region as an idealized
truss. The truss model shall contain struts, ties, and nodes. The truss model shall be
capable of transferring all factored loads to the supports or adjacent B-regions. Strut-
and-tie models represent strength limit states and designers should also comply with
the requirements for serviceability in the code. Deflections of deep beams or similar
136
members can be estimated using an elastic analysis to analyze the strut-and-tie
model. Besides ACI 318-06, Strut and Tie Model has been included in the following
building codes as well.
a. AASHTO-LRFD Bridge Design Specifications 3.3
b. CSA A23.3-94 (Canadian Code)
c. NZS 3101:1995 (New Zealand Code)
d. FIB Recommendations (Eurocode-2)
Brown (2005) worked on the use of STM for the shear design in reinforced concrete
in his doctoral studies and made the following important conclusions:
i. When the STM provisions of the ACI-318 and AASHTO RLFD of Bridge
design are applied to the test data, they provide less conservative results.
ii. Beams subjected to uniform loads exhibit increased shear strength
compared with beams with concentrated loads.
iii. Current ACI 318 provisions for sectional design result in un-conservative
estimates of strength for beams with concentrated loads between 2 and 6
times the effective depth from the support.
iv. Shear span-to-depth ratio has a large effect on shear strength.
v. The strength of beams with shear span-to-depth ratios less than two are
better represented by a direct strut mechanism.
vi. Nearly parallel shear cracks were observed just prior to failure of the beam
specimens.
vii. Failure occurs due to crushing of the strut at node-strut interfaces.
137
He has made a number of recommendations for further research in the application of
STM to shear design of beams.
The Strut and Tie Model (STM) will continue as design option for both disturbed and
beam regions in concrete structures. However the experience of designers to use
the STM to various design problems and modern research shall play an active role
in standardizing the design techniques on the basis of STM. Extensive research is
therefore required to generalize the STM as an equally acceptable design tool for
concrete structures, particularly for the disturbed regions in concrete structures.
138
Chapter No. 5
Provisions of International Building and Bridges Codes for
Design shear Reinforcement of Normal and High Strength
Concrete Beams.
Chapter Introduction: This chapter gives a brief summary of the provision in the selected international building and bridges codes for shear design of concrete beams. Certain research on the comparison of building codes for shear design has been given with some important findings.
The shear design of the normal and high strength concrete beams is usually done by
adopting the provisions of different codes based on various rationales. These
provisions are expressed in the form of empirical equations. Some of the most
commonly used design equations for the shear design of RC structures are as
follows;
i. British Standards (BS-8110)
ii. ACI Code 318 (American Concrete Institute)
iii. Canadian Standards for design of Concrete structures. CSA A-
23.3-94
iv. AASHTO LRFD (Load Reduction Factor Design) Bridge Design
Specifications -2004
v. European Code EC2-2003.
vi. Empirical methods for beams without shear reinforcement.
The design equations and relevant code provisions are illustrated in the following
sections.
139
5.1 British Standards (BS-8110)
A semi-empirical approach was developed by Regan in 1967 and used in CP
110:1972, and also used in BS 8110. According to BS-8110, the characteristics
design equation for shear capacity is given as (BS, 2005).
4/13/121 400
100.79.0
dbd
AKKR s
mLW
c (Ref: BS 3.4.5.4, Table 3.8) ……. (5.1)
k1 is the enhancement factor for support compression, and is conservatively taken as 1, (Ref. BS 3.4.5.8)
0.125/ 3/12 cufK (Ref. BS 3.4.5.4, Table 3.8)
m = 1.25 (Ref. BS 2.4.4.1)
sA is area of tensile steel
0.50≤
bd
As100 ≤ 3.0 (Ref BS 3.4.5.4, Table 3.8)
d
400≥ 1.0 and fcu ≤ 40 N/mm2 (for calculation purpose only). (BS 3.4.5.4, Table 3.8).
The Eq. 5.1 is subject to the following conditions.
If ≤ c provide minimum stirrups defined by yvv
sv
f
b
S
A
95.0
4.0 (Ref. BS 3.4.5.3)
If c +40 ≤ ≤ max provide shear steel as yv
c
v
sv
f
b
S
A
95.0
)( (Ref. BS 3.4.5.3)
If ≥ max , a shear failure is declared. (Ref. BS 3.4.5.2, 3.4.5.12)
5.2 European Code EC2-2003.
Shear is dealt with by clause 4.3.2 and 4.3.4 of Eurocode EC2. The following four
cases are given in the code for shear design of RC structures ( Eurocode, 2003).
i. Members without shear reinforcement
ii. Strength of members with shear reinforcement
iii. Maximum shear strength that can be carried by a member.
iv. Behaviour of section close to supports.
140
1RDV : The shear strength for members without shear reinforcement
2RDV : The upper limit of the shear strength to prevent web crushing failures.
3RDV : The shear strength for members with shear reinforcement.
dbkV wlrdRD )402.1(1 ( SI units) (5.2)
)51(,5.2
x
d an enhancement factor can be applied if the member is loaded by a
concentrated load situated at a distance dx 5.2 from the face of the support
RD : Basic design shear strength = 05.025.0 ckf
05.0ckf : Lower 5% fractile characteristic tensile strength= ctmf7.0
ctmf : mean value of the tensile concrete strength=3/2)(3.0 ckf
0.1)1000/6.1( dk
ckf = characteristic cylinder compressive strength of concrete '9.0 cf
dbA wsll / wb= effective web width, = effective depth
Thus, the above equation can be simplified to the following equation.
dbfkV wlckRD)402.1()(0525.0 3/2
1 (SI units) (5.3)
)9.0(5.01
dbfV wcdRD
(SI units) (5.4)
cdf Factored design strength= 15/ckf
(For analysis purpose cdf = ckf is considered to be appropriate)
: The effectiveness factor 5.0200
7.0 kf c (may taken as 0.6)
141
5.3 ACI Code 318-06 (American Concrete Institute)
The ACI building code 318-06 is no doubt the most widely applied Code for the
shear design of concrete. The nominal shear capacity of reinforced concrete beam
Vn, is given as the sum of Concrete contribution Vc, and contributions of stirrups Vs
.i.e.
Vn = Vc + Vs
For beams without shear reinforcement, the shear capacity is given as;
Reinforced Concrete Members: (limit < 70 MPa)
dbf
Vc wc
6
'
(SI units) (ACI 11.3) (5.5
7
120' db
M
dVfVc w
u
uc
(SI units) (ACI 11.5)
Or dbM
dVfVc w
u
uwc
25009.1 '
(English units) (5.6)
When 0.1.0.5ln/ u
u
M
dVandd
Where '
cf = Compressive strength of concrete
w = Longitudinal steel ratio
wb = Width of the beam web.
uM Factored Moment at the section
dV
M
u
uExpression for the shear span to depth ratio a/d
ACI code limits the value of cf to 100 psi (8.3 MPa) unless the amount of web
reinforcement is increased as per Clause 11.1.2.1 by the ratio.
cf /3.45 ≤ 3.0 (ACI 11.1.2.1)
142
y
wv f
sbA 345.0 (ACI 11.13) (5.7)
For High Strength Concrete (HSC), the minimum shear reinforcement is given as
y
ww
y
cv
y
wv f
sbsb
f
fA
f
sbA 035.1
01.0345.0
'
for 69MPa≤ 'cf (ACI 11.1.2.1) (5.8)
For members subjected to compression, the shear strength is given as;
dbfA
NVc wc
g
u '
200012
( English units) ( ACI 11.3.1.2) (5.9)
Subject to maximum value of
dbfA
NVc wc
g
u '
50015.3
( English units) (ACI 11.3.2.2) (5.10)
Where Nu is ultimate compression force and Ag is gross area of the compression member.
For members subjected to significant axial tension, the shear strength is given as
dbfA
NVc wc
g
u '
50012
( ACI 11.3.2.3) (5.11)
For members having pre-stressed forces;
dbM
dVfVc w
u
puc
7006.0 '
( ACI 11.4.2) (5.12)
143
5.4 Canadian Standards for design of Concrete structures. CSA A- 23.3-94
The General design method of Canadian Code has been based on Modified
Compression Field Theory (MCFT) and applies to concrete up to 81 MPa (16000
psi). The factored shear resistance of non pre-stressed section is given as
sgcgrg VVV (5.13)
vwcccg dbfV ')12(3.1 (CSA 11.18) (5.14)
sdfAV vyvssg /)cot(cot (CSA 11.20) (5.15)
rgV ; Factored shear strength of RC member
;cgV Concrete contribution
sgV ; Steel contribution
=Angle of inclined stirrup to longitudinal axis.
= Factor accounting for shear resistance of cracked concrete.
= Factor accounting for density of concrete; 1 for normal density concrete
= Angle of inclination of diagonal compressive stresses to longitudinal axis member
sc , = material factor for concrete and steel 85.0,60.0, sc
y
wcv f
sbfA '06.0min (CSA 11.1) (5.16)
The method for determining of the values of has already been explained in the
MCFT.
A more simplified design method gives
dbfdbfd
V wcwcc
1.0
100
260 (5.17)
if y
wc
v f
sbfA
06.0 d≥300mm ( CSA 11.7) (5.18)
144
s
dfAs
V yvs i.e. =45 degrees, 90 degrees (CSA 11.20) (5.19)
5.5 AASHTO LRFD (Load Reduction Factor Design) Bridge Design Specifications -
1996.
It is based on MCFT applicable to both non-pre stressed and pre-stressed concrete,
for 16 cf 70 MPa range.
The nominal shear strength of RC beams is given as
Vn = Vc + Vs.
vvcvvcc dbfdbfV '' 25.0083.0 (5.20)
. indicates ability of diagonally cracked concrete to transmit tension.
=2 for d≤400mm
This is 38% more than the Canadian Code.
For culverts less than 600 mm or more fill,
Where 0.1u
u
M
dV
y
vvcv f
sdbfA '083.0
(5.21)
5.6 Empirical methods for beams without shear reinforcement
Zsutty’s (1968) proposed the following empirical equation on the basis of regression
analysis of database of shear strength of 151 beams;
vvcc dba
dfV 3/12.2 for a/d ≥ 2 (5.22)
The empirical equation of Okumura (1986), Niwa (1986) included all the important
parameters for the shear strength of beams without shear reinforcement.
Vc = 0.20 ρ 1/3 /d ¼ (fc′)1/3 (0.75 + 1.40 bw d) fc' in MPa. …………………… (5.23)
145
Table5.1 Summary of Major Code Expressions for the Concrete Contribution to Shear Resistance
Codes or Researcher
Equations Factors
Accounted ACI 318-95 (1995) db
fVc w
c
6
'
(SI units) Simplified Equation
dbfdb
M
dVfVc wc
w
u
u
wc
'' 3.07
120
(SI units)- Detailed Eq.
f'c, (a/d),
AASHTO LRFD 1996 vvcvvcc dbfdbfV '' 25.0083.0
f'c, (d), (a/d), (), agg
Canadian Standard
CSA A23.3-94 (1994)
dbfVc wc'2.0 (SI Units) if sb
f
fA w
y
cv
'06.0 or mmd 300
dbfdbfd
V wcwcc
1.0
1000
260 (SI units) if
sbf
fA w
y
cv
'06.0 , d>300mm
f'c, d, a/d
Eurocode EC2, Part 1 (1990) dbkV wlrdRD )402.1(1 where ,5
5.21
x
d
,0.1)1000/6.1( dk 5.0200/7.0 ylf
f'c, d, a/d,
British Standard BS 8110 (1985)
4/13/121 400
100.79.0
dbd
AKKR s
mLW
c (SI units) for a/d ≥ 2
f'c, d,
Zsutty’s equation (1968)
vvcc dba
dfV 3/12.2 (SI units) for a/d ≥ 2
146
5.7 Research on high strength concrete beams and its comparison with building
codes at different Universities in near past.
A number of researchers have dealt with shear problem in different ways and the
experimental results vary from case to case. Some of research results have been
given in Table 5.2, which shows the findings of these researches on high strength
concrete beams and its comparison with the building codes. The results of Cornel
University tests and Purdue University tests have given some important results for
further verification. The findings of the earlier tests, describing the provisions of ACI
Code for shear strength of HSC as un-conservative by 10-30% are a significant
outcome. Similarly the later findings at Purdue University, requiring increase in the
minimum web reinforcement for compressive strength more than 10,000 psi are also
an important recommendation.
147
Table 5.2 Summary of Research Results conducted at various Universities.
University Researchers Design Parameters Results/Findings.
Cornel University
Elzanti and Nilson ( 1987) Nilson and Slalte ( 1986)
Beams with web Reinf. fc’ = 9100 psi ( 1 beam) fc’ = 5800 psi ( 1 beam)
ACI Code equation is un-conservative by 10% to 30% for high strength and medium strength Concrete
Connecticut University
Maphonde and Frantz ( 1984) Rotter and Russel ( 1990)
Beams without web steel. fc’≤ 6500 psi ( 11 beams) at a/d = 3.6 Beams with web reinf. fc’ ≥ 6000 psi ( 12 beams) fc’= 5000 psi to 10500 psi
The following equation was proposed as compared with Zustty’s equation; Vu = 63.40 ( ρ fc’ d/a) 1/3 - Proposed the following equation on the basis of Regression analysis. Vn= 1.15 √fc’ +90+1.6Avfy/bs - The ACI 318-99 provisions require an increase in the minimum reinforcement for fc’≥ 10,000 psi
Purdue University
Johnson and Rameriz ( 1989)
fc’= 5000 psi to 10500 psi The result justified the ACI code provisions to limit fc’ to 10,000 psi and increase in the min web reinf by fc’/5000
Norwegian University
Torenfeldt Draug Sholt ( 1990)
Beams without web reinf. (Under two point load) fc’ = 7800 to 14200 psi a/d = 2.30 to 4.0 ρ = 1.8 % to 3.20 %
The tensile strength remained constant after fc’= 10,000 psi. The ultimate shear strength increased with increase in the shear span to depth ratio.
Korean Test Kim amd Park (1994)
Beams with web reinf fc’ = 5300psi ( 20 Nos) Beams without web reinf fc’ = 5300psi ( 06 Nos)
- CEB- FIB gave the closer results. - BS 8110 is excessively conservative. - ACI equation is safe for large beams. Effect of ρ and a/d ration has not been significantly improved by concrete strength - The size effect for HSC is same as NSC.
Comparison of ACI and AASTO LRFD method
Shahwy and Batchelor( 1996)
AASHTO type II ( 1989) 20 pre-tensioned beams and compared it with AASHTO LRFD specs.
- AASHTO 1989 based on ACI 1994 provisions give excellent predictions of girders having shear reinf, between 1 and required by the Code ( R) The LRFD code over estimates the shear strength of over designed girders (2R<ρv<3R) and under estimates the under designed girders (0<ρv<R/2> - For ρv = R , ASSHTO 1989, provide better estimates.
University of Japan
Tagaki and Kanoh ( 1991)
-Used high strength steel of 800 MPa ( 116000 psi) as shear reinforcement.
- Use of high strength stirrups is efficient for high strength concrete.- The effectiveness of high strength stirrups in NSC is not ensured yet.
148
5.8 National Cooperative Highway Research Program (USA), for the evaluation of
shear design for simplified shear design method for structural concrete members
(NCHRP, 2005)
National Cooperative Highway Research Program (NCHP) was initiated by
Transportation Research Board (National Academies -USA) in 2005 to develop
“Simplified Shear Design of Structural Concrete Members”. This research was
performed by the University of Illinois at Urbana-Champaign. The objectives of the
research program are given as follows;
i. The simplified provisions must be directly usable without iteration for shear
design and evaluation of the shear capacity of the members.
ii. Must be useful in conducting field evaluation to estimate the failure loads for
the shear cracking by the site Engineers.
iii. Must be easy to explain by the Engineers to others.
iv. Allow reliable and hand based design method.
v. Provide safe and accurate estimates for the RC members in the selected test
database.
vi. The shear reinforcement as a result must be reasonable.
5.8.1 Evaluation of shear design methods using test database
The shear test database of 1359 beams of NSC was studied by National
Cooperative Highway Research Program (NCHRP),USA which consisted of 878 RC
beams and 481 pre-stressed concrete beams. In total of 1359, those containing
shear reinforcement were 160 whereas 718 did not contain shear reinforcement.
The test results Vtest were compared with the code values Vcode for the six different
codes and the mean and coefficient of variation ( CoV) were determined from the
Vtest/Vcode values as tabulated below in Table 5.3
149
Table 5.3 Comparison of test values and Codes values based on shear data base (NCHRP; 2006)
Member Type All RC-beams Pre-stressed beams. Inferences
With or
without AV
No
Both
RC
No Av
RC
With Av
Both PC
No Av
PC
With Av
- CSA and
LRFD has given
best results
particularly
Nos.
Code
1359 878 718 160 481 321 160
ACI
Mean 1.44 1.51 1.54 1.35 1.32 1.38 1.21
CoV 0.371 0.404 0.418 0.277 0.248 0.247 0.221
LRFD
Mean 1.38 1.37 1.39 1.27 1.40 1.44 1.32
CoV 0.262 0.262 0.266 0.224 0.261 0.290 0.154
CSA
Mean 1.31 1.25 1.27 1.19 1.41 1.46 1.31
CoV 0.275 0.274 0.282 0.218 0.261 0.287 0.147
JSCE
Mean 1.51 1.36 1.35 1.38 1.80 1.85 1.70
CoV 0.321 0.28 0.293 0.216 0.292 0.297 0.272
EC2
Mean 1.85 1.75 1.75 1.70 2.06 2.13 1.91
CoV 0.409 0.328 0.328 0.373 0.470 0.43 0.687
DIN
Mean 2.05 2.10 2.10 1.25 2.25 2.59 1.58
CoV 0.395 0.327 0.327 0.267 0.413 0.345 0.357
Av; stands for beams with shear reinforcement.
The following results have been inferred from the analysis of database.
CSA (Canadian Code) and LRFD (AASHTO) have given best results
particularly for Prestressed concrete (PC) beams with shear reinforcement.
ACI provisions are poor predictor of shear for reinforced concrete (RC) &
prestressed concrete (PC) beams with no transverse reinforcement but for
beams with Av, these are reasonably good, hence Av is required where
Vu≥φVC/2
DIN (Denmark Code) is very poor predictor followed by JSCE (Japan Society
of Civil Engineers Code).
150
The following major variations were reported by NCHRP the shear design
provisions of building codes;
i. The amount of shear reinforcement calculated by different codes widely
varies and in some cases, it is even two to three times for the same
section and external forces, when calculated with different codes.
ii. The minimum shear reinforcement also varies substantially from code to
code and in some cases it is double, than the others.
iii. Some codes require providing the minimum shear beyond the section,
where the factored deign shear force is half the design strength of
concrete while others require it when the factored shear exceeds the
design strength of the section.
iv. Again there is great variation in the maximum allowed shear reinforcement
by different codes.
v. The depth effect also called the size effect has been taken into account by
very few codes. In practice, the depth has substantial effect on the shear
capacity
vi. The shear design provisions are based on the experimental data,
equilibrium conditions, and comprehensive behaviour of the model for
capacity.
5.8.2 Simplified shear design method proposed by NCHRP
As a result of the project, an alternative and simplified shear design method was
proposed to overcome the limitations of the LRFD sectional design model. The
following simplified provisions were recommended by the project.
1. The web shear cracking Vcw was simplified as
pvvpcccw VdbffV )30.09.1( ' ( in English System) (5.24)
151
pcf ; Compressive stress in concrete after all pre-stress losses have occurred either at
centroid of the cross-section resisting live load or at the junction of the web and flange when
the centroid lies in the flange.
Vp is the pre-stress force applied to the section.
2. Flexure-shear cracking strength is given as
pcri
dvvcci VM
MVVdbfV
max
'632.0 (5.25)
Vd = shear carried by dowel action; shear force at section due to un-factored dead load.
Vi =factored shear force at section due to externally applied loads occurring
simultaneously with Mmax
Mcr = cracking moment.
3. Contribution of shear reinforcement is given as
cots
fAV yv
s
( 5.26)
For value of
8.1095.00.1'
c
pc
f
fCot
When Vcw < Vct ( where stresses given in psi)
Hence for no pre-stress, fpc = 0 and cotθ= 1.0 and θ = 45degree
4. vvcsc dbfVVVn '25.0 ( 5.27)
Where vvcc dbfV ' ( 5.28)
sfAV yvs
sin)cot(cot
( 5.29)
3102.0
)(2
__5.0/
ppss
pspspuuvux AEAE
fAVVNdM (5.30)
152
For members having less than Avmin, the equivalent crack spacing is determined as;
)63.0(
38.1
g
xex a
SS
(5.31)
Where ga is maximum aggregate size.
To determine the shear resistance of the cracked concrete, ß = 4.8
The minimum shear reinforcement is given as :
y
vcv f
sbfA
'
min (5.32)
For members having at least minimum shear reinforcement, the angle of diagonal
compression is given as;
x 700029
and is obtained as above.
The simplified design method of NCHRP, when applied to the shear data base gave
relatively less coefficient of variation. The flow chart for the simplified design method
is given in Figure 5.1
5.8.3 Recommendation of the NCHRP
The research project of NCHRP further made the following recommendations;
i. Web based national data base of shear test results may be established.
ii. More research is required on the types of beams and material, for which
very little or no database is available.
iii. Data on high strength concrete beams is very little and further research is
required.
iv. The range of applicability of size effect and its relationship to minimum
shear reinforcement needs to be better understood. The size effect rather
than the compressive strength of concrete may be the best estimator for
minimum shear reinforcement.
153
No
Figure 5.1 Flowchart for use of the NCHRP simplified design method (NHRP, 2006).
START
Given bv,dv,Mmax,fc’, Vp, Nu, fp, Ap…
Where f'c in Ksi
Vn/φ>0.25f’cbvdv+Vp?
RC members
vvcc dbfV ,06.0 Prestressed Members (PC)
cV is lesser of
vvccri
dvvcci dbfM
MVVdbfV ,
max
, 06.006.0
and
pvvpocc VdbffV )3.006.0( ,
Required shear strength for shear reinforcement:
9.0
whereVV
Vs cu
Calculate required shear reinforcement:
min)/(cot
/ sAordf
VsA v
vy
sv
Where y
vcv f
bfsA '
min 0316.0)/(
For value of cot : For RC members 0.1cot For PC members if cwci VV or 0.1 CotMM cru
Otherwise 8.130.1' pe
f
fCot
End
Increase the section
154
5.8.4 Summary of literature review of shear strength of beams ( B-region) and
Disturbed (D-region) in RC members.
The literature reviewed in this study has revealed the following broader facts in the
contemporary research on the shear design of reinforced concrete structures,
including beams (B- region) and Disturbed region (D-region).
i. The shear strength of HSRC beams is not increasing in the same
magnitude as in case of the NSRC beams with the increase in
compressive strength of the concrete. This may be mainly due to loss
in the shear resistance due to aggregate interlocking.
ii. The shear failure of HSRC beams is relatively brittle. This failure
becomes even more brittle and sudden with further increase in the
compressive strength of concrete. Hence the HSRC beams without
web reinforcement poses severe design threat.
iii. The research data on the shear strength of HSRC beams is limited and
cannot be used for some major changes in the provisions of building
codes.
iv. The minimum web reinforcement for the shear design of beams is
required to be rationalized to ensure ductile failure of the HSRC
beams.
v. The Codes provisions provide very little information about the shear
design of HSRC beams with compressive strength of 90Mpa and
more.
vi. There are wide variations in the required web reinforcement under
different building codes for external applied loading conditions. Hence
joint work is required to rationalize and generalize the provisions of
these building codes. Even for minimum web reinforcement, the values
proposed by various building codes vary significantly.
155
vii. The latest concepts of shear design of HSRC beams are relatively
rational but complicated and require more simplification for the use of
designers.
viii. Though extensive research has been carried out in the last six
decades to comprehend the shear behaviour of RC beams, yet the
shear strength of HSRC structures is relatively newer area and more
research is required in this direction.
ix. The design of disturbed region in concrete structures requires serious
considerations in the contemporary research.
x. The Strut and Tie Model (STM) has been used to analyze and design
both the beam and disturbed regions in structural concrete more
frequently in the latest research around the world. However the use of
STM for practicing structural engineers in the field still poses many
challenges.
xi. The selection of an appropriate STM is the most difficult part of the
design decision of disturbed region with STM. Some computerized
models have been developed, which uses a number of iterations to
reach at identifying the most critical truss path. However application of
the STM tool for the design of shear critical structures would require
more experimental research.
156
Chapter No. 6
Experimental programme and discussion of test results
Of HSC beams (B-region)
Chapter Introduction: This chapter is based on the experimental program of research in the HSC beam, followed by the discussion of the experimental results, failure modes of the beams and effects of various parameters on the shear strength of HSC beams. At the end general comments on the shear behaviour of HSRC beams with and without web reinforcement has been given.
6.1 Introduction to experimental programme
Experimental programme to better understand the shear behaviour of high strength
concrete was conducted at Structural Engineering Laboratory of Civil Engineering
Department, University of Engineering & Technology, Taxila Pakistan.
6.1.1 Objectives of the experimental programme are as follows;
1. To study the effect of shear span to depth ratio on the shear capacity
of HSRC beams.
2. To study the effect of longitudinal steel on the shear capacity of HSRC
beam.
3. To identify the minimum longitudinal steel level for ensuring the shear
failure of the HSRC beams.
4. To study the contribution of shear reinforcement in HSC beams for
various levels of longitudinal steel.
5. To compare the provisions of building Codes for shear design with the
observed values and their degree of safety.
6. To check the effectiveness of minimum shear reinforcement towards
increasing the shear capacity of the beams and avoiding the brittle
failure of HSRC beams.
7. To make further recommendations on the basis of observations for
future research in Pakistan in the shear design of HSC.
157
6.1.2 Material used 6.1.2.1 Concrete.
The details of mix design of concrete and average 28 days compressive strength
obtained are given in Table. 6.1
Table 6.1 Mix Proportioning/ Designing of High Strength Concrete.
Constituent Mix proportioning Type- I Cement 628 kg/m3
Fine aggregates 484 kg/m3
Coarse aggregates 1128 kg/m3
HRWR @ by weight of cement 10.70 kg/m3
Water @ 0.25 w/c ratio 157 kg/m3
Average Cylinder Compressive strength ( 28 days) fc′
52.0 MPa
6.1.2.2 Reinforcing Steel
Deformed steel bars of specified 60,000 psi (414 MPa) yield stress were used. The
sizes of bars used are English sizes #3, #4, #6 and #7 and corresponding metric
sizes as #10, #13, #19 and #22 respectively. The bars were selected from the same
batch of re-bars.
For web reinforcement plain bars of size #2 ( 2/8 in), equal to 6mm sizes of specified
yield strength of 40,000 psi( 275MPa) were provided at 15cm c/c ( 6 in c/c).
Various parameters of the steel bars are given in Table 6.2
Table 6.2 Details of reinforcing bars used in the beams.
Nominal size Nominal Areas Average
Yield strength
Average
Ultimate strength
% elongation
US Metric in2 mm2 psi MPa psi MPa
#2 #6 0.05 32 45240 312 60121 414.62 13.21
#3 #10 0.11 71 66752 460 106653 735.54 12.50
#4 #13 0.20 129 67648 466 106483 734.37 15.63
#6 #19 0.44 284 67025 462 106836 736.80 14.06
#7 #22 0.60 387 64340 444 100678 694.33 15.23
158
6.1.3 Details of test specimens
In total 70 beams of size 23 cm x 30 cm were cast, out of which 35 beams were cast
without web reinforcement called as series-I beams and designated as B1,B2,B3,B4
and B5. 35 beams in series-II, were cast with web reinforcement and designated as
Bs1,Bs2,Bs3,Bs4 and Bs5. The shear reinforcement in series-II, was used as per
minimum web reinforcement required under ACI-318.
As the longitudinal steel was one of the basic variable of study, therefore five values
of ρ=0.33%, 0.75%, 1%, 1.5% and 2% were used. The second variable of the study
was shear to span ratio a/d. Hence for each values of ρ, seven values of a/d were
used as a/d=3,3.5,4,4.5,5,5.5 and 6. The third variable of study was web
reinforcement. Min web reinforcement was used in 35 beams of series-II, while
keeping the values of ρ and a/d same as used in 35 beams of series-I.
The details of test specimen and steel bars used as longitudinal and transverse
reinforcement are given in Figure 6.1and Table 6.3.
6.1 (a) Without web reinforcement.
6.1(b) With web reinforcement
Figure 6.1 Details of beams used in the testing.
159
Table 6.3 Reinforcement details of beams
7 (SI units of bar # given in Metric sizes and corresponding US sizes given in parenthesis) ρ=As/bd, Where As= Area of steel bars in the cross section of the beam, b is width of the beam=23 cm, d is effective depth of the beam =30 cm.
As the effective depth of the test specimen was constant, therefore the span of the
beams were selected such that the shear span to depth ratio of the beams to be
tested for each set of longitudinal steel ratio becomes a/d =3,3.5,4,4.5,5,5.5,6
(seven values). The details of span of the beams are given in Table 6.4
Table 6.4 shear span to depth ratio and corresponding span of seven beams in each set of
longitudinal reinforcement.
a/d 3.0 3.5 4.0 4.5 5.0 5.5 6.0
clear span of
beams ( cm) 150 178 203 229 254 280 305
The final details of 70 beams , 35 each in series-I and 35 in series-II are given in
Table 6.5 and Table 6.6 respectively.
Beams without stirrups Beams with stirrups
Bea
m T
itle
Steel
Bea
m T
itle
Tension steel Comp Steel
Transverse steel Tension
Bars ρ% Bars ρ(%) Bars ρ (%)
Stirrups ρv (%)
B1 1#10+1#13 (1#3+1#4)
0.33 Bs1
1#10+1#13 (1#3+1#4)
0.33
2#10 (2#3)
0.25 #6@15 cm (#2@6in)
0.16
B2 2#10+2#13 (2#3+2#4)
0.73 Bs2
2#10+2#13 (2#3+2#34)
0.73
2#10 (2#3)
0.25 #6@15 cm (#2@6in)
0.16
B3
2#19 (2#6)
1.00 Bs3
2#19 (2#6)
1.00 2#10 (2#3) 0.25 #6@15 cm
(#2@6in) 0.16
B4
3#19 (3#6)
1.50
Bs4
3#19 (3#6)
1.50 2#10 (2#3)
0.25 #6@15 cm (#2@6in)
0.16
B5
2#22 (2#7)
2.00
Bs5
2#22 (2#7)
2.00
2#10 (2#3)
0.25 #6@15 cm (#2@6in)
0.16
160
Table6.5 Details of Series-I beams without web
reinforcement ( 35 Nos)
am
Title
b
cm
h
cm
Steel
ratio
ρ
Span
cm
a/d
B1-1 23 30 0.0033 152.40
3.0
B1-2 23 30 0.0033 177.80
3.5
B1-3 23 30 0.0033 203.20
4.0
B1-4 23 30 0.0033 228.60
4.5
B1-5 23 30 0.0033 254.00
5.0
B1-6 23 30 0.0033 279.40
5.5
B1-7 23 30 0.0033 304.80
6.0
B2-1 23 30 0.0073 152.40
3.0
B2-2 23 30 0.0073 177.80
3.5
B2-3 23 30 0.0073 203.20
4.0
B2-4 23 30 0.0073 228.60
4.5
B2-5 23 30 0.0073 254.00
5.0
B2-6 23 30 0.0073 279.40
5.5
B2-7 23 30 0.0073 304.80
6.0
B3-1 23 30 0.010 152.40
3.0
B3-2 23 30 0.010 177.80
3.5
B3-3 23 30 0.010 203.20
4.0
B3-4 23 30 0.010 228.60
4.5
B3-5 23 30 0.010 254.00
5.0
B3-6 23 30 0.010 279.40
5.5
B3-7 23 30 0.010 304.80
6.0
B4-1 23 30 0.015 152.40
3.0
B4-2 23 30 0.015 177.80
3.5
B4-3 23 30 0.015 203.20
4.0
B4-4 23 30 0.015 228.60
4.5
B4-5 23 30 0.015 254.00
5.0
B4-6 23 30 0.015 279.40
5.5
B4-7 23 30 0.015 304.80
6.0
B5-1 23 30 0.020 152.40
3.0
B5-2 23 30 0.020 177.80
3.5
B5-3 23 30 0.020 203.20
4.0
B5-4 23 30 0.020 228.60
4.5
B5-5 23 30 0.020 254.00
5.0
B5-6 23 30 0.020 279.40
5.5
B5-7 23 30 0.020 304.80 6.0
Table 6.6 Details of Series-II beams with web
reinforcement ( 35 Nos)
Beam
Title
b
cm
h
cm
Steel
ratio
ρ
Span
cm
a/d) Shear
steel
(ρv)
Bs1-1 23 30 0.0033 152.4
3.0 0.16 %
Bs1-2 23 30 0.0033 177.8
3.5 0.16 %
Bs1-3 23 30 0.0033 203.2
4.0 0.16 %
Bs1-4 23 30 0.0033 228.6
4.5 0.16 %
Bs1-5 23 30 0.0033 254.0
5.0 0.16 %
Bs1-6 23 30 0.0033 279.4
5.5 0.16 %
Bs1-7 23 30 0.0033 304.8
6.0 0.16 %
Bs2-1 23 30 0.0073 152.4
3.0 0.16 %
Bs2-2 23 30 0.0073 177.8
3.5 0.16 %
Bs2-3 23 30 0.0073 203.2
4.0 0.16 %
Bs2-4 23 30 0.0073 228.6
4.5 0.16 %
Bs2-5 23 30 0.0073 254.0
5.0 0.16 %
Bs2-6 23 30 0.0073 279.4
5.5 0.16 %
Bs2-7 23 30 0.0073 304.8
6.0 0.16 %
Bs3-1 23 30 0.010 152.4
3.0 0.16 %
Bs3-2 23 30 0.010 177.8
3.5 0.16 %
Bs3-3 23 30 0.010 203.2
4.0 0.16 %
Bs3-4 23 30 0.010 228.6
4.5 0.16 %
Bs3-5 23 30 0.010 254.0
5.0 0.16 %
Bs3-6 23 30 0.010 279.4
5.5 0.16 %
Bs3-7 23 30 0.010 304.8
6.0 0.16 %
Bs4-1 23 30 0.015 152.4
3.0 0.16 %
Bs4-2 23 30 0.015 177.8
3.5 0.16 %
Bs4-3 23 30 0.015 203.2
4.0 0.16 %
Bs4-4 23 30 0.015 228.6
4.5 0.16 %
Bs4-5 23 30 0.015 254.0
5.0 0.16 %
Bs4-6 23 30 0.015 279.4
5.5 0.16 %
Bs4-7 23 30 0.015 304.8
6.0 0.16 %
Bs5-1 23 30 0.020 152.4
3.0 0.16 %
Bs5-2 23 30 0.020 177.8
3.5 0.16 %
Bs5-3 23 30 0.020 203.2
4.0 0.16 %
Bs5-4 23 30 0.020 228.6
4.5 0.16 %
Bs5-5 23 30 0.020 254.0
5.0 0.16 %
Bs5-6 23 30 0.020 279.4
5.5 0.16 %
Bs5-7 23 30 0.020 304.8
6.0 0.16 %
161
Proper curing of the beams was ensured with wet sand and the spaces around the
beams were filled with sand, which was kept continuously wet with clean water.
Curing was continued for 28 days. In Figure 6.2, all the 70 beams and wet sand
curing of these beams have been given.
Figure 6.2 Wet sand filled around the beams for curing.
162
6.3 Test set up
6.3.1 Loading arrangement and supports conditions
For application of loads at the mid span of the beams, the loading frame fabricated
at the structural laboratory of Engineering University Taxila-Pakistan was used.
Loads were applied through the hydraulic system attached to proving ring and the
readings were taken accordingly. The ends of the beams were placed at roller
supports, so that the beams can behave like simply supported beams without lateral
constraint. The details of loading arrangements and supports are shown in Figure
6.3 and Figure 6.4 respectively.
Figure 6.3 Details of loading arrangement for the testing of RC beams.
Figure 6.3 Details of roller supports and deflection gauges used for the beams.
Clear Span
Proving Ring
Hydraulic Jack
Deflection Gauge
Shear Span=a=L/2
163
6.3.2 Instrumentation for loading and deflection measurement For measurement of loads applied at the mid span of the beams, high tensile
proving ring was used and the loads were applied through the hydraulic system at
the mid span of the beams. The reading taken from the proving ring was converted
into equivalent “kN” from the conversion chart provided by the manufacturer of the
proving ring. Deflection gauges were provided at the mid span and critical sections
for shear at distance “d” from both supports.
6.3.3 Loading procedure
After locating the midpoint of the beams, the beams were placed under the
concentrated load of the hydraulic system and the loads were applied as shown in
Figure 6.2. Loads were applied at uniform rate of about 5kN per second, so that the
application of load is gradual and monotonic. The readings of the calibrated proving
rings were taken after every increment of 5kN, the beams were continuously
observed from both sides. The deflection of the gauges placed under the mid span
and critical sections for shear were taken for each 5kN increment of load. The cracks
appearing in the beams were carefully observed and the corresponding load applied
was recorded at the point, where the crack initiated. The crack path and the
respective loading were also marked on both sides of every beam. The application
of loading was continued till the failure of the beams.
Proper curing of the beams was ensured with wet sand and the spaces around the
beams were filled with sand, which was kept continuously wet with clean water.
Curing was continued for 28 days. In Figure 6.4, all the 70 beams and wet sand
curing of these beams have been given.
164
6.4 Experimental results
6.4.1 Failure loads of the beams
The failure load for beams without web reinforcement for various values of
longitudinal steel ratio and shear span to depth ratio a/d, has been shown in Table
6.7, which includes the self weight of the beams and external loads applied at the
point of failure of the beams. Similarly the values of failure load for 35 beams with
web reinforcement beams has been shown in Table 6.8
Table 6.7 Total applied failure load at the beams without web reinforcement.
Pu(tets); Total failure load
Beam Title
Revised
Beam Title
Steel ratio (ρ)
a/d Applied Failure
load (KN)
Self weight (KN)
Total applied load at failure
( KN) Pu(test)
B1
B0.33,3
0.33
3.0 68.08 2.40 70.48 B0.33.3.5 3.5 57.56 2.98 60.54 B0.33,4 4.0 47.02 3.20 50.22 B0.33,4.5 4.5 44.24 3.60 47.84 B0.33,5 5.0 38.12 4.00 42.12 B0.33,5.5 5.5 33.36 4.40 37.76 B0.33,6 6.0 27.28 4.80 32.08
B2
B0.73,3
0.73
3.0 120.48 2.40 122.88 B0.73.3.5 3.5 110.46 2.98 113.44 B0.73,4 4.0 100.28 3.20 103.48 B0.73,4.5 4.5 89.96 3.60 93.56 B0.73,5 5.0 80.02 4.00 84.02 B0.33,5.5 5.5 69.54 4.40 73.94 B0.73,6 6.0 48.68 4.80 53.48
B3
B1,3
1.0
3.0 155.64 2.40 158.04 B1,3.5 3.5 132.94 2.98 135.92 B1,4 4.0 117.52 3.20 120.72 B1,4.5 4.5 111.12 3.60 114.72 B1,5 5.0 97.38 4.00 101.38 B1,5.5 5.5 95.12 4.40 99.52 B1,6 6.0 72.12 4.80 76.92
B4
B1.5,3
1.50
3.0 228.98 2.40 231.38 B1.5.3.5 3.5 203.64 2.98 206.62 B1.5,4 4.0 175.96 3.20 179.16 B1.5,4.5 4.5 155.56 3.60 159.16 B1.5,5 5.0 135.06 4.00 139.06 B1.5,5.5 5.5 120.64 4.40 125.04 B1.5,6 6.0 105.46 4.80 110.26
B5
B2,3
2.0
3.0 292.98 2.40 295.38 B2,.3.5 3.5 244.98 2.98 247.96 B2,4 4.0 200.02 3.20 203.22 B2,4.5 4.5 187.9 3.60 191.5 B2,5 5.0 167.36 4.00 171.36 B2,5.5 5.5 149.22 4.40 138.26 B2,6 6.0 134.48 4.80 125.35
165
Table 6.8 Total applied failure load at the beams with web reinforcement
Revised Beam notation is shows with two subscripts. The first one showing the longitudinal steel ratio in % ( As/bd x100) and the second one shows the a/d ratio. For example B0.33,3, shows beam B-1 with longitudinal steel ratio of 0.33% and a/d ratio as 3.
Beam Title
Revised Beam Title
Steel ratio (ρ)
a/d Applied Failure load (KN)
Self weight (KN)
Total applied load at failure ( KN)
Pu(test)
Bs1
Bs0.33,3
0.33
3.0 77.96 2.40 80.36
Bs0.33.3.5 3.5 71 2.98 73.98
B0.33,4 4.0 60.6 3.20 63.8
B0.33,4.5 4.5 65.24 3.60 68.84
Bs0.33,5 5.0 59.16 4.00 63.16
Bs0.33,5.5 5.5 44.54 4.40 48.94
Bs0.33,6 6.0 38.78 4.80 43.58
Bs2
Bs0.73,3
0.73
3.0 161.14 2.40 163.54
Bs0.73.3.5 3.5 152.42 2.98 155.4
Bs0.73,4 4.0 131.34 3.20 134.54
Bs0.73,4.5 4.5 121.6 3.60 125.2
Bs0.73,5 5.0 111.36 4.00 115.36
Bs0.33,5.5 5.5 101.62 4.40 106.02
Bs0.73,6 6.0 91.42 4.80 96.22
Bs3
Bs1,3
1.0
3.0 188.98 2.40 191.38
Bs1,3.5 3.5 166.64 2.98 169.62
Bs1,4 4.0 154.08 3.20 157.28
Bs1,4.5 4.5 151.46 3.60 155.06
Bs1,5 5.0 141.84 4.00 145.84
Bs1,5.5 5.5 125.12 4.40 129.52
Bs1,6 6.0 101.02 4.80 105.82
Bs4
Bs1.5,3
1.50
3.0 247.96 2.40 250.36
Bs1.5.3.5 3.5 229.22 2.98 232.2
Bs1.5,4 4.0 188.64 3.20 191.84
Bs1.5,4.5 4.5 160.82 3.60 164.42
Bs1.5,5 5.0 139.68 4.00 143.68
Bs1.5,5.5 5.5 127.46 4.40 131.86
Bs1.5,6 6.0 112.72 4.80 117.52
Bs5
Bs2,3
2.0
3.0 318.68 2.40 321.08
Bs2,.3.5 3.5 267.64 2.98 270.62
Bs2,4 4.0 228.76 3.20 208.76
Bs2,4.5 4.5 221.72 3.60 202.79
Bs2,5 5.0 194.74 4.00 178.87
Bs2,5.5 5.5 77.96 2.40 80.36
Bs2,6 6.0 71 2.98 73.98
166
6.4.2 Shear strength and failure angles of the beams.
The shear strength of beams is taken as half of the total load carried by the beams
at the failure point, as the beams are simply supported. The angles of diagonal
cracks causing failure of the beams were measured to the nearest 5 degrees. The
shear strength of beams and corresponding angles are shown in Table 6.9 and
Table 6.10 for both sets of beams.
Table 6.9 Shear Strength and failure angles of 35 HSC beams, without web reinforcement
Beam Title Revised Beam Title
Steel ratio (ρ)
a/d Shear taken by the beam at the failure
Vtest = Pu(test)/2 ( KN)
Approximate Failure angle.
(degrees)
B1
B0.33,3
0.33
3.0 35.24
60-70 B0.33.3.5 3.5 30.27 B0.33,4 4.0 25.11 B0.33,4.5 4.5 23.92 B0.33,5 5.0 21.06 B0.33,5.5 5.5 18.88
75-80 B0.33,6 6.0 16.04 B2
B0.73,3
0.73
3.0 61.44
55-65 B0.73.3.5 3.5 56.72 B0.73,4 4.0 51.74 B0.73,4.5 4.5 46.78 B0.73,5 5.0 42.01 B0.33,5.5 5.5 36.97
75-80 B0.73,6 6.0 26.74 B3
B1,3
1.0
3.0 79.02 40-55 B1,3.5 3.5 67.96
B1,4 4.0 60.36
45-65 B1,4.5 4.5 57.36 B1,5 5.0 50.69 B1,5.5 5.5 49.76 B1,6 6.0 38.46
B4
B1.5,3
1.50
3.0 115.69 35-50 B1.5.3.5 3.5 103.31
B1.5,4 4.0 89.58 B1.5,4.5 4.5 79.58
40-60 B1.5,5 5.0 69.53 B1.5,5.5 5.5 62.52 B1.5,6 6.0 55.13
B5
B2,3
2.0
3.0 147.69 30-50 B2,.3.5 3.5 123.98
B2,4 4.0 101.61 B2,4.5 4.5 95.75
35-60 B2,5 5.0 85.68 B2,5.5 5.5 76.81 B2,6 6.0 69.64
167
Table 6.10 Shear Failure mode of 35 beams with web reinforcement
The angle of shear crack rounded to the nearest 5 degrees measure.
Beam Title Revised Beam
Title
Steel ratio (ρ)
a/d Shear taken by the beam at the
failure Vtest = Pu(test)/2
( KN)
App. Failure angle.
(degrees)
Bs1
Bs0.33,3
0.33
3.0 40.18
60-55 Bs0.33.3.5 3.5 36.99 B0.33,4 4.0 31.90 Bs0.33,4.5 4.5 34.42 B0.33,5 5.0 31.58 Bs0.33,5.5 5.5 24.47
70-80 Bs0.33,6 6.0 21.79 Bs2
Bs0.73,3
0.73
3.0 81.77
50-65 Bs0.73.3.5 3.5 77.70 B0.73,4 4.0 67.27 Bs0.73,4.5 4.5 62.60 Bs0.73,5 5.0 57.68 Bs0.33,5.5 5.5 53.01
70-80 Bs0.73,6 6.0 48.11 Bs3
Bs1,3
1.0
3.0 95.69 30-50 Bs1,3.5 3.5 84.81
Bs1,4 4.0 78.64
40-65 Bs1,4.5 4.5 77.53 Bs1,5 5.0 72.92 Bs1,5.5 5.5 64.76 Bs1,6 6.0 52.91
Bs4
Bs1.5,3
1.50
3.0 125.18 30-45 Bs1.5.3.5 3.5 116.10
Bs1.5,4 4.0 95.92 B1.5,4.5 4.5 82.21
35-60 Bs1.5,5 5.0 71.84 Bs1.5,5.5 5.5 65.93 Bs1.5,6 6.0 58.76
Bs5
Bs2,3
2.0
3.0 160.54 30-50 Bs2,.3.5 3.5 135.31
Bs2,4 4.0 115.98 Bs2,4.5 4.5 112.66
30-50 Bs2,5 5.0 99.37 Bs2,5.5 5.5 95.03 Bs2,6 6.0 77.77
168
6.5 Discussion of results
6.5.1 Cracking pattern and failure modes of beams.
The cracking pattern and failure mode of the beams was closely observed. When
loads were applied to beams without web reinforcement vertical cracks appeared in
the mid span region. Initially the cracks were of small width and concentrated in the
mid span region with angles being more or less vertical. However with further
increase of load, the depth and width of cracks increased. The angles of cracks
became shallower and turned diagonal. The change in the angle of cracks can be
attributed to the cantilever action of the cracked concrete restrained by the
longitudinal reinforcement in the tension zone. When load was further increased, the
depth of some of the diagonal cracks further enhanced and crossed into the
compression zone of the beams, which ultimately caused the failure of the beams as
the cracks extended further towards the point of application of loads. The typical
shear failure of beams has been shown in Figure 6.5. This kind of failure is also
called “diagonal tension” failure, which was observed in the beams having a/d of 3
and more.
For beams having a/d>5.0 the failure has been observed predominantly due to
flexural cracks, which are also called the shear flexure failure as shown in Figure
6.6. Here the flexural cracks are dominant in the middle third region and the angle of
failure is large. These represent the values of a/d, where the beams are about to
achieve the flexural strength before shear failure on the upper boundary of famous
“Kani’s shear valley”. The theoretical flexural values and shear strength of such
beams are falling very closer to each other in this region.
169
(B0.33, 3, ρ = 0.33 %, a/d = 3.00 , span = 152 cm)
( B073, 3: ρ = 0.73 % , a/d = 3 span = 152 cm)
(B1.0, 3, ρ = 1 %, a/d = 3 span = 152 cm)
Figure 6.5 Failure of beams without web reinforcement due to diagonal tension shear failure mode of the beam.
(B0.33,5.5, ρ = 0.33 % , a/d = 5.5 span = 279.4 cm)
(B0.73, 5.5 ρ = 0.73 % , a/d = 5.5 span = 279.4 cm)
(B0.73,6.0 ρ = 0.73% , a/d = 6.0 span = 279.4 cm)
Figure 6.6 Flexural shear failure of beams without web reinforcement having a/d>5.
170
The failure of beams without web reinforcement and longitudinal steel of 1% or more
has been observed due to shear failure as shown in Figure 6.7. However the pattern
of shear crack has been changed with the increase of longitudinal steel and a/d
ratio. At lower values of a/d (3, 3.5 and 4.0), the failure is more like a pure shear
crack, in the form of arch action compression failure. The cracks originate closer to
the supports and gradually extend towards the mid span at relatively shallower
angle, in the range of 40 to 50 degrees. When the crack further extends to the mid
span and a clearer shear crack observed starting from the region near the supports
and reaching at the mid span to crack the beam, across a well defined path. This
kind of failure is more typical for high strength concrete beams, particularly, where
the longitudinal steel is more. This brittle failure phenomenon of the HSC beam
must be surely a point of concern in the contemporary research.
In all the three illustrations of Figure 6.8, the failure has taken place along a well
defined shear crack, very abrupt in nature and the inclination not more than 50
degrees. The research was also focused on reducing the chances of brittle failure of
HSC beams without web reinforcement, by adding minimum web reinforcement as
per ACI-381 in these beams.
(B1.0, 5 , ρ = 1% , a/d = 5 span = 254 cm)
(B1.5, 5.5, ρ = 1% , a/d = 5.5 span = 280 cm)
(B2,6ρ = 2% , a/d = 6 span = 305 cm)
Figure 6.7 Typical shear failures of beams without web reinforcement. The failure is more brittle and sudden amongst all. The crack causing failure of the beam was not noticed in the beginning and beams failed very suddenly due to tension shear failure
171
The failure angles of beams, corresponding shear strength and failure modes for 35
beams without web reinforcement is shown in Table 6.11.
Table 6.11 Shear Strength, failure angles and failure modes of 35 HSC beams, without web reinforcement
Beam Title
Revised
Beam Title
Steel ratio (ρ)
a/d
Shear taken by the beam at the failure
Vtest ( KN)
Failure mode Approximate Failure angle.
(degrees)
B1
B0.33,3
0.33
3.0 35.24 Beam failure/Diagonal tension failure
60-70 B0.33.3.5 3.5 30.27 Beam failure/ Diagonal tension failure B0.33,4 4.0 25.11 Beam failure/Diagonal tension failure B0.33,4.5 4.5 23.92 Beam failure/ Diagonal tension failure
B0.33,5 5.0 21.06 Beam failure/Diagonal tension failure
B0.33,5.5 5.5 18.88 Flexural shear failure dominant 75-80 B0.33,6 6.0 16.04 Flexural shear failure dominant
B2
B0.73,3
0.73
3.0 61.44 Beam failure/Diagonal tension failure
55-65 B0.73.3.5 3.5 56.72 Beam failure/ Diagonal tension failure
B0.73,4 4.0 51.74 Beam failure/Diagonal tension failure
B0.73,4.5 4.5 46.78 Beam failure/ Diagonal tension failure
B0.73,5 5.0 42.01 Beam failure/Diagonal tension failure
B0.33,5.5 5.5 36.97 Flexural shear failure dominant 75-80 B0.73,6 6.0 26.74 Flexural shear failure dominant
B3
B1,3
1.0
3.0 79.02 Arch failure/Compression shear failure 40-55 B1,3.5 3.5 67.96 Arch failure/Compression shear failure
B1,4 4.0 60.36 Beam failure/Diagonal tension failure
45-65 B1,4.5 4.5 57.36 Beam failure/ Diagonal tension failure
B1,5 5.0 50.69 Beam failure/Diagonal tension failure
B1,5.5 5.5 49.76 Beam failure/ Diagonal tension failure
B1,6 6.0 38.46 Beam failure/Diagonal tension failure
B4
B1.5,3
1.50
3.0 115.69 Arch failure/Compression shear failure 35-50 B1.5.3.5 3.5 103.31 Arch failure/Compression shear failure
B1.5,4 4.0 89.58 Arch failure/Compression shear failure
B1.5,4.5 4.5 79.58 Beam failure/Diagonal tension failure 40-60 B1.5,5 5.0 69.53 Beam failure/ Diagonal tension failure
B1.5,5.5 5.5 62.52 Beam failure/Diagonal tension failure
B1.5,6 6.0 55.13 Beam failure/ Diagonal tension failure
B5
B2,3
2.0
3.0 147.69 Arch failure/Compression shear failure 30-50 B2,.3.5 3.5 123.98 Arch failure/Compression shear failure
B2,4 4.0 101.61 Arch failure/Compression shear failure
B2,4.5 4.5 95.75 Beam failure/Diagonal tension failure 35-60 B2,5 5.0 85.68 Beam failure/ Diagonal tension failure
B2,5.5 5.5 76.81 Beam failure/Diagonal tension failure
B2,6 6.0 69.64 Beam failure/ Diagonal tension failure
172
For beams with shear reinforcement, the cracking pattern has been considerably
affected by the addition of web reinforcement in HSC beams. The number of cracks
has been increased but their widths have been decreased. The failure angles have
also been reduced. The shear resisted by the beam with web reinforcement, their
failure modes and approximate failure angle for 35 beams with web reinforcement
are given in Table 6.12. The failure modes of some of the HSRC beams with web
reinforcement are shown in Figure 6.8
Table 6.12 Shear strength, failure mode and failure angles for 35 HSRC beams with web reinforcement.
Beam Title
Revised
Beam Title
Steel ratio (ρ)
a/d
Shear taken by the beam at the failure
Vtest ( KN)
Failure mode App. Failure angle.
(degrees)
Bs1
Bs0.33,3
0.33
3.0 40.18 shear Compression failure
60-55 Bs0.33.3.5 3.5 36.99 shear Compression failure B0.33,4 4.0 31.90 shear Compression failure Bs0.33,4.5 4.5 34.42 Beam failure/ Diagonal tension failure B0.33,5 5.0 31.58 Beam failure/ Diagonal tension failure Bs0.33,5.5 5.5 24.47 Beam failure/Diagonal tension failure
70-80 Bs0.33,6 6.0 21.79 Beam failure/ Diagonal tension failure Bs2
Bs0.73,3
0.73
3.0 81.77 shear Compression failure
50-65 Bs0.73.3.5 3.5 77.70 shear Compression failure B0.73,4 4.0 67.27 Beam failure/Diagonal tension failure Bs0.73,4.5 4.5 62.60 Beam failure/ Diagonal tension failure Bs0.73,5 5.0 57.68 Beam failure/Diagonal tension failure Bs0.33,5.5 5.5 53.01 Beam failure/Diagonal tension failure
70-80 Bs0.73,6 6.0 48.11 Beam failure/Diagonal tension failure Bs3
Bs1,3
1.0
3.0 95.69 Arch failure/Compression shear failure 30-50 Bs1,3.5 3.5 84.81 Arch failure/Compression shear failure
Bs1,4 4.0 78.64 Beam failure/Diagonal tension failure
40-65 Bs1,4.5 4.5 77.53 Beam failure/ Diagonal tension failure Bs1,5 5.0 72.92 Beam failure/Diagonal tension failure Bs1,5.5 5.5 64.76 Beam failure/ Diagonal tension failure Bs1,6 6.0 52.91 Beam failure/Diagonal tension failure
Bs4
Bs1.5,3
1.50
3.0 125.18 Arch failure/Compression shear failure 30-45 Bs1.5.3.5 3.5 116.10 Arch failure/Compression shear failure
Bs1.5,4 4.0 95.92 Arch failure/Compression shear failure B1.5,4.5 4.5 82.21 Beam failure/Diagonal tension failure
35-60 Bs1.5,5 5.0 71.84 Beam failure/ Diagonal tension failure Bs1.5,5.5 5.5 65.93 Beam failure/Diagonal tension failure Bs1.5,6 6.0 58.76 Beam failure/ Diagonal tension failure
Bs5
Bs2,3
2.0
3.0 160.54 Arch failure/Compression shear failure 30-50 Bs2,.3.5 3.5 135.31 Arch failure/Compression shear failure
Bs2,4 4.0 115.98 Arch failure/Compression shear failure Bs2,4.5 4.5 112.66 Beam failure/Diagonal tension failure
30-50 Bs2,5 5.0 99.37 Beam failure/ Diagonal tension failure Bs2,5.5 5.5 95.03 Beam failure/Diagonal tension failure Bs2,6 6.0 77.77 Beam failure/ Diagonal tension failure
173
(Bs0.73, 3, ρ = 0.73 % , a/d = 3 span = 152 cm)
(Bs1, 3 , ρ = 1 % , a/d = 3 span = 152 cm)
Figure 6.8 Beam shear failure or diagonal tension shear failure in beams with web reinforcement.
6.5.2 Effect of longitudinal steel ratio on the shear strength of beams
When the longitudinal steel ratio has been increased, the shear strength of beams in
both sets has been increased. However this increase in more in case of beams with
web reinforcement. The increase in shear strength of beams with the increase of
longitudinal steel is also referred as “dowel action”.
For a constant a/d ratio, when the longitudinal reinforcement was increased, the
number of cracks, their widths and failure angle reduced as shown in Figure 6.6.
This verifies the concept of bond between concrete and longitudinal steel given in
“Kani tooth model”. Due to increase in the longitudinal steel, the bond force between
the cracked concrete at the cantilever end also increased, thereby applying more
action at the free end of cracks and reducing the failure angle. The phenomena is
well illustrated by Modified Compression Filed Theory ( MCFT) of Vecchio and
Collins (1986), where the steel provided on the tension face of beams plays a
significant role in restraining the cracks and improve the shear strength of beams.
The width of the shear crack and their spacing has been observed as an important
parameter for shear failure of HSC beams. The reduction in failure angle due to
increase in the longitudinal steel is given in Figure 6.9.
174
( B1.5, 3 : ρ = 1.5 % , a/d = 3 span = 152 cm)
(B2.0, 3: ρ = 2 % , a/d = 3 span = 152 cm)
Figure 6.9 Failure of beams without web reinforcement due to diagonal tension shear failure mode of the beam. The failure angles have been reduced with the increase in longitudinal steel.
The increase in the shear strength due to increase in longitudinal steel is given in
Table 6.12 and Figures 6.9 and 6.10. The increase in shear strength with the
increase of longitudinal steel ratio in beams with web reinforcement is relatively
more due to better packing of the steel bars by the stirrups.
175
Table 6.12 Effect of the longitudinal steel on the shear strength of beams for constant a/d values.
Beams without web steel Beams with web steel
ρ (%) a/d Beam Title Shear Strength (KN) Beam Title Shear Strength (KN) 0.33 3.0 B0.33,3 35.24 Bs0.33,3 40.18
0.733 3.0 B0.733,3 61.44 Bs0.733,3 81.77
1.0 3.0 B1.0,3 79.02 Bs1.0,3 95.69
1.5 3.0 B1.5,3 115.69 Bs1.5,3 125.18 2 3.0 B2 3 147.69 Bs2 3 160.54
0.33 3.5 B0.33,3.5 30.27 Bs0.33,3.5 36.99
0.733 3.5 B0.733,3.5 56.72 Bs0.733,3.5 77.70
1.0 3.5 B1.0,3.5 67.96 Bs1.0,3.5 84.81
1.5 3.5 B1.5,3.5 103.31 Bs1.5,3.5 116.1
2 3.5 B2,3.5 123.98 Bs2,3.5 135.31
0.33 4.0 B0.33,4 25.11 Bs0.33,4 31.90
0.733 4.0 B0.733,4 51.74 Bs0.733,4 67.27
1.0 4.0 B1.0,4 60.36 Bs1.0,4 78.64
1.5 4.0 B1.5,4 89.58 Bs1.5,4 95.92
2 4.0 B2,4 101.61 Bs2,4 115.98
0.33 4.5 B0.33,4.5 23.92 Bs0.33,4.5 34.42
0.733 4.5 B0.733,4.5 46.78 Bs0.733,4.5 62.60
1.0 4.5 B1.0,4.5 57.36 Bs1.0,4.5 77.53
1.5 4.5 B1.5,4.5 79.58 Bs1.5,4.5 82.21
2 4.5 B2,4.5 95.75 Bs2,4.5 112.66
0.33 5.0 B0.33,5 21.06 Bs0.33,5 31.58
0.733 5.0 B0.733,5 42.01 Bs0.733,5 57.68
1.0 5.0 B1.0,5 50.69 Bs1.0,5 72.92 1.5 5.0 B1.5,5 69.53 Bs1.5,5 71.84 2 5.0 B2,5 85.68 Bs2,5 99.37
0.33 5.5 B0.33,5.5 18.88 Bs0.33,5.5 24.47
0.733 5.5 B0.733,5.5 36.97 Bs0.733,5.5 53.01
1.0 5.5 B1.0,5.5 49.76 Bs1.0,5.5 64.67 1.5 5.5 B1.5,5.5 62.52 Bs1.5,5.5 65.93
2 5.5 B2,5.5 76.81 Bs2,5.5 95.03
0.33 6.0 B0.33,6 16.04 Bs0.33,6 21.79 0.733 6.0 B0.733,6 26.74 Bs0.733,6 48.11
1.0 6.0 B1.0,6 38.46 Bs1.0,6 52.91
1.5 6.0 B1.5,6 55.13 Bs1.5,6 58.76 2 6.0 B2,6 69.64 Bs2,6 77.70
176
a/d=5.0
a/d=4.0
a/d=3.0
a/d=3.5
a/d=3
a/d=5.5
a/d=6
a/d=4.5
0
20
40
60
80
100
120
140
160
0.33% 0.73% 1% 1.50% 2%
She
ar s
tren
gth
of
bea
m (
KN
)
Longitudinal steel ( %)
Effect of longitudinal steel on the shear strength of HSC beams without stirrups
a/d=3
a/d=3.5
a/d=4
a/d=4.5
a/d=5
a/d=5.5
a/d=6
Figure 6.9 Effect of longitudinal Steel ratio on the shear strength of concrete beams without stirrups for same value of a/d.
a/d=5.0
a/d=4.0
a/d=3.0
a/d=3.5
a/d=3
a/d=5.5 a/d=6
a/d=4.5
0
20
40
60
80
100
120
140
160
180
0.33% 0.73% 1% 1.50% 2%
Sh
ear str
en
gth
of b
eam
( K
N)
Longitudinal steel ( %)
Effect of longitudinal steel on the shear strength of HSC beams with web reinforcement.
a/d=3
a/d=3.5
a/d=4
a/d=4.5
a/d=5
a/d=5.5
a/d=6
Figure 6.10 Effect of longitudinal Steel ratio on the shear strength of concrete beams with web reinforcement for same value of a/d.
177
6.5.3 Effect of shear span to depth (a/d) ratio on the shear strength of beams
The shear span to depth a/d ratio has a strong influence on the shear strength of
HSRC beams like NSRC beams. The shear strength decreases with the increase of
a/d values for the same longitudinal steel. However the decrease is relatively more
in case of HSRC beams without web reinforcement. The increase in shear span
increases the number of cracks formed and as result more cantilever force applied
at the cracked concrete, reducing the shear strength of concrete to greater extent.
The effect of a/d values on the shear strength of HSRC beams has been shown in
Figures 6.11 and 6.12.
The increase in shear span for a constant section of beam leads to increase in the
shear span to depth ratio. When the shear span increases, the deflection under
external loads also increases and flexural cracks are formed at relatively lower
values of external loads. The crack widths also increases which leads to reduction in
interface shear transfer and larger cracks are formed. These cracks also reduce the
depth of compression zone responsible for resisting the tensile stresses in the un-
cracked part of concrete web. The phenomena can also be explained in terms of the
cracked concrete, when the depth of the concrete cracks increases due to more
deflection of the beams, the lever arm of the cracked concrete cantilever also
increases, leading to more diagonal force on the un cracked part of the concrete
web, forcing it to fail at relatively lower value of applied load. Hence the “Tooth
model of Kani” and “ Failure of Compression zone” of Kosovo can explain this
phenomena.
The effect of a/d on the shear strength is prominent at lower values of longitudinal
steel, where the tendency of flexural cracking is more, due to early bond failure at
one hand and large deflection of beams at other hand. Thus larger tensile strain due
to small longitudinal steel can lead to reduced shear capacity of the RC beams
without web reinforcement.
178
In the corresponding beams with web reinforcement, the reduction in shear strength
due to increase in the a/d ratio is relatively less, as the diagonal shear cracks are
intercepted by the stirrups in the compression zone. Thus the effect of a/d ratio on
reduction of the shear strength of beams is more pronounced in the HSC beams
without web reinforcement.
2%
1.5%
1%
0.73%
0.33%
0
20
40
60
80
100
120
140
160
a/d=3 a/d=3.5 a/d=4 a/d=4.5 a/d=5 a/d=5.5 a/d=6
Sh
ear s
tren
gth
of b
eam
( K
N)
Shear span to depth ratio ( a/d)
Ef fect of shear span to depth ration on the shear strength of HSC beams without web reinforcement
0.33%
0.73%
1%
1.50%
2%
Figure 6.11 Effect of shear span to depth ratio on the shear strength of concrete beams
without stirrups for same value of longitudinal steel ratio.
0.33%
0.73%
1%
1.5%
2%
0
20
40
60
80
100
120
140
160
180
a/d=3 a/d=3.5 a/d=4 a/d=4.5 a/d=5 a/d=5.5 a/d=6
She
ar s
tren
gth
of b
eam
( K
N)
Shear span to depth ratio ( a/d)
Effect of shear span to depth ration on the shear strength of HSC beams with web reinforcement 0.33%
0.73%
1%
1.50%
2%
Figure 6.12 Effect of shear span to depth ratio on the shear strength of concrete beams
without stirrups for same value of longitudinal steel ratio.
179
6.5.4 Effect of web reinforcement on shear strength of HSC beams.
The use of web reinforcement has increased the shear strength of all beams of, as
shown in Table 6.14.
Table 6.14 Increase in the shear of HSC beams strength due to addition of web steel.
Beams without web reinforcement Beams with web reinforcement Increase in the shear strength of beams. (kN)
% increase in the shear with stirrups
Beam tiles
Steel ratio (ρ)
a/d
Shear at critical section
(kN)
Beams with web reinforcement
Shear at critical section
(kN)
B1
B0.33,3 0.33
3.0 35.24
Bs1
Bs0.33,3 40.18 4.94 14 B0.33.3.5 3.5 30.27 Bs0.33.3.5 36.99 6.72 22 B0.33,4 4.0 25.11 B0.33,4 31.90 6.79 27 B0.33,4.5 4.5 23.92 Bs0.33,4.5 34.42 10.50 44
B0.33,5 5.0 21.06 B0.33,5 31.58 10.52 50
B0.33,5.5 5.5 18.88 Bs0.33,5.5 24.47 5.59 30
B0.33,6 6.0 16.04 Bs0.33,6 21.79 5.75 36
Average 7.30 31 B2
B0.73,3
0.73
3.0 61.44
Bs2
Bs0.73,3 81.77 20.33 33
B0.73.3.5 3.5 56.72 Bs0.73.3.5 77.70 20.98 37
B0.73,4 4.0 51.74 B0.73,4 67.27 15.53 30
B0.73,4.5 4.5 46.78 Bs0.73,4.5 62.60 15.82 34
B0.73,5 5.0 42.01 Bs0.73,5 57.68 15.67 37
B0.33,5.5 5.5 36.97 Bs0.33,5.5 53.01 16.04 43
B0.73,6 6.0 26.74 Bs0.73,6 48.11 21.37 80
Average 17.96 42 B3
B1,3
1.0
3.0 79.02
Bs3
Bs1,3 95.69 16.67 21
B1,3.5 3.5 67.96 Bs1,3.5 84.81 16.85 25
B1,4 4.0 60.36 Bs1,4 78.64 18.28 30
B1,4.5 4.5 57.36 Bs1,4.5 77.53 20.17 35
B1,5 5.0 50.69 Bs1,5 72.92 22.23 44
B1,5.5 5.5 49.76 Bs1,5.5 64.76 15.00 30
B1,6 6.0 38.46 Bs1,6 52.91 14.45 38
Average 17.66 32 B4
B1.5,3
1.50
3.0 115.69
Bs4
Bs1.5,3 125.18 9.49 8
B1.5.3.5 3.5 103.31 Bs1.5.3.5 116.10 12.79 12
B1.5,4 4.0 89.58 Bs1.5,4 95.92 6.34 7
B1.5,4.5 4.5 79.58 B1.5,4.5 82.21 2.63 3
B1.5,5 5.0 69.53 Bs1.5,5 71.84 2.31 3
B1.5,5.5 5.5 62.52 Bs1.5,5.5 65.93 3.41 5
B1.5,6 6.0 55.13 Bs1.5,6 58.76 3.63 7
Average 5.80 6 B5
B2,3
2.0
3.0 147.69
Bs5
Bs2,3 160.54 12.85 9
B2,.3.5 3.5 123.98 Bs2,.3.5 135.31 11.33 9
B2,4 4.0 101.61 Bs2,4 115.98 14.37 14
B2,4.5 4.5 95.75 Bs2,4.5 112.66 16.91 18
B2,5 5.0 85.68 Bs2,5 99.37 13.69 16
B2,5.5 5.5 76.81 Bs2,5.5 95.03 18.22 24
B2,6 6.0 69.64 Bs2,6 77.77 8.13 12
Average 13.64 15
180
The addition of web steel has increased the failure loads of the beams, thereby
reducing the gap between the flexural moment ( Mf ) and ultimate cracking moment
(Mu) on the famous “Kani’s valley of shear failure”. The Kani’s explanation to
increase in the shear strength with stirrups has already been given in the relevant
section. One of the arguments put forward by Kani (1967), for this increase is due to
providing supports to the part of compression arches, which are otherwise
unsupported and transfer their reactions to the supports, thereby avoiding their
failure at lower loads. This explanation is no doubt in contrast to the famous “Parallel
chord truss model of Ritter (1899)”, where the role of stirrups was assumed to lift the
load to the compression zone. The later argument seems more relevant to explain
the increase in shear strength with stirrups. The same rationale was also supported
by Kotsovos (1984) in his later work.
The increase in the shear strength is however not uniform for same values of web
reinforcement. This has supported some of the research by Shehata et al (2000),
where the shear strength provided by the addition of shear reinforcement has been
described as a complex phenomena and merely addition of the equivalent shear
strength of stirrups with the concrete shear strength would not predict the total shear
strength of RC beams. The results also support the fact that the role of web
reinforcement in increasing the shear strength of beams is still not fully understood
and there are contrasting explanations as given by Kani and Kotsovos to this
phenomenon.
The existing building and bridges Codes in most of the cases use a uniform value of
shear for certain level of web reinforcement, which is often taken as independent of
the compressive strength of concrete, longitudinal steel ratio and shear span to
depth ratio a/d. The test results have not supported this basic consideration of the
codes, as the increase in shear strength is random for uniform value of web
reinforcements. This observation deserves further research.
181
The average increase in the shear strength of HSC beams with addition of web steel
is relatively more at low level of longitudinal steel ratio. In beams without web
reinforcement and lower values of longitudinal steel, flexural cracks are developed at
early stage and the depth of cracks and concrete teeth increase with further increase
of loads. The resultant cantilever action also increases and the diagonal cracking
due diagonal tension failure of beams happens. However, when the stirrups are
added to such beams, the propagation of cracks is avoided at lower values of loads
due to resistance of web steel in the compression zone, thereby increasing the
resistance to the diagonal cracking, leading to increase in the shear strength of HSC
beams.
At higher values of longitudinal steel, the diagonal cracks are formed at relatively
higher values of loads and hence the role of stirrups comes into play at later stage at
higher loads, which leads to yielding of web reinforcement with less increase in the
applied load. Thus when the longitudinal steel is increased, a decrease in the
contribution of web steel to resist the diagonal failure can be expected. This further
supports the earlier argument that the role of web reinforcement towards increase in
shear strength of HSC beams cannot be thought as independent of the longitudinal
steel.
182
6.5.5 Load deflection curves for beams:
The load deflection curves of the beams have been developed by observing to study
the general behaviour of HSRC beams. Some of the load deflection curves has been
given in Figure 6.14 and Figure 6.15.
0
20
40
60
80
100
120
140
160
0 1 2 3 4 5 6
She
ar s
tren
gth
of b
eam
( K
N)
mid span defeltion ( mm)
load defelction curve for beam (B0.73,3)
020406080
100120140160
0 0.5 1 1.5 2 2.5 3 3.5
She
ar s
tren
gth
of b
eam
(
KN
)
mid span defeltion ( mm)
load defelction curve for beam (B0.73,5)
020406080
100120140160
0 0.5 1 1.5 2 2.5She
ar s
tren
gth
of b
eam
(
KN
)
mid span defeltion ( mm)
load defelction curve for beam (B0.73,6)
Figure 6.14 load deflection curves for beams without web reinforcement and ρ=0.0073
Load ( kN) Deflection (mm)
10 0.2
20 0.3
25 0.71
35 1.0
45 1.21
55 1.55
65 2.02
75 2.39
85 2.88
90 3.3 100 3.8
110 4.28
Load ( kN) Deflection (mm)
10 0.32
20 0.42
25 0.98
35 1.65
45 1.89
55 2.12
Load ( kN) Deflection (mm)
10 0.28
20 0.39
25 0.85
35 1.35
45 1.48
55 1.75
65 2.35
75 2.90
183
0
20
40
60
80
100
120
140
160
0 0.5 1 1.5 2 2.5 3
She
ar s
tren
gth
of b
eam
( K
N)
mid span defeltion ( mm)
load defelction curve for beam (B2.0,,3)
020406080
100120140160
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
She
ar s
tren
gth
of
beam
( K
N)
mid span defeltion ( mm)
load defelction curve for beam (B2.0,5)
Figure 6.15 load deflection curves for beams without web reinforcement and ρ=0.02
Load ( kN) Deflection (mm) 20 0.10 40 0.16 60 0.25 80 0.46 100 0.66 120 0.75 140 1.01 160 1.15 180 1.26 200 1.80 220 2.03 240 2.50 260 2.80
Load ( kN) Deflection (mm)
10 0.05 20 0.20 35 0.25 55 0.35 75 0.60 100 0.89 120 1.09 140 1.25 150 1.45
184
Chapter No.7
Experimental programme and discussion of
results of two way HSC corbels.
Chapter Introduction: This chapter is based on the experimental program of research on the D-region based on Strut and Tie Model (STM). Two way corbels were designed on the basis of Strut and Tie Model against assumed external loads. Later 9 two way corbels were experimentally investigated. The results were then compared with the theoretical values of the shear strength.
7.1 Experimental Programme for testing of disturbed region in concrete ( D-
region)
To study the shear strength of the disturbed region (D-region) in concrete
structures, the Strut and Tie Model (STM) was selected for the design, which was
applied to high strength concrete two way corbels.
The objectives of the testing of two way corbels as D-region are as follows;
- To study the behaviour of two way corbels in shear.
- Modeling of corbels with Strut and Tie Model (STM), under assumed service
load.
- Checking the shear strength of these corbels and comparing the actual and
theoretical values.
- To check the reasonability of the STM for the selected structures.
185
7.1.1 Geometry of the corbels
The assumed geometry of the proposed two way cobles is given in Figure 7.1
Figure 7.1 Geometry of the proposed two way corbel and proposed STM.
The details of form work used for the test specimen is given in Figure 7.2
Figure 7.2 Reinforcement Form work used for the two way corbels.
186
7.1.2 Materials used
The details of mixed design for HSC used for two way corbels is given in Table
7.1
Table 7.1 Mix Proportioning/ Designing of High Strength Concrete Double Corbels
Title Nominal mix
ratio
Cement:sand:Aggt
Cme
( kg)
Sand
( kg)
Agg
( Kg)
w/c
ratio
HRWR
(lit/100kg
of cement
Slump
(cm )
Gauge
position
Comp
strength
(MPa)
DCB-1 1: ½: 1 100 500 100 0.25 1.10 Collapse Strut 60
DCB-2 1: ½: 1 100 500 100 0.29 0.70 5 Tie 58
DCB-3 1: ½: 1 100 500 100 0.29 0.70 5 - 60
DCB-4 1: 1: 1½ 100 100 150 0.25 0.60 6 - 50
DCB-5 1: 1: 1 ½ 100 100 150 0.32 0.60 6 Tie 48DCB-6 1: 1: 1½ 100 100 150 0.28 0.60 5 - 51
DCB-7 1: 1½:2 ½ 100 150 250 0.30 0.350 6 - 43
DCB-8 1: 1½:2 ½ 100 150 250 0.35 0.300 5 - 46
DCB-9 1: 1½:2 ½ 100 150 250 0.40 0.250 5 Strut 45
For steel reinforcement deformed steel bars of specified yield strength of 60,000
psi (414MPa) was used.
7.1.3 Loading arrangement
Load was applied to the overhanging parts on both end of the corbel at 4.5 in
from the face of support. The conceptual diagram for load arrangement is given
in Figure 7.3. Loads were applied through the hydraulic system placed on high
tensile steel rail at the centre of the corbel. The steel rail rested on two rollers
placed at the point of application of loads at the corbel ends, attached to the
proving ring, which gives reading for the equivalent applied load in kN. The total
load applied through the hydraulic system was read from proving ring. The high
tensile steel section further transmits the total applied load in two equal parts at
the ends of corbels.
187
Figure 7.3 Loading arrangement for HSC two way corbels
7.1.4 Embedment Strain Gauge.
To measure the strain in concrete or steel inside the concrete, embedment strain
gauges are used. In two way corbels, embedment strain gauges were used.
Further details about the installation etc of the gauges are given as follows;
Installation of embedment strain gauge in reinforced beam
In reinforced or pre-stressed concrete applications, the embedment strain gauge
is usually tied to the reinforcing cage. In mass concrete applications, the gauge
may be installed either before or immediately after replacement of the concrete.
Gauges may be configured in a rosette either by direct placement in the soft
concrete or by attachment to a rosette adapter. The details of embedment gauge
have been shown in Figure
7.4. In our case the embedment strain gauge was tied to the steel along the
direction of struts and ties.
188
Figure 7.4 Details of embedment strain gauge.
7.1.5 Strain Data Logger
Three terminals coming out of the embedment stain gauge were attached to
strain data logger as shown Figure 7.5, and strain was recorded for various
levels of applied loads.
Support Bars
Cable Ties
Data Cable
189
Figure 7.5 Strain Data Logging system used.
7.2 Design of the two way corbel by Strut and Tie Model ( STM)
Strut and Tie Model was used for the design of two way corbels. The detailed
design steps adopted for the corbels are given in Appendix-A. The final member
forces under assumed external loads on each end, for the assumed STM have
are given in Figure 7.6.
A A '
B
C
B '
C '
V u = 8 0 k ip s V u = 8 0 k ip s
85.3
8 ki
ps
85.3
8 ki
ps
80 k
ips
2 9 . 8 4 k ip s
80 k
ips
8 0 k ip s8 0 k ip s
2 9 . 8 4 k ip s
Figure 7.6 Member Force in strut and Tie model for two way corbel.
190
Nine HSC corbels were cast in three sets and the details of assumed external
loads, concrete compressive strength, member forces, reinforcement and other
relevant details are given in Table 7.2.
Table 7.2 Details of technical parameters and member forces in assumed STM
Corbel CB-1 to CB-3 CB-4 to CB-6 CB-7 to CB-9
Technical Parameters
fc′ (ksi)
8.5 (59MPa)
7.50 ( 52Mpa)
6.5 ( 45 MPa)
As provided (in2) ( SI units)
3#4 ( 0.60) (3#10)
3#4+1#3 ( 0.71)
( 3#10+1#13)
4#4 ( 0.81) ( 4#13)
Theoretical Shear capacity (kips)
60 ( 269 kN)
70 ( 311kN)
80 ( 356kN)
Truss Forces & Geometry
Strut (kips) 65.84 ( 292kN)
76.84 ( 342kN)
87.84 ( 390kN)
Tie (kips) 27.11 ( 120kN)
31.70 ( 141kN)
36.28 ( 161kN)
Strut AA width (in) 2.05 ( 5.2cm)
2.32 (5.89cm)
2.68 ( 6.8cm)
Strut A width (in) 1.80 ( 4.57cm)
1.79 ( 4.5cm)
1.77 ( 4.49cm)
The actual reinforcement of the corbels is given in Figure 7.6
191
Figure 7.7 Details of reinforcement, formwork and embedment gauges.
7.3 Test results and discussion of two way corbel testing: The results are given in Table 7.3. Table 7.3 Comparison of theoretical and actual failure loads of HSC double corbels
Corbel CB-1 to CB-3 CB-4 to CB-6 CB-7 to CB-9
Technical Parameters fc′
(ksi) 8.5
(59MPa) 7.50
( 52Mpa) 6.5
( 45 MPa) As provided (in2)
( SI units) 3#4
( 0.60) (3#10)
3#4+1#3 ( 0.71)
( 3#10+1#13)
4#4 ( 0.81) ( 4#13)
Theoretical Shear capacity (kips)
60 ( 269 kN)
70 ( 311kN)
80 ( 356kN)
Truss Forces & Geometry
Strut (kips) 65.84 ( 292kN)
76.84 ( 342kN)
87.84 ( 390kN)
Tie (kips) 27.11 ( 120kN)
31.70 ( 141kN)
36.28 ( 161kN)
Strut AA width (in) 2.05 ( 5.2cm)
2.32 (5.89cm)
2.68 ( 6.8cm)
Strut A width (in) 1.80 ( 4.57cm)
1.79 ( 4.5cm)
1.77 ( 4.49cm)
Strut Angle Theoretical 67.55 67.58 67.62
Actual 67 66.97 72.75 Failure Shear Loads
Theoretical (kN) 60 ( 269 kN)
70 ( 311kN)
80 ( 356kN)
Actual 63 68 72
192
( 280kN) ( 302kN) ( 320kN) % variation 5% 3% 10%
Strain in tie ( 10-6 , µs) 209 285 300
The HSC two way corbels were tested under two point loads applied at the
centre line of bearing plats at both ends. The loads were gradually increased at
5kN/sec and the cracks developed in the corbels were closely observed. The
cracks start from the edge of the plates and gradually extended down towards
the connection of corbel and column, showing a typical shear cracks. With further
increase of applied load, the crack surface widened and became more
prominent, ultimately caused failure of the corbel. The corbels have been failed
mainly due to failure of compression struts as shown in Figure 7.8. The
inclination of the struts, causing the failure of the corbels was measured, which
are falling in the range of 670 to 720, against the theoretical values of 670. The
failure is more brittle and sudden. The theoretical shear capacity of the corbels
for given reinforcement, material stresses and truss geometry was compared
with the actual values in Table 7.3.
The theoretical and observed values of the shear capacity are relatively closer for
fc′=6500 psi (45MPa), but the variation is more in case of high strength concrete
fc′=7500, 8500 psi (50,55MPa). The strut angle in case of high strength concrete
corbels is also steeper. This may be mainly due to reduction in the aggregates
interlocking at higher strength of concrete. The concrete has been cracked
across the aggregates rather than at the contract points of the aggregates and
mortar and the failure is more brittle and sudden.
193
vii. Corbel CB-1
viii. Corbel CB-3
Figure 7.8 Shear failure of Corbels CB-1 and CB-2, fc′ = 59 MPa, As= 387 mm2 (0.71in2
). The failure is more sudden and brittle typical for shear failure of HSC structure in shear. Figure 7.8 Cont’d
194
Figure 7.8 (c) CB-4 fc′ = 52 MPa, As= 477 mm2 (0.71in2)
Figure 7.8 (d) CB-5, fc′ = 52 MPa, As= 477 mm2 (0.71 in2 )
Figure 7.8 (e) CB-5, fc′ = 45 MPa, As= 522 mm2 (0.81in2)
195
Chapter No. 8
Comparison of the shear strength of HSC beams with
the provisions of international building and bridges
codes.
Chapter Introduction: In this chapter, the observed test values of the shear strength of HSC beams have been compared with the provision of the International Building and bridges Codes for shear design of concrete beams and the results have been evaluated in the light of recent research.
As already described in the literature review, different Codes have proposed
certain empirical equations for the shear design of high strength concrete based
on various approaches to the shear failure of concrete. To compare the observed
values with these provisions, the following most commonly used codes are
considered.
American Concrete Institute building code 318-08
Canadian Standards: Design of Concrete Structures,1994
AASHTO’s LRFD (Load Reduction Factor Design) Bridge design
Specification.
Eurocode-2002
New theory proposed in ACI structural Journal (March, April 2003)
8.1 American Concrete Institute (ACI) Code 318-08
The relevant equation of ACI-318 are already given as;
Eq (5.5) dbf
Vc wc
6
'
(SI units) –ACI Simplified equation
Eq (5.6) dbfdb
M
dVfVc wc
w
u
u
wc
'' 3.07
120
(SI units) –ACI detailed
equation.
196
The comparison of actual and ACI-318 results for shear strength of HSC beams
with and without web reinforcement has been given in Tables 8.1 and Table 8.2
respectively.
Table No. 8.1 Comparison of the shear strength of beams without web reinforcement with the provisions of the ACI 318-08
Beam Title
ρ (%)
a/d
Shear Strength ( KN)
Vtest/VACI Vtest VACI
B1
B0.33,3
0.33
3.0 35.24 45.28 0.78
B0.33.3.5 3.5 30.27 45.16 0.67
B0.33,4 4.0 25.11 45.07 0.56
B0.33,4.5 4.5 23.92 45.00 0.53
B0.33,5 5.0 21.06 44.94 0.47
B0.33,5.5 5.5 18.88 44.90 0.42
B0.33,6 6.0 16.04 44.86 0.36
B2
B0.73,3
0.733
3.0 61.44 46.29 1.33
B0.73.3.5 3.5 56.72 46.03 1.23
B0.73,4 4.0 51.74 45.83 1.13
B0.73,4.5 4.5 46.78 45.68 1.02
B0.73,5 5.0 42.01 45.55 0.92
B0.33,5.5 5.5 36.97 45.45 0.81
B0.73,6 6.0 26.74 45.37 0.59
B3
B1,3
1.00
3.0 79.02 44.70 1.77
B1,3.5 3.5 67.96 44.66 1.52
B1,4 4.0 60.36 44.64 1.35
B1,4.5 4.5 57.36 44.61 1.29
B1,5 5.0 50.69 44.60 1.14
B1,5.5 5.5 49.76 44.58 1.12
B1,6 6.0 38.46 44.57 0.86
B4
B1.5,3
1.50
3.0 115.69 48.22 2.40
B1.5.3.5 3.5 103.31 47.68 2.17
B1.5,4 4.0 89.58 47.28 1.89
B1.5,4.5 4.5 79.58 46.96 1.69
B1.5,5 5.0 69.53 46.71 1.49
B1.5,5.5 5.5 62.52 46.50 1.34
B1.5,6 6.0 55.13 46.33 1.19
B5
B2,3
2.0
3.0 147.69 49.48 2.99
B2,.3.5 3.5 123.98 48.76 2.54
B2,4 4.0 101.61 48.22 2.11
B2,4.5 4.5 95.75 47.80 2.00
B2,5 5.0 85.68 47.46 1.81
197
B2,5.5 5.5 76.81 47.19 1.63
B2,6 6.0 69.64 46.96 1.48
Table No. 8.2 Comparison of the shear capacity of beams with web reinforcement with the provisions of the ACI 318-08
Beam Title
a/d
ρ
(%)
a/d
Vtest ( KN)
Shear capacity as ACI provisions
( KN)
VACI = Vc+Vs
( KN)
Vntest/VACI
Vc
Vs Bs1
Bs0.33,3
0.33
3.0 40.18 45.28 11.65 56.93 0.75 Bs0.33.3.5 3.5 36.99 45.16 11.65 56.81 0.69 Bs0.33,4 4.0 31.90 45.07 11.65 56.72 0.60 B0.33,4.5 4.5 34.42 45.00 11.65 56.65 0.65 Bs0.33,5 5.0 31.58 44.94 11.65 56.59 0.60 Bs0.33,5.5 5.5 24.47 44.90 11.65 56.55 0.46 Bs0.33,6 6.0 21.79 44.86 11.65 56.51 0.41
Bs2
Bs0.73,3
0.733
3.0 81.77 46.29 11.65 57.94 1.51 Bs0.73.3.5 3.5 77.70 46.03 11.65 57.68 1.44 Bs0.73,4 4.0 67.27 45.83 11.65 57.48 1.25 Bs0.73,4.5 4.5 62.60 45.68 11.65 57.33 1.16 Bs0.73,5 5.0 57.68 45.55 11.65 57.20 1.08 Bs0.33,5.5 5.5 53.01 45.45 11.65 57.10 0.99 Bs0.73,6 6.0 48.11 45.37 11.65 57.02 0.90
B3
Bs1,3
1.00
3.0 95.69 44.70 11.65 56.35 1.81 Bs1,3.5 3.5 84.81 44.66 11.65 56.31 1.61 Bs1,4 4.0 78.64 44.64 11.65 56.29 1.49 Bs1,4.5 4.5 77.53 44.61 11.65 56.26 1.47 Bs1,5 5.0 72.92 44.60 11.65 56.25 1.38 Bs1,5.5 5.5 64.76 44.58 11.65 56.23 1.23 Bs1,6 6.0 52.91 44.57 11.65 56.22 1.00
Bs4
Bs1.5,3
1.50
3.0 125.18 48.22 11.65 59.87 2.23 Bs1.5.3.5 3.5 116.10 47.68 11.65 59.33 2.09 Bs1.5,4 4.0 95.92 47.28 11.65 58.93 1.74 Bs1.5,4.5 4.5 82.21 46.96 11.65 58.61 1.50 Bs1.5,5 5.0 71.84 46.71 11.65 58.36 1.31 Bs1.5,5.5 5.5 65.93 46.50 11.65 58.15 1.21 Bs1.5,6 6.0 58.76 46.33 11.65 57.98 1.08
Bs5
Bs2,3
2.0
3.0 160.54 49.48 11.65 61.13 2.80 Bs2,.3.5 3.5 135.31 48.76 11.65 60.41 2.39 Bs2,4 4.0 115.98 48.22 11.65 59.87 2.07 Bs2,4.5 4.5 112.66 47.80 11.65 59.45 2.02 Bs2,5 5.0 99.37 47.46 11.65 59.11 1.79 Bs2,5.5 5.5 95.03 47.19 11.65 58.84 1.72
198
Bs2,6 6.0 77.77 46.96 11.65 58.61 1.42
The increase in the shear strength of beams with the additions of stirrups is given
in Table 8.3
Table No. 8.3 Comparison of increase in shear strength due to stirrups and ACI-318
provision for stirrups contribution.
Beam Title
ρ
(%)
a/d Vtest of beams ( KN)
Vs
VsACI
Without shear steel
With shear steel
1 2 3 4 5 6 7=6-5 8
Bs1
Bs0.33,3
0.33
3.0 35.24 40.18 4.94 11.65 Bs0.33.3.5 3.5 30.27 36.99 6.72 11.65 Bs0.33,4 4.0 25.11 31.90 6.79 11.65 B0.33,4.5 4.5 23.92 34.42 10.5 11.65 Bs0.33,5 5.0 21.06 31.58 10.52 11.65 Bs0.33,5.5 5.5 18.88 24.47 5.59 11.65 Bs0.33,6 6.0 16.04 21.79 5.75 11.65
Bs2
Bs0.73,3
0.733
3.0 61.44 81.77 20.33 11.65 Bs0.73.3.5 3.5 56.72 77.70 20.98 11.65 Bs0.73,4 4.0 51.74 67.27 15.53 11.65 Bs0.73,4.5 4.5 46.78 62.60 15.82 11.65 Bs0.73,5 5.0 42.01 57.68 15.67 11.65 Bs0.33,5.5 5.5 36.97 53.01 16.04 11.65 Bs0.73,6 6.0 26.74 48.11 21.37 11.65
B3
Bs1,3
1.00
3.0 79.02 95.69 16.67 11.65 Bs1,3.5 3.5 67.96 84.81 16.85 11.65 Bs1,4 4.0 60.36 78.64 18.28 11.65 Bs1,4.5 4.5 57.36 77.53 20.17 11.65 Bs1,5 5.0 50.69 72.92 22.23 11.65 Bs1,5.5 5.5 49.76 64.76 15 11.65 Bs1,6 6.0 38.46 52.91 14.45 11.65
Bs4
Bs1.5,3
1.50
3.0 115.69 125.18 9.49 11.65 Bs1.5.3.5 3.5 103.31 116.10 12.79 11.65 Bs1.5,4 4.0 89.58 95.92 6.34 11.65 Bs1.5,4.5 4.5 79.58 82.21 2.63 11.65 Bs1.5,5 5.0 69.53 71.84 2.31 11.65 Bs1.5,5.5 5.5 62.52 65.93 3.41 11.65 Bs1.5,6 6.0 55.13 58.76 3.63 11.65
Bs2,3
3.0 147.69 160.54 12.85 11.65 Bs2,.3.5 3.5 123.98 135.31 11.33 11.65
199
Bs5
Bs2,4 2.0
4.0 101.61 115.98 14.37 11.65 Bs2,4.5 4.5 95.75 112.66 16.91 11.65 Bs2,5 5.0 85.68 99.37 13.69 11.65 Bs2,5.5 5.5 76.81 95.03 18.22 11.65 Bs2,6 6.0 69.64 77.77 8.13 11.65
Comparison of actual test results of shear strength of HSC beams with the
results given by ACI-318 in Tables 8.1, 8.2 and 8.3 equations gives the following
general observations;
1. The ACI-318 provision for shear strength of HSC beams is un-
conservative for small values of longitudinal steel both for beams with and
without web reinforcement.
2. The shear strength given by ACI-318 for ρ≥1% are however reasonably
good and becomes more conservative for ρ =1.5% and 2%.
3. The increase in shear strength of HSC beams due to addition of same
amount of stirrups in all beams is not same, as given by ACI-318. Hence
the superposition of the two contributions (concrete and transverse steel)
has not been shown in the results.
8.2 Canadian Standards for the design of concrete structures (CSA A23.3-94).
The general design method of CSA A23.3-94 is based on the Modified
Compression Filed Theory (MCFT), which requires the maximum longitudinal
strain in the concrete, which depends on the factored shear force at the section,
crack angle and longitudinal steel. However the simplified design method is
based on the concrete compressive strength only. The relevant equations are
given below;
Eq (5.17) dbfVc wc
'2.0 (SI Units) if sbf
fA w
y
cv
'06.0 or mmd 300
200
Eq(5.16) dbfdbf
dV wcwcc
1.0
1000
260 if sb
f
fA w
y
cv
'06.0 ,d>300mm
All terms have been already defined.
Since detailed MCFT has been discussed in next sections, therefore only
simplified design method of Canadian Standards has been used for comparison
here and the results have been given in Table 8.4 and 8.5.
Table No. 8.4 Comparison of the shear Strength of beams without web reinforcement with the provisions of the Canadian Standards (Simplified Method)
Beam Title
ρ (%) a/d
Shear Strength Vtest/VCSA Vtest
( KN) VCSA
(KN)
B1
B0.33,3
0.33
3.0 35.24 40.303 0.87 B0.33.3.5 3.5 30.27 40.303 0.75 B0.33,4 4.0 25.11 40.303 0.62 B0.33,4.5 4.5 23.92 40.303 0.59 B0.33,5 5.0 21.06 40.303 0.52 B0.33,5.5 5.5 18.88 40.303 0.47 B0.33,6 6.0 16.04 40.303 0.40
B2
B0.73,3
0.733
3.0 61.44 40.303 1.52 B0.73.3.5 3.5 56.72 40.303 1.41 B0.73,4 4.0 51.74 40.303 1.28 B0.73,4.5 4.5 46.78 40.303 1.16 B0.73,5 5.0 42.01 40.303 1.04 B0.33,5.5 5.5 36.97 40.303 0.92 B0.73,6 6.0 26.74 40.303 0.66
B3
B1,3
1.00
3.0 79.02 40.303 1.96 B1,3.5 3.5 67.96 40.303 1.69 B1,4 4.0 60.36 40.303 1.50 B1,4.5 4.5 57.36 40.303 1.42 B1,5 5.0 50.69 40.303 1.26 B1,5.5 5.5 49.76 40.303 1.23 B1,6 6.0 38.46 40.303 0.95
B4
B1.5,3
1.50
3.0 115.69 40.303 2.87 B1.5.3.5 3.5 103.31 40.303 2.56 B1.5,4 4.0 89.58 40.303 2.22 B1.5,4.5 4.5 79.58 40.303 1.97 B1.5,5 5.0 69.53 40.303 1.73 B1.5,5.5 5.5 62.52 40.303 1.55 B1.5,6 6.0 55.13 40.303 1.37
B5
B2,3
2.0
3.0 147.69 40.303 3.66 B2,.3.5 3.5 123.98 40.303 3.08 B2,4 4.0 101.61 40.303 2.52 B2,4.5 4.5 95.75 40.303 2.38 B2,5 5.0 85.68 40.303 2.13
201
B2,5.5 5.5 76.81 40.303 1.91 B2,6 6.0 69.64 40.303 1.73
Table No. 8.5 Comparison of the shear Strength of beams with web reinforcement with the provisions of the Canadian Standards (Simplified Method)
Beam Title
a/d
ρ (%)
Vtest ( KN)
Shear capacity as per CSA provisions
( KN)
VCSA = Vc+Vs
( KN)
Vtest/VCSA
Vc
Vs Bs1
Bs0.33,3
0.33
3.0 40.18 40.303 16.67 56.98 0.71 Bs0.33.3.5 3.5 36.99 40.303 16.67 56.98 0.65 Bs0.33,4 4.0 31.90 40.303 16.67 56.98 0.56 B0.33,4.5 4.5 34.42 40.303 16.67 56.98 0.60 Bs0.33,5 5.0 31.58 40.303 16.67 56.98 0.55 Bs0.33,5.5 5.5 24.47 40.303 16.67 56.98 0.43 Bs0.33,6 6.0 21.79 40.303 16.67 56.98 0.38
Bs2
Bs0.73,3
0.733
3.0 81.77 40.303 16.67 56.98 1.44 Bs0.73.3.5 3.5 77.70 40.303 16.67 56.98 1.36 Bs0.73,4 4.0 67.27 40.303 16.67 56.98 1.18 Bs0.73,4.5 4.5 62.60 40.303 16.67 56.98 1.10 Bs0.73,5 5.0 57.68 40.303 16.67 56.98 1.01 Bs0.33,5.5 5.5 53.01 40.303 16.67 56.98 0.93 Bs0.73,6 6.0 48.11 40.303 16.67 56.98 0.84
Bs3
Bs1,3
1.00
3.0 95.69 40.303 16.67 56.98 1.68 Bs1,3.5 3.5 84.81 40.303 16.67 56.98 1.49 Bs1,4 4.0 78.64 40.303 16.67 56.98 1.38 Bs1,4.5 4.5 77.53 40.303 16.67 56.98 1.36 Bs1,5 5.0 72.92 40.303 16.67 56.98 1.28 Bs1,5.5 5.5 64.76 40.303 16.67 56.98 1.14 Bs1,6 6.0 52.91 40.303 16.67 56.98 0.93
Bs4
Bs1.5,3
1.50
3.0 125.18 40.303 16.67 56.98 2.20 Bs1.5.3.5 3.5 116.10 40.303 16.67 56.98 2.04 Bs1.5,4 4.0 95.92 40.303 16.67 56.98 1.68 Bs1.5,4.5 4.5 82.21 40.303 16.67 56.98 1.44 Bs1.5,5 5.0 71.84 40.303 16.67 56.98 1.26 Bs1.5,5.5 5.5 65.93 40.303 16.67 56.98 1.16 Bs1.5,6 6.0 58.76 40.303 16.67 56.98 1.03
Bs5
Bs2,3
2.0
3.0 160.54 40.303 16.67 56.98 2.82 Bs2,.3.5 3.5 135.31 40.303 16.67 56.98 2.37 Bs2,4 4.0 115.98 40.303 16.67 56.98 2.04
202
Bs2,4.5 4.5 112.66 40.303 16.67 56.98 1.98 Bs2,5 5.0 99.37 40.303 16.67 56.98 1.74 Bs2,5.5 5.5 95.03 40.303 16.67 56.98 1.67 Bs2,6 6.0 77.77 40.303 16.67 56.98 1.36
The increase in the shear strength of beams with the additions of stirrups and its
comparison with CSA, is given in Table 8.6
Table No. 8.6 Comparison of increase in shear strength due to stirrups and CSA
provision for stirrups contribution.
Beam Title
ρ
(%)
a/d Vtest of beams ( KN)
Vs-test
VsACI
Without shear steel
With shear steel
1 2 3 4 5 6 7=6-5 8
Bs1
Bs0.33,3
0.33
3.0 35.24 40.18 4.94 16.67 Bs0.33.3.5 3.5 30.27 36.99 6.72 16.67 Bs0.33,4 4.0 25.11 31.90 6.79 16.67 B0.33,4.5 4.5 23.92 34.42 10.5 16.67 Bs0.33,5 5.0 21.06 31.58 10.52 16.67 Bs0.33,5.5 5.5 18.88 24.47 5.59 16.67 Bs0.33,6 6.0 16.04 21.79 5.75 16.67
Bs2
Bs0.73,3
0.733
3.0 61.44 81.77 20.33 16.67 Bs0.73.3.5 3.5 56.72 77.70 20.98 16.67 Bs0.73,4 4.0 51.74 67.27 15.53 16.67 Bs0.73,4.5 4.5 46.78 62.60 15.82 16.67 Bs0.73,5 5.0 42.01 57.68 15.67 16.67 Bs0.33,5.5 5.5 36.97 53.01 16.04 16.67 Bs0.73,6 6.0 26.74 48.11 21.37 16.67
B3
Bs1,3
1.00
3.0 79.02 95.69 16.67 16.67 Bs1,3.5 3.5 67.96 84.81 16.85 16.67 Bs1,4 4.0 60.36 78.64 18.28 16.67 Bs1,4.5 4.5 57.36 77.53 20.17 16.67 Bs1,5 5.0 50.69 72.92 22.23 16.67 Bs1,5.5 5.5 49.76 64.76 15 16.67 Bs1,6 6.0 38.46 52.91 14.45 16.67
Bs4
Bs1.5,3
1.50
3.0 115.69 125.18 9.49 16.67 Bs1.5.3.5 3.5 103.31 116.10 12.79 16.67 Bs1.5,4 4.0 89.58 95.92 6.34 16.67 Bs1.5,4.5 4.5 79.58 82.21 2.63 16.67 Bs1.5,5 5.0 69.53 71.84 2.31 16.67 Bs1.5,5.5 5.5 62.52 65.93 3.41 16.67 Bs1.5,6 6.0 55.13 58.76 3.63 16.67
203
Bs5
Bs2,3
2.0
3.0 147.69 160.54 12.85 16.67 Bs2,.3.5 3.5 123.98 135.31 11.33 16.67 Bs2,4 4.0 101.61 115.98 14.37 16.67 Bs2,4.5 4.5 95.75 112.66 16.91 16.67 Bs2,5 5.0 85.68 99.37 13.69 16.67 Bs2,5.5 5.5 76.81 95.03 18.22 16.67 Bs2,6 6.0 69.64 77.77 8.13 16.67
Comparison of actual test results of shear strength of HSC beams with the
results given by CSA equations in Tables 8.4, 8.5 and 8.6, gives the following
general observations;
1. Like ACI-318 provision, the provisions of CSA for shear strength of
HSC un-conservative for small values of longitudinal steel both for
beams with and without web reinforcement.
2. The shear strength given by CSA for ρ≥1,1.5% are however
reasonably good and becomes more conservative for 2%.
3. The increase in shear strength of HSC beams due to addition of same
amount of stirrups in all beams is not same, as given in CSA like ACI-
318. Hence the superposition of the two contributions (concrete and
transverse steel) has not been shown in the results by CSA as well.
8.3 AASHTO’s LRFD DESIGN SPECIFICATION (1994)
(Based on Modified Compression Field theory-MCFT)
The values of shear strength are given by the following equations;
Eq (5.20) vvcvvcc dbfdbfV '' 25.0083.0
The actual and calculated values of shear strength are compared with the values
proposed by AASHTO’s LRFD design specification based on Modified
Compression Field theory in Tables 8.7 & 8.8.
204
Table 8.7 Comparison of the shear Strength of beams without web reinforcement with the provisions of MCFT( LRFD Method)
Beam Title ρ (%) a/d
Shear Strength Vtest/VMCFT Vtest
( KN) VMCFT
(KN)
B1
B0.33,3
0.33
3.0 35.24 46.94 0.75 B0.33.3.5 3.5 30.27 46.94 0.64 B0.33,4 4.0 25.11 46.94 0.53 B0.33,4.5 4.5 23.92 46.94 0.51 B0.33,5 5.0 21.06 46.94 0.45 B0.33,5.5 5.5 18.88 46.94 0.40 B0.33,6 6.0 16.04 46.94 0.34
B2
B0.73,3
0.733
3.0 61.44 46.94 1.31 B0.73.3.5 3.5 56.72 46.94 1.21 B0.73,4 4.0 51.74 46.94 1.10 B0.73,4.5 4.5 46.78 46.94 1.00 B0.73,5 5.0 42.01 46.94 0.90 B0.33,5.5 5.5 36.97 46.94 0.79 B0.73,6 6.0 26.74 46.94 0.57
B3
B1,3
1.00
3.0 79.02 46.94 1.68 B1,3.5 3.5 67.96 46.94 1.45 B1,4 4.0 60.36 46.94 1.29 B1,4.5 4.5 57.36 46.94 1.22 B1,5 5.0 50.69 46.94 1.08 B1,5.5 5.5 49.76 46.94 1.06 B1,6 6.0 38.46 46.94 0.82
B4
B1.5,3
1.50
3.0 115.69 47.54 2.43 B1.5.3.5 3.5 103.31 47.54 2.17 B1.5,4 4.0 89.58 47.54 1.88 B1.5,4.5 4.5 79.58 47.54 1.67 B1.5,5 5.0 69.53 47.54 1.46 B1.5,5.5 5.5 62.52 47.54 1.32 B1.5,6 6.0 55.13 47.54 1.16
B5
B2,3
2.0
3.0 147.69 50.04 2.95 B2,.3.5 3.5 123.98 50.04 2.48 B2,4 4.0 101.61 50.04 2.03 B2,4.5 4.5 95.75 50.04 1.91 B2,5 5.0 85.68 50.04 1.71
205
B2,5.5 5.5 76.81 50.04 1.53 B2,6 6.0 69.64 50.04 1.39
Table 8.8 Comparison of the shear Strength of beams with web reinforcement with the provisions of MCFT ( LRFD Method)
Beam Title
a/d
ρ (%)
Vtest ( KN)
Shear capacity as MCFT provisions
( KN)
VMCFT = Vc+Vs
( KN)
Vtest/VMCFT
Vc
Vs Bs1
Bs0.33,3
0.33
3.0 40.18 46.94 19.62 66.56 0.60 Bs0.33.3.5 3.5
36.99 46.94 19.62 66.56 0.56 Bs0.33,4 4.0 31.90 46.94 19.62 66.56 0.48 B0.33,4.5 4.5
34.42 46.94 19.62 66.56 0.52 Bs0.33,5 5.0 31.58 46.94 19.62 66.56 0.47 Bs0.33,5.5 5.5
24.47 46.94 19.62 66.56 0.37 Bs0.33,6 6.0 21.79 46.94 19.62 66.56 0.33
Bs2
Bs0.73,3
0.733
3.0 81.77 46.94 19.62 66.56 1.23
Bs0.73.3.5 3.5 77.70 46.94 19.62 66.56 1.17 Bs0.73,4 4.0
67.27 46.94 19.62 66.56 1.01 Bs0.73,4.5 4.5 62.60 46.94 19.62 66.56 0.94 Bs0.73,5 5.0
57.68 46.94 19.62 66.56 0.87 Bs0.33,5.5 5.5 53.01 46.94 19.62 66.56 0.80 Bs0.73,6 6.0
48.11 46.94 19.62 66.56 0.72 B3
Bs1,3
1.00
3.0 95.69 46.94 19.62 66.56 1.44 Bs1,3.5 3.5 84.81 46.94 19.62 66.56 1.27 Bs1,4 4.0
78.64 46.94 19.62 66.56 1.18 Bs1,4.5 4.5 77.53 46.94 19.62 66.56 1.16 Bs1,5 5.0
72.92 46.94 19.62 66.56 1.10 Bs1,5.5 5.5 64.76 46.94 19.62 66.56 0.97 Bs1,6 6.0
52.91 46.94 19.62 66.56 0.79 Bs4
Bs1.5,3
1.50
3.0 125.18 47.54 19.62 67.16 1.86
Bs1.5.3.5 3.5 116.10 47.54 19.62 67.16 1.73
Bs1.5,4 4.0 95.92 47.54 19.62 67.16 1.43
Bs1.5,4.5 4.5 82.21 47.54 19.62 67.16 1.22
Bs1.5,5 5.0 71.84 47.54 19.62 67.16 1.07
Bs1.5,5.5 5.5 65.93 47.54 19.62 67.16 0.98
Bs1.5,6 6.0 58.76 47.54 19.62 67.16 0.87
Bs2,3
3.0 160.54 50.04 19.62 69.66 2.30
Bs2,.3.5 3.5 135.31 50.04 19.62 69.66 1.94
206
Bs5 Bs2,4 2.0
4.0 115.98 50.04 19.62 69.66 1.66
Bs2,4.5 4.5 112.66 50.04 19.62 69.66 1.62
Bs2,5 5.0 99.37 50.04 19.62 69.66 1.43
Bs2,5.5 5.5 95.03 50.04 19.62 69.66 1.36
Bs2,6 6.0 77.77 50.04 19.62 69.66 1.12
The increase in the shear strength of beams with the additions of stirrups and its
comparison with MCFT, is given in Table 8.9
Table No. 8.9 Comparison of increase in shear strength due to stirrups and MCFT
provision for stirrups contribution.
Beam Title
ρ
(%)
a/d Vtest of beams ( KN)
Vs-test
VsMCFT
Without shear steel
With shear steel
1 2 3 4 5 6 7=6-5 8
Bs1
Bs0.33,3
0.33
3.0 35.24 40.18 4.94 19.62 Bs0.33.3.5 3.5 30.27 36.99 6.72 19.62 Bs0.33,4 4.0 25.11 31.90 6.79 19.62 B0.33,4.5 4.5 23.92 34.42 10.5 19.62 Bs0.33,5 5.0 21.06 31.58 10.52 19.62 Bs0.33,5.5 5.5 18.88 24.47 5.59 19.62 Bs0.33,6 6.0 16.04 21.79 5.75 19.62
Bs2
Bs0.73,3
0.733
3.0 61.44 81.77 20.33 19.62 Bs0.73.3.5 3.5 56.72 77.70 20.98 19.62 Bs0.73,4 4.0 51.74 67.27 15.53 19.62 Bs0.73,4.5 4.5 46.78 62.60 15.82 19.62 Bs0.73,5 5.0 42.01 57.68 15.67 19.62 Bs0.33,5.5 5.5 36.97 53.01 16.04 19.62 Bs0.73,6 6.0 26.74 48.11 21.37 19.62
B3
Bs1,3
1.00
3.0 79.02 95.69 16.67 19.62 Bs1,3.5 3.5 67.96 84.81 16.85 19.62 Bs1,4 4.0 60.36 78.64 18.28 19.62 Bs1,4.5 4.5 57.36 77.53 20.17 19.62 Bs1,5 5.0 50.69 72.92 22.23 19.62 Bs1,5.5 5.5 49.76 64.76 15 19.62 Bs1,6 6.0 38.46 52.91 14.45 19.62
Bs4
Bs1.5,3
1.50
3.0 115.69 125.18 9.49 19.62 Bs1.5.3.5 3.5 103.31 116.10 12.79 19.62 Bs1.5,4 4.0 89.58 95.92 6.34 19.62 Bs1.5,4.5 4.5 79.58 82.21 2.63 19.62 Bs1.5,5 5.0 69.53 71.84 2.31 19.62 Bs1.5,5.5 5.5 62.52 65.93 3.41 19.62 Bs1.5,6 6.0 55.13 58.76 3.63 19.62
Bs2,3 3.0 147.69 160.54 12.85 19.62
207
Bs5
Bs2,.3.5
2.0
3.5 123.98 135.31 11.33 19.62 Bs2,4 4.0 101.61 115.98 14.37 19.62 Bs2,4.5 4.5 95.75 112.66 16.91 19.62 Bs2,5 5.0 85.68 99.37 13.69 19.62 Bs2,5.5 5.5 76.81 95.03 18.22 19.62 Bs2,6 6.0 69.64 77.77 8.13 19.62
Comparison of actual test results of HSC beams with the values given by LRFD
(MCFT) methods in Table 8.6 ,Table 8.7 and Table 8.9, give the following
general observations;
1. Except for minimum value of ρ, LRFD gives reasonably good prediction of
the shear strength of HSC beams for ρ≤1%.
2. For ρ = 1.5% and 2%, LFRD gives conservative results.
3. Uniform increase in the shear strength due to addition of stirrups as given
in the Code, is not experimentally demonstrated.
208
8.4 Comparison of observed values with the provisions of Eurocode-02 The relevant equations of EC-02 are given as follows;
Eq (5.2) dbkV wlrdRD )402.1(1 Where ,5
5.21
x
d
,0.1)1000/6.1( dk 5.0200/7.0 ylf
The results are given in Tables 8.10 and 8.11 Table 8.10 Comparison of the shear Strength of beams without web reinforcement with the provisions of EC-02
Beam Title
ρ (%) a/d
Shear Strength Vtest/VEC02 Vtest
( KN) VEC-02
(KN)
B1
B0.33,3
0.33
3.0 35.24 36.70 0.96 B0.33.3.5 3.5 30.27 36.70 0.82 B0.33,4 4.0 25.11 36.70 0.68 B0.33,4.5 4.5 23.92 36.70 0.65 B0.33,5 5.0 21.06 36.70 0.57 B0.33,5.5 5.5 18.88 36.70 0.51 B0.33,6 6.0 16.04 36.70 0.44
B2
B0.73,3
0.733
3.0 61.44 41.14 1.49 B0.73.3.5 3.5 56.72 41.14 1.38 B0.73,4 4.0 51.74 41.14 1.26 B0.73,4.5 4.5 46.78 41.14 1.14 B0.73,5 5.0 42.01 41.14 1.02 B0.33,5.5 5.5 36.97 41.14 0.90 B0.73,6 6.0 26.74 41.14 0.65
B3
B1,3
1.00
3.0 79.02 44.08 1.79 B1,3.5 3.5 67.96 44.08 1.54 B1,4 4.0 60.36 44.08 1.37 B1,4.5 4.5 57.36 44.08 1.30 B1,5 5.0 50.69 44.08 1.15 B1,5.5 5.5 49.76 44.08 1.13 B1,6 6.0 38.46 44.08 0.87
B4
B1.5,3
1.50
3.0 115.69 49.59 2.33 B1.5.3.5 3.5 103.31 49.59 2.08 B1.5,4 4.0 89.58 49.59 1.81 B1.5,4.5 4.5 79.58 49.59 1.60 B1.5,5 5.0 69.53 49.59 1.40 B1.5,5.5 5.5 62.52 49.59 1.26 B1.5,6 6.0 55.13 49.59 1.11
209
B5
B2,3
2.0
3.0 147.69 55.10 2.68 B2,.3.5 3.5 123.98 55.10 2.25 B2,4 4.0 101.61 55.10 1.84 B2,4.5 4.5 95.75 55.10 1.74 B2,5 5.0 85.68 55.10 1.56 B2,5.5 5.5 76.81 55.10 1.39 B2,6 6.0 69.64 55.10 1.26
Table No. 8.11 Comparison of the shear Strength of beams with web reinforcement with the provisions of EC-02
Beam Title
a/d
ρ
(%)
Vtest ( KN)
Shear capacity as per EC-02 provisions
( KN)
V EC02 = Vc+Vs
( KN)
Vtest/VEC02
Vc Vs
Bs1
Bs0.33,3
0.33
3.0 40.18 36.70 17.08 53.77 0.75 Bs0.33.3.5 3.5 36.99 36.70 17.08 53.77 0.69 Bs0.33,4 4.0 31.90 36.70 17.08 53.77 0.59 B0.33,4.5 4.5 34.42 36.70 17.08 53.77 0.64 Bs0.33,5 5.0 31.58 36.70 17.08 53.77 0.59 Bs0.33,5.5 5.5 24.47 36.70 17.08 53.77 0.46 Bs0.33,6 6.0 21.79 36.70 17.08 53.77 0.41
Bs2
Bs0.73,3
0.733
3.0 81.77 41.14 17.08 58.21 1.40 Bs0.73.3.5 3.5 77.70 41.14 17.08 58.21 1.33 Bs0.73,4 4.0 67.27 41.14 17.08 58.21 1.16 Bs0.73,4.5 4.5 62.60 41.14 17.08 58.21 1.08 Bs0.73,5 5.0 57.68 41.14 17.08 58.21 0.99 Bs0.33,5.5 5.5 53.01 41.14 17.08 58.21 0.91 Bs0.73,6 6.0 48.11 41.14 17.08 58.21 0.83
B3
Bs1,3
1.00
3.0 95.69 44.08 17.08 61.16 1.56 Bs1,3.5 3.5 84.81 44.08 17.08 61.16 1.39 Bs1,4 4.0 78.64 44.08 17.08 61.16 1.29 Bs1,4.5 4.5 77.53 44.08 17.08 61.16 1.27 Bs1,5 5.0 72.92 44.08 17.08 61.16 1.19 Bs1,5.5 5.5 64.76 44.08 17.08 61.16 1.06 Bs1,6 6.0 52.91 44.08 17.08 61.16 0.87
Bs4
Bs1.5,3
1.50
3.0 125.18 49.59 17.08 66.67 1.88 Bs1.5.3.5 3.5 116.10 49.59 17.08 66.67 1.74 Bs1.5,4 4.0 95.92 49.59 17.08 66.67 1.44 Bs1.5,4.5 4.5 82.21 49.59 17.08 66.67 1.23 Bs1.5,5 5.0 71.84 49.59 17.08 66.67 1.08 Bs1.5,5.5 5.5 65.93 49.59 17.08 66.67 0.99 Bs1.5,6 6.0 58.76 49.59 17.08 66.67 0.88
Bs5
Bs2,3
2.0
3.0 160.54 55.10 17.08 72.18 2.22 Bs2,.3.5 3.5 135.31 55.10 17.08 72.18 1.87 Bs2,4 4.0 115.98 55.10 17.08 72.18 1.61 Bs2,4.5 4.5 112.66 55.10 17.08 72.18 1.56 Bs2,5 5.0 99.37 55.10 17.08 72.18 1.38
210
Bs2,5.5 5.5 95.03 55.10 17.08 72.18 1.32 Bs2,6 6.0 77.77 55.10 17.08 69.66 1.08
The increase in the shear strength of beams with the additions of stirrups and its
comparison with EC-02, is given in Table 8.12
Table No. 8.12 Comparison of increase in shear strength due to stirrups and EC-02
provision for stirrups contribution.
Beam Title
ρ
(%)
a/d Vtest of beams ( KN)
Vs-test
VsEC-02
Without shear steel
With shear steel
1 2 3 4 5 6 7=6-5 8
Bs1
Bs0.33,3
0.33
3.0 35.24 40.18 4.94 17.08 Bs0.33.3.5 3.5 30.27 36.99 6.72 17.08 Bs0.33,4 4.0 25.11 31.90 6.79 17.08 B0.33,4.5 4.5 23.92 34.42 10.5 17.08 Bs0.33,5 5.0 21.06 31.58 10.52 17.08 Bs0.33,5.5 5.5 18.88 24.47 5.59 17.08 Bs0.33,6 6.0 16.04 21.79 5.75 17.08
Bs2
Bs0.73,3
0.733
3.0 61.44 81.77 20.33 17.08 Bs0.73.3.5 3.5 56.72 77.70 20.98 17.08 Bs0.73,4 4.0 51.74 67.27 15.53 17.08 Bs0.73,4.5 4.5 46.78 62.60 15.82 17.08 Bs0.73,5 5.0 42.01 57.68 15.67 17.08 Bs0.33,5.5 5.5 36.97 53.01 16.04 17.08 Bs0.73,6 6.0 26.74 48.11 21.37 17.08
B3
Bs1,3
1.00
3.0 79.02 95.69 16.67 17.08 Bs1,3.5 3.5 67.96 84.81 16.85 17.08 Bs1,4 4.0 60.36 78.64 18.28 17.08 Bs1,4.5 4.5 57.36 77.53 20.17 17.08 Bs1,5 5.0 50.69 72.92 22.23 17.08 Bs1,5.5 5.5 49.76 64.76 15 17.08 Bs1,6 6.0 38.46 52.91 14.45 17.08
Bs4
Bs1.5,3
1.50
3.0 115.69 125.18 9.49 17.08 Bs1.5.3.5 3.5 103.31 116.10 12.79 17.08 Bs1.5,4 4.0 89.58 95.92 6.34 17.08 Bs1.5,4.5 4.5 79.58 82.21 2.63 17.08 Bs1.5,5 5.0 69.53 71.84 2.31 17.08 Bs1.5,5.5 5.5 62.52 65.93 3.41 17.08 Bs1.5,6 6.0 55.13 58.76 3.63 17.08
Bs2,3 3.0 147.69 160.54 12.85 17.08
211
Bs5
Bs2,.3.5
2.0
3.5 123.98 135.31 11.33 17.08 Bs2,4 4.0 101.61 115.98 14.37 17.08 Bs2,4.5 4.5 95.75 112.66 16.91 17.08 Bs2,5 5.0 85.68 99.37 13.69 17.08 Bs2,5.5 5.5 76.81 95.03 18.22 17.08 Bs2,6 6.0 69.64 77.77 8.13 17.08
Comparison of actual test results of HSC beams with the values given by EC-02,
in Table 8.10, 8.11 and Table 8.12 give the following general observations;
1. Except for minimum value of ρ, EC-02 gives reasonably good prediction of
the shear strength of HSC beams for ρ up to 1%.
2. For ρ = 1.5% and 2%, EC-02is conservative. However the degree of
safety is relatively less when compared with the other equations already
discussed.
3. Uniform increase in the shear strength due to addition of stirrups as given
in EC-02 Code, is not exhibited.
8.5 New Theory Proposed by Prodromos D.Zararis (2003) P. D.Zararis (2003), has reported a new equation for the design of shear strength
and minimum shear reinforcement of RC beams, in ACI structural Journal March-
April, 2003. The actual and theoretical values given by the proposed equations
are compared in Table 8.13 and Table 8.14.
212
Table No. 8.13 Comparison of the shear Strength of beams without web reinforcement with equation proposed in new theory of Zararis,P.D.
Beam Title
ρ (%) a/d
Shear Strength Vtest/VNew Theo Vtest
( KN) VNew Theo
(KN)
B1
B0.33,3
0.33
3.0 35.24 41.50 0.85 B0.33.3.5 3.5 30.27 38.31 0.79 B0.33,4 4.0 25.11 30.62 0.82 B0.33,4.5 4.5 23.92 24.35 0.98 B0.33,5 5.0 21.06 21.02 1.00 B0.33,5.5 5.5 18.88 17.75 1.06 B0.33,6 6.0 16.04 10.83 1.48
B2
B0.73,3
0.733
3.0 61.44 41.50 1.48 B0.73.3.5 3.5 56.72 38.31 1.48 B0.73,4 4.0 51.74 34.93 1.48 B0.73,4.5 4.5 46.78 31.58 1.48 B0.73,5 5.0 42.01 28.36 1.48 B0.33,5.5 5.5 36.97 24.95 1.48 B0.73,6 6.0 26.74 18.05 1.48
B3
B1,3
1.00
3.0 79.02 43.17 1.83 B1,3.5 3.5 67.96 41.93 1.62 B1,4 4.0 60.36 38.48 1.57 B1,4.5 4.5 57.36 35.23 1.63 B1,5 5.0 50.69 31.88 1.59 B1,5.5 5.5 49.76 28.66 1.74 B1,6 6.0 38.46 25.25 1.52
B4
B1.5,3
1.50
3.0 115.69 81.13 1.43 B1.5.3.5 3.5 103.31 73.33 1.41 B1.5,4 4.0 89.58 70.66 1.27 B1.5,4.5 4.5 79.58 60.41 1.32 B1.5,5 5.0 69.53 57.21 1.22 B1.5,5.5 5.5 62.52 50.84 1.23 B1.5,6 6.0 55.13 46.90 1.18
B5
B2,3
2.0
3.0 147.69 73.21 2.02 B2,.3.5 3.5 123.98 69.41 1.79 B2,4 4.0 101.61 67.39 1.51 B2,4.5 4.5 95.75 59.75 1.60 B2,5 5.0 85.68 53.51 1.60 B2,5.5 5.5 76.81 50.47 1.52 B2,6 6.0 69.64 46.90 1.48
213
Table No. 8.14 Comparison of the shear Strength of beams with web reinforcement with equation proposed in new theory of Zararis,P.D (2003).
Beam Title
ρ
(%)
a/d Vtest ( KN)
Shear capacity as New theory eq of
Zararis ( KN)
Vnew theory
= (Vc+Vs) ( KN)
Vtest/Vnew theory
Vc
Vs Bs1
Bs0.33,3
0.33
3.0 40.18 41.50 13.62 55.12 0.73 Bs0.33.3.5 3.5 36.99 38.31 14.98 53.29 0.69 Bs0.33,4 4.0 31.90 30.62 16.34 46.96 0.68 B0.33,4.5 4.5 34.42 24.35 17.70 42.05 0.82 Bs0.33,5 5.0 31.58 21.02 19.06 40.08 0.79 Bs0.33,5.5 5.5 24.47 17.75 20.42 38.17 0.64 Bs0.33,6 6.0 21.79 10.83 21.78 32.61 0.67
Bs2
Bs0.73,3
0.733
3.0 81.77 41.50 13.62 55.12 1.48 Bs0.73.3.5 3.5 77.70 38.31 14.98 53.29 1.46 Bs0.73,4 4.0 67.27 34.93 16.34 51.27 1.31 Bs0.73,4.5 4.5 62.60 31.58 17.70 49.28 1.27 Bs0.73,5 5.0 57.68 28.36 19.06 47.42 1.22 Bs0.33,5.5 5.5 53.01 24.95 20.42 45.38 1.17 Bs0.73,6 6.0 48.11 18.05 21.78 39.83 1.21
B3
Bs1,3
1.00
3.0 95.69 43.17 13.62 56.79 1.69 Bs1,3.5 3.5 84.81 41.93 14.98 56.91 1.49 Bs1,4 4.0 78.64 38.48 16.34 54.82 1.43 Bs1,4.5 4.5 77.53 35.23 17.70 52.93 1.46 Bs1,5 5.0 72.92 31.88 19.06 50.94 1.43 Bs1,5.5 5.5 64.76 28.66 20.42 49.08 1.32 Bs1,6 6.0 52.91 25.25 21.78 47.04 1.12
Bs4
Bs1.5,3
1.50
3.0 125.18 81.13 13.62 94.75 1.32 Bs1.5.3.5 3.5 116.10 73.33 14.98 88.30 1.31 Bs1.5,4 4.0 95.92 70.66 16.34 87.00 1.10 Bs1.5,4.5 4.5 82.21 60.41 17.70 78.12 1.05 Bs1.5,5 5.0 71.84 57.21 19.06 76.27 0.94 Bs1.5,5.5 5.5 65.93 50.84 20.42 71.26 0.93 Bs1.5,6 6.0 58.76 46.90 21.78 68.69 0.86
Bs5
Bs2,3
2.0
3.0 160.54 73.21 13.62 86.83 1.85 Bs2,.3.5 3.5 135.31 69.41 14.98 84.38 1.60 Bs2,4 4.0 115.98 67.39 16.34 83.73 1.39 Bs2,4.5 4.5 112.66 59.75 17.70 77.45 1.45 Bs2,5 5.0 99.37 53.51 19.06 72.57 1.37
214
Bs2,5.5 5.5 95.03 50.47 20.42 70.90 1.34 Bs2,6 6.0 77.77 46.90 21.78 68.69 1.13
The increase in the shear strength of beams with the additions of stirrups and its
comparison with EC-02, is given in Table 8.12
Table No. 8.15 Comparison of increase in shear strength due to stirrups and New theory
of Zararis.P (2003) for stirrups contribution.
Beam Title
ρ
(%)
a/d Vtest of beams ( KN)
Vs-test
Vsnew theory
Without shear steel
With shear steel
1 2 3 4 5 6 7=6-5 8
Bs1
Bs0.33,3
0.33
3.0 35.24 40.18 4.94 13.62 Bs0.33.3.5 3.5 30.27 36.99 6.72 14.98 Bs0.33,4 4.0 25.11 31.90 6.79 16.34 B0.33,4.5 4.5 23.92 34.42 10.5 17.70 Bs0.33,5 5.0 21.06 31.58 10.52 19.06 Bs0.33,5.5 5.5 18.88 24.47 5.59 20.42 Bs0.33,6 6.0 16.04 21.79 5.75 21.78
Bs2
Bs0.73,3
0.733
3.0 61.44 81.77 20.33 13.62 Bs0.73.3.5 3.5 56.72 77.70 20.98 14.98 Bs0.73,4 4.0 51.74 67.27 15.53 16.34 Bs0.73,4.5 4.5 46.78 62.60 15.82 17.70 Bs0.73,5 5.0 42.01 57.68 15.67 19.06 Bs0.33,5.5 5.5 36.97 53.01 16.04 20.42 Bs0.73,6 6.0 26.74 48.11 21.37 21.78
B3
Bs1,3
1.00
3.0 79.02 95.69 16.67 13.62 Bs1,3.5 3.5 67.96 84.81 16.85 14.98 Bs1,4 4.0 60.36 78.64 18.28 16.34 Bs1,4.5 4.5 57.36 77.53 20.17 17.70 Bs1,5 5.0 50.69 72.92 22.23 19.06 Bs1,5.5 5.5 49.76 64.76 15 20.42 Bs1,6 6.0 38.46 52.91 14.45 21.78
Bs4
Bs1.5,3
1.50
3.0 115.69 125.18 9.49 13.62 Bs1.5.3.5 3.5 103.31 116.10 12.79 14.98 Bs1.5,4 4.0 89.58 95.92 6.34 16.34 Bs1.5,4.5 4.5 79.58 82.21 2.63 17.70 Bs1.5,5 5.0 69.53 71.84 2.31 19.06 Bs1.5,5.5 5.5 62.52 65.93 3.41 20.42 Bs1.5,6 6.0 55.13 58.76 3.63 21.78
Bs2,3
3.0 147.69 160.54 12.85 13.62 Bs2,.3.5 3.5 123.98 135.31 11.33 14.98
215
Bs5
Bs2,4 2.0
4.0 101.61 115.98 14.37 16.34 Bs2,4.5 4.5 95.75 112.66 16.91 17.70 Bs2,5 5.0 85.68 99.37 13.69 19.06 Bs2,5.5 5.5 76.81 95.03 18.22 20.42 Bs2,6 6.0 69.64 77.77 8.13 21.78
The comparison of values given by the proposed new equation with the actual
values of shear strength of HSC beams shows that the new equation of Zararis
gives very closer values for almost all level of longitudinal steel, except for
minimum longitudinal steel. The equations proposed are the best estimators of
the discussed methods.
The overall comparison of Vtest/VCode for ACI, CSA, MCFT, EC-02 and New
Equation for beams without web reinforcement has been shown in Table 8.16,
whereas the values for beams with web reinforcement have been shown in Table
8.17. The final comparison of Means, Standards deviations and Coefficient of
Variation of Vtest/VCode for ACI, CSA, MCFT, EC-02 have been shown in Table
8.18 and 8.19.
216
Beam Title
ρ %
a/d Vtest/VCode.
ACI CSA MCFT ( LRFD)
EC-02 New Eq.
B1
B0.33,3
0.33
3 0.78 0.87 0.75 0.96 0.75 B0.33.3.5 3.5 0.67 0.75 0.64 0.82 0.64 B0.33,4 4 0.56 0.62 0.53 0.68 0.53 B0.33,4.5 4.5 0.53 0.59 0.51 0.65 0.51 B0.33,5 5 0.47 0.52 0.45 0.57 0.45 B0.33,5.5 5.5 0.42 0.47 0.40 0.51 0.40 B0.33,6 6 0.36 0.40 0.34 0.44 0.34
Mean 0.54 0.60 0.52 0.66 0.52 Stand Dev 0.28 0.150 0.13 0.166 0.130
CoV(%) 21.17 25.15 25.09 25.29 25.09 B2
B0.73,3
0.733
3 1.33 1.52 1.31 1.49 1.31 B0.73.3.5 3.5 1.23 1.41 1.21 1.38 1.21 B0.73,4 4 1.13 1.28 1.10 1.26 1.10 B0.73,4.5 4.5 1.02 1.16 1.00 1.14 1.00 B0.73,5 5 0.92 1.04 0.90 1.02 0.90 B0.33,5.5 5.5 0.81 0.92 0.79 0.90 0.79 B0.73,6 6 0.59 0.66 0.57 0.65 0.57
Mean 1.00 1.14 0.98 1.12 0.98 Stand dev 0.236 0.274 0.235 0.268 0.235 CoV 23.59 24.05 23.99 23.95 23.99 B3
B1,3
1.00
3 1.77 1.96 1.68 1.79 1.68 B1,3.5 3.5 1.52 1.69 1.45 1.54 1.45 B1,4 4 1.35 1.50 1.29 1.37 1.29 B1,4.5 4.5 1.29 1.42 1.22 1.30 1.22 B1,5 5 1.14 1.26 1.08 1.15 1.08 B1,5.5 5.5 1.12 1.23 1.06 1.13 1.06 B1,6 6 0.86 0.95 0.82 0.87 0.82
Mean 1.29 1.43 1.23 1.31 1.23 Stand Dev 0.273 0.305 0.26 0.278 0.260 CoV 21.19 21.35 21.15 21.17 21.15 B4
B1.5,3
1.50
3 2.40 2.87 2.43 2.33 2.43 B1.5.3.5 3.5 2.17 2.56 2.17 2.08 2.17 B1.5,4 4 1.89 2.22 1.88 1.81 1.88 B1.5,4.5 4.5 1.69 1.97 1.67 1.60 1.67 B1.5,5 5 1.49 1.73 1.46 1.40 1.46 B1.5,5.5 5.5 1.34 1.55 1.32 1.26 1.32 B1.5,6 6 1.19 1.37 1.16 1.11 1.16
Mean 1.74 2.04 1.73 1.66 1.73 Stand Dev 0.408 0.504 0.427 0.411 0.42
CoV 23.50 24.73 24.67 24.75 24.67 B5
B2,3
2.0
3 2.99 3.66 2.95 2.68 2.95 B2,.3.5 3.5 2.54 3.08 2.48 2.25 2.48 B2,4 4 2.11 2.52 2.03 1.84 2.03 B2,4.5 4.5 2.00 2.38 1.91 1.74 1.91 B2,5 5 1.81 2.13 1.71 1.56 1.71 B2,5.5 5.5 1.63 1.91 1.53 1.39 1.53 B2,6 6 1.48 1.73 1.39 1.26 1.41
217
Table No. 8.16 Comparison of Vtest/VCode for ACI, CSA, MCFT, EC-02 and New Equation for beams without web reinforcement. Table No. 8.17 Comparison of Vtest/VCode for ACI, CSA, MCFT, EC-02 and New Equation for beams with web reinforcement.
Mean 2.08 2.49 2.00 1.82 2.00 Stand Dev 0.490 0.629 0.510 0.462 0.506
CoV 23.57 25.78 25.49 25.41 25.32
Beam Title
ρ %
a/d Vtest/VCode.
ACI CSA MCFT( LRFD) EC-02 New Eq.
Bs1
Bs0.33,3
0.33
3 0.75 0.71 0.60 0.75 0.73 Bs0.33.3.5 3.5 0.69 0.65 0.56 0.69 0.69 Bs0.33,4 4 0.60 0.56 0.48 0.59 0.68 B0.33,4.5 4.5 0.65 0.60 0.52 0.64 0.82 Bs0.33,5 5 0.60 0.55 0.47 0.59 0.79 Bs0.33,5.5 5.5 0.46 0.43 0.37 0.46 0.64 Bs0.33,6 6 0.41 0.38 0.33 0.41 0.67
Mean 0.59 0.55 0.48 0.59 0.72 Stand Dev 0.11 0.108 0.090 0.11 0.06 CoV 19.07 19.62 18.82 18.94 8.5 Bs2
Bs0.73,3
0.733
3 1.51 1.44 1.23 1.40 1.48 Bs0.73.3.5 3.5 1.44 1.36 1.17 1.33 1.46 Bs0.73,4 4 1.25 1.18 1.01 1.16 1.31 Bs0.73,4.5 4.5 1.16 1.10 0.94 1.08 1.27 Bs0.73,5 5 1.08 1.01 0.87 0.99 1.22 Bs0.33,5.5 5.5 0.99 0.93 0.80 0.91 1.17 Bs0.73,6 6 0.90 0.84 0.72 0.83 1.21
Mean 1.19 1.12 0.96 1.10 1.30 Stand Dev 0.20 0.20 0.17 0.19 0.11 CoV 17.56 18.19 18.09 17.78 8.74 Bs3
Bs1,3
1.00
3 1.81 1.68 1.44 1.56 1.69 Bs1,3.5 3.5 1.61 1.49 1.27 1.39 1.49 Bs1,4 4 1.49 1.38 1.18 1.29 1.43 Bs1,4.5 4.5 1.47 1.36 1.16 1.27 1.46 Bs1,5 5 1.38 1.28 1.10 1.19 1.43 Bs1,5.5 5.5 1.23 1.14 0.97 1.06 1.32 Bs1,6 6 1.00 0.93 0.79 0.87 1.12
Mean 1.43 1.32 1.13 1.23 1.42 Stand Dev 0.24 0.22 0.193 0.206 0.16 CoV 16.91 16.93 17.09 16.82 11.28 Bs4
Bs1.5,3
1.50
3 2.23 2.20 1.86 1.88 1.32 Bs1.5.3.5 3.5 2.09 2.04 1.73 1.74 1.31 Bs1.5,4 4 1.74 1.68 1.43 1.44 1.10 Bs1.5,4.5 4.5 1.50 1.44 1.22 1.23 1.05 Bs1.5,5 5 1.31 1.26 1.07 1.08 0.94 Bs1.5,5.5 5.5 1.21 1.16 0.98 0.99 0.93 Bs1.5,6 6 1.08 1.03 0.87 0.88 0.86
Mean 1.59 1.54 1.31 1.32 1.37 Stand Dev 0.409 0.41 0.35 0.35 0.34 CoV 25.75 26.86 26.78 26.75 24.98 Bs5
Bs2,3
2.0
3 2.80 2.82 2.30 2.22 1.85 Bs2,.3.5 3.5 2.39 2.37 1.94 1.87 1.60 Bs2,4 4 2.07 2.04 1.66 1.61 1.39 Bs2,4.5 4.5 2.02 1.98 1.62 1.56 1.45 Bs2,5 5 1.79 1.74 1.43 1.38 1.37 Bs2,5.5 5.5 1.72 1.67 1.36 1.32 1.34
218
Table 8.18 Summary of means of the ratios of observed values and different code
values For shear strength of beams without web reinforcement
Statistical
Parameters Vtest/Vcode
ACI Canadian MCFT Eurocode New theory ρ = 0.33%
Mean 0.54 0.60 0.52 0.66 0.52 Stand Dev 0.28 0.150 0.13 0.166 0.130 CoV 21.17 25.15 25.09 25.29 25.09 ρ = 0.73%Mean 1.00 1.14 0.98 1.12 0.98 Stand Dev 0.236 0.274 0.235 0.268 0.235 CoV 23.59 24.05 23.99 23.95 23.99 ρ = 1%Mean 1.29 1.43 1.23 1.31 1.23 Stand Dev 0.273 0.305 0.26 0.278 0.260 CoV 21.19 21.35 21.15 21.17 21.15 ρ = 1.5% Mean 1.59 1.54 1.31 1.32 1.37 Stand Dev 0.409 0.41 0.35 0.35 0.34 CoV 25.75 26.86 26.78 26.75 24.98 ρ = 2% Mean 2.08 2.49 2.00 1.82 2.00 Stand Dev 0.490 0.629 0.510 0.462 0.506 CoV 23.57 25.78 25.49 25.41 25.32 Mean of Means 1.300 1.44 1.208 1.246 1.22
Table 8.19 Summary of means of the ratios of observed values and different code
values for shear Strength of beams with web reinforcement.
Statistical Parameters
Vtest/VcodeACI Canadian MCFT Eurocode New theory
ρ = 0.33%
Mean 0.59 0.55 0.48 0.59 0.72 Stand Dev 0.11 0.108 0.090 0.11 0.06 CoV 19.07 19.62 18.82 18.94 8.5 ρ = 0.73%Mean 1.19 1.12 0.96 1.10 1.30 Stand Dev 0.20 0.20 0.17 0.19 0.11 CoV 17.56 18.19 18.09 17.78 8.74 ρ = 1%Mean 1.43 1.32 1.13 1.23 1.42 Stand Dev 0.24 0.22 0.193 0.206 0.16 CoV 16.91 16.93 17.09 16.82 11.28 ρ = 1.5%Mean 1.59 1.54 1.31 1.32 1.37
Bs2,6 6 1.42 1.36 1.12 1.08 1.13 Mean 2.03 2.00 1.63 1.58 1.45 Stand Dev 0.42 0.45 0.36 0.35 0.21 CoV 20.82 22.34 22.23 22.09 14.41
219
Stand Dev 0.409 0.41 0.35 0.35 0.34 CoV 25.75 26.86 26.78 26.75 24.98 ρ = 2% Mean 2.03 2.00 1.63 1.58 1.45 Stand Dev 0.42 0.45 0.36 0.35 0.21 CoV 20.82 22.34 22.23 22.09 14.41 Mean of Means 1.37 1.31 1.102 1.164 1.252
From the comparison of Vtest/Vcode given in tables 8.16 thru 8.19, the following
general comments can be made;
1. All the equation of various codes and methods discussed in the study are
not safe for the shear design of HSC beams for minimum longitudinal steel
ratio for both beams with and without web reinforcement.
2. For beams without web reinforcement, the corresponding values Vtest/Vcode
have increased for ρ=0.33% and 0.73% for all equations but these are still
un-conservative. Hence the equations given by most of the codes and
methods discussed are un-conservative for HSC beams with ρ<1% for both
the cases with and without web reinforcement.
3. The equations are giving reasonably good predictions of the shear strength
of HSC beams without web reinforcement for ρ = 1% and 1.5%.
4. For large values of ρ = 2%, all the equations are giving half of the actual test
values. Hence the equations considered are over conservative for ρ=2%.
5. The additions of stirrups have increased the shear strength of all HSC
beams, as generally given by the equations of shear design. But the
increase is random and irregular. The basic assumption of most of the
building codes and shear design equation for summing up the individual
contribution of concrete and stirrups has not been observed.
220
6. The shear design of HSC beams at low level of longitudinal steel by all the
methods and equations considered need further work and verification of the
improvement of the equations.
7. The shear strength of HSC beams with web reinforcement is generally
considered as the sum of the individual contributions of the concrete and
web steel but in fact is more complicated phenomena as their roles are not
independent. The addition of web steel also affects the role of concrete
contribution and its capacity, as pointed out by Kani( 1964,1969) and
Kotsovos (1984,1989).
8. MCFT and New Equation proposed by Zararis,P.D give reasonably good
estimate of the HSC beams with web reinforcement.
221
Chapter No.9
Statistical Model for the prediction of shear strength of
High Strength Concrete beams.
Chapter Introduction: In this chapter, efforts have been made to develop the non linear regression models for the shear strength of HSC beams based on the observed data. Coefficient of variation has been worked out for the observed values and predicted values by the proposed statistical model. These are also compared with the ACI models and some other models.
9.1 Regression model and its application in Civil Engineering.
In statistics, regression analysis is a collective name used for techniques and
methods to model and analyze the numerical data consisting of values of a
dependent variable (also called response variable or measurement) and of one
or more independent variables (also known as explanatory variables or
predictors). The dependent variable in the regression equation is modeled as a
function of the independent variables, corresponding parameters ("constants"),
and an error term. The error term is treated as a random variable. It represents
unexplained variation in the dependent variable. The parameters are estimated
so as to give a "best fit" of the data. Most commonly the best fit is evaluated by
using the least squares method, but other criteria have also been used.
Regression equation has been used extensively in Civil Engineering for
predicting the compressive strength and shear strength of RC members, besides
other engineering properties of concrete in the fresh and hardened form. Zain et
al (2006) developed a multiple regression model for predicting the compressive
strength of High Performance Concrete (HPC). The equation used the following
parameters for predicting the compressive strength of HPC at various ages of
concrete.
i. Mix proportioning of HPC, including the proportioning of cement, fly ash,
silica fumes, water, fine and coarse aggregates.
222
ii. Slump test of the concrete in fresh form.
iii. Density of concrete.
They used their proposed model for the prediction of the compressive strength of
HPC, which gave a high correlation of 99%.
As already discussed in the literature review, Prodromos D.Zararis (2003) has
reported the following models for the shear strength of beams without web
reinforcement
Eq(2.46) bdf
d
cd
d
aV ctcr )()(2.02.1
The shear Strength of RC beams in complete form is as follows:
Eq (2.49) bdf
d
af
d
cd
d
ayvvct )25.05.0()..2.02.1 .
Karim et al (2000) proposed the following equation for prediction of ultimate shear
stress in beams without web reinforcement.
adfbd
Vc
uc /4.0 dA310 ( SI Units)
(9.4)
Where Ad = d
a for 1.0 < a/d<2.5 and 2.5 for a/d > 2.5
Guray Arsalan, used the Zsutty’s equation vvcc dba
dfV 3/12.2 for a/d ≥ 2
for multiple regression of the values of shear strength of RC beams and deduced
the following models;
223
65.050.0 )(02.0)(15.0 ccc ff (For normal strength of concrete).
(9.5)
65.050.0 )(02.0)(15.0 ccc ff (For high strength concrete). (
9.6)
Razak and Wong ( 2004) used the data of 750 specimen of HPC and developed
a regression equation on the basis of best fit relationship between tensile
strength &compressive strength as well as stiffness & compressive strength of
HPC. They also reported that for HPC, the square root function recommended by
most of the codes for the tensile strength of HPC is not valid.
9.2 Regression equation for HSC beams without web reinforcement
The shear strength of HSC beams without stirrups has studied in this research
depends on three following parameters, studied in the experimental program;
Longitudinal steel ρ = bd
As expressed in percentage (%)
Shear span to depth ratio ( a/d)
Compressive strength of concrete; fc' ( Mpa)
Regression models have been developed on the basis of the test data of 70
beams, by using the trial version 9 of DtatFit software of Oakdale Engineering.
The following three models were tried for the shear stress in beams without web
reinforcement.
)/(' dacbaflc
(9.7)
ddacbaflc )/(
(9.8)
ddacbaflce
)/( (9.9)
The results generated for three model 1 has been given as follows;
Model1. )/( dacbaflc )/(208.067.50026.0 ' daf
lc
(9.10)
224
Number of observations = 35 Number of missing observations = 0 Solver type: Nonlinear Nonlinear iteration limit = 250 Diverging nonlinear iteration limit =10 Number of nonlinear iterations performed = 11 Residual tolerance = 0.0000000001 Sum of Residuals = -3.20488143509065E-02 Average Residual = -9.15680410025899E-04 Residual Sum of Squares (Absolute) = 0.564293097380102 Residual Sum of Squares (Relative) = 0.564 Standard Error of the Estimate = 0.132793671886608 Coefficient of Multiple Determination (R^2) = 0.90 Proportion of Variance Explained = 90.4097492% Adjusted coefficient of multiple determination (Ra^2) = 0.8981035852 Durbin-Watson statistic = 0.448319312620612 Regression Variable Results: a = .026, b =50.678 , c= -0.208
The software has also generated the scatter diagram and proposed regression line passing through the data,
shown in Figure 9.1
Figure 9.1 Plot of the proposed model generated by the software.
Model 2 gdacbaflc )/( 765.0)/(204.032.480407.0 ' daf
lc
(9.11)
Solver type: Nonlinear Nonlinear iteration limit = 250 Diverging nonlinear iteration limit =12 Number of nonlinear iterations performed = 13 Residual tolerance = 0.0000000001 Sum of Residuals = -2.2048814350 E-02
225
Average Residual = -9.1532410025899E-04 Residual Sum of Squares (Absolute) = 0.564293097380102 Residual Sum of Squares (Relative) = 0.664 Standard Error of the Estimate = 0.132793671886608 Coefficient of Multiple Determination (R^2) = 0.94 Proportion of Variance Explained = 90.4097492% Adjusted coefficient of multiple determination (Ra^2) = 0.8981035852 Durbin-Watson statistic = 0.448319312620612 Regression Variable Results: a = 0.0407, b =48.32 , c= -0.204 , g= -0.765
Model 3. 174.3)/(189.031.44065.0 ' daf
lce (9.12)
Solver type: Nonlinear Nonlinear iteration limit = 250 Diverging nonlinear iteration limit =12 Number of nonlinear iterations performed = 13 Residual tolerance = 0.0000000001 Sum of Residuals = -2.2048814350 E-02 Average Residual = -8.1532410025899E-04 Residual Sum of Squares (Absolute) = 0.564293097380102 Residual Sum of Squares (Relative) = 0.664 Standard Error of the Estimate = 0.132793671886608 Coefficient of Multiple Determination (R^2) = 0.94 Proportion of Variance Explained = 90.4097492% Adjusted coefficient of multiple determination (Ra^2) = 0.8981035852 Durbin-Watson statistic = 0.448319312620612 Regression Variable Results: a = 0.065, b =44.31 , c= -0.189 , d= -3.174
Table 9.1 Comparison of actual and predicated values of shear stress of High Strength concrete
beams without web reinforcement for three proposed models.
226
The validity of the proposed model can be questioned on the basis of limited test
data and very few parameters considered in the model, however it is expected
that the initial work shall be further elaborated and new equations shall be
Beam Title fc΄
(MPa)
(ρ)
( %)
a/d Shear Stress
( MPa)
Absolute Residual
Actual Predicted Model 1
Model 2
Model 3
Model 1
Model 2
Model 3
B1
B0.33,3 50
0.33
3.00 0.95 0.84 0.82 0.74 0.09 0.13 0.21 B0.33.3. 50 3.50 0.88 0.74 0.72 0.67 0.03 0.16 0.21 B0.33,4 50 4.00 0.70 0.63 0.61 0.61 0.20 0.09 0.09 B0.33,4. 50 4.50 0.56 0.53 0.51 0.55 0.17 0.05 0.01 B0.33,5 50 5.00 0.48 0.43 0.41 0.50 0.14 0.07 0.02 B0.33,5. 50 5.50 0.41 0.32 0.31 0.46 0.11 0.10 0.05 B0.33,6 50 6.00 0.25 0.22 0.21 0.42 0.08 0.04 0.17
B2
B0.73,3 54
0.73
3.00 0.95 1.15 1.18 1.14 0.06 0.23 0.19 B0.73.3. 54 3.50 0.88 1.05 1.07 1.04 0.12 0.19 0.16 B0.73,4 54 4.00 0.80 0.94 0.97 0.95 0.19 0.17 0.15 B0.73,4. 54 4.50 0.73 0.84 0.87 0.86 0.12 0.14 0.13 B0.73,5 54 5.00 0.65 0.73 0.77 0.78 0.09 0.12 0.13 B0.33,5. 54 5.50 0.57 0.63 0.67 0.71 0.06 0.10 0.14 B0.73,6 54 6.00 0.41 0.53 0.56 0.65 0.04 0.15 0.24
B3
B1,3 50
1.00
3.00 0.99 1.18 1.14 0.99 0.00 0.15 0.00 B1,3.5 50 3.50 0.96 1.08 1.04 0.90 0.02 0.08 0.06 B1,4 50 4.00 0.88 0.97 0.94 0.82 0.29 0.06 0.06 B1,4.5 50 4.50 0.81 0.87 0.84 0.75 0.22 0.03 0.06 B1,5 50 5.00 0.73 0.77 0.73 0.68 0.26 0.00 0.05 B1,5.5 50 5.50 0.66 0.66 0.63 0.62 0.14 0.03 0.04 B1,6 50 6.00 0.58 0.56 0.53 0.56 0.16 0.05 0.02
B4
B1.5,3 55
1.50
3.00 1.86 1.57 1.59 1.72 0.12 0.27 0.14 B1.5.3.5 55 3.50 1.68 1.46 1.49 1.56 0.14 0.19 0.12 B1.5,4 55 4.00 1.62 1.36 1.38 1.42 0.09 0.24 0.20 B1.5,4.5 55 4.50 1.39 1.25 1.28 1.29 0.07 0.11 0.10 B1.5,5 55 5.00 1.31 1.15 1.18 1.18 0.01 0.13 0.13 B1.5,5.5 55 5.50 1.17 1.05 1.08 1.07 0.09 0.09 0.10 B1.5,6 55 6.00 1.08 0.94 0.98 0.97 0.12 0.10 0.11
B5
B2,3 53
2.00
3.00 1.68 1.77 1.75 1.88 0.09 0.07 0.20 B2,.3.5 53 3.50 1.59 1.66 1.65 1.71 0.06 0.06 0.12 B2,4 53 4.00 1.55 1.56 1.54 1.55 0.09 0.01 0.00 B2,4.5 53 4.50 1.37 1.46 1.44 1.41 0.03 0.07 0.04 B2,5 53 5.00 1.23 1.35 1.34 1.29 0.20 0.11 0.06 B2,5.5 53 5.50 1.16 1.25 1.24 1.17 0.17 0.08 0.01 B2,6 53 6.00 1.08 1.14 1.14 1.06 0.14 0.06 0.02
227
developed by other graduate students at UET Taxila. This preliminary work will
pave way for further research in this direction. The comparison of the actual
values of shear stress and predicted values by the three proposed models has
shown in Table 9.1 for HSC beams without web reinforcement.
9.3 Regression Models for shear strength of beams with web reinforcement
The shear strength of the HSC beams with web reinforcement was additionally
assumed to depend on the transverse steel ratio. Hence the three models
worked out by the mentioned software are as follows;
)()/('vgdacbaf
lc
(9.13)
EDdacbaf vc l )()/(
(9.14)
EDdaCBAf vlce )()/( (9.15)
Out of the three models run by the software, first model gave somehow
reasonable values, as the coefficient of determination was very less for the
remaining two models ( R2< 50%) . The actual and predicted values have been
given in Table 9.2
)(64.7)/(256.059.00107.0
)()/('
'
v
v
daf
gdacbaf
lc
lc
(9.16)
Solver type: Nonlinear Nonlinear iteration limit = 250 Diverging nonlinear iteration limit =12 Number of nonlinear iterations performed = 13 Residual tolerance = 0.0000000001 Sum of Residuals = -2.2048814350 E-02 Average Residual = -8.1532410025899E-04 Residual Sum of Squares (Absolute) = 0.564293097380102 Residual Sum of Squares (Relative) = 0.664 Standard Error of the Estimate = 0.132793671886608 Coefficient of Multiple Determination (R^2) = 0.74 Proportion of Variance Explained = 74.4097492% Adjusted coefficient of multiple determination (Ra^2) = 0.7381 Durbin-Watson statistic = 0.448319312620612 Regression Variable Results: a = 0.0107, b =0.59 , c= -0.256 , g= 7.64
228
Table: 9.2 Comparison of actual and predicted values of shear stress of High Strength
concrete beams with web reinforcement.
229
9.4 Comparison of the proposed models with ACI-318 Code and other models:
Beam Title fc΄
(MPa)
(ρ)
(%)
(ρv)
(%)
a/d Shear Stress ( MPa) Residual details
Actual Predicted
Model 1
Absolute
Residual
Absolute
Residual % Bs1
Bs0.33,3 50
0.33
0.16
3.00 0.99 1.19 0.21 0.20
Bs0.33.3 50 3.50 0.88 1.06 0.21 0.18
Bs0.33,4 50 4.00 0.8 0.94 0.09 0.14
B0.33,4.5 50 4.50 0.73 0.80 0.01 0.07
Bs0.33,5 50 5.00 0.57 0.68 0.02 0.11
Bs0.33,5 50 5.50 0.46 0.55 0.05 0.09
Bs0.33,6 50 6.00 0.41 0.42 0.17 0.01
Bs2
Bs0.73,3 54
0.73
0.16
3.00 1.7 1.47 0.19 0.23
Bs0.73.3 54 3.50 1.38 1.34 0.16 0.04
Bs0.73,4 54 4.00 1.22 1.22 0.15 0.00
Bs0.73,4 54 4.50 1.06 1.09 0.13 0.03
Bs0.73,5 54 5.00 0.98 0.96 0.13 0.02
Bs0.33,5 54 5.50 0.82 0.83 0.14 0.01
Bs0.73,6 54 6.00 0.75 0.70 0.24 0.05
Bs3
Bs1,3 50
1.00
0.16
3.00 1.7 1.59 0.00 0.11
Bs1,3.5 50 3.50 1.54 1.46 0.06 0.08
Bs1,4 50 4.00 1.45 1.34 0.06 0.11
Bs1,4.5 50 4.50 1.39 1.20 0.06 0.19
Bs1,5 50 5.00 1.23 1.08 0.05 0.15
Bs1,5.5 50 5.50 1.07 0.95 0.04 0.12
Bs1,6 50 6.00 0.91 0.82 0.02 0.09
Bs4
Bs1.5,3 55
1.50
0.16
3.00 1.86 1.94 0.14 0.08
Bs1.5.3. 55 3.50 1.8 1.81 0.12 0.01
Bs1.5,4 55 4.00 1.71 1.68 0.20 0.03
Bs1.5,4. 55 4.50 1.6 1.55 0.10 0.05
Bs1.5,5 55 5.00 1.56 1.43 0.13 0.13
Bs1.5,5. 55 5.50 1.24 1.30 0.10 0.06
Bs1.5,6 55 6.00 1.08 1.17 0.11 0.09
Bs5
Bs2,3 53
2.00
0.16
3.00 2.19 2.21 0.20 0.02
Bs2,.3.5 53 3.50 1.94 2.09 0.12 0.15
Bs2,4 53 4.00 1.88 1.95 0.00 0.07
Bs2,4.5 53 4.50 1.8 1.83 0.04 0.03
Bs2,5 53 5.00 1.73 1.70 0.06 0.03
Bs2,5.5 53 5.50 1.57 1.57 0.01 0.00
Bs2,6 53 6.00 1.24 1.45 0.02 0.21
230
9.4.1 Beams without shear reinforcement.
The ACI building Code is widely applied for the shear design of concrete. The
nominal shear capacity of reinforced concrete beam Vn, is given as the sum of
Concrete contribution Vc, and contributions of stirrups Vs .i.e. scn VVV
The shear stress of the beams was worked out with the ACI equation and other
models proposed by, Bazant and Kim (1984) and Russo et al.( 2004) have for
the tested beams. Regression equations were worked out for the shear stress of
HSC beams based on the test data. The actual shear stress and values given by
proposed model and other equations have been compared in Table 9.3 and
Figure 9.2
9.4.2 Beams with shear reinforcement.
For beams with web reinforcement, the proposed model for shear stress has
been compared with ACI equation and model proposed by G.Russo et al. (2004)
and shown Table 9.4 and Figure 9.3.
Table 9.3 Comparison of proposed model ACI equation and model proposed by G.Russo et al. (2004) for beams without web reinforcement.
231
Beam Title
fc΄
(ρ) (%) a/d
Shear Stress ( MPa) ACI
Bazant et al
Russo G. et al Actual Predicted
B0.33,3 50
0.33
3.00 0.95 0.84 0.94 0.80 0.49 B0.33.3.5 50 3.50 0.88 0.74 0.94 0.77 0.46 B0.33,4 50 4.00 0.70 0.63 0.94 0.76 0.45 B0.33,4.5 50 4.50 0.56 0.53 0.94 0.75 0.43 B0.33,5 50 5.00 0.48 0.43 0.94 0.74 0.43 B0.33,5.5 50 5.50 0.41 0.32 0.94 0.73 0.42 B0.33,6 50 6.00 0.25 0.22 0.94 0.73 0.42 B0.73,3 54
0.73
3.00 0.95 1.15 0.98 1.14 0.80 B0.73.3.5 54 3.50 0.88 1.05 0.98 1.08 0.74 B0.73,4 54 4.00 0.80 0.94 0.98 1.05 0.70 B0.73,4.5 54 4.50 0.73 0.84 0.98 1.03 0.67 B0.73,5 54 5.00 0.65 0.73 0.98 1.01 0.66 B0.33,5.5 54 5.50 0.57 0.63 0.98 1.00 0.64 B0.73,6 54 6.00 0.41 0.53 0.98 0.99 0.63 B1,3 50
1.00
3.00 0.99 1.18 0.94 1.26 0.93 B1,3.5 50 3.50 0.96 1.08 0.94 1.18 0.85 B1,4 50 4.00 0.88 0.97 0.94 1.14 0.80 B1,4.5 50 4.50 0.81 0.87 0.94 1.11 0.76 B1,5 50 5.00 0.73 0.77 0.94 1.09 0.74 B1,5.5 50 5.50 0.66 0.66 0.94 1.08 0.73 B1,6 50 6.00 0.58 0.56 0.94 1.07 0.71 B1.5,3 55
1.50
3.00 1.86 1.57 0.99 1.55 1.24 B1.5.3.5 55 3.50 1.68 1.46 0.99 1.45 1.12 B1.5,4 55 4.00 1.62 1.36 0.99 1.39 1.04 B1.5,4.5 55 4.50 1.39 1.25 0.99 1.35 0.99 B1.5,5 55 5.00 1.31 1.15 0.99 1.32 0.96 B1.5,5.5 55 5.50 1.17 1.05 0.99 1.30 0.93 B1.5,6 55 6.00 1.08 0.94 0.99 1.29 0.91 B2,3 53
2.00
3.00 1.68 1.77 0.97 1.74 1.45 B2,.3.5 53 3.50 1.59 1.66 0.97 1.61 1.29 B2,4 53 4.00 1.55 1.56 0.97 1.53 1.20 B2,4.5 53 4.50 1.37 1.46 0.97 1.48 1.14 B2,5 53 5.00 1.23 1.35 0.97 1.44 1.09 B2,5.5 53 5.50 1.16 1.25 0.97 0.80 1.06 B2,6 53 6.00 1.08 1.14 0.97 0.77 1.04
232
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
3 3.50 4.00 4.50 5.00 5.5 6
Shea
r stre
ss o
f bea
m (M
Pa)
shear span to depth ratio a/d
Comparison of actual and predicted values of shear stress of beams without web reinforcement bypropsoed model and other models for ρ= 0.33% and fc'=50Mpa
Actual
Predicted
ACI
Bazant
Russo
Figure 9.2 (a) =0.33%
0
0.2
0.4
0.6
0.8
1
1.2
1.4
3 3.50 4.00 4.50 5.00 5.5 6
She
ar s
tress
of b
eam
(MP
a)
shear span to depth ratio a/d
Comparison of actual and predicted values of shear stress of beams without web reinforcement bypropsoed model and other models for
ρ= 0.73% and fc'=54Mpa
Actual
Predicted
ACI
Bazant
Russo
Figure 9.2 (b) =0.73%
233
Figure 9.2 Comparison of actual values of shear stress with the predicted values by proposed regression model and other models for HSC beams without web reinforcement. Figure 9.2 Cont’d
0
0.2
0.4
0.6
0.8
1
1.2
1.4
3 3.50 4.00 4.50 5.00 5.5 6
Sh
ea
r str
ess
of b
ea
m (M
Pa
)
shear span to depth ratio a/d
Comparison of actual and predicted values of shear stress of beams without web reinforcement bypropsoed model and other models for ρ= 1%. and
fc'=50Mpa
Actual
Predicted
ACI
Bazant
Russo
Figure 9.2 (c) =1 %
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
3 3.50 4.00 4.50 5.00 5.5 6
Sh
ear s
tres
s o
f bea
m (M
Pa)
shear span to depth ratio a/d
Comparison of actual and predicted values of shear stress of beams without web reinforcement bypropsoed model and other models for
ρ= 1.5%. and fc'=55Mpa
Actual
Predicted
ACI
Bazant
Russo
Figure 9.2 (d) =1.5 %
234
Figure 9.2 cont’d
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
3 3.50 4.00 4.50 5.00 5.5 6
She
ar s
tress
of b
eam
(Mpa
)
shear span to depth ratio a/d
Comparison of actual and predicted values of shear stress of beams without web reinforcement by propsed models and other models for
ρ= 2% and fc'=53Mpa
Actual
Predicted
ACI
Bazant
Russo
Figure 9.2 (e) =1.5 %
235
Table9.4 Comparison of actual and predicted values of shear stress of High Strength
concrete beams having stirrups with the models proposed by ACI and Russo.
Beam fc΄ (ρ) a/d (ρV) ρVfy Shear Stress ACI Russo
236
Title Mpa
( %) (%) ( MPa) G.et al Actual Predicted
1 2 3 4 5 6 7 8 9 10 Bs0.33,3 50 0.33 3.00 0.16 0.27 0.99 1.19 1.14 1.00
Bs0.33.3. 50 0.33 3.50 0.16 0.27 0.88 1.06 1.14 0.99
Bs0.33,4 50 0.33 4.00 0.16 0.27 0.8 0.94 1.14 0.99
B0.33,4.5 50 0.33 4.50 0.16 0.27 0.73 0.80 1.14 0.98
Bs0.33,5 50 0.33 5.00 0.16 0.27 0.57 0.68 1.14 0.98
Bs0.33,5. 50 0.33 5.50 0.16 0.27 0.46 0.55 1.14 0.98
Bs0.33,6 50 0.33 6.00 0.16 0.27 0.41 0.42 1.14 0.98
Bs0.73,3 54 0.73 3.00 0.16 0.27 1.7 1.47 1.18 1.36
Bs0.73.3. 54 0.73 3.50 0.16 0.27 1.38 1.34 1.18 1.31
Bs0.73,4 54 0.73 4.00 0.16 0.27 1.22 1.22 1.18 1.28
Bs0.73,4. 54 0.73 4.50 0.16 0.27 1.06 1.09 1.18 1.26
Bs0.73,5 54 0.73 5.00 0.16 0.27 0.98 0.96 1.18 1.26
Bs0.33,5. 54 0.73 5.50 0.16 0.27 0.82 0.83 1.18 1.25
Bs0.73,6 54 0.73 6.00 0.16 0.27 0.75 0.70 1.18 1.24
Bs1,3 50 1.0 3.0 0.16 0.27 1.7 1.59 1.14 1.50
Bs1,3.5 50 1.0 3.5 0.16 0.27 1.54 1.46 1.14 1.44
Bs1,4 50 1.0 4.0 0.16 0.27 1.45 1.34 1.14 1.40
Bs1,4.5 50 1.0 4.5 0.16 0.27 1.39 1.20 1.14 1.37
Bs1,5 50 1.0 5.0 0.16 0.27 1.23 1.08 1.14 1.36
Bs1,5.5 50 1.0 5.5 0.16 0.27 1.07 0.95 1.14 1.35
Bs1,6 50 1.0 6.0 0.16 0.27 0.91 0.82 1.14 1.34
Bs1.5,3 55 1.5 3.0 0.16 0.27 1.86 1.94 1.19 1.87
Bs1.5.3.5 55 1.5 3.5 0.16 0.27 1.8 1.81 1.19 1.76
Bs1.5,4 55 1.5 4.0 0.16 0.27 1.71 1.68 1.19 1.69
Bs1.5,4.5 55 1.5 4.5 0.16 0.27 1.6 1.55 1.19 1.64
Bs1.5,5 55 1.5 5.0 0.16 0.27 1.56 1.43 1.19 1.62
Bs1.5,5.5 55 1.5 5.5 0.16 0.27 1.24 1.30 1.19 1.60
Bs1.5,6 55 1.5 6.0 0.16 0.27 1.08 1.17 1.19 1.58
Bs2,3 53 2.0 3.0 0.16 0.27 2.19 2.21 1.17 2.12
Bs2,.3.5 53 2.0 3.5 0.16 0.27 1.94 2.09 1.17 1.96
Bs2,4 53 2.0 4.0 0.16 0.27 1.88 1.95 1.17 1.88
Bs2,4.5 53 2.0 4.5 0.16 0.27 1.8 1.83 1.17 1.82
Bs2,5 53 2.0 5.0 0.16 0.27 1.73 1.70 1.17 1.78
Bs2,5.5 53 2.0 5.5 0.16 0.27 1.57 1.57 1.17 1.75
Bs2,6 53 2.0 6.0 0.16 0.27 1.24 1.45 1.17 1.73
237
0
0.2
0.4
0.6
0.8
1
1.2
1.4
3 3.50 4.00 4.50 5.00 5.5 6
Sh
ea
r str
ess
of
be
am
(Mp
a)
shear span to depth ratio a/d
Comparison of actual and predicted values of shear stress of beams with web reinforcement by propsed models and other models for
ρ= 0.33% and fc'=50Mpa
Actual
Predicted
ACI
Russo
Figure 9.3 (a) =0.33 %
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
3 3.50 4.00 4.50 5.00 5.5 6
Sh
ear s
tres
s o
f bea
m (M
pa)
shear span to depth ratio a/d
Comparison of actual and predicted values of shear stress of beams with web reinforcement by propsed models and other models for
ρ= 0.73% and fc'=50Mpa
Actual
Predicted
ACI
Russo
Figure 9.3 (b) =0.73 %
Figure 9.3 Comparison of actual shear stress of beams having stirrups with the proposed regression model and other models.
238
Figure 9.3 cont’d
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
3 3.50 4.00 4.50 5.00 5.5 6
Sh
ear s
tres
s o
f bea
m (M
pa)
shear span to depth ratio a/d
Comparison of actual and predicted values of shear stress of beams with web reinforcement by propsed models and other models for
ρ= 1% and fc'=54Mpa
Actual
Predicted
ACI
Russo
Figure 9.3 (c) =1 %
0
0.5
1
1.5
2
2.5
3 3.50 4.00 4.50 5.00 5.5 6
Sh
ear s
tres
s o
f bea
m (M
Pa)
shear span to depth ratio a/d
Comparison of actual and predicted values of shear stress of beams with web reinforcement bypropsoed model and other models for ρ= 1.5% and fc'=55Mpa
Actual
Predicted
ACI
Russo
239
Figure 9.3 (d) =1.5 %
Figure 9.3 cont’d
0
0.5
1
1.5
2
2.5
3 3.50 4.00 4.50 5.00 5.5 6
Sh
ea
r str
en
gth
of b
ea
m (M
pa
)
shear span to depth ratio a/d
Comparison of actual and predicted values of shear stress of beams without web reinforcement by propsed models and other models for ρ= 2% and
fc'=53Mpa
Actual
Predicted
ACI
Russo
Figure 9.3 (d) =2%
The ratios of actual values of shear stress and the values predicted by various
models have been compared and the coefficient of variation are given in Table
9.5 and Table 9.6 for beams with and without web reinforcement respectively.
The proposed model gives least CoV for beams without web reinforcement.
Whereas for HSC beams with web steel Russo Model gives minimum variation.
The coefficient of variation for keeping a/d constant and varying is given in
Table 9.7 and for keeping constant and a/d varying in Table 9.8. The
proposed model gives minimum variation for beams without web steel, whereas
for beams with web steel, Russo Model gives best prediction.
Beam
Title
%
a/d Shear Stress ( MPa)
ACI Bazant et al Russo
G. et al
test/pred test/ACI test/Bazan test/Rus
240
Table 9.5 Comparison of test/pred by the proposed model and other models for beams without shear reinforcement ( 35 Nos). ( For constant steel ratio and variable a/d) Table 9.6 Comparison of test/pred by the proposed model and other models for beams with shear reinforcement ( 35 Nos) ( For constant steel ratio and variable a/d)
0.33
Actual Predicted
B0.33,3 3 0.95 0.84 0.94 0.80 0.49 1.13 1.01 1.19 1.94 B0.33.3.5 3.5 0.88 0.74 0.94 0.77 0.46 1.19 0.94 1.14 1.91 B0.33,4 4 0.70 0.63 0.94 0.76 0.45 1.11 0.74 0.92 1.56 B0.33,4.5 4.5 0.56 0.53 0.94 0.75 0.43 1.06 0.60 0.75 1.30 B0.33,5 5 0.48 0.43 0.94 0.74 0.43 1.12 0.51 0.65 1.12 B0.33,5.5 5.5 0.41 0.32 0.94 0.73 0.42 1.28 0.44 0.56 0.98 B0.33,6 6 0.25 0.22 0.94 0.73 0.42 1.14 0.27 0.34 0.60 B0.73,3
0.73
3 0.95 1.15 0.98 1.14 0.80 0.83 0.97 0.83 1.19 B0.73.3.5 3.5 0.88 1.05 0.98 1.08 0.74 0.84 0.90 0.81 1.19 B0.73,4 4 0.80 0.94 0.98 1.05 0.70 0.85 0.82 0.76 1.14 B0.73,4.5 4.5 0.73 0.84 0.98 1.03 0.67 0.87 0.74 0.71 1.09 B0.73,5 5 0.65 0.73 0.98 1.01 0.66 0.89 0.66 0.64 0.98 B0.33,5.5 5.5 0.57 0.63 0.98 1.00 0.64 0.90 0.58 0.57 0.89 B0.73,6 6 0.41 0.53 0.98 0.99 0.63 0.77 0.42 0.41 0.65 B1,3
1
3 0.99 1.18 0.94 1.26 0.93 0.84 1.05 0.79 1.06 B1,3.5 3.5 0.96 1.08 0.94 1.18 0.85 0.89 1.02 0.81 1.13 B1,4 4 0.88 0.97 0.94 1.14 0.80 0.91 0.94 0.77 1.10 B1,4.5 4.5 0.81 0.87 0.94 1.11 0.76 0.93 0.86 0.73 1.07 B1,5 5 0.73 0.77 0.94 1.09 0.74 0.95 0.78 0.67 0.99 B1,5.5 5.5 0.66 0.66 0.94 1.08 0.73 1.00 0.70 0.61 0.90 B1,6 6 0.58 0.56 0.94 1.07 0.71 1.04 0.62 0.54 0.82 B1.5,3
1.5
3 1.86 1.57 0.99 1.55 1.24 1.18 1.88 1.20 1.50 B1.5.3.5 3.5 1.68 1.46 0.99 1.45 1.12 1.15 1.70 1.16 1.50 B1.5,4 4 1.62 1.36 0.99 1.39 1.04 1.19 1.64 1.17 1.56 B1.5,4.5 4.5 1.39 1.25 0.99 1.35 0.99 1.11 1.40 1.03 1.40 B1.5,5 5 1.31 1.15 0.99 1.32 0.96 1.14 1.32 0.99 1.36 B1.5,5.5 5.5 1.17 1.05 0.99 1.30 0.93 1.11 1.18 0.90 1.26 B1.5,6 6 1.08 0.94 0.99 1.29 0.91 1.15 1.09 0.84 1.19 B2,3
2
3 1.68 1.77 0.97 1.74 1.45 0.95 1.73 0.97 1.16 B2,.3.5 3.5 1.59 1.66 0.97 1.61 1.29 0.96 1.64 0.99 1.23 B2,4 4 1.55 1.56 0.97 1.53 1.20 0.99 1.60 1.01 1.29 B2,4.5 4.5 1.37 1.46 0.97 1.48 1.14 0.94 1.41 0.93 1.20 B2,5 5 1.23 1.35 0.97 1.44 1.09 0.91 1.27 0.85 1.13 B2,5.5 5.5 1.16 1.25 0.97 0.80 1.06 0.93 1.20 1.45 1.09 B2,6 6 1.08 1.14 0.97 0.77 1.04 0.95 1.11 1.40 1.04
Mean 1.01 1.02 0.86 1.19
CoV 12.80% 40.09% 7.69% 24.2%
Beam Title
% a/d Shear Stress ( MPa)
ACI Russo G.et al
test/pred test/ACI test/Russo
241
Table 9.7 Comparison of test/pred by the proposed model and other models for beams without shear reinforcement ( 35 Nos). ( For constant a/d and variable steel ratio)
Actual Predicted
Bs0.33,3
0.33
3 0.99 1.19 1.14 1.00 0.83 0.87 0.99 Bs0.33.3.5 3.5 0.88 1.06 1.14 0.99 0.83 0.77 0.89 Bs0.33,4 4 0.8 0.94 1.14 0.99 0.85 0.70 0.81 B0.33,4.5 4.5 0.73 0.80 1.14 0.98 0.91 0.64 0.74 Bs0.33,5 5 0.57 0.68 1.14 0.98 0.84 0.50 0.58 Bs0.33,5.5 5.5 0.46 0.55 1.14 0.98 0.84 0.40 0.47 Bs0.33,6 6 0.41 0.42 1.14 0.98 0.98 0.36 0.42 Bs0.73,3
0.73
3 1.7 1.47 1.18 1.36 1.16 1.44 1.25 Bs0.73.3.5 3.5 1.38 1.34 1.18 1.31 1.03 1.17 1.05 Bs0.73,4 4 1.22 1.22 1.18 1.28 1.00 1.03 0.95 Bs0.73,4.5 4.5 1.06 1.09 1.18 1.26 0.97 0.90 0.84 Bs0.73,5 5 0.98 0.96 1.18 1.26 1.02 0.83 0.78 Bs0.33,5.5 5.5 0.82 0.83 1.18 1.25 0.99 0.69 0.66 Bs0.73,6 6 0.75 0.70 1.18 1.24 1.07 0.64 0.60 Bs1,3
1
3 1.7 1.59 1.14 1.50 1.07 1.49 1.13 Bs1,3.5 3.5 1.54 1.46 1.14 1.44 1.05 1.35 1.07 Bs1,4 4 1.45 1.34 1.14 1.40 1.08 1.27 1.04 Bs1,4.5 4.5 1.39 1.20 1.14 1.37 1.16 1.22 1.01 Bs1,5 5 1.23 1.08 1.14 1.36 1.14 1.08 0.90 Bs1,5.5 5.5 1.07 0.95 1.14 1.35 1.13 0.94 0.79 Bs1,6 6 0.91 0.82 1.14 1.34 1.11 0.80 0.68 Bs1.5,3
1.5
3 1.86 1.94 1.19 1.87 0.96 1.56 0.99
Bs1.5.3.5 3.5 1.8 1.81 1.19 1.76 0.99 1.51 1.02
Bs1.5,4 4 1.71 1.68 1.19 1.69 1.02 1.44 1.01
Bs1.5,4.5 4.5 1.6 1.55 1.19 1.64 1.03 1.34 0.98 Bs1.5,5 5 1.56 1.43 1.19 1.62 1.09 1.31 0.96 Bs1.5,5.5 5.5 1.24 1.30 1.19 1.60 0.95 1.04 0.78 Bs1.5,6 6 1.08 1.17 1.19 1.58 0.92 0.91 0.68 Bs2,3
2
3 2.19 2.21 1.17 2.12 0.99 1.87 1.03 Bs2,.3.5 3.5 1.94 2.09 1.17 1.96 0.93 1.66 0.99 Bs2,4 4 1.88 1.95 1.17 1.88 0.96 1.61 1.00 Bs2,4.5 4.5 1.8 1.83 1.17 1.82 0.98 1.54 0.99 Bs2,5 5 1.73 1.70 1.17 1.78
1.02 1.48 0.97 Bs2,5.5 5.5 1.57 1.57 1.17 1.75 1.00 1.34 0.90 Bs2,6 6 1.24 1.45 1.17 1.73
0.86 1.06 0.72
Mean 0.99 1.11 0.88
CoV 9.60% 34.9% 4.07%
Beam
Title
a/d
% Shear Stress
( MPa) ACI Bazant
et al Russo G. et al
test/pred test/ACI test/Bazan test/Rus
Actual Predicted
242
Table 9.8 Comparison of test/pred by the proposed model and other models for beams with shear reinforcement ( 35 Nos) ( For constant a/d and variable steel ratio)
B0.33,3
3
0.33 0.95 0.84 0.94 0.80 0.49 1.13 1.01 1.19 1.94
B0.73,3 0.73 0.95 1.15 0.98 1.14 0.80 0.83 0.97 0.83 1.19
B1,3 1 0.99 1.18 0.94 1.26 0.93 0.84 1.05 0.79 1.06
B1.5,3 1.5 1.86 1.57 0.99 1.55 1.24 1.18 1.88 1.20 1.50
B2,3 2 1.68 1.77 0.97 1.74 1.45 0.95 1.73 0.97 1.16
B0.33.3.5
3.5
0.33 0.88 0.74 0.94 0.77 0.46 1.19 0.94 1.14 1.91
B0.73,3.5 0.73 0.88 1.05 0.98 1.08 0.74 0.84 0.90 0.81 1.19
B1..3.5 1 0.96 1.08 0.94 1.18 0.85 0.89 1.02 0.81 1.13
B1,5,3.5 1.5 1.68 1.46 0.99 1.45 1.12 1.15 1.70 1.16 1.50
B2,.3.5 2 1.59 1.66 0.97 1.61 1.29 0.96 1.64 0.99 1.23
B0.33,4
4.0
0.33 0.70 0.63 0.94 0.76 0.45 1.11 0.74 0.92 1.56
B0.73,4 0.73 0.80 0.94 0.98 1.05 0.70 0.85 0.82 0.76 1.14
B1,4 1 0.88 0.97 0.94 1.14 0.80 0.91 0.94 0.77 1.10
B1.5,4 1.5 1.62 1.36 0.99 1.39 1.04 1.19 1.64 1.17 1.56
B2,4 2.0 1.55 1.56 0.97 1.53 1.20 0.99 1.60 1.01 1.29
B0.33,4.5
4.5
0.33 0.56 0.53 0.94 0.75 0.43 1.06 0.60 0.75 1.30
B0.73,4.5 0.73 0.73 0.84 0.98 1.03 0.67 0.87 0.74 0.71 1.09
B1,4.5 1 0.81 0.87 0.94 1.11 0.76 0.93 0.86 0.73 1.07
B1.5,4.5 1.5 1.39 1.25 0.99 1.35 0.99 1.11 1.40 1.03 1.40
B2,4.5 2.0 1.37 1.46 0.97 1.48 1.14 0.94 1.41 0.93 1.20
B0.33,5
5.0
0.33 0.41 0.32 0.94 0.73 0.42 1.28 0.44 0.56 0.98
B0.73,5 0.73 0.48 0.43 0.94 0.74 0.43 1.12 0.51 0.65 1.12
B1,5 1 0.73 0.77 0.94 1.09 0.74 0.95 0.78 0.67 0.99
B1,5.5 1.31 1.15 0.99 1.32 0.96 1.14 1.32 0.99 1.36 1.5
B2,5 2.0 1.23 1.35 0.97 1.44 1.09 0.91 1.27 0.85 1.13
B0.33,5.5
5.5
0.33 0.41 0.32 0.94 0.73 0.42 1.28 0.44 0.56 0.98
B0.73,5.5 0.73 0.57 0.63 0.98 1.00 0.64 0.90 0.58 0.57 0.89
B1,.5.5.5 1 0.66 0.66 0.94 1.08 0.73 1.00 0.70 0.61 0.90
B1.5,5.5 1.5 1.17 1.05 0.99 1.30 0.93 1.11 1.18 0.90 1.26
B2,,5.5 2 1.16 1.25 0.97 0.80 1.06 0.93 1.20 1.45 1.09
B0.33,6
6.0
0.33 0.25 0.22 0.94 0.73 0.42 1.14 0.27 0.34 0.60
B0.73,6 0.73 0.41 0.53 0.98 0.99 0.63 0.77 0.42 0.41 0.65
B1,6 1 0.58 0.56 0.94 1.07 0.71 1.04 0.62 0.54 0.82
B1.5,6 1.5 1.08 1.14 0.97 0.77 1.04 0.95 1.11 1.40 1.04
B2,6 2.0 1.08 0.94 0.99 1.29 0.91 1.15 1.09 0.84 1.19
Mean 1.01 1.01 0.86 1.19 CoV(%) 2.3 10.11 6.8 5.1
a/d Beam Title
% a
%/d
Shear Stress ( MPa)
ACI Russo G.et al
test/pred test/ACI test/Russo
Actual Predicted
Bs0.33,3 0.33 0.99 1.19 1.14
1.00 0.83 0.87 0.99
243
9.5 Discussion on the proposed regression models.
The models proposed in the above arguments have inherent weaknesses of
limited data and few parameters. A polynomial regression model incorporating
more parameters can be developed on the basis of available test results on
Bs0.73,3
3
0.73 1.7 1.47 1.18
1.36 1.16 1.44 1.25 Bs1,3 1 1.7 1.59 1.1
41.50 1.07 1.49 1.13
Bs1.5,3 1.5 1.86 1.94 1.19
1.87 0.96 1.56 0.99 Bs2,.3.5 2 2.19 2.21 1.1
72.12 0.99 1.87 1.03
Bs0.33.3.5
3.5
0.33 0.88 1.06 1.14
0.99 0.83 0.77 0.89 Bs0.73.3.5 0.73 1.54 1.46 1.1
41.44 1.05 1.35 1.07
Bs1,3.5 1 1.54 1.46 1.14 1.44 1.05 1.35 1.07
Bs1.5.3.5 1.5 1.8 1.81 1.19
1.76 0.99 1.51 1.02 Bs2,3.5 2 1.94 2.09 1.17 1.96 0.93 1.66 0.99
Bs0.33,4
4.0
0.33 1.38 1.34 1.18
1.31 1.03 1.17 1.05 Bs0.73,4 0.73 1.22 1.22 1.18 1.28 1.00 1.03 0.95
Bs1,4 1 1.45 1.34 1.14
1.40 1.08 1.27 1.04 Bs1.5,4 1.5 1.71 1.68 1.19 1.69 1.02 1.44 1.01
Bs2,4 2.0 1.88 1.95 1.17 1.88 0.96 1.61 1.00
B0.33,4.5
4.5
0.33 0.73 0.80 1.14
0.98 0.91 0.64 0.74 Bs0.73,4.5 0.73 1.06 1.09 1.18 1.26 0.97 0.90 0.84
Bs1,4.5 1 1.45 1.34 1.14 1.40 1.08 1.27 1.04
Bs1.5,4.5 1.5 1.71 1.68 1.19 1.69 1.02 1.44 1.01
Bs2,4.5 2.0 1.8 1.83 1.17 1.82 0.98 1.54 0.99
Bs0.33,5
5.0
0.33 0.57 0.68 1.14
0.98 0.84 0.50 0.58 Bs0.73,5 0.73 0.98 0.96 1.18 1.26 1.02 0.83 0.78
Bs1,5 1 1.23 1.08 1.14 1.36 1.14 1.08 0.90
Bs1,5 1.5 1.56 1.43 1.19 1.62 1.09 1.31 0.96
Bs2,5 2.0 1.73 1.70 1.17 1.78 1.02 1.48 0.97
Bs0.33,5.5
5.5
0.33 0.55 1.14
0.98 0.84 0.40 0.47 0.46 Bs0.73,5.5 0.73 0.82 0.83 1.18 1.25 0.99 0.69 0.66
Bs1,5.5 1.0 1.07 0.95 1.14 1.35 1.13 0.94 0.79
Bs1.5,5.5 1.5 1.24 1.30 1.19 1.60 0.95 1.04 0.78
Bs1.5,5.5 2.0 1.57 1.57 1.17 1.75 1.00 1.34 0.90
Bs0.33,6
6.0
0.33 0.41 0.42 1.14
0.98 0.98 0.36 0.42 Bs0.73,6 0.73 0.75 0.70 1.18 1.24 1.07 0.64 0.60
Bs1,6 1 0.91 0.82 1.14 1.34 1.11 0.80 0.68
Bs1.5,6 1.5 1.08 1.17 1.19 1.58 0.92 0.91 0.68 Bs2,6 2.0 1.24 1.45 1.17 1.73 0.86 1.06 0.72
Mean 1.15 1.09 0.84
CoV 11% 12% 6%
244
shear strength of HSC beams from the database in the future research work.
However the following results are derived from the above discussions.
i. ACI equation for shear strength of high strength concrete beams is
conservative for both the cases with and without web reinforcement.
ii. The proposed model for beams with web and without web reinforcement
has given less values of coefficient of variations as compared to ACI
equation and hence the models may better estimates the shear strength of
the beams for given range of compressive strength of concrete, than ACI
equation.
a. The Coefficient of variation of Bazant Model for beams without
shear reinforcement is less of all the models. Hence Bazant model
may better estimates the shear strength of beams without shear
reinforcement.
b. The Russo model for shear strength of beams with shear
reinforcement has given the least coefficient of variation as
compared to other equations. Hence the Russo model may better
estimate the shear strength of beams with shear reinforcement.
c. The proposed two non linear regression models with reasonable
coefficient of variation for both the cases can be further generalized
through experimental work.
d. For constant a/d and variable longitudinal steel ratio, the proposed
model gives the least coefficient of variation whereas for beams
with web steel, Russo Model gives least variation.
245
e. For constant longitudinal steel ratio and variable a/d , the proposed
model gives least variation for beams without web steel and Russo
model best predict the shear stress of HSC beams tested for
beams with web steel
246
Chapter No.10
Conclusions and recommendations
Chapter Introduction: The chapter at last but not the least gives the conclusions and results of the research findings and venues for future research have been identified. This will surely help the researchers of the new generations to explore more realistic and accurate models for the shear design of high strength concrete structures.
Conclusions
On the basis of testing of 70 HSC-RC beams for five values of longitudinal steel
and seven values of shear span to depth ratio, the following specific conclusions
are drawn. 1. The failure in most of the beams has been caused due to diagonal tension
cracking; however it was more dominant failure mode for beams without web
reinforcement and having ρ≥1%. For beams with ρ<1%, flexural shear failure
was obvious failure mode.
2. For beams without web reinforcement and having large values of longitudinal
steel (ρ=1% and 1.5%), the shear failure is more brittle and sudden, giving
no sufficient warning.
3. The HSC beams with web reinforcement, the failure has been caused mainly
by diagonal tension shear cracking even for small longitudinal steel ratio,
instead.
4. The shear strength of all the beams has been increased for all values of
longitudinal steel and “a/d” ratio with the addition of stirrups, but in a very
247
random manner. Hence uniform increase in shear strength of beams as
given in most of the Codes was not observed. This increase is more
prominent at lower steel ratios.
5. The addition of web reinforcement has avoided the brittle failure of the beam
at higher values of longitudinal steel and the ductility of beams has increased.
6. The shear strength of the HSC beams has been increased with the increases
of longitudinal steel in both the caes without and with web reinforcement. This
increase is relatively more in case of beams with web reinforcement.
7. For both types of beams, the shear strength of HSC beams has been
decreased with the increase of a/d ratio. However this decrease is relatively
more in the beams without web reinforcement.
8. The shear equation of ACI-318, Canadian Code (CSA), EuroCode (EC-02) ,
LRFD( MCFT) studied in this research are less give the shear strength more
than the actual test values for beams with ρ<1.0 and hence these are not
safe. But these equation are reasonably good predictor of shear strength of
beams for ρ= 1% and 1.5%. However for ρ=2%, most of the equations are
over conservativ.
9. Modified Compression Field Theory (MCFT) and new Equation proposed by
Zararis (2003) gave reasonably good estimates of the HSC beams with web
reinforcement.
10. The models proposed in the research have inherent weaknesses of limited
data and few parameters. However the proposed model estimates the shear
strength of HSC beams with and without web reinforcement better than ACI-
318 equation.
248
11. Bazant model better estimates the shear strength of beams without shear
reinforcement and Russo model may better estimate the shear strength of
beams with shear reinforcement.
Conclusions on the work in disturbed region
1. For design of two HSC corbels, Strut and Tie Model used in the study gives
reasonable prediction of the shear strength of corbels.
2. The variation in actual and theoretical values of strut angles and shear
strength is less for HSC corbels.
3. The shear failure of HSC two way corbels is sudden and brittle as in case of
HSC beams without web reinforcement, due to loss in the aggregates
interlocking and failure of the HSC corbles long smooth plane.
249
Recommendations for future work
1. The International building and bridges codes provisions for shear
strength of HSRC beams needs to be rationalized further particularly for
the cases where shear span to depth ratio is more than 5 and
Compressive strength of concrete is 70 Mpa or more.
2. Most of the equations given in the Building and Bridges Codes for shear
design of HSC beam having ρ≤1% are less conservative and further
research is required to work out the reduction factor as function of
concrete compressive strength.
3. Further research is required to generalize the Strut and Tie Model for
typical disturbed region in concrete structures like deep beams, pile caps,
dapped ended beams and corbels. This will require extensive
experimental evidence before generalizing the model.
4. It is proposed that a research group may be developed at the Department
of Civil Engineering, University of Engineering and Technology-Taxila,
Pakistan represented by researchers from the faculty and field to work
further on the shear design of disturbed region in reinforced concrete as
part of graduates research.
5. Graduate level research is required to develop polynomial regression
models on the basis of shear database of HSRC beams, which can be
tested and later adopted with more validity and reliability.
250
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Appendix-A: Design of two way corbel using Strut and Tie Model ( STM)
The step by step approach given at the website of Strut and Tie Resources (2005) was
used for the design of two way cobles as follows;
1. Geometry of the two way corbels
The geometry of the proposed two way corbel has been shown in Figure A-1
Figure A-1 : geometry of Two way corbel.
2 Design of two way corbels.
The corbels were designed against an assumed external load of 160 kips and the
following design steps were adopted using STM as per ACI-318-06.
2.1 Determine the bearing plate dimension:
Bearing plate measuring 9inx 6in is selected to transfer the load evenly to the corbel
projection, the bearing area of plate is 54 in2 .The corbel is designed to carry a load of
80Kips at each end.
The bearing stress = 80x1000/54 = 1481 psi
Thus the allowed limits of the bearing stress = φ ( 0.85ßn fc′ ) =
261
C C'
B B'
A A'
= 0.75 x 0.85x 0.80x 4.76 = 2427 psi
The bearing stress of 1481 psi is less than the allowed 2427 psi and Ok
2.2 Choosing the Corbel dimension:
The depth of column face is 9 in ACI requires that the depth outside the bearing must be
at least half of the depth of column. In our case the load is applied at 4.5 in from the face
of corbel which is half the column dimension. Hence the minimum requirements are
fulfilled.
2.3 Establish Strut and Tie Model.
The geometry of assumed truss is shown in Figure A-2. The centre of the tie is assumed
to be 2 in below the top of the corbel. Hence d = 18-2 = 16 in. The horizontal strut BB’ is
assumed to lie in the horizontal line at the corbel column joint.
Figure A-2 Geometry of assumed Strut and Tie Model ( STM)
The location of strut CB centerline can be found by calculating the required compressive
force in strut CB, NCB, and the strut stress limit to obtain the strut width a. The strut CB
force is
NCB = 80 kips
The limit stress on the nodal zone B (also strut CB) is
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Here is strength reduction factor =0.75
n ; Capacity reduction factor of struts =0.85 for straight type nodes
cuf = 0.75 x (0.85 x 1 x 5,600) = 3.03 ksi
Thus, we have
bf
Na
cu
CB
= 80 / (0.75 x 3.03 x 9) = 2.93 in
This fixes the geometry of the truss and means that member AB has a horizontal
projection of 4.5 + 2.93 / 2 = 4.97 in.
2.4 Determine truss member forces:
The forces in all the members of the truss are given in Table A-1. The positive sign
indicates tension, negative compression. The finally analyzed truss is given in Figure A-
3.
Table A-1 Forces in Truss of double corbel after analysis.
Member AA' AB BB' BC
Force (kips) 29.84 -85.38 -80 -29.84
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A A '
B
C
B '
C '
V u = 8 0 k ip s V u = 8 0 k ip s
85.3
8 ki
ps
85.3
8 ki
ps
80 k
ips
2 9 . 8 4 k ip s
80 k
ips
8 0 k ip s8 0 k ip s
2 9 . 8 4 k ip s
Figure A-3 Member Force in strut and Tie model for two way corbel.
2.5 Design of Tie.
The area of reinforcement required for tie AA’ is y
AArequired f
NAs
= 29.84 / (.75 x 40,000)
=0.995 in2
And the minimum area of reinforcement is
=0.04 x 5,600/40,000 x (9 x 16)
Choose 5 # 4 bars,
As Provided = 5 x 0.2 =1 in
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2.6 Check the struts capacity
The struts will be checked by computing the strut widths and checked whether they will
fit in the space available.
The stress of the diagonal strut AB is limited to
= 0.75 x (0.85 x 0.75 x 5,600)
= 2.27 ksi
Hence, the required width for strut AB is
bf
Na
cu
AB
= = 85.38 / (0.75 x 2.27 x 9) == 4.18 in
The stress of the vertical struts CB and horizontal strut BB’ is limited to
= 0.75 x (0.85 x 1 x 5600)
= 3.03 ksi.
Hence, the required widths for strut BB’ is bf
Na
cu
BB
= 80 / (0.75 x 3.03 x 9) =
= 1 .09 in
The required width for strut CB is equal to a, i.e. 2.93 in.
As shown in Figure A-3, all the strut widths fit into the outline of the corbel region. Thus,
this solution is accepted.
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2.8 Design of Nodal zone and check for anchorages.
The width (a) of nodal zone A was chosen to satisfy the stress limits on the nodal zone.
To satisfy the nodal zone stress limit, the tie reinforcement must engage an effective
depth of concrete at least equal to
= ( 29.83 x 1000) / (0.75 x 0.85 x 0.8 x 5600 x 9)
= 1.16 in
This limit is easily satisfied since the nodal zone available is 2.93 in.
The required anchorage length for tie AA’ is
= 0.7 x 1200 x 0.5)/5600
= 5.613 in
Since this is less than the available length, i.e. 9 – 1.5 – 1/2 = 7 in., the anchorage length
is adequate. The final reinforcement has been given in Figure A-4
Figure A-4 Reinforcement details of two way corbel designed for 80 Kips (355KN) load
by STM.