UNIVERSITY OF CALGARY. Reinforced Concrete Beam Design for Shear. Hongge (Gordon) Wang A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

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1 UNIVERSITY OF CALGARY Reinforced Concrete Beam Design for Shear by Hongge (Gordon) Wang A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING CALGARY, ALBERTA NOVEMBER, 2002 Hongge (Gordon) Wang 2002

2 The author of this thesis has granted the University of Calgary a non-exclusive license to reproduce and distribute copies of this thesis to users of the University of Calgary Archives. Copyright remains with the author. Theses and dissertations available in the University of Calgary Institutional Repository are solely for the purpose of private study and research. They may not be copied or reproduced, except as permitted by copyright laws, without written authority of the copyright owner. Any commercial use or re-publication is strictly prohibited. The original Partial Copyright License attesting to these terms and signed by the author of this thesis may be found in the original print version of the thesis, held by the University of Calgary Archives. Please contact the University of Calgary Archives for further information: uarc@ucalgary.ca Telephone: (403) Website:

3 ABSTRACT The two methods for design of shear adopted by the present CSA Standard A23.3 are either too simple or too complicated. That presents the need for ongoing research to establish a new design guideline for shear design. Recent studies by Dr. Loov and others have shown that shear design can be based on the shear resistance along potential inclined crack and slip planes. Because the basic equations for this shear design method are derived from "shear friction" theories, we call it "the shear friction method". In this thesis an entire review of shear design methods has been given and a method of shear design based on shear friction theories has been introduced. From comparison calculations with present code methods it is proved that "the shear friction method" provides a simpler and more accurate approach for shear design. iii

4 ACKNOWLEDGMENTS I am extremely grateful to my supervisor, Dr. Robert E. Loov for his endless patience and guidance throughout the course of this program. I would also like to thank my current employer Kassian Dyck & Associates for giving me the chance to finish this thesis. Finally I wish to thank my wife Candy for her support and encouragement. iv

5 TABLE OF CONTENTS Cover Page Approval page Abstract Acknowledgements Table of Contents List of Tables List of Figures Notation i ii iii iv v viii ix xiii CHAPTER ONE: INTRODUCTION General Code Review Scope of Study Thesis Organization 3 CHAPTER TWO: BASIC SHEAR THEORIES Homogeneous Beam Beam Cracking Modes Shear Transfer Mechanisms Shear Failure Modes Beams without Shear Reinforcement Beams with Shear Reinforcement Factors Affecting the Shear Strength Tensile Strength of Concrete Longitudinal Reinforcement Shear Span-to-depth Ratio, a/d Size of Beams 19 v

6 2.5.5 Axial Forces Web Reinforcement 20 CHAPTER THREE: SHEAR DESIGN - CSA STANDARD A General Simplified Method Shear Supported by Concrete, V c Shear Supported by Stirrups, V s General Method Shear Supported by Concrete, V c g Shear Supported by Stirrups, V s g 26 CHAPTER FOUR: SHEAR DESIGN - SHEAR FRICTION METHOD General General Equations for Beams Based on Shear Friction Shear Friction Strength Basic Shear Design Equations Based on Work by Loov Approximate Shear Capacity of Concrete Approximate Shear Capacity of Stirrups Approximate Shear Design Equations for Beams with T>T opl Critical Shear Failure Angle Beams with Longitudinal Reinforcement T<T op, 44 CHAPTER FIVE: EXPERIMENTAL STUDIES AND COMPARISON Application of Shear Friction Method Test Results in Literature Yoon, Cook and Mitchell's Tests, Saram and Al-Musawi's Tests, vi

7 5.2.3 Summary of Tests from Literature Beams with Shear Reinforcement Beams without Shear Reinforcement 93 CHAPTER SIX: PROPOSED CODE CLAUSES FOR SHEAR DESIGN Proposed Code Clauses for Shear Design Required Shear Resistance Factored Shear Resistance Determination of V c s f Determination of V s s f Determination of Determination of \ / Limiting Shear Failure Angle Design Examples 108 CHAPTER SEVEN: DISCUSSION AND CONCLUSION Conclusions and Recommendations Ill 7.2 Future Research 112 BIBLIOGRAPHY 113 vii

8 LIST OF TABLES TABLE 5.1 Details of Beam Specimens (Yoon) 5.2 Test Results and Comparison of Predictions (Yoon) 5.3 Details of Beam Specimens (Sarsam) 5.4 Details of Materials (Sarsam) 5.5 Test Results and Comparison of Predictions (Sarsam) 5.6 Details of Specimens with Stirrups 5.7 Details of Specimens without Stirrups 5.8 Comparison of Predictions for Beams with Stirrups 5.9 Comparison of Predictions for Beams without Stirrups viii

9 LIST OF FIGURES FIGURE 2.1 Internal Forces in Beam Distribution of Flexural Shear Stresses Principal Stresses 7 2.4a Stress Trajectories 8 2.4b Potential Crack Pattern A Cracked Beam without Shear Reinforcement (MacGregor, 2000) A Cracked Beam with Shear Reinforcement (Peng, 1999) Internal Forces in a Cracked Beam Effect of a/d on Shear for Beams Without Shear Reinforcement (MacGregor, 2000) Shear Failure Modes (Pillai, 1983) Shear Strength vs. Longitudinal Reinforcement (MacGregor, 2000) Shear Strength vs. a/d (Kani, 1979) Influence of Member Size on Shear Strength (CSA A ) Effect of Axial Loads in Inclined Cracking Shear (MacGregor, 2000) Distribution of Internal Shears of Beam with Shear Reinforcement (MacGregor, 2000) Comparison of Shear Design Methods and Test Results (CSA A ) Shear Friction Concept (CSA A ) Reinforcement Inclined to Potential Failure Cracks (CSA A ) Push-off Test Results (Loov, 1998) Free Body Diagram of Beam (Loov, 1998) Shear Strength vs. Crack Angle (Loov, 1998) 34 ix

10 4.6 Three-dimensional surface of shear strength along all possible failure planes for beam 544 (Loov, 1998) Possible Critical Shear Failure Planes (Loov, 1999) A Tested Beam with Critical Shear Cracks (Peng, 1999) Shear Strength vs. coto Shear Strength vs. cot9 by Eq and Eq Shear Strength vs. Longitudinal Reinforcement of Beam The Shear Contributions of Concrete and Discrete Stirrups (Loov, 1998) Details of Beam Specimens and instrumentation (Yoon, 1996) Effect of Concrete Strength on the Shear Friction Method (Yoon) Effect of Concrete Strength on the Simplified Method (Yoon) Effect of Concrete Strength on the General Method (Yoon) Effect of Stirrup Spacing on the Shear Friction Method (Yoon) Effect of Shear Reinforcement on the Shear Friction Method (Yoon) Effect of Stirrup Spacing on the Simplified Method (Yoon) Effect of Shear Reinforcement on the Simplified Method (Yoon) Effect of Stirrup Spacing on the General Method (Yoon) Effect of Shear Reinforcement on the General Method (Yoon) Effect of Concrete Strength on the Shear Friction Method (Yoon) Effect of Concrete Strength on the Simplified Method (Yoon) Effect of Concrete Strength on the General Method (Yoon) Details of Beam Specimens and Instrumentation (Sarsam,1992) Effect of the Ratio of Shear span on the Shear Friction Method (Sarsam) Effect of the Ratio of Shear span on the Simplified Method (Sarsam) Effect of the Ratio of Shear span on the General Method (Sarsam) Effect of Concrete Strength on the Shear Friction Method (Sarsam) Effect of Concrete Strength on the Simplified Method (Sarsam) Effect of Concrete Strength on the General Method (Sarsam) 65 x

11 5.21 Effect of Stirrup Spacing on the Shear Friction Method (Sarsam) Effect of Shear Reinforcement on the Shear Friction Method (Sarsam) Effect of Stirrup Spacing on the Simplified Method (Sarsam) Effect of Shear Reinforcement on the Simplified Method (Sarsam) Effect of Stirrup Spacing on the General Method (Sarsam) Effect of Shear Reinforcement on the General Method (Sarsam) Effect of Longitudinal Reinforcement on the Shear Friction Method (Sarsam) Effect of Longitudinal Reinforcement on the Simplified Method (Sarsam) Effect of Longitudinal Reinforcement on the General Method (Sarsam) Predicted Results by the Shear Friction Method (with stirrups) Predicted Results by the Simplified Method (with stirrups) Predicted Results by the General Method (with stirrups) Effect of the Ratio of Shear Span on the Shear Friction Method (with stirrups) Effect of the Ratio of Shear Span on the Simplified Method (with stirrups) Effect of the Ratio of Shear Span on the General Method (with stirrups) Effect of Concrete Strength on the Shear Friction Method (with stirrups) Effect of Concrete Strength on the Simplified Method (with stirrups) Effect of Concrete Strength on the General Method (with stirrups) Effect of Stirrup Spacing on the Shear Friction Method (with stirrups) Effect of Stirrup Spacing on the Simplified Method (with stirrups) Effect of Stirrup Spacing on the General Method (with stirrups) Effect of Shear Reinforcement on the Shear Friction Method (with stirrups) Effect of Shear Reinforcement on the Simplified Method (with stirrups) Effect of Shear Reinforcement on the General Method (with stirrups) Effect of Longitudinal Reinforcement on the Shear Friction Method (with stirrups) 90 xi

12 5.46 Effect of Longitudinal Reinforcement on the Simplified Method (with stirrups) Effect of Longitudinal Reinforcement on the General Method (with stirrups) Effect of Beam Depth on the Shear Friction Method (with stirrups) Effect of Beam Depth on the Simplified Method (with stirrups) Effect of Beam Depth on the General Method (with stirrups) Predicted Results by the Shear Friction Method (without stirrups) Predicted Results by the Simplified Method (without stirrups) Predicted Results by the General Method (without stirrups) Effect of the Ratio of Shear Span on the Shear Friction Method (without stirrups) Effect of the Ratio of Shear Span on the Simplified Method (without stirrups) Effect of the Ratio of Shear Span on the General Method (without stirrups) Effect of Concrete Strength on the Shear Friction Method (without stirrups) Effect of Concrete Strength on the Simplified Method (without stirrups) Effect of Concrete Strength on the General Method (without stirrups) Effect of Longitudinal Reinforcement Ratio on the Shear Friction Method (without stirrups) Effect of Longitudinal Reinforcement Ratio on the Simplified Method (without stirrups) Effect of Longitudinal Reinforcement Ratio on the General Method (without stirrups) Effect of Longitudinal Reinforcement Strength on the General Method (without stirrups) Effect of Beam Depth on the Shear Friction Method (without stirrups) Effect of Beam Depth on the Simplified Method (without stirrups) Effect of Beam Depth on the General Method (without stirrups) 105 xii

13 NOTATION a a c shear span, distance from centre of support to point load clear shear span, distance between outer edge of plate for concentrated load and inner edge of plate at support A A s A v b w c Cb area of potential shear failure plane area of longitudinal reinforcement in tension zone area of one stirrup width of beam web coefficient of the cohesion between a potential shear failure plane concrete cover at top of beam c, concrete cover at bottom of beam C C r concrete strength of beam factored concrete strength of beam C. O. V. coefficient of variation d distance from the extreme compression fibre to the centroid of the longitudinal tension reinforcement d b d v diameter of a reinforcing bar distance measured perpendicular to the neutral axis between the resultants of the tensile and compressive forces due to flexure d ev fry f y f' c h k effective length of stirrup in the shear friction method specified yield strength of the stirrups specified yield strength of the longitudinal reinforcement or stirrups specified compressive concrete strength overall height of member factor for relating shear strength and normal stress determined from experiments n R number of stirrups crossed by a potential shear failure plane normal force acting on potential shear failure plane xiii

14 5 spacing of stirrups S T T opi T r T v T vr v V c V cg shear force on potential shear failure plane longitudinal reinforcement strength of beam force in longitudinal reinforcement for peak shear strength factored longitudinal reinforcement strength of beam tension force in a stirrup factored tension resistance in a stirrup average shear stress on potential shear failure plane according to Loov's equations factored shear resistance attributed to the concrete factored shear resistance attributed to the concrete for the CSA general method V CS f factored shear resistance attributed to the concrete for the shear friction method V d Vf V g V p V r V rg V s V s f V sg dowel force in the longitudinal reinforcement factored shear force at section shear resistance of beam using CSA A general method factored transverse component of prestress of beam factored shear resistance of beam factored shear resistance of beam using CSA A general method factored shear resistance provided by the shear reinforcement factored shear resistance for the shear friction method factored shear resistance provided by the stirrups for the CSA general method V sim V s V ss f shear resistance of beam using CSA A simplified method shear resistance provided by one stirrup factored shear resistance attributed to the reinforcement for the shear friction method xiv

15 V t V 4S a ultimate shear resistance of beam measured from test shear resistance of concrete on a 45 plane angle between transverse reinforcement and the shear plane a/ angle between shear friction reinforcement and longitudinal axis (8 factor that depends on the average tensile strains in the cracked concrete using CSA general method j3 v calibration factor for shear friction method 6 angle between longitudinal axis and potential shear failure plane 9 min minimum shear failure angle for the shear friction method X factor to account for low density concrete \i coefficient of friction p p v o <j) c longitudinal tension reinforcement ratio transverse reinforcement ratio average normal stress on potential shear failure plane resistance factor for concrete. resistance factor for reinforcement y/ factor that depends on the ratio of longitudinal reinforcement strength to optimum tension for the shear friction method xv

16 1 CHAPTER 1 INTRODUCTION 1.1 General Failure in shear of reinforced concrete takes place under combined stresses resulting from an applied shear force, bending moments and, where applicable, axial loads and torsion as well. Because of the non-homogeneity of material, non-uniformity and non-linearity in material response, presence of cracks, presence of reinforcement, combined load effects, etc., the behavior of reinforced concrete in shear is very complicated, and the current understanding of and design procedures for shear effects are based on analyses of results of extensive tests and simplifying assumptions rather than on an exact universally acceptable theory. The best-known model for the expression of the behavior of beams with web reinforcement failing in shear is the truss model. The truss model is a helpful tool to visualize the nature of stresses in the stirrups and in the concrete, and to base simplified design concepts and methods on. It may also be used to derive equations for the design of shear reinforcement. However, it does not recognize fully the actual action of web reinforcement and its effect on the various types of shear transfer mechanisms. A shear-friction model has been developed to predict the shear strength of beams by Loov (17)(18)(I9) and many others OWiXWMWWWS). Because shear friction works well for composite beams, it might also predict the shear strength of beams which also have potential major cracks along which slip can occur. Stirrups and longitudinal reinforcement provide a clamping force thereby increasing the friction force which can be transferred across a crack along a potential failure plane. This model is based on the shear strength after cracking so that no diagonal tension strength is included. In this thesis, the shear friction model has been investigated and developed for the purpose of shear design of beams.

17 2 1.2 Code Review Prior to its 1984 revision, CSA Standard A23.3 recommended a method for shear and torsion design based on the traditional method adopted by the ACI code. The procedure is called the "V c + V" approach. The term V c is referred to as the "shear carried by the concrete", while the term V s is referred as the "shear carried by the stirrups". A23.3 assumes that V c is equal to the shear strength of a beam without stirrups and further simplifies V c to equal the shear at inclined cracking. V s relies on the tensile strength of the transverse reinforcement. The stirrups and the inclined compressive struts are assumed to act as members of a 45-degree truss and the term V s is calculated based on this model. The 1984 revision of the Canadian Standard, CAN3 A23.3-M84, recommended two alternative methods for shear design. The first of these, termed the "simplified method" (CAN3 A23.3-M84 (11.3)) is a shortened version of the traditional method followed by ACI and previous Canadian codes. In the simplified method, the transverse reinforcement is designed for the combined effect of shear and axial load if any, while the longitudinal reinforcement is designed for the combined effect of flexural and axial load. The second method is termed as "general method" for shear design (CAN3 A23.3-M84(l 1.4)). In this method, the truss analogy has been used in a more direct manner to account for the influence of diagonal tension cracking on the diagonal compressive strength of concrete, and the influence of shear on the design of longitudinal reinforcement. The code requires that deep beams, parts of members with deep shear span, brackets and corbels, and regions with abrupt changes in cross-section (such as regions of web openings in beams) be designed by the general method only. But we will find later in this thesis that the general method is not suited to the design of deep beams, brackets and corbels. CSA Standard A recommends three alternative methods for shear design. Regions of members in which it is reasonable to assume that plane sections remain plane shall be proportioned for shear and torsion using either the general method or the simplified method (if member is not subjected to significant axial tension) or the strut-

18 3 and-tie model. Regions of members in which the plane section assumption of fiexural theory is not applicable shall be proportioned for shear and torsion using the strut-and-tie model. The simplified method of shear design described in CSA Standard A is not simple. The designer is required to check numerous equations and limits. On the other hand, the general method is extremely complex so engineers rarely use it in engineering practice. 1.3 Scope of Study The main objective of this study is to introduce the shear-friction method for engineering design. After reviewing shear design theories and shear design methods which are used by recent CSA Standard A , a method of shear design based on shear friction theories has been applied to predict the shear capacity of reinforced concrete beams. A comparison of the shear-friction method and recent code methods with the test results of beams from the literature has been presented in this thesis. Proposed code clauses for shear design based on the shear-friction method have been developed and design examples based on the shear friction method are also included in this thesis. 1.4 Thesis Organization Chapter 2 contains the review of basic shear theories. The factors of shear strength are listed and shear failure mechanisms and modes are discussed in this chapter. In Chapter 3 CSA Standard A for shear design has been introduced and the design methods have been discussed. Chapter 4 introduced the shear friction methods by Loov and others. In Chapter 5 a modified equation of the shear friction method has been introduced and a comparison of the shear-friction method and recent code methods with the test results of beams from the literature has been presented in this chapter.

19 4 Proposed code clauses for shear design based on the shear-friction method with design examples have been put in Chapter 6. Conclusions and recommendations are given in Chapter 7.

20 5 CHAPTER 2 BASIC SHEAR THEORIES 2.1 Homogeneous Beam In order to gain an insight into the causes of shear failure in reinforced concrete, the stress distribution in a homogeneous elastic beam of rectangular section will be reviewed briefly. From the free-body diagram as shown in Fig.2-1, it can be seen that Where dm = the bending moment change from section to section dx = the distance between sections V = the shear force on the section M+dM Fig. 2-1, Internal Forces in Beam By the traditional theory for homogeneous-elastic-uncracked beams, the shear stresses, v, and the flexural stress, f x, at a point in the section distant y from the neutral axis are given by (2-2)

21 6 (2-3) Where Q = the first moment about the neutral axis of the part of the cross-sectional area above the depth y I - the moment of inertia of the cross section b = the width of the beam The distribution of these stresses is as shown in Fig Considering an element at depth y (Fig. 2-3), the fiexural and shear stresses can be combined using Mohr's circle into equivalent principal stresses, f and f 2, acting on orthogonal planes inclined at an angle a, where i ÍZ7 (2-4) f u ~ 2 f x ± Í{2 f x and tan(2«) = (2-5) Fig. 2-2, Distribution of Fiexural Shear Stresses

22 Fig. 2-3, Principal Stresses The principal stress trajectories in the uncracked beam are plotted in Fig. 2-4a. Stress trajectories are a set of orthogonal curves, whose tangent at any point is in the direction of the principal stress at that point. The compressive stress trajectories are steep near the bottom of the beam and flatter near the top. In concrete, which is weak in tension, tensile cracks would occur at right angles to the tensile stresses and hence the compressive stress trajectories indicate potential crack patterns (see Fig.2-4b). (Note that if in fact a crack is developed, the stress distributions assumed here are no longer valid in that region and redistribution of the internal stresses takes place.) The location of the absolute maximum principal tensile stress will depend on the variation off x and v, which in turn depends on the shape of the cross section and on the span and loading. It is seen that the general influence of shear is to induce tensile stresses on an inclined plane. Failure of concrete beams in shear is triggered by the development of these inclined cracks under combined stresses. To avoid a failure of the concrete in compression, it is also necessary to ensure that the principal compressive stress,/^, is less than the compressive strength of concrete under the biaxial state of stress.

23 Fig. 2-4a, Stress Trajectories Fig. 2-4b, Potential Crack Pattern

24 9 Although several theories of failure have been used for concrete shear design, for the traditional method of shear and torsion design, the principal tensile stress theory has been followed. 2.2 Beam Cracking Modes The cracking pattern in a test beam is shown in Fig.2-5. Two types of cracks can be seen. The vertical cracks occurred first, due to fiexural stresses. These start at the bottom of the beam where the fiexural stresses are the largest. The inclined cracks at the ends of the beam are due to combined shear and flexure. These are commonly referred to as inclined cracks or shear cracks. Such cracks must exist before a beam can fail in shear. Several of the inclined cracks have extended along the reinforcement toward the support, weakening the anchorage of the reinforcement. Fig. 2-5, A Cracked Beam without Shear Reinforcement (Ref. 27) Although there is a similarity between the planes of maximum principal tensile stresses and the cracking pattern, fiexural cracks generally occur before the principal tensile stresses at midheight become critical. Once such a crack has occurred, the

25 10 principal tensile stresses across the crack drops to zero. To maintain equilibrium, a major redistribution of stresses is necessary. As a result, the onset of inclined cracking in a beam cannot be predicted from the principal stresses unless shear cracking precedes flexural cracking. This very rarely happens in reinforced concrete but does occur in some prestressed beams (such as I-section beam). The cracking pattern in a test beam with shear reinforcement is shown in Fig.2-6. It is obvious that inclined cracks are almost straight lines instead of curves that we have seen in the test beam without shear reinforcement in Fig.2-5. Another evidence we can see is that inclined cracks bypass as many stirrups as possible. These evidences are useful to predict possible beam shear failure planes. Fig. 2-6, A Cracked Beam with Shear Reinforcement (Ref. 35) 2.3 Shear Transfer Mechanisms There are several mechanisms by which shear is transmitted between two planes in a concrete member. Fig. 2.7 shows a free body of one of the segments of a reinforced concrete beam separated by an inclined crack. The major components contributing to the shear resistance are:

26 11 (1) The shear strength, V cz, of the uncracked concrete; (2) The vertical component, V^, of the aggregate interlock shear, V a ; (3) The dowel force, V d, in the longitudinal reinforcement; (4) The shear, V s, carried by the shear reinforcement. Fig. 2-7, Internal Forces in a Cracked Beam The aggregate interlock, V a, is a tangential force transmitted along the plane of the crack, resulting from the resistance to relative movement (slip) between the two rough interlocking surfaces of the crack, much like frictional resistance and transverse rebar dowel effects. So long as the crack is not too wide, the force V a may be very significant. The dowel force in the longitudinal tension reinforcement is the transverse force developed in these bars functioning as a dowel across the crack, resisting relative transverse displacements between the two segments of the beam.

27 12 Each of the components of this process except V s has a brittle load-deflection response. So it is difficult to quantify the contributions of V cz, V, and V ay. In design, these are lumped together as V c, referred to as "the shear carried by the concrete". Thus the nominal shear strength, V n, is assumed to be V n = V c + V s (2-6) In North American design practice, V c is traditionally taken equal to the failure capacity of a beam without stirrups. 2.4 Shear Failure Modes Beam without Shear Reinforcement In beams without shear reinforcement, the breakdown of any of the shear transfer mechanisms may lead to failure. In such beams there are no stirrups enclosing the longitudinal bars and restraining them against splitting failure and the value of V d is usually small. The component V ay also decreases progressively due to the unrestrained opening up of the crack. The spreading of the crack into the compression zone decreases the area of uncracked concrete section contributing to V cz. However, in relatively deep beams (a/d < 1), tied-arch action may develop following inclined cracking (see Fig. 2-9 (b)), which in turn will transfer part or all of the shear load at the section directly to the supports thereby the shear capacity of the beam does not totally rely on V ay and V cz. Because of the uncertainties in all these effects, it is difficult to predict precisely the behavior and strength beyond diagonal cracking of beams without shear reinforcement. In beams without shear reinforcement, the shear failure load may equal or exceed the load at which inclined cracks develop, depending on several variables such as the ratio M/(Vd), thickness of web, influence of vertical normal stresses, concrete cover and resistance to splitting (dowel) failure. Further, the margin of strength beyond diagonal cracking fluctuates considerably. Hence, for beams of normal proportions (M/(Vd) > about 2.5), as a design criterion, the shear force, V cr, causing the formation of the first

28 13 inclined crack is generally considered as the usable ultimate strength for beams without shear reinforcement. The moments and shears at inclined cracking and failure of rectangular beams without web reinforcement are plotted as a function of the shear span, a, to the depth, d, in Fig.2-8. The shaded areas in this figure show the reduction in strength due to shear, so web reinforcement has to be provided to ensure that the full fiexural capacity can be developed. Typical shear failure modes of reinforced concrete beams, and the influence of the a/d ratio, are illustrated in Fig. 2-9 with reference to a simply supported rectangular beam subjected to symmetrical two-point loading. In very deep beams (a/d < 1) without web reinforcement, inclined cracking transforms the beam into a tied-arch (Fig. 2-9b). The tied-arch can fail by either a breakdown of its tension element, or by a breakdown of the concrete compression chord by crushing. In relatively short beams, with a/d in the range of 1 to 2.5 (Fig. 2-9c), the failure is initiated by an inclined crack, usually a flexural-shear crack. The actual failure may take place by crushing of the reduced concrete section above the head of the crack under combined shear and compression, or cracking along the tension reinforcement resulting in loss of bond and anchorage of the tension reinforcement. This type of failure usually occurs before the fiexural strength of the section is attained. Normal beams have a/d ratios in excess of about 2.5. Such beams may fail in shear or in flexure. The limiting a/d ratio above which fiexural failure occurs depends on the tension reinforcement ratio, yield strength of reinforcement and concrete strength.

29 a V v a (a) Beam. c 03 O re c E o 2 Deep * H Slender, y/a/ery slender Very.Shorty short 1 1 " ' V ^¾¾^ / / / / ^ ^ Flexural capacity ^^C^ ^ Failure ^ s ^ ^ ^*». Inclined cracking i a/d (b) Moments at cracking and failure. Inclined cracking and failure <T3 Ct> JO CO Flexural capacity nclined cracking and failure a/d (c) Shear at cracking and failure. Fig. 2-8, Effect of a/d on Shear for Beams Without Shear Reinforcement (Ref. 27)

30 15 i (a) Shear-tension failure ( c) l«a/d<2 5 Shear-compression failure Diagonal tension failure (d) 2.5 <a/d <**6 T (e) Web-crushing failure Fig. 2-9, Shear Failure Modes (Ref. 36)

31 16 For beams with a/d ratios in the range of 2.5 to 6, fiexural tension cracks develop early on; however, before the ultimate fiexural strength is reached the beam may fail in shear by the development of inclined flexure-shear cracks, which, in the absence of web reinforcement, rapidly extend right through the beam as shown in Fig. 2-9d. This type of failure is usually sudden and without warning and is termed diagonal-tension failure. Addition of web reinforcement in such beams leads to a shear-compression failure or a fiexural failure. In addition to these different modes, thin webbed members such as I-beams with web reinforcement may fail by crushing of the concrete in the web portion between inclined cracks under the diagonal compression forces (Fig. 2-9e) Beam with Shear Reinforcement In members with shear reinforcement the shear resistance continues to increase even after inclined cracking until the shear reinforcement yields and V s can increase no more. Any further increase in applied shear force leads to increases in V cz, V d, and V^. With progressively widening crack width (which is no longer restrained because of yielding of the shear reinforcement), begins to decrease forcing V cz and V d to increase at a faster rate until either a splitting (dowel) failure occurs, or the concrete in the compression zone fails under the combined shear and compression forces. Thus, in general, the failure of shear-reinforced members is more gradual (ductile). 2.5 Factors Affecting the Shear Strength Tensile Strength of Concrete The inclined cracking load is a function of the tensile strength of the concrete. The stress state in the web of the beam involves biaxial principal tension and compression stresses as discussed before. A similar biaxial state of stress exists in a split cylinder tension test and the inclined cracking load is frequently related to the strength from such test.

32 Longitudinal Reinforcement Fig shows the shear capacities of simply supported beams without stirrups as a function of the steel ratio, p. When the steel ratio, p, is small, flexural cracks extend higher into the beam and open wider, as a result inclined cracking occurs earlier and the beam shear strength tends to be lower Shear Span-to-depth Ratio, a/d As discussed earlier, the shear span-to-depth, a/d, has effects on the inclined cracking shears and ultimate shears of "deep" beam, while for longer shear spans with a/d greater than 3 it has little effect on the inclined cracking shear Fig. 2-10, Shear Strength vs. Longitudinal Reinforcement (Ref. 27)

33 Fig. 2-11, Shear Strength vs. a/d (Ref. 14)

34 Size of Beam As the overall depth of a beam increases, the shear stress at inclined cracking tends to decrease for a given f' c, p, and a/d. As the depth of the beam increases, the crack widths at points above the main reinforcement tend to increase. This leads to a reduction in aggregate interlock across the crack, resulting in earlier inclined cracking. In beams with web reinforcement the web reinforcement holds the crack faces together so that the aggregate interlock is not lost as much as that of beams without web reinforcement. Fig. 2-12, Influence of Member Size on Shear Strength (Ref. 7)

35 Axial Forces Axial tensile forces tend to decrease the inclined cracking load, while axial compressive forces tend to increase it. As the axial compressive force is increased, the onset of fiexural cracking is delayed and the fiexural cracks do not penetrate as far into the beam. So a larger shear is required to cause principal tensile stresses equal to the tensile strength of the concrete. Vu ^Fcbwd _ -/- ^**" Eq. 6-17a / - (ACI Eq. 11-4) Eq. 6-17b (ACI Eq. 11-8) Compression Axial stress, NJA g (psi) Fig. 2-13, Effect of Axial Loads in Inclined Cracking Shear (Ref. 27) Web Reinforcement Prior to inclined cracking, the strain in the stirrups is equal to the corresponding strain of the concrete. Since concrete cracks at a very small strain, the stress in the stirrups prior to inclined cracking will be very small. Thus stirrups do not prevent inclined cracks from forming. They come into play only after the cracks have formed. Following the development of inclined cracking, stirrups intercepted by the cracks resist a portion of the shear. The web reinforcement contributes significantly to the

36 21 overall shear strength by the direct contribution of V s to the shear strength. Secondly, web reinforcement crossing the inclined cracks restricts the widening of the crack and thereby helps maintain the aggregate interlock resistance of shear. The web reinforcement also can improve the longitudinal tension reinforcement dowel action and provide another dowel action of itself crossing inclined cracks. Flexural Inclined Yield of Failure cracking cracking stirrups Applied shear Fig. 2-14, Distribution of Internal Shears of Beam with Shear Reinforcement (Ref. 27)

37 22 CHAPTER 3 SHEAR DESIGN - CSA STANDARD A General The CSA Standard A recommends two alternative methods for shear design. The "Simplified Method" is a short version of the traditional method followed by ACI and previous Canadian Codes. In the Simplified Method, a 45-degree truss model has been used and the transverse reinforcement is designed based on that. The second method is the "general method" for shear design. In this method, the truss analogy has been used in a more direct manner to account for the influence of diagonal tension cracking on the diagonal compressive strength of concrete, and the influence of shear on the design of longitudinal reinforcement. Both simplified method and general method are sectional methods and can be applied only to the flexural region of beams, in which it is reasonable to assume that plane sections remain plane and that shear stresses are distributed in a reasonably uniform manner over the depth of the beam. Because of this, both methods are not appropriate for regions of members near static or geometric discontinuities, the code requires regions with abrupt changes in cross-section (such as regions of web openings in beams) and brackets and corbels, to be designed by the strut-and-tie method, which is capable of more accurately modeling the actual flow of forces in these regions. 3.2 Simplified Method For flexural members not subjected to significant axial tension, the Canadian code allows shear design based on the simplified method. Required shear resistance for beam is: V r >V, (3-1)

38 23 Where Vf is the factored shear force at a section, and V r is the sum of the contribution attributed to the concrete and transverse reinforcement. V r = V c + V, (3-2) But V r is limited to: V,iV e +0.8ty e JfJb w d (3-3) This upper limit is intended to ensure that the stirrups will yield prior to crushing of the web concrete and that diagonal cracking at specified loads is limited Shear Supported by Concrete, V c V c =0.2ty e Jf b w d (3-4) This equation can be used only for beams with minimum transverse reinforcement given by Clause if ^exceeds 0.5 V c + <j) p V p : A v =0.06jf c^f (3-5) J V The minimum transverse reinforcement restrains the growth of inclined cracking and increases ductility to provide a warning of failure. For beams without transverse reinforcement, Clause shall be used to account for the reduced strength of beams deeper than 300 mm.

39 24 V, d At c Jfìb w d>0.la&4fìb w d (3-6) Studies have shown that the equations for V c above are more appropriate for beam with a/d ratios greater than three. It results in overly conservative design for beams with a/d ratios less than 2.5 (see Fig. 3-1) «a» a»»j //.illi.l f 610mm bdf, Clause 11.5: Strut-and-tie model Experimental result fc' = 27.2 MPa max. agg. = 19 mm d = 538 mm b= 155 mm A, = 2277 mm Clause 11.4: Sectional model ry=372 MPa A v=0 3 4 a/d Fig. 3-1, Comparison of Shear Design Methods and Test Results (Ref.7)

40 Shear Supported by Stirrups, V s <l> s Avf y d s (3-7) Here the transverse reinforcement is assumed to be perpendicular to the longitudinal axis of the member. Additional maximum spacing (Clause ) and minimum transverse reinforcement requirement (Clause ) have been patched onto the basic equation in order to obtain satisfactory behavior under various conditions. 3.3 General Method Shear resistance for beam is: (3-8) Where V cg is the factored shear resistance contributed by concrete at a section, and V sg is the factored shear resistance contributed by transverse reinforcement. But V shall not exceed V c = 0.25 Áfcf b w d v (3-9) Where d v is the distance measured perpendicular to the neutral axis between the resultants of the tensile and compressive forces due to flexure, but need not be taken less than 0.9d. This upper limit is intended to ensure that the stirrups will yield prior to crushing of the web concrete and that diagonal cracking at specified loads is limited.

41 Shear Supported by Concrete, V cg V c =UA&ft 4f!b w d v (3-10) Where p is determined in accordance with Clause Shear Supported by Stirrups, V sg S Where 0is given in Clause Obviously if 0 = 45 both simplified method and general method will have the same shear resistance contributed by transverse reinforcement. Again assume that the transverse reinforcement is perpendicular to the longitudinal axis of the member. For members with transverse reinforcement inclined at an angle a to the longitudinal axis, V sg shall be computed from _ faa v f y d v (cot0 + cota)sina

42 27 CHAPTER 4 SHEAR DESIGN - SHEAR FRICTION METHOD 4.1 General The Clause in CSA Standard A states that shear friction shall be used to design "Interfaces between elements such as webs and flanges, between dissimilar materials, and between concrete cast at different times or at existing or potential major cracks along which slip can occur..." Because beam shear failure normally comes with a major crack and slip between the crack, it would seem that shear friction can also be applied to predict the shear strength of beams. In 1997, Loov presented the rudiments of a procedure (19), which applied this concept to the shear design of beam. In recent years, Loov, Peng, Tozser, Kriski, and others, have shown that it is possible to use a simpler method for shear design that is based on the shear friction theory. (16)(17)(18)(21)(23)(24){25)(26)(35) It is encouraging that some of the resulting equations derived by Loov match those equations derived by a number of people, including Braestrup (5), Nielsen (33) and Zhang (45) based on theories of plasticity. 4.2 General Equations for Beam Shear Based on Shear Friction: The shear-friction concept for concrete-to-concrete interfaces is based on the assumption that a crack will form and shear will be transferred across the interface between the two parts that can slip relative to one another. If the crack faces are rough and irregular, this slip is accompanied by separation of the crack faces. The separation will stress the reinforcement crossing the crack until the reinforcement reaches its yield point. Thus the reinforcement provides a clamping force across the crack interface.

43 28 Shear displacement t î t î î t î 1111 Compression in concrete = T i Shear stress (i) Shear Tension Causing Crack Opening Tension in reinforcement = T (ii) Free-Body-Diagram Fig. 4-1, Shear Friction Concept (Ref.7) Shear Friction Strength: There are many equations that have been developed for predicting shear friction strength. Fig.4-1 illustrates the shear friction concept for the case where the reinforcement is perpendicular to the potential failure plane. Because the interface is rough, shear displacement will cause a widening of the crack. This crack opening will cause tension in the reinforcement balanced by compressive stresses, a, in the concrete across the crack. The shear resistance of the face is often assumed to be equal to the cohesion, c, plus the coefficient of friction, ju, times the compressive stress, a, across the face. That is, v r =À&(c+Mff) (4-1) If inclined reinforcement is crossing the crack, part of the shear can be directly resisted by the component, parallel to the shear plane, of the tension force in the reinforcement. See Fig.4-2. Clause 11.6 of CSA Standard A23.3-M94 suggests that the factored shear stress resistance of the shear plane shall be computed as: v r =A,fc(c+fia)+ûp v fcosa/ (4-2)

44 29 Where a f is the angle between the shear friction reinforcement and the shear plane. \ \ \ Fig. 4-2, Reinforcement Inclined to Potential Failure Cracks (Ref.7) CSA Standard A23.3-M94 also gives an alternative equation for shear friction strength, which is based on the work of Loov and Patnaik (20)(22). (4-3) Where & = 0.5 for concrete placed against hardened concrete k = 0.6 for concrete placed monolithically. In this method, the shear resistance is a function of both the concrete strength and the amount of reinforcement crossing the failure crack. Fig. 4-3 shows how this equation compares with the results from various push-off tests.

45 a Mattock (uncracked) Mattock (cracked) A Walraven (cracked) v/0~fhmpa) Fig. 4-3, Push-off Test Results (Ref.22)

46 31 Fig. 4-4 shows a free body diagram of the end portion of a simple beam with loads applied somewhere to the right of the section. Two equilibrium equations relate the normal force, R, and the shear force, S, to T, the force in the main tension reinforcement, nt v, the total force in the stirrups crossing the plane and V, the end reaction. The forces on a potential failure plane vary with the angle 0 between the axis of the beam and the plane. When the loads between the reaction and the plane in question are negligible, then V is equal to the vertical shear on the inclined plane. R = Tsin0-(V-ZT v )cosd (4-4) n S = Tcos0-(V-ZT v )sin0 (4-5)

47 32 Where T = Af y and T v = AJ^. Here A s is the area of longitudinal reinforcement and f y is its yield strength, while A v is the total area of all legs of a stirrup and f vy is the stirrup yield strength. Using the relationship from Eq. 4-3, the shear friction stress is v = While kjtf (4-6) and a = R Where A is the area of the inclined failure plane, A = b wh sine? (4-7) Where b w is the width of web, h is the total depth, and 6 is the angle between the longitudinal reinforcement and the crack. The shear force is therefore proportional to the square root of the normal force, R S = k4rf^4 (4-8) The equations shown above (Eq. 4-4 to Eq. 4-8) can be combined to give a general equation for the shear strength V = 0.5k 2 C Where 0.25 k 2 C + cot2 0-cotO (1 + cot 2 6)-Tcot0 + Yjv ( 4 " 9 ) C - f'xh (4-10)

48 33 This equation is similar to that derived by Braestrup (5) and by Nielsen (33) with plasticity theory. For design, the factored values should be used thus V r =0.5k 2 C, Where 0.25k'C. + cot2 0-cotO (1 + cot 2 0)-T r cot0 + J]T vr (4-11) n (4-12) T r=<t>asfy (4-13) T vr = <t>s A vf,y (4-14) All planes between the inside edge of the support and the edge of the load to a maximum angle of 90 should be considered to be potential failure planes. The shear strength on each plane is calculated and the lowest strength, when comparing all possible failure planes, is the governing shear strength. Under some circumstances it may be extremely unlikely that a crack will form along particular failure planes so that choosing the absolute lowest strength without regarding to location may be excessively conservative. This aspect has been investigated by Zhang (45). Fig. 4-5 shows the change in predicted shear strength as the failure plane angle is varied. When a crack intercepts a stirrup, the shear strength increases by T v, the force that can be developed in the stirrup. Fig. 4-6 shows a three-dimensional surface plot, which was obtained by analyzing beam tests by Kani (14). The test beams had only one stirrup but in different locations to determine the effects of stirrup location. The test result shows that it is not necessary to check every potential failure plane. The planes with the lowest strength have the flattest possible angle while intersecting a minimum number of stirrups.

49 Fig. 4-6, Three-dimensional surface of shear strength along all possible failure planes for beam 544 (Réf. 18)

50 35 Fig. 4-7 shows a beam with possible critical shear failure planes. Fig. 4-8 is a photograph of a beam indicating that the actual cracks correspond to the expected failure planes. essala V B - 7 T - Fig. 4-8, A Tested Beam with Critical Shear Cracks (Ref. 35)

51 Approximate Shear Capacity of Concrete If the shear failure plane bypasses the stirrups, the strength along the weakest plane depends on the longitudinal reinforcement and the angle of the failure plane, but is unaffected by the stirrup strength. From Eq. 4-9 we can obtain V = 0.5k 2 C 0.25k 2 C + cot2 6 -cot6 (1 + cot 2 0)-Tcot6 (4-15) Beams depend on longitudinal reinforcement and the anchorage of longitudinal reinforcement to develop shear capacity. The optimum tension in the longitudinal reinforcement, by which the maximum shear capacity will be developed, can be obtained by differentiating Eq dv c (l + cot 2 6) / - =. ' =-coto (4-16) dt 4t /0.25k 2 C +cot 2 0 T opt = 0.25k 2 C(2 +tan 2 6) (4-17) Substitute Eq into Eq. 4-15, the shear strength of beams will be V c = 0.25k''Ctond (4-18) Eq.4-18 gives the shear capacity of beams with longitudinal reinforcement tension capacity f y A s >T opt k 2 C(2+tan 2 9). It is assumed that anchorage for longitudinal reinforcement to develop such tension capacity is sufficient. Fig. 4-9 shows VjC vs.cot# for different ratios of longitudinal reinforcement. It is clear that Eq. 4-8 represents the upper bound value of shear capacity of beams. For beams with longitudinal reinforcement tension capacity less than T op t> the ^ e s s the longitudinal reinforcement tension capacity, the less the shear capacity.

52 > Fig. 4-9, Shear Strength vs. coto For beams with longitudinal reinforcement tension capacity f y A s less than J opt, Eq can be substituted approximately by a simple equation as following: V, = 0.25k 2 sin n T Ctand (4-19) 2 T \ t* J

53 38 Fig. 4-10, Shear Strength vs. coto by Eq and Eq The curves from Eq and Eq have been plotted on Fig for comparison. The graph shows that Eq is a useful approximation for the shear capacity of concrete. For factored design, we should use: V cr = 0.25 k 2 C r tane When T>T opt (4-20) V = 0.25k 2 sin n T 2 T V p> J C r tan6 When T<T apt (4-21) Where (4-22)

54 39 T = <t> s AJ y (4-23) T opt = 0.25k 2 C r (2 + tan 2 9) (4-24) Fig plots the beam shear strength of concrete vs. the beam longitudinal reinforcement for a particular plane in the beam based on shear-friction equations of Eq and Eq It shows that the variation of the beam shear strength of concrete increases as the beam longitudinal reinforcement increases. When the beam longitudinal reinforcement reaches f y A s = 0.25k 2 C(2 +tan 2 9), the beam shear strength of concrete reaches its peak value and will not increase even though the beam longitudinal reinforcement increases. A sf y(kn) Fig. 4-11, Shear Strength vs. Longitudinal Reinforcement of Beam

55 Approximate Shear Capacity of Stirrups The usual equations for the shear strength of stirrups are overly optimistic. Fig.4-7 shows several possible failure planes with zero, one and two stirrups crossing them. Fig. 4-8 is a photograph of a beam indicating that the actual cracks correspond to the expected failure planes. To ensure a conservative prediction, the number of stirrups that are considered to cross the shear plane should be the number of stirrup spaces crossed by the crack minus one. Marti (28) correctly accounted for this in his work. Therefore, because of the nature of shear failure planes that tend to avoid stirrups the proper estimate of the stirrup contribution may be ÚL cote (4-25) For factored design, we shall use: V - V y sr Where r si 'd ev cote } vy (4-26) (4-27) Equation 4-27 is one of the most significant discoveries by Marti (28) and Loov (18) in shear design, because this corrects a basic mistake that has been used for years in shear design Approximate Shear Design Equations for Beams with T>T opt Using the " V c + V s " approach, the approximate shear strength along a plane at an angle 0 to the beam axis is

56 41 V r =V 4S tanû Where 'décote + V sl -1 v s j (4-28) V 4} = 0.25k 2 C r (4-29) Further, Eq can be written as: V 45 =WJ v 4TXh (4-30) Where J3 V = 0.25k 2 4/: (4-31) The coefficients k and fi v are calibration factor that can be adjusted to match the equation with test results. The shear strength of beams without stirrups is governed by the first term in Equation (4-28), where 0 is the angle of the failure plane with the lowest slope that can be expected to occur. V r =V 45 tang (4-32) Critical Shear Failure Angle Although theoretically we have only an integer number of possible shear failure planes such as 1, 2 and 3 in Fig. 4-7 and Fig. 4-8, it is convenient to treat Eq as a continuous function of 6 when deriving the critical shear failure angle. It is notable that the effects of stirrup spacing will be ignored and Eq will form a lower bound of the shear capacity, when Eq is considered to be a continuous function of 0 (see Fig. 4-12).

57 42 Fig. 4-12, The Shear Contributions of Concrete and Discrete Stirrups (Ref. 18) The critical angle 0 corresponding to the minimum strength can be found by differentiating Eq ñ V V d ev V = ½ " d0 r cos 2 0 sin 2 0 s =0 (4-33) IK, tane =i^l^- (4-34) ] v a s Substituted Eq into Eq. 4-28,

58 43 V. = V 45 \v 45 + v sl -i (4-35) y45 s So d V r =2AV 45 V sl^--v s (4-36) Eq is a direct solution for the shear strength of reinforced concrete beams. It combines the contribution of the web stirrups and concrete corresponding to the minimum strength of the combination. From Eq we can solve directly to obtain the maximum stirrup spacing. s< (Vf+K,) 2 (4-37) Eq can be used for design of stirrup spacing, while Eq is used to calculate the shear capacity of a beam with known stirrup spacing. Eq and Eq do not apply in cases where the shear failure angle is not determined by Eq The shear failure crack can only be formed between the beam support and load, so the beam shear span limits the minimum shear failure angle to: tano> a.. (4-38) The strength along this steeper plane can be obtained directly using Eq However, Eq and Eq are conservative if the shear failure angle becomes steeper under the limitation of beam shear span.

59 Beams with Longitudinal Reinforcement T <T. opt For beams without stirrups, the shear capacity can be derived from Equation (4-21) as following: K =yv 45 tano When T<T opt (4-39) Where y/ - sin TV T 2 T (4-40) Accordingly, Eq and Eq need to be modified as follows: V; = 2 \ V 45 V sl -^-V sl V s (4-41) s< (vf +v slr (4-42) It is worthy to notice that Eq and Eq may generate conservative results for beams with short shear span.

60 45 CHAPTER 5 EXPERIMENTAL STUDIES AND COMPARISONS By testing the proposed shear friction method against available experimental results from different authors, the shear friction method for shear design of beams will be evaluated in this chapter. A comparison study of the simplified method and the general method is also conducted in this chapter to choose a more accurate method for shear strength prediction. 5.1 Application of Shear Friction Method a beam is: Using the "V c + V" approach as discussed in Chapter 4, the total shear capacity of V =V +V Y y sf csf T y ssf (5-1) The shear capacity of concrete, V cs^ can be calculated from Vcs f = V 45 tane (5-2) Where (5-3) w = i When T>T, opt (5-4) if/ = sin n T 2 T When T<T opt (5-5) tano = V 4S s (5-6) T = *,AJ y (5-7) T o =V 45 (2 + tan 2 0) (5-8)

61 46 IT (5-9) The value of V ss / shall be computed from Y ssf y si d., cot 6 -I (5-10) Eq. 5-2 is derived from the equation in Chapter 4 with some modifications. The value of Pv from Eq.4-37 is: P v =0.25k 2 4fl (5-11) It has been found that k becomes smaller as the concrete strength increases (22)(37). The equation found from a least-squares fit of tests is: k=2.0(/:)- 0 4 (5-12) Substituted Eq into Eq and get: / \0.30 '30^ A =0.36 \f'c J (5-13) To consider the effects of beam depth as discussed in Chapter 2, Eq needs to be modified. According to the researches by Tozser and Loov (25)(26), the shear strength of beams decreases when the depth of beams increases in proportion to h~ 025. Finally, the equation for calculating fi v is presented by Eq

62 47 /3 V = h (5-14) There are two limitations for the cracking angle 0. First, for beams with short shear spans the shear cracking angle may be limited by the ciç/h ratio as mentioned in Chapter 4. Second, from pictures of crack patterns of specimens from literature (14)(45), it is observed that when the shear span is greater than 2.5, the shear cracking angle 0 stays at a limiting angle even with increasing shear span. Based on the analysis of the test results from literature, the minimum shear cracking angle 9 is about 2if-^/^J degree. So the two limitations for the failure angle are: tano> (5-15) 6>21 'fit" \30) (5-16) 5.2 Test Results in Literature: Experimental data from the literature were examined to verify whether the shear friction method is a rational approach for estimating the shear capacity of beams. Tests from two series of tests from the literature are presented and discussed in detail. The results predicted by the shear friction method were then compared with the test results from a total of 113 beams with stirrups and 105 beams without stirrups. All selected beams were simply supported rectangular beams subjected to a symmetrical single or two-point load. The effects of concrete strength, shear span ratios, amount of longitudinal reinforcement and stirrup spacing are discussed. Notice that the limitation on maximum stirrup spacing by the CSA A clauses for the simplified method and the general method was ignored during the analysis.

63 Yoon, Cook and Mitchell's Tests, 1996 (44) : Yoon, Cook and Mitchell investigated six full-scale beam specimens (44). The six beams having different amounts of shear reinforcement at each end were tested to provide a total of 12 shear tests. Fig. 5-1 shows the details of the 375 mm wide x 750 mm deep specimens that were tested with a clear shear span of 2000 mm and shear span ratio of a/d = 3.28 and a^h = The fiexural tension reinforcement for all of the specimens consisted of 10-No.30 bars in two layers, giving a reinforcement ratio of p = A symmetrical single point load had been applied at midspan. Table 5-1 lists the details of the beam specimens mm dear 150-1» ELEVATION VIEW 1/¾ y 2150^ 350i 1 2-No.lO Stirrup rwrtforoefnent vanea 10- No 30 cover MOmm r"*""""4 J a/2 I 650 J6JL Strain 9«9e«on fwnrafonimflt LVOT» on concreto SECTION A-À INSTRUMENTATION Fig. 5-1, Details of Beam Specimens and instrumentation (Yoon, Cook and Mitchell)

64 49 Table. 5-1, Details of Beam Specimens (, Shear reinforcement Specimen //. MPa Stirrup size and spacing, mm b w s* MPa Comments N-Se ríes: Nl-S Nl-N N2-S N2-N mm diameter at mm diameter at mm diameter at No stirrups Min /4,, s - d/2 Min A,, s = 0.7d > Min A,, s = d/2 M-Series: Ml-S Ml-N M2-S M2-N mm diameter at mm diameter at mm diameter at No stirrups AC1 83, ACI 89,* CSA 84 Min A^ s = d/2 CSA 94 min /1,, s = d/2 ACI 89t mm A,, s < d/2 H-Series: Hl-S Hl-N H2-S H2-N mm diameter at mm diameter at mm diameter at No stirrups ACI 83, ACI 89,* CSA 84 Min Ay, s = d/2 CSA 94 min A s < d/2 ACI 89t min A* s < d/2 Lower amount of minimum *4 V provided when *jf c ií a/69 MPa in design. tupper amount of minimum A v provided when Jf/ > a/69 MPa in design. Noterj^. for all stirrups is 430 MPa; area of 8.0-mm-diamcter bar = 50 mm 2 ; area of 9.5-mm-diameter bar = 7! mm 2. The purpose of the paper was to evaluate the minimum shear reinforcement requirements in normal, medium, and high-strength reinforced concrete beams. Therefore the tested beams were reinforced with minimum shear reinforcement, except three

65 50 specimens without shear reinforcement (Nl-S, Ml-S and Hl-S). Here these test data are used for evaluation of shear design methods under the effects of concrete strength, stirrup spacing and shear reinforcement, Table 5-2 gives the test results and a comparison of predicted and measured shear capacities of specimens. The predictions using the shear friction method and the simplified method agree well with the experimental results with value of C.O.V. 6.8% and 12.5% respectively, while the prediction using the general method results in a higher value of C.O.V. 23.7%. Table. 5-2, Test Results and Comparison of Predictions V, V S f Vsim Vg Specimen (kn) (kn) (kn) (kn) Vt/V,, v t /v sim Vt/V g Nl-S Nl-N N2-S N2-N Ml-S Ml-N M2-S M2-N Hl-S Hl-N H2-S H2-N m a C.O.V. 6.8% 12.5% 23.7%

66 51 The analyses of the 9 beams with shear reinforcement are illustrated from Fig. 5-2 to In Fig. 5-2, the ratios of test results to the results predicted by the shear friction method against concrete strength, f' c, are plotted to demonstrate the effect of concrete strength on the shear friction method. It shows no obvious trend in the prediction of shear capacity for beams with different concrete strength. Fig. 5-3 and Fig 5-4 present the analysis results of the effects of concrete strength using the CSA simplified method and general method respectively. A downward trend exists for both of methods V t 1.5 Vrf 1 -tr f c (MPa) Fig. 5-2, Effect of Concrete Strength on the Shear Friction Method

67 2.5r V t 1.5- Vsim fc (MPa) 90 Fig. 5-3, Effect of Concrete Strength on the Simplified Method 2.5r V t B- -s f c (MPa) 90 Fig. 5-4, Effect of Concrete Strength on the General Method

68 53 The ratios of test results to the results predicted by the shear friction method against the ratios of s/d and the web reinforcement index Pyfyy are plotted in Fig. 5-5 and Fig. 5-6 respectively. The shear friction method demonstrates a consistent accuracy of the prediction of shear capacity for beams with different stirrup spacing and different amounts of shear reinforcement. In Fig. 5-7 to Fig. 5-10, the measured/calculated ratios of shear capacity versus the ratios of s/d and the web reinforcement index /Vvy by m e CSA simplified method and general method are plotted. There is a larger scatter for these results than the scatter when shear strength is predicted by shear friction. Notice that the scatter gets significantly larger around /Vvy = 0.3 ~ 0.4. The reason is that some specimens are just under the minimum shear reinforcement requirement by the code and the application of different equations creates inconsistent conservative results. Fig.5-9 also shows that when the ratio of s/d increases the general method tends to be more conservative Vrf 1 o o s d Fig. 5-5, Effect of Stirrup Spacing on the Shear Friction Method

69 2.5r 1 I : I r i r i r V t 15 Vsf D J I I I I I I L p v - f vy(mpa) Fig. 5-6, Effect of Shear Reinforcement on the Shear Friction Method 2.51 V t 15 V v sim 1 -a s d Fig. 5-7, Effect of Stirrup Spacing on the Simplified Method

70 55 2.5r i r v t 1.5- V v sim J L J L J I P v - f vy (MPa) Fig. 5-8, Effect of Shear Reinforcement on the Simplified Method 2.5r V t 1-5 -B Fig. 5-9, Effect of Stirrup Spacing on the General Method

71 56 2.5r v, 14 ODD 1 S Q- O.5- _!_ Pv-fyy (MPa) 1.1 Fig. 5-10, Effect of Shear Reinforcement on the General Method Fig. 5-11, Fig and Fig 5-13 present the analysis results of the effects of concrete strength for 3 beams without shear reinforcement using shear friction method, the CSA simplified method and the general method respectively. The results by shear friction method are slightly more conservative than the other methods. Both the simplified method and the general method show a slightly larger downward trend when f' c increases.

72 2.5r V t 1.5- V f o' f c (MPa) 90 Fig. 5-11, Effect of Concrete Strength on the Shear Friction Method 2.5 V t 1-5 V v sim f c (MPa) 90 Fig. 5-12, Effect of Concrete Strength on the Simplified Method

73 58 2.5r V t 1.5- V * 1 0.5" f c (MPa) Fig. 5-13, Effect of Concrete Strength on the General Method Sarsam and Al-Musawi's Tests, 1992 (40) : A total of 14 beams had been tested and all failed in shear. Fig shows the details of the 180 mm wide x 270 mm deep specimens. All beams have the same 4 mm diameter stirrups with different spacings. The shear span ratios of a/d = 2.5 and 4 had been tested and different concrete strengths had been used. Different fiexural tension reinforcement had been provided to test the effects on prediction of shear capacity. A symmetrical two-point load had been applied at midspan. Table 5-3 and Table 5-4 list the details of beam specimens. The tests were designed to evaluate the effects of concrete strength, shear span ratios, amount of longitudinal reinforcement and stirrup spacing.

74 59 25mm cross bars with positive wtlding to alt As bars ( typ.) 25*100*180 mm plat* (typ) 4 mm stirrups /d s spacing through out î 150mm j 22 $ mm (a/d * A), /SOmoi 4' fm//> J * /5«/ - /575 mm ( a/d :2.5 ) i (min ) î for As 2-10mm bars: 25 mm cover on a: mm stirrups for As bars: 3-20 mm or or 2.25mm +/_ 16 mm 3.25mm 2 5 mm cover on As Section A-A Fig. 5-14, Details of Beam Specimens and Instrumentation (Sarsam and Al-Musawi)

75 60 Table. 5-3, Details of Beam Specimens: Spacing of 4- mm ft* stirrups, Specimen mm aid N/mm* mm p» mm N/mm kn 2 2 AL2-N AL2-H AS2-N AS2-H AS3-N AS3-H BL2-II BS2-H BS3-H BS4-H CL2-H CS2-H CS3-H CS4-H Table. 5-4, Details of Materials: Reinforccmeni diameter, mm Description Application N/mm 2 4 High-yield, colddrawn smooth wire 10 Hot-rolled deformed bar 16 Hot-rol led deformed bar Stirrups 820 Top reinforcement Tension reinforcement Hot-rolled deformed bar 25 Hot-rolled deformed bar Tension reinforcement Tension reinforcement

76 61 Table 5-5 shows that all methods are conservative with high values of C.O.V. from 21.0% to 23.1%. The small scale of the specimens may cause the somewhat greater variation. Comparing the two same size beams CS2-H and CS4-H, they have the same ratio of a/d, the same longitudinal reinforcement, while CS4-H with higher concrete strength and two times more shear reinforcement (half of the stirrup spacing of CS2-H), but the test results show that CS2-H has higher shear capacity than CS4-H. The same thing happened on beam BS2-H and BS4-H. Because of the accuracy of the tests this group of test samples will not be included in the further analysis, even though it is a good example for demonstrating variables of beam shear capacity. Table. 5-5, Test Results and Comparison of Predictions: v t V s f Vsim Vg Specimen (kn) (kn) (kn) (kn) Vt/V rf v t /v sim Vt/V g AL2-N AL2-H AS2-N AS2-H AS3-N AS3-H BL2-H BS2-H BS3-H BS4-H CL2-H CS2-H CS3-H CS4-H m a C.O.V. 21.2% 21.0% 23.1%

77 62 In Fig. 5-15, the ratios of V/V s f against the ratios of shear span a/d are plotted. The figure shows the prediction of shear capacity tends to be more conservative and more scattered when the ratios of shear span a/d = 2.5. For the specimens with a/d = 4.0 the prediction by the shear friction method agrees well with the test results. The prediction results by the CSA simplified method and general method are plotted in Fig and Fig respectively for comparison. The figures also show that the predictions tend to be more conservative and more scattered for both methods when the ratios of shear span a/d = Vrf a Fig. 5-15, Effect of the Ratio of Shear span on the Shear Friction Method d

78 i 1 r i r ft sim a J L J I L J L a 4.2 Fig. 5-16, Effect of the Ratio of Shear span on the Simplified Method d il ft a 4.2 Fig. 5-17, Effect of the Ratio of Shear span on the General Method d

79 64 In Fig. 5-18, the ratios of V/V s f against concrete strength, f' c, are plotted to demonstrate the effect of concrete strength on the shear friction method. It shows no obvious trend in the prediction of shear capacity for beams with different concrete strength. The prediction results by the CSA simplified method and general method are plotted in Fig and Fig respectively for comparison. There is also no obvious trend f c (MPa) Fig. 5-18, Effect of Concrete Strength on the Shear Friction Method

80 l i l i v sim DDO 2 o n f c (MPa) Fig. 5-19, Effect of Concrete Strength on the Simplified Method b 1 o fc (MPa) Fig. 5-20, Effect of Concrete Strength on the General Method

81 66 The ratios of test results to the results predicted by the shear friction method against the ratios of s/d and the web reinforcement index pjvy are plotted in Fig and Fig respectively. The shear friction method demonstrates no obvious trend in the prediction of shear capacity for beams with different stirrup spacing and different amounts of shear reinforcement. The comparison results by the CSA simplified method and general method are plotted in Fig to Fig There is no obvious trend 't V s f OD n a D a o s d Fig. 5-21, Effect of Stirrup Spacing on the Shear Friction Method

82 67 V. t V s f Pv,f vy(mpa) Fig. 5-22, Effect of Shear Reinforcement on the Shear Friction Method 3i i r v c sim s d Fig. 5-23, Effect of Stirrup Spacing on the Simplified Method

83 i i i i i i v sim B a i i i i i i i i P v - f vy (MPa) Fig. 5-24, Effect of Shear Reinforcement on the Simplified Method d s d Fig. 5-25, Effect of Stirrup Spacing on the General Method

84 69 I I I I I I I 1 I Pv' f vy (MPa) Fig. 5-26, Effect of Shear Reinforcement on the General Method In Fig. 5-27, the measured/calculated ratios of shear capacity versus the ratios of beam longitudinal reinforcement p by the shear friction method are plotted. There is a slight trend up when beams are reinforced with more bottom reinforcement. The prediction results by the CSA simplified method and general method are plotted in Fig and Fig respectively for comparison. There is also an up-trend shown in both Fig and Fig when increasing beam longitudinal reinforcement.

85 P (%) Fig. 5-27, Effect of Longitudinal Reinforcement on the Shear Friction Method Vt V c sim I P (%) Fig. 5-28, Effect of Longitudinal Reinforcement on the Simplified Method

86 71 (). I I 1 I I I I P (%) Fig. 5-29, Effect of Longitudinal Reinforcement on the General Method Summary of Tests from Literature: Test data from a total of 113 beams with stirrups and 105 beams without stirrups are selected from literature. All selected beams were simply supported rectangular beams subjected to a symmetrical single or two-point load. The selected beams had different concrete strengths from about 20 MPa up to more than 100 MPa. The selected beams also had different shear span ratios, different amounts of longitudinal reinforcement, different shear reinforcement and different stirrup spacing. Table 5-6 gives the details of specimens with stirrups and Table 5-7 gives the details of specimens without stirrups.

87 Table 5-6, Details of Specimens with Stirrups h d a fc P fy f vy S NAME SPECIMEN (mm) (mm) (mm) (mm) (mm) (mm) (MPa) (%) (MPa) (mm 2 ) (MPa) (mm) (kn) Rodriguez E2A et al E2A E2A C2A C2A E3H E3H C3H C3H Debaiky & Al Eliniema Bl CI Dl D F F Mphonde B B B B B B B B B B B B

88 Table 5-6, Details of Specimens with Stirrups (cont.) h b w d d ev a a c f'c P fy fvy S NAME SPECIMEN (mm) (mm) (mm) (mm) (mm) (mm) (MPa) (%) (MPa) (mm 2 ) (MPa) (mm) (kn) Elzanaty G et al G G Johnson No.l & Ramirez No No No No No Roller & Russell Bresler & A-l Scordelis A B-l B C-l C Xie NNW NHW et al NHW-3a NHW-3b NHW