U N I V E R S I T Y OF C A L G A R Y Reinforced Concrete Beam Design for Shear by Hongge (Gordon) Wang A THESIS S U B M I T T E D TO T H E F A C U L T Y OF G R A D U A T E STUDIES IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF E N G I N E E R I N G D E P A R T M E N T OF CIVIL E N G I N E E R I N G CALGARY, ALBERTA N O V E M B E R , 2002 © Hongge (Gordon) Wang 2002 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: E-mail: uarc@ucalgary.ca Telephone: (403) 220-7271 Website: http://archives.ucalgary.ca ABSTRACT The two methods for design of shear adopted by the present C S A 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 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 T A B L E OF C O N T E N T S Cover Page i Approval page ii Abstract iii Acknowledgements iv Table of Contents v List of Tables viii List of Figures ix Notation xiii CHAPTER ONE: INTRODUCTION 1 1.1 General 1 1.2 Code Review 2 1.3 Scope of Study 3 1.4 Thesis Organization 3 C H A P T E R T W O : B A S I C S H E A R THEORIES 5 2.1 Homogeneous Beam 5 2.2 Beam Cracking Modes 9 2.3 Shear Transfer Mechanisms 10 2.4 Shear Failure Modes 12 2.4.1 Beams without Shear Reinforcement 12 2.4.2 Beams with Shear Reinforcement 16 2.5 Factors Affecting the Shear Strength 16 2.5.1 Tensile Strength of Concrete 16 2.5.2 Longitudinal Reinforcement 17 2.5.3 Shear Span-to-depth Ratio, a/d 17 2.5.4 Size of Beams 19 v 2.5.5 Axial Forces 20 2.5.6 Web Reinforcement 20 C H A P T E R T H R E E : S H E A R DESIGN - C S A S T A N D A R D A23.3-94 22 3.1 General 22 3.2 Simplified Method 22 3.2.1 Shear Supported by Concrete, V 3.2.2 Shear Supported by Stirrups, V 23 c 25 s 3.3 General Method 25 3.3.1 Shear Supported by Concrete, V 3.3.2 Shear Supported by Stirrups, V 26 c g 26 s g C H A P T E R F O U R : S H E A R DESIGN - S H E A R FRICTION M E T H O D 27 4.1 General 27 4.2 General Equations for Beams Based on Shear Friction 27 4.2.1 Shear Friction Strength 28 4.2.2 Basic Shear Design Equations Based on Work by Loov 31 4.2.3 Approximate Shear Capacity of Concrete 36 4.2.4 Approximate Shear Capacity of Stirrups 40 4.2.5 Approximate Shear Design Equations for Beams with T>T 40 opl 4.2.6 Critical Shear Failure Angle 41 4.2.7 Beams with Longitudinal Reinforcement T<T , op C H A P T E R F I V E : E X P E R I M E N T A L STUDIES A N D C O M P A R I S O N 44 45 5.1 Application of Shear Friction Method 45 5.2 Test Results in Literature 47 5.2.1 Yoon, Cook and Mitchell's Tests, 1996 48 5.2.2 Saram and Al-Musawi's Tests, 1992 58 vi 5.2.3 Summary of Tests from Literature 71 5.2.3.1 Beams with Shear Reinforcement 79 5.2.3.2 Beams without Shear Reinforcement 93 C H A P T E R SIX: P R O P O S E D C O D E C L A U S E S F O R S H E A R D E S I G N 106 6.1 Proposed Code Clauses for Shear Design 106 6.1.1 Required Shear Resistance 106 6.1.2 Factored Shear Resistance 106 6.1.3 Determination of V c s f 106 6.1.4 Determination of V s s f 107 6.1.5 Determination of 0 107 6.1.6 Determination of \|/ 108 6.1.7 Limiting Shear Failure Angle 108 6.2 Design Examples 108 C H A P T E R S E V E N : DISCUSSION A N D C O N C L U S I O N 7.1 Conclusions and Recommendations 7.2 Future Research BIBLIOGRAPHY 111 Ill 112 113 vii LIST OF T A B L E S 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 LIST OF FIGURES FIGURE 2.1 Internal Forces in Beam 5 2.2 Distribution of Flexural Shear Stresses 6 2.3 Principal Stresses 7 2.4a Stress Trajectories 8 2.4b Potential Crack Pattern 8 2.5 A Cracked Beam without Shear Reinforcement (MacGregor, 2000) 9 2.6 A Cracked Beam with Shear Reinforcement (Peng, 1999) 10 2.7 Internal Forces in a Cracked Beam 11 2.8 Effect of a/d on Shear for Beams Without Shear Reinforcement (MacGregor, 2000) 14 2.9 Shear Failure Modes (Pillai, 1983) 15 2.10 Shear Strength vs. Longitudinal Reinforcement (MacGregor, 2000) 17 2.11 Shear Strength vs. a/d (Kani, 1979) 18 2.12 Influence of Member Size on Shear Strength (CSA A23.3-94) 19 2.13 Effect of Axial Loads in Inclined Cracking Shear (MacGregor, 2000) 20 2.14 Distribution of Internal Shears of Beam with Shear Reinforcement (MacGregor, 2000) 21 3.1 Comparison of Shear Design Methods and Test Results ( C S A A23.3-94) 24 4.1 Shear Friction Concept (CSA A23.3-94) 28 4.2 Reinforcement Inclined to Potential Failure Cracks ( C S A A23.3-94) 29 4.3 Push-off Test Results (Loov, 1998) 30 4.4 Free Body Diagram of Beam (Loov, 1998) 31 4.5 34 Shear Strength vs. Crack Angle (Loov, 1998) ix 4.6 Three-dimensional surface of shear strength along all possible failure planes for beam 544 (Loov, 1998) 34 4.7 Possible Critical Shear Failure Planes (Loov, 1999) 35 4.8 A Tested Beam with Critical Shear Cracks (Peng, 1999) 35 4.9 37 Shear Strength vs. cotO 4.10 Shear Strength vs. cot9 by Eq. 4-15 and Eq. 4-19 38 4.11 Shear Strength vs. Longitudinal Reinforcement of Beam 39 4.12 The Shear Contributions of Concrete and Discrete Stirrups (Loov, 1998) 42 5.1 Details of Beam Specimens and instrumentation (Yoon, 1996) 48 5.2 Effect of Concrete Strength on the Shear Friction Method (Yoon) 51 5.3 Effect of Concrete Strength on the Simplified Method (Yoon) 52 5.4 Effect of Concrete Strength on the General Method (Yoon) 52 5.5 Effect of Stirrup Spacing on the Shear Friction Method (Yoon) 53 5.6 Effect of Shear Reinforcement on the Shear Friction Method (Yoon) 54 5.7 Effect of Stirrup Spacing on the Simplified Method (Yoon) 54 5.8 55 Effect of Shear Reinforcement on the Simplified Method (Yoon) 5.9 Effect of Stirrup Spacing on the General Method (Yoon) 55 5.10 Effect of Shear Reinforcement on the General Method (Yoon) 56 5.11 Effect of Concrete Strength on the Shear Friction Method (Yoon) 57 5.12 Effect of Concrete Strength on the Simplified Method (Yoon) 57 5.13 Effect of Concrete Strength on the General Method (Yoon) 58 5.14 Details of Beam Specimens and Instrumentation (Sarsam,1992) 59 5.15 Effect of the Ratio of Shear span on the Shear Friction Method (Sarsam) 62 5.16 Effect of the Ratio of Shear span on the Simplified Method (Sarsam) 63 5.17 Effect of the Ratio of Shear span on the General Method (Sarsam) 63 5.18 Effect of Concrete Strength on the Shear Friction Method (Sarsam) 64 5.19 Effect of Concrete Strength on the Simplified Method (Sarsam) 65 5.20 Effect of Concrete Strength on the General Method (Sarsam) 65 x 5.21 Effect of Stirrup Spacing on the Shear Friction Method (Sarsam) 66 5.22 Effect of Shear Reinforcement on the Shear Friction Method (Sarsam) 67 5.23 Effect of Stirrup Spacing on the Simplified Method (Sarsam) 67 5.24 Effect of Shear Reinforcement on the Simplified Method (Sarsam) 68 5.25 Effect of Stirrup Spacing on the General Method (Sarsam) 68 5.26 Effect of Shear Reinforcement on the General Method (Sarsam) 69 5.27 Effect of Longitudinal Reinforcement on the Shear Friction Method (Sarsam) 70 5.28 Effect of Longitudinal Reinforcement on the Simplified Method (Sarsam) 70 5.29 Effect of Longitudinal Reinforcement on the General Method (Sarsam) 71 5.30 Predicted Results by the Shear Friction Method (with stirrups) 80 5.31 Predicted Results by the Simplified Method (with stirrups) 81 5.32 Predicted Results by the General Method (with stirrups) 82 5.33 Effect of the Ratio of Shear Span on the Shear Friction Method (with stirrups) 84 5.34 Effect of the Ratio of Shear Span on the Simplified Method (with stirrups) 85 5.35 Effect of the Ratio of Shear Span on the General Method (with stirrups) 85 5.36 Effect of Concrete Strength on the Shear Friction Method (with stirrups) 86 5.37 Effect of Concrete Strength on the Simplified Method (with stirrups) 86 5.38 Effect of Concrete Strength on the General Method (with stirrups) 87 5.39 Effect of Stirrup Spacing on the Shear Friction Method (with stirrups) 87 5.40 Effect of Stirrup Spacing on the Simplified Method (with stirrups) 88 5.41 Effect of Stirrup Spacing on the General Method (with stirrups) 88 5.42 Effect of Shear Reinforcement on the Shear Friction Method (with stirrups) 89 5.43 Effect of Shear Reinforcement on the Simplified Method (with stirrups) 89 5.44 Effect of Shear Reinforcement on the General Method (with stirrups) 90 5.45 Effect of Longitudinal Reinforcement on the Shear Friction Method (with stirrups) 90 xi 5.46 Effect of Longitudinal Reinforcement on the Simplified Method (with stirrups) 91 5.47 Effect of Longitudinal Reinforcement on the General Method (with stirrups) 91 5.48 Effect of Beam Depth on the Shear Friction Method (with stirrups) 92 5.49 Effect of Beam Depth on the Simplified Method (with stirrups) 92 5.50 Effect of Beam Depth on the General Method (with stirrups) 93 5.51 Predicted Results by the Shear Friction Method (without stirrups) 95 5.52 Predicted Results by the Simplified Method (without stirrups) 96 5.53 Predicted Results by the General Method (without stirrups) 97 5.54 Effect of the Ratio of Shear Span on the Shear Friction Method (without stirrups) 99 5.55 Effect of the Ratio of Shear Span on the Simplified Method (without stirrups) 99 5.56 Effect of the Ratio of Shear Span on the General Method (without stirrups) 100 5.57 Effect of Concrete Strength on the Shear Friction Method (without stirrups) 100 5.58 Effect of Concrete Strength on the Simplified Method (without stirrups) 101 5.59 Effect of Concrete Strength on the General Method (without stirrups) 101 5.60 Effect of Longitudinal Reinforcement Ratio on the Shear Friction Method (without stirrups) 102 5.61 Effect of Longitudinal Reinforcement Ratio on the Simplified Method (without stirrups) 102 5.62 Effect of Longitudinal Reinforcement Ratio on the General Method (without stirrups) 103 5.63 Effect of Longitudinal Reinforcement Strength on the General Method (without stirrups) 103 5.64 Effect of Beam Depth on the Shear Friction Method (without stirrups) 104 5.65 Effect of Beam Depth on the Simplified Method (without stirrups) 104 5.66 Effect of Beam Depth on the General Method (without stirrups) 105 xii NOTATION a shear span, distance from centre of support to point load a clear shear span, distance between outer edge of plate for concentrated c load and inner edge of plate at support A area of potential shear failure plane A area of longitudinal reinforcement in tension zone A area of one stirrup s v b width of beam web c coefficient of the cohesion between a potential shear failure plane Cb concrete cover at top of beam c, concrete cover at bottom of beam C concrete strength of beam w C factored concrete strength of beam C. O. V. coefficient of variation r d distance from the extreme compression fibre to the centroid of the longitudinal tension reinforcement d diameter of a reinforcing bar d distance measured perpendicular to the neutral axis between the resultants b v of the tensile and compressive forces due to flexure d effective length of stirrup in the shear friction method fry specified yield strength of the stirrups f specified yield strength of the longitudinal reinforcement or stirrups ev y f' specified compressive concrete strength h overall height of member k factor for relating shear strength and normal stress determined from c experiments n number of stirrups crossed by a potential shear failure plane R normal force acting on potential shear failure plane xiii 5 spacing of stirrups S shear force on potential shear failure plane T longitudinal reinforcement strength of beam T opi force in longitudinal reinforcement for peak shear strength T r factored longitudinal reinforcement strength of beam T v tension force in a stirrup T vr factored tension resistance in a stirrup v average shear stress on potential shear failure plane according to Loov's equations V factored shear resistance attributed to the concrete V factored shear resistance attributed to the concrete for the C S A general c cg method V f CS factored shear resistance attributed to the concrete for the shear friction method V dowel force in the longitudinal reinforcement Vf factored shear force at section d V shear resistance of beam using C S A A23.3-94 general method V factored transverse component of prestress of beam V factored shear resistance of beam V rg factored shear resistance of beam using C S A A23.3-94 general method V s factored shear resistance provided by the shear reinforcement Vf factored shear resistance for the shear friction method V factored shear resistance provided by the stirrups for the C S A general g p r s sg method V sim shear resistance of beam using C S A A23.3-94 simplified method V s¡ shear resistance provided by one stirrup Vf factored shear resistance attributed to the reinforcement for the shear ss friction method xiv V ultimate shear resistance of beam measured from test V shear resistance of concrete on a 45° plane a 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 t 4S using C S A general method j3 calibration factor for shear friction method 6 angle between longitudinal axis and potential shear failure plane v 9 minimum shear failure angle for the shear friction method X factor to account for low density concrete \i coefficient of friction p longitudinal tension reinforcement ratio p transverse reinforcement ratio o average normal stress on potential shear failure plane min v <j) resistance factor for concrete ¢¡. resistance factor for reinforcement y/ factor that depends on the ratio of longitudinal reinforcement strength to c optimum tension for the shear friction method xv 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 ( 1 7 ) ( 1 8 ) ( I 9 ) and many others O W i X W M W W W S ) . 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. 2 1.2 Code Review Prior to its 1984 revision, C S A Standard A23.3 recommended a method for shear and torsion design based on the traditional method adopted by the A C I code. The procedure is called the "V + V" approach. The term V is referred to as the "shear carried c c by the concrete", while the term V is referred as the "shear carried by the stirrups". A23.3 s assumes that V is equal to the shear strength of a beam without stirrups and further c simplifies V to equal the shear at inclined cracking. V relies on the tensile strength of the c s transverse reinforcement. The stirrups and the inclined compressive struts are assumed to act as members of a 45-degree truss and the term V is calculated based on this model. s The 1984 revision of the Canadian Standard, C A N 3 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 A C I and previous Canadian codes. In the simplified method, the transverse reinforcement is designed for the combined effect of shear and axial load i f 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. C S A Standard A23.3-94 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- 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 C S A Standard A23.3-94 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 C S A Standard A23.3-94, 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 C S A Standard A23.3-94 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. 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. 5 CHAPTER 2 B A S I C S H E A R 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 , at a point in the section distant y from the neutral x axis are given by (2-2) 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. 2-2. 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 , acting on orthogonal planes inclined at an 2 angle a, where ÍZ7 i f u ~ 2 f x ± Í{2 f (2-4) x and (2-5) tan(2«) = — Fig. 2-2, Distribution of Fiexural Shear Stresses 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 and v, which x 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. Fig. 2-4a, Stress Trajectories Fig. 2-4b, Potential Crack Pattern 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 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: 11 (1) The shear strength, V , of the uncracked concrete; cz (2) The vertical component, V^, of the aggregate interlock shear, V ; a (3) The dowel force, V , in the longitudinal reinforcement; d (4) The shear, V , carried by the shear reinforcement. s Fig. 2-7, Internal Forces in a Cracked Beam The aggregate interlock, V , is a tangential force transmitted along the plane of the a 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 may be very significant. a 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. 12 Each of the components of this process except V has a brittle load-deflection s response. So it is difficult to quantify the contributions of V , V¿¡, and V . In design, cz ay these are lumped together as V , referred to as "the shear carried by the concrete". Thus c the nominal shear strength, V , is assumed to be n V =V +V n c (2-6) s In North American design practice, V is traditionally taken equal to the failure c capacity of a beam without stirrups. 2.4 Shear Failure Modes 2.4.1 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 is d 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 , causing the formation of the first cr 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. V v a a (a) Beam. Deep * H Slender Very.Shorty short c 1 re ^¾¾^ c Failure 03 O E o , y / A / e r y slender " ' V 1 / / / / ^ ^ C ^ ^ ^ s ^ ^ ^ ^ Flexural capacity Inclined cracking and failure ^*». 2 • 1.0 Inclined cracking i 2.5 6.5 a/d (b) Moments at cracking and failure. <T3 C>t Flexural capacity JO CO nclined cracking and failure 1.0 2.5 6.5 a/d (c) Shear at cracking and failure. Fig. 2-8, Effect of a/d on Shear for Beams Without Shear Reinforcement (Ref. 27) 15 i (a) Shear-tension failure ( ) c Diagonal tension failure l«a/d<2 5 Shear-compression failure (d) 2.5 < a / d < * * 6 T ( e ) Web-crushing failure Fig. 2-9, Shear Failure Modes (Ref. 36) 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). 2.4.2 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 can increase no s more. Any further increase in applied shear force leads to increases in V , V , and V^. cz d With progressively widening crack width (which is no longer restrained because of yielding of the shear reinforcement), begins to decrease forcing V and V to increase cz d 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 2.5.1 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. 17 2.5.2 Longitudinal Reinforcement Fig. 2-10 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. 2.5.3 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. 0.005 0.010 0.015 0.020 0.025 0.030 0.035 Fig. 2-10, Shear Strength vs. Longitudinal Reinforcement (Ref. 27) 0.040 Fig. 2-11, Shear Strength vs. a/d (Ref. 14) 19 2.5.4 Size of Beam As the overall depth of a beam increases, the shear stress at inclined cracking tends to decrease for a given f' , p, and a/d. As the depth of the beam increases, the c 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) 20 2.5.5 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 - 1 7 a / (ACI Eq. 11-4) 1500 1000 • • -/-— ^**" Eq. 6 - 1 7 b (ACI Eq. 11-8) 500 Compression A x i a l stress, NJA g (psi) Fig. 2-13, Effect of Axial Loads in Inclined Cracking Shear (Ref. 27) 2.5.6 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 21 overall shear strength by the direct contribution of V to the shear strength. Secondly, web s 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 cracking Inclined cracking Yield of stirrups Failure Applied shear Fig. 2-14, Distribution of Internal Shears of Beam with Shear Reinforcement (Ref. 27) 22 CHAPTER 3 S H E A R DESIGN - C S A S T A N D A R D A23.3-94 3.1 General The C S A Standard A23.3-94 recommends two alternative methods for shear design. The "Simplified Method" is a short version of the traditional method followed by A C I 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 >V, r (3-1) 23 Where Vf is the factored shear force at a section, and V is the sum of the r contribution attributed to the concrete and transverse reinforcement. V = V + V, r (3-2) c But V is limited to: r V,iV +0.8ty JfJb d e e (3-3) w 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. 3.2.1 Shear Supported by Concrete, V c V =0.2ty Jf¡b d c e (3-4) w This equation can be used only for beams with minimum transverse reinforcement given by Clause 11.2.8.4 i f ^exceeds 0.5 V + <j) V : c p p A =0.06jf ^f v (3-5) c 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 11.3.5.2 shall be used to account for the reduced strength of beams deeper than 300 mm. 24 V, 260 1000+ d (3-6) At Jfìb d>0.lA&4fÌb d c w w Studies have shown that the equations for V above are more appropriate for c 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). 0.30 |« a 0.25 »| |——a »»j //.illi.l f 0.20 Clause 11.5: Strut-and-tie model bdf, - 610mm • f ' = 27.2 MPa c max. agg. = 19 mm 0.15 d = 538 mm Experimental result b= 155 mm 0.10 A, = 2277 m m Clause 11.4: Sectional model r = 3 7 2 MPa y A =0 v 0.05 3 4 a/d Fig. 3-1, Comparison of Shear Design Methods and Test Results (Ref.7) 2 25 3.2.2 Shear Supported by Stirrups, V s <l> Avf d s y (3-7) s Here the transverse reinforcement is assumed to be perpendicular to the longitudinal axis of the member. Additional maximum spacing (Clause 11.2.11) and minimum transverse reinforcement requirement (Clause 11.2.8) 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 is the factored shear resistance contributed by transverse reinforcement. sg But V„ shall not exceed V = 0.25 c Áfcf¿b d w v (3-9) Where d is the distance measured perpendicular to the neutral axis between the v 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. 26 3.3.1 Shear Supported by Concrete, V V =UA&ft cg 4f!b d c w (3-10) v Where p is determined in accordance with Clause 11.4.4. 3.3.2 Shear Supported by Stirrups, V sg S Where 0is given in Clause 11.4.4. Obviously i f 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 shall be computed from sg _ faA f d (cot0 v y v + cota)sina 27 CHAPTER 4 S H E A R D E S I G N - S H E A R FRICTION M E T H O D 4.1 General The Clause 11.1.3 in C S A Standard A23.3-94 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 (19) presented the rudiments of a procedure , 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 (5) Braestrup , 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. 28 Shear displacement t î t î î t î 1111 Compression in concrete = T (i) Shear Tension Causing Crack Opening i Shear stress Tension in reinforcement = T (ii) Free-Body-Diagram Fig. 4-1, Shear Friction Concept (Ref.7) 4.2.1 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 =À&(c+Mff) (4-1) r 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 C S A Standard A23.3-M94 suggests that the factored shear stress resistance of the shear plane shall be computed as: v =A,fc(c+fia)+ûp fcosa/ r v (4-2) 29 Where a is the angle between the shear friction reinforcement and the shear f plane. \ \ \ Fig. 4-2, Reinforcement Inclined to Potential Failure Cracks (Ref.7) C S A Standard A23.3-M94 also gives an alternative equation for shear friction strength, which is based on the work of Loov and P a t n a i k (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. a Mattock (uncracked) • Mattock (cracked) A Walraven (cracked) v/0~fHMPa) Fig. 4-3, Push-off Test Results (Ref.22) 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 , the total force in the stirrups crossing the plane and V, the end reaction. The forces v 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 )cosd n (4-4) S = Tcos0-(V-ZT )sin0 (4-5) v v 32 Where T = Af y and T = AJ^. Here A is the area of longitudinal reinforcement v s and f is its yield strength, while A is the total area of all legs of a stirrup and f y v vy is the stirrup yield strength. Using the relationship from Eq. 4-3, the shear friction stress is v= kjtf (4-6) While R and a = Where A is the area of the inclined failure plane, A= bh w (4-7) sine? Where b is the width of web, h is the total depth, and 6 is the angle between the w 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 2 V = 0.5k C 2 2 0.25 k C + cot 0-cotO (1 + cot 6)-Tcot0 2 + Yjv 4 9 (") Where C - f'Xh (4-10) 33 This equation is similar to that derived by Braestrup ( 5 ) and by Nielsen ( 3 3 ) with plasticity theory. For design, the factored values should be used thus 2 V r 2 =0.5k C, 0.25k'C. • + cot 0-cotO (1 + cot 0)-T cot0 2 r + J]T vr (4-11) n Where (4-12) (4-13) T r=<t>Asfy T vr (4-14) A = <t>s vf,y A l l 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 ( 4 5 ) . 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 , the force that can be developed in the stirrup. v Fig. 4-6 shows a three-dimensional surface plot, which was obtained by analyzing beam tests by Kani ( 1 4 ) . 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. Fig. 4-6, Three-dimensional surface of shear strength along all possible failure planes for beam 544 (Réf. 18) 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 T- Fig. 4-8, A Tested Beam with Critical Shear Cracks (Ref. 35) B - 7 36 4.2.3 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 2 V = 0.5k C 2 2 0.25k C 2 + cot 6 -cot6 (1 + cot (4-15) 0)-Tcot6 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. 4-15 2 dV (l + cot 6) — - = . ' c / 2 dT =-cotO (4-16) 2 4t /0.25k C +cot 0 2 T opt (4-17) 2 = 0.25k C(2 +tan 6) Substitute Eq. 4-17 into Eq. 4-15, the shear strength of beams will be V = 0.25k''Ctond (4-18) c Eq.4-18 gives the shear capacity of beams with longitudinal reinforcement tension capacity f A y s >T 2 opt 2 - 0.25k C(2+tan 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 t> the ^ op longitudinal reinforcement tension capacity, the less the shear capacity. e s s the 37 0.50 0.40 0.30 > 0.20 0.10 0.00 Fig. 4-9, Shear Strength vs. cotO For beams with longitudinal reinforcement tension capacity f A y s less than Jopt, Eq. 4-15 can be substituted approximately by a simple equation as following: 2 V, = 0.25k sin \ n T 2T °t* J Ctand (4-19) 38 Fig. 4-10, Shear Strength vs. cotO by Eq. 4-15 and Eq. 4-19 The curves from Eq. 4-15 and Eq. 4-19 have been plotted on Fig. 4-10 for comparison. The graph shows that Eq. 4-19 is a useful approximation for the shear capacity of concrete. For factored design, we should use: 2 V cr = 0.25 k C tane r 2 V„ = 0.25k sin n T C tan6 r 2T V °p> When T>Topt (4-20) When T<Tapt (4-21) J Where (4-22) 39 T = <t> AJ s T opt (4-23) y 2 = 0.25k C (2 r 2 + tan 9) (4-24) Fig. 4-11 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. 4-18 and Eq. 4-19. It shows that the variation of the beam shear strength of concrete increases as the beam longitudinal reinforcement increases. When the beam longitudinal 2 reinforcement reaches f A y s 2 = 0.25k C(2 +tan 9), the beam shear strength of concrete reaches its peak value and will not increase even though the beam longitudinal reinforcement increases. A f (kN) s y Fig. 4-11, Shear Strength vs. Longitudinal Reinforcement of Beam 40 4.2.4 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. M a r t i ( 2 8 ) 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 y sr - V r 'd cote } ev (4-26) si Where (4-27) vy Equation 4-27 is one of the most significant discoveries by M a r t i ( 2 8 ) and Loov ( 1 8 ) in shear design, because this corrects a basic mistake that has been used for years in shear design. 4.2.5 Approximate Shear Design Equations for Beams with T>T opt Using the " V + V " approach, the approximate shear strength along a plane at an c s angle 0 to the beam axis is 41 V =V tanû r 4S 'décote + V sl v s -1 (4-28) j Where 2 V = 0.25k C 4} (4-29) r Further, Eq. 4-31 can be written as: V =WJ 4TXh 45 (4-30) v Where 2 J3 = 0.25k 4/: (4-31) V The coefficients k and fi are calibration factor that can be adjusted to match the v 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 =V tanG r 45 (4-32) 4.2.6 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. 4-28 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. 4-28 will form a lower bound of the shear capacity, when Eq. 4-28 is considered to be a continuous function of 0 (see Fig. 412). 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. 4-28. ñ —V d0 r V =—½ cos 0 2 V d " =0 sin 0 s ev IK, tane =i^L^- v ] a (4-33) 2 s Substituted Eq. 4-34 into Eq. 4-28, (4-34) 43 +v V. = V45 \v -i sl 45 y45 (4-35) s So d„ V =2AV V ^--V r 45 sl s¡ (4-36) Eq. 4-36 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. 4-36 we can solve directly to obtain the maximum stirrup spacing. s< (4-37) (Vf+K,) 2 Eq. 4-37 can be used for design of stirrup spacing, while Eq. 4-36 is used to calculate the shear capacity of a beam with known stirrup spacing. Eq. 4-36 and Eq. 4-37 do not apply in cases where the shear failure angle is not determined by Eq. 4-34. 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> (4-38) a.. The strength along this steeper plane can be obtained directly using Eq. 4-28. However, Eq. 4-36 and Eq. 4-37 are conservative i f the shear failure angle becomes steeper under the limitation of beam shear span. 44 4.2.7 Beams with Longitudinal Reinforcement T <T.opt For beams without stirrups, the shear capacity can be derived from Equation (421) as following: K =yV tanO When 45 T<Topt (4-39) Where TV T y/ - sin (4-40) 2 T Accordingly, Eq. 4-36 and Eq. 4-37 need to be modified as follows: V; = 2 \ V V -^-V s ¥ V s< 45 sl (v +v r f sl (4-41) (4-42) sl It is worthy to notice that Eq. 4-41 and Eq. 4-42 may generate conservative results for beams with short shear span. 45 CHAPTER 5 E X P E R I M E N T A L STUDIES A N D C O M P A R I S O N S 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 Using the "V + V" approach as discussed in Chapter 4, the total shear capacity of c a beam is: =V V Y y sf csf T (5-1) +V ssf y The shear capacity of concrete, V ^ can be calculated from cs (5-2) Vcs =¥V tane f 45 Where (5-3) =i w if/ = sin tanO = n T 2T o When T<Topt (5-5) (5-7) y 45 (5-4) s *,AJ T =V (2 T>T,opt (5-6) V 4S T = When 2 + tan 0) (5-8) 46 (5-9) IT The value of V / shall be computed from ss d„., cot 6 Y ssf y si -I (5-10) Eq. 5-2 is derived from the equation in Chapter 4 with some modifications. The value of P from Eq.4-37 is: v P =0.25k 4fl 2 (5-11) v It has been found that k becomes smaller as the concrete strength increases ( 2 2 ) ( 3 7 ) . The equation found from a least-squares fit of tests is: k=2.0(/:)- 0 (5-12) 4 Substituted Eq. 5-12 into Eq. 5-11 and get: / A =0.36 \0.30 '30^ \f'c (5-13) J To consider the effects of beam depth as discussed in Chapter 2, Eq. 5-13 needs to be modified. According to the researches by Tozser and Loov ( 2 5 ) ( 2 6 ) , the shear strength of 025 beams decreases when the depth of beams increases in proportion to h~ . Finally, the equation for calculating fi is presented by Eq. 5-14. v 47 0.30 /3 =0.36 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 ( 1 4 ) ( 4 5 ) , 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 2 i f - ^ / ^ 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. A l l 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 C S A A23.3-94 clauses 11.2.11 for the simplified method and the general method was ignored during the analysis. 48 5.2.1 Yoon, Cook and Mitchell's Tests, 1996 ( 4 4 ) : Yoon, Cook and Mitchell investigated six full-scale beam specimens ( 4 4 ) . 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 = 2.67. The fiexural tension reinforcement for all of the specimens consisted of 10-No.30 bars in two layers, giving a reinforcement ratio of p = 0.028. A symmetrical single point load had been applied at midspan. Table 5-1 lists the details of the beam specimens. 75 11 |350| 1 2000 mm dear - 150 1—» 1/¾ 2150 2150^ 5000 y 1 |350i ELEVATION VIEW 2-No.lO Strain 9«9e« on Stirrup rwrtforoefnent vanea fWNrafONIMflt r"*""""4 J a/2 LVOT» on concreto 10- No 30 cover MOmm I SECTION A-À 650 J6JL INSTRUMENTATION Fig. 5-1, Details of Beam Specimens and instrumentation (Yoon, Cook and Mitchell) 49 Table. 5-1, Details of Beam Specimens (, Shear reinforcement / / . Stirrup size and spacing, MPa mm Specimen 36 Comments 8.0 mm diameter at 325 9.5 mm diameter at 465 9.5 mm diameter at 325 0.00 0.35 0.35 0.50 No stirrups Min /4,, s - d/2 Min A,, s = 0.7d > Min A,, s = d/2 8.0 mm diameter at 325 0.00 0.35 M-Series: Ml-S Ml-N 67 M2-S M2-N 9.5 mm diameter at 325 9.5 mm diameter at 230 No stirrups AC1 83, ACI 89,* CSA 84 Min A^ s = d/2 0.50 CSA 94 min /1,, s = d/2 0.70 ACI 89t mm A,, s < d/2 8.0 mm diameter at 325 0.00 0.35 H-Series: Hl-S Hl-N H2-S H2-N 87 w MPa N-Se ríes: Nl-S Nl-N N2-S N2-N b s* 9.5 mm diameter at 270 9.5 mm diameter at 160 No stirrups ACI 83, ACI 89,* CSA 84 Min Ay, s = d/2 0.60 CSA 94 min A„ s < d/2 1.00 ACI 89t min A* s < d/2 •Lower amount of minimum *4 provided when *jf ií a/69 MPa in design. V tUpper amount of minimum A provided when Jf/ v c > a/69 MPa in design. 2 Noterj^. for all stirrups is 430 MPa; area of 8.0-mm-diamcter bar = 50 mm ; area of 2 9.5-mm-diameter bar = 7! mm . 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 50 specimens without shear reinforcement ( N l - S , M l - S and H l - 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, Vf Vsim Vg Specimen (kN) (kN) (kN) (kN) Vt/V,, v /v Nl-S 249 208 232 204 1.199 Nl-N N2-S N2-N Ml-S Ml-N M2-S M2-N Hl-S Hl-N H2-S H2-N 457 363 483 296 405 552 689 376 354 436 270 411 468 567 302 432 534 712 318 318 418 316 403 525 576 360 209 208 404 278 269 455 504 1.216 1.025 1.108 1.096 0.985 1.180 1.214 1.075 1.436 316 1.082 447 606 708 300 526 625 1.117 1.119 1.012 m a 327 483 598 721 S C.O.V. t sim 1.143 1.156 0.937 1.006 1.051 1.196 0.908 1.082 0.986 Vt/V g 1.223 2.188 1.741 1.194 1.066 1.508 1.213 1.367 1.034 1.018 1.612 1.136 1.154 1.11 1.08 1.37 0.075 6.8% 0.135 12.5% 0.325 23.7% 51 The analyses of the 9 beams with shear reinforcement are illustrated from Fig. 5-2 to 5-10. In Fig. 5-2, the ratios of test results to the results predicted by the shear friction method against concrete strength, f' , are plotted to demonstrate the effect of concrete c 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 C S A simplified method and general method respectively. A downward trend exists for both of methods. 2.5 2 V t Vrf ••• 1.5 • • • 1 • • -tr 0.5 30 40 50 60 70 80 f c (MPa) Fig. 5-2, Effect of Concrete Strength on the Shear Friction Method 90 2.5r 1.5- • Vsim ••• 1 • V t • -B- 0.5 0 30 40 50 60 70 80 90 f c (MPa) Fig. 5-3, Effect of Concrete Strength on the Simplified Method 2.5r V t ••• -s• • 1-5 1 0.5 0 30 40 50 60 f c 70 80 (MPa) Fig. 5-4, Effect of Concrete Strength on the General Method 90 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 / V v y by m e C S A 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 / V v y = 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. 2.5 2 • Vrf • •• o 1 • • o 0.4 0.5 • 0.5 0 0.2 0.3 0.6 s d Fig. 5-5, Effect of Stirrup Spacing on the Shear Friction Method 0.7 1 2.5r V t : I r r i r i 15 • • • Vsf • •D I • 1 0.5 0 0.2 J I I I I I I L 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 f p - vy(MPa) v Fig. 5-6, Effect of Shear Reinforcement on the Shear Friction Method 2.51 V t 15 • • Vs i•m ••• 1 v -a- 0.5 0 0.2 0.3 0.4 0.5 0.6 s d Fig. 5-7, Effect of Stirrup Spacing on the Simplified Method 0.7 55 2.5r v r i 1.5- t Vs i•m • • v ••• 1 0.5 0 0.2 J L 0.3 0.4 0.5 0.6 J L 0.7 0.8 J 0.9 I 1 f P - v y (MPa) v Fig. 5-8, Effect of Shear Reinforcement on the Simplified Method 2.5r V t ••• • • 1-5 -B- 1 0.5 0.2 0.3 0.4 0.5 0.6 Fig. 5-9, Effect of Stirrup Spacing on the General Method 0.7 1.1 56 2.5r • • • —Q- v, 14 S ODD 1 • O.5- 0 _!_ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Pv-fyy (MPa) Fig. 5-10, Effect of Shear Reinforcement on the General Method Fig. 5-11, Fig. 5-12 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 C S A 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. 2.5r V 1.5- t V¡f ••• 1 0.5 o' 1 30 1 40 1 50 1 60 f c (MPa) 1 70 80 90 Fig. 5-11, Effect of Concrete Strength on the Shear Friction Method 2.5 V t 1-5 Vs i•m ••• 1 v • 0.5 0 30 40 50 60 f c 70 80 (MPa) Fig. 5-12, Effect of Concrete Strength on the Simplified Method 90 58 2.5r V t • •*• V 1.5• 1 1 • 0.5" 30 40 50 60 f c 70 80 90 (MPa) Fig. 5-13, Effect of Concrete Strength on the General Method 5.2.2 Sarsam and Al-Musawi's Tests, 1 9 9 2 (40) : A total of 14 beams had been tested and all failed in shear. Fig. 5-14 shows the details of the 180 mm wide x 270 mm deep specimens. A l l 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. 59 25mm wtlding cross bars with to alt As bars stirrups /d s positive ( typ.) 25*100*180 mm plat* (typ) 4 mm spacing through out î j 22 $4 -2280 150mm mm (a/d 4'fm//> J * / 5 « / - /575 mm for As ( a/d for bars: As 3-20 or î on mm 2.25mm 2 5 mm +/_ 16 mm or ) i (min ) a: 2 - 10mm stirrups :2.5 25 mm cover bars: ¿mm , /SOmoi * A) on 3.25mm Section cover As A-A Fig. 5-14, Details of Beam Specimens and Instrumentation (Sarsam and Al-Musawi) 60 Table. 5-3, Details of Beam Specimens: Specimen AL2-N AL2-H AS2-N AS2-H AS3-N AS3-H BL2-II mm 235 235 235 232 235 235 233 BS2-H BS3-H BS4-H CL2-H CS2-H CS3-H CS4-H ft* 4 4 2.5 2.5 2.5 2.5 4 N/mm* 40.4 75.3 39.0 75.5 40.2 71.8 75.7 mm 943 943 943 943 943 943 1181 p» 0.0223 0.0223 0.0223 0.0226 0.0223 0.0223 0.0282 233 233 233 233 2.5 2.5 2.5 4 73.9 73.4 80.1 70.1 233 233 233 2.5 2.5 2.5 70.2 74.2 75.7 1181 1181 1181 1470 1470 1470 1470 0.0282 0.0282 0.0282 0.0351 0.0351 0.0351 0.0351 aid 2 Spacing of 4mm stirrups, mm N/mm 150 0.76 150 0.76 150 0.76 150 0.76 100 1.14 100 1.14 150 0.76 150 100 75 150 0.76 1.14 1.53 0.76 kN 114.7 122.6 189.3 201.0 199.1 199.1 138.3 223.5 228.1 206.9 147.2 150 100 75 0.76 1.14 1.53 247.2 247.2 220.7 2 Table. 5-4, Details of Materials: Reinforccmeni diameter, mm Description Application N/mm 4 High-yield, colddrawn smooth wire Stirrups 820 10 Hot-rolled deformed bar Top reinforcement 450 16 Hot-rol led deformed bar Tension reinforcement 525 20 Hot-rolled deformed bar Tension reinforcement 495 25 Hot-rolled deformed bar Tension reinforcement 543 2 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 Vf s Vsim Vg Specimen (kN) (kN) (kN) (kN) Vt/V AL2-N 114.7 97.1 86.1 75.5 1.182 AL2-H AS2-N AS2-H AS3-N AS3-H BL2-H BS2-H 122.6 189.3 201.0 104.6 97.3 106.2 123.5 132.1 104.7 106.4 105.8 85.2 104.5 102.2 120.2 104.9 104.0 92.7 70.5 90.1 90.3 106.0 91.8 86.1 1.172 1.945 1.894 BS3-H BS4-H 228.1 206.9 147.2 247.2 247.2 220.7 132.4 157.6 119.8 139.0 102.1 102.2 120.1 103.7 124.1 CL2-H CS2-H CS3-H CS4-H 199.1 199.1 138.3 223.5 103.7 105.7 132.6 156.6 136.9 88.2 84.8 104.0 122.3 m a C.O.V. rf 1.613 1.508 1.321 2.100 v /v t sim 1.331 1.159 2.222 1.923 1.948 1.656 1.319 2.149 Vt/V g 1.520 1.323 2.685 2.231 2.204 1.878 1.506 2.596 2.340 1.865 1.409 1.905 1.489 1.442 2.420 2.058 1.612 2.199 1.668 1.669 2.914 1.63 1.76 2.04 0.35 21.2% 0.37 0.47 21.0% 23.1% 1.723 1.313 1.419 2.377 1.804 62 In Fig. 5-15, the ratios of V/V f against the ratios of shear span a/d are plotted. s 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 C S A simplified method and general method are plotted in Fig. 5-16 and Fig. 5-17 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 = 2.5. 1 1 1 1 1 1 1 1 • • 1 • • • • • 2 Vrf 2.2 1 • • • 1 1 1 1 1 1 1 1 1 2.4 2.6 2.8 3 3.2 a 3.4 3.6 3.8 4 d Fig. 5-15, Effect of the Ratio of Shear span on the Shear Friction Method 4.2 i 1 i r r ft • a • • sim 2.2 J L 2.4 2.6 2.8 J I L 3 3.2 a 3.4 3.6 J L 3.8 4 4.2 d Fig. 5-16, Effect of the Ratio of Shear span on the Simplified Method 1 • 1 1 1 1 1 1 1 1 • • • il ft • • • 2.2 1 1 1 1 1 1 1 1 1 2.4 2.6 2.8 3 3.2 a 3.4 3.6 3.8 4 d Fig. 5-17, Effect of the Ratio of Shear span on the General Method 4.2 64 In Fig. 5-18, the ratios of V/V f against concrete strength, f' , are plotted to s c 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 C S A simplified method and general method are plotted in Fig. 5-19 and Fig. 5-20 respectively for comparison. There is also no obvious trend. 35 40 45 50 55 60 65 70 75 80 f c (MPa) Fig. 5-18, Effect of Concrete Strength on the Shear Friction Method 85 l i l 1 i 1 1 1 1 • • • • 2 o • • n • v sim DDO 35 • • • • • 1 1 1 1 1 1 1 1 1 40 45 50 55 60 65 70 75 80 f c 85 (MPa) Fig. 5-19, Effect of Concrete Strength on the Simplified Method 1 1 1 1 1 1 1 b • • • • • • 2 • o • 35 1 • • • • 1 1 1 1 1 1 1 1 1 40 45 50 55 60 65 70 75 80 f c (MPa) Fig. 5-20, Effect of Concrete Strength on the General Method 85 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. 5-21 and Fig. 5-22 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 C S A simplified method and general method are plotted in Fig. 5-23 to Fig. 5-26. There is no obvious trend. 1 1 • • s 1 1 1 1 • n a D • • • • 't V f 1 a o • • OD 0 0.3 1 1 1 1 1 1 1 0.3 0.4 0.4 0.5 s 0.5 0.6 0.6 d Fig. 5-21, Effect of Stirrup Spacing on the Shear Friction Method 0.7 67 V.t V f s 1 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 ,f Pv vy(MPa) Fig. 5-22, Effect o f Shear Reinforcement on the Shear Friction Method 3i 1 1 1 i r • v c sim 0.3 0.3 0.4 0.4 0.5 0.5 0.6 s d Fig. 5-23, Effect of Stirrup Spacing on the Simplified Method 0.6 0.7 68 1 i 1 i i i i i • B • • v sim a • • • • • 0.7 i 0.8 i 0.9 i 1 i 1.1 i 1.2 i 1.3 i 1.4 i 1.5 1.6 f P - v y (MPa) v Fig. 5-24, Effect of Shear Reinforcement on the Simplified Method 1 1 1 1 1 d 1 • • 2 • • • • • 0.3 1 1 1 1 1 1 1 0.3 0.4 0.4 0.5 0.5 0.6 0.6 s d Fig. 5-25, Effect of Stirrup Spacing on the General Method 0.7 69 0.7 I I I I 0.8 0.9 1 1.1 I I I 1 I 1.2 1.3 1.4 1.5 1.6 f P v ' v y (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 C S A simplified method and general method are plotted in Fig. 528 and Fig. 5-29 respectively for comparison. There is also an up-trend shown in both Fig. 5-28 and Fig. 5-29 when increasing beam longitudinal reinforcement. 70 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 P (%) Fig. 5-27, Effect of Longitudinal Reinforcement on the Shear Friction Method 1 1 1 1 1 1 1 • • • • • Vt V • • • c sim 0 • • • • 1 1 1 1 1 1 I 2.2 2.4 2.6 2.8 3 3.2 3.4 P (%) Fig. 5-28, Effect of Longitudinal Reinforcement on the Simplified Method 3.6 71 ( ) . I I 1 I I I I 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 P (%) Fig. 5-29, Effect of Longitudinal Reinforcement on the General Method 5.2.3 Summary of Tests from Literature: Test data from a total of 113 beams with stirrups and 105 beams without stirrups are selected from literature. A l l 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 M P a up to more than 100 M P a . 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. Table 5-6, Details of Specimens with Stirrups h NAME SPECIMEN Rodriguez et al E2A1 E2A2 E2A3 C2A1 C2A2 E3H1 E3H2 C3H1 C3H2 Al Bl CI Dl D2 F5 F6 B50-3-3 B50-7-3 B50-11-3 B50-15-3 B100-3-3 B100-7-3 B100-11-3 B100-15-3 Debaiky & Eliniema Mphonde B150-3-3 B150-7-3 B150-11-3 B150-15-3 (mm) 368 368 368 368 368 368 368 368 368 300 300 300 300 300 300 300 337 337 337 337 337 337 337 337 337 337 337 337 d (mm) 152 152 156 154 157 152 152 152 152 120 120 120 120 120 120 120 150 150 150 150 150 150 150 150 150 150 150 150 (mm) 318 321 321 318 321 318 311 318 323 260 260 260 260 260 260 260 298 298 298 298 298 298 298 298 298 298 298 298 a (mm) 274 274 274 274 274 267 267 267 267 234 234 234 234 224 232 232 282 282 282 282 276 276 276 276 276 276 276 276 (mm) 1295 1295 1295 1295 1295 864 864 864 864 975 775 975 975 975 975 975 1067 1067 1067 1067 1067 1067 1067 1067 1067 1067 1067 1067 (mm) 1193 1193 1193 1193 1193 762 762 762 762 900 700 900 900 900 900 900 965 965 965 965 965 965 965 965 965 965 965 965 fc P fy (MPa) 25.5 19.3 20.1 22.6 22.1 24.8 27.5 22.6 22.8 24.5 24.5 28.0 29.8 30.6 20.2 20.5 22.1 39.8 59.7 83.0 27.9 47.1 68.6 81.9 28.7 46.6 69.5 82.7 (%) 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 1.57 2.89 2.89 2.89 2.89 2.89 2.41 1.92 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 (MPa) 320 325 331 318 324 380 377 385 410 408 408 408 408 408 408 408 448 448 448 448 448 448 448 448 448 448 448 448 2 (mm ) 142 142 142 142 142 253 253 253 253 47 47 47 47 100 57 57 16 16 16 16 36 36 36 36 51 51 51 51 fvy S (MPa) 346 347 351 354 349 331 316 318 318 318 318 318 318 318 314 314 303 303 303 303 266 266 266 266 284 284 284 284 (mm) 254 254 254 254 254 152 191 152 191 200 200 200 100 200 200 200 89 89 89 89 89 89 89 89 89 89 89 89 (kN) 130.5 119.7 128.9 99.8 122.8 213.7 189.3 189.3 173.9 72.1 67.5 71.1 81.9 73.5 66.2 61.3 76.1 93.9 97.9 111.2 95.2 120.5 151.7 115.7 139.0 133.4 161.5 149.9 Table 5-6, Details of Specimens with Stirrups (cont.) NAME SPECIMEN Elzanaty et al G4 G5 G6 No.l No.2 No.4 No.5 No.7 No.8 1 2 3 4 5 6 7 9 10 A-l A-2 B-l B-2 C-l C-2 NNW-3 NHW-3 NHW-3a NHW-3b NHW-4 Johnson & Ramirez Roller & Russell Bresler & Scordelis Xie et al h b d d a a f'c (mm) 305 305 305 610 610 610 610 610 610 635 679 718 718 743 870 870 870 870 561 559 556 561 559 559 254 . 254 254 254 254 (mm) 178 178 178 305 305 305 305 305 305 356 356 356 356 356 457 457 457 457 307 305 231 229 155 152 127 127 127 127 127 (mm) 267 267 267 539 539 539 539 539 539 559 559 559 559 559 762 762 762 762 466 464 (mm) 242 242 242 535 535 535 535 535 535 502 520 547 547 572 724 724 724 724 436 434 461 466 464 464 203 198 198 198 198 431 436 434 434 185 185 185 185 185 (mm) 1067 1067 1067 1670 1670 1670 1670 1670 1670 1397 1397 1397 1397 1397 2286 2286 2286 2286 1827 2288 1821 2286 1831 2289 610 594 594 594 792 (mm) 967 967 967 1505 1505 1505 1505 1505 1505 1194 1194 1194 1194 1194 2083 2083 2083 2083 1611 2072 1605 2070 1615 2073 508 492 492 492 690 (MPa) 62.7 40.0 20.7 36.4 36.4 72.3 55.8 51.3 51.3 120.1 120.1 120.1 120.1 120.1 72.4 72.4 125.3 125.3 24.1 24.3 24.8 23.2 29.6 23.8 42.9 103.4 94.7 108.7 104.0 w ev c P (%) 3.3 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 1.65 3.04 4.56 6.08 6.97 1.73 1.88 2.35 2.89 1.8 2.28 2.43 2.43 1.8 3.66 3.2 4.54 4.54 4.54 4.54 fy (MPa) 434 434 434 525 525 525 525 525 525 472 431 431 431 462 464 483 483 464 555 555 555 555 555 555 421 421 421 421 421 2 (mm ) 63 63 63 63 63 63 63 63 63 63 253 396 396 396 143 143 143 143 63 63 63 63 63 63 63 63 63 63 63 fvy S (MPa) 379 379 379 479 479 479 479 479 479 407 448 458 458 458 445 445 445 445 325 325 325 325 325 325 324 324 324 324 324 (mm) 191 191 191 133 267 267 133 267 267 216 165 127 89 64 381 197 197 133 210 210 191 191 210 210 102 99 76 64 99 (kN) 147.2 113.2 77.6 338.5 221.9 315.9 382.7 280.8 258.1 297.3 1097.4 1655.0 1940.0 2234.5 665.4 787.9 749.4 1172.2 233.5 244.7 222.4 200.2 155.7 162.4 87.1 102.4 108.2 122.5 93.7 Table 5-6, Details of Specimens with Stirrups (cont.) NAME SPECIMEN Anderson & Ramirez Wl W2 W3 NIN N2S N2N MIN M2S M2N H1N H2S H2N 1 3 5 7 8 9 10 B-l B-2 B-3 B-4 B-5 B-6 B-7 BM100 BM100D Yoon et al Kriski Peng Podgorniak & Stanik h b w d dey (mm) 406 406 406 750 750 750 750 750 750 750 750 750 400 400 400 400 400 400 400 320 320 320 320 320 320 320 1000 1000 (mm) 406 406 406 375 375 375 375 375 375 375 375 375 360 360 360 360 360 360 360 280 280 280 280 280 280 280 300 300 (mm) 343 343 343 655 655 655 655 655 655 655 655 539 345 345 345 345 345 345 345 274 274 274 274 274 274 274 925 925 (mm) 292 292 292 638 632 632 638 632 632 638 632 632 327 327 327 327 327 327 327 247 247 247 247 225 225 225 870 870 a (mm) 914 914 914 2150 2150 2150 2150 2150 2150 2150 2150 2150 1050 1050 900 1050 900 1050 900 950 950 950 950 950 950 950 2700 2700 a c (mm) 812 812 812 2000 2000 2000 2000 2000 2000 2000 2000 2000 900 900 750 900 750 900 750 848 848 848 848 848 848 848 2548 2548 f'c (MPa) 29.2 32.2 32.3 36.0 36.0 36.0 67.0 67.0 67.0 87.0 87.0 87.0 28.9 28.9 30.1 74.3 77.8 77.0 76.3 31.3 31.8 32.7 33.0 32.4 29.3 32.2 47.0 47.0 P (%) 2.275 2.275 2.275 2.85 2.85 2.85 2.85 2.85 2.85 2.85 2.85 2.85 2.01 2.01 2.01 2.01 2.01 2.01 2.01 2.7 2.7 2.7 2.7 2.7 2.7 2.7 0.76 0.76 fy (MPa) 434 434 434 400 400 400 400 400 400 400 400 400 433 433 433 433 433 433 433 478 478 478 478 478 478 478 550 550 2 (mm ) 285 285 143 100 142 142 100 142 142 100 142 142 51 51 51 51 51 51 51 51 51 51 51 200 200 200 142 142 fvy S v, (MPa) 544 544 544 430 430 430 430 430 430 430 430 430 600 600 600 600 600 600 600 587 587 587 587 456 456 456 508 508 (mm) 178 178 89 325 465 325 325 325 230 325 270 160 150 150 150 150 150 150 150 355 300 250 195 355 300 250 (kN) 458.2 548.9 504.4 457.0 363.0 483.0 405.0 552.0 689.0 483.0 598.0 721.0 249.0 224.5 293.0 304.5 391.0 242.0 390.5 114.0 119.0 121.0 143.0 181.0 191.0 187.0 600 600 342.0 461.0 Table 5-6, Details of Specimens with Stirrups (cont.) NAME SPECIMEN Clark Al-1 Al-2 Al-3 Al-4 Bl-1 Bl-2 Bl-3 Bl-4 Bl-5 Cl-1 Cl-2 Cl-3 Cl-4 Dl-1 Dl-2 Dl-3 YB2000/9 YB2000/6 YB2000/4 DB120M DB140M DB165M DB180M DB0.530M SE100A-M-69 SE100B-M-69 SE50A-M-69 SE50B-M-69 Yoshida Angelakos Collins & Kuchma h b w d dev a a f'c (mm) 457 457 457 457 457 457 457 457 457 457 457 (mm) 203 203 203 203 203 203 203 203 203 203 203 203 203 203 203 203 300 300 300 300 300 300 300 300 295 295 169 169 (mm) 390 390 390 390 390 390 390 390 390 390 390 390 390 390 390 390 1890 1890 1890 925 925 925 925 925 920 920 459 459 (mm) 335 335 335 335 335 335 335 335 335 335 335 335 335 335 335 335 1814 1814 1814 860 860 860 860 860 896 896 447 447 (mm) 914 914 914 914 762 762 762 762 762 610 610 610 610 457 457 457 5400 5400 5400 2700 2700 2700 2700 2700 4600 4600 2500 2500 (mm) 825 825 825 825 673 673 673 673 673 521 521 521 521 368 368 368 5100 5100 5100 2548 2548 2548 2548 2548 4448 4448 2424 2424 (MPa) 24.6 23.6 23.4 24.8 23.4 25.4 23.7 23.3 24.6 25.6 26.3 24.0 29.0 26.2 26.1 24.5 36.0 36.0 36.0 21.0 38.0 65.0 80.0 32.0 71.0 75.0 74.0 74.0 457 457 457 457 457 2000 2000 2000 1000 1000 1000 1000 1000 1000 1000 500 500 c P (%) 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 2.07 2.07 2.07 2.07 1.63 1.63 1.63 0.74 0.74 0.74 1.01 1.01 1.01 1.01 0.50 1.03 1.03 1.03 1.03 fy (MPa) 321 321 321 321 321 321 321 321 321 321 321 321 321 335 335 335 455 455 455 550 550 550 550 550 483 483 475 475 A v 2 (mm ) 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 645 284 129 142 71 71 71 71 200 200 51 51 f vy S (MPa) 331 331 331 331 331 331 331 331 331 331 331 331 331 331 331 331 467 467 467 508 508 508 508 508 522 522 593 593 (mm) 183 183 183 183 191 191 191 191 191 203 203 203 203 152 152 152 2700 1350 590 600 300 300 300 300 440 440 276 276 (kN) 222.5 209.2 222.5 244.7 278.9 256.6 284.8 268.1 241.4 277.7 311.1 245.9 285.9 301.1 356.7 256.6 474.0 551.0 659.0 282.0 277.0 452.0 395.0 263.0 516.0 583.0 139.0 152.0 76 Table 5-7, Details of Specimens without Stirrups h NAME SPECIMEN Kani et al 24 25 26 85 87 94 100 27 28 b w d a a fc v P fy (MPa) 27.9 24.5 27.1 25.5 27.2 25.3 27.2 29.8 29.2 (%) 1.88 1.88 1.88 2.80 2.80 2.80 2.80 1.88 1.88 24.5 25.2 26.1 26.1 27.5 27.4 27.4 27.4 30.3 25.3 25.3 27.2 26.2 1.88 1.88 1.88 1.88 2.80 2.80 2.80 2.80 2.80 2.80 2.80 2.80 2.80 (MPa) 396 396 396 381 366 352 366 396 396 350 350 491 491 343 343 342 364 372 338 335 366 366 155 285 425 555 695 985 1235 1505 29.6 29.6 29.6 29.6 29.6 29.6 31.0 31.0 2.07 2.07 2.07 2.07 2.07 2.07 2.01 2.01 465 465 465 465 465 465 465 465 388.3 259.9 147.1 81.6 60.3 60.8 62.3 65.7 699 699 699 699 699 699 762 762 30.3 31.0 31.0 31.5 21.2 21.6 2.17 2.15 2.22 2.37 1.62 1.63 313 313 313 313 313 313 60.1 66.7 75.6 71.2 56.3 60.1 36.7 25.8 1.89 313 1.89 313 57.8 52.3 30.7 1.89 313 52.0 30.9 1.89 313 51.2 c 29 30 35 36 81 83 84 91 93 95 96 97 99 (mm) 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 305 Leonhardt & Walther 1 2 3 4 5 6 7 8 320 320 320 320 320 320 320 320 Moody et al Al-A A2-A A3-A A4-A Bl-A B2-A 305 305 305 305 305 305 Bl-B B3-B 305 305 (mm) (mm) (mm) 152 271 407 152 271 543 152 271 543 154 274 272 154 269 272 153 273 543 153 270 544 152 271 678 152 271 678 152 271 1221 152 271 1221 155 269 953 153 273 953 153 274 1628 156 271 814 151 271 1085 154 269 1628 155 273 1763 153 275 678 153 275 1085 152 276 815 152 272 679 190 270 270 190 270 400 190 270 540 190 270 670 190 270 810 190 270 1100 190 278 1350 190 278 1620 178 262 800 178 267 800 178 268 800 178 270 800 178 800 267 178 268 800 152 268 914 152 268 914 B5-B 305 152 268 914 762 B7-B 305 152 268 914 762 (mm) 305 441 441 170 170 441 442 577 577 1119 1119 851 851 1526 712 983 1526 1662 577 983 714 578 t (kN) 181.9 104.1 78.1 233.5 239.5 110.5 111.9 51.4 54.3 42.9 46.3 44.9 51.6 51.2 64.9 55.4 51.0 53.8 72.7 56.3 62.5 77.2 77 Table 5-7, Details of Specimens without Stirrups (cont.) h NAME SPECIMEN Van Den Berg A4-1 A4-2 A4-3 A4-4 A4-5 A4-6 A4-7 A4-8 . A4-9 A4-10 A4-12 Dl D2 D3 D4 D5 D6 D7 D8 DIO DU D12 D13 D14 D15 D16 D17 D18 D19 D20 Mphonde & Frantz AO-3-3b AO-3-3c AO-7-3a AO-7-3b AO-11-3a AO-11-3b AO-15-3a AO-15-3b AO-15-3c b w d a a c fc (mm) (mm) (mm) (mm) (mm) ( M P a ) 419 229 359 991 889 43.6 419 229 359 1372 1270 38.9 419 229 359 1448 1346 41.8 419 229 359 1524 1422 38.9 419 229 359 1257 1156 39.6 419 229 359 1524 1422 44.9 419 229 359 1257 1156 50.3 419 229 359 1600 1499 42.8 419 229 359 1676 1575 47.6 419 229 359 1753 1651 35.4 419 229 359 991 889 44.0 419 229 359 1257 1156 49.8 419 229 359 1257 1156 43.0 419 229 359 1257 1156 36.1 419 229 359 1257 1156 35.5 419 229 359 1257 1156 43.0 419 229 359 1257 1156 41.3 419 229 359 1257 1156 32.2 419 229 359 1257 1156 25.5 419 229 359 1257 1156 26.7 419 229 359 1257 1156 19.1 419 229 359 1257 1156 23.3 419 229 359 1257 1156 20.8 419 229 359 1257 1156 23.9 419 229 359 1257 1156 22.3 419 229 359 1257 1156 25.9 419 229 359 1257 1156 22.2 419 229 359 1257 1156 24.4 419 229 359 1257 1156 27.4 419 229 359 24.2 1257 1156 337 152 298 1074 973 20.8 337 152 298 1074 973 27.1 337 152 298 1074 973 37.7 337 152 298 1074 973 41.6 337 152 298 1074 973 74.9 337 152 298 1074 973 74.6 337 152 298 1074 973 81.3 337 152 298 1074 973 93.7 337 152 298 1074 973 91.8 P fy v (%) 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 (MPa) (kN) 310 310 310 310 310 310 310 310 310 310 310 310 310 310 310 310 310 310 310 177.9 133.4 134.3 135.0 133.4 142.3 142.3 124.5 131.2 122.3 177.9 151.2 131.2 129.0 144.6 131.2 140.1 140.1 117.9 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 310 310 310 310 310 310 310 310 310 310 310 126.8 109.0 106.8 99.2 106.8 102.3 111.2 104.5 104.5 115.7 106.8 3.36 2.32 3.36 3.36 3.36 414 414 414 414 414 3.36 3.36 3.36 414 414 414 64.6 66.8 82.2 82.8 89.7 89.4 93.5 100.0 3.36 414 97.9 t Table 5-7, Details of Specimens without Stirrups (cont.) h NAME SPECIMEN Ahmad et al Al A2 A3 A7 A8 A9 Bl B2 B3 B7 B8 B9 Cl C2 C3 C7 C8 C9 Bresler & Scordelis OA-1 OA-2 OA-3 Xie et al Yoon et al K d a a v fc P fy (mm) (mm) 813 711 610 508 549 447 832 731 624 522 562 460 807 705 605 503 545 443 832 731 624 522 562 460 737 635 552 451 497 396 826 724 620 518 558 456 (MPa) (MPa) 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 414 (kN) 60.8 60.8 60.8 60.8 60.8 60.8 67.0 67.0 67.0 67.0 67.0 67.0 64.3 64.3 64.3 64.3 64.3 64.3 (%) 3.93 3.93 3.93 1.77 1.77 1.77 5.04 5.04 5.04 2.25 2.25 2.25 6.64 6.64 6.64 3.26 3.26 3.26 1829 2286 3200 1613 2070 2985 22.5 23.7 37.6 1.81 2.27 2.74 555 555 555 166.8 177.9 189.0 648 648 546 546 39.7 104.2 2.07 2.07 421 421 36.7 45.7 655 655 2150 2150 2000 2000 36.0 67.0 2.80 2.80 400 400 249.0 296.0 655 2150 2000 87.0 2.80 400 327.0 NNN-3 NHN-3 (mm) (mm) (mm) 254 127 203 254 127 203 254 127 203 254 127 208 254 127 208 254 127 208 254 127 202 254 127 202 254 127 202 254 127 208 254 127 208 254 127 208 254 127 184 254 127 184 254 127 184 254 127 207 254 127 207 254 127 207 556 310 461 561 305 466 556 307 462 254 127 216 254 127 216 N1S MIS 750 750 375 375 HIS 750 375 c t 57.8 68.9 68.9 46.7 48.9 80.1 51.2 68.9 100.1 44.6 46.7 80.1 54.3 75.6 68.9 45.4 44.5 45.4 79 5.2.3.1 Beams with Shear reinforcement: Table 5-6 gives the details of specimens with stirrups. Table 5-8 gives the summary of the comparison of predictions by three different methods for beams with stirrups. The predictions by the shear friction method are slightly conservative. The average value of V/V fis s 1.09 and the coefficient of variation (C.O.V.) is 16.6%. The predictions by the simplified method have a larger scatter. The average value of V/V sim is 1.16 and the coefficient of variation (C.O.V.) is 20.4%. The most conservative predictions with the largest scatter are obtained by the general method. The average value of V/V is 1.35 and the coefficient of variation (C.O.V.) is 24.6%. g Table. 5-8, Comparison of Predictions for Beams with Stirrups v /v t vw sf t NAME No. m a C.O.V m Rodriguez 9 1.18 0.098 8.3% Debaiky 7 1.07 0.113 Mphonde 12 1.09 Elzanaty 3 1.11 Johnson v /v t sim e C.O.V m a C.O.V 1.04 0.112 10.7% 1.05 0.160 15.2% 10.6% 1.24 0.143 11.6% 1.39 0.195 14.0% 0.106 9.7% 1.23 0.116 9.4% 1.47 0.184 12.5% 0.223 20.0% 1.19 0.140 11.8% 1.37 0.172 12.6% 6 0.94 0.081 8.6% 1.02 0.076 7.5% 1.16 0.130 11.2% Roller 9 1.13 0.208 18.3% 1.07 0.228 21.4% 1.26 0.314 25.0% Bresler 6 1.12 0.047 4.2% 1.33 0.070 5.2% 1.61 0.079 4.9% Xie 5 0.96 0.035 3.7% 1.08 0.053 4.9% 0.049 4.1% Anderson 3 1.09 0.115 10.5% 1.11 0.074 6.7% 1.20 1.12 0.078 6.9% Yoon 9 1.11 0.081 7.3% 1.12 0.130 11.6% 1.46 0.326 22.4% Kriski 7 1.10 0.189 17.1% 1.20 0.190 15.9% 1.39 0.230 16.5% Peng 7 1.16 0.170 14.7% 1.07 0.058 5.4% 1.17 0.980 8.4% Podgorniak 2 0.99 0.147 14.8% 1.09 0.162 14.8% 1.75 0.260 14.8% Clark 16 1.29 0.152 11.7% 1.49 0.184 12.4% 1.57 0.252 16.0% Yoshida 3 0.86 0.021 2.4% 0.94 0.298 31.7% 1.74 0.814 46.7% Angelakos 5 0.78 0.128 16.3% 0.80 0.180 22.7% 1.09 0.410 37.7% Collins 4 0.90 0.049 5.5% 0.80 0.039 4.9% 0.91 0.045 4.9% Total 113 1.09 16.6% 1.16 0.236 20.4% 1.35 0.331 24.6% 0.182 a 80 Fig. 5-30 to Fig. 5-32 illustrate the results of predictions by the shear friction method, the C S A simplified method and the C S A general method respectively. Comparing the figures it is clear that the shear friction method gives the best prediction of the three methods. 2500 (kN) V r f 1250 •oo a 0 1250 2500 V (kN) t Fig. 5-30, Predicted Results by the Shear Friction Method 81 2500 (kN) • • simi250 • • » D D • • o > 1250 2500 V (kN) t Fig. 5-31, Predicted Results by the Simplified Method 82 2500 (kN) V g 1250 o • • • • 1250 v. Fig. 5-32, Predicted Results by the General Method 2500 (kN) 83 In Fig. 5-33, the ratios of V/V f against the ratios of shear span a/d are plotted. s Generally speaking, the prediction by the shear friction method agrees well with the test results. In Fig. 5-34 and Fig. 5-35, the measured/calculated ratios of shear capacity versus the ratios of shear span a/d by the C S A simplified method and general method are plotted respectively. There is a larger scatter of the results predicted by the general method compared to that predicted by shear friction. In Fig. 5-36, the ratios of V/V j- against concrete strength, f' , are plotted to s c 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 predicted results by the simplified method and the general method against concrete strength, f' , are plotted in Fig. 5-37 and Fig. 5-38 respectively. c 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 f\f y V are plotted in Fig. 5-39 and Fig. 5-42 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. Fig. 5-40 and Fig. 5-43 give the results by the simplified method against the ratios of s/d and the web reinforcement index / V y y - The results by the general method against the ratios of s/d and the web reinforcement index Pyf y are shown V in Fig. 5-41 and Fig. 5-44. It is notable that the lowest ratios occur with small / V v y t 0 a ^ three methods. In Fig. 5-45, Fig. 5-46 and Fig. 5-47, the measured/calculated ratios of shear capacity versus the ratios of beam longitudinal reinforcement p by the shear friction method, the simplified method and the general method are plotted respectively. When p > 1.2%, all three methods show that there is no obvious trend in the predictions of shear capacity for beams with different amounts of longitudinal reinforcement. But, when p < 1.2%, the lowest ratios occur to all three methods while the general method has a larger scatter in this range. 84 Fig. 5-48 to Fig. 5-50 illustrate the effects of beam depth to predictions of shear capacity by the three methods. There is no obvious trend in the predictions of shear capacity for beams with different beam depth for all three methods. Fig. 5-33, Effect of the Ratio of Shear Span on the Shear Friction Method 85 Fig. 5-34, Effect of the Ratio of Shear Span on the Simplified Method • • li • a • • • B S • • • UP m • • a B° S Q 1.5 2.5 • • • • • • • • • • o • 3 3.5 • 4.5 a d Fig. 5-35, Effect of the Ratio of Shear Span on the General Method • • 5.5 86 60 80 100 f c (MPa) Fig. 5-36, Effect of Concrete Strength on the Shear Friction Method 1 V 1 1 1 1 1 B t • _ O g1 0 0^ § • • • n Bü Q• % m ° g3 sim D • • ^ • • ft • ° • ° • 0 1 i i i 20 40 60 80 f c i 100 i 120 (MPa) Fig. 5-37, Effect of Concrete Strength on the Simplified Method • 140 87 1 1 1 1 1 1 • B • n Dn u rfio V g • • k i o • • D • a a D • a „ an Û On o ° • * • rj 11 OO D œ • 0 D D D g 1=1 • a • u ° • n • • 1 1 1 1 1 1 20 40 60 80 100 120 f c (MPa) Fig. 5-38, Effect of Concrete Strength on the General Method Fig. 5-39, Effect of Stirrup Spacing on the Shear Friction Method 140 88 1 1 1 1 1 1 1 2 • • sim g f Bn 1 í • LP • • U • a n 1 1 ! U n • • • • D Bo • 0 1 1 1 1 1 1 1 0.2 0.4 0.6 0.8 s 1 1.2 1.4 d Fig. 5-40, Effect of Stirrup Spacing on the Simplified Method Fig. 5-41, Effect of Stirrup Spacing on the General Method 1.6 89 1 31 v 1 1 1 1 1 1 rn 2 3 4 5 6 7 8 t " 0 1 Pv-fvy(MPa) Fig. 5-42, Effect of Shear Reinforcement on the Shear Friction Method 31 I 0 1 I 2 I 3 I 1 4 5 1 6 1 7 8 f p - v y (MPa) v Fig. 5-43, Effect of Shear Reinforcement on the Simplified Method • a • •M ŒJ 1 • OD • • -O • B- • • 0 0 1 2 3 4 5 6 7 f P - v y (MPa) v Fig. 5-44, Effect of Shear Reinforcement on the General Method Fig. 5-45, Effect of Longitudinal Reinforcement on the Shear Friction Method 91 1 sim 1 • 1• D D 13 Í 1 • a s s Ha¡3l u D 3 1 1 • • PS • i • • a 1 0 1 • j • • 1 1 2 i i i 3 4 5 1 i 7 i 6 p (%) Fig. 5-46, Effect of Longitudinal Reinforcement on the Simplified Method • -Qv u • B D t • • Beg • B-O • • • a 2 3 4 P (%) Fig. 5-47, Effect of Longitudinal Reinforcement on the General Method 92 r i v t °B Vsf 1 a B -cr l • • 0 200 J L 400 600 J 800 1000 L 1200 1400 h (mm) J 1600 L 1800 2000 Fig. 5-48, Effect of Beam Depth on the Shear Friction Method i E B v sim c • Ë • D -B- r • • • a • 0 200 400 J L J 600 800 1000 I 1200 1400 h (mm) 1600 1800 Fig. 5-49, Effect of Beam Depth on the Simplified Method 2000 93 3 1 1 1 1 1 1 1 1 1 • • •on HfitWmnmnrrrn 2 B a • • • 200 U • n • • • n B " D D ° o • °b • n • • Ü • 1 1 1 1 400 600 800 1000 • 1 1 1200 1400 h (mm) 1 1 1 1600 1800 2000 Fig. 5-50, Effect of Beam Depth on the General Method 5.2.3.2 Beams without Shear reinforcement: Table 5-9 gives the summary of the comparison of predictions by three different methods for beams without stirrups. The predictions by the shear friction method are conservative. The average value of the measured/calculated ratios of shear capacity, V/V A is 1.41 and the coefficient of variation (C.O.V.) is 15.9%. The predictions by the S simplified method and by the general method are more conservative than the shear friction method and have much larger scatters. The average value of the measured/calculated ratios of shear capacity by the simplified method, V/V , is 1.58 sim and the coefficient of variation (C.O.V.) is 59.2%. The average value of the measured/calculated ratios of shear capacity by the general method, V/V , is 1.80 and the g coefficient of variation (C.O.V.) is 62.1%. 94 Table. 5-9, Comparison of Predictions for Beams without Stirrups NAME No. m V,/V a Kani 22 1.35 0.193 Leonhardt 8 1.29 Moody 10 1.26 0.226 17.6% 0.082 6.5% Van Den Berg 30 1.58 0.101 Mphonde 9 1.55 0.142 Ahmad Bresler 18 1.35 3 Xie sf C.O.V m 14.3% 2.01 2.51 V,/V a v,/v sim e m 1.328 66.2% 2.26 1.500 66.4% 3.13 1.39 2.523 80.5% 1.26 2.049 81.7% 0.090 7.2% 0.096 6.9% 6.4% 1.40 0.124 8.8% 1.50 0.132 8.8% 9.2% 1.29 0.168 13.1% 1.69 0.185 10.9% 0.265 1.49 0.399 26.7% 1.62 0.449 27.8% 1.47 19.6% 0.080 5.4% 0.096 7.1% 1.90 0.133 7.0% 2 0.96 0.030 3.1% 1.35 0.94 1.09 3 1.13 0.052 4.6% 0.97 1.11 0.076 0.083 7.0% Yoon 13.1% 0.073 7.5% Total 105 1.41 0.225 15.9% 1.58 0.936 59.2% 1.80 1.119 62.1% 0.123 a C.O.V C.O.V 7.5% Fig. 5-51 to Fig. 5-53 illustrate the results of predictions by the shear friction method, the C S A simplified method and the C S A general method respectively. Comparing the figures it is clear that the shear friction method gives the best prediction of the three methods. 95 200 V t Fig. 5-51, Predicted Results by the Shear Friction Method 400 (kN) 96 400 (kNJ • V • • • s i m 200 • o • a poo en • 200 400 v. (kN) Fig. 5-52, Predicted Results by the Simplified Method 97 400 (kN) • • V 200 • na g 200 400 (kN) Fig. 5-53, Predicted Results by the General Method 98 In Fig. 5-54, the ratios of V/V s against the ratios of shear span a/d are plotted. s The figure shows that the predictions of shear capacity by the shear friction method are within a very narrow range to the test results. Notice that the ratios of V/V f does not s change significantly even when the ratios of shear span, a/d, are less than 2.5. The measured/calculated ratios of shear capacity versus the ratios of shear span, a/d, by the C S A simplified method and the C S A general method are plotted in Fig. 5-55 and Fig. 5-56. There are larger scatters of the results than the scatter predicted by the shear friction method. Notice that the ratios of V/V sim a n ¿ V/V get significant larger g when the ratios of shear span, a/d, are less than 2.5. In Fig. 5-57, the ratios of V/V f against concrete strength, f' , are plotted to s c demonstrate the effect of concrete strength on the shear friction method. It shows no obvious trend in the prediction o f shear capacity for beams with different concrete strength. The predicted results by the simplified method and the general method against concrete strength, f' , are plotted in Fig. 5-58 and Fig. 5-59 respectively. There is no c obvious trend in the prediction of shear capacity for both of the methods. The ratios of test results to the results predicted by the shear friction method, the simplified method and the general method against the ratios of longitudinal reinforcement ratio, p, are plotted in Fig. 5-60, Fig. 5-61 and Fig. 5-62 respectively. N o obvious trend can be found in any of the figures. In Fig. 5-63, the ratios of V/V j- against the ratios of longitudinal reinforcement s strength, T/T , are plotted. The figure shows that the predictions of shear capacity by opt the shear friction method are within a very narrow range to the test results and there is no obvious trend against T/T . opt Fig. 5-64 to Fig. 5-66 show the effects of beam depth to predictions o f shear capacity by the three methods. There is no obvious trend in the predictions of shear capacity for beams with different beam depth for all three methods. 9 8 7 6 v.i 5 Vrf 4 • • • 3 2 •co D A 1 3 O 4 a Fig. 5-54, Effect of the Ratio of Shear span on the Shear Friction Method 9 8 7 -o- 6 h 5 ^sim • • • 4 • • 3 • 2 1 00 1 2 3 4 5 6 a d" Fig. 5-55, Effect of the Ratio of Shear span on the Simplified Method ' 100 9 8 7 • • 6 5 4 • • • 3 TO 2 -B faff i 1 0 BD t • n jp fln° a 0 a 7 Fig. 5-56, Effect of the Ratio of Shear span on the General Method 9 i 1 r r 8 7 6 v t Vrf • • a 5 4 3 • 2 0 10 20 30 40 o -EL • 1 50 J L 60 70 f (MPa) c 80 -tr J I 90 100 Fig. 5-57, Effect of Concrete Strength on the Shear Friction Method 110 101 9 8 7 6 _ Ï L ^sim • • • 5 4 dpoü 3 o 2 fea a l i 1 0 10 20 30 40 50 • • _Q •— 60 70 f (MPa) c 80 90 100 110 Fig. 5-58, Effect of Concrete Strength on the Simplified Method 9 8 7 -QCL 6 v t • 5 4 • • D • • -on— 3 2 • o r • 10 20 30 40 in Ü-|J 1 0 • 50 60 70 f (MPa) c • 80 _L _L 90 100 Fig. 5-59, Effect of Concrete Strength on the General Method 110 102 8 7 6 v t Vrf • on 5 4 3 2 1 1 -B• • _L 0 2 3 4 5 6 7 P (%) Fig. 5-60, Effect of Longitudinal Reinforcement Ratio on the Shear Friction Method 9 8 -B- 7 6 5 Vs i•m v 4 ODD 3 • • -¾—cr 2 1 0 s 9 | DCL • • • • • J 4 P (%) Fig. 5-61, Effect of Longitudinal Reinforcement Ratio on the Simplified Method 103 9 8 7 6 • 5 4 • • • 3 • 2 -rJ3„ qj n -e- • "XT -R- 1 • "ET • • _D_ 0 Fig. 5-62, Effect of Longitudinal Reinforcement Ratio on the General Method 9 8 7 6 V.1 5 Vrf • • • 4 3 2 • 1 Q 0.5 cP •1 • 1.5 an IB • a 2 2.5 ^ • • • 3.5 T opt Fig. 5-63, Effect of Longitudinal Reinforcement Strength on the Shear Friction Method 1 300 %0 400 500 h (mm) 600 800 700 Fig. 5-64, Effect of Beam Depth on the Shear Friction Method -B- § a • I -BJL >00 300 400 500 600 700 h (mm) Fig. 5-65, Effect of Beam Depth on the Simplified Method 800 105 • • • ~o— -B-fl- %0 300 400 500 600 700 h (mm) Fig. 5-66, Effect of Beam Depth on the General Method 800 106 CHAPTER 6 PROPOSED C O D E C L A U S E S FOR S H E A R D E S I G N 6.1 Proposed Code Clauses for shear design 6.1.1 Required Shear Resistance Member subjected to shear shall be proportioned so that V >V sf (6-1) f 6.1.2 Factored Shear Resistance: The factored shear resistance shall be determined by (6-2) r -V +V++V, ¥ t¥ But V shall not exceed sf Kr=^JXh +V (6-3) p 6.1.3 Determination of V csf : The value of V shall be computed from csf V^=y/V tane 45 (6-4) Where #is given in Clause 6.1.5, i//is given in Clause 6.1.6 and (6-5) 107 / \ 0.30 , 0.25 i 30) (500^ (6-6) P =0.36 v \fc J 6.1.4 Determination of V : ssf To determine V. ss/ (a) For members with transverse longitudinal axis, the value o f V ssf V y ssf =V r (décote reinforcement perpendicular to the shall be computed from (6-7) -1 si Where #is given in Clause 6.1.5 and (6-8) Ki = AAAvy (b) For members with transverse reinforcement inclined at an angle a to the longitudinal axis, the value of V ssf shall be computed from d (cot 0 + cot a) ev sina (6-9) 6.1.5 Determination o f &. The value o f 0 for each design section of a member shall be computed from tan 6 K d e v V 4i s (6-10) 108 Where a is the distance from the face of the support to the applied concentrated c load location, or from the face of the support to the design section for uniform load. 6.1.6 Determination of yr. The value of y/for each design section of a member shall be computed from 1// = 1 When T>T (6-11) When T<T (6-12) opi f - \ y/ = sin n T opt 2 T Where T = AA f s (6-13) y 2 T =V (2 + tan 0) opl 4S (6-14) 6.1.7 Limiting Shear Failure Angle: The failure angle 0 shall satisfy the following conditions: tan6> (6-15) a,. IL 6>2l K (6-16) 30j 6.2 Design Examples Design a simply supported beam for shear to span 4 m. The beam carries one 300 k N concentrated live load at midspan. Use 25 M P a concrete and 400 M P a steel. The beam will have 6-No.25 longitudinal reinforcing bars at bottom and will use No. 10 stirrups. 109 (1) Factored shear force: Vf = 1.5x300/2 = 225 k N (2) Beam size: Choose beam size b x h = 400 mm x 600 mm . n 25 \ OJO , / 30 ¡3 =0.36 0.36 x V V n ) V 45 = ^c¡3v^TcbJi ^30^ ( y 25 y \600 j 500^ = 0.36 = 1.0x0.6x0.36x425x400x600x 3 10~ =259 k N (3) Check maximum shear capacity: 3 Ka* = -<Pcf'cKh = -x0.6x25x400x600x 4 4 >V = 10~ = 900 k N 225 k N , ok! f (4) Stirrup spacing: Assume T > T opt d ev = h-(c,+c +4d ) = 600-(40 + 40 + 4x11.3) = 475 mm b b 3 Ki =<¡> AJ = s 4V s 4sV d s<— sl 0.85 x 100 x2x 400 xlO' vy 4x259x68x475 , = —; -— ev = 390 mm 2 (Vf+Vsi) (225 + 68) Choose stirrup spacing at 350 mm. (5) Calculate shear angle 6 : x tanO V 45 6 = 31.0° s 259 = 0.60 350 3 f ì n =68 m (6) Calculate yr. T = fcAJ =0.85x6 x 500x400 xl0~ 3 =1020 k N y 2 2 T =V (2 + tan 0) = 259x(2 + tan (31.0)) = 612 k N r>r o p opl 45 „ S o i// = i (7) Check shear failure angle: V 68 ev tan0 = ' " ^ 5 \ 1x259 0 = 31.0° >21 v 475 •x 350 600 a, 7600 = 20.3° 30j y h 1 5 2f] 27 x 50 =o.60> K (8) Check shear capacity of beam: v =v sf csf + v +v ssf p y/V tan0 + Vsi 45 (d cot0 ^ ev i = 1x259 x 0.60 + 68x1 I 4 7 5 x 1 6 7 - ? 350 = 155.4 + 86.1 = 241.5 k N > V, = 225 kN, ok! Ill CHAPTER 7 CONCLUSIONS A N D RECOMMENDATIONS The following conclusions and recommendations were made based on the analyses of the test data from a total of 113 beams with stirrups and 105 beams without stirrups from the literature. A l l 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 120 MPa. The selected beams also had different shear span ratios from about 1 up to 7, different amounts of longitudinal reinforcement, different shear reinforcement and different stirrup spacing. 7.1 Conclusions and Recommendations: 1. The shear friction method changes the way of beam shear design by simplifying design procedure and increasing accuracy. The results of the comparison show that the predictions by the shear friction method had less scatter with test results compared to the predictions by the C S A simplified method and general method. 2. The simplified method and the general method are only suitable for beams with shear span, a/d, greater than 2.5. But the shear friction method may be used for short shear span situations, such as deep beams. 3. According to the analysis in Chapter 5, stirrup spacing of beams has no obvious effects on the predictions of the shear capacity of beams by all three methods. Notice that the limitation on maximum stirrup spacing by the C S A A23.3-94 clauses 11.2.11 for the simplified method and the general method was ignored during the analysis. Further investigation should be conducted to determine whether the limitation on maximum stirrup spacing needs to be revised. 112 4. The calibration factor, f3 , accurately predicts the effects of concrete strength v for the ranges of 20 M P a to 125 M P a and beam height for the ranges of 250 mm to 2000 mm. 5. The shear capacity of beams relies on not only the longitudinal reinforcement of beams, but also the anchorage of such reinforcement. So any excessive bar cut offs near beam supports shall be avoided. 6. The equation for calculating minimum shear failure angle, 0 , mm is appropriate in the prediction of the shear capacity of beams without stirrups. It is not necessary for shear calculations for beams with the usual amounts of shear reinforcement. 7.2 Future Research: 1. 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