SHEAR DESIGN OF PILE CAPS AND OTHER MEMBERS WITHOUT TRANSVERSE REINFORCEMENT By Zongyu Zhou B.Eng. Tongji University 1982, M.Eng. Tongji University 1987 A ThESIS SUBMITTED IN PARTIAL FULFILLMENT OP THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1994 © Zongyu Zhou, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of V L The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract This thesis deals with the shear design of structural concrete members without trans verse reinforcement. The three major parts of this study are the transverse splitting of compression struts confined by plain concrete, the development of a rational design procedure for deep pile caps, as well as a general study of the shear transfer mechanisms of concrete beams. Three-dimensional compression struts that are unreinforced and confined by plain concrete, as occur in deep pile caps, were studied both analytically and experimentally. Based on the study results, bearing stress limits are proposed to prevent compression struts from transverse splitting. The maximum bearing stress depends on the amount of confinement, as well as the aspect ratio (height to width) of the compression strut. The proposed bearing stress limit was incorporated into a strut-and-tie model to develop a rational design procedure for deep pile caps. Two methods are proposed. The first method is a direct extension of two-dimensional strut-and-tie models used for deep beams. The second method is presented in a more traditional form in which “flexural design” and “shear design” are separated. The shear design is accomplished by limiting the bearing stress at the columns and the piles. The first method is more appropriate for analysis, while the second method is more appropriate for design. The rationality and accuracy of the proposed methods are demonstrated by the comparison with previous test results. In the final part of this study, the influence of bond between concrete and longitudinal reinforcement upon the load transfer mechanism of both deep members and slender members without stirrups are investigated. An interpretation of an important shear 11 failure mechanism is presented. 111 Table of Contents Abstract ii List of Tables viii List of Figures ix Acknowledgement 1 2 3 xvi Introduction 1 1.1 Background 1 1.2 Objectives and Outlines of Thesis 4 Analytical Study of Transverse Splitting 8 2.1 Finite Element Modelling 8 2.2 Internal Stress Distributions within Cylinders 9 2.3 Definitions of Geometrical Parameters and First Cracking 10 2.4 Location of First Cracking 10 2.5 Bearing Stress at First Cracking 11 2.6 Summary 12 Experimental Study of Transverse Splitting 17 3.1 Description of Test Program 17 3.2 Specimen Preparation 17 3.3 Instrumentation and Data Acquisition 18 . . iv 3.4 4 19 3.4.1 Internal Cracking 19 3.4.2 Influence of Concrete Confinement 20 3.4.3 Failure Mode • • • . Bearing Strength of Concrete Compression Struts 4.1 Comparison of Measured and Predicted Cracking Loads 4.2 Comparative Study of Ultimate Bearing Strength 4.3 5 Test Observations 20 29 • • . . 29 30 4.2.1 Previous Studies of Bearing Strength 4.2.2 Influence of Concrete Compressive Strength 4.2.3 Influence of Size Effect 4.2.4 Influence of Loading Geometry 34 4.2.5 Summary 34 • . . 30 32 • Design Recommendations for Concrete Compression Struts . • . • . . . . Shear Design of Deep Pile Caps 33 35 45 5.1 Introduction 45 5.2 ACI Code Approach for Pile Cap Design 47 5.2.1 Code Procedure for Shear Design 47 5.2.2 Inadequacies of ACI Code Procedure for Shear Design of Deep Pile 5.2.3 Caps 49 Comparison of Different ACI Code Editions 50 5.3 CRSI Approach for Shear Design of Deep Pile Caps 51 5.4 Strut-and-Tie Model Approach 53 5.4.1 CSA Approach for Deep Beams 53 5.4.2 Proposed Design Method (1) 54 5.4.3 Proposed Design Method (2) 57 V 5.5 Summary of Previous Pile Cap Tests 5.6 Comparative Study of Different Design Procedures 5.7 6 7 58 • . . . 63 5.6.1 Comparison of Design Methods 63 5.6.2 Comparison of Prediction Results 64 Conclusions 66 Shear Failure of Beams Without Stirrups 99 6.1 Introduction 99 6.2 Brief Review of the Literature 100 6.2.1 Transition from Deep Beam to Slender Beam 101 6.2.2 Behaviour of Slender Beams 102 6.3 One Interpretation of Shear Failure of Beams without Stirrups 105 6.4 Bond Effect in Longitudinally Reinforced Concrete Beams 108 . 6.4.1 Bond Influence in Uncracked Beams 109 6.4.2 Bond Influence in Cracked Beams 112 6.5 Shear Displacements along Cracks 113 6.6 Load Transfer Mechanism 114 6.7 Interpretation of Some Beam Test Results 116 6.8 Conclusions 117 Bond Splitting Failure 143 7.1 Introduction 143 7.2 General Bond Action 144 7.3 Previous Experimental Studies 146 7.4 Previous Studies of Ultimate Splitting Failure 147 7.5 Previous Studies of Splitting Initiation 149 7.6 Proposed Design Equation for Bond Splitting Initiation 151 vi 7.7 8 Some Comments and Conclusions . Brief Summary and Further Research 153 168 Bibliography 173 Appendices 183 A Measured Bearing Stresses and PUNDIT Readings 183 B ACI Code and CRSI Handbook Predictions 193 C Predictions from Proposed Design Method (1) 237 D Predictions from Proposed Design Method (2) 250 VII List of Tables 3.1 Summary of experimental results 22 3.1 (cont’d) Summary of experimental results 23 5.1 Summary of pile cap test results 68 5.1 (con’t) Summary of pile cap test results 69 5.2 Summary of ACT Building Code and CRSI Handbook predictions (flexure and bearing) 5.2 70 (con’t) Summary of ACT Building Code and CRST Handbook predictions (flexure and bearing) 71 5.3 Summary of ACT Building Code and CRSI Handbook predictions (shear). 72 5.3 (con’t) Summary of ACT Building Code and CRST Handbook predictions (shear) 5.4 73 Comparison of ACT Code and CRSI Handbook predictions: ratio of mea sured capacity to predicted capacity and failure mode 5.4 74 (con’t) Comparison of ACT Code and CRST Handbook predictions: ratio of measured capacity to predicted capacity and failure mode 75 5.5 Comparison of proposed design methods with experimental results. 5.5 (con’t) Comparison of proposed design methods with experimental results. viii . . . 76 77 List of Figures 1.1 Strut-and-tie model for a deep beam or pile cap: (a) the idealized loadresisting truss; (b) linear elastic stress trajectories with transverse tension due to spreading of compression and (c) refined truss model with concrete tension tie to resist transverse tension 1.2 6 Maximum bearing stress to cause transverse splitting in a biaxial stress field, from Schlaich et al.[3} 2.1 7 Structural modelling of an idealized compression strut: (a) geometry; (b) finite element mesh; (c) isometric view showing stresses produced by ax ially symmetric loading and (d) an axially symmetric finite element of rectangular cross section. Figs.(c) and (d) are from Ref.[15] 2.2 13 Typical internal stress distributions along the central Z axis within the cylinders: (a) tall cylinder and (b) short cylinder 2.3 14 Influence of D/d on the bearing stress to cause cracking for the case of H/d=2 2.4 15 Analytical study of the bearing stress at first cracking: (a) influence of confinement and (b) influence of height 16 3.1 Schematic of test set-up 24 3.2 Photograph of test set-up 25 3.3 Typical relationships of load vs. PUNDIT readings (transit time incre ments) for the case of H/d 3.4 = 2 26 Typical relationships of load vs. axial deformation for the case of H/d ix = 2 27 3.5 Photograph of the typical failure mode of a concrete cylinder 4.1 Comparison of experimental results and analytical predictions: influence 28 of confinement for various heights 4.2 37 Comparison of experimental results and analytical predictions: influence of height for various degrees of confinement 4.3 Comparison of ACT Building Code bearing strength prediction with ex perimental results: (a) present investigation and (b) previous tests. 4.4 38 . . Comparison of ACT Building Code bearing strength with Hawkins’s sug gested equation for various concrete strengths 4.5 39 40 Influence of specimen height on ultimate bearing strength: (a) the force flow in a single-punch test; (b) the force flow in a double-punch test and (c) the force flow in a tall double-punch test 4.6 41 Influence of height on accuracy of ACT Building Code bearing strength prediction of double-punch tests 42 4.7 Comparison of suggested design equation with the analytical results. 4.8 Comparison of predictions from suggested design equation with experi . . mental results 5.1 43 44 ACT Code specified critical sections for flexure and shear investigation of pile cap 78 5.2 A deep two-pile cap 79 5.3 A deep three-pile cap 80 5.4 Comparison of two-way punching shear calculations: (a) the cap with square column; (b) the cap with circle column x 81 5.5 CR51 approach for shear design of deep pile caps: (a) allowable shear stress, v, for one-way shear while w/d < 1.0; (b) allowable shear stress, v, for two-way shear while wfd < 0.5, from Ref.[36) 82 5.6 Flow chart for ACT and CRST design procedures for pile caps 83 5.7 Strut-and-tie model for deep beam, from Ref.[39] 84 5.8 Loading geometry of compression struts with linearly varying cross section 85 5.9 Various layouts of main reinforcing bars used by Blévot and Frémy, Ref.[32] 86 5.10 Various anchorage lengths used by Clarke, Ref.[44] 87 5.11 Comparison of one-way shear design methods for two-pile caps: (a) plan view of pile cap; (b) to (d) influence of pile cap depth on column load for various pile cap widths 88 5.11 (cont’d) Comparison of one-way shear design methods for two-pile caps: (a) plan view of pile cap; (b) to (d) influence of pile cap depth on column load for various pile cap widths 89 5.12 Comparison of two-way shear design methods for a typical four-pile cap: (a) plan view of pile cap; (b) influence of pile cap depth on column load 90 5.13 Comparison of ACT ‘77 predictions with experimental results 91 5.14 Comparison of ACT ‘83 predictions with experimental results 92 5.15 Comparison of ACT ‘[11.8] predictions with experimental results, clause 11.8 considered 93 5.16 Comparison of CRST predictions with experimental results 94 5.17 Comparison of proposed method (1) predictions with experimental results of all specimens in Table 5.1 95 5.18 Comparison of proposed method (1) predictions with experimental results of specimens with bunched reinforcement in Table 5.1 xi 96 5.19 Comparison of proposed method (2) predictions with experimental results of all specimens in Table 5.1 97 5.20 Relationship of measured ultimate bearing stress and confinement on col umn zones of pile caps in Table 5.1 6.1 98 Distribution of transverse compressive stress for various shear span ratios, from Mau and Hsu, Ref. [52] 6.2 120 Load near the support: transition from deep beam to slender beam, from Schlaich et al., right side simple models; left side refined models, Ref.[3]. 6.3 121 Predictions of shear strength versus a/d ratio for tests reported by Kani[13J, from Collins and Mitchell, Ref.[39]. 122 6.4 Truss model developed by Adebar, Ref.[54J 123 6.5 Truss model developed by Reineck, Ref.[55j 123 6.6 Truss models developed by Al-Nahlawi and Wight, Ref.[53]. 124 6.7 Structural model developed by Muttoni and Schwartz, Ref.[56]. 124 6.8 Structural model developed by Kotsovos, Ref.[57] • • • . 6.9 Crack pattern of a beam tested by Kani, Ref.[13] • . • . 125 125 6.10 Geometric relationship of a crack at reinforcement level 126 6.11 A simply supported and central loaded beam 127 6.12 Linear elastic finite element analysis: modelling of (a) perfect bond and (b) no bond case 128 6.13 Internal force flows in uncracked beams: (a) perfect bond case and (b) no bond case 129 6.14 Modelling of bond effect upon transverse splitting of compression struts in deep beam 130 XII 6.15 Bond influence upon transverse tensile stresses of compression struts in deep beam 131 6.16 Internal stress distributions of cracked beams due to pure beam action, from Ref.[39J 132 6.17 Shear stress distributions of cracked beams due to combined beam action and arch action: (a) modelling of combined beam and arch actions; (b) tension force in longitudinal reinforcement; (c) shear stress distribution at section n — n 133 6.18 Shear displacement along cracks in shear span: (a) vertical cracks and (b) inclined cracks 134 6.19 Shear transfer at cracks by aggregate interlock 135 6.20 A beam with inclined cracks in shear span: (a) strut force due to arch ac tion; (b) one assumed tension force in reinforcement; (c) another assumed tension force in reinforcement; (d) measured concrete strain and tension force in reinforcement from Ref.[71] 136 6.21 Measured force in bar, bond stress and crack locations, from Ref.[59]. . 137 6.22 A load transfer mechanism just before the occurrence of splitting along longitudinal reinforcement: (a) truss model and (b) tension force in rein forcement 138 6.23 Design details of beams tested by Kotsovos: (a) a/d = 1.5 and (b) a/d> 2.5, from Ref.[73j 139 6.24 Load-deflection curves of beams shown in Figure 6.5: (a) a/d aid = 3.3 and (c) a/d = 4.4, from Kotsovos, Ref.[73J = 1.5; (b) 140 6.25 Beam with internal stirrups tested by Chana, from Ref.[58] 141 6.26 Beam with external stirrups tested by Kim et al., from Ref.[74] 141 6.27 Beams tested by Muttoni et al., from Ref.[56j 142 xlii 6.28 Beams by Kuttab et al., from Ref.[75] . 7.1 Bond stress—slip relationship, from Gambarova et al., Ref.[80] 7.2 Pullout tests: (a) concentric pullout-test specimen; (b) commonly used ec 142 155 centric pullout-test specimen and (c) eccentric pullout-test specimen used by Ferguson, Ref.[60] 7.3 156 The University of Texas beam tests: (a) possible tension force distribution along the top test bar after the occurrence of inclined crack; (b) side view of specimen; (c) plain view of top reinforcement; and (d) moment diagram, adapted from Ferguson and Thompson, Ref.[61] 157 7.4 Stub-beam or cantilever bond specimen, from Gergely, Ref.[77] 158 7.5 Mechanism representation for bond, from Tepfers, Ref.[84] 159 7.6 Bond Splitting Failure Patterns, from Oranguri et al., Ref.[82] 160 7.7 Analysis of Bond Splitting Stresses: (a) bursting and bond stresses; (b) uncracked elastic state; (c) uncracked plastic state and (d) partly cracked elastic state. Adapted from Tepfers, Ref.[84} 7.8 161 Comparison of test and prediction for bond splitting initiation, from Tepfers, Ref.[84] 7.9 162 Comparison of Equation 7.7 with test data on cracking loads carried out by Kemp and Wilhelm Ref.[66] 163 7.10 The relationship of bond splitting strength and ld/d proposed by Teng and Ye Ref.[86] for concentric pullout tests 164 7.11 Comparison of Equation 7.2 arid 7.10 with test results from Kemp and Wilhelm Ref. [66] 165 7.12 Comparison of Equation 7.10 with test results from Tepfers Ref.[84]. . . . 166 7.13 Relationship of bond strength versus ld/db, presented by Eqs. 7.2 and 7.10. 167 xiv 8.1 Load transfer mechanisms of structural concrete members without trans verse reinforcement: (a) very deep members (aid < 1) as well as more slender members with no bond between concrete and reinforcement; (b) slender members with normal bond between concrete and reinforcement. xv 172 Acknowledgement The author would like to express his sincerest gratitude to his supervisor Dr. Perry Adebar for his constant guidance, criticism, and encouragement throughout the course of research work and in the preparation of this thesis. Financial support to undertake this study was provided by the Natural Sciences and Engineering Research Council of Canada. Finally, this thesis is dedicated to the author’s family for their understanding and support, and especially to his wife, Jialing, and son, George, for their patience through the duration of this study. xvi Chapter 1 Introduction 1.1 Background Plain concrete is strong and reasonably ductile in compression, but is weak and brittle in tension. Thus reinforcing steel bars are usually provided in concrete members to resist any tension force. In a reinforced concrete beam subjected to bending, for example, the flexural tension force is resisted by longitudinal reinforcing bars. For more complex loading conditions, such as combined shear and bending, structural concrete members are usually reinforced with both longitudinal and transverse reinforcement. Structural concrete members reinforced with both longitudinal and transverse rein forcement can be designed using simple truss models or in the more general case strutand-tie models [1-3]. It has been suggested by Schlaich et al. [3] that structures be subdivided into B-regions and D-regions for the purpose of design. B-regions are where the Bernoulli hypothesis of plane strain distribution is reasonable (B standing for beam or Bernoulli). Internal stresses in B-regions can be derived from the sectional forces with out considering the details of how the forces are applied. Truss models for B-regions with stirrups involve compression and tension chords to model the flexural stresses, tension ties to model the stirrups and diagonal compression struts to model the cracked concrete web. Truss models that have been developed for the shear design of beams with stirrups are rational and work well [4-8]. In D-regions (D standing for discontinuity or disturbance) the strain distribution is 1 Chapter 1. Introduction 2 significantly nonlinear, and the details of how the forces are applied are very important. The more general form of truss models, strut-and-tie models can be used for both B- and D-regions. The theoretical justification for strut-and-tie (truss) models comes from the theory of plasticity (limit analysis) [1]. Strut-and-tie models fall within the domain of lower-bound limit analysis. As strut-and-tie models allow designers to freely choose the load path, a sufficient amount of uniformly distributed reinforcement must be provided to distribute cracks, thereby ensuring adequate ductility (plastic deformation) for internal redistribution of stresses so that the assumed strut and tie system can eventually develop. Unfortunately, it is not always practical to provide the reinforcement needed for crack control and ductility. Traditionally many structural concrete members are designed and built without transverse reinforcement. In North America, both the ACT Building Code [9] and the Canadian Concrete Code [10] allow the following structural concrete members to be designed without stirrups: slabs and footings; concrete joists; beams with a total depth less than 10 in. (254 mm); wide beams having a total depth less than the width of the web; and tee beams having a total depth less than 2.5 times the thickness of the flange. Many of the current design procedures for members without stirrups are less than satisfactory. For example, the design procedures currently specified in the ACT Building Code and the Canadian Concrete Code for deep pile caps (footings) are inappropriate. The ACT Building Code procedure for the shear design of footings supported on piles (pile caps) is the same sectional approach used for footings supported on soil and for slabs. The procedure involves determining the section thickness which gives a concrete contribution V greater than the shear force applied on the code defined critical section. While this approach is reasonable for slender footings supported on numerous pile, it is not appropriate for deep pile caps which are entirely a D-region. On the other hand, ap plying strut-and-tie (truss) models to deep pile caps, which are usually without transverse Chapter 1. Introduction 3 reinforcement and have very limited ductility, is also questionable. The internal force flow within a deep pile cap can be represented by the simple strut-and-tie model shown in Figure 1.1(a). The column load is transmitted directly to the piles by compression struts. In order to prevent the piles from being spread apart, tension ties must be provided. The results of an experimental study on pile caps [11] has demonstrated that such a simple strut-and-tie model is a better model for deep pile caps than the basis of the traditional design procedures for pile caps. A prismatic compression strut is a highly idealized representation of the stress state between a column and pile in a deep pile cap. In reality, the compressive stresses in the compression strut spread out and transverse tensile stresses are created due to strain compatibility [see Figure 1.1(b)]. A refined strut-and-tie model for the compression strut, which includes a tension tie to model the transverse tension, is shown in Figure 1.1(c). As the compression struts which transfer load in deep pile caps are usually unreinforced, the tension tie must be provided by concrete tensile strength which is only about a tenth to a fifteenth of the concrete compressive strength. It is obvious that the failure of unreinforced compression struts could be initiated by transverse splitting. Figure 1.2 from Schlaich et al. [3] shows the influence of the “amount of spreading” on the bearing stress to cause transverse splitting for a plane stress field (appropriate for the case of a deep beam). Based on Figure 1.2 Schlaich et al. suggest that the concrete compressive stresses within an entire unreinforced D-region can be considered safe if the maximum bearing stress in all nodal zones is limited to O.6f (or in unusual cases O.4f) [3]. However what bearing stress limit is appropriate for compression struts confined by surrounding plain concrete, such as occurs in deep pile caps, is not known. This hinders the application of strut-and-tie models for the design of deep pile caps. While some truss models have been developed for slender members without transverse reinforcement (i.e., B-regions), there is still no generally accepted interpretation of the Chapter 1. Introduction 4 shear failure mechanism for such members. Experimental investigations have shown that the bond between concrete and longitudinal reinforcement has a large influence upon the shear capacity of reinforced concrete beams without transverse reinforcement [12-14]. Truss models that have been developed for members without transverse reinforcement do not adequately account for the influence of bond, particularly bond splitting failures at the base of the critical diagonal crack. 1.2 Objectives and Outlines of Thesis This thesis deals with the shear design of structural concrete members without stirrups, and includes three major parts. The first part is a study of the transverse splitting of compression struts confined by plain concrete as occurs in deep pile caps. The second part involves the development of a rational design procedure for deep pile caps, which makes use of the bearing stress limit developed in the first part. In deep pile caps the column load is transmitted to the piles by direct compression struts. In more slender members a direct compression strut cannot form and the load transfer mechanism is influenced by the bond between concrete and reinforcing bars. A more general study of the load transfer mechanisms in members without transverse reinforcement is undertaken in the third part of this thesis. In Chapter 2 three-dimensional compression struts confined by plain concrete are studied analytically to investigate the internal stress distributions. The objective is to determine the bearing stress at which first cracking occurs due to the transverse tension. The influence of various geometrical parameters upon the bearing stress at first cracking is presented. In Chapter 3 the transverse splitting of compression struts is investigated experimen tally. The results from tests on plain concrete cylinders of various sizes loaded over a Chapter 1. Introduction 5 constant bearing area are presented. Design recommendations for the bearing stress limit of unreinforced concrete com pression struts based on a combination of the analytical and experimental results are proposed in Chapter 4. Comparisons are also made with previous studies for ultimate bearing strength. In Chapter 5, the proposed bearing stress limit of compression struts is incorporated into a simple strut-and-tie model to develop a design procedure for deep pile caps. Two design procedures are proposed. The first design method is appropriate for a detailed analysis, while the second design method is more suitable for design. A comparison of predictions from the proposed procedures, as well as the ACT Code and CRSI Handbook procedures are carried out for available pile cap test results. In Chapter 6, a general study of the load transfer mechanisms in members without stirrups is carried out. Recently developed truss models are reviewed. The shear failure mechanism of beams without stirrups is studied from the compatibility of the critical diagonal cracking propagation. The effect of bond between concrete and reinforcing bars upon shear resistance mechanism are also studied. Based on the study in this chapter, some interesting conclusions are drawn. Finally in this chapter, experimental results from beam shear tests are interpreted. As a natural extension to Chapter 6, the bond splitting failure along reinforcing bars is investigated in Chapter 7. Based on previous studies on bond, an empirical equation is proposed to predict the tension force causing bond splitting in beams without transverse reinforcemeilt. Finally, some major conclusions from this study and a few recommendations for fur ther investigations are summarized in Chapter 8. Chapter 1. Introduction 6 V V comPressio/,.,.N , I/ nocal zone J. / N/ Ti 1 tie Lension e T = V I tan D=V/sine (a) (b) a /// (c) Figure 1.1: Strut-and-tie model for a deep beam or pile cap: (a) the idealized load-resisting truss; (b) linear elastic stress trajectories with transverse tension due to spreading of compression and (c) refined truss model with concrete tension tie to resist transverse tension. Chapter 1. Introduction fb Failure f, C 1 Transverse Cracking 0 1 2 3 4 5 6 7 8 b 9 a 1. b Figure 1.2: Maximum bearing stress to cause transverse splitting in a biaxial stress field. from Schlaich et al.[3J. Chapter 2 Analytical Study of Transverse Splitting In order to better understand the transverse splitting phenomenon in deep pile caps and develop an appropriate strength criterion for plain concrete compression struts, idealized three-dimensional compression struts are studied analytically. The study focuses on the initial cracking within struts rather than the post-cracking stage, which is a much more complicated nonlinear problem. 2.1 Finite Element Modelling The compression struts are idealized as cylinders of exterior diameter D, and height H, subjected to concentric axial compression over a constant size circular bearing area of diameter d, both on top and bottom. See Figure 2.1(a). Linear elastic finite elements were used to study the internal triaxial stress distributions, and to determine when and where first cracking occurs within the cylinders. For the problem at hand, the geometry, material properties and external loading are all axis-symmetric so that the problem is mathematically two-dimensional. The static displacements and stresses are independent of angle 0, circumferential displacement v is zero and material points have only u(radial) and w(axial) displacement components. Non-zero stress components are and o, r and Tzr O, o, o,. and Tzr. The stress o is a principal stress are components of the two other principal stresses. In the analysis, axis-symmetric 4-node quadrilateral elements were used [15]. There are two integration points in each one of radial, axial and circumferential directions. Stresses at the central 8 Chapter 2. Analytical Study of Transverse Splitting 9 point of each element were calculated. See Figure 2.1(c) and (d). The finite element mesh used is shown in Figure 2.1(b), which also shows the geom etry, grid layout and boundary conditions. Due to symmetry of the problem and the characteristic of the elements used, only a quarter of the cylinder is shown. The radial displacement u = 0 was prescribed at all nodes that lie on the Z axis and the rigid body motion along Z direction was restrained by prescribing w = 0 at the mid-height of the symmetrical cylinder. But no restrictions were imposed for vertical displacement on the Z axis and horizontal displacement along the radial direction with the exception of one location at the corner. By adding more elements either horizontally or vertically (i.e., increasing D or H), the influence of various loading geometry on the internal stress distributions was investigated. 2.2 Internal Stress Distributions within Cylinders Based on the numerical analysis results the internal stress distributions along the central Z axis are shown qualitatively in Figure 2.2. The vertical stress component r has its maximum value beneath the loading plate and decreases gradually with increasing distance from the loading plate. The stress o is distributed uniformly on the transverse cross section at a certain distance away from the loading plate. The stress components 0 and ,. on the Z axis are equal and may be negative (compressive), or positive (tensile), as well as zero at locations where the stress cr becomes uniformly distributed. The internal stress distribution within the cylinder can be divided into three zones [see Figure 2.2(a)]. A zone of triaxial compression exists immediately below the loading plate. Further down from the loading plate is a zone with horizontal biaxial tension and vertical uniaxial compression. At the mid-height of relatively tall cylinders is a zone of vertical uniaxial compression. In short cylinders, the third zone (of uniaxial compression) Cha p 0 ter 2. Analytical Study of Transverse Splitting 10 does not exist. See Figure 2.2(b). 2.3 Definitions of Geometrical Parameters and First Cracking The geometry of the problem can be summarized in terms of two parameters, namely the ratio of the cylinder diameter to the loaded area diameter, D/d, and the ratio of the cylinder height to the loaded area diameter, H/d. The stress field within a cylinder can be expressed in terms of the four stress compo nents oo, o, o,. and re,. or by three principal stresses and corresponding principal stress directions. The stress O is always a principal stress. It was assumed that cracking will occur when the maximum principal tensile stress reaches the concrete tensile strength. The location of the maximum principal tensile stress defines where the first cracking oc curs. The influence of the other principal stress components upon the cracking strength was neglected for simplicity. 2.4 Location of First Cracking After the internal stress distributions within the cylinders were examined, it was found that the maximum tensile stress would occur either on Z axis (at the centre of the cylinder) or near the exterior surface of the cylinder depending on the geometry. Hence, the location of first cracking could be located either in the inside of the cylinder or near the outside of the cylinder. The influence of loading geometry upon the location of first cracking is shown in Figure 2.3, where the predicted bearing stress at cracking versus the ratio of Did are summarized for the case of H/ d = 2. There are two curves in the figure: one indicating cracking inside, the other cracking outside. It can be seen that for small ratios of Did, first cracking occurs on the surface, while for D/d > 1.5 the first cracking occurs at the Chapter 2. Analytical Study of Transverse Splitting 11 centre of the cylinder. As cracking outside occurs only when there is very little confinement, which is not of much practical interest, oniy cracking inside was considered below. 2.5 Bearing Stress at First Cracking Figure 2.4 summarizes the analytical results regarding the the influence of D/ d ratio and H/d ratio upon the bearing stress at first cracking (inside). In order to plot the bearing stress as a function of the compressive strength of concrete, the concrete tensile strength f’ was assumed to one-fifteenth of the concrete compressive strength f (i.e., f’ = f/l5). Figure 2.4(a) shows the relationship between bearing stress at first cracking and D/d for specific H/d values. For small ratios of D/d (i.e., close to 1.0), the bearing stress at first cracking is independent of height and tends to be very large. In this range the uniaxial compression is more critical than the transverse tension. In the range of D/d > 1.5 2.0, increasing the amount of confinement (i.e., increasing D/d further) will increase the bearing stress at first cracking, but not very much for small and medium H/ d values. The significant transverse tension is introduced due to the spreading-out of compressive force. The transverse tensile stresses will reach concrete tensile strength at relatively low external load in this range. Figure 2.4(b) shows the relationship between bearing stress at first cracking and H/d for specific D/ d values. Figure 2.4(b) illustrates a similar trend to Figure 2.4(a). In the range of H/ d < 1, the bearing stress at first cracking is independent of confinement and tends to be very large. The localized uniaxial compressive stress field between two loading plates is dominant within the compression strut. For H/d > 1.5, increasing height will increase the bearing stress at first cracking. But increasing height further (i.e., Hid > 4 ‘-‘ 6) does not have significant influence upon the bearing stress at first Chapter 2. Analytical Study of Transverse Splitting 12 cracking. It can be seen that the minimum bearing stress at first cracking is approximately O.6f at D/d = 2 and H/d = 1.5. More discussions about aspect ratio (H/d) upon the bearing stresses at first cracking will be given in the comparative study of ultimate bearing strength in Chapter 4. 2.6 Summary The numerical investigation indicates that the loading geometry has a significant influence upon the internal stress distribution within a confined compression strut. See Figure 2.2. Qualitatively, the response of an idealized compression strut (cylinder) to external load can be described as follows. For a compression strut without confinement (D/d = 1), the strut is uniaxial stressed. The compressive stress is constant over the height and is equal to the bearing stress. However, if there is confinement outside the loaded area (D/d> 1), the additional concrete will be mobilized to sustain the external load, and the axial compressive stress along the central axis of the strut will decrease continually with increasing distance away from the loading plate. As expected, higher bearing stresses may be possible due to the confinement. On the other hand, as a result of strain compatibility the transverse tensile stresses are introduced. Because concrete tensile strength about a tenth to a fifteenth of concrete compressive strength f, f’ is internal cracking of the concrete compression strut (cylinder) may occur due to the weak tensile strength at relatively low compression stresses. Chapter 2. Analytical Study of Transverse Splitting 13 d/2 TT H H12 J 1c D12 (a) (b) z,w 4 3x’ \; Axis—r,u 2 (C) (d) Figure 2.1: Structural modelling of an idealized compression strut: (a) geometry; (b) finite element mesh; (c) isometric view showing stresses produced by axially symmetric loading and (d) an axially symmetric finite element of rectangular cross section. Figs.(c) and (d) are from Ref.[15]. Chapter 2. Analytical Study of Transverse Splitting 14 z 5.r Fy 2 — - H/2 bd fb Compression Tension H (a) z I I e.J D ZS(L Tension Compression (b) Figure 2.2: Typical internal stress distributions along the central Z axis within the cylinders: (a) tall cylinder and (b) short cylinder. Chapter 2. Analytical Study of Transverse Splitting 15 / / f 3- H/d=2 fc’/ft’=15 2 Cracking Outside Cracking Inside 0 I 1 2 3 I D d Figure 2.3: Influence of D/d on the bearing stress to cause cracking for the case of H/d=2. Chapter 2. Analytical Study of Transverse Splitting 16 4 3 H/d = 10 H/d—8 HId =6 H/d =4 H/d =3 H/d—2 0 (a) B D 4 Did =7 D/d—6 D/d =5 D/d—4 2 D/d =3 I D/d =2 0 (b) 12 H d Figure 2.4: Analytical study of the bearing stress at first cracking: (a) influence of confinement and (b) influence of height. Chapter 3 Experimental Study of Transverse Splitting In order to confirm the transverse splitting phenomenon within compression struts, and to investigate the influence of H/d and D/d ratios upon the bearing stress at first cracking experimentally, a series of experiments were conducted on plain concrete cylinders. 3.1 Description of Test Program Table 3.1 summarizes the properties of the 60 different specimens which were tested. The diameter of the concrete cylinders varied from 6 inches (152 mm) up to 24 inches (610 mm), while the diameter of the circular bearing plate was constant at 6 inches (152 mm). The height of the cylinders varied from 9 inches (230 mm) up to 36 inches (914 mm). The variation of geometrical parameters covers the range of 1 < Did 1.5 Hid 4 and 6, which is believed to be the practical range. Note that D6-l2 in Table 3.1 denotes a 6 inch (152 mm) diameter cylinder which is 12 inch (305 mm) high. 3.2 Specimen Preparation The specimens were constructed from two batches of concrete. Both batches were sup plied by a local ready-mix supplier (25 MPa was specified) and had crushed stone as aggregate with a maximum size of 3/4 inch (19 mm). The specimens were cast in sono tube forms with plywood bases, which were stripped after about 7 days. The specimens continued to cure in the laboratory for an additional 50 to 60 days before being tested. 17 Chapter 3. Experimental Study of Transverse Splitting 18 The specimens were always covered by plastic sheets during curing time. As it turned out, Batch A concrete had a cylinder compressive strength of 30 MPa (4350 psi), while Batch B concrete had a compressive strength of only 20 MPa (2900 psi) at testing. 3.3 Instrumentation and Data Acquisition The test set-up used is shown in Figure 3.1 and 3.2. The specimen was placed on the testing table of a 400 kip capacity Baldwin testing machine and loaded through 6 inch di ameter, half-inch thick steel discs. Special care was taken to ensure the specimen was not loaded eccentrically. Two Linear Variable Differential Transducer (LVDT) displacement transducers were mounted on two steel bars which were connected to the top loading (steel) disc. Two transducers (one transmitter and one receiver) of a Portable Ultrasonic Non-Destructive Digital Indicating Tester (PUNDIT) [16] were put at the height where the maximum tensile stresses were expected based on the analytical results. Two small wood pieces were glued to the side face of the specimen to act as supports for the two transducers. An elastic rope was used to tighten the transducers to the specimen, and at same time to give as little transverse constraint to the specimen as possible. The axial deformations of specimens were measured by the two LVDTs. The axial deformation versus the applied loads were plotted with the help of an X- Y plotter. The measured axial deformation included the deformations of the steel discs, plaster and the concrete cylinder so the deformation of the steel discs and plaster had to be eliminated before obtaining the true axial deformation of a concrete cylinder. An important piece of equipment used in the experimental investigation was the PUNDIT [16]. The PUNDIT measures the travelling time of an ultrasonic pulse whose velocity is proportional to the density of the material. The transit time is displayed by three ‘in-line’ numerical indicator tubes. The transit time was recorded by hand at Chapter 3. Experimental Study of Transverse Splitting 19 predetermined load values. If the last digit either remained at a constant value or would oscillate between two adjacent-values with a bias for one of the values, this value was recorded. If the last digit oscillated between two adjacent-values evenly, the mean of the two values was recorded. In this way, an accuracy of 0.05 microsecond was possible. 3.4 Test Observations Load was applied at an approximate rate of 2.75 MPa every minute, continuously until failure. Major test observations are summarized below. 3.4.1 Internal Cracking The internal cracking within concrete cylinders could not been seen by eye so that the PUNDIT was used to detect it. By observing the change in ultrasonic pulse transit time, the internal cracking was determined indirectly. While load was being increased, the transit time of ultrasonic pulse passing through the specimens (i.e., from transmitter to receiver) also increased. The rate of change of the ultrasonic pulse travel time varied at different loading stages. The transit time increased very little or remained almost stable at low load levels, and then increased steadily at medium load levels. Finally, the transit time increased very quickly just before failure. The quick increase of transit time was always a sign that failure would soon occur. Typical load PUNDIT reading (transit time increments) relationships for specimens with H/d = 2 are shown in Figure 3.3. The relationship between load and PUNDIT reading (transit time increment) is ini tially linear. As the velocity of ultrasonic pulses travelling in a solid material depends on the density and elastic properties of the material, the transit time depends on both the path length and the quality of concrete. In the linear stage, the increase in transmit time Chapter 3. Experimental Study of Transverse Splitting 20 is believed to be a result of the transverse deformation of cylinders due to the Poisson’s ratio effect (i.e., the small increase in path length). In the post-linear stage, both the quality of the concrete and the transverse deformation of concrete are believed to have an influence upon the transmit time increment. First cracking is believed to be indicated by the onset of nonlinearity. The non-linearity in the PUNDIT readings for 6 inch (152 mm) specimens is believed to be due to the micro-cracking that occurs when concrete is subjected to uniaxial compression. 3.4.2 Influence of Concrete Confinement It is obvious that the confinement has an influence on the structural behaviour of concrete cylinders subjected local compression on both the top and bottom surfaces. Typical bearing stress versus axial deformation relationships for specimens with H/ d = 2 are shown in Figure 3.4, in which the deformations coming from the steel discs and plaster have been eliminated and the applied loads have been converted to bearing stresses. It can be seen that increased confinement will make the axial stiffness of the spec imens larger and allow higher ultimate bearing strengths to be achieved. In contrast, confinement has a minor influence on the bearing stress at first cracking. See both Figs. 3.3 and 3.4. 3.4.3 Failure Mode In general, the failure of specimens occurred suddenly and with a loud noise. Typically, three or four radial cracks split the specimen into approximately equal segments. Con crete “cones” usually formed in the triaxial compressed zone below the loading plates. Figure 3.5 shows a typical concrete cylinder after failure. The 6 inch (152 mm) diameter and 8 inch (203 mm) diameter specimens were less brittle than other specimens. See Figure 3.4. In the 8 inch (203 mm) specimens surface Chapter 3. Experimental Study of Transverse Splitting 21 cracks were observed before failure. That is, the specimens resisted additional load after exterior cracks were observed. Note that the 8 in (203 mm) specimens were predicted to crack on the outside before the inside. See Figure 2.3. The 6 inch (152 mm) specimens were standard cylinders that failed in compression. : ISZ C’ t3 t’. .3 C I— L’3 CAZ C C C C I--s I t’. L3 I--i t’.Z I--i I--i I , I— —1 - t.Z I.- I_ t\1 : t.. I.-. I— .• C C i— ) . C I—i I- C : t’Z t’. I— I--i :‘ 00 00 I—i : CT .) k1 C t’ I—i ;‘ © © —4 © 00 ‘3 I — I— ,— I—i 00 00 00 t’3 t t’.) C — C.3 CZ C3 I— I I ç C.C c ,— oo i— I. .• t. —1 t.) 00 00 ) t’Z : t\) I C —4 . t c I t\) t t t’.D —.‘ ‘.— —‘ C) CD C]) O:s CD , CI) c-I I-s I-” L2 I-s ‘CD CD o * s CD CI) CD I-s I CD tN3 F ei Chapter 3. Experimental Study of Transverse Splitting 23 Table 3.1: (cont’d) Summary of experimental results. Specimen Designation* D14-9-1 D14-9-2 014-9-3 014-18-1 014-18-2 D14-18-3 D14-36-1 014-36-2 D14-36-3 018-9-1 018-9-2 D18-9-3 D18-12-1 018-12-2 D18-12-3 D18-12-4 D18-18-1 018-18-2 018-18-3 D18-36-1 D18-36-2 018-36-3 024-12-1 D24-12-2 024-12-3 Concrete Batch B B B B B B B B B B B B A A A A B B B B B B A A A Bearing Stress at Cracking (MPa) 15.9 14.6 17.1 26.8 25.6 24.4 34.2 28.1 29.3 26.8 19.5 20.7 26.8 26.8 29.3 29.3 34.2 31.7 29.3 37.8 37.8 39.1 32.9 35.4 34.1 *D612 denotes 0 = 6 in; H 1 MPa = 145 psi 1 in = 25.4 mm = Max. Bearing Stress (MPa) 32.5 31.7 33.2 48.1 50.0 48.8 58.1 53.2 51.7 44.4 42.2 44.4 40.2 43.4 42.0 41.7 61.0 55.6 56.9 71.5 68.3 68.3 59.8 65.5 62.0 12 in Chapter 3. Experimental Study of Transverse Splitting 24 d=6 in. Displacement r1 ansducer Plaster Specimen UNDIT Steel disc D =varies Figure 3.1: Schematic of test set-up. H=Varies PUNDIT receiver Chapter 3. Experimental Study of Transverse Splitting Figure 3.2: Photograph of test set-up. 25 Chapter 3. Experimental Study of Transverse Splitting 26 80 70 Cl) Cracking • 60- D24-12-- G) -4-i C,) 50 - D18-12 40 / - 30 D12-12 D8-12 - D6-12 20 - 1 io4 (: 0 I 0 1 2 I 3 4 5 6 7 PUNDIT Transmit Time Increment (microseconds) Figure 3.3: Typical relationships of load vs. PUNDIT readings (transit time increments) for the case of H/d = 2. Chapter 3. Experimental Study of Transverse Splitting 27 80 60U) Cl) 50 D24-12 - Dl 8-12 ctI 40 - D12-12 30 D8-12 - D6-12 20 - 10 - 0 0 I 0.5 I 1 I 1.5 Displacement (mm) Figure 3.4: Typical relationships of load vs. axial deformation for the case of H/d = 2. Chapter 3. Experimental Study of Transverse Splitting I Figure 3.5: Photograph of the typical failure mode of a concrete cylinder. 28 Chapter 4 Bearing Strength of Concrete Compression Struts 4.1 Comparison of Measured and Predicted Cracking Loads Figure 4.1 and 4.2 compare the measured cracking loads with the linear elastic predic tions. Figure 4.1 shows the influence of confinement (D/d) for various height, while Figure 4.2 shows the influence of height (H/d) for various degrees of confinement. The tensile strength used in the prediction, sidering that only one parameter f = f/13, was chosen to give the best fit. Con (f/f) is adjusted, there is a very good correlation between the analytical prediction and the experimental results regarding the influence of Did and H/d on the bearing stress to cause first cracking. It should be pointed out that the large deviation of some experimental data points from analytical prediction in the case of H/cl = 2 of Figure 4.1 is not surprising. As discussed in Section 2.4 (Figure 2.3) the analytical results predict that there is a range of first cracking inside as well as first cracking outside. The ratio of D/d = 1.33 for the 8 inch (203 mm) diameter and 12 inch (305 mm) height specimens lies in the range of first cracking outside. Test observation also confirmed that some external cracking occurred before the specimens failed. The early external cracking is believed to be the main reason for this deviation from the analytical prediction. It should be remembered that the predictions shown in Figure 4.1 represent that of first cracking inside. 29 Chapter 4. Bearing Strength of Concrete Compression Struts 4.2 30 Comparative Study of Ultimate Bearing Strength While this study is concerned primarily with the bearing stress to cause initial transverse cracking, the test results also give valuable information for ultimate bearing strength. A comparison of the bearing stress which caused failure in the present tests and the bearing strength predicted by the ACT Building Code [9] was therefore carried out. According to the ACT Building Code the maximum bearing strength of concrete is 0.85f, except when the supporting surface area A 2 is wider on all sides than the loaded area A , the bearing 1 strength is multiplied by , 1 / 2 A A but not more than 2. Note that D/d = i.JA / 2 Ai. Surprisingly it was found that the ACT approach is unconservative in the range 1.5 < D/d < 3.5. See Figure 4.3(a). Thus, it seems that further study on ultimate bearing strength is warranted. 4.2.1 Previous Studies of Bearing Strength As bearing strength is important in the design of many concrete structures, a great number of investigations have been done on the bearing strength of concrete. It is generally believed by most investigators that the principal variables influencing the concrete bearing strength are: 1. The geometry of loaded area and specimen; 2. the nature of the supporting bed under the specimen; 3. the supported area of specimen; 4. the concrete strength, and 5. the specimen size. The main conclusions based on the previous experimental work are that: Chapter 4. Bearing Strength of Concrete Compression Struts 31 1. The failure is due to the punching out of an inverted cone of concrete beneath the loaded area and the radial pressures exerted by this cone split the specimen; 2. The ratio of bearing strength over concrete compressive strength increases contin uously for increasing confinement, ie. increasing the ratio of D/d; 3. There are cracking loads and failure loads which are sometimes different and depend upon the size of bearing plates and the height of specimens; 4. The larger the loading plate size (smaller D/d) and smaller the relative height of specimen, the greater is the difference between failure load and external cracking load; 5. Specimens supported on a compressible bed give less bearing strength than for similar specimens on a rigid support medium; 6. Specimens subjected to localized forces from opposite ends exhibit lower bearing strength compared to localized compression from one end only; 7. The higher the compressive strength of concrete, the lower the ratio of ultimate bearing strength to concrete compressive strength; 8. The ratio of bearing strength over concrete compressive strength decreases with the increase in specimen size, Conclusions (3)—(8) are mainly from Niyogi’s tests [17]. However, there are different opinions regarding the influence of specimen height. Some authors considered that bearing strength over concrete compressive strength fb/f’ is independent of specimen height [18-26]. But Niyogi [17] concluded that for double punch tests the ratio f&/f increases with the increase in the specimen height and supporting Chapter 4. Bearing Strength of Concrete Compression Struts 32 area. For single punch tests the bearing strength decreases with increasing specimen height in some ranges of geometrical parameters, and in the reverse (the bearing strength decreases with decreasing specimen height) in other ranges of geometrical parameters. Double-punch tests involves the specimens being loaded symmetrically using bearing plates on both the top and bottom surfaces. In single-punch tests the base of the specimen is set directly on the lower platen of the testing machine and the upper platen is used to load the bearing plate. Figure 4.3(b) compares the ACT Building Code bearing strength prediction with pre vious tests conducted by Shelson [19], Au and Baird [20], Douglas and Trahair [23], Middendorf [24], Hawkins [26], Chen and Trumbauer [27], as well as Niyogi [17]. As shown in Figure 4.3(a) for the present investigation, this figure also indicates that the ACT prediction is unconservative for quite a number of data points. After the unconser vative data points were carefully examined, it was found that the major reasons for the lower ratios of bearing strength to concrete compressive strength f are the influence of concrete compressive strength itself, size effect and loading geometry. These are discussed in detail below. 4.2.2 Influence of Concrete Compressive Strength The Commentary to ACT 318-89 Section 10.15 makes reference to Hawkins [26] who suggested the following expression for the bearing strength of concrete fb where f = J1(/A 50 / 2 A f+1 — 1) (4.1) is in psi units. That is, the enhancement in bearing strength due to confining concrete is proportional to the tensile strength of concrete. In the ACT Building Code this equation has been simplified so that the bearing strength enhancement is proportional to Chapter 4. Bearing Strength of Concrete Compression Struts 33 the concrete compressive strength. Figure 4.4 compares the ACT approach with Equation 4.1 for different concrete strengths. Note that when the concrete compressive strength is high than 5,000 psi (34.5 MPa), the original expression suggested by Hawkins [26] gives lower bearing strengths than the ACT Building Code. The low strength reduction factor of 0.70 used in the ACT Building Code for bearing on concrete partly compensates for this over-simplification. 4.2.3 Influence of Size Effect A second factor which contributes to lower bearing strengths in Figure 4.3 is size effect. The ACT Building Code approach is based on tests of relatively small specimens, while data points shown in Figure 4.3(a) are from large specimens. Since bearing failures involve fracture of concrete or crack propagation process due to indirect tension, there is a significant size effect involved. Based on the result of 42 double-punch tests, in which the specimens had the geo metrical proportions D = H = 4d, Marti [28] concluded that the size effect was in good agreement with Baant’s (nonlinear fracture mechanics) size effect relation [29, 30]. Thus the bearing strength of concrete is proportional to fo cc (4.2) )Da where D is the maximum aggregate size and .X is an empirical constant, which was found to be 38.0 and 68.5 in Marti’s two series of tests [28]. For the geometrically identical specimens that were made with the same concrete, f = 3425 psi (23.6 MPa), and ranged in size from 8 to 128 times the maximum aggregate size (i.e., the size varied by a factor of 16), the bearing strength decreased by a factor of 1.6 from the smallest specimen to the largest specimen. Chapter 4. Bearing Strength of Concrete Compression Struts 4.2.4 34 Influence of Loading Geometry The most important factor to contribute to the actual bearing strength being lower than the ACT Building Code prediction is the geometry of loading. The majority of the test results, upon which the ACT Building Code procedure is based, are from single-punch tests. In the present investigation, as well as some previ ous studies, the specimens were loaded by double-punch method. In single punch tests transverse tensile stresses are created at one end only and the lower platen of the testing machine restrains the expansion of the specimen. In double-punch tests the compression at both ends of the specimen results in higher tensile stresses at the mid-height of the specimen as shown in Figure 4.5(b) and the restraint is eliminated. When the specimens are relatively tall [Figure 4.5(c)], the double-punch tests produce similar results as the single-punch tests except for the restraint issue [Figure 4.5(a)]. Figure 4.6 illustrates that the accuracy of the ACT bearing strength depends on the H/d ratio for double-punch tests. In addition to the influence of specimen height discussed above, the data points in Figure 4.3 also show that the effect of confinement is not as large as the ACT procedure predicts in the range of small D/d ratios, where quite a number of test results either are unsafe or have a very low safety margin. Tt is interesting to compare Figures 4.3 and 4.6 with Figure 2.4. Most lower bearing strength data points are located in the range 1 < H/d < 3 and 1.5 < D/d < 3.5, exactly where the analytical results indicate the least bearing stress to cause first cracking. 4.2.5 Summary On the basis of present investigation, some conclusions which supplement the conclusions of previous studies (Section 4.2.1) are that: Chapter 4. Bearing Strength of Concrete Compression Struts 35 1. Bearing failure is due to transverse splitting, provided that the supporting surface area is wider on all sides than the loaded area; 2. The bearing stress at external visible cracking could be less than or equal to the bearing stress at failure depending upon whether first cracking occurs outside or inside; 3. Bearing strength depends, to a major extent, upon the loading geometry, which includes the loading method, surrounding confinement (D/d) as well as specimen heights (H/d); 4. Due to either higher concrete compressive strength or size effect or loading geome try, the bearing strength according to the ACT Code can be unconservative. 4.3 Design Recommendations for Concrete Compression Struts Based on the results of the analytical and experimental studies presented in Chapter 2 and 3, it is suggested that when designing deep members (disturbed regions) without sufficient distributed reinforcement to ensure redistribution after initial cracking, the maximum bearing stress should be limited to fb O.6f(1 + 2a/3) (4.3) where = O.33(/ 0.33( — — 1) 1) 1.0 1.0 The ratio h/b, which represents the aspect ratio (height/width) of the compression strut, should not be taken less than 1.0 (i.e., 3 0). The parameter a accounts for Chapter 4. Bearing Strength of Concrete Compression Struts 36 the amount of confinement, while the parameter j3 accounts for the geometry of the compression stress field. The lower bearing stress limit of O.6f is appropriate if there is no confinement (i.e. 2 /A /A 1 1) regardless of the height of the compression strut, as well as when the com pressive strut is relatively short (i.e. h/b The upper limit of Equation 4.3, O.6f x 3 1) regardless of the amount of confinement. = l.8f, is chosen to correspond approximately to the upper limit of bearing strength given in the ACT Building Code. The interaction of confinement and geometry (aspect ratio) is chosen so as to give a reasonably simple expression and yet correspond well with the finite element predictions and the exper imental results. Figure 4.7 compares Equation 4.3 with the finite element prediction, while Figure 4.8 compares predictions from Equation 4.3 with the experimental results. As mentioned previously, concrete bearing strength is actually proportional to the concrete tensile strength. If the concrete compressive strength is significantly greater than 5,000 psi (34.5 MPa), a more appropriate limit for the bearing stress is fb where f in MPa units. by 72a/3. 0.6f + 6aç8fl (4.4) If psi units are used, the 6cr6 in Equation 4.4 should be replaced Chapter 4. Bearing Strength of Concrete Compression Struts 37 80 60 - Cl) U) Experiment&: Cracking • Failure o 40 Cl) 20 Q) H/d=1.5 I Predicted Cracking ft’=1.52MPa 4 • 0 2 4 s 80 3 H/d=2.0 40 20 01 2 Predicted Cracking ft’=2.3OMPa (Concrete Batch A) 4 3 80 o 60- 0 H/d=3 8 Predicted Cracking ft’=1.52 MPa 01 2 4 3 80 8 60 H/d=6 - 40 20 0 Predicted Cracking ft’=1.52MPa - 2 3 Did Figure 4.1: Comparison of experimental results and analytical predictions: influence of confinement for various heights. Chapter 4. Bearing Strength of Concrete Compression Struts 38 80 60 - Cl) Cl) Cl) 40 Experimental: • Cracking o Failure D/d=1.67 - 20edicted Cracking ft=1.52MPa 00 1 2 3 4 5 6 7 8 80 60 40 D/d=2.0 - 0 6 0 - 20 - “%-“edied Cracking 1.52 MPa 1 2 3 4 5 6 7 8 80 8 D/d =3.0 60- 0 8 40- 20 \_JedictedCracking 1.52 MPa 1 2 3 4 5 6 7 8 H/d Figure 4.2: Comparison of experimental results and analytical predictions: influence of height for various degrees of confinement. Chapter 4. Bearing Strength of Concrete Compression Struts 39 6 f,C 54. 3- . II 2- $ $ I S 11--, (a) 01 2 3I 2 3 ACt Code 4 5 6 f,C 5 4 3 2 1 (b) 0 1 1 / 2 A A Figure 4.3: Comparison of ACT Building Code bearing strength prediction with experi mental results: (a) present investigation and (b) previous tests. Chapter 4. Bearing Strength of Concrete Compression Struts f3 -‘b / 40 A B C 2 D ACI code S S S S S F •1 —— I S S S S F A B C D = = = = 2,500 psi (17 PAPa) 5,000 psi (34.5 PAPa) 7,500 psi (51.7 PAPa) 10,000 psI (69 MPa) 0 1 2 3 Figure 4.4: Comparison of ACT Building Code bearing strength with Hawkins’s suggested equation for various concrete strengths. Chapter 4. Bearing Strength of Concrete Compression Struts 41 — — — II (a) (b) (c) Figure 4.5: Influence of specimen height on ultimate bearing strength: (a) the force flow in a single-punch test; (b) the force flow in a double-punch test and (c) the force flow in a tall double-punch test. Chapter 4. Bearing Strength of Concrete Compression Struts 42 3 b(exp) b(AcI) 2- 1 0 I I 0 1 2 3 4 I I 5 6 7 8 9 10 H/d Figure 4.6: Influence of height on accuracy of ACT Building Code bearing strength pre diction of double-punch tests. Chapter 4. Bearing Strength of Concrete Compression Struts 43 — f; H/d =4 2 -j H/d=3 Hid = 1 o I I I 4 3 2 1 A JA 1 D/d= / 2 — —. ,.-13Id4_-— 2 - - D/cI= 1 0 0 I I I 1 2 4 3 H/d Figure 4.7: Comparison of suggested design equation with the analytical results. Chapter 4. Bearing Strength of Concrete Compression Struts 44 3 (exp) (pred) 2 . • : .4 I • • • • . : . . . . I . . . : S I 01 2 3 4 5 A A 1 J 4 D/d= / 2 Figure 4.8: Comparison of predictions from suggested design equation with experimental results. Chapter 5 Shear Design of Deep Pile Caps 5.1 Introduction This chapter deals with shear design of deep pile caps. The main objective is to de velop a rational design procedure for deep pile caps based on the design recommendation proposed in Chapter 4 for concrete compression struts. The methods currently used in the design of reinforcement for deep pile caps are mainly classified as two categories. Some designers assume that a cap acts as a beam spanning between the piles and design the reinforcement on the basis of simple bending theory [9]. Others assume that the cap acts as a truss, the column load is transmitted to the piles through axial struts and the tension force required for equilibrium is taken by reinforcement [3], 32]. A literature survey for previous theoretical and experimental studies on pile caps can be found in Ref.[33]. Bending theory and truss analogy may lead to similar amounts of steel, but suggest different arrangements in plan: the former leads to a uniform grid whereas the latter favours bands of steel running between the piles. In addition, the two methods have different anchorage requirements. Nominal anchorage is required beyond the centre-lines of the piles for bending theory. Truss analogy, on the other hand, requires that full anchorage lengths are provided beyond the centre-lines of the piles. A sectional method is usually used for the shear design of deep pile caps. The shear stresses on the code defined critical sections are limited to the shear strength contributed 45 Chapter 5. Shear Design of Deep Pile Caps 46 by concrete. However a previous experimental study [11] revealed that a sectional method is not appropriate for the shear design of deep pile caps. Further investigations on the use of sectional methods for the design of deep pile caps are carried out in this chapter. This chapter is composed of six parts. In the first part, the ACT Code approach for design of pile caps is summarized. Some practical design examples are presented, which show that the ACT Code approach is inadequate for shear design of deep pile caps. The definition of deep pile caps is also given in this part. The background of the CR51 (Concrete Reinforcing Steel Institute) approach for shear design of deep pile caps is then discussed. The CRST suggested allowable concrete shear stresses for critical beam sections across the width of the pile cap at the face of the column and for two-way slab punching shear on the periphery (faces) of the column are summarized. Thirdly, the Canadian Code (CSA A23.3-M84) approach of using strut-and-tie mod els to design deep members is summarized. However, there is actually no shear design procedure for deep pile caps in the Canadian concrete code, i.e., there is no appropriate strength criterion for compression struts confined by plain concrete. By applying the de sign equation developed in Chapter 4 for the bearing stress limit of concrete compression struts, two design methods are proposed for the shear design of deep pile caps. Previous experimental work for pile caps and some important conclusions suggested by various investigators are summarized. A comparative study is carried out in the fifth part of this chapter. The predictions using various design procedures (ACT Code, CRSI and proposed methods) are compared with the previous experimental results. Finally, some conclusions are drawn at the end of this chapter. Chapter 5. Shear Design of Deep Pile Caps 5.2 47 ACT Code Approach for Pile Cap Design The ACT Building Code (ACT 318) [9, 34-35] uses a sectional force approach for the design of pile caps regardless of the depth. The procedure involves three separate steps: (1) shear design, in which the depth of the slab or pile cap is chosen so that the concrete contribution to shear resistance is greater than the shear applied on the code defined critical section; (2) fiexural design, in which the usual procedures for beams are used to determine the required amount of longitudinal reinforcement at the critical section for flexure, and; (3) a check that the bearing stresses on the column and piles do not exceed the bearing strength. The procedure to check bearing strength is to satisfy that bearing on concrete at contact surface (e.g., the top of the pile cap under the column, the bottom of the pile cap above the pile) does not exceed the code bearing strength (ACT 318-89: 15.8.1.1) [9]. This check is to ensure that the concrete does not crush at these locations. For moment and shear calculations the pile force may be considered concentrated at the centre of the pile. Flexural design is straight forward. The critical moment is equal to the pile reaction times distance from pile centre to face of a concrete column (Clause 15.4.2). For example, the moment at section m — m in Figure 5.1 is equal to the product of the reactions from three left side piles and the distance s. The required amount of longitudinal reinforcement is provided to resist this moment, similar to the procedure used for the fiexural design of slabs or beams. 5.2.1 Code Procedure for Shear Design Shear design involves checking one-way beam shear and two-way punching shear (see Figure 5.1). For one way beam shear, the critical section is measured at a distance d from the face of supported member (column) (Clauses 15.5.2 and 11.12.1.1 of ACI 318-89). For Chapter 5. Shear Design of Deep Pile Caps 48 two way punching shear, the critical section is taken at a distance d/2 out from perimeter of the column, or the pile (Clause 11.12.1.2), where d is effective height measured from extreme compression fibre to centroid of longitudinal tension reinforcement. Any pile whose centre is located d/2 or more outside the critical section shall be considered as producing shear on that section, where d is diameter of pile. In Figure 5.1 the shear on the section n-n is introduced by the reactions from three right side piles. All eight side piles produce shear on the critical section for two-way shear at d/2 from the perimeter of the column. Reaction from any pile whose centre is located d/2 or more inside the critical section shall be considered as producing no shear on that section (Clause 15.5.3). Figure 5.2 shows a two pile cap, for which the pile produces no shear on the section n-n. For intermediate positions of pile centre, the pile reaction is assumed as distributed linearly across the pile diameter d in the direction shear is accumulated (Clause 15.5.3.3). For pile caps where the pile reactions are located either under the column or outside the critical sections (d from the face of the column for one-way shear or d/2 for two-way shear calculation), the code procedure for shear design is straight forward. It is proposed herein that such a pile cap, as shown in Figure 5.1, can be defined as a slender pile cap. However, a great number of pile caps used in practice are not slender pile caps. These pile caps are herein called ‘deep’ pile caps. The effective height d of deep pile caps is equal to or greater than the distance from the centre line of the closet pile to the face of the supported column. The ACT Code approach becomes less transparent and inadequate for the shear design of deep pile caps. This inadequacy is illustrated in the following examples. Chapter 5. Shear Design of Deep Pile Caps 5.2.2 49 Inadequacies of ACI Code Procedure for Shear Design of Deep Pile Caps Figure 5.2 shows a deep two-pile cap, for which the piles are located within the code defined critical section from the face of the column (i.e., within the distance d from the column face). There are no code provisions specifically for the shear design of such a pile cap. Hence ACT 318-89 Commentary Section 15.5.3 says: “when piles are located inside the critical sections d or d/2 from face of column, analysis for shear in deep flexural members in accordance with Section 11.8 needs to be considered.” In Clause 11.8 some guidance for deep beams is given. A critical section is established at 0.5 a from face of support, where a is shear span and is defined as a distance from a concentrated load to face of support. It becomes hard to define what is the support (the pile or the column?). Based on the guidance for deep beams, it could be considered that piles are supports and column load is concentrated load. In Figure 5.2, the critical section j-j is defined by this interpretation. The code equations 11-29 or 11-30 are applicable and thus shear strength on that section can be calculated. However, if the cap depth is increased, this defined critical section could eventually cut through the column so that again no code provisions can be applied. Figure 5.3 shows a deep three-pile cap. If the piles are still considered as supports, it is difficult to determine the shear span a (i.e., where is the concentrated load ?). It should also be noted that there are no code provisions whatsoever applicable to perform punching shear calculations when the three piles are located within the square critical section shown in the figure. Figure 5.4 shows another example that demonstrates the inadequacy of the ACT Code approach. Two four-pile caps are identical except that one has a square column and the other has a circular colunm. The two columns have the same areas. For the pile cap Chapter 5. Shear Design of Deep Pile Caps 50 that has the square column and four circular piles, as shown in Figure 5.4(a), the square critical section d/2 away from the perimeter of the column just has its corners at the centres of piles. The ACT Code assumption that the pile reaction is distributed linearly across the depth, d (pile diameter) (Clause 15.5.3.3), can be utilized for punching shear calculation. If the code is followed literally, half of all pile reactions will introduce shear on the critical section. On the other hand, if the square column is replaced by a circular column with the same area and correspondingly the critical section is circular rather than square, [see Figure 5.4(b)], the pile reactions are now totally located outside the critical section so that the critical section is subjected to the shear from the full value of pile reactions. Consequently, the prediction for load capacity of the pile cap with the square column is as much as almost two times that of the pile cap with the circle column, based on the two way punching shear calculation. The sensitivity of the predicted capacity to the shape of the colunm is a demonstration that a sectional approach is not appropriate for deep pile caps. 5.2.3 Comparison of Different ACI Code Editions In the above section the 1989 ACT Code [9] procedures for shear design of pile caps have been summarized, and it has also been pointed out that these procedures are transparent and directly applicable to slender pile caps. From the 1977 edition to the 1989 edition, the ACT Code have made some changes in shear design clauses which do not affect the shear design of slender pile caps, but do affect the shear design of deep pile caps. These changes are summarized below. In ACT 318-77 [34], Clause 11.1.3.1 stated that for nonprestressed members, sections located less than a distance d from face of support may be designed for the same shear as that computed at a distance d. In the accompanying commentary to ACT 318-77, a statement is made that if the shear at sections between the support and a distance d Chapter 5. Shear Design of Deep Pile Caps 51 differs radically from the shear at distance ci, the shear at the face of the support should be used. In ACT 318-83 [35], the contents of the cormnentary have been incorporated into the formal body of the code, which means that the requirement that the shear capacity at the face of the support should be evaluated is imposed when there is a concentrated load between the face of support and the code defined critical section. From ACT 318-83 to ACT 318-89 no additional changes have been made to Clause 11.1.3. In comparison with the previous ACT Code editions, more explanations about Clause 11.1.3 have been added to the Commentary (ACT 318R-89). Something new in the ACT 318R-89, which is relevant to this study, is that the shear at the face of the column should be investigated for footings supported on piles when the pile reaction is located within a distance d from the face of the column. The ACT Building Code procedures for two-way shear have not been modified recently. The critical section remains at d/2 from the perimeter of the column regardless whether there is a concentrated load applied within the critical section. 5.3 CRSI Approach for Shear Design of Deep Pile Caps As there are actually no specific procedures in the ACT Code for the shear design of deep pile caps [36], formulas are presented in the CR51 Handbook [36, 37] for allowable shear on concrete as a function of the ratio w/d, where w is the horizontal distance from face of column to centre of pile reaction and ci is effective depth of pile cap. Based on recognition of considerable reserve shear strength observed in deep beam tests [38], allowable shear strength on concrete is assumed to increase rapidly from ‘diagonal tension’ to ‘pure shear’ as the distance w varies from d to 0 for one-way shear and from d/2 to 0 for two-way punching shear. In addition, the critical sections for shear calculation of deep pile caps Chapter 5. Shear Design of Deep Pile Caps 52 are specified. For one-way shear of deep pile caps, the critical section is at the face of the column. The formula for shear strength calculation, which is a modification of the ACT Equation 11-30, is V = (--)(3.5 — 2.5)(1.9iTh+ 2500p where (w/d) < 1.0, 1.0 > M/Vd> 0 and f is in psi the formula is further reduced for the case of p V which gives (d/w) = (-)(3.5 — )bd 10Jbd (5.1) unit. In the CRSI handbook [37] 0.002 and f’ = 3000psi to 2.5#)(1.9.Th + O.l/)bd 1O/bd when the product of the last bracketed expressions (5.2) = 2’J and 1. Figure 5.5(a) summarizes the one-way shear calculations for deep pile caps. The CRSI critical section for two-way shear in deep pile caps is defined at the perime ter of the column face. The formula for the allowable shear stress at the critical section, which is a modification of the ACT Equation 11-36, is V. = d 0 (-)(1 + )(4/) <32./jb where for a square column of dimension c, b 0 equals 4 x c. When d/2w (5.3) = 1, the CRST formula leads to the same expression as the ACT approach where the critical section is at d/2 from the column perimeter (b 0 equals 4 x (c + d)). Figure 5.5(b) summarizes the CRSI two-way shear calculations for deep pile caps. The flow chart summarizing ACT Code and CRSI approaches is presented in Figures 5.6. Chapter 5. Shear Design of Deep Pile Caps 5.4 53 Strut-and-Tie Model Approach An alternative approach which can overcome the drawbacks of ACI approach for the shear design of deep pile caps is strut-and-tie models which consider the complete flow of forces within a pile cap rather than the forces at any one particular section. The internal force flow indicates that the vertical column load is transmitted to the piles by inclined compression struts and in order to prevent the piles from spreading apart, tension ties (reinforcement) must be provided. Assuming reinforcing steel bars are properly anchored in the nodal zone, it is believed that a “shear failure” of a deep pile cap will occur when a concrete compression strut fails prior to yielding of tension ties. It should be noted that truss models previously proposed for the design of pile caps, as mentioned in the introduction of this chapter, have been used only for “flexural design”. 5.4.1 CSA Approach for Deep Beams Strut-and-tie models have been adopted in the Canadian Concrete Code (CSA A23.3M84 [10]) for the design of deep beams [39]. The internal flow of forces in a simply supported deep beam can be approximated by the truss model shown in Figure 5.7. The truss is composed of several components. The zones of unidirectional compressive stress in the concrete are modelled as compression struts, while tension ties are used to model the principal reinforcement. The regions of concrete subjected to multi-directional stresses, where the struts and the ties meet (the joints of the truss) are represented by ‘nodal zones’. The Canadian Code [10] requires that the concrete compressive stresses in the nodal zones not exceed the following limits (Section 11.4.7.5): (a) 0.85f in nodal zones bounded by compressive struts and bearing areas; Chapter 5. Shear Design of Deep Pile Caps 54 (b) O.75f in nodal zones anchoring only one tension tie; and (c) 0.60f in nodal zones anchoring tension ties in more than one direction, The tension tie reinforcement is distributed and anchored over an effective area of concrete at least equal to the tensile tie force divided by the stress limits given above (Sections 11.4.7.4 and 11.4.7.6). The key part of this approach is the establishment of stress limit for the compression struts based on the work by Vecchio and Collins [40]. The code requires that the concrete compressive stress f2 in the struts shall not exceed f2 f2rnax = f2ma (Section 11.4.7.3), i.e. 0.8 +170c 1 (5.4) where f2ma shall not exceed f unless the concrete is triaxial confined and e is determined by considering the strain conditions of the concrete and the reinforcement in the vicinity of the strut. This approach is rational for deep beams, however extending this approach to deep pile caps seems questionable. In deep pile caps, compression struts are usually unreinforced and confined by surrounding plain concrete. Thus the Canadian Code has no provisions specifically for the shear design of deep pile caps. In the Canadian Concrete Design Handbook [41], Suter and Fenton have ignored shear calculations for deep pile caps and incorporated a more conservative flexural model to predict deep pile cap behaviour. 5.4.2 Proposed Design Method (1) The stress limit for compression struts in the Canadian Code was established by testing reinforced concrete panels [40], which were subjected to in-plane stresses. The stress limit is appropriate for planar reinforced concrete members and is not applicable to three dimensional pile caps where compression struts are confined by plain concrete. Chapter 5. Shear Design of Deep Pile Caps 55 In Chapter 4 the design equations for the bearing stress limit of compression struts confined by surrounding plain concrete were proposed, based on the analytical and ex perimental investigation. For convenience, the equations are again written here: = 0.6f(1 + 2afl) (5.5) where a 0.33(J4- 1 = = O.33( — 1) 1.0 1.0 If the concrete compressive strength is significantly greater than 5,000 psi (34.5 MPa), the limit for the bearing stress is fb where f 0.6f + 6a/3/ (5.6) is in MPa units. If psi units are used, the 6cq3 should be replaced by 72a. These equations are directly applicable to the idealized compression struts as studied in Chapter 2 to Chapter 4. See Figure 5.8(a). However, the geometry of compression struts in deep pile caps may be different from that of the idealized struts. Some factors which may affect the determination of geometrical ratios of D/d and h/b have to be taken into consideration. One situation that could occur in design of deep pile caps is shown in Figure 5.8(b) and (c). In addition to different loading areas on two ends, the compression strut does not have a constant cross-section along its length. Hence new ratios of both h/b and D/b are needed in using the design equations. Herein a weighted average method is proposed to calculate the new geometrical ratios. These are Chapter 5. Shear Design of Deep Pile Caps D b — — h b 56 /b + 2 1 (D 2 ) /b (D ) D + 1 / 2 b D 5 7 (h/b + (h/b 2 ) 1 ) 2 +h/b 1 h/b 2 58 — — where b , b 1 2 and D 2 are loading geometry of the compression struts. See Figure 5.8. , D 1 The allowable bearing stress limit of the compression strut in Figure 5.8(a) is greater than that in Figure 5.8(b). And again the allowable bearing stress limit of the one in Figure 5.8(b) is less than that in Figure 5.8(c). It should be noted that the predicted bearing stress fb for each compression strut is on the end with smaller loaded area. As the force in a strut is constant, the bearing stress on the other end can be determined from equilibrium. By incorporating the developed bearing stress limit for compression struts confined by plain concrete into a simple three-dimensional strut-and-tie model, one design procedure is proposed. As strut-and-tie models suggested in the Canadian Concrete Code (CSA A23.3-M84) [39] for deep beam design, the first design procedure proposed here for deep pile caps does not split the procedure into “flexural design” and “shear design,” which are traditionally used for the design of slender members. The procedure involves several steps to develop an equilibrium force system. The initial pile cap dimension can be chosen based on previous design experience or with the help of published design aids. The effective depth d of a pile cap can be determined from the concrete cover requirement and then the location of reinforcement can also be determined. The strut-and-tie model is drawn and the geometric parameters of compression struts are then calculated. The bearing stress limit on the nodal zones and the corresponding force in the compression struts can be calculated by using Equations 5.5 or 5.6, 5.7 and 5.8. The horizontal components of the inclined compression strut forces are resisted by providing properly anchored reinforcement. The Chapter 5. Shear Design of Deep Pile Caps 57 sum of vertical components of the inclined compression strut forces is the calculated load capacity of the pile cap. Prediction examples can be found in Appendix C. 5.4.3 Proposed Design Method (2) The second design method is a simplified procedure of the first design method and is presented in a more traditional way with a separated “flexural design” and “shear design”. In the second design method, the calculation of the bearing stress limit is based on some simple parameters such as the effective depth of the pile cap and the dimensions of the column and the pile rather than more complicated geometric parameters of the compression strut. As a result, this simplified procedure is more easy to apply in the design of deep pile caps. The traditional truss analogy principle is used for “flexural design”. The fiexural capacity depends strongly on the inclination of the compression strut, which is defined by the location of the nodal zones. The truss model used in this procedure is that the lower nodal zones are located at the center of the piles at the level of the longitudinal reinforcement, while the upper nodal zone is assumed to be at the top surface of the pile cap at the column quarter point. The “shear design” of a deep pile cap involves applying Equation 5.5 or 5.6 to limit the concrete stresses in compression struts and nodal zones to ensure the tension tie reinforcement yields prior to any significant diagonal cracking in the plain concrete com pression struts. The ratio A 1 / 2 A is identical to that used in the ACT Code to calculate bearing strength. To calculate the maximum bearing stress for the nodal zone below a column, where two or more compression struts meet, the aspect ratio of the compression strut can be approximated as (5.9) Chapter 5. Shear Design of Deep Pile Caps 58 where d is the effective depth of the pile cap and c is the dimension of a square column. For a round column, the diameter may be used in place of c. To calculate the maximum bearing stress for a nodal zone above a pile, where only one compression strut is anchored, the aspect ratio of the compression strut can be approximated as (5.10) where 4, is the diameter of a round pile. A general shear design procedure for deep pile caps can be accomplished as follows. First, the initial pile cap depth is chosen using the ACT Code one-way and two-way shear design procedures. In the case of one-way shear, the critical section should be taken at d from the column face, and any pile force within the critical section should be ignored (i.e., the ACT procedure prior to 1983). Secondly, the nodal zone bearing stress should be checked using above described procedure. If necessary, the pile cap depth may be increased (/3 increased), or the pile cap dimensions may be increased in order to increase the confinement of the nodal zones (increase a), or else the bearing stresses may need to be reduced by increasing the column or pile dimensions. Prediction examples showing this method can be found in Appendix D. 5.5 Summary of Previous Pile Cap Tests In order to evaluate the various pile cap design procedures quantitatively, these pro cedures will be used to predict previously tested pile caps. In this section previous experimental work on pile caps is summarized. Hobbs and Stein (1957) [42] tested 70 one-third scale models of two-pile caps to verify their design method, which is based on analytically determined bending stress distribu tions on vertical planes through two-pile caps. The specimens had various amounts of Chapter 5. Shear Design of Deep Pile Caps 59 either straight or curved reinforcing bars, which were anchored by a number of different methods. During the test, the first cracks consistently appeared at or close to the pile cap mid-span. Later, diagonal cracks, which originated at the top of the piles, propagated to the loading plate and were usually followed directly by failure. Deutsch and Walker (1.963) [43] tested four two-pile caps. The pile caps were designed with the same centre to centre pile spacing 42 inches (1067 mm), the same width 15 inches (381 mm), the same stub pile dimensions 10 x 10 in. (254 x 254 mm) and the same stub column dimensions 6.5 x 6.5 in. (165 x 165mm). The depth of the pile caps and amount of reinforcing steel were varied in an attempt to investigate the structural action of pile caps and to compare different design methods (sectional force methods and the truss analogy with compression struts inclined at 35 and 45 degrees). Owing to the limitation of the loading capacity of the testing apparatus, only two pile caps (No.3 and No.4) were tested to failure, which were designed on the truss analogy with compression struts inclined at 45 and 35 degrees respectively. None of the pile caps were actually thought to have failed in shear. All pile caps behaved similarly with one main crack forming at the vertical centre line, extending to within one inch from the top of the pile cap at failure. Blévot and Frémy (1967) [32] tested two series of pile caps. First series specimens included 51 four-pile caps, 37 three-pile caps and 6 two-pile caps and were all of about half scale size. Second series specimens were of full scale size and had 8 four-pile caps, 8 three-pile caps and 6 two-pile caps. The main objectives was to determine the influence of different reinforcing steel layouts and to verify their proposed truss models, which were used to design reinforcement. The height of the pile caps and the layouts of reinforcing steel were varied in the study. Figure 5.9 shows the five different reinforcing steel layouts investigated in the four-pile caps of first series specimens. These layouts can be named bunched square, bunched diagonal, bunched hybrid, and grid. The three-pile caps of the first series specimens had similar layouts to those in the four-pile caps. In the second Chapter 5. Shear Design of Deep Pile Caps 60 series specimens, three-pile caps and four-pile caps had two kinds of layouts, bunched hybrid, and bunched square or triangle plus grid. Layouts in the two-pile caps of both series specimens are similar to that in simply supported deep beams. Clarke (1973) [44] tested 15 four-pile caps, with the objective to compare the strength of caps with values predicted by various design methods. Moreover the effect of different steel layouts and different anchorage lengths upon the behaviour of pile caps at service loads and ultimate loads were also investigated. The specimens included two types of pile caps, namely type A and type B. A pile diameter of 200 mm, a column dimension of 200 x 200 mm and the total depth of 450 mm were used throughout, giving plan dimensions of 950 mm square for the type A caps and 750 mm square for the type B. Three different steel layouts (grid, bunched square and bunched diagonal) and four different anchorage lengths (nil, nominal, full and full-plus-bob) were considered. See Figure 5.10. Most pile caps failed in shear after reinforcement yielded. Four pile caps were thought to have failed in flexure. Clarke and Taylor (1974) [45] also tested a number of eight-pile caps at 1:4, 1:15 and 1:38 scale to investigate the influence of pile stiffness upon distribution of the column load in pile caps. The uncracked pile caps were found to distribute pile loads as predicted by an elastic solution. When the concrete cracked, load carried by the outer piles was considerably decreased. In order to confirm their proposed shear design approach for deep pile caps, Sabnis and Gogate (1980, 1984) [46, 47] tested nine four-pile caps, all at a one-fifth scale. The first three specimens, one of which was of plain concrete, served to finalize the test set-up. The remaining six specimens were used to study the effect of the amount of uniformly placed reinforcement upon the shear capacity of deep pile caps. All the tested pile caps had the same dimensions: 3 inch (76 mm) diameter piles and columns, 13 inch (330 mm) square caps and 6 inch (152 mm) total depth of caps. The reinforcement ratio was varied Chapter 5. Shear Design of Deep Pile Caps 61 between 0.0014 and 0.0079 for the main six specimens. In order to prevent anchorage failure, the reinforcement was hooked and further extended vertically the full depth of the pile cap. Punching shear was thought to be the predominant failure mechanism for all the specimens tested. Adebar et al. (1990) [11] tested six full scale pile caps in an attempt to investigate the applicability of strut-and-tie models (truss models) to the design of pile caps and to evaluate the validity of the current ACT design procedures for pile caps. Of six tested pile caps, four caps were of diamond shapes, one cap of rectangular shape and one cap of cruciform shape. The pile caps all had an overall depth of 600 mm and were loaded through 300 mm square cast-in-place reinforced concrete columns. They were supported by 200 mm diameter precast reinforced concrete piles. The layouts of reinforcing steel used were bunched bars, grid, and bunched plus grid. Full straight anchorage lengths were provided for reinforcing bars passing over the piles. Of six pile caps, four are believed to have failed in shear while two are thought to have failed in flexure. From above mentioned experimental investigations for pile caps, some important con clusions obtained by the investigators are summarized in the following: 1. Truss models are more appropriate for deep pile caps than simple bending theory. Blévot and Frémy [32] confirmed their truss models by carrying out a comprehensive series of tests. Best results were obtained with the imaginary struts inclined at between 45° and 55° to the horizontal. Clarke [44] concluded that the truss analogy was an adequate method of analyzing a four-pile cap in order to ascertain its flexural capacity and to determine the required amount of tensile reinforcement. Adebar et al. [11] concluded that the ACT Building Code (sectional method) failed to capture the trend of the experimental results. 2. Layouts of reinforcing steel have a great effect on the behaviour of deep pile caps Chapter 5. Shear Design of Deep Pile Caps 62 under service loading and on the ultimate capacity of deep pile caps. The most desirable behaviour was obtained by the layouts of the bunched hybrid or bunched steel plus a relative light grid of steel. Blévot and Frémy [32] observed that the bunched steel gave approximately 20% higher strength than the same weight of steel spread out in a grid pattern. Clarke [44] concluded, from his four-pile cap tests, that bunching the steel in the form of a square increases the load by about 14%, while bunching it along the diagonals only increases the strength by a negligible amount. It was reported by both Blévot and Clarke that the crack control was very much improved by using bunched hybrid or bunched hybrid plus grid, instead of using bunched diagonal only. 3. Only nominal anchorage of reinforcing steel is required. Deutsch and Walker [43] showed analytically and confirmed by their tests that only nominal anchorage was required past the edges of the piles. Blévot and Frémy [32] recommended the use of standard hooks as regular requirement for anchorage based on their experimental results. Clarke [44] concluded from his four pile cap tests that caps with bunched square steel with nil anchorage was the most efficient if the efficiency is defined as the load carried by the cap divided by the total weight of steel used. The reason that the anchorage requirement can be reduced was, suggested by Clarke, that reinforcing bars bunched over piles are subjected to high lateral bearing stresses which help to lock the bars into place. It was also shown in Clarke’s tests that the full anchorage resulted in a 30% increase in load capacities. Clarke thought that this substantial increase was mostly due to the vertical portion of the reinforcement providing reinforcing across shear cracks. These vertical portions of the reinforcing bars acted as stirrups. It should be noted that the reinforcing bars in the specimens tested by Sabnis and Gogate [47] were also full anchorage with the bars extended Chapter 5. Shear Design of Deep Pile Caps 63 vertically the full depth of the pile cap. 5.6 Comparative Study of Different Design Procedures 5.6.1 Comparison of Design Methods To compare the one-way shear design procedures, Figure 5.11 summarizes the relationship between the maximum column load and the width b and depth d of a two-pile cap. When the width of the pile cap is the same as the column width (b = c), the pile cap is essentially a deep beam. See Figure 5.11(b). When the width of the pile cap is increased, larger shear forces can be resisted by the increased concrete area at the critical section, and the limiting column load due to bearing strength is increased as a result of confinement, see Figure 5.11(c) and (d). Three different ACT Code predictions for one-way shear are given in Figure 5.11. The least conservative prediction, entitled “ACI ‘77,” is what designers of pile caps would have used prior to the 1983 edition of the ACT Building Code (any pile within d of the column face is assumed to produce no shear on the critical section). The “ACT ‘83” prediction, in which the critical section is at the face of the column, gives very conservative predictions. The predicted column load based on one-way shear calculation at the critical section half way (a/2) between the face of the column and the centre of the closest pile, “ACT [11.8],” gives an intermediate result. The CRST method, in which the critical section is also at the face of the column, is much less conservative than “ACT ‘83” due to an enhanced concrete contribution, but is more conservative than “ACT ‘77.” Note that all methods predict that as the pile cap becomes very deep, the maximum column load is limited by bearing strength. When the pile cap is twice as wide as the column (b 2c) the ACT Code predicts that the confinement is sufficient so that the bearing strength has reached the upper limit of 2 x O.85f. Chapter 5. Shear Design of Deep Pile Caps 64 Figure 5.12 compares the influence of pile cap depth on two-way shear strength predic tions for a typical four-pile cap. Although the CRSI Handbook method gives a concrete contribution which is significantly larger for deep pile caps, the maximum column load is always smaller than the ACT method. This is because in the ACT method the critical section is at d/2 from the column face and any pile which intercepts the critical section is assumed to transmit part of the load directly to the column. For example, if a pile is centred on the critical section, only half of the pile reaction must be resisted by the critical section. 5.6.2 Comparison of Prediction Results Table 5.1 summarizes the 48 specimens which were chosen for the comparative study. These include two pile caps tested by Deutsch and Walker [43], eighteen pile caps tested by Blévot and Frémy [32] in their second series, fourteen pile caps tested by Clarke [44], eight pile caps tested by Sabnis and Gogate [47] as well as six pile caps tested by Adebar et at. [11]. Specimens not considered in the comparative study include the small wide beam models tested by Hobbs and Stein, the small-scale specimens (first series) tested by Blévot and Frémy, and the one specimen tested by Clarke and two specimens tested by Deutsch and Walker which did not fail. Table 5.2 and Table 5.3 give the ACT Code and CRST predictions. The predicted flexural strengths and bearing strengths are summarized in Table 5.2, while the predicted shear capacities are summarized in Table 5.3. In the case of one-way shear, three different predictions are given from the ACT Building Code: the 1977 edition of the ACT Building Code (critical section at d from the column face); the 1.983 ACT Building Code (critical section at the column face); and the special provisions for deep flexural members (Clause 11.8, critical section at half way between the column face and the closest pile centre). Tn the case of two-way shear, ACT Chapter 5. Shear Design of Deep Pile Caps 65 Code procedures involve critical section at d/2 from the column face and the pile face. Details of the predictions can be found in Appendix B. Table 5.4 presents the ratio of measured pile cap capacity to predicted capacity for the three ACT Code predictions as well as the CRSI prediction. As mentioned before, actually there are no shear design procedures for deep pile caps in ACT ‘77, therefore most pile caps are predicted to fail in flexure. If Clause 11.8 is considered, the load capacity of some pile caps is limited by shear (especially specimens tested by Blévot and Frémy). Tf the 1983 ACT Building Code is applied, it can be seen that the load capacity of almost all pile caps is controlled by shear due to the lower predicted shear capacities. It is very interesting to examine the CRST predictions. The original objective of the CRSI equations is to reflect the considerable reserve strength observed in the tests of deep beams. Thus the CRSI equations generally give higher shear capacities. This results in the change of predicted failure modes for most of the pile caps to flexural failure as in the ACT ‘77 predictions. The comparisons are illustrated graphically in Figure 5.13 to 5.16. Table 5.5 summarizes the predictions from the proposed design methods and compares the predictions with the experimental results. The comparisons of measured pile cap capacities and the predictions from the pro posed design method (1) are shown in Figure 5.17 and 5.18. The predictions for specimens listed in Table 5.1 are shown in Figure 5.17, while the predictions for the pile caps with bunched reinforcement are shown in Figure 5.18. The comparison of measured test results with the predictions from the proposed design method (2) is shown in Figure 5.19. Chapter 5. Shear Design of Deep Pile Caps 5.7 66 Conclusions Some conclusions can be arrived at from the study in this chapter. The ACT Code design procedures (especially shear design) are not adequate for deep pile caps. The ACT Code 1977 edition gives scattered and not conservative predictions. The ACT Code 1983 edition, in contrast with 1977 edition, gives overly conservative predictions, obviously due to the equation used for shear calculation on the critical section of the face of the column which underestimates the shear capacity of deep pile caps. Though this problem has been realized in the development of CRST shear design method, the problem is not totally solved. In addition, the ACT Code design procedures or CRSI procedures are sometimes very complex and tedious, involving many calculations in order to meet the code requirement. For example, predicting the three-pile cap specimens tested by Blévot is difficult. The ACT Code flexural strength predictions are unconservative for deep pile caps. These fiexural strength procedures are meant for lightly reinforced beams which are able to undergo extensive fiexural deformations (increased curvatures) after the reinforcement yields. As the curvature increases, the fiexural compression stresses concentrate near the compression face of the member. Deep pile caps are too brittle to undergo such deforma tions, therefore, assuming the fiexural compression is concentrated near the compression face is inappropriate. Assuming the flexural compression is uniform across the entire pile cap, which strain measurements have shown to be incorrect [11], leads to a further overprediction of the flexural capacity. In Figure 5.20, the bearing capacities on column zones for previously tested pile caps are illustrated in terms of the amount of confinement at column zones. The dashed line shown in the figure demonstrates a similar trend as explored analytically and experi mentally on the double punch loaded concrete cylinders. This indirectly further confirms Chapter 5. Shear Design of Deep Pile Caps 67 that the transverse splitting phenomenon is predominant for the shear failures of deep pile caps. Both proposed design methods herein capture the physical behaviour of deep pile caps, and produce conservative and reasonably accurate predictions. The proposed de sign method (1) is a direct extension of the two-dimensional strut-and-tie model of the Canadian concrete code. The proposed design method (2) is a simplified method which treats “fiexural design” and “shear design” separately. This is the concept traditionally accepted by current engineering practice. Chapter 5. Shear Design of Deep Pile Caps 68 Table 5.1: Summary of pile cap test results. Number of Piles Blévot & Frémy 2N1 2 2Nlb 2 2N2 2 2N2b 2 2N3 2 2N3b 2 3N1 3 3Nl 3 3N3 3 3N3 3 4N1 4 4N1 4 4N2 4 4N2b 4 4N3 4 4N3b 4 4N4 4 4N4b 4 Deutsch & Walker No.3 2 No.4 2 d Specimen (mm) Pile Size (mm) 495 495 703 698 894 892 447 486 702 736 674 681 660 670 925 931 920 926 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 533 373 254 sq. 254 sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. Column Size (mm) f (MPa) Reinf. Layouts Failure Load (kN) sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. 23.1 43.2 27.3 44.6 32.1 46.1 44.7 44.5 45.4 40.1 36.5 40.0 36.4 33.5 33.5 48.3 34.7 41.5 bunched bunched bunched bunched bunched bunched bunched bunched bunched bunched b.& g. b.& g. bunched bunched b.& g. b.& g. bunched bunched 2059 3187 2942 5100 4413 5884 4119 4904 6080 6669 6865 6571 6453 7247 6375 8826 7385 8581 165 sq. 165 sq. 23.8 23.6 bunched bunched 596 289 350 350 350 350 350 350 450 450 450 450 500 500 500 500 500 500 500 500 Chapter 5. Shear Design of Deep Pile Caps 69 Table 5.1: (con’t) Summary of pile cap test results. Specimen Number of Piles d (mm) Clarke Al 4 400 A2 4 400 A3 4 400 A4 4 400 A5 4 400 A6 4 400 A7 4 400 A8 4 400 A9 4 400 AlO 4 400 All 4 400 A12 4 400 Bl 4 400 B3 4 400 Sabnis & Gogate SS1 4 111 SS2 4 112 4 SS3 111 SS4 4 112 4 109 SS5 SS6 4 109 5G2 4 117 4 SG3 117 Adebar, Kuchma and Collins A 4 445 B 4 397 C 6 395 D 4 390 E 4 410 F 4 390 Pile Size (mm) Column Size (mm) (MPa) Reinf. Layouts Failure Load (kN) f’ 200 200 200 200 200 200 200 200 200 200 200 200 200 200 rd. rd. rd. rd. rd. rd. rd. rd. rd. rd. rd. rd. rd. rd. 200 200 200 200 200 200 200 200 200 200 200 200 200 200 sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. sq. 20.9 27.5 31.1 20.9 26.9 26.0 24.2 27.5 26.8 18.2 17.4 25.3 26.9 36.3 grid bunched bunched grid bunched bunched grid bunched grid grid grid grid grid grid 1110 1420 1340 1230 1400 1230 1640 1510 1450 1520 1640 1640 2080 1770 76 76 76 76 76 76 76 76 rd. rd. rd. rd. rd. rd. rd. rd. 76 76 76 76 76 76 76 76 rd. rd. rd. rd. rd. rd. rd. rd. 31.3 31.3 31.3 31.3 41.0 41.0 17.9 17.9 grid grid grid grid grid grid grid grid 250 245 248 226 264 280 173 177 200 200 200 200 200 200 rd. rd. rd. rd. rd. rd. 300 300 300 300 300 300 sq. sq. sq. sq. sq. sq. 24.8 24.8 27.1 30.3 41.1 30.3 grid bunched bunched bunched bk g. bunched 1781 2189 2892 3222 4709 3026 Chapter 5. Shear Design of Deep Pile Caps 70 Table 5.2: Summary of ACT Building Code and CRSI Handbook predictions (flexure and bearing). Specimen Name 2N1 2Nlb 2N2 2N2b 2N3 2N3b 3N1 3Nlb 3N3 3N3b 4N1 4Nlb 4N2 4N2b 4N3 4N3b 4N4 4N4b No.3 No.4 Flexural Capacity (kN) 2197 3756 3429 5553 5413 7259 3825 5290 6014 7982 7969 8171 7812 8546 8283 10788 9864 10866 512 270 Bearing Capacity Column Pile (kN) (kN) 5492 2746 10270 5135 3244 6488 10620 5310 7632 3816 5485 10970 15378 23860 15311 23757 15631 24255 21424 13808 26083 15525 28570 17005 15464 25977 14234 23913 23913 14234 20549 34521 14734 24753 17631 29621 1100 3907 1093 3881 Chapter 5. Shear Design of Deep Pile Caps 71 Table 5.2: (con’t) Summary of ACT Building Code and CRSI Handbook predictions (flexure and bearing). Specimen Name Al A2 A3 A4 A5 A6 A7 A8 A9 AlO All A12 Bi B3 SS1 SS2 553 SS4 SS5 SS6 SG2 SG3 A B C D E F Flexural Capacity (kN) 1258 1266 1256 1258 1265 1252 1262 1266 1264 1252 1252 1262 2022 1528 133 116 194 157 317 455 302 628 2256 2790 4008 5646 7428 5324 Bearing Capacity Column Pile (kN) (kN) 1421 3908 1870 5140 2115 5812 1421 3908 5028 1829 1768 4860 1646 4524 1870 5140 1822 5008 3404 1238 1183 3252 4728 1720 5028 1829 2468 6784 242 808 242 808 242 808 242 808 1060 318 318 1060 463 139 139 463 5296 3794 5296 3794 4146 8682 6472 4636 6288 8780 6472 3091 Chapter 5. Shear Design of Deep Pile Caps 72 Table 5.3: Summary of ACT Building Code and CRSI Handbook predictions (shear). Specimen Name 2N1 2Nlb 2N2 2N2b 2N3 2N3b 3N1 3Nlb 3N3 3N3b 4Nl 4Nlb 4N2 4N2b 4N3 4N3b 4N4 4N4b No.3 No.4 One-Way Shear (kN) ACT CRSI 1977 1983 [11.8] 1053a 316 947 775 1438a 432 1296 898 b 488 1463 2438 b 616 1846 2634 b 3364 673 2018 b 804 2414 4023 2130a 1583a 4490 2130 2707a 1715a 4737 2639 b 2504a 7496 9321 b 2468a 7385 9871 b 2436 7308 12004 b 2575 7702 12114 b 2377 7130 11319 b 2318 6955 10672 b 3201 9603 16005 b 3858 11286 19292 b 3236 9695 16178 b 3566 10437 17831 1881a 329 925 558 285 230 504 d Two-Way Shear Column ACT CRSI c c c c c c c c c c c c 3722a 6594 4397a 8062 b 21249 b 22297 11831a d 12824a d 10959a d 11103a d 56759a 13338 70248a 16111 54102a 13425 63794a 14870 c c c c (kN) Pile c c c c c c 3789 4157 b b b b b b b b b b c c Chapter 5. Shear Design of Deep Pile Caps 73 Table 5.3: (con’t) Summary of ACT Building Code and CRSJ Handbook predictions (shear). Specimen Name One-Way Shear (kN) Two-Way Shear (kN) ACI CRSI Column Pile 1977 1983 [11.8] ACT CRSI Al b 1646 578 2714 2916a 1458 1996 A2 b 662 1848 3078 3344a 1672 2288 A3 b 704 1934 3246 3556a 1778 2432 A4 b 1646 578 2714 2916a 1458 1996 A5 b 1830 654 3046 3308a 1654 2264 A6 b 644 1790 2988 3252a 1626 2224 A7 b 1750 620 2902 3138a 1569 2148 A8 b 662 1848 3078 3344a 1672 2288 A9 b 654 1828 3042 3302a 1651 2260 AlO b 1554 538 2550 2722a 1361 1860 All b 526 1526 2498 2660a 1330 1820 A12 b 634 1784 2962 3208a 1604 2196 Bl b 516 2066a 2584 b 3308 b B3 b 600 2344a 3002 b 3843 b SS1 b 69 186 254 122 d 228 SS2 b 178 68 249 122 d 228 SS3 b 181 68 121 250 d 226 SS4 b 71 190 122 258 d 228 b SS5 84 229 296 134 d 251 SS6 b 229 89 315 d 134 251 SG2 b 164 65 261 101 d 185 b SG3 164 84 273 d 101 185 A 3248 2400 6056 6352 2360a d 6250 B 3408 2084 5306 4278 d 1839 2764 6304 1820 4938 C 3734 d 1899 2989 D 3770 2432 6348 4726 d 1968 3107 4478 3076 8140 E 7078 2475 d 3970 F 1604 572 1224 1618 c c c a = Increased capacity since piles partially within critical section; b = Infinite capacity since piles totally within critical section; c = Procedure not applicable; d = CRSI prediction not applicable use ACT. - Chapter 5. Shear Design of Deep Pile Caps 74 Table 5.4: Comparison of ACT Code and CRSI Handbook predictions: ratio of measured capacity to predicted capacity and failure mode. Specimen Name ACT ‘77 ACT ‘83 2N1 2Nlb 2N2 2N2b 2N3 2N3b 3N1 3Nlb 3N3 3N3b 4Nl 4Nlb 4N2 4N2b 4N3 4N3b 4N4 4N4b No.3 No.4 1.96 s 1 2.22 s 1 0.91 b 0.96 b 1.16 b 1.07 b 1.93 s 1.81 s 1.01 f 0.84 f 0.86 f 0.80 f 0.83 f 0.85 f 0.77 f 0.82 f 0.75 f 0.79 f 1.16 f 1.07 f 6.52 7.38 6.03 8.28 6.56 7.32 2.60 2.86 2.43 2.70 2.82 2.55 2.71 3.13 1.99 2.29 2.28 2.41 1.81 1.26 s i s i .5 i s s s s 1 s s, L 8 s 1 s s s i s 1 s ACT [11.8] CRSI 2.17 i 2.46 s 1 2.01 s 1 2.76 Sj 2.19 s 1 2.44 s 1.11 2 1.04 i 1.01 f 0.90 s 0.94 s 0.85 i 0.91 i 1.04 si 0.77 f 0.82 f 0.76 5j. 0.82 s 1.16 f 1.07 f 2.66 s 1 1 3.55 s 1.21 i 1.94 i 1.31 i 1.46 s 1 1.93 s 1 1.86 si 1.01 f 0.84 f 0.86 f 0.80 f 0.83 f 0.85 f 0.77 f 0.82 f 0.75 f 0.79 f 1.16 f 1.07 f Reported Failure Mode s s s S S s s s s s s s S s s s 5 s f f Chapter 5. Shear Design of Deep Pile Caps 75 Table 5.4: (con’t) Comparison of ACT Code and CRSI Handbook predictions: ratio of measured capacity to predicted capacity and failure mode. Specimen Name ACT ‘77 ACT ‘83 ACT [11.8] CRSI Reported Failure Mode s s s s s s s s s f f f s f Al 0.88 f 1 1.92 s 0.88 f 0.88 f A2 1.12 f 1 2.15 s 1.12 f 1.12 f A3 1.07 f 1 1.90 s 1.07 f 1.07 f A4 0.98 f 2.13 s 0.98 f 0.98 f 1.11 f A5 2.14 s 1.11 f 1.11 f A6 0.98 f 1 1.91 s 0.98 f 0.98 f A7 1.30 f 2.65 s 1.30 f 1.30 f A8 1.19 f 1 2.28 s 1.19 f 1.19 f A9 1.15 f 2.31 s 1.15 f 1.15 f AlO 1.23 b 2.83 s 1.23 b 1.23 b All 1.39 b 3.12 i 1.39 b 1.39 b A12 1.30 f 2.59 s 1.30 f 1.30 f Bl 1.14 b 1 1.14 b 4.03 s 1.14 b 1.16 f B3 1 2.95 s 1.16 f 1.16 f SS1 2.05 2 3.62 i 2.05 s 2 2.05 2 S 2.11 f SS2 2.11 f 3.60 i 2.11 f s 2.05 s SS3 2 2.05 2 3.65 s 2.05 s 2 s SS4 1.85 2 1 3.18 s 1.85 s 2 1.85 s 2 s 1.97 2 SS5 1.97 2 3.14 i 1.97 2 S 2.09 s SS6 2 1 2.09 2 3.15 s 2.09 s 2 s SG2 1.71 2 1 2.66 s 1.71 2 1.71 2 S 1.75 2 SG3 1 2.11 s 1.75 2 1.75 2 A 0.79 f 0.79 f 0.79 f 0.79 f f 1.19 2 B 1.19 2 1.19 s 2 1.19 s 2 s 1.52 2 1 C 1.59 s 1.52 1.52 2 S D 1.64 2 2 1.64 s 1.64 2 1.64 2 S B 1.90 2 2 1.90 s 1.90 2 1.90 2 S F 1.89 s 5.29 si 1.87 i 2.47 s s f=flexure; b=column bearing; s =one-way shear; 2 1 s = two-way shear; s=shear. Chapter 5. Shear Design of Deep Pile Caps 76 Table 5.5: Comparison of proposed design methods with experimental results. Specimen Name 2N1 2Nlb 2N2 2N2b 2N3 2N3b 3N1 3Nlb 3N3 3N3b 4N1 4Nlb 4N2 4N2b 4N3 4N3b 4N4 4N4b No.3 No.4 Predicted (1) (kN) 1663 2921 2581 4186 3893 5317 3207 4107 5060 6170 5025 5398 4876 4864 6041 7018 6704 7704 467 250 Predicted (2) Flexure Shear (kN) (kN) 2128 1053a 3570 1438a 3109 2148 5051 3504 4835 2552 6443 3622 3256 2l30a 2707a 4531 7493 5070 6767 6869 6041 9037 6178 9790 5933 8868 6512 8393 10604 6208 13993 7010 7414 10825 8150 12450 480 732 254 285 Experimental (kN) 2059 3187 2942 5100 4413 5884 4119 4904 6080 6669 6865 6571 6453 7247 6375 8826 7385 8581 596 289 Exp. Pred.(1) 1.24 1.09 1.14 1.22 1.13 1.11 1.28 1.19 1.20 1.08 1.37 1.22 1.32 1.49 1.06 1.26 1.10 1.11 1.28 1.16 Exp. Pred.(2) 1.96 2.22 1.37 1.46 1.73 1.62 1.93 1.81 1.20 0.99 1.14 1.06 1.09 1.11 1.03 1.26 1.00 1.05 1.24 1.14 s s s s s s s s f f f f f f f f f f f f Chapter 5. Shear Design of Deep Pile Caps 77 Table 5.5: (con’t) Comparison of proposed design methods with experimental results. Predicted Predicted (2) Experimental Exp. Exp. Flexure Shear (1) Pred.(l) Pred.(2) (kN) (kN) (kN) (kN) Al 792 1030 1420 1110 1.40 1.08 f A2 1031 928 1716 1.53 1420 1.38 f A3 992 1020 1868 1340 1.35 1.31 f A4 792 1420 1030 1230 1.19 f 1.55 A5 916 1031 1688 1400 1.53 1.36 f A6 900 1648 1020 1230 1.21 f 1.37 A7 1030 868 1572 1640 1.59 f 1.89 A8 1031 928 1716 1510 1.63 1.47 f A9 1030 916 1684 1450 1.59 1.41 f AlO 696 1030 1296 1520 2.18 1.48 f All 668 1256 1030 1640 1.59 f 2.46 A12 888 1620 1030 1640 1.59 f 1.85 Bl 1008 1592 1374 2.06 2080 1.51 f B3 1030 1028 1972 1770 1.72 1.72 f SS1 98 122a 97 250 2.55 2.58 f SS2 84 122a 85 245 2.92 2.88 f SS3 116 121a 144 248 2.14 2.05 s SS4 112 116 122a 226 1.95 f 2.02 149 SS5 238 134a 264 1.77 1.97 s 149 SS6 347 134a 280 2.09 s 1.88 SG2 71 lOla 231 173 2.44 1.71 s SG3 71 543 lOla 177 1.75 s 2.49 A 1448 1915 1445 1781 1.23 1.23 f B 1659 1662 1696 2189 1.32 1.32 f 1499 1684 C 1502 2892 1.93 f 1.93 D 3454 2320 1968a 3222 1.64 s 1.39 E 3505 5084 2475a 4709 1.90 s 1.34 F 1774 3472 1303 3026 2.32 s 1.71 ACI ‘77 prediction critical; s=snear critical; f=flexure critical Specimen Name Chapter 5. Shear Design of Deep Pile Caps 78 Flexural Calculation / s d h L m I I / I I \ I I j I / I I I J \ \ Two Way 0n Way Shear Punching Shear ! / I \ /\ ‘ I Ijl I I I I I I I I — / d/2 - H I / I dP) I L m n . . . Figure 5.1: ACT Code specified critical sections for fiexure and shear investigation of pile cap. Chapter 5. Shear Design of Deep Pile Caps 79 d Deep Beam Behaviour N___ H One Way Shear ,__/ dpi )%_ I I “— a/2 Figure 5.2: A deep two-pile cap Chapter 5. Shear Design of Deep Pile Caps 80 One WayShear? -- — / ,--‘ I / r———--—-—-——-7—I Two Way d/2 Punching Shear - —, ÷ I’ / / I—’ Figure 5.3: A deep three-pile cap Chapter 5. Shear Design of Deep Pile Caps 81 / I I \ Two Way Punching Shear I / I 4— / _, _Hd12m__ (a) / / / N’ / Two Way Punching Shear (b) Figure 5.4: Comparison of two-way punching shear calculations: (a) the cap with square column; (b) the cap with circle column. Chapter 5. Shear Design of Deep File Caps 82 0.1 0.3 4 2 U. )[a.z.s (a) .4 0 0 0.Z 0.4 0. w-/d 0.8 I ‘ : 3 —— C,. — — ‘_‘ 1 •_Ic rut ‘1 IC k l }c) k Ii ‘JcI÷d/cZ s I U 0.1 — 1 4 r {(I+d1c_..I (b) -_w. 0.Z 03 0.4 0.5 O. Figure 5.5: CRSI approach for shear design of deep pile caps: (a) allowable shear stress, v, for one-way shear while while w/d < 0.5, w/d from Ref.[36}. < 1.0; (b) allowable shear stress, v, for two-way shear Chapter 5. Shear Design of Deep Pile Caps 83 C ) Given: Column size Pile group Pile capacity fy pile cap? No Yes (ACI 77) Select d: Have no appropriate design procedure for shear One-way shear (11.11.1.1, 11.1, 11.2, 11.3and 11.5) Two-way shear (11.11.1.2, 11.11.2 and 15.5) Minimum footing depth (15.7) One-way shear (11.1—3) Have no appropriate design procedure for two-way shear Select d:(ACI 83) Select d: (CRSI) Find As: One-way shear (CRSI) Two-way shear (CRSI) Moment calculation (15.4.1 and 15.4.2) Detailing (15.4.3, 15.4.4 and 12.2) Compute cap thickness h: (7.7.1) Shear check: (for Individual pile) Perimeter shear (11.11.1.2 arid 11.11.2) Beam shear (11.11.1.1 and 11.3,1.1) Bearing check: (10.16) Figure 5.6: Flow chart for ACT and CRSI design procedures for pile caps I -a 9- C— Cl) CD 0 CD 0 U) C’, H C-) -4. f-Th * -. CD 0(n CD D QCD 0 :3 Cl) CD 0) 0 3c 00 T1 CD D ci m 0 C) CD C’, 0 0 —i, 0 . 10 CD CD CD0 0 0)’ D ‘0 CD CD ‘C) D ( CD — — :3-a CD — cD(D < Cf) 0 01 0- C) 01 co p 0 D CD N D 0 0. Q) CD CD Chapter 5. Shear Design of Deep Pile Caps 85 1 D H h (a) (b) (c) Figure 5.8: Loading geometry of compression struts with linearly varying cross section. Chapter 5. Shear Design of Deep Pile Caps 86 (a) (b) N :, / (c) N (d) - - (e) Figure 5.9: Various layouts of main reinforcing bars used by Blévot and Frémy, Ref.[32j. Chapter 5. Shear Design of Deep Pile Caps 87 7 L (1) Nil (2) Nominal — L (3) Full ‘p I (4) Full-plus-bob Figure 5.10: Various anchorage lengths used by Clarke, Ref. [44]. Chapter 5. Shear Design of Deep Pile Caps 88 36” ‘I d=1O” ,-— I / i (a) ‘i c=1O” b=c=1O (ki p) ° 40 lAd ‘77 300 / 200 CRSL— Proposed (2) - ACI [11.8] - 100 (b) ——ZC 183 0 12 14 16 18 20 22 24 26 28 30 d(in.) Figure 5.11: Comparison of one-way shear design methods for two-pile caps: (a) plan view of pile cap; (b) to (d) influence of pile cap depth on column load for various pile cap widths. Chapter 5. Shear Design of Deep Pile Caps 89 b=2c=20” p 700 (kip) 600 500 400 300 200 100 (c) 01 d (in.) b=4c=40” 700 Proposed (2) (kip) 600 500 400 300 200 100 (d) 0 12 14 16 18 20 d (in.) Figure 5.11: (cont’d) Comparison of one-way shear design methods for two-pile caps: (a) plan view of pile cap; (b) to (d) influence of pile cap depth on column load for various pile cap widths. Chapter 5. Shear Design of Deep Pile Caps 90 /dp8 15” / 36” f.. 14’ —-. I — I I (a) 15” 15” —. 36” 15” .-4 — P 1800 1600 (kip) 1400 1200 1000 800 600 400 200 (b) 12 16 20 24 28 32 36 d - (in.) Figure 5.12: Comparison of two-way shear design methods for a typical four-pile cap: (a) plan view of pile cap; (b) influence of pile cap depth on column load. Chapter 5. Shear Design of Deep Pile Caps 91 4 Pexp ACI ‘77 I I pred. 0 3 A 2 Flexure One-way shear Two-way shear A A 0 1 * U 0 0 Mean 1.31 CCV 34.8% 2000 4000 .0_000 0 0 6000 8000 10000 12000 Pexp (kN) Figure 5.13: Comparison of ACT ‘77 predictions with experimental results. Chapter 5. Shear Design of Deep Pile Caps 92 4 Pexp ACI ‘83 Ppred. • 2 : - One-way shear Two-way shear : A A 1 C Mean 3.09 C0V55.1% 2000 4000 6000 8000 10000 12000 Pexp (kN) Figure 5.14: Comparison of ACI ‘83 predictions with experimental results. Chapter 5. Shear Design of Deep Pile Caps Pexp 93 ACI [11.8] Ppred - One-way shear Two-way shear • A 2 A A * 00 ,C*A A 1 U 10 0 Mean 1.42 CCV 38.9% 2000 4000 6000 8000 10000 12000 Pexp (kN) Figure 5.15: Comparison of ACT ‘[11.8] predictions with experimental results, clause 11.8 considered. Chapter 5. Shear Design of Deep Pile Caps 94 4 Pex CRSI pred. • A Flexure One-way shear Two-way shear Bearing . I • • • I A • A . 0 1 0 0 Mean 1.40 CCV 40.7% 0 2000 4000 0&00 0 0 6000 8000 10000 12000 Pexp. (kN) Figure 5.16: Comparison of CRSI predictions with experimental results. Chapter 5. Shear Design of Deep Pile Caps 95 4 Pexp. Ppred. Proposed (1) 3. 3 . . 2 • $ • • •. ••, • • . . •• • • • •• • . • • • •• • • •• • Mean 1.57 CCV 29.6% 0 I 0 2000 4000 6000 Pexp 8000 10000 12000 (kN) Figure 5.17: Comparison of proposed method (1) predictions with experimental results of all specimens in Table 5.1. Chapter 5. Shear Design of Deep Pile Caps 96 4 Pexp ppred. Proposed (1) 3 (bunched reinforcement) 2 . . . . . . . •. . . Mean 1.32 CCV 16.5% 0 0 2000 4000 6000 8000 10000 12000 Pexp (kN) Figure 5.18: Comparison of proposed method (1) predictions with experimental results of specimens with bunched reinforcement in Table 5.1. Chapter 5. Shear Design of Deep Pile Caps 97 4 Pexp. Proposed (2) I Ppred 3 “Flexure” t “Shear ° o 0 L . . S 2 . • 0 • o 0 0 0 . 0. o 0 o 1 00 • 0 0 Mean 1.55 CCV 27.8% I o 2000 4000 I 6000 8000 10000 12000 Pexp. (kN) Figure 5.19: Comparison of proposed method (2) predictions with experimental results of all specimens in Table 5.1 Chapter 5. Shear Design of Deep Pile Caps f 3 b • — .c I 98 ° • ‘ Adebar et a elévot and Frémy Clarke Deutsch and Walker Sabns and Gogate 8 2 . 0 . .. — 0 D 0 0 — 0 0 C C 2 3 4 5 Figure 5.20: Relationship of measured ultimate bearing stress and confinement on column zones of pile caps in Table 5.1. Chapter 6 Shear Failure of Beams Without Stirrups 6.1 Introduction In the previous chapters the discussion focused on the shear resistance of deep pile caps. The model that was proposed assumes that the load is transmitted from the column to the piles by direct compression struts, and that the tensile stresses which cause cracking of the compression struts is due only to the transverse spreading of compression stresses. As pile caps become more slender, other mechanism will be involved in creating tensile stresses which may lead to a shear (diagonal tension) failure. Note that this issue was accounted for in the proposed design method (Section 5.4) by suggesting that the ACT Building Code (empirical) method be used for slender pile caps. In order for this thesis to be a relatively comprehensive treatment of pile caps and other members without transverse reinforcement, it is necessary to consider the mecha nisms involved in the shear failure of more slender members. Towards that end, a study was made of the shear resisting mechanisms in slender beams. While this study is really a separate topic, it is closely related to the first part of this thesis (transverse splitting in deep members). In fact, as will be shown in Chapter 8, the truss model which is associated with the transverse splitting mechanism in deep members is in fact very sim ilar to the truss model associated with what is believed to be an important shear failure mechanism in slender beams without transverse reinforcement. It should be pointed out however, that unlike the first part of this thesis, which was a 99 Chapter 6. Shear Failure of Beams Without Stirrups 100 complete study of transverse splitting (it resulted in the development of a new improved design method for pile caps), this study on shear in slender beams is much more of a pilot study. In this and the next chapter considerable information is given about the shear resisting mechanisms in slender beams, but additional research is needed before the concepts presented here can be implemented into a shear design procedure which can be used by practising engineers. This chapter is divided into seven parts. A brief review of the literature is presented and some important conclusions about the current knowledge of shear failures in beams without web reinforcement are summarized first. Then, an interpretation of an important shear failure mechanism is presented based on the deformation compatibility of critical diagonal crack propagation. Next, bond influence upon internal stress distributions of both uncracked and cracked beams are investigated by using linear elastic finite elements and the concepts of arch action and beam action. This is followed by a study on shear displacements along vertical cracks and inclined cracks in beams, and the presentation of a load transfer mechanism based on studies in this chapter and the experimental measurements of previous work. As an application of the understanding obtained in this study, the test results carried out by various researchers are explained in the sixth part. Finally, the conclusions arrived at are summarized. 6.2 Brief Review of the Literature State-of-the-art summaries of the shear resistance of structural concrete members without transverse reinforcement have been reported previously in Ref. [38, 48-51]. The literature review presented here focuses on more recently developed models for shear resistance of slender beams and understanding the transition from deep beams to slender beams. Chapter 6. Shear Failure of Beams Without Stirrups 6.2.1 101 Transition from Deep Beam to Slender Beam The distribution of the transverse compression stresses within the shear span estimated by Mau and Hsu [52] is shown in Figure 6.1. When a/h is maximum at the line of actions. While a/h = = 0, transverse compression stress 0.25 and 0.5, the maximum values of transverse compression stress still occur at the centre of the line in. connection with load point and support point, but the magnitudes decrease with increasing a. While a/h = 1, the distribution of transverse stress shows the characteristics of two humps. These twohumps become more distinct and the transverse compressive stresses approach zero at the centre of the shear span when a/h = 2. Hence, it is seen that direct compressive stress field between the load and the support is formed when a/h < 1. Following the internal force flow, Schlaich et al. [3] gave a description of the transition from deep beams to slender beams using truss models for uncracked beams. See Figure 6.2. If a single load is applied at a distance a < h near the support, the load is carried directly to the support by a compressive stress field as simulated by a simple compression strut. A further examination indicates that the transverse tensile stresses will be intro duced in the compression strut as shown by the refined truss model and as discussed in Chapter 1. With increasing shear span a, the compressive member C 1 joins the part of the tensile force T 1 and simultaneously the transverse tensile ties blend into the vertical ties of the truss model, which are now needed for hanging up the shear forces in slender beam. Thus the beam having a < h, where the load can be transmitted directly to the support, is treated as a deep beam. Collins and Mitchell [39] compared the shear strengths of a series of simply supported beams tested by Kani [13] with the predicted capacities from both sectional and strut and-tie analyses for assumed cracked beams. In Kani’s tests the shear span to depth ratio a/d varied from 1 to 7 and no web reinforcement was provided. The comparison Chapter 6. Shear Failure of Beams Without Stirrups 102 indicated that the shear resistance is governed by strut-and-tie action, which assumes the loads are carried to the supports with direct compression struts. The failures are governed by a strut-and-tie model (crushing of the compression strut) at a/d less than about 2.5, but are governed by a sectional model for aid greater than 2.5. See Figure 6.3. Their conclusions suggest that beams with aid < 2.5 can be treated as deep beams. The truss models developed by Al-Nahlawi and Wight [53] illustrate that when a/d> 1, the shear failure of the beams without transverse reinforcement is characterized by the failure of concrete tension ties. Their work suggested that the beams with a/d> 1 could be considered as slender beams. In the next section more details of Al-Nahlawi and Wight’s truss models will be presented. 6.2.2 Behaviour of Slender Beams Based on the modified compression field theory which suggests that concrete tensile stress is present in cracked reinforced concrete members, a simple truss model with concrete tension ties was developed by Adebar [54] for a diagonally cracked beam which is simply supported, subjected to point loads and reinforced with only longitudinal reinforcement. See Figure 6.4. The inclination of compression struts in the web is equal to half of the inclination of the uniformly spaced diagonal cracks, and concrete tension ties are perpen dicular to the compression struts. The shear capacity of the beam depends primarily on the crack width, which is strongly influenced by the crack inclination. A mechanicaI model was developed by Reineck [55] based on a tooth model con sidering the shear carrying actions of friction along cracks, dowel force of longitudinal reinforcement, cantilevering action of tooth from the compression zone and shear force component in the compression chord. After a detailed analysis of the shear carrying actions involved, a stress field for a beam without transverse reinforcement was proposed and a truss model representing this stress field was developed. See Figure 6.5. This truss Chapter 6. Shear Failure of Beams Without Stirrups 103 model is similar to the one given by Adebar [54] except that the inclination of uniformly spaced diagonal cracks are assumed to be 60 degrees. The failure is characterized by a crack further propagating into the compression zone and breaking off the tooth, and is defined by the critical slip along the inclined crack which reaches a critical crack width. Based on an idea that a truss model must include a concrete tension member that will fail in tension before yielding of the flexural reinforcement, Al-Nahiawi and Wight [53] developed the truss models shown in Figure 6.6. The truss models have a 45-deg compression strut originating at the node under the applied load and a 35-deg compres sion strut originating at the support. The shear capacity of beams without stirrups is controlled by the failure of a concrete tension tie when its maximum tensile stress reaches the tensile strength of concrete subjected to transverse compression. They consider the failure of a concrete tensile tie as corresponding to the unchecked propagation of an in clined crack. One of conclusions from their models is that for beams with a/d> 1, shear failure modes are similar and indicated by the failure of concrete tensile ties. Muttoni and Schwartz [56] developed a structural model for the loading stage when a typical crack pattern has formed. See Figure 6.7. They indicated that the form of this critical crack pattern is accompanied by a collapse of three shear carrying actions, which are cantilever, interlocking and dowelling actions, and then the direct transfer of the load to the support also becomes impossible because of the wider crack. Their model shows that the compression strut turns in the central region of the beam and acts together with a concrete tie. Failure occurs either when the tensile strength in the region D is reached or when the strength in zone E, which is subjected to both tension and compression, is exceeded. After carrying out tests designed to test the validity of current design concepts, Kotsovos [57] has concluded that aggregate interlock and dowel action make a negli gible contribution to the load-carrying capacity and the strength of compressive zones Chapter 6. Shear Failure of Beams Without Stirrups 104 increases due to triaxial stress/strain states, thus making a significant contribution to shear capacity. In an attempt to summarize the experimental information, the concept of the ‘compressive force path’ has been developed by Kotsovos. The concept considers that the load-carrying capacity of beams without transverse reinforcement is associated with the strength of uncracked concrete in the region of the paths along which compres sive forces are transmitted to the supports. See Figure 6.8 (it should be noted that the original model presented by Kotsovos in this figure does not satisfy equilibrium). The shear failure is believed to be related to the development of tensile stresses mainly in the region of the path. The tensile stresses may be developed due to changes in the path direction, the varying intensity of compression stress field along the path and bond fail ure at the level of the tension reinforcement between two consecutive fiexural or inclined cracks, etc. More discussion about Kotsovos’ experimental work is given in Section 6.7. In summary, it is generally accepted that point loads can be transmitted directly to the support in beams either uncracked or cracked while a < d. However, there are different opinions about load transfer mechanisms for cracked beams when 1 <a/d < 2.5. Secondly, all the investigators, who tried to give rational interpretations of shear fail ure mechanism of reinforced concrete beams without transverse reinforcement, believe that the tensile strength of concrete plays an important role in the shear resistance for both uncracked beams and cracked beams, which is illustrated by various truss mod els composed of tension ties. However the interpretations of where and how the tensile strength of concrete is mobilized for shear resistance are different. Adebar [54], Reineck [55], A1-Nahlawi and Wight [53] considered that concrete tensile strength is needed for hanging up the shear force in the lower part of beams. Muttoni and Schwartz [56] con sidered that concrete tensile strength is needed in the upper part of beams to help the compression strut to deviate from the critical diagonal crack. In Kotsovos’ compressive force path concept [57], concrete tensile strength mainly contributes to the change of Chapter 6. Shear Failure of Beams Without Stirrups 105 the compressive force path direction. Thus it is evident that there is still no univer sally accepted shear failure mechanism for reinforced concrete beams without transverse reinforcement. 6.3 One Interpretation of Shear Failure of Beams without Stirrups In order to investigate the shear failure mechanism, the behaviour of a slender, longi tudinally reinforced concrete beam simply supported and subjected to two symmetrical point loads is examined herein. See Figure 6.9. The vertical flexural cracks first form in the pure bending region and then the additional cracks form in the shear spans between the concentrated loads and supports. The vertical cracks in the pure bending region keep propagating upward vertically with increasing load. However the flexural cracks formed in the shear spans, after extending vertically to longitudinal reinforcement level, will become slightly inclined toward the load. Traditionally it is believed that these cracks become inclined because of shear. As the loading is further increased, a characteristic inclined crack, the so-called critical diagonal tension crack, forms. Its propagation into the compression zone of the beam near the section of maximum moment is usually rel atively sudden, splitting the beam into two pieces and causing collapse. Above shear failure observations are well known and quite general. As reviewed briefly in Section 6.2, the various interpretations for shear failure mech anism given by previous studies are generally based on force transfer concept, which emphasizes the various actions involved in shear carrying mechanism and on the satis faction of equilibrium conditions. On the contrary, an alternate interpretation of shear failure mechanism is based on the deformation compatibility of a critical diagonal crack. It is very reasonable to choose the crack opening of the critical diagonal crack at the reinforcement level as an important geometric parameter to keep track of the crack Chapter 6. Shear Failure of Beams Without Stirrups 106 propagation. See Figure 6.10. If the crack opens horizontally only, the crack propa gates vertically, which is the case in the pure bending region. If the crack opening has both horizontal and vertical components, the crack propagation will deviate from verti cal direction and become inclined. This is the situation in the shear span range. The orientation of stable crack propagation can be approximately determined by the ratio of horizontal and vertical opening components. Figure 6.10 shows this geometric rela tionship. Test observations indicate that there are several inclined cracks in the shear span, all of which have the potential to become the critical diagonal crack. The shear failure observations show that only one of the inclined cracks, the critical diagonal crack, penetrates into the compression zone and extends horizontally to load point before the beam collapses. This means that of all inclined cracks, the critical diagonal crack that distinguishes itself from other inclined cracks must have a significant vertical opening component, which is geometrically compatible with this crack propagation, at the rein forcement level or the other locations along the crack. Then it is logical to infer that the formation of a horizontal crack, which introduces the significant vertical opening compo nent of the critical diagonal crack, is a critical stage in the shear failure of a reinforced concrete beam without transverse reinforcement. It is of interest that more information can be found for the phenomenon described above. The shear failure descriptions given in ACT 426 Committee report [38] are: Beams may exhibit a number of different modes of shear failure, the most common of which is the crushing or shearing of the compression flange over the inclined crack which is often accompanied or initiated by splitting along the tension reinforcement. It can be seen from this description that there are some uncertainties of which one occurs first in this statement, i.e., the crushing of the compression flange or splitting along the Chapter 6. Shear Failure of Beams Without Stirrups 107 tension reinforcement. Fortunately, there is valuable experimental evidence that can help to clarify this point. The evidence comes from the experimental investigation carried out by Chana [58]. Chana used a high speed tape recorder in conjunction with electrical demountable strain transducers to continuously monitor crack widths at critical locations while the tested beams were approaching failure. His test results clearly show that the beam shear failure was preceded by splitting along the longitudinal reinforcing bars. Another factor which is of help to understand this mechanism is the crack control efficiency of longitudinal reinforcement. The major function of longitudinal reinforcement in concrete beams is to compensate for the weakness of concrete low tensile strength and provide enough tensile strength to have high concrete compressive strength fully utilized. This function is well obtained in the pure bending region of the beam. After the vertical cracks have occurred in this region, their propagation are most efficiently controlled by the longitudinal reinforcement, which is at right angle to them. The crack opening at the reinforcement level is proportional to the deformation of the reinforcement if there is proper bond between concrete and reinforcement. Under a same load level a larger percentage of reinforcing bars results in smaller stresses in the bars (or smaller crack opening). On the other hand, the force system in the pure bending region also favours checking crack propagation. If more reinforcement is arranged at the bottom of the beam, more compressive force (or a larger compression zone) can be developed at the top of the beam. This compressive force effectively restrains the crack from penetrating vertically into the compression zone. However in the shear spans of the beam, the cracks are diagonal rather than vertical. The reinforcement is skew to the cracks. In order to check the propagation of the diag onal cracks, not only axial action but also dowel action of the reinforcement should be mobilized. The crack control of longitudinal reinforcement to inclined cracks is much less Chapter 6. Shear Failure of Beams Without Stirrups 108 efficient than to vertical cracks. The flatter the inclined cracks are, the less the crack con trol efficiency of longitudinal reinforcement. The crack control efficiency of longitudinal reinforcement is least when the cracks are horizontal. From the above discussion, an intuitive and rational conclusion about shear failure mechanism of slender, longitudinal reinforced concrete beams is that splitting along the longitudinal reinforcement, which produces a considerable vertical opening component to the critical diagonal crack, is the immediate cause of shear failure. In general, a distinct characteristic of slender beams without transverse reinforcement is lack of the ability to control horizontal crack propagation and the brittleness of shear failure results from the occurrence of either flat inclined or horizontal cracks. Should the defect have been overcome by any mechanism, the shear resistance behaviour will be greatly improved. Traditionally, the transverse reinforcement in conjunction with longitudinal reinforcement is provided for shear design of structural concrete members. 6.4 Bond Effect in Longitudinally Reinforced Concrete Beams It was concluded above that the horizontal splitting along longitudinal reinforcement, which precedes the propagation of a critical diagonal crack into compression zone, indi cates shear failure of a beam without transverse reinforcement. In this section the load carrying mechanism before the occurrence of this horizontal splitting and the cause of the horizontal splitting are investigated. The influence of bond between concrete and reinforcing bars upon the load transfer mechanism is focused on as so much information from previous studies [13, 59-67] indicates that the bond has a significant effect, which was either ignored or considered to a less extent in previous studies of shear failure of beams without transverse reinforcement. Kani [13] tested a series of point loaded and simply supported beams without web Chapter 6. Shear Failure of Beams Without Stirrups 109 reinforcement to investigate the influence of various bond qualities on the shear failure mechanism. By introducing an intermediate layer of a vermiculite-cement mix with different elastic properties and strengths between the reinforcement and the concrete in the shear span, the varying bond quality (varying average ultimate bond stress) was obtained. The tests showed that “the better the bond, the lower is the load capacity of the beam.” As the extreme case, the no bond beam presented flexural failure with no single crack occurring in the shear spans. Though the test results did not tell the whole story about the shear failure mechanism of beams without transverse reinforcement, they did reveal the fact that bond plays an important role in load sustaining mechanism. The second example showing the interaction of bond and shear failure is the bond tests done by Ferguson et al. [61, 62] and Kemp et al. [66]. Ferguson et al. tested beams to investigate development length of reinforcing bars in bond. They found that there were always the combinations of diagonal tension failure and bond splitting failure for the narrow beams that were tested. Kemp et al. [64, 66] observed that the shear crack formed only after a longitudinal bond crack and bond slip had occurred for the cantilever-type bond specimens tested (more discussions on these tests are given in the next chapter). The third example is the experimental work by Mains [59]. His measurement of bond stresses along reinforcing bars in beams indicated that the distribution was not uniform and that there was a significant localization after the occurrence of the critical diagonal crack. This evidence conflicts with the basic assumption adopted by many authors that the stress variation of longitudinal reinforcement in cracked beams has a similar shape as the moment diagram. 6.4.1 Bond Influence in Uncracked Beams In order to investigate the effect of bond on the load transfer mechanism of reinforced concrete beams without web reinforcement, a simply supported and centrally loaded Chapter 6. Shear Failure of Beams Without Stirrups 110 beam was studied to find the internal stress distributions in the uncracked concrete. See Figure 6.11. Based on the work of Schlaich et al. [3], this beam can be divided into B-regions and D-regions, which are also shown in the figure. The prediction using classical beam theory, which assumes that cross sections remain plane during deformation (Bernouli assumption), is only valid for B-regions and not valid for D-regions so that the linear elastic finite element method was used to investigate the effect of bond on the internal stress distributions. Nine node Lagrange quadratic plane elements were used in analysis. The biaxial stresses in x and y directions and principal stresses were calculated at each node of the element. First, perfect bond and no bond cases for slender beams were studied. The beam was assumed to already have a single flexural crack formed at mid-span. Because of the symmetry of the problem, only half of the beam was modeled. See Figure 6.12. In Figure 6.12(a) the force T modelling the tension force in the reinforcement is applied at the the front of the beam, which is the simulation of the case with perfect bond between the longitudinal reinforcement and concrete. In Figure 6.12(b), the corresponding force T is applied at the end of the beam to represent the no bond case. Final analysis results, showing the internal stress distributions, are also given in Figure 6.12. The shear stresses, in both the perfect bond case and the no bond case, give similar distributions in the B-region. The shear stresses are confined within a certain part along the cross sections in the D-regions and distributed uniformly in the B-regions. The shear stress distributions demonstrate the shear transfer mechanism in the uricracked beams, but do not indicate the load transfer mechanism. The fundamental difference between these two internal stress distributions is that the whole cross sections are subjected to compression for the no bond case whereas the cross sections are subjected to tension on the bottom and compression on the top for the perfect bond case. The load transfer mechanism can be found by looking at the internal force flows in Chapter 6. Shear Failure of Beams Without Stirrups 111 the beam. Figure 6.13 shows two principal stress trajectories for perfect bond and no bond situations respectively. In Figure 6.13(b), the load is transferred to the support directly by a compression stress field (compression strut). The internal force flow is consistent with the externally applied loads. The resultant of forces at the support and the resultant of forces at the upper compression zone act along the same line but in opposite directions. In Figure 6.13(a), the force system is different from that in Figure 6.13(b). As a requirement of static equilibrium, the action line of the resultant of forces at the upper compression zone will still go through the point where the support reaction and the tension force in the reinforcement intersect. However the force flow in the beam is curved rather than straight so that the tensile strength is mobilized to make the force flow change direction. Though the force flow in Figure 6.13(a) is different from that in Figure 6.13(b), this force flow is consistent with the corresponding external force system. As there is no horizontal force applied at the bottom corner of the beam, the force flow originating from the support is almost vertical. Then the additional internal force (tension force) is required to make the force flow from the upper load point to the support possible. It has been mentioned before that the load can be directly transferred to the support in deep beams (a/d < 1). The transverse splitting of compression struts dominates shear failures of deep beams. In this section the influence of various bond stress distributions upon the transverse splitting of compression struts is also investigated. Various bond stress distributions are shown in Figure 6.14. The analysis results are summarized in Figure 6.15. The bond influence on the transverse tension of the compression strut is concentrated at the lower end, where the transverse tensile stresses are amplified due to bond. For the no bond case, the transverse tensile stress is almost uniformly distributed along the inclined compression strut. The shear stress distributions along the cross sections for various bond cases are similar to those in Figure 6.12. Chapter 6. Shear Failure of Beams Without Stirrups 112 Bond Influence in Cracked Beams 6.4.2 In order to investigate the bond effect upon the load transfer mechanism of cracked beams, the traditional concepts of arch action and beam action [68] are used in a sim ple equilibrium analysis procedure. It is believed that the bond effect can be better understood by using these simple concepts. For a cracked beam in the shear span, the relationship between external moment and internal moment of resistance, at a distance x from the support, can be approximated as [68] M where T ment, j = = = Tjd (6.1) T(x) =tensile force resultant acting at the centroid of longitudinal reinforce j(x) =variable coefficient and d =effective depth measured from the extreme compression fibre to the centroid of longitudinal reinforcement. The shear force may be expressed as V = dM/dx. Hence, by means of Equation 6.1 one obtains V = jd + T- (6.2) where V is the constant shear force acting through the shear span. The first term at the right-hand side of Equation 6.2 is usually termed the “beam action”, which reflects the change of the force in reinforcement, and the second term is called the “arch ac tion”, which represents the inclination of the internal thrust force. The simultaneous occurrence of both actions require the corresponding internal stress distribution and the compatibility of the deformation within the beam. A half beam with idealized flexural cracks is shown in Figure 6.16. The shear resis tance modes of the varying tension force in reinforcement with the constant lever arm and the constant tension force in reinforcement with the varying lever arm are the two Chapter 6. Shear Failure of Beams Without Stirrups 113 extreme cases, which could be termed the “pure beam action” and the “pure arch ac tion” respectively. The internal stress distributions for the “pure beam action” are well known and first described by Mörsch [69]. See Figure 6.16. The shear stress is calcu lated by horizontal equilibrium of a piece of beam element and is found to be uniformly distributed along the cross section. However, the shear stress for the “pure arch action” is concentrated as shown in Figure 6.12(b). The shear stresses combine with horizontal compression stresses to produce a resultant which travels straight from the load point to the support. The internal shear stress distribution, for combined “beam action” and “arch action,” is somewhat different from that of either one action. The same half beam, as shown in Figure 6.16, is again given in Figure 6.17, and a plot of the tension force in the longitudinal reinforcement is also given. It is assumed that the combined beam action and arch action exists in the range of from the load point to section m — m with the resultant thrust force shown in Figure 6.17(a). Correspondingly the variation of the force in the longitudinal reinforcement is shown in Figure 6.17(b). Then the shear stress distribution along section n. — n., coming from the contributions of both actions, can be determined. One part of shear stresses is calculated by equilibrium consideration, in which the shear stress is related to the bond stress between the concrete and the reinforcement, as shown in Figure 6.16. Another part can be determined from the vertical component of inclined thrust force. Summing up these two parts should be equal to external applied shear force. See Figure 6.17(c). Equation 6.2 is the mathematical expression of Figure 6.17. 6.5 Shear Displacements along Cracks After the shear stress distributions along the idealized vertical cracks have been examined above, the corresponding shear displacement along the cracks is investigated in this Chapter 6. Shear Failure of Beams Without Stirrups 114 section. It is generally believed that there are two kinds of deformations in cracked beams, which could introduce the relevant shear displacements along the cracks [70]. One is the flexural rotation of the compression zone and another is the bending within a concrete tooth caused by bond force LIT. For the idealized vertical flexural cracks in the shear span, the compatibility of de formations illustrates that only the bending of the concrete teeth causes the shear dis placement along the cracks and thus develop uniform shear stress distributions along the cracks, which is consistent with the assumed beam action. See Figure 6.18(a). The rota tion of the compression zone makes the vertical crack opening and does not introduce the shear displacement along the crack. This latter deformation for idealized vertical cracks is only consistent with the beam elements subjected to pure bending moment, where no bond force and no shear stresses will be developed. However, the existence of the shear force in the shear span makes the cracks inclined rather than vertical so that the both deformations have contributions to the shear dis placement. See Figure 6.18(b). The shear displacement will still be uniform along the inclined crack, if only the bending of the concrete teeth is considered. The shear dis placement due to the rotation of the compression zone will not be uniform, with the largest shear displacement occurring at the mouth of the crack. One of the important conclusions about the shear displacement along the inclined cracks is that the flatter the cracks are, the more shear displacement will be introduced along the cracks for the same compression zone deformation. 6.6 Load Transfer Mechanism Based on the previous detailed analysis, an insight into the load transfer mechanism can be obtained by examining the shear failure process of a simply supported beam subjected Chapter 6. Shear Failure of Beams Without Stirrups 115 to the point loading. When the beam is loaded to a certain stage, the obvious aggregate interlock at the critical inclined crack is mobilized with the company of the occurrence of considerable shear displacement along the crack due to the increasing load and the reduced compres sion zone (i.e., the increasing bending of the tooth and the increasing rotation of the compression zone). This can be related to the experimental observation that only the critical crack keeps opening and the widths of other cracks remain almost unchanged under increasing loading. In addition, the shear displacement along the crack is always accompanied by the crack opening for shear transfer by aggregate interlock action due to the rough contacting surfaces. See Figure 6.19. At this stage, it can be considered that both beam action and arch action already exist for shear resistance, based on the previous analysis. With the loading being further increased, in addition to the shear stresses, the compression stresses along the crack will also be introduced as a result of the longitudinal reinforcement’s constraint to the crack opening. These shear stresses and compression stresses acting on the crack are one of the direct reasons for the occurrence of the second diagonal crack, which is commonly observed in shear failures of longitudinal reinforced concrete beams. See Figure 6.20(a). As the loading is approaching this stage, just before the occurrence of the second diagonal crack, the full arch action can be approximately assumed in the range from the load point to the critical crack. See Figure 6.20. This assumption is justified by measurements of reinforcement strain and concrete strain in the shear span [59, 71, 72], an example of which is also shown in Figure 6.20(d). If full arch action is assumed in the range from the load point to the critical crack, and full beam action with a reduced lever arm is assumed in the range from the critical crack to the support, the variation of tension force in reinforcement is shown in Figure 6.20(b). Because the magnitude of bond stress is proportional to the gradient of the tension force in reinforcement, larger bond stresses will be developed near the support. Chapter 6. Shear Failure of Beams Without Stirrups 116 If the variation of the tension force is further assumed as a solid line shown in Figure 6.20(c), the bond stress will be amplified locally in the zone near the critical crack. This assumption can be fully confirmed by Mains’ measurements [59]. See Figure 6.21. It can be concluded that the second diagonal crack or even splitting along reinforcement partly results from this high bond stress. A possible load transfer model is shown in Figure 6.22. The full arch action is assumed in the range from the load point to the cross section n — n where the critical crack occurs. Then the resultant compression force on the cross section n — n, F, should be as shown in the figure. Because the compression force flow N originating from the support is not along the same line as F, the tension force T is required for equilibrium. The F. has two components, one meeting with the force N and another going downward to intersect the longitudinal reinforcement. 6.7 Interpretation of Some Beam Test Results With the help of the model presented above, it is possible to give an alternative interpre tation of the experimental information presented by Kotsovos in Ref.[57, 73] and other researchers in Ref.[56, 58, 74, 75]. Kotsovos [73] tested a series of simply supported beams with various arrangements of shear reinforcement, and subjected to two-point loading with various shear span to depth ratios (aid). See Figure 6.23. The main test results are given in Figure 6.24 which shows the load-deflection curves of the beams tested. Series C and D beams were found to have a load-carrying capacity significantly higher than that of series A beams which had no shear reinforcement throughout their span. Series D beams, in all cases, exhibited a ductile behaviour, which is indicative of a flexural mode of failure, and their load-carrying capacity was higher than that of series A beams by an amount varying from 40 to 100% Chapter 6. Shear Failure of Beams Without Stirrups 117 depending on a/d. In addition, near the peak load the inclined crack of series D beams had a width in excess of 2 mm. Kotsovos considers that the test results are conflict with the concept of shear capacity of critical sections, the view that aggregate interlock makes a significant contribution to shear resistance, and even the truss analogy concept. Chana [58] tested beams with traditional internal stirrups, but locally arranged as shown in Figure 6.25. Kim et al. [74] tested beams with external stirrups. See Figure 6.26. The prestressed external stirrups were provided at the outer third sections of beams. Muttoni [56] and Kuttab [75] tested beams with the arrangements of reinforcement shown in Figure 6.27 and 6.28 respectively. All beams failed in ductile modes and reached failure loads which are almost the full flexure capacities of the beams. The shear capacity of these beams was about double the capacity of similar beams without the nonconventional transverse reinforcement. The test results from Kotsovos or the other researchers are very interesting, but not that surprising. The common point of these tests was that the beams were provided with a mechanism that gave effective constraint to the development and propagation of flat (or horizontal) cracks in the critical zones, making the critical diagonal crack more stable. In reality, the arrangements of reinforcement used by these researchers can be considered as various modifications of the conventional form in which the stirrups are placed throughout the whole beam and anchored in the compression zone. 6.8 Conclusions Based on the study in this chapter, some conclusions are arrived at: 1. A distinct characteristic of slender beams without transverse reinforcement is the lack of ability to control horizontal crack propagation and the brittleness of shear failure results from the occurrence of either flat inclined cracks or horizontal cracks. Chapter 6. Shear Failure of Beams Without Stirrups 118 2. The shear stresses in both the perfect bond case and the no bond case have similar distributions along the cross sections in the B region of the beam. This suggests that simply focusing on on the shear stress distributions is not adequate for the investigation of shear failure mechanism. How a load transfers is much more im portant than shear stress distributions. 3. As the entire cross section of a beam is subjected to compression stress and the shear stress in the no bond case, no inclined cracks will be developed in the shear span. This was illustrated in the experimental work by Kani [13]. 4. Bond has an influence upon the transverse splitting of compression struts of very deep beams (a/d < 1); however this influence can be ignored because the deteri orated bond in deep beams with inclined cracks will reduce this influence signifi cantly. 5. Generally, the shear stress distributions over the cross section of cracked beams are not uniform. The localized shear stress distributions can be considered as the direct result of arch action or vice versa. 6. The ideal direct load transfer mechanism or “pure arch action” can only be realized in the no bond condition, regardless of the shear span ratio a/d. The form of the direct compression strut between the point load and the support depends on the resultant forces acting on the both ends of the strut. These resultant forces on two ends of the strut should act along the same line and in opposite directions. Consequently, the inclination of the strut originating from the support cannot be arbitrarily chosen when the truss models are developed. 7. In addition to dowel action, which must be introduced due to the shear displacement along the crack, the locally amplified bond stress at the zone near the critical crack Chapter 6. Shear Failure of Beams Without Stirrups 119 contributes to the occurrence of the second diagonal crack and the splitting parallel to the longitudinal reinforcement. This bond problem has also been recognized by other researchers [76] but from a different point of view. Chapter 6. Shear Failure of Beams Without Stirrups 120 h/4 — ‘I’ S. —-S. I’ / I — hi I i h/2 g I’ ‘1’ 1 , I I I ‘ i = (b) a/h 0 = , , - = Isostatic Compressive Curve 0.5 , , — — .- .. Z’’ 1 F / a=2h a=h Cd) a/h -r — Cc) a/h 0.25 ._._— / i ‘ / I, h/2 I / ‘ 5---, (a) a/h Distribution ot Transverse Compression — /7 ....7—-—j--------4.F hCT> I I : / , , 7 7 ,•• , I F ‘ .. ‘- — — / F I Ce) a/h = 2 Figure 6.1: Distribution of transverse compressive stress for various shear span ratios. from Mau and Hsu, Ref.[52]. Chapter 6. Shear Failure of Beams Without Stirrups 121 F” j .‘ h IS,,, Lrc f , ‘I -*k-/ / I , \ , , I j, z-v. Figure 6.2: Load near the support: transition from deep beam to slender beam, from Schlaich et al., right side simple models; left side refined models, Ref.[3]. Chapter 6. Shear Failure of Beams Without Stirrups • 6 x 6 x o 25 69 - 1 122 in, (152 x 152 o V •67 25 mm) plate 6 *9 x 2 in, (152 x 229 x 51 mm) plate 3*0.38 in. (152* 76* 95mm)plate 6 V ia 0.20 }m) V 0.15 l 3940 psi (272 MPa) bdf max. aag.- 3/4 in. (19mm) d21.2in. (538mm) b6.1in. (155mm) ) 2 2 (2277 mm As 3.53 in 0.10 - f.=53.9 ksi (372 MPa) • 72 081 65 .76 U.V.J 0 71 063 sectionaI model strut and tie model 0 066 I 0 1 2 3 4 5 6 7 aid Figure 6.3: Predictions of shear strength versus a/d ratio for tests reported by Kani[13j, from Collins and Mitchell, Ref.[39}. Chapter 6. Shear Failure of Beams Without Stirrups 123 V vJ I (a) Geometry and Loading r iiV 1 d, f V4j 2 J Ib) Truss Mode! Figure 6.4: Truss model developed by Adebar, Ref.[54}. F — / / \Nc I’- _\_. — — 1L fv Figure 6.5: Truss model developed by Reineck, Ref.{55}. Chapter 6. Shear Failure of Beams Without Stirrups h T * O 450 * 450 Figure 6.6: Truss models developed by Al-Nahiawi and Wight, Ref.{53j. Figure 6.7: Structural model developed by Muttoni and Schwartz, Ref.[56J. 124 Chapter 6. Shear Failure of Beams Without Stirrups 125 ‘4 I I Figure 6.8: Structural model developed by Kotsovos, Ref.[57J. 24j Z/f/)f Figure 6.9: Crack pattern of a beam tested by Kani, Ref.[13}. Chapter 6. Shear Failure of Beams Without Stirrups 126 1 HHA Figure 6.10: Geometric relationship of a crack at reinforcement level. Chapter 6. Shear Failure of Beams Without Stirrups Figure 6.11: A simply supported and central loaded beam. 127 o o I CD CD c-l I-. C, C’, CD CD b-’. ij 0 0 C -‘ cD (I) Cl) 0 Cl) Cl) z 0 ZQfffl 1IIP -1 I -uvi11J rrrIrJnrrr1a 11]y- 0 -S cD 7 IUI[EW cfIfjIL1LIw r-- ±: - -- . - - -. -. .111 .III L’3 I-’ I CD Chapter 6. Shear Failure of Beams Without Stirrups 129 I — — — — — C. — / / ,/ — — — — — —, - .- — — — / — — — ‘- — — 1’! I / I I I — K — c_ — .‘ / — / — / / / i’, / 7 1 / I I I I I I I I — — ---------- I 1 — — 7//il — / — — / — C C /7/ / / / — — -, / 1 ,) / / ///‘ 7 / I --- .— — — / ------ - (a) ----- - — — -, — — — — c — — — — — — .-. — — , - — — — .- .- — / — t.t. — ./ — -- — —, . ... — ,, — / — 2 — — — •_ / — — —. — — — _ — — _ — / — — — — / — — _• — / / — — ,, / ; / — — — — .. / - H — - — .- .- — — C C — — — — — — ; — — — ./. (b) Figure 6.13: Internal force flows in uncracked beams: (a) perfect bond case and (b) no bond case. Chapter 6. Shear Failure of Beams Without Stirrups 130 -1 - - - - - - - - — - -i - ———----—--q - - - —- - — — - — — — ———- - - — - — — - (a) - - (b) - ————-- —1 - - - - - -— - - - - ————- - -——-- - - -——- - ———- LU LU (c) (d) Figure 6.14: Modelling of bond effect upon transverse splitting of compression struts in deep beam. Chapter 6. Shear Failure of Beams Without Stirrups 131 1T ff/ / 9. /\/ \____ (a) (b) (c) (d) Figure 6.15: Bond influence upon transverse tensile stresses of compression struts in deep beam. Chapter 6. Shear Failure of Beams Without Stirrups 2 T 132 b 2 1 ;5z:--- —I—- ___ 1 T—’ C+AC jd T+AT b id — a T T+ AT 1 Ax longitudinal equilibrium longitudinal stresses shear stresses Figure 6.16: Internal stress distributions of cracked beams due to pure beam action, from Ref. [39]. Chapter 6. Shear Failure of Beams Without Stirrups m 133 m (a) pure arch action (b) + (c) beam arch combined Figure 6.17: Shear stress distributions of cracked beams due to combined beam action and arch action: (a) modelling of combined beam and arch actions; (b) tension force in longitudinal reinforcement; (c) shear stress distribution at section n n. — Chapter 6. Shear Failure of Beams Without Stirrups 7 7 (a) 134 // /77 (b) Figure 6.18: Shear displacement along cracks in shear span: (a) vertical cracks and (b) inclined cracks. Chapter 6. Shear Failure of Beams Without Stirrups 135 V+dV VtOV V+dV Figure 6.19: Shear transfer at cracks by aggregate interlock. Chapter 6. Shear Failure of Beams Without Stirrups 136 n C second (a) lv n jT (b) (C) Cracked Concrete rr. ‘ 4. NJ Cracked Steel Figure 6.20: A beam with inclined cracks in shear span: (a) strut force due to arch action; (b) one assumed tension force in reinforcement; (c) another assumed tension force in reinforcement; (d) measured concrete strain and tension force in reinforcement from Ref.[71]. Chapter 6. Shear Failure of Beams Without Stirrups 137 / Measured \ steel tension Calculated teéI tension I Critical flexural 4 crack section (a) •l IRLI&ILP\JeL/ I. I I (b) Figure 6.21: Measured force in bar, bond stress and crack locations, from Ref.[59]. Chapter 6. Shear Failure of Beams Without Stirrups 138 n -4 (a) (b) Figure 6.22: A load transfer mechanism just before the occurrence of splitting along longitudinal reinforcement: (a) truss model and (b) tension force in reinforcement. I I I V Cl) C o Cl) 0 CDI cJ CD C’) CD o CD CD c.. JCD CD O 0 0 U’ C a, D C) W W 0 ’L 0 WII iiI iooI •1____ a p UI 1• II II Ii i i I I II ‘ I II Iiii 0 0 III I II I ‘I I 01W o I I I a, TI II. ii i 0’ 1W I-. I,IC I• • I I I I I I w I—. II.. I I I I I I It II • I W I!0 ‘Tn 100 H 90 —i I-’- I. ‘. yb (ID 0 CD cJ- Chapter 6. Shear Failure of Beams Without Stirrups a) a) 1 2 2 I x ci) — — 140 -\ZE 0 0S 0 06 C - 0 002- 00? 0 c .2 /c5:— 0 I • 5 10 15 20 25 I I 2 0 I 1. 6 8 10 l2 deflection mm deflection mm - - (b) (a) ci) I- 12 B 2 - D x 0 •0 Ca .2 V a) 0 0.2 - 0 V .2 0 0 2 L 5 8 10 12 central deflection mm - (c) Figure 6.24: Load-deflection curves of beams shown in Figure 6.5: (a) a/d a/d = 3.3 and (c) a/d = 4.4, from Kotsovos, Ref.[73]. = 1.5; (b) Chapter 6. Shear Failure of Beams Without Stirrups 141 35 Linknos 321 30 lW 25 Lnk 2/ / Failure of span without links at 9SkN 20 15 Lnk I - /k3 1/ 10. Failure load =158kN 0 20 40 60 80 120 100 TOTAL SHEAR FORCE - 140 160 kN Figure 6.25: Beam with internal stirrups tested by Chana, from Ref.{58]. + L . 4 J — L 1 25 f -—r- I I IO7 II .iLIPS S I I PS 24.1 *IPS 4. NUT SECT ION A—A Figure 6.26: Beam with external stirrups tested by Kim et al., from Ref.[74j. Chapter 6. Shear Failure of Beams Without Stirrups S ‘ —— — — ,/. 142 / (b) Figure 6.27: Beams tested by Muttoni et aL, from Ref.[56]. Path of Compressive Force / n-i 11 Path of Compressive Force ‘ I [I_ujH1_rH I I I I aid = 2 aid = 3.6 .1 I I I Figure 6.28: Beams by Kuttab et al., from Ref.[75]. Chapter 7 Bond Splitting Failure 7.1 Introduction The main objective of this chapter is to deal with the question of what causes the horizon tal splitting along longitudinal reinforcement in a beam without transverse reinforcement. It is generally believed that splitting of concrete along reinforcing bars in a beam occurs primarily due to the combined effect of wedging action of bar deformations (bond) and dowel action of reinforcement. Ferguson et al. [61, 62j focused on bond splitting. They noticed that the existence of a diagonal crack resulted in lower bond strengths. Based on an extensive experimental study, Cergely [77] concluded that the dowel force is the most important factor producing splitting in beams without stirrups, and that the dowel effect overshadows the pure bond effect in most situations, especially in beams with small bar spacing. Jimenez et al. [78] found that bond strength and dowel capacity are independent of each other. Kemp et al. [66] observed from their experimental work that there is a weak interaction between dowel force and bond resistance until approximately 80 percent of the pure dowel capacity is reached at which time bond capacity decreases very rapidly. They suggested a 20 percent reduction in design ultimate bond to include the effect of dowel action. The conclusions arrived at in Chapter 6 about the load transfer mechanism prior to the occurrence of splitting, are consistent with above mentioned experimental observations. Dowel action and severity of bond stresses (i.e., amplification and localization) will both 143 Chapter 7. Bond Splitting Failure 144 be introduced due to the occurrence of the critical diagonal crack. Consequently, dowel action and bond action both will make contributions to splitting along reinforcement. To date no satisfactory quantitative analysis results are available to describe the interaction of these two actions (bond and dowel). Traditionally bond and shear have been dealt with separately so that previous investigations on bond splitting were mostly concerned with ultimate bond splitting failure. However, the conclusions arrived at in Chapter 6 make it evident that the occurrence of horizontal splitting along reinforcement in a critical zone indicates shear failure of beams without transverse reinforcement. Then it seems that the initiation of bond splitting could be the more appropriate criterion than the ultimate bond splitting strength for the study of shear capacity of beams without transverse reinforcement. In this chapter, the attention is focused on bond splitting with an objective to develop a tentative design criterion for the initiation of longitudinal bond splitting. The chapter is divided into three parts. In the first part, previous studies are briefly reviewed. This includes general bond actions, experimental studies, as well as bond splitting strength (ultimate and initiation). A proposed design equation for bond splitting initiation is presented in the second part. Finally, some comments are given about Ferguson’s [61, 62] experimental work and a number of conclusions are drawn. 7.2 General Bond Action Studies of bonding forces for plain reinforcing bars and deformed bars by Lutz and Gergely [79] showed that bond for plain bars is made up of three components: (1) chemical adhesion, (2) friction and (3) mechanical interaction between concrete and steel. When plain bars without surface deformations are used, bond depends mainly upon chemical adhesion, and after slip, upon friction. There is also some negligible mechanical action Chapter 7. Bond Splitting Failure 145 due to the roughness of the bar surface. With use of deformed bars, the main reliance is changed to bearing of lugs on concrete and to shear strength of concrete sections between lugs. Because slip of deformed bars can occur in two ways namely: (1) the lugs can split the concrete by wedging action and (2) the hugs can crush the concrete, two types of bond failures can occur. If the surrounding concrete resistance is moderate, as it is for ordinary concrete cover, the lugs of large steel bars can split the concrete without crushing it. With small bars or with large cover over the bars, the lugs will shear the concrete and pull out without splitting the concrete. Bond between concrete and a deformed reinforcing bar that is subjected to a pull-out force as well as is with a moderate concrete cover, can be characterized by the relationship between averaged bond stress along the embedment length and slip at the loaded end, with four different stages as shown in Figure 7.1 [80]. In Stage 1. (small values of the bond stress), bond is assured by chemical adhesion, and no bar slip occurs. In stage 2 (larger bond stress values), the chemical adhesion breaks down and bonding is assured by bearing action or wedging action of the bar lugs. In Stage 3 (still larger values of bond stress), the first longitudinal cracks form as a result of the increasing wedge action of lugs and more bond stress can be sustained by the interlock between concrete and reinforcement lugs. Once the longitudinal cracks break out through the whole cover, failure occurs abruptly in this stage if no transverse reinforcement is provided. If enough transverse reinforcement is provided, the confinement exerted by the reinforcement would allow the bond stress to reach a larger value in spite of concrete splitting. See Stage 4 in Figure 7.1. For bond related shear failures of structural concrete members without shear rein forcement, the confinement provided by transverse reinforcement is not available. Hence only Stage 3 described above is relevant to the problem to be dealt with. The beginning Chapter 7. Bond Splitting Failure 146 and end of Stage 3 indicate splitting initiation and ultimate splitting failure. Both split ting initiation and splitting failure are investigated in the following. But it is believed that the bond related shear failure is more relevant to splitting initiation than to ultimate splitting failure, as dowel action, which can aggravate the bond splitting problem, will always be introduced due to the occurrence of diagonal cracks in beams. 7.3 Previous Experimental Studies In the history of bond research, the problem most considered has been bar development length d, 1 which is the embedment length necessary to assure that a bar can be stressed to its yield strength without failing in bond. In order to determine bond stresses or development length, a variety of test methods have been used. Typically there are ordinary pull-out tests (also denoted as concentric pull-out test), eccentric pull-out tests, full-beam tests and semi-beam tests (often called stub-beam or cantilever tests). In Figure 7.2 ordinary pull-out test specimens and eccentric pull-out test specimens commonly used by researchers are shown. It was believed by Ferguson et al. [60] that the ordinary pull-out test is not entirely realistic as the measure of bond strength in beams, because it carries no shear on the splitting plane. In a beam, a short 1x length of beam, as shown in Figure 6.16, transfers the change in bar tension by bond stress into a horizontal shearing stress, a considerable part of which acts on the section through the level of bars. Also the loaded end of a concentric pull-out specimen is in compression and is restrained due to friction forces. Hence the concentric pull-out test gives higher bond strengths than those expected in beams, especially where splitting is an important factor. The eccentric pull-out test [Figure 7.2(b) and (c)] greatly improves the disadvantages mentioned for the ordinary pull-out test and keeps the advantage of simplicity. Chapter 7. Bond Splitting Failure 147 Full-beam tests are considered most reliable because the influences of both transverse shear stresses and flexural tension cracks are included. One kind of full-beam test used at the University of Texas [61, 62] is shown in Figure 7.3. In order to eliminate the influence of support reaction upon bond strength, the development length of the bar is placed in a negative moment region, where the L” shown in the figure is the development length investigated. However the full-beam tests are very expensive test procedures and difficult to generalize for different variables. Semi-beam specimens (or stub cantilever specimens), have been utilized at Cornell University [77] and West Virginia University [66], as well as in Japan [81]. See Figure 7.4. This type of specimen still provides a realistic strain gradient through the depth of the specimen as in beams, but the cost is much lower than that of full-beam specimen tests. In addition, it is quite versatile since ratios of bond, shear and flexure can be easily varied from one specimen to another. In summary, in order to get correct information about bond splitting in beams without transverse reinforcement, full-beam test procedures are most preferred. Semi-beam tests and eccentric pull-out tests are satisfactory substitutes for full-beam tests. The test results from semi-beam specimens and eccentric pull-out specimens can be considered applicable to real beams. 7.4 Previous Studies of Ultimate Splitting Failure The generally accepted bond mechanism, when wedging action has been mobilized, is that reinforcing bar force is transferred to surrounding concrete by inclined compressive forces radiating out from lugs on reinforcing bars and making an angle with the bar axis. The inclined compressive forces can be decomposed into radial and tangential components. The radial components (bursting forces) are balanced by circumferential tensile (ring) stress in the surrounding concrete. For the case of concern to this study, Chapter 7. Bond Splitting Failure 148 longitudinal cracks (splitting) appear when the tensile rings, which are usually weakest in the thinnest concrete cover protecting the reinforcement, are stressed to the tensile strength. See Figure 7.5. Several factors can affect bond splitting strength. These are concrete cover or clear spacing of bars, concrete tensile strength (which is related to the compressive strength f), embedment length and bar diameter etc. Orangun et al. [82] proposed an approach for determining bond strength or develop ment length, that included all the variables mentioned above. The approach is based on a physical bond model. The radial forces, generated between the lugs and the surrounding concrete, can be regarded as water pressure acting against a thick—walled cylinder with an inner diameter equal to the bar diameter and a thickness c that is the smaller of the clear bottom or side cover cb or 1/2 the clear spacing c 8 between adjacent bars. See Figure 7.6. The capacity of the cylinder depends on the tensile strength of the concrete. With cb greater than 3 c / 2, a horizontal split develops at the level of the bars and is termed a “side split failure.” With 3 c / 2 greater than cb, a “face-and-side split failure” forms with longitudinal cracking through the cover followed by splitting through the plane of the bars. When c/2 is much greater than cb, a “V-notch failure” forms with longitudinal splitting followed by inclined cracks that separate a V-shaped segment of cover from the member. From the results of 62 beam tests, an equation of the bond stress was developed by using a nonlinear regression analysis. That is u = 1.22 + 3.23c 53db + db (7.1) in which u is the ultimate bond strength; c is the smaller of c 8 or c; c is the smaller of one half of clear spacing or side cover; cb is the concrete cover; d 6 is the bar diameter; and id is the development length or splice length (all units are in psi and inches). This equation was further modified by rounding the coefficients to obtain a somewhat Chapter 7. Bond Splitting Failure 149 more conservative value for u, denoted as uj d, (7.2) d 1 Orangun et al. [82] compared the bond stresses calculated by Eq. 7.2 to test results obtained from a total of nine studies (over 500 tests) of splice and development strength for bars not confined by transverse reinforcement. The predicted strengths gave a close match with the test results. According to the recent work by Darwin et al. [83], the dimensionless equations Eqs. 7.1 and 7.2 for bond strength can better be expressed as the tension force in reinforcement in terms of the same variables. They believed that bond force provides a better measure of member response than bond stress, since bond strength can be considered as being a structural property rather than a material property. The recommended equation is = where Ab is bar area and 7.5 f 3ld(c + 0.4db) + 200Ab (7.3) is steel stress at ultimate bond strength. Previous Studies of Splitting Initiation It should be noted that the previous study results presented above are interesting but only related to ultimate bond splitting strength. Equations 7.1 and 7.2 are valuable for bond development design if shear and bond problems are dealt with separately. As a matter of fact, shear and bond are always co-existing and interactive, therefore the initiation of bond splitting is most concerned in this study. Unfortunately little work has been carried out on bond splitting initiation in comparison with the work done on ultimate bond splitting failure, especially experimental investigations. Chapter 7. Bond Splitting Failure 150 Using the thick-walled pipe analogy, Tepfers [84] performed an analysis of bond crack ing for the concrete in uncracked elastic state, plastic state and partly cracked elastic state. See Figure 7.7. By setting the known maximum tensile hoop stress equal to ma terial tensile strength and approximating the angle of inclined compression forces as 45 degrees, expressions to predict splitting bond stress were developed for above mentioned different states. These expressions are: ft i 12 -r) + 1)2 + ( — u12 ) 7 ()2 (7.5) db = (0.3 + 0.6-) (7.6) where Equations 7.4, 7.5 and 7.6 are for the uncracked elastic state, uncracked plastic state and partly cracked elastic state respectively, and tt is the bond stress indicating crack initiation, c is the thickness of cover, db is the diameter of a deformed reinforcing bar and f is concrete tensile strength. Figure 7.8 compares Tepfers’ [84] predictions with his eccentric pull-out test results and other test results [85]. It can be seen that the plastic prediction gives the upper bound solution, while the partly cracked elastic prediction gives a lower bound to the results. Kemp and Wilhelm [66] conducted a linear regression analysis of their experimental data from semi-beam tests (see Figure 7.4) and suggested the following equation: = (2.64 + 2.37-) (7.7) Chapter 7. Bond Splitting Failure 151 where c is concrete cover and db is diameter of test bars, all units in psi and inches. A comparison of Equation 7.7 and test data on cracking loads is shown in Figure 7.9. 7.6 Proposed Design Equation for Bond Splitting Initiation There is a significant difference between the equations for bond cracking and ultimate bond splitting strength. The embedment length d 1 of reinforcing bars is not included in the Equations 7.4—7.7 for initial bond cracking in comparison with the Equations 7.1 and 7.2 for ultimate bond strength. The uncertainties about the validity of Equations 7.4—7.7 to predict bond splitting initiation strength in real beams exist due to lack of enough test data. The ld/d ratios of the specimens used by Kemp et al. and Tepfers were 11.34 and 3.13 respectively. Because the bond stresses at which first concrete crack was visible were averaged along the embedment lengths in the studies of Kemp et al. and Tepfers, none of the equations developed by them can be readily applied to different beams with different beam sizes. Teng and Ye [86] have carried out a series of concentric pull-out tests to study bond and slip relationships for deformed reinforcing bars. Both ultimate bond splitting loads and bond splitting initiation loads were recorded during testing. Based on the statistical regression analysis of test data, the following design equations were developed: (1.106 + 1.3)- (7.8) (1.162 + 1.8O2)- (7.9) = = where u, is bond cracking stresses and u is ultimate bond splitting strengths, all units in kg/cm 2 and cm. These two Equations 7.8 and 7.9 are shown graphically in Figure 7.10. It can be seen that the ultimate bond strength and cracking bond strength have Chapter 7. Bond Splitting Failure 152 a similar trend. They both are a function of concrete cover, bar size, concrete strength as well as embedment length. Although the two equations cannot be directly applied to beams, as discussed in Section 7.2, the evidence revealed by the two equations can be used as a basic assumption to develop an empirical equation for the prediction of bond splitting initiation strength in beams. The following equation is proposed for the bond splitting initiation strength ——=1.2+3--+15--. (7.10) d 1 where all units of stress are in psi. This equation is actually a simple modification of Equation 7.2, whose accuracy of predictions for ultimate bond splitting strength was recently confirmed by Darwin et al. [83]. Equations 7.2 and 7.10, as well as the experi mental results measured by Kemp and Wilhelm [66], are shown in Figure 7.11. Equation 7.10 results from Equation 7.2 by shifting a parallel displacement downwards for the tested idid ratio. The comparison of Equation 7.10 and experimental results measured by Tepfers [84] is shown in Figure 7.12. The close agreement of the predictions with test results is illustrated. Figure 7.13 demonstrates Equations 7.2 and 7.10 for the c/d& of 0.5. Equations 7.10 can also be modified to express bar force at cracking normalized with respect to as recommended by Darwin [83]. That equation is = where Ab is bar area and 3 f 1 7 3 d (c r + 0.4db) + 6OAb is steel stress at cracking. (7.11) Chapter 7. Bond Splitting Failure 7.7 153 Some Comments and Conclusions From Figure 7.13, it can be seen that the bond splitting initiation strength is much lower than ultimate bond splitting strength. The combination of this lower bond cracking strength and dowel action can lead to lower shear strengths as well as lower ultimate bond splitting strengths. The University of Texas beam [61] is shown in Figure 7.3 with a potential diagonal crack within development length L”. In experimental studies carried out by Ferguson et al. [61, 62], diagonal cracks always developed in narrow beams. These diagonal cracks hastened the bond splitting process, particularly with longer L” values, and led to lower bond strengths (splitting along whole development length). Sometimes, diagonal cracks dominated failures in shear with the ends of the bars still fully bonded near the inflection point. In both cases, the occurrence of diagonal cracks led to lower shear capacities. Based on the study in Chapter 6, after the occurrence of a diagonal crack the variation of tension force in a reinforcing bar may be shown by the solid line in Figure 6.20(b). As discussed before, dowel action will be introduced and bond stress will be locally amplified due to the development of a diagonal crack. Based on the study in this chapter, it can be seen that the bond problem is further aggravated by the reduced development length <L” by referring to Equation 7.3 and lower allowable tension force in reinforcement by the comparison of Equation 7.3 and Equation 7.11. See Figure 7.3. Then lower shear capacity or lower bond splitting strength can be expected. When wider beams are used for testing, the unfavourable factors for bond splitting strength will be eliminated with the disappearance of a critical diagonal crack. The higher shear capacities and the higher bond splitting strengths can be expected. Another issue is about whether dowel force or bond force dominates splitting along reinforcing bars in beams without transverse reinforcement. If a single bar is put in a wide Chapter 7. Bond Splitting Failure 154 beam (small dowel force and large dowel capacity), bond splitting could be prominent as the low steel percentage and the large steel stresses cause splitting on the bottom face, Ref. [62]. If a couple of bars are put in the same dimension beam with small spacing in one layer (large dowel force and little dowel capacity), dowel action could dominate splitting as the splitting occurs on the sides under comparable load, Ref.[77]. Both situations, experimentally indicated, lead to lower shear capacities. The shear resistance mechanism and the bond strengths between concrete and longi tudinal reinforcing bars are very complicated in a reinforced beam without stirrups. The interaction of shear and bond seems much more complicated. Another example showing this complexity is Hall’s discussion [67] of the experimental work carried out by Baant and Kazemi [87] for the investigation of size effect on diagonal shear failure of beams without stirrups. Hall considered that the observed results are due to the location and distribution of the reinforcement, which implies bond failure. In summary, the study in this chapter has led to a tentative equation for bond splitting initiation and an improved understanding of the interaction of shear and bond in a beam without transverse reinforcement. Further work on this topic is necessary, however, the problem is very complicated and many factors are involved. It is not possible to complete this topic as part of the present study. Suggestions for further study are given in the next chapter. Chapter 7. Bond Splitting Failure 155 Cl) C’) ci) Cl) -o C 0 co inadequate confinement Bar Slip Figure 7.1: Bond stress—slip relationship, from Gambarova et aL, Ref.[80]. Chapter 7. Bond Splitting Failure 156 Reaction Each Side Reacrion (a) (c) (b) Figure 7.2: Pullout tests: (a) concentric pullout-test specimen; (b) commonly used ec centric pullout-test specimen and (c) eccentric pullout-test specimen used by Ferguson, Ref. [60]. Chapter 7. Bond Splitting Failure 157 (a) IIIIIIIII I I I iF Potential inclined crack 1 (b) L” P.’. (c) (d) Figure 7.3: The University of Texas beam tests: (a) possible tension force distribution along the top test bar after the occurrence of inclined crack; (b) side view of specimen; (c) plain view of top reinforcement; and (d) moment diagram, adapted from Ferguson and Thompson, Ref.[61]. (I) CD cc CJ( F 0 Chapter 7. Bond Splitting Failure ZzrzE Figure 7.5: Mechanism representation for bond, from Tepfers, Ref.[84]. 159 Chapter 7. Bond Splitting Failure 160 Failure plane C/2. C=C,/2 C > 6 I? Just before failure Side split failure C >z - - I I - At failure I I Lz V-Notch failure >Cb C,/ > 2 Face-and-side split failure C,/2>Cb Figure 7.6: Bond Splitting Failure Patterns, from Orangun at aL, Ref.[82]. Chapter 7. Bond Splitting Failure 161 1 a (a) / Or+dr (b) edb/2 (c) (d) Figure 7.7: Analysis of Bond Splitting Stresses: (a) bursting and bond stresses; (b) uncracked elastic state; (c) uncracked plastic state and (d) partly cracked elastic state. Adapted from Tepfers, Ref.[84j. Chapter 7. Bond Splitting Failure 162 6 .-—--________ 0 5— 0 0 0 / / 4: Icbc . .. 8° • 0 00 0 0c £ I • • o Ct e 0.% •. 2 • - ° x x 1 elastic stage (equabon 5) - X 0 1 • ordinary concrete 0 Iightweght concrete I I I 2 3 4 5 6 c/d Figure 7.8: Comparison of test and prediction for bond splitting initiation, from Tepfers, Ref. [84]. Chapter 7. Bond Splitting Failure 163 15 uc 00 05 4 5 Figure 7.9: Comparison of Equation 7.7 with test data on cracking loads carried out by Kemp and Wilhelm Ref. [66]. Chapter 7. Bond Splitting Failure 164 U fl 6 — — — — Ultimate Strength (Eq. 7.9) Cracking Strength (Eq. 7.8) 4 2=20 db 2- c —=1.5 db • db db 0 I 0 5 10 15 20 Id db Figure 7.10: The relationship of bond splitting strength and ld/d proposed by Teng and Ye Ref.[86] for concentric pullout tests. Chapter 7. Bond Splitting Failure _!_ 165 . Cracking o Ultimate ( 15 - Equation 7.2 10 :osFE7b0T Figure 7.11: Comparison of Equation 7.2 and 7.10 with test results from Kemp and Wilhelm Ref.[66]. Chapter 7. Bond Splitting Failure 166 uc 7r Cracking • 15 . . . Equation 7.10 (---=3.13) 0 I 0 0.5 1 I 1.5 2 C db Figure 7.12: Comparison of Equation 7.10 with test results from Tepfers Ref.[84j. Chapter 7. Bond Splitting Failure 167 U 25 1 fl Ultimate Strength Cracking Strength 20 - 15 - Equation 7.2 10 - =0.5 db 5- Equation 7.10 0 0 I I I I 5 10 15 20 d db Figure 7.13: Relationship of bond strength versus ld/db, presented by Eqs. 7.2 and 7.10. Chapter 8 Brief Summary and Further Research This study, on the shear design of structural concrete members without transverse rein forcement, includes three main topics: transverse splitting of compression struts, devel opment of a rational design procedure for deep pile caps, as well as a more general study of the load transfer mechanism in concrete beams without stirrups. For deep pile caps, the compression strut that transmits column load to a pile is usually unreinforced and confined by surrounding plain concrete. The compression in a strut will spread out thereby introducing transverse tension near mid-height of the strut due to strain compatibility. As there is no reinforcement provided to resist this transverse tension, the concrete tensile strength must be mobilized. A refined truss model includes a concrete tension tie to model the transverse tension. It is believed that the brittle shear failure of deep pile caps is initiated by internal cracking due to this transverse tension. That is, the shear failure of a deep pile cap results from transverse splitting rather than crushing of compression struts. In this study, the compression struts have been idealized as concrete cylinders of various diameter D, and height H, subjected to concentric axial compression over a constant size circular bearing area of diameter d. Linear elastic finite elements were used to determine the triaxial stresses at first cracking within cylinders (Chapter 2). The numerical study has indicated that the bearing stress at first cracking within cylinders depends on the amount of confinement, the aspect ratio (height/width), as well as the ratio of concrete compressive strength to concrete tensile strength. In order to confirm the 168 Chapter 8. Brief Summary and Further Researth 169 transverse splitting phenomenon within compression struts a series of experiments were conducted on large size concrete cylinders (Chapter 3). A good correlation was found between the analytical prediction and the experimental results regarding the influence of D/d and H/d on the bearing stress to cause first cracking. Based on the results of the analytical and experimental studies, a bearing stress limit was proposed in terms of the amount of confinement and the aspect ratio (height/width) of the compression strut, as well as concrete strength (Chapter 4). The proposed bearing stress limit is given for the maximum nodal zone bearing stress to prevent diagonal tension (shear) failures in deep pile caps with unreinforced compression struts. In contrast, the ACT Building Code bearing stress limit is intended to prevent crushing of concrete in nodal zones and does not preclude a shear failure due to transverse splitting of a compression strut. By incorporating the proposed bearing stress limit into a strut-and-tie model that emphasizes internal force flow rather than the “shear stress” on any prescribed section, two rational design methods for deep pile caps were proposed (Chapter 5). The two methods are similar except for the details of how the bearing stress limit is applied. The first design method is a direct extension of the two dimensional strut-and-tie model for deep beams (CSA approach). The procedure involves defining an equilibrium force system in a deep pile cap. The forces in the compression struts are calculated from the proposed bearing stress limit. The horizontal components of the compression strut forces must be equilibrated by tension forces in the provided reinforcement. The sum of the vertical components of the compression strut forces gives the designed load capacity of the deep pile cap. The second design method is presented in a more traditional way, in which “flexural design” and “shear design” are separated. A truss model is used for “flexural design” (i.e., to calculate required longitudinal reinforcement). “Shear design” is accomplished Chapter 8. Brief Summary and Further Research 170 by limiting the maximum bearing stress (between pile and cap or column and cap) below the proposed bearing stress limit. Although a similar force flow is implied in the second method as in the first, the details of the strut geometry are not needed in the more simplified second method. Comparison of predictions from the proposed design method with predictions from ACT Code procedure and CRST Handbook procedure for 48 previously tested pile cap specimens demonstrated that the proposed method is more rational and more accurate than what is presently used to design deep pile caps (Chapter 5). Tn deep pile caps the shear stress is concentrated in zones (compression struts) between the column and piles, and is not uniform over the height making it difficult to calculate a meaningful average shear stress. The sectional methods of the ACT Code and CRST Handbook are not appropriate for the shear design of deep pile caps. For example, the one-way shear design provisions of the 1983 ACT Building Code (and subsequent edition) are excessively conservative for deep pile caps. It was found that the traditional ACT Building Code flexural design procedures are unconservative for deep pile caps. These flexural strength procedures are meant for lightly reinforced beams which are able to undergo extensive flexural deformations (increased curvature) after the reinforcement yields. Deep pile caps are large blocks of plain concrete which cannot undergo significant flexural deformations without triggering a brittle shear failure. In the third part of this study, the load transfer mechanisms of beams, which unlike pile caps transmit the load in one direction, without stirrups were investigated. Addi tional considerations herein are the crack propagation of discrete diagonal cracks and the influence of bond between concrete and longitudinal reinforcing bars. Based on a study of the compatibility of the displacements of a critical inclined crack, an interpretation of an important shear failure mechanism of slender beams without Chapter 8. Brief Summary and Further Research 171 stirrups is presented (Chapter 6). The interpretation suggests it is the occurrence of either very flat or horizontal cracks near the critical inclined crack that is most indicative of shear failure in beams. In order to get a better understanding of this interpretation, the influence of bond upon the load carrying mechanism of the beam was studied. In this study, two mechanisms of shear resistance in structural concrete members without stirrups have been identified. In a very deep member (a/d < 1) or somewhat more slender member with no bond between concrete and reinforcement, loads are trans mitted directly to supports by compression struts [see Figure 8.1(a)]. The shear failure of such members is characterized by transverse splitting of compression struts. For more slender members with normal bond between concrete and longitudinal reinforcement, the load transfer mechanism is as shown in Figure 8.1(b). A concrete tension tie is needed to transfer load to the support. In this case, the capacity of the concrete tension tie relies upon the bond strength between concrete and reinforcement. An empirical equation for the strength of bond splitting initiation has been devel oped based on previous experimental results (Chapter 7). However, the shear resistance mechanisms in slender beams are very complicated, involving both shear and bond. A design procedure which can be used by practising engineers cannot be developed in this pilot study which is only a small part of this thesis. Additional concentrated research is necessary. For example, more analytical and experimental research is required on the bond splitting initiation in beams without stirrups. The interaction of dowel action and bond splitting to cause cracking along reinforcement should be studied further both ex perimentally and analytically with the objective to develop a quantitative relationship of this interaction. Based on a better understanding of the splitting phenomenon along longitudinal reinforcing bars, a rational design procedure for beams without stirrups can be developed, in which bond splitting is avoided by limiting the maximum tension force in longitudinal reinforcement. Chapter 8. Brief Summary and Further Research 172 H ** F__I \ d (a) $1 ? (b) Figure 8.1: Load transfer mechanisms of structural concrete members without transverse reinforcement: (a) very deep members (a/d < 1) as well as more slender members with no bond between concrete and reinforcement; (b) slender members with normal bond between concrete and reinforcement. Bibliography [1] Marti, P., “Basic Tools of Reinforced Concrete Beam Design,” ACT Journal, Proceedings, V. 82, No. 1, Jan.-Feb. 1985, pp. 46-56. [2] Collins, Michael P., and Mitchell, Denis, “Rational Approach to Shear DesignThe 1984 Canadian Code Provisions,” ACT Journal, Proceedings, V. 83, No. 6, Nov.-Dec. 1986, pp. 925-933. 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[43] Deutsch, G.P., and Walker, D.N.O., “Pile Caps,” Fourth Year Civil Engineering Research Project, University of Melbourne, 1963, 75 pp. [44] Clarke, J.L., “Behaviour and Design of Pile Caps with Four Piles,” Cement and Concrete Association, London, Report No. 42.489, November 1973, 19 pp. [45] Clarke, J.L., and Taylor, H.P.J., “Model Tests to Determine the Influence of Sup port Stiffness Upon the Distribution of Pile Loads on an Eight-Pile Cap,” Magazine of Concrete Research, Vol. 26, No. 86, March 1974, pp. 39-46. [46] Gogate, A.B., and Sabnis, G.M., “Design of Thick Pile Caps,” ACI Journal, pro ceedings, Vol. 77, No. 1, Jan.-Feb. 1980, pp. 18-22. Bibliography 178 [47] Sabnis, G.M., and Gogate, A.B., “Investigation of Thick Slab (Pile Cap) Be haviour,” ACT Journal, Proceedings, Vol. 81, No. 1, Jan.-Feb. 1984, pp. 35-39. [48] ACT-AS CE Committee 326, “Shear and Diagonal Tension,” ACT Journal, proceedings V. 59, Jan. and Feb., 1962, pp. 1-30, 277-333, and 353-395. 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[66] Kemp, E.L., and Wilhelm, W.J., “Investigation of the Parameters Influencing Bond Cracking,” ACT Journal, January 1979, pp. 47-71. [67] Hall, R.T., discussion of “Size Effect on Diagonal Shear Failure of Beams without Stirrups,” by Bazant, Z.P., and Kazemi, M.T., ACI Structural Journal, MarchApril 1992, pp. 211-212. [68] Park, P., and Paulay, T., “Reinforced Concrete Structures,” John Wiley & Sons, New York, 1975. [69] Morsch, E., “Concrete Steel Construction,” English Translation E.P. Goodrich, McGraw-Hill, New York, from 3rd edn of Der Eiseribetonbau (1st edition 1902). [70] Fenwick, R.C., and Paulay, T., “Mechanisms of Shear Resistance of Concrete Beams,” JournaloftheStructuralDivision, ASCE, V. 94, ST1O, Oct. 1968, pp. 2325-2350. [71] dePaiva, H.A.R., and Siess, C.P., “Strength and Behaviour of Deep Beams in Shear,” Journal of the Structural Division, proceedings of the ASCE, ST5, Oc tober 1965, pp. 19-41. 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E., “A Reevaluation of Test Data on Development Length and Splices,” ACT Journal, Proceedings V. 74, No. 3, March 1977, pp. 114- 122. [83] Darwin, D., McCabe, S.L., Idun, E.K., and Schoenekase, S.P., “Development Length Criteria: Bars Not Confined by Transverse Reinforcement,” ACT Struc tural Journal, V. 89, No. 6, November-December 1992, pp. 709-720. [84] Tepfers, Ralejs, “Cracking of concrete cover along anchored deformed reinforcing bars,” Magazine of Concrete Research, Vol. 31, No. 106, March 1979, pp. 3-12. [85] Tilantera, T., and Rechardt, T., “Bond of reinforcement in light-weight aggregate concrete,” Otaniemi, Helsinki University of Technology, Division of Structural En gineering, 1977. Publication 17. pp. 1-36. [86] Teng, Z.M., and Ye, Z.M., “An experimental study of bond and slip relationship of deformed reinforcing bars,” Department of Civil Engineering, Tsinghua University, Beijing, China, 1984. [87] Baant, Z. P., and Kazemi, M. T., “Size Effect on Diagonal Shear Failure of Beams without Stirrups,” ACI Structural Journal, May-June 1991, pp. 268-276. Appendix A Measured Bearing Stresses and PUNDIT Readings D6-12-4 D6-12-3 60 60 -50 50 Cl) Cl) 40 a) 30 30 — — — ( C 1 20 15.9 I 10 2 3 - I2 00 4 D6-1 2-5 D8-12-1 60 60 50 50 40 40 30 20 10 r 00 2 3 . 30 .. 24.4 20 10 4 5 r______ 0 2 3 Transmit Time Increments (microseconds) 183 4 5 Appendix A. Measured Bearing Stresses and PUNDIT Readings D8-1 2-2 184 D8-1 2-3 c 60 60 -5o 50 Cl) 040 40 C) ( ,zz •1- (I) C 30 20 1o 10 00 3 2 1 4 r 00 5 3 60 50 50 40 40 20 10 r 00• 1’ 2 3 10 4 5 Dl 0-9-2 1:IJ———— 50 50 40 40 30 n 20 20 4 2 5 20 Dl 0-9-1 00 4 30 60 10 5 D8-1 2-5 D8-12-4 60 30 4 10 4 5 / Oo 13.4 1 2 3 Transmit Time Increments (microseconds) Appendix A. Measured Bearing Stresses and PUNDIT Readings 010-9-3 D10-18-1 60 60 50 50 40 40 .I C,) 30 C 20 30 0 Cl, Cl, G) 20 ci) 10 10 00 15 fZ.3 00 1 2 D10-18-2 60 50 50 40 40 20 10 ( 20 10 4 5 00 Dl 0-36-1 Dl 0-36-2 60 60 50 50 40 40 30 30 10 /c4T 00 4 30 00 20 3 D10-18-3 60 30 185 2 20 10 3 4 5 ....................................................................... 4 19.5 1 2 3 Transmit Time Increments (microseconds) 5 Appendix A. Measured Bearing Stresses and PUNDIT Readings 010-36-3 Dl 2-9-1 60 60 50 50 40 40 30 ,u C) 20 20 10 4 1% I I 0 U) U) ci) ci m 0 ) 12 3 186 4 5 I Cl Dl 2-9-2 12.2 2 34 5 Dl 2-9-3 60 60 50- 50 40 30 6 4 001 50 4 5 D12-12-2 D12-12-1 60 23 60 50 —-.-— 40 30 20 10 2 3 Transmit Time Increments (microseconds) 4 - 5 Appendix A. Measured Bearing Stresses and PUNDIT Readings 187 D12-12-4 012-12-3 60 0 -50 Cl) cn4O a) 4-.. C’) c,) C ci) 2 3 D12-12-6 D12-12-5 60 60 50 50 40 40 30 /1 30 23.2 20 20 10 10 oc ) 1 3 4 D12-18-2 012-18-1 60 bU 50 50 40 4 30 30/ 20 122.0 20-- 7/ ‘ 24.4 0• 10 00 — —. n_________— 2 3 4 5 I 2 3 Transmit Time Increments (microseconds) 4I 5 Appendix A. Measured Bearing Stresses and PUNDIT Readings 188 D12-18-3 Dl 2-36-1 Dl 2-36-2 Dl 2-36-3 D14-9-1 Dl 4-9-2 Cu 0 Cl) Cl) G) C’) cz) C I- Cu CL) 60 50 40 30 20 10 60 60 50 50 40 40 30 20 10 00 19 2 30 20 10 3 4 5 1 2 3 Transmit Time Increments (microseconds) 4 5 Appendix A. Measured Bearing Stresses and PUNDIT Readings Dl 4-9-3 189 D14-18-1 60 a 50 50 C’) C,) 40 ci 30 30 c,) 20 /7’”26.8 C o 10 0 ) 2 3 4 5 4 5 D14-18-3 D14-18-2 60 50 40 30 /‘6 20 10 34 5 z 1r0 Dl 4-36-2 Dl 4-36-1 60 40 7 ——- 30 28.1 20 10 .10 2 3 4 5 00 1 2 3 Transmit Time Increments (microseconds) Appendix A. Measured Bearing Stresses and PUNDIT Readings 190 Dl 8-9-1 Dl 4-36-3 60 Ca a -50 U) U) a) 30 c,) C 50 4 40 30 29.3 20 26.8 10 1 2 3 4 5 U 0 1 Dl 8-9-2 60— 60— z—— 50 3 4 2 34 Dl 8-9-2 - 40 30 20 10 S D18-12-1 n ‘0 20.7 1 2 3 D18-12-2 60— 50 40 30 26.8 20 0 - - n____ 2 3 4 5 Transmit Time Increments (microseconds) 4 5 Appendix A. Measured Bearing Stresses and PUNDIT Readings D18-12-3 D18-12-4 60 60 50 50 40 40 Ci) 30 30 20 20 C) 10 10 Cu 0 U) U) a) C m .10 I 2 3 4 5 00 12Th4 5 D18-18-1 D18-18-2 60 60 50 50 40 40 30 30 20 10 .K: 00 4 20 10 3 00 1 ‘ D18-18-3 80 70 60 50 40 30 20 10 50 40 29.3 20 10 2 3 2 3 4 5 4 5 Dl 8-36-1 60 30 191 4 5 4ZJ 00 2 3 Transmit Time Increments (microseconds) Appendix A. Measured Bearing Stresses and PUNDIT Readings Dl 8-36-2 192 Dl 8-36-3 80 c3) 30 C 2O 1o Oü Kz 2 4 D24-1 2-2 D24-1 2-1 60 50 40 30 20 10 00 2 3 D24-1 2-3 60 50 40 30 20 10 Transmit Time Increments (microseconds) 4 5 Appendix B ACT Code and CRSI Handbook Predictions ?er/. I Coo. jyo),. c-f ecre’€. le& r&L 2c’ J. To-hC- .i o- a — o1 O P-L CQ c Cc’: Zê’(’. 4 o- - or—- cc— €_ot J± + O—vj — -&4e- co&m. Eedt-+ec& o O€—v-&, SheCvrL3) CIL4. — CdV&’. , 2 c’c’ Q O -&‘JC o- ,Cp U20 c’ 1r ?kC/3. ts) +L . - he---i L4) — cJc,}-v c’v i _-C4LaL o* -cC_ Of ‘“ “. ‘ -- —‘- h€cy . — a—vJu - -i Th.e_j evy-C’2) LcL’k — C r- cR —r---- j93 , +o— C —frs -p €-.,-—- Appendix B. ACI Code and CRSI Handbook Predictions —t —- 194 vv - JJ OAVtA () —— Coi’-. o’’r. nc ;OQ 14.’;L. _w.& - cit:. 1=’4 c4— Ori€—’j ‘1J ZJf, cpa) — J t4f’D 1 aij = — - <a.zS?J LH?’.) (oo3 r= = = o2J’ ... £IJ c3_2. (3—24 (35—..5 )C o43 )(o -t-z5’-) —25 C-) C3_2 rs) 1)1:. 4 o3 = = (?-) ) -- to c) 7744 )(°.°S jj (‘e?E) i L1?’) ( = (}‘3) = - Appendix B. ACI Code and CRSI Handbook Predictions 195 ftL&J4 <4±: S az-c, Lt t t 1 z3 /cA — -c3-L3I49.c J1 4 L_J too 4cM_ 42. = 112 L) z) 2 -rz = 27 LkU) _-vj jLI) -- 103’ = L o34J,’3 xos-4’f.S = 1caQ one—-JJ = — 2’iT4 t) 3.7t) = = 1.t le3 L4) - skeL7.J: .5 4s1’.1 23 yo4J øi— w 1 j (t) I .-___---1 ‘H-’ L_ )4 Q 4 ==- 2 V ’32.2.Lt) = beo-’- C3; 2l.25 122 10= O—k)O. &le.&ri3) /O32M z.f (35—Z o.43) 2. 244 Q—iQ .aLt) > 3 Lt) 1c : 4 ’ iN = 7&.6 it-) heOX L.4): “%CQQ (35 —25cQ. vc, = = - \J3.5C-t) 0 •-“ = .35 ( ) t 3q.L 4’-): cñi’• 3/4;Q.€? 2OC) a.c39j -- t71 Q.O244 __7)4oe 49.; . 47 L& ) z.togS c4o4’,5’ O.L’f) = h€c 274i (4 J 3 ) srr& ZY7it) 7L’k$) 2 ‘•t-) = 2 L2-): — )= Appendix B. ACI Code and CHSI Handbook Predictions i96 ç4c4 }€tL fL) A=43.21e. -F= -44’?.az’ 3O4 Yc= = °/ (:) 73 ct) V. = — = Ct) = 2? = 33 c= c-c) Qk) =- O—v)- SeocL-) Vt: = 0. )0$(3) o /Lk0)(4’5) = °2o3, p t) 2 t4ts49.5 = 04 Ct) (-2. ,43= z.+3 4J) 44Ct) 4 ‘/= 24’( . = 25 5e3q, -i75.qo2o3)44 > = zv= ia L&)= QP€ beojn eL4): O.O203. 40 = %‘ eo’r (3—2.-’) VC. 3(o.MJ = 4tt) — 44’? •=HO.l Ct) . ii) ca. 3Z c4 3(-)= 3L&) 4it2). z.CCt) = = 04r1.1 Ct) a2Yjo c&’_) v=4u) —‘,it i(LAL =7Q3cA A=42’crnt) tjl31 -;=Z7 r’e Y/L \— 743xOZ/ rtL = 2iL 4 i74 74;3 4. ‘4= ct) X)= 4Lt 14 ck) che3vc 3) = kec) ).OcZ 473 42 , Yz9o-t .OIr2 75.!,Q7- 2. 1 L) Appendix B. ACI Code and CRSI Handbook Predictions cr. fr J 4 ( — 42.5 7 t 0 .gç = /_ (3—25o.. 197 k’ex’i c-he-, )= ÷i75.sIai72-)4ø73 ;:; 33Ut)=3z44 c) g?c 7 ,2 c) 33Oä) Ut) > = V tt4.3Lt) c.= 2z4.3,(1iJ,) 3 ‘j. 2Z .1.: 4 ev+ fL G€—)Rj k’L) nst On.,—io Jo.4 )=I2o3tt-r) 4 — 4 o.53o4-J Vc L3 3.4 it) 422 G 3 3tt) / = -)‘= 53e+) 1=b1.tt) ikt-’) — ‘e’.’’C) Ofl€LJ 43.l/3)o.O14 C3_1)(3) = 2. C2-): 2.737 2. (aof 31.Lt) p =aoi4, (3. oc4cL3 - V >. — U= Ya= t 2.5LO)’ t j .3g(o 3J4;t; -I- = I4..3Ct) ‘4=4iit) %4343Lb?, Q= 2V zsi’4(4) Ltoi-j s4€.+kL IL..tc,L = Q.- )L?:SZ= = = = Z2-7 2LI rJ,) 30 1Ut) cO-= (c4 Co- 041 z.a) = 31/ ‘ t173(t-m) ii’7.3 o>4i,= z’ L) Appendix B. ACI Code and CRSI Handbook Predictions 198 r€—’i) ,oi: P= c-L2..): Or—”Wj OL93, -= oz4- (3.5 — 2.424.)= 2. >2.5 (odj +S4os-.4= 34.3 tt ..t •>. = — ‘/=azLt, Lt) 93(kJ skerc4) one— -‘-°4 bem.r’j 23(O3’ t$10.°I3)4ot4 V71.5(.) j7 2 ) Q..Q5_ix33Pct) 4 tsJC (3—2.5?L44) — 2.3 = 24 Lt..) t 4 <.r•’ ectr’r ?.0L3, kCD2205.th.2.L, .1j-iL2) 3?J et) = lLQL 3g (.) 7.2C)= %32L4e-) 1 r , @ne—’ seor-) =4O.24c4 $q=4q, = dSI.tqirt o—’J-J chea,LZ) 4Jo )= 7, )#é = 2)sipe t74Lt) 7251 = Ye... arie—w e1Vr Szt)= 4-t-?..) (4..) 4, 4 ‘O, 2J ‘4cr2 (3.—2.5Q.24-)’ -‘= 2. )Z.5 = xS’2 > ‘ie- = 2IJp ckiJ) ofl€—i. V, = 4o.2 =ALO 1t) v= o53c4f = 370.1 Ct-) — t 0 W t z.5Ct) ) - V zV = z44.’Z (t) I Ct) s4-H (..) 5 53It)= SLJ-) -*41lo35 Ci-) Q.3tt) - Z(t’J 205iCt), NccZY4I0.2L4O23(J) Appendix B. ACI Code and CRSI Handbook Predictions 199 3U As = 2.7cr.\ , 1 297 V/cm 1 3 co334j.7t -A i-tt L__. 92 f —2- 419qX2 ,g 1 _ — - a L_J c?&-7 I )(-t1t) — I lo 45ci. = 44) l2.LO .ç. = , • 37.S l.)L45I3’ = I3°° 1*) 29/- sf 3 1 M= — =43. 3.13 - A 4’7.°’ e’? IO.S297O+32252O r — 47 = / 2529x45 1’°t’’ .f-& 47’lx57€ = S (II4T1+ 7 5 q 3 = .ZO/ — = 49 ) = 32 t-) I4i. c-t) LJ =4c4’i A. $.252c “ •ttj= = 1 fGq tE4 4 9 (L4 )= 3Ct) — I = /j 2 !.3)O 22.Lt) = 3L3O.O qocL= 3Z5(L) -= Appendix B. ACI Code and CRSI Handbook Predictions I 200 ° Vt = S3a.J s%4474o5 (;t) = = b —: 43 = = — I47.t) tt)z13oL4i) 3x2.4 3I Oi1.—J vco3L441 = (t) 12 —&; 30 3Lk) Zt4L+’6, — Ct), rJjI4 = 3 eUc3): = — s—zo.)= 2.’L = _4 — O.o1S z(o31fi -+ 75l : IS.75 .75 P +3/( i ‘447) = 3S (-t) v=( y=’.2a) (t +Ci&, , iork D2.L±) > \/ Appendix B. AOl Code and CRSI Handbook Predictions 201 o€—w k’r (4): b 4T’S/çi.r G, I hO ri ae—j’-” —b C icrL,irore) icì. = C 72kt) zo k; 3 tjxi n-i) 4 (4 + 447) f 37. 35 C.ft v=iJ;- x3 - 3• L) V .zi4?t = = 1jo—,Ja eCC2.) = v’ = tt) 372Ju-i) V= 1 = CW-, = = 44 Z.14/ o,o4-G ,72.4 Ct) V= 4jj ?x4+!7J7.2tt) . 4 4) 72.+C) shei’u.) --7.4€ — =. M’i i.°’7ja1 3.i-Lt) 37 (-kiJ) 1 X2 -i4- (z) L9 4e 3 tq-33p L+) z3 &) Appendix B. ACI Code and CRSI Handbook Predictions 202 1 1 :2ct A 44 ‘---7 444 — , - 4OZ+!%3 =43. c/ ‘, 3t i —--- — S4-. f44n A= A -t ‘ —43(4—--)— 3L,Lt—) — 23) L), = - z-i Lt) - -4 3Lt) g4-x Z3C) f7t) zc (:&eJ) -—M he—c’) vc=’*_ *JL , V — .3 Appendix B. AOl Code and CHSI Handbook Predictions 203 e—c h!G-) P 432/( 24=Q3 o&4.I’) r t 4 ) II.O t) 2.(O3Jç ‘4i6T.Q’i) ( —25XO4) z.32.5 -‘/c t—c lAu/= ‘4 .; (o3 -7cgLYøz44 -)ot VL21%,+Lt), , =32.o (3w— 2.’:2—’ 3.2,-aS goxs2. z4-) t) >- Jz2cMt) q =3Mb1.o = 43L±=4737 &ki 4 t’ jrec) bri- C i ••••) One—’Jej heo’L4-) ) = y .(o3J --t7$ “c’L —37t) — O77J v (° c—c. 5°31 - l2.I4/=o2 f3 (asJ L2 v 4q4t) fD 31 (3_2so7’fl 4)°’4+ 75.I i.’j Lt) 4o.z/,=o4b, +jT . 5S’-Z.5 c L3—2.5)=2.2. d443X4b)2ZI.71t) > 2 J4t I Z7L) 6 i-p’c V -t’o—’i he) *-4 = 411 .o t) ‘4 = .r2.ijTh /\ J / I t /r L_J — \ = - t = - ,22.l(±) S2 243 Lt) Appendix B. ACT Code and CR51 Handbook Predictions O-’. 204 ‘—r () 7I---4 1 J = J_ ‘4 . cM, t l2c5 7 = 4qL4) lk.t) = = F1T 4S() JfT’. =2 .L43S3’? = 2.?t = fl€-L o’Ct) &) &h-ø+K — = ‘2.43o COO A.qs& - , S3a 2 7t 7 ZPc = 3 $ 73.m, z 4a zS3 (7o’z 7 t1r4.73 I7 coO°= A = 1 °=4m, -1’ t80 C Oi— WCM 3zD44 ‘17.7 Ct’) 3 .21t)= 1 £ , 14 JpIe= xlO/ tt) .% 4 3 ,l4L-kL) /A ) SieuC2): 0—: V b— V= , J 0 ‘j3c2€37 3iS’ X5S = 94(.3 71 — ‘z4-c =W7 33k ot3Z+331 O.O23 (70.0— j4p9 cc’2& , A= cA6 -= 2q13 l/ =3 23?( /4 =to44ct) e 7 t--) — A4+4oco .‘T3 44.3 z37 = = = . cl 3 = ‘e,S(t) 304 3’44C3 xt4-x-70 a.o(t)i1S5.tLt) c—cr = 3w =2)2O44f.N) Appendix B. ACI Code and CRSI Handbook Predictions O—vJQ S’.) +$/ Q 205 tJAj) rJ’OD, — V=4Lt) z3A-’ c 2 3 (3—2T5?25 , ) ) , , C-—c V 2.5 v) -j 71 = V=432.!) ZJp€ = 32L2.Lt) — aç4S Ct+4-L, orc&) &e—vJc 744Ct) = 74C-kJ) çhCOV((+) J— 4’,7g/T ’ 2 ’L, _44 O -- 25(o V P _ jr P 3J4 I75 = 9z.3t) — “-): Ct) . 039 = 4 /LIsos 7Y2) 3 . , ij 9 3 & 4 772IPtt) 4z . 5 t) 3 3l 3 = O.4-Ct 3 = 2! (k) /A) LA)/ 9 2S4/ n 0 SCL’) °“‘4+, )M.-7o = , 3,Lt) 4 _)3c7O.2 / o 7 r 4 ) i.i(o o3jZ- (cl ‘44 1A)/ V. p_4. 2E(5oj .jt75 = C—; oQ2 4Z.Ct) =zbI.ILt) > a O 7 I 34.) 4çJ C/A ) Z) t’e&i r °‘‘ = 3It) iP_i€ =a.7 C7- 4.3>&31 34f.•• Appendix B. AOl Code and CRSI Handbook Predictions 206 -kLr&,L d,74.t lSe A2)c3C0S3/4J 7 d. ==2cAt, A= . 5 ±=4°’? - = = C3.C 43U $02 -z z7.3t. 3±3o= =‘74, A7e 34’ 4-31 i)= 34c! )4-it &73. — 4- _ . 37 IlO.7?d/ CVI) tz7-)) = J 1 3 L3 7 3z $I%(t)- ‘7LL4d-) OflP_—OjerC4) O —VAJ t/L2.) oI-JiSZ73’’ o+.z(t)= k—i,: C—C: J€ WP&. V= O4j 3 I’T 3 2i(t) 1 , = = z5 Lt) ct) 1 —vc .hoL3): ) c,ct73 z4Lt) ‘/2.(°34 ‘4=. x13’ -> 2.GA)= Q—w -— -U) = 42 2) 21-3; Appendix B. ACI Code and CRSI Handbook Predictions ‘! (øi ‘‘c .s;c’o4.a -i-’, 3L1)= 1? 2.j3(r3. 53 -k) 335.S.) ) 51,e3J<J) 207 C ecrC2) 1 t—P’ iJ/ o= 4o.t; o—- = V o 24/3 4-L V z c-) 73.,c.;= a+fox73= 2Z7t) .- i2ZLkF) () (243 )(4? 140s Ott) t€ )(43;2= i 7 z (t) — 4IJ 1: f I I I L L. -:1-LtJ 1—. L1 14p7 cm’ .3c72.s/# 4= 23 ,o 4 s/c A 2 , 7 g +4Z7bZX.Lg+ i4o7.x Z3 )•‘ 5 9 4 ,cZ7b = 272(47.4- = 4iZI — .-f)’ 3(&1) = e71—uXAJ c-eci) E1 o—)o’’-j i__I L_1 heo-r2): v= o54-f xi’7.4- 244.C±) Z.4W = Z43 (ki)) = Ofle—Jck1 TT Z rr h€o-’(3) 32 P= >2.5 I ) 7 OOO5)I(W7.4 V j .G I 120 l° = 4z.3 L) > .s ;J ,ø7.4= 372. Ct) Appendix B. ACI Code and CHSI Handbook Predictions 208 = 3(7,Ct)= aiJL :I?e= 74a*)=73o S€t2C(4-) fl1—W /6ci-= O52 =5 V. 2,p03J ÷tiod74 — = (3.’5—Z5)OS) 22 ,IZ.O(t) ‘. ‘Jt=V2PC±;) I 2 J., cot 44J.1 I24(k04{) 4-(sV 7.4) 94 V = 2O4Lt) 3( (-I&) -o---,JO.. 1ie?’-Cz) = c wi- acot) t,eOrift — L5ZtC441) S S3.I 2 ofZY3 etr4C): 35=Lt) —-: QZ 32 = 4t€’ fLLVL = 4o.ocm 44’O ‘8/ — , 1 4z.4t =e-f=4 4’ d t-F=5L7 4443a &B5÷924.Sit7?cbb = 4• 44455 ( 44LL= 2 1k t-), — . 2Lw = 2()= 7!(&I) ‘-= c€o’) fl€—VJ. C/Pt) A’L-): VL= 4 Z Øox’. j’31.3Lt), ztJJe = = Z24C+) 7kS) ce—-O- t’xL): /,a) /= a.5IS j— Y3Q 7/6 Zb, L322) .. 2.5 )‘2,5 34-o C&) Appendix B. ACT Code and CRSI Handbook Predictions ct) = = = g5.4t) 7 ‘ = 209 7732(&) OAi0j sie(C4) (3—25XO22.’ = = kCt) 7 &ai= 6i 223(o.5° V = 67.1 ) = = = /e& 3 v.1= 4t) v= 37.)= = ( Ij4 z4Lk) l2352CU 4= 4o-IMi-): 2. z24&t.) 4 rio erLS): _ioj Ci); ?axfn 1 c4os3 r(23tt) 3+°t) 7 1t= saxo5= I 2.I3.Z.Lt)2S57 () 4= = +t - = -c”= s ‘Th /em? QS4 Z .7C?t’f? ‘3 7 2795 S/- c L,Cm’, /k 27 7&G2x28b L.)L19o37I pZ2,65 =,•a±), oJ-U): one—t)Jc’ ) i3.4 ‘Iot /A) zH VL = 53o4j o = i’Z z(tr) ziJr V. cQ 2iJpe z+z.s1-ct 239j k.43) , ofl€.—-j ieL3) AAvh&a L7/ 4 = { vc.= 5(±) ore—vja (it) (39.—2,5)L027) 1O?(b5. 411.1-ct)> .5 J = 27 ‘}27.Oct) = jxjZ0Xc.9 = 73C L-JJ) OfC4) = .53, (3S_2.5)2. 791 V Vc7JCt) 24L, )iox6. = 77.l 2.3’2.5 tk). 42Ct)lLJJ) 3C35 c-t) Appendix B. ACI Code and CHSI Handbook Predictions —oj ‘4 210 €&r) 62’f.’ZL), ô& J 7 j _t14 = Wt= /s.’i = o53 ‘ tlJ.c5(t)= 1ocC&t) nci a , G—U ceckr(3): i,u—irw +‘4-, 1e+Cz): izt37 x = &= l5 6.(t) 7 = 4o= 4$Jpt 2L4.)Z17CkJ) 4a r-L 4 f{ k3oc, -cc=-34- -F = c-*t Az 4’so /c4.42, A= +S.11cm> d’4 ..f — +ibi4 x4 2.t--W) 4l((7 — = 52.O/ =439 Ct)) 7i4C4. (j.) Icøt one—- heov) one v’JL ¶4i€OYC?) — a7= fl2 Lt), o 3ô4j zN,,t= = 44 = 23b.4 Ct) ‘/. 23) (-?iJ) erC3) =4 /rp °‘-°. i75/, /(/a) = 4-.1)= 34(±) ,i3Ct)> ‘4= 2o5D5 + 1tJR. 4JpL ‘7.2 C1) ) or —icu.j 4ec’cL4); =o.i, 1 21(O5Of 4- 0 V 441() z 3-23-) t)L )(J=9c41Lt)< 2.!5i9 o44 9o(t) a,t)=jaZ) -tuio—w& bI4Lt), ‘4=tC&77 b-wj ?heo.rL7-) 4€X) eA” ¶d7€+k(t) QQ 0 ‘J .JZL LO’t = 2M-3.4-’tt= zc1 r3 -&ii) Appendix B. ACI Code and CRSI Handbook Predictions 211 iqi-: i= lj;’ z’i/-, 0/cmj f 4l5 c-= qsi 2W = =- 2L 7 2j , 44L±) = 132 G) , V4-.tt) cfl’L-uJWJ = = qj. t7! oQo3’ --)!ox25 0 ‘4= 2S ( V ‘JeL = Y = Zt’JpJ(l2.) ç(±)32 4JJ €,A’(): /cv..A) —i 2S3(k3) e.r(.Z): VO.3C4S L72 = — 27o 4A.rC) _vo&Lj Oflf—WO- l3,OX23lsO “Ltt ) = 47. LtW) — 1Oz/ ‘4’223Lt) 147. o—uX (2S 6t (3! 7 —z = oCW ojG., o.: t) C3._2.5)cIV37,2.5 4\7’.2i) = ,c(€o25 - !:t) = Q3(kFi) hXC4); z.O3J4.t5 +!72is°37 —)o25 !ze7 Lt) > O(72= &(*) J2= 4 = —o—W.J hetML): V t. J4 7 -+1*30 — -t-kD92-5= 774.i)= o3ft), (-JJ) JCA o4-(i-i V 4X 923D(±) . 4-Jj 2Ox25=2o!.I (t) JL-= 1 ke t) .j /f) 45l,4Lt) J,L — c,L 24.3g.4ct) = a3’I3 (.J) Appendix B. AOl Code and CHSI Handbook Predictions 212 PS 43 -çL 4 Skv -c=4s4 /- =‘935 k=2.L’-, k3,I4’ AI2.Gt ,;+Z /(A.t-, 44/ Ac. ‘43 J =cW øC (Zo+S4+ p24x414o)+ 3 r tj Ar 94 °(‘1_ = = c’heo..rci): ‘- 43) 192. t4) J-j= 1IgCt-)= = 4 .ot-), p7 (-?dJ) (/F) t74XL2) OhE_—v) v= rI I tV- = i%.’7 Lt) 2 j Jt= v’4 — ?j3.4(t)’ , tkii) w&j —, c’= = ‘4= S7.4Lt) one_—W’- = 1150.8(.t) L — €-A4(32So.3)2.’ = v 5. I L) ) Loj2. = 75.4-W) ‘-2. + I752 .Lt) 7 4JL= jq V’3.&Lt) = cga t et) = 1371.9 ct) ‘ji ot,28 -= 2.S L9Z1Z(iJ) o—J& S1eosA’O: “4= i.o1o7j4oi.1z)Xc12. 7 .OCt) 24,4Lt) 9OZ4(4tJ) —bo — &tj ie12) ‘92 LA)/cA vc Q3 _c.4-<j -÷ il1-Z. L) -_J8vXC3): 34-17.0(t) 4xD)92= k2.8 =Y 4-ZCt,= (QJJ) C’/A) 4 e M1Ii C7-) e”s I’Jc.Q = OS 4I3xo = oL(-kN) 2D1S.3(t) O = 4A3 X? 2kO(t) = 320oCt)342 I (&) Appendix B. AOl Code and CHSI Handbook Predictions 213 441+ hec-tk Pcs T.24 c Z4 )= 23cS’=27.7LC, 72I4/eA y= J 1 ô/= tzLt) 2J.jt , L’ 4.” (‘/A) V=omo+Jtoc1;.L1= iS3t&) 1 t’ne—w k12 =vI 3O.) 2I’(k-I) 3 , 5I1er(3): /j,J) = v= , oie—v 4j (t) )fQX 2.(3gJ + I 5.oc44 7 L3S-2SX9)3O3>2, =9) = 443 ±) > — Me—rc4): . __L y o3J +i75flx&ôt4 N/S4CtE) = ,Lt) 7 j)o[ =jZ3 ‘f’7&) — 5)6.7Lt) ( io2 (*±J) o— ewrC) 3/o W/a Vt. e.4f4- 3tt) (- = tt. Vc r(3): /A) bexr 1-tJ t(35. = 1473+ (k.i) seMHC,) 2t) 1 3 JJ. 1 4 9 , Lt) 73 L&t) t ‘25Z’ t)=A- Appendix B. ACI Code and CRSI Handbook Predictions 214 ‘44* i 9 cm€’ 1 -€tt -fk-1/, =T4c 92.w , /CA%j 1 134 344’4 4r k3 93O1-= e—Lc’j ;50Ct) e-LU —j ‘ —---) = 3.1 -) 44( 1tO3pC)=)O/,CkiJ) co 1 , (/A) ç1iec’-U-) y=oo1-Jv’z.&=’ J.t) = 2-V 33.1t) 3,bc&i) ,‘he-rt3) 6 O.OZ = ‘4 I - 3jLt), o2.L, L a. L = kco’1cI42L) 32 L1) LR92. = lo437L-ki) iec+) o—iioj = 3.S-25XI)=3p3 25 , (PD1 PJ I’(5.& Z V o9.ILt) o—u J€ , = 021g) IC/J ) J.4L4)7 2-V jI2Lt)= ‘—fl 73I(-&) ectv- LU: I2,3t) lJ,L — Z. = 4:9 Li)=’ J) &rL2): - ‘4 = o,3o4 ‘6 — (I4 — 13Lt) T4-x T= lb3Lt) kcoL —je--W ie—[3); = II.3L) < 48i 23Z3Lt) l’f7O (JJ) ‘-VA) he’f eM-rL2) =çI3)7q7.gIt) = 7’ -) .) 2 O 4 tt)9&2-(4 R= ) Appendix B. ACI Code and CRSI Handbook Predictions 215 o.3 S: LLtJ = 2I 4iol,1:, AS2?.i ,fc A4 p 4.%cYD 574 = 1I,2= 5i2() = 4eoii) a iL= LZ1 rii U 1-2L9 p = iiCfr) kci; II V =2js1 2t”d — l’Lr 2V 740kr 32() 4,e&fC3): an—i ‘)7/O4Z (35_2,XO12) = 4z.5 -) . LJ D 2Z?r25C1J) (fi5JSQSi a—vJ° eaA-c4-): = 47#I_ !7/2 -S) C3-Q.X°25) i oq—w cheorci jg J z?eq37 O . i4 t 0 = 2V 54 eu?r’ rjr1 = -I z45 +ku-t the p+ti: ‘=)47”, )= O= 2. 3’?O’It4JJ) 1o.1- =a.92, =4,oD1z - ) iX1) ?(52i#i = Lr L.iJ) 375 -iji-) = JcoQ = i t 1 = 3Z/= k’ Appendix B. AOl Code and ORSI Handbook Predictions 216 ene—Aj -heacL) = øfl€— .4 z5. jc -LV — — 4e2) = , —‘ GZ/ 23okkJ) h’3) 5?(4vT) J u +2 &7P a4Z =zJj3.44’(4&) on€_w&i ietrL4) + _ beoir ç-c) Q 0 ki eoMYUj &c(2 X34Q2 )L LS= 24 -r c.4--eL7) =g X34 \O .43,,3 ep 1clt 21J 3kt) — A1&A4 200 () A )L3i0 =5.4mr, jMoH1’si., =4oon) 4ai )= )2.7(—M.) 2ipJ = 5r)LG/ =19 k) =4)=t2S8c4) ( J7p) v= ‘=z9 ØkJ) =zV= 7 — ofle—WR ie-r(’) ‘°° 7VfxoDZi FI±T f / L 3S9 2V Q—)O 1eo4) L_F_H y=29o/ =o V= I 600 IL7 = L3,5_1.)21S - 9)4O(O M-3(&JJ Appendix B. ACI Code and CRSI Handbook Predictions 217 Wco= V ‘€r-i) - ‘1% —+-ja-4 ce(I) G—1JO1 h-eAr(3): zv (2o+4o)4ov= I4sQJJ, 33Z •5.. V = ki4i kt= 44=?q6(kM) ‘v3322J = 17eoJin ‘9’: +kL2) $ ç A & A = fLQsr.L - 410 Iz66&iJ) , ,33 &t-1 C N/n) exO) VOIJ5X944VJJ jJ) 2 ( = NcoL2V z.87E,2S OQ-’AJ ?4L3) ‘4= 25(57jE5 - ‘424-J ane— -keoJ(4) v 4ct,= 21--bJ 4,OPclZl 1 2 7 a-J . S = QVlC*) z.zc(I57Sf i724 >OOZ.1 97?c4OD j53’ ki) VJ, \I3 4 9 —o—-. SoL): —)---Jo heoc). -Jo—1)1c.Jj 4At3). y=. Q33fl z4’o .lAJ/ =• = LVc,= tCft3) V= 1j2-4&, S2I mi’-j V 3344 c-ku) 2Y = c72 J, 33zzg2.1 = 0 Q 4’4 z( J) A A= 77SPrftf4Ih1&, ‘2 )4= 774l’’— are—’ 1Jpl€ o4 r c 4 4’,-f3’-1 z-{ 4iJ €O(U): = vj= OflPJ€X(3): = O2-5 ct,4’a) ? / 7 V ch4 ‘4= = Z5 (o I J - I7z4 7 ‘/i3C) 1 2O = 7o4c1) (c-zsxo.zs)=2.75 ‘25 )95D?4O W7 * 2.i978JIt = qc o—Jaj 12O ‘ 4fS3j D.x4o O5é -k Sk -k) ) ox4oO= ‘c324(,’-kJ < -a3t) oj 7(frjj) Appendix B. ACI Code and CRSI Handbook Predictions *‘eR’( ) — \= V bert = I7f ii, 7 v= - J/=o., —ha— 240D x4or 3 ‘3322j- J, 218 ‘icQ 5(J) 2-V lI S 4 i 0 c f 4tJ 332zSfl zc4o-= aiJ= JJ L7) Jir k C’): S+-€AC + 1 243zL-kJ) c& x t Q(2;’-3LI 2C2I5C4&) 0 k z 73413 M= 7g4)L4o (40o ccz) — 35 J ‘j= coQVfc one—wjseLtr3): 11= ‘z-V, one —ioj e’X(4) = I’JcoQ O—AWv —W ‘c3): ,e.CLFin 04 ’(IJ) = e 3 225(o.IJ * 7.2cDPe2 )so4oo=1z3 &t )J QJIQ=3 L 7 (4)) zV. = c,&=2V33O&k)J) ‘WLI) H-oio—woj c’1Z). 3 )5Ox4OD= 2.5(PI57Ji + = =V=54t) L4i7=o., 2I4’’ +—F) V.= Q.332ZJ 4( mip (J): ske° ’ 4 i 1 a z,c-2L,i .2c’D= 2kJ) , {jZ6pK?ci. 2 A17Smst , t2i-tt), ZkiL{ 1r (/A ) 0—Wj b5x4OO322 e4rc2) o— 1xi3)’ Vt95& co2V7(42J-) 3—o.%j SeiAL4): 17.z4 = —-v.o—vi&q c*XC) z,s4y= % jJ 2 4J’ “4 = 2.5( J Q32Z cøt’ 2-Vt Zxi4 o)95rs4o= 4L4 g3oJ - 1 L48’zi) ct ZV 3QZL&) -ki’) Appendix B. ACI Code and CRSI Handbook Predictions J/ w—i- se&rc2); — -f.-L’) =oE 219 2CJ) y= tJJ 14t= 4 (t), \4 zztiC&k1) a -h’en -F+. (). Wco 4I(R) øZ-X2Oø 215(k) 1 22&L2aO’ 0g ? 1 A j+{\; 7rn 24Zl4? 4o’ o1€—W — , lZ2(’LtJ) — ]{ - —__--—-—-)=124.1J--nt) (3 (k’i) heci. €-r(2); )5Dx4oo 24LO2 7 ‘4= = 2 Vt = _J zcV p L4i) = 2Y LO2.L) 2V 3 i31J) Jco.t ‘V’)/o.s, ecaAL); •-IiWQ- ]Cj 43J t3Jj* 5 e%C): sheoca) yt (JJ) = O—IA3AJ -V Nct= 332zj2C2\)t’O 537 Lk.i), 4lJLc z4 U). ZG= 131 s4Te1L1-; _ p& 9 22 4t IZ,4LIJ) , 75-)(43 -z4(J) tCJ ) $J-), z’1= 12’O/,, 32 kiJ (J/) 4-LktD kli( C2). v= z;(7SS 71 o’z1 ) 1’c rx4o q 4*) q3 ,cjc14pO=78P r-) 2-Vc Iz v= I&-(kJ) V2.iL), ,5 o-t.1C’Aj se-rL) —-VJ e4-ct) i/ —O5) L(-ki), = 333Z L-k) Appendix B. ACI Code and CRSI Handbook Predictions iLt) €i H 9 - l€w h+; y 1 y 7 - A EOJL) - 78;x4I 1 ZJ?& 52 I22L41—ff) 4&) LkI-) i’4= *iO—iOA.i .1.eLkYL1) icj /= O. 25Va(-kJ’)) QV I3’ kiJ 4 &)=+J, V3t2-3 j b’ (-kg) = 12) e’-’ fr+t) QO.W7Z2Y,2X2Ot 3J 4oo 134L&bJ) =}75L4). ‘4= -2 cQ -Yio- 53C-)) c.,L= l2O.21 )9 o4o= 77’7(-) . V = 2S Co.I7i Wct=2V;= flf—vJc- 1eL1-4C4 4i) u-• /A) 29L+)J), o€—c .he&rc3): 125Z LJi) c4pt -cL=\e.2 My 74ioC4— -J 4pI 5D(4J) oS ±!!x2,2 4x2oo l22(1Ji) s2O 2iJJtJ) c O,-WQ’M OQS(*JJJ) 84 %. V= J \g=32 -; 220 )23(k1J) cJ 4)Jiie 4O4 (-) ø A1LO MI m, 9 5 7s4IO O-Wo 4iecx-(e9 7? L4oo )= — I ()J-fli-) zJt /. (oi’37i4 2 ; cZ’ OWOj i€/L4) = eo.tL1) —kJo—’?J-4 2 *E -ecxrC3) z)4 — - cot - Z( --°i -)9ox4= 73L-b). ia’ (PJ) 33x4vqkJ) ki’i) . o3o5 )(cW)(4po= L3i6LJ) Lzç = -+Wo—W’ zJ £-kiJ) C 1bI45X4 23(k)) (3)- 25 .XI°= i= ck) 4c I YcZ’ C’) 2O?&) Appendix B. ACI Code and CHSI Handbook Predictions bead +flq+1L) 221 rL2) Jt= z.31PoL H l C= 4.=32Lk1J) xL74xX23(.i) fl,3&fr1) 1cot = 0,S5’c22 75x4l 0 7x4ko(40o— 4$JQ 116.21’kii-P’9) l22X10/ 4CkJ) , (,3) 27S \/2’12-(k) o—iaj +.rX0D0ZI Z4ao =zL ki.3 9 3s. o,g3j g-(&1i) iVC 7 lct C’57i “4= ,2L+-) q4x=iIs)’ 3o5x4oo’’15tkJJ) - , V=°33flj MOV’O —-io—Mcj shea.rL2): to_V’.Oj ck€o-L3: bica 1 24odx=/,c14(J2P) — — 2O (ki)) W/=oc, &) r “4 = 32aj 1 - = ztc1L cJ) c-høi- L2) eo”n OJ= 14Q)) 4ce =o.x ZxZ.3 2 r-- ozs!s4 x2t= fl3zLkJ) I1 = 4 l-i 1Q= 4 L+j.J) -çj=z,.9 HP 1 44o tr CD s-410C400— $2. )(75•D 9 l. f t / 3 2Jhl1so l 0 1-kJJ, ç’Ø )“ 101. (’ 2 t”4Jp 2O22.,&) oe—9Aw” (H-) i4ct2Y (H-) iL L4’&) one—A3t - 4c 950S02) = O ;ga)=;; 125 (35_25XI2) )-Z - \ •/ \ •1 Po 24,coZ1)0M0O 7 ‘4=. 2(ol73j 31 i I I -40O r 1 V’ (fJ’)> L2JJ), 9 974o= o4’3fj Zp2o ?v L41SJ) 033(k1J), =4Ck-i) Appendix B. ACI Code and CRSI Handbook Predictions o—u 222 (3S- z?02.7’2. 4) = (o S J -i-i7z4-oo2I 25 7 lct = —-uo-- -L): -4,—W- C/A) i/ =O2 ct2XCZ-) V - —i-- — eo &&i i O25 = -3o_vjcod(L2): 2V,’ 25.4L4&) V;c&) /A) i1\(2 9 ,cZOt 2O (27(b) L21 2.CF) =42N) co4L A4”, 2Mp$( ,c. 3 5 D ‘i 9 Ofl€—”3Oj Shec) 333g (-ji) . fmioi41LI): ).Ico-Q = S 7 3 x —i- 12S) — ( /A) ic,Q on L2): 7 ‘/c1. c>) 7 ot€-wcj 2V./ UJ) O1é = v,= 25cI7w? V = 7’ ki4), ZIL Gflj h€L1-c(#) sJ 7 v=5 (5 xOO - *ecu) oI25 )T 49t— a344.&iJ) _L) r 4tr )J= oS 3, 3cn’2(kI) = (t’/A) 575Jgoox4oz V = vG ax(3) O-37L> e.4’3 )c4 — ,4x°”L’ Ye IDI (kiJ), c24 ±cio—vJaj SiecuU) , -— 5I24(R) (7A) .32,z1SL-kJJ), x33 =ib% tkiJ) Jce474-LkI) 2 xo Appendix B. ACI Code and CHSI Handbook Predictions 223 5EZ—3 - A12, 4E3 c )=-si& i- —- 41 P L7b 0 / I ?- o.O77 7 ‘779 97b(4.34— 20 -t = 2 3 Z I3k-1p, t4ae2Lpkp= n,tLJ) L43 - \ / 3)23O /.4i’.. 4 ‘. 12 - — .4c t ) — 4 . a3233(5+ 25 A r14’I077O (44 _--) = 44 3 I-Iit — — B” += DT’& k =z 427. =- )=8Q7 S— 7 z?63Q(4Z -fD, A49’737i6’j 5ItL, 7 =4Z /4-, -P-= “‘7i i243 (+.27V— M 2t’’ ?) As — o.33 °.33 4S J &1, 4 joi- 1’ = =i.i -lJp çpt, J) 4.lz’k I7hd) 1 47L2k L-k) -=z&’s- A.=°i’7 , 1 = 2’o ib -;. . 2iJJ O2Z-k - 9o’35 èZp, ZPJ$ jOCO’ = z3 L&) 47 24’2 ? H Appendix B. ACI Code and CRSI Handbook Predictions kvf&) : .ii 224 /A) V,4.3T&= .c, p = 3.IiZ/ 4375 ) L) — 4.E;y: ‘.‘I4 f= or1r7/( = l’7) 5 4 94 Yd/H=43 1’. 12 3t Vc) Q 0 Jc 13x47S344.c i1. 35 I V=(I .j <3. J ,x434-= 3’-7 ik. 441 3 V/M=43$/2=I.74I y=i_ -j- ao,7x i.74)3 ‘4/= 47 43q4. I.9 I1L sc 3.5S /’1 4 /K x34-34c= j 7IL4) ‘ = :142., I’ 3.5J MJ=V = 4, ILC.Q 44Q zock 5:!_ = q34 I. ‘Y= V= 4 .3c-) LQ47 +z- oiz.) 4= 3?c462 4a-•= ‘ 5(-kN) )(42S) p= 35J’?( I3X47’ iL = /o,x4,z) \4, °1 /xi..zY 737 OZZ 2t4k= J5 4La= ‘ftJ) = zk= 374 I 4/3?(4)a.O2.5 ’) 4i4 793B 7 3 -- ‘= - L1I’Jr1.= i4&’() zL’v, $4.: (‘7Ot, fl. 93 = OOiZ 3i I2’ I6 Appendix B. ACT Code and CRSI Handbook Predictions 225 3flij S4€OA”3): 4375/. = 3, (3, — = 2.9 V 25(9j, +2ODOZI 3YT) ia35= zo S fl,. 9 = ‘4,, = = t= 4;J_ I3xb3875= () = t4 c’kt) 25 =352) (3 25/) 4c/’, = 29 2c,36 ii,. 2Q \/, Vd/ Q 43/ 2 ( 5 ç 2J V, 1 t 4 r)= (.3,5 = ÷2v UCG.9 1 r 444 (35 1= - 2/) Z.7’T .c 2 \4 = 3 ,s _2s/ 2S2Ll 4$L? , b. 2.5/.) 4 7 /,zs=.2 j+2vX3.Z J/.j= 7)(34z1I5= ‘. S = x3)(4Z= ‘2572.1 I1. -‘r 2.y (J) “zS’ c74r( ‘ G .f 51.4 1c. 34275 ‘r2. Z5’Z( , (i) t3c’2 / 5 )= 2,72,S .b 4 xE5’c3.7) 3x4625 V = 4- c’ 35 , v/= x3k.3’?+523ag . ‘2 5I. 2=V7’272) ( .1c49 S92: li’. > 25 2(rr7 -7 vd/ 2Z931 4k=4z.S4v’ 1cbJ) 4’3T5/j ,ss. ‘. r 1 i 2 /- Vc 2-(4tVc I3L13b5 iI, ( 2a344 2.)3430’3 V. =25( 1ij4s-k 4 Z3o 278 ‘25’ — 40.’7 34sq4 ‘- 1Jg() ko)97x34)13 (145 25 43q4-5,/ l, z’z2- i. ‘‘J (3 I L4 -‘ 71 , 2 4bzc% .=”3 3Z5’7 2i’ V j )1cQ’ I. > J(4i2= I1-(k) 3’5 17. Appendix B. ACI Code and CRSI Handbook Predictions 226 Seoc4); 1J/ = v.= o57 i.° 4375 2S, -x2o9ic.78Z4 (—2’7)= 2.OS 2.3.b1r 4t= , SSZ- ) = = Z.O 0c 3414 = ZWk=\4, j3 = 25/, 7 .i . 3—l,7) 2.Dg -f’ ‘/e’ 2pS77oo= 2) 4 J= 5’- 4314 z.,/ 4 1J/ y,=- 517 < 44L= c2*fl25O(ktI) LO, c ‘21.O 1 2p3,r t-4- L-.5o5=- z.O *ae 2 _L 333z <oJ i*t vs., ç<- (- 2s)-=2DS Lf’ = \/ e.2k:r 142’(5 42. &? 4L L4?’ 2’bLi4) ‘L 5 42..9 3-’ <tGjj’4 = ‘I = =e1E4 O) 341 T 25/4= 2.4)2.I5 = tt= = L) 21 = 4=&’r -k’J) --G Vc30.’7 f 4 ? (,5—25O5k= &, Appendix B. ACI Code and CRSI Handbook Predictions 227 e.cct) two—QvJ 3+ 3S7) r( SSI. 23.Z V , 4j4.4 23.2 37=’ 27.4-k = V =z7.4”= t6 ii =i(43) Z3J IJcQVt . 7544 , 2l(&)J) 11(- 4s4.5)= 232 SS+: 23.2s-+3q4 Vc=4 zai., — V — 4 j .i 7-Z3,2t 4s94- \4. 1*.. - ‘kr 22. ES= <.sS(2 b(34275)= 240 S-i12) - toX=• —ie—WCj (‘/ ‘4= •J 3 v jr., o) V’22Jt? a-JJ hc(a) ‘ o+ -a-bt) 4 V 4 g 3 t(5E o > e%L): . (4.•375) = t) — SS2: ba +149)t° Ij 442L’= -- 4-V V -I’k.? =s (-kzp)= =4J ke-) 2Z6 2Z ‘kt) S2 S-S; 4z.:) 2lJ) (3- 4.’5)O.t V -J= 4V= *-‘P 434S 4i3 (3-4275)i, 4 =4V V4i 5&’A-’ {3t4.625)= W°J SG2SGi3: = — 4-’/ — o?27=4 () =sik y 4J = t.ox4i o.4’k Appendix B. ACI Code and CRSI Handbook Predictions 4-k () S)S2= 3= 228 -itz eir1g 4 scs2=Ss=- -: 1-536 WGQ L-kT)= r4 242(4W) Ss5 SS,: 57J,(k) = cGrz= 7i.(-ki)= 313 (-kiJ) = Sz= SS: 2__fop 4 t3S2 = = c-ki) I3 4)J.pLL r) 4 4- = ‘-‘ (:&p) c&) A: I6O i€wk: $ 44nn’, -L43-W. H=9ox4J7(44o— 7%.3 = ‘11-’.S) ki 234D) I 0 47L1 141’o. —- z4.L1oJ) )_o j630 k a=o)z4r = 3J,. (.çJ3) r()= d 41I)= © ‘-‘/b3 21kF)) 2(222561i) iof__ 0 4r9 ne-W’- k€O.((-I) 576 ÷(—. )Ciz4) I2-O 7ojOPOZ, ‘‘ —-= 25 ‘4 = (I51iJj -t-i724-x25) 2’4V4-S7 (*s) Tt \4= 417c_ = 1coZ -r fl3](-kN) 2( I3-4-) 1 324(-kJ) tiu-*S) Appendix B. ACI Code and CHST Handbook Predictions z6O 229 o€-Jj Sot b’ 234Drrn)-) p - V( o.i7-H7z+ )Z3&V4O = 9 Z’o44c heorC3) 3q6 cj tp: ‘/= o_j4 o3C4}-1) r1,-crr’ -1-f e4) — Wc ane—Joj $ke-): i@t = .3 -- p —j) (_34)52 —= 2Z ‘°‘ Q4L3j ØI 4 t ‘4= 2344O2p -dXO1 €-k-) c4) eA1) =C-kN) z)20 Lt- 2qO- zI2O) ) 4 boE( eccL4-) ei: iS Yo vZ- c= ca-)= l,3(-&) -z = z f(1ox+)4-*- V= °-33J* ---o = 2 oj 44 22 2J4L-kI) I31& 4 = - = . a. 5- -. - cyk c> L,t-o), 3LJ) Appendix B. ACT Code and CHSI Handbook Predictions +_ WOj co-t 230 &FQ3); LZ: = 2(4W(I- = 33 ) 4-?q2J33 ‘j- = 2 is --= 37 ji) -fr- #- L-ft 24-7 - b2vt&1) 2(3Z)j c- = zP 3Ornfl, dt-. ) — = 22k ,22q’ U&-n., , 2 A=’l 22.C49j (4o45— ‘2 7÷6) )(49 z4S Z7’o(k) 1Z 7 (L) ---4’j ‘W 1)- 41i 6& 3b Ot2/ f 3 OP 3 --= N0h-R((2): v 7(W) Q2O7J 23O?(3’TOj33Z (kJ) ti\ — — zL)-Y “-‘Q): v24L+-)-) -(—:)vz1.= iot I.zTzmnl /az2i445)°r7- Mt g 7 V= (ai çl.ot 737(k3) ,LI’N) Ph2oz;I) .irect = =2.(3 l/= 33, 7’j - C3.W—2,3S) = 7z4xi3 )23t3o=qi )o.4*3. 1-eon; oe -c c4-) sI,ot €tü: °‘ =3373°i1) \/ I4t)’ I 35 2. OT7) sq&= z244(kJ.) Appendix B. AOl Code and CRSI Handbook Prediciions 231 Jo—J0’J V03322ç[ 3G—W l’et,%c7): (LJ/ t 4L *i)o A B171ss9 kJ) = = kiJ = > a.3) o--r(oo ÷)= 15O 0 b hfs AArec1’ti 4cQY c4 =2(4J 2(3)7+(kJ) on rectL2() 32ZJ2 x V = &)j 4 2ll c& f.1Z = C: 2600 I t f — o 0 0 o ii- ) . o4- = NCQZ)(l0*&Z004fi5 tkiJ) crj Q 0 4cZ 4oo CjJ) Jk( b= z-4tjJ 0—AcIj aom’n =°‘ j, =oooZ ML4. 4-)’7ci4o= 4 j-2 ‘Ic (P7iV7 (iii) = IVL = — I24 - Appendix B. AOl Code and CELSI Handbook Predictions 232 oe_w&ç cket1: V Z6°)i (O7 JoN) V/ç ° 0 3Q/ O 3 ZOc’i ,3 =3(-i4) = —tDAJ cJe&c (3): sko 2omn — ?04_ , /‘ = = zC’ LkJ) V = 2.(O.1578J 41’z4xo.,34-x Z. )2&t)(39O = c’L iV A&f) Ofl€—j çhrr 43k) az: j V=.)(qIa= L35_25,r77) .a )7 L4jJ) < ,Ic..f= J 0 4 9 vV3 ) -f 2 o)4+C4--3OV)CJ = oooo ht bA z ft3’?o -W SoW): jq&kt \4=c22jxqq(kI4) ko—oj c11e(2): nc = ) 4 GCøi Ol’J0J heic3: — —co-r1i€r.:O = OO--2lOO?(4O =&1,m’*., fleo-2l br(2b)4n V,= q3 2j74?. 37 = 127c-bi) .20i2+ — 2SV)7L4Qt) m’t c2) ce- 3322] Vc L3< W71 Lki3) ‘l€iv j+ 1 , 4 lZx3O+2W0x4 l2OD-i2’ = (-& 2.303 4)o7—= 2.o ) J Ir..LL), 3b.4Y)ftt ,=--b.4-)= 4-)c)y3fl7fli< 17L mn1 v=43zJz7 3•w1. 7 ;g ,74 i,9a) o-- 2’3om, 4vc4. = 3omM -f,= 33.4&) t,i?o. )=43.z,o(&4-m) cu)= 43)44(3) 33(¼) 2i+ 1 Appendix B. ACI Code and CRSI Handbook Predictions Ofle—VJJ *e&r () 233 0 2 4 q7440)&3 4ret.,v’: 4o/ 1’74 \4=(oi 78!+L7 Fx0o3xI ‘Jot ‘Ic — 4’tRs) reetoi b = 23O 7q374oo27 Lo oAre-$ = 2134 + 33t) tcQ Ofl—1XAJ V ,IqeL-rL3) 3 220 I20 L4()’) 4p) 7 r 37 0 -i-;) ot 74 2s7Jc1Oo -- = )z3iG3 -33z(bJ) 3Z = = Q7 3S0 o.z9or1J $23.)(3B0 1- J) = = .Y j-o I’ M ==ZS3 P 23[.I, 2.>2.S 3) z3I I $3o=zI72L V= Zi57J -z4 27 ree4-vt Wt heC) V= 2JZ i:7 S3Z — 1017 74d,1kII) 47) C’( 744) 2 o%v my !bi = iC \/o.3Z2j o 3 3.c1o7i,p bC) (1)/ hLG A30&AJ 1L3): A%so.71>0s-j Shor& rt; I24 Z)+60 ÷r1)o Ar€1Zc’i-.; b= 3107 2X4Ot LkiJ) co-h-Q 4oo 5& *1Z32J s.44’c eI (4L1) 210 Z’!(1 101 r — LQ3 2?(O3-+ ZI0€ — 2205 \/o 3 3 2ZJ )4L= V. 0 I2’4 c3S0 = V 322..j3G? 9 pn LS—2.%, ) 1 =l.3 L-kM) — v= i7 Lc. cret-Dv’. €QvTt)) 22- O zCtc3Z+2(Z)=’ V .27s -+?Jo—1.d O2(&)J) ---‘ *rt rt.-: czt q4?33J9L23IlX3&,= zd&) )J9tk) 4) Appendix B. ACI Code and CRSI Handbook Predictions S1JC Short ritt;on: 234 4? 33on’n z4oo’43-i- o,c4’7q 3 4#.i (411 4 — 4ML&4’9, 3O tar, k41,!, (Y’Y’t ‘ - 53X’ S — an AIr€t..: 2i 99= is177 1J) 74zSL-ki.i) t=2-L2I37-N97)= frl€—WC i1eoc.4-u). J 9i”O, I - 57L,t f(4o?ci2l) 1 4e7A/ VAd/ 21 Vc(0.’78iTh --.24.xooIeo — t 4+7S&ki)) Zøt O1I(€et v(o.Is7JI = 3Q43 Y=7 --=f1t) ritot $hort ) 2.! -4’7.1 =kR7L&lJ)2qoJ4 x/238. b4o79 = 3 o z 2 ._L) i 44 ,,( 4 1724 3 oreet Lon -- /P &&JJ) <Q29O7J .234’x4-J I ‘?73 )o (*MJ 474i) = 2_= 2(Jo7)a7Ji) O—WO *€cLF(3); 1?=23Iø.)mn, sret 1i4 ‘ ’94)22 2 C3 ‘4 Z5(°.7kW +7.z rtEoY ‘4= 2.o3- = ).JZ(J2Jo7±ZS3) oe—J Iheut4): 4 N = 247(&l) 1.37 lpnecto= *0—’j 4eocc [ iloz.) = JLZ). (-7I = < 9070 L4ü) 441)4 \4=33Zl4i.I Xl llTZS7 = ±0AIC’j i.37 ’flbZ1) 3 VZ47 2C2.4-37 -r 2 l’fo (k)1) /444t/ horétw. ll344-(*j 4 274)2316i 3/ 7S).o5 300 -4o7q)4J = L= 0 c V tpt) = 24-&) Appendix B. ACI Code and CHSI Handbook Predictions *(AJO_j 1..c-(3) 1’o•-t redt: 2A- 235 2O4. tr -iT3- 9.w’ = L3(d7&k) Ne lJ3rL Vt 2(I37(1-1 = z4b+ir(w 0 b ,)rea = 3322J )=qc€i) 3 Z -Q 35 I G ‘I23t,2v)%M IO7+(,JJ) I23.2.L4o7. = ) = 69e c-?J) = 2o94(. I I 6O +Ie1S+(: reeefl: Aieomn olE 3BOl1Wt, -=44-1? )377z (-kil-4v1) 4 2 17 3a.3 x4L = A 4emi = 5 4OO Mr 44,(4— 4gooSb (UJ-+) — 1s7 1\iL) t’4caQ = 5 b’!.% = II53(h) ZCI 1153) = Gn-VJQLj gLl) ireci’ j 4 3O I4’ - ,)=q33 4 ,X 40 /( b=4”’, .74. 23 &iJ S 7 V (ol 25& i) -> oo4CO NY2&L)) 22S&(1+ 1.J ort ree+t -= i.sz 4= V l.4A ‘4= g.÷l (l z 7 4 jxt.2)4ot)L3S 2.k3Lk)J) Vt I’icZ = 195 2L--95)’ 57zLk) I ckw) Appendix B. ACI Code and CRSI Handbook Predictions one—ij& 4hecA (3) odrL; = \425(o.I573ji-l- 24 236 4oc’mn, —23,C3_2 > 3 2)25 253)4,Bo.... L4) re: coL= l t v=’ q54)2-tk) SOr-tE recfii d/w3°/, jt I9 — S2) L)J) < C-kF4) 1n4r’ect,1: zcs) be.&r1 4 = AL I3C-kt) jq4- o52 X 2Sx beoc 3T+kJ) 2 A: = : -1ct= O.)L2Xi7. = ]c.L C,: E ! = NL4M32’ &. ! c-fr-ik ez) (bi) 4,324 zC () tO,ZIz20 Ncd = i44J7 = 7C-ki) 5 l44 ii) 4 £S2 L qSz3X2ottIui) L 0 ‘J 4c b47Z ck) 4ki L= 0 lL1c 4)(25 J??.L= 53xvisek 4-42 7 kJ) Appendix C Predictions from Proposed’ Design Method 1. I I (1) etZ x.m?i€ ° I t I i I I ml I I L_1 4- I I I ,= E1 4O.2 c, 4t, 3/2 I = I I the I I L__J I eJt & -rt- ‘net -i=. r- r’ —t tL 0(2 I e=c( j ç ) 7 =3a2 oi( Q2= ‘= — e—c= 600 —J’7.+ = 4o3 S’vt2 237 cf Appendix C. Predictions from Proposed Design Method (1) -tue ‘1t eii 5347 30.27 rA Qf -*k -—--- 6€ co s-h-d 40 53.b7 = c0lrrn’L cd 0n 6 COSS.15 ‘2?( Z. h = .77 - -r = esi?e. +mct ok 0 i.34 Ø3 Z ?— c -tt cj r3 22O € 9 ±kQ. W€r VOL h — T — 4- -. ‘ Z.I3’-+ r(L 7 T 2.l.77 = —n\_ i-u OJ9fl.9 033(J43_l) 4 OCI.ii--—t)=Lo4 (S = o ccl Q.3L137— 2elL’e x1rt. Fb’ )= = 0kf(I i-r 20 0.32) o.af (j€rw,’t) CoS° -c- . ‘_ — 0A)2 ve-co-t = ±. 2 CS3’1 cs4b.3° toa. Qt L’eI. e-.d- of = o.s4f 52 - caci I75 ? 35 9 C0s46.S o-%zL cao- o 4- . .2 -frut = — C) 333 x440= ij Lt) -coc IicLn— fm = ct) 238 Appendix C. Predictions from Proposed Design Method (1) coL V 2q7,Lt)- kkt)= Scmt. c 4S -LF 239 2L1 -kt) 2. , A , 2 ZCW 1 ’ 2J° 2,4-L,cmt, 53.2.CMt /= .7S, b (j4vt43 /=i92 .o47. =o.z Fv S4Stt) F , — — 1l11: 23/2 bzffo.7f ii.aut) <- lb.L’U L,W3(1) C = Z7.tt) ‘9ZI(&) !I=.3efrt, , 2 25ck A= , 3 q=24.3 4 7 2.3’ b /=2.o4, 32,OeM o<=3°, f—iZ.5° (J)—’.*3 (J) t 1 .i4 = o= 1S3S3, b °ft, f O.iYfc’ F= ‘3C) ia= ‘z.F= .O’, 4.2J 4= E= }.{= I 23.9i, = 33.7GeAt, h/= 97 :—= ZIML, 2 b 4t/,=z.3. (=L,lz-3, c=oo47, F lqo,ct) = Zr3.+Ct), v 47/z, ?4O.2lC rl,ft, i4—z,.%tt) . (f) = f3c- 4k 4zcmt, CAt, 227/1, I7.T. 4= 23.7 1 = 3.O2t7i , ‘L 3327 1r/. b3., t.)j1fS Appendix C. Predictions from Proposed Design Method (1) 3 J-= 2j4, F tx=oc47, F= I,1t) = 240 t°-7 4I.oL1)< 37.ott)= 3S3(-bi) f4 c 7 / • t 9= L.ZCA?t = 7b.+e- x— l5’ = = , S2.19cA’., )yzLs J=ts F =1-7t.1() = 311 2f -f emt, = I .4’2t / =o.o47 =z7 /c4.t2, 4+7em., A-= O(- = 5.S — 4°, Q = 37.9 b,z3.3ct, -k/E,=24 (J)=.7 - , k/8) 5’c.’t, O- 9.imt, c4I,I k=,bjøt, ob 4C.øt?, ott, &= 7.1 b=iz.1em 0 33c, 443 qi° ci . f’O.22fc’ 19.r, - i-’ q= 3V° (J)=22 f3f 4.2(44= 794Lt) 1’?3Lt) 3.t) =4ss, ISO(t =zS =i7 ) --O = ‘= /i , , /=7°, !-=.o -CdQ 4 7’) =scA-i 2 b = = i4Lt)= )t4h-’7ai4’JLt2 3F7• )1= . (i*)22 (‘O3,3 J4=ZP loq.o’± fv= =51f =o42,f =j.9lct, ‘icQ (J.)=4.3 /b’23, = E = 37 CO) = 1=°•- Fv —. = 4s/ Appendix C. Predictions from Proposed Design Method (1) =0c_1) 241 3h,cM, =4’M°, c=,3a°, 7l0, zZ71’, l3.o OI3e, = = h/b = 330, Z3 h/b , z7.’, l7I , h/=.zc , O3 Z,a2 = 3.b = 2 f Q7p = ,=Z0.7Lt) 3F, = 7,n c&= S, h=/e-’, h/ 7.IJ O( 1 1 =a.7 = 33.G, CI f;7z.’/c i’7.7 X h/,=L5 b=43M = , Z7b L4ce, Ac= (j3 f ’ f= 2 —F o7b-f Oo3 Pj33LL) = 4Fv o25kFi) iZ.41t) 4S/ Ac=*.ie , 6 &= 41S 1= X 32’ =4j,et , h=l7e4 = kl. 4z; =-? r1 c’22.7 , c])3 i23, h/tvJ4, a2S, =L79h, = , = I37 2-Lt) ‘- f=o4 i54() 3cg kt*) = 7cøi, cL= p= •qe+tt, fL=37 O(-334° O(ZO.0°, CL=23.C41&, 4jmt, 1 k= LM-, I4i J I ‘L+. 3 Ct) ==i75 ‘/b7 h/b b , S/ = 4o Ct) 4P = 4-7.2it)= 47’k1’)) I-u, çSf 54a L) Appendix C. Predictions from Proposed Design Method (1) 9.oew. 4: 0.. = — ko4r?4.t, 3 c 4 - = 3L,.S, b=4z4cwi J*=2.1 h/= 3) j=Oi4-h Fh=.Lt’ 43: 4Fv 3.3° cA 14$1, Lrp (j)=3. =o33 , J4.9Ct) As’ 2727, +20 k4-s$ -,=l.o2$L’, t =1f (4() -4bZ 7.7x4b2oL) lz9.c-tz= I gtt) 7 7iUz) 7aL-kJ) 47:3ekM 2 , 1 cJ.I.9e4fl =ji’ ,=i7.9°, = 4g.57ct, 9334cmt. oizh4°, 42° q=.z,.O° j-)1= i7I (J) = -oent. 0.’3, Pk= .°)= ..Q1lfL’ -‘=o,aF =l3tt) b361t)704) 4F, =2flt j k 34-/c4 iLi)iO4L-kiJ) I1c- 4F ‘‘ . LX’1°, X=ki’7, Fk 0, j=IOOeM., 1 2,7cJ%q’- /b= 3.°4, ‘• I41) i’4P 4k .. b.bLt) o(=’•9° , F -L= (J). b= & f=o.7I. 43CAI 925O’).-, &= IOOCA) _ 4%.0C±)4LAC-ktJ) let, — 4-3.4, (J)ii (j)= 3o K%=t52 ) = JeZ Of’- — S3c’$t. /b=L.44- 242 -A-tcn, As , 1 toe.4& /c/t ob2 4=2o.7 k/= %=2:17, 22° ()= 243 Appendix C. Predictions from Proposed Design Method (1) (j-)= 2, 24-3 F= I&4-Lt), kQ °IJ Lt) <Lt) 7 5O. f= 4Fv = =A13 Ct)z fflQ4d’J) 4!DJ 2wt. , 44ø, O(2-I.°, 121fl., &I.32i. 1 Zj° O&!Ll° b° 1/,= ro, (}&) c=O3L’7 = = 25Lk) F , Ioo L41’) = 4t7 (-k1) NL = = I 4I? ., Q= O2ifrt, = 37.0° , , q’= 33,’2.°, o=3S = 7, 5 O=o2 = ‘21 = 4wt -=‘fL O. 4 2r,k, A2 & A Q , oL=o)7,4-, 44f: A= 4o e=.’ 7- o(= = 4-to t4?’ - 2.&°, I7.’) =449,5bwt O(29o, /=2t 15 .z mn’t, , t.c’, =z&€i) = 4 mM, = = -f c9 03 h/ =2o,7rAm) 1 b , 4WMi’tc, 22., ) 1 4R 37.7 ,k 4iv Mis, =4ic2m%, (=.o F 37.3(p) ‘ t.(p)= zt,(kii) = )4-it, ‘ 41 k/ qi, , (-r) C? cj)=i b= -, Al &A4: =o[Llfc’ -‘ -=,3’L , 1 h/ =764-, 21) -=t.07f , -cL’ = 27. M j0 1 L Appendix C. Predictions from Proposed Design Method (1) = 227, ((r)) ‘1 = p ls1= 4F— SckiJ) O=-477, 45 , 4Io4P, z 2o 39 nI P1 1 319 )J.E=33 = 4 h 4Fv , 2 / 1 r b )= (j= +‘!1U, t,zb- , F=.7Lk) -=D-.3L’kI’J) ‘24-iJ) coL i97 1’32. X ’ 1 2° = ‘‘‘ P= 2V77(J) =4’viflt )—4ionia k= 244 TfZ L) or0Mi 5 l1=4 1fm, -4o?-’W, f24’fr1 OmY0t, O 3 I?i L,t O4°, 44mj b= Za4Sr11t, h •?z= L,Qra1fl) ,j-=3.32. X0’b4 — Fv Fk= -j) 4F J-2 t7 = 1 h/zy, (J4 , 4c4, -Ii,= 7,z()27.7(k) qit4ci”) 4oo1W1, H=4°, = 9 (,vnt, 44-9° A1T o = i-L°, q = =2I.Mt’ h44.Z0t 21GM?c. ia’, h/k= 2,fl , = 0.744, 3Z, 42 = A7 -F )= 4’ -= io4f’, - = o3’7f i- = ‘o’(-kIJ) 1 d=4ooit, 450 rrtt OVLW(, -h= 49 zVC4&) = i34l, (J)= k/ , 1 = z7, k/= c.36o-j U 443.Un, As -4io-1P, 430 (4° c?—l97 ,= tilo.44flt h/ . 2 L} ., .L Z4’ç2MP S 4 Lj) = 1 h (j)=4o =S- Fv= 27 (L), F=7 1 )2277(kO -R’=sf’ s]coL F 4 / 8(&tJ) Appendix C. Predictions from Proposed Design Method (1) *=4oiwl, d=4omo = ot, k — 446 P cp= az-’ ) .3 H=.4om1rt (j)l (7&+, 49 F= 4F cc4J) 227.7LeJJ) a= 4cxD’niifl, 1 k 4OQfl1fl A5n. ‘4-io HF, -=1ZHF I2= -•‘ h/= t9q) 24P /= =$32..wm, 4kJ) 1e )(=4omnt, c=4o-mit, O=4• h = 4o8.o1ii f=-myvj, kiL f=of =o46f’ k(1&) =4-i&-P 4i1€ =—t° 2 q? (j) = b = 3.3Z, lVlLkrJ) -yc:= 4z3.I0, o=S1-.8’, O(,=2b, z43tn1M) = 1 R 4o F 2.3Lk)277&k))), , h/b = !, (J) 7c = 1 , =ou)cf, At2: -i.øE, FO3 , zon’w, =4o.°, ci= Aii c - k/= Z.7’(, , , 12 c= 32., = , k F= c=41oHP, f2SMPa 2 A4 = 245 O( 764 ‘/,=I6S, = (J)=4- O27 ?. ) 4Pv =4.qw1 = jJ) 1 =4niw Ach’ f4omi1 37.7° 2&°, 13J° &7fln) k= 44.4jPint = II3I, €,.511%t, 7 = 33332 h/ = zzC4iJ) , 1= , /=2z, (f)?4-o c=.7 , =2cL?,L-ct)) j- zzY].7(-o) 4 F iL4Fv4-Z2t’ H=4omm, k, , 1 = .u gc-ki) =4oo,rn1., 2.CO1M-, 1:= 39.7Sn1fl =4%, 1 c% Ø = vzjo’ 1 b 1/,= -5° Appendix C. Predictions from Proposed Design Method (1) ‘2nt )a3.7c /bi37 = 78 , =33c: 8-Io) 4=iooS(kJ) jt= -(=4r3fl1M, 4ç= 1 t.=4ooYnnt = = +°9°-, , V52L-kIJ) F,v 246 3331 i-L= A- ‘Z P° 3b3pU’ 333.2 440 = 3k 4J=. 02.S (4I) j= £SI d=47t. cu=o 4 A ‘r3°? 43’Y- AoPS4a43=.It2) , P= Si)ct4t.2 C4) • 45 s a=o f GOI 4., 4 tj l- 97b7 DOt)(9,tJW1 = $€ .31-kp) t&4I.g°= 47t2) R’-LO= )= 4ck) 1 -.kF=’ .4d ‘r1=.in, o= AOP7OS4) Z3OD1S, 436iv1., Q35.S°, c=47Se, q= 67°. , h/ i.’v 4, ‘ h= . = IS— b-?) t’i= S4: -= t.. , x= (J) = 2 ° 0764., (3 _4,9D = 4.9 t, I=3 , ro99?, fc-z4f L:=436F c= q,= /= 3;- 32° 147 (j4 b F=Lk) 39 4-fL:) , ck As°9 c I o39 9r) , F’)= # t., c° , , .t-z, f’43 OO76+, -= a4 ,f’ow- Appendix C. Predictions from Proposed Design Method (1) c&= 1t=, k= z.i4-’t., = A-.= 4-= p, 1 4= 7.9’. c. O2lI a 3 = 331 = 4’T.°, h/b, = , c=o.7M-, 72O D 0 t.- f= . , 2 h/= /= L7 j= i 247 ()=+o 3’’ 2-8, -f=ciS4-f(’, ,“cz4fj R = 33.& (ktp)’ (-ktJ) = 4Z75 A=o,o13 2 4-Pg i-= B= 331 15lat., 4o = Cj-)=irj = 1 b 21 k/. 1>_ i.73 4Pj L&p) 1 k=4-snt. -= LQ.3’Z6 14-9 C4J) /= I.75, sz -r q=i, I71 =—2.° 393I b= -71 =332, fl•) h/,= 1 (3= 437 , -=o.S3cc=o.7”r = Yt-kp) < F 4(kF) 4P’-° = j = 43=?z.5.2-°, 1’ JcJL o.S+ f=4-’ F zSkj’) c , = (J=4o 3.31, O(=O.7LA-, = ki 9z43c o.-g+’ r L-&) 1Ick)J) 71 — t’ a.= 0, -= 44ovr&, =471P = Asfj = qo 47q •= = , 2 Ont 4-3,.I4’1.6° . 0 =zr = a7c4vJ3 JcoI= 1tZc t39Qr, 4’i9-, Appendix C. Predictions from Proposed Design Method (1) o.= a , 248 vz,vx47’ — 57 ( 7 = ii.J) F )= 1 = zq7 IQi’J) ;ht iret-o; f: 4r7 -)= pii, ct= . A’c -(,i°, f7 frP 5’4L?&) 2o41 = = ko-.t& .$q7(I-i. a€Z- ..hoi.tr H=Ynnt Z1O)1VJ ô3On1M-. a=iO’’, = 1 H?R, 1 h/ 3I2.O h/.. h/bh92..J r3i4i H4t42J1) 1 Tq4( T 2x —i- = a 11 i7O) O=S7.2°i b fo.siPo cJ)=3S7 , h io4+1kJ) Z3’Lo 4 j Ck,) Skoit dreetvi 14? 1 -1riWt 1 -f=Ut’irnn j=4Zk°, h=4S.G4n11vt, c=41.T°, q c4°, 0 g b,zb.3rn1., %= • , 7 (J)=3.s z.nwt, 1 /=L4-I, 3 ,j= M?. , P ,-.°4fL’ (’kJ) 7 F1o(&t)< sc O3i - ± -)=&t) t=rIYtn, k=9Oflltt A Z4’Z O• —25, 1 &—)fl p)2 , 3-j? Appendix C. Predictions from Proposed Design Method (1) J4 = (Q242, Fk= 0tL&) WL vb44(i -- 2O) 1774 4eJ-L.) 249 7-PL’ c Appendix D Predictions from Proposed Design Method (2) 1 Freet 2tfl 35 A&’ C4t’, 1 t f44o/c “ 1.’s -= /t fr4 . 9 5t.1t ffL •‘; .4 L_1 -&=-:;-= coV= zAPj±o4 = 2 .4O.Zl (44 = ‘ c=33(-L)047 -o4o-tiqo 275.3 1iLRL%) )C = 3 0b03J444 3i = o.047 o4 x4f-ij -+ I =34) ka.r fr’+k” C.oLan: irr.j - 600 = = 337.2 Lt) 33(—)o.(S7 x o37 )4-33 () IeLt3= isf33(f&) 250 3337 L-) = 5/c_i.2 26.6 ieo..fi) 4 Ofl.-4)0j , ‘a. = 275.3 AiieJx . Appendix D. Predictions from Proposed Design Method (2) 251 =223/, = = =-4o.2its, 5 A 4gg/- = .Z /‘7tt) 2I2 ki) z4.Z52323. =4 ‘= 2A-ci’8 P0 4 ., =i’ 3t L 7 t)c 2’Z 473s/ctl, A4O2Ie43, A1 d= .4tiM 4.25&, , = A () 2 - 5L.2CA4 43 L±) = 43 kIJ) z7 4køt 4 7Lt) = zs41 = 0 j I c4e)= 4 + the.- c.c. =441c), = of vttkMk Crcl—= o.Sciii c-e- t= 447ewt, +68jT =7sr3/eA.,t.L 32±) — 325 (-k14) N; = 27/4, A=43 = ! x445 - =7O2, = coL , oO (ki; =6w 34o4-3L,Lt) 7”’ /‘- 4At47’ J J 59 Ct) z.S39 A= 4.ld tj 4 caL” 4”(*)= 1- €- Hlj = t:;= o(j Wf.+ Appendix D. Predictions from Proposed Design Method (2) =3y-, k 4Lt) 43. 7 t Z3 x -ic2. bZS: 4 3(’k) 90c4 0 1 ,4t) 42 Lh 7’ = 1 o 7 =a S/2 =43.o3, 43t 252 ‘2S KL= , — -) Lr;2c.g iJ) )•‘‘ A47c 4M4: Z91/ 44F732j4 —-j !7c±)= ri4i4Lk1J) -t /‘ntj ct= 4c3 = f= 2I.2’7 co) No. 7 C-) = 4iofr1? , A1’P4: eat A2AS: -b 1o3aL4) = As=t’’i fj 4toMVt ,=4cDri =33’, toSi(k) =° Mo 4X ni, pii I =A4=A7=i=MO AW’AZ: = AA1; h1V, 17 4 4 44oJ) 44o— A = j=(2oo-)JZi2.lmfk. Appendix D. Predictions from Proposed Design Method (2) : = = = T2 ‘ ?- 4. og444- 7.. 72430 2! .S 14’) = 253 i, ‘ 7 (&J) 9= ft’, 43Q4 L 4=)23 1’ 2) A— = 4.4t Th -1t= ?‘-°‘T s90’77o o.2fl , , 4A kJ) =7zr 4i -[ 3.1(r)=3t) 4c5.) A33-?, 39’3)L9ZOj J aç sqS: 4 j.!t’) 4-Lo2flx92os. MoL l44Cd-) Y 42il-, 141i 4.!iz- 4?cz2 ,6c’ocD( -L4l4, Ao.-’ 4o54-,o’eD 122Lr)E4t.k1J) 4cfl .ç= 41 fri?, 6{=44o’, 440 ) 4 = !O Z L 2. X9G) ‘ 49 Ot3ThJ - _37m !012..,L r.(L) 4 l -- 534 = 433 tl€ii) c=1O1I-rk33I4i-C) J395mM, 2acz 2GOThM.., jOT1”, 1 Ic h=37m’) fl -. 32 ck1) ‘=---- 3= =_ Appendix D. Predictions from Proposed Design Method (2) 254 di ZA1 =OYt) 4°f 31]r5 234C&, = 23C.4 ZrlJL) 32O LOO1JJ) = eCtO /=3,vrrn, 242441P 330 = 3flmrt’) 4-ltii t’! LkI’), zJ Jtt 1 — 3 3DJ S3 4LkJJ), 2,L) L2OL41, L= 24i 0 I’ic seJw 375P.) 3,-I?€&, A=Z4’, i)OS ±0A 234 LJ) ca =23/i, CO1fl: )=°“ , IJ_ 33(iY0P47 , -f= o6x25S t DO4 O3j &2.355 Pft47 J23 ane—ij he-orLI) CILI) N,nz) ki’1) 35O.9tY344otk) NCOU3* lo3COi) =MoJ/nt, 45At , -S, =obx44 1 - = o,Arc n—w&j gt= -\3 04 - .bO3o •2’7:3 oi37 4-o 2Jb. netC() ef= (37= oi37 1433(&N) f/ 37.2;o7(kt) WZL)= z. 3x2 31t)4) Appendix D. Predictions from Proposed Design Method (2) 255 d L= 27g =o33(--_-1)= o =33(----l)o3,; c6c3oT, — 31. /,,;-) 5=222.7L) 21 = Z7I X5= 429 (;t)= “)_ (ê) nei 22 - = 4 —l) o=o”47, oc) JAj O)1— = 3a 7 . 7 4 ct—.O (-n.— 22.4 3’ zo,2.L) 4t7 L-&J4) ( 212.4 i-3 k,( W ptLCLtbL) oi.°ti) , N.t /‘- =295 22(&J), 1.L= H)11, z52 = 4ii (k)) Ckt) q ; cottI) N7)= /2 2ZI.C Lt) 7oi4L.) he’O o- apt,b) f’= 45 l/ Ot 3I5Zj- k/ =p4’7, ofl€-M f ,2J i/ o eki) 7 4,t) O327 1)=o.i3, ShQRC(i) irt z71z 8.4 e t)>i.ô Ofl-JOj ziZ7. i’ti Ø=bO47) ‘Q33(- N c> 323j nt sh4) 7 j+5=. C4f’ 323, ct)o1o 3z = =a.b4*o —I’7’7 O3 o.33(-- —l).2) f 44]e, C Zoitc Qf 33(Ii)O’2cl, o4S-1- P’ =33(4_4)0Q4.3 1’ Crt.& = 3.5Lt) = 22(kJ)) 7 3.,x4-. O€—l-4 kc(i): HC .cLftJ-E 4!b2 )(OX4O43T-. flr? 3’1.541) N(L2) C.L(3)O(ki’J), I.o1&)i) Iun= o.t) = 3x —‘J SJ’cLt)- NcL4)72Z(klJ) Appendix D. Predictions from Proposed Design Method (2) 256 L = 33(— c4I, ) 4S3. 3(-E—--D=op7, - xoo7Io,zz’ —2774 ,?35 Ui)= 4-7 eiJ) 1 r’ O—’ c4Oj435323 6s43. O3O -4j—Jo-4 tL o2&(t) OD7cLJ) eas’); 4.(4it) 27O7’) =7om 3N3; =4i )=o2, (- O.234, —I )oz, f=obs.43 ictLu’ 3773 kS 3(), 7 7+.° Lt)74 .cheo--u . 4L = nnn = - 3z.c = , 97X2 (-) he’-rc.’) = 4-ent O33(1 )Q49, 33(-——1)’0233, _ =o)L37254 N/ , z3233 j3 4z4-3)L3I-k±) 1 t 7 33’ AtFi) 3 i/’. y 33_)oZ3, = (fJ) z7!.4 7004 t)SLkIi)) e*-c o4 voJt g ±) &‘-j fthX4Oi -PP XOZ3,X 2T CrCO ‘rt) II 41o OJ,4O A)?)3J -= 3O2.b’3 L2) fl3,e 33{l o25, = ? 1-ki.i) (3=o33(-—)=ob2 OzL3.3 .O52j 44,3= 3773 + €o4’t) t— t74 4o -fb4’3- o—’w e3rC) tJc.Lq.) 1k) ;f3’/: , rTt; 2& ‘O) r 23 Appendix D. Predictions from Proposed Design Method (2) )—)L 224 ( /) s-e’-. J) = 257 kM) 42: ?33(4 —)= .44- = j) , 373 = , 22 3L7 -- ‘5922x 3 ’ 2 tb = = =2’23 432.3 (-ki) 3b’(k’) = bS 112 G33( )O# 09 0Z3t,, j &34i \ O.3(2- —O23 34z= ±22. )a.732 —I) tZ3b, 35= O9) O(J) c9(4)= 1 3 3t— 5°4r’Th3 -‘z 023IZ32Yr =g- ;.i= tWO—i)) x=o.’’5S, 3 , 44- j= 4• x5 — 2 -fb 3+IW- P> 424 ‘.tt’ l’icoLU 5 1 c 5 23 443 I244 /e 1 4ct) S ‘ie , /t tobo4(kN) c9a.Seh 4’: 1 c=oZ =O,23, 33() 9 4 Tb°’ 33(—l)=O.4k5E, = =o4-t tlP I33()) 3OV’1 €r1C): L cc.2(i) = 23o44f 4X-4OS 3” /t j/ ’ 5tt)=j353f-) T Ibb = 7o24C4zJ , -o353.4CO23b, I7Ø7 = ‘.0 °.33 — lj=.’g, f =b 354-p ccLL) ‘ 1 4-zLkN) 5 r 45 o7 := 243g}. 43 i22LLt) 44 J/2 ‘z.4.2 f75 r) (&‘)) Appendix D. Predictions from Proposed Design Method (2) 44: 2 258 42 F/f 5L = 32=3(-j_1)=a444-, -,&4z-- o9,5= 4s24-.9 1 t ?) c-tt4) W{’)- — 374(), 24j 3= 44ct)= — -4p-k, = fl3>ç}>.) 3 O 0ft°33f XIbXR363 ij- z 32(ki’i) 7 1 (&) z’23 ?$Z 4okr) 4c-fcy &rt) I31 L-kN) = = 21(kIJ) 732(kI1) F=3+2.p ci= .4;z 1 34 3kZ234? ) ‘47 o•3! —i)Q NCØ-Lr 3ø’] -9=aQM?&, Nc= 321k) -G’ p$ vzLkt) ftJcJ. c=4oOmrt o33(—>t, (,= o33(— L)O.3S —j-(, $ ( ct nie,r e vi€,,f -t)= 033 =e•33( )2+3) ‘= 2 , cett 2t-k)J), Se4-(I) A1&A1- 2) 9z(-*jJ) I3.t c 4 p) g 3 3SJi = 3.5 M?& = 355I2O’ I’f 20 (4 t’ b ct.) -—ijc shec(I)- 4 c *3 —o- 2 G33o 4.3 M1’ bo kij K2= SeeL3): tq’ (-ku) = ii= I4’ZoC$) &A - (13S, =24.5 o33, -f=-o.x27-i- a24So33=p& rfr,()4) jo—Jc Sr(): 24k) 44Lkk1) mJz= 7:(-kt) . hAJc sbexc3) 2ZS(NJ Appendix D. Predictions from Proposed Design Method (2) 259 HP. Oi.°, , - =a,7c3I.L± .) d— 2 4&, 24G7933Jñ — ZhLk) coL(ij = +J1”- iO)’--i Sie(3) , = Z432CkN) . j,,= a.33S, = icLLLY -,= ob-- &3j’2 1 f o. ibSG4J) 422 £ z4xo:3j )c24.9 I.4-2= 4cL2 , .9 MI -kt*io-tj Seut3) = ‘rf1 Jn X2-4j 24Ø, + xo2A ‘ 4i -S (-ki.3), zøo= kUU) AS4.) CcJ + 2?44CdJ) 43’-, o(i-°, L t Ns = =‘2&OM?i, = (t) 4i.2 I4? 33jzh. = 11 4s = i3.l x 2OI5 Z2Y *30-WOj 5kcL2xt3). V2±4}J) 22/ (e) lkL?N) = =oz4z.-i035, tL)= —-- rab4Z + 320 (ZC&4)) r): 3’.3 -i?o. tL4) oa433J eJj) 3t3(k)} 4&9 ZJJj z4Lk) j_3VJ =z-, =4e =4z1M? =I0,=38, =a.24 =O33 -‘ ,c2,.S-+ / .32(k 2433J 4- =1 V?t’ TX2 •O-0-’ ((1.) 233’L&) ‘-AcJc) Appendix D. Predictions from Proposed Design Method (2) = A1O: 7r 4g4 = 4Xi -= = =oS2+ IQ, 4= (O3, o.2-t 32.4)c2’’ i29t&) Nc) -) Sk€cx) 4t -4}-, -f,= 74--i-f ou7 4-i- b$O.Z4 G()0) =o2J S 1 , (3= o.33, 2 (-1) Jt) Th1- Al2; c 4-c= 7I (-k) e-): 4)Zki, —= GX2,-S3SE aC-ki) lzoL&), — -); NcJ4 4o. fri. 7b )(Z3 t 2=3) 2DO 2 =4etmrk -L-°, ‘ic-&L) o3jE4-j2. 4?c (-kN) = ‘z.Sl?&, ,3+Ck) x4xL 2 2fr ‘4c’(1) , (4v) = = b )=ar22J.i) 4 L Sb_tLr(); GAL 4 All: 260 ) 17 —L2= -+UjO—&A3’ -r) zzt(ki e)= 2 (kJ) 20 l2 11 c=zTM ‘ =4on 33(’)33’, =cb2ti - =ck33 L33Zl)) = = = (S 2 = 7 xooco38J qzci) , “33, , 3--(C 3’33= 44? 4f7 HP = - =‘ .s Appendix D. Predictions from Proposed Design Method (2) 434y o22 , 4.;r5 11 43gf 51 = -,- oI 261 PZ/.X. =é43+ 7ZX2Z(°I53j4 otpst) 32= 2t) 2(kL))J Nc-b 4Io(k) o— ,hetct). 1 aj,35 lC-(4)= 22.LkJJ) 1c.t1> tfL): O0V€ ZZ() = 434 -? 3(kJ). 4-x3 4-i. I tkp) WLI) = —tuo-ij heof ‘i raz = 2 P) L?) = ., gJ.t-kL-) - -o—-w he’—t3 S Qi’22(tJJ) 33I)O0 a 1T7 b _eArL1) 4c.Jk) 0 (? ( 2 2J ‘‘ )2I&kJ) , te-W0j eL4-3) CI) Z2, ) L&J) Th= ZI4ei.)) c( ‘: .0 O(t , 4.1i., +?Z$ 1] N L€fl) =4tZLk?)2I t ) 4 — ShW1): SGz= s3= -= .2_,c7 p • y 40X €‘-ft 2 2lFI4.5 = ks’74I 374 LP) IoS(k)= CL5)2I() x= o-z 4o& LP) = c.-= ctzy4 JCAJ&J.5h1X(3) QI.I) 4 =,o, 5V +2 SC-) (4p) -‘kN) q7 j= -, .IckIz) rø 4Y7 o7 Pit) 7 42k)= l’5) Appendix D. Predictions from Proposed Design Method (2) oc dr3o; 262 Lt+, o4&, = —2S+kW = 2) , 27(?) x2’== 3Lj()) iCO.e.L) 34.(V± = I’15 1Jc) 23 (-kM) -÷ eJ) Oe—ix 32’fLk), MZL)= hearc3) = k) €X’) —4A .J.j4) (-kJ) = 2 ,o(=’ = 31O z÷432j O43Z o, = 27.C M?) = 23L1P&) )zo t4vJ), = a a- )= L) ceQcz) X j 3 [ i, i’v1l (I t 2Z.(4t)J) 7 IZ 3kJL), -±wo-J&j shxi ) — — SbGrt 26(kN) iLL+) 2,4(4UJ) (&I3) 2j M?m.. t: 9e nt o3(i)=o4z. cO33 I 2 )> sq f= Z7.1-i-xt42jj= LPR) - °, 34-I c-ku), = 2.I A2( f2ø= Ofl—t’J 6h’erU): ‘wo—t3J shzrL3) >. jy= 3o4jJ) lg4-ck) 34- (i-+ = A-° (I -‘——j- )= za (kiJ) ie.tr 9kJ) 3-±)= b1fl $JL 3-IP, = c(= 03Li)3ii2, 1°, =o33(-j_l )=o2, = ;4,(-k), WD= 7(ki), 4-C-) c)T(— Appendix D. Predictions from Proposed Design Method (2) —) fLf:) c )37/€?J) 3 ( cL) = r 1 { Mc’k )11I” 70 r 5 u—WM , 263 Sare).. k (kIJ) (iP) /=oc44, t &z= 4-I = I 2 ) *20 2339 c) hEzVf(j) ‘-co.L) 40 33(_t) O1Yt 33’1 i)=0441, o, 3%. 3_)2T) •7s’= Ncts = &r€—IAIcj L 0 347 (-k4) 9cD ks)) 03M’& rtt\ IJtc) 4 = -il)O_1Jj IAau. 4c.-t) &1eaJ L3) 27sqck) -+- —‘ -÷ NkL2-) 3° (1#) ii’fJ z4L-ki’) ‘J--t = — 3â= (4z)J) = 39.0 ) -to—iuj L1OJJ c4c4-) C-k3I), 7 82 2I42ZD ec3rLI): coW r34(-k), -cC3) I104.(&J) iI4= ic-ki- f3o.3- 37(I + M( jj=’zI.4 I3ok1) + 5L-kJJ) 7 q (4&)
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Shear design of pile caps and other members without transverse reinforcement Zhou, Zongyu 1994
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Title | Shear design of pile caps and other members without transverse reinforcement |
Creator |
Zhou, Zongyu |
Date Issued | 1994 |
Description | This thesis deals with the shear design of structural concrete members without trans verse reinforcement. The three major parts of this study are the transverse splitting of compression struts confined by plain concrete, the development of a rational design procedure for deep pile caps, as well as a general study of the shear transfer mechanisms of concrete beams. Three-dimensional compression struts that are unreinforced and confined by plain concrete, as occur in deep pile caps, were studied both analytically and experimentally. Based on the study results, bearing stress limits are proposed to prevent compression struts from transverse splitting. The maximum bearing stress depends on the amount of confinement, as well as the aspect ratio (height to width) of the compression strut. The proposed bearing stress limit was incorporated into a strut-and-tie model to develop a rational design procedure for deep pile caps. Two methods are proposed. The first method is a direct extension of two-dimensional strut-and-tie models used for deep beams. The second method is presented in a more traditional form in which “flexural design” and “shear design” are separated. The shear design is accomplished by limiting the bearing stress at the columns and the piles. The first method is more appropriate for analysis, while the second method is more appropriate for design. The rationality and accuracy of the proposed methods are demonstrated by the comparison with previous test results. In the final part of this study, the influence of bond between concrete and longitudinal reinforcement upon the load transfer mechanism of both deep members and slender members without stirrups are investigated. An interpretation of an important shear failure mechanism is presented. |
Extent | 5985302 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-04-08 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050429 |
URI | http://hdl.handle.net/2429/6970 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1994-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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