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21 and therefore used in developing the resulting empirical enables the designer to consider the differing contributions approaches. As will be discussed in Section 2.3, there are of the various mechanisms of resistance to shear capacity and large deficiencies in what has been tested experimentally; the factors that can influence these mechanisms of resistance. most experiments have been on small, rectangular, simply There are two principal shortcomings with this approach. supported members that are over-designed in flexure, loaded First, in developing this equilibrium diagram, many assump- by one or two point loads, and supported on bearings posi- tions are made that cannot be fully substantiated. For exam- tioned underneath the member. In addition, most tests have ple, it is typical that these approaches focus on only one of the been on members that do not contain shear reinforcement. By multiple mechanisms of resistance (e.g., shear in compression contrast, most members in practice are continuous and large, zone, interface shear transfer, dowel action, arch action, and have top flanges, contain shear reinforcement, are acted on by direct transmission of tensile stress across cracks) that exist. distributed loads, and are built integrally into supports at their Second, these approaches then assume that mechanism is the ends. Because what has been tested does not represent what dominant mechanism for all loading and material conditions. is designed with provisions, there is no reason to believe that No single equilibrium diagram can capture accurately the crit- empirically derived provisions will provide a reasonable and ical condition for all types of members at any point along the conservative design procedure for members that fall outside design span and for any combination of loading. the range of the experimental database used in developing the A further complication is that the experimentally measured empirical provisions. This fact was illustrated in Section 1.2.4 concrete contribution to shear resistance used to calibrate this where new types of tests illustrated that the effect of depth, type of model also requires an assumption for the angle of diag- concrete strength, and axial effects were not reasonably onal compression to be used in calculating the concrete contri- accounted for in traditional U.S. design practice. bution to shear resistance. Thus, the concrete contribution to A further complication is that only a limited selection of shear strength Vc cannot be clearly established by this approach. experimental test data has previously been available to code Although the angle of cracking may seem to be a clear committees in developing or validating empirical design indicator of the direction of diagonal compression, many approaches. The database effort being led by Professors researchers contend that substantial shear stress is transferred Reineck and Kuchma is attempting to overcome this problem across these shear cracks with the effect that the true angle of by assembling most of the published test results. A remain- diagonal compression is typically smaller than the angle of ing challenge is in selecting which test results to use in eval- diagonal cracking. In NCHRP Project 12-56, shear tests on uating provisions because even within the narrower range of large bulb-tee girders were conducted from which the angle what has been tested there is a bias toward members of par- of diagonal compression was often somewhat larger than the ticular types. Furthermore, not all tests are equally reliable angle of diagonal cracking near the end regions of members and those classified as shear tests may actually have included because of the introduction of the large anchorage force from beams failing in flexure, because of anchorage failures, or the strands. A further complication is in counting how many tests deficient in their setups or members deficient in their stirrups cross the line of diagonal compression. Some detailing. Therefore, to use this database effectively for researchers argue that cracks often do not cross stirrups and developing shear provisions, a means of selecting and are likely to run from the top of one stirrup to the base of weighting test data still needs to be developed. another. Thus, these researchers propose that the number of An example of provisions that are effectively empirical is the stirrups that should be considered to cross the plane of equi- AASHTO standard provisions for reinforced concrete mem- librium in these models should be taken as dvcot/s - 1. bers. These provisions are empirical because the angle of diag- To describe more accurately how shear is carried, some of onal compression is assumed to be 45 degrees and because the these provisions provide two different relationships for Vc, concrete contribution is taken as the diagonal cracking strength one for members with shear reinforcement and one for mem- which is not physically related to the concrete contribution at bers without shear reinforcement. the ultimate limit state. It is only through validation with exper- The truss model with crack friction is an example of a imental test data that these provisions can be justified as effec- model based on an equilibrium diagram of a member in its tive. The AASHTO standard provisions are not based on a fully ultimate limit state. Additional information on this method is consistent mechanistic model of shear behavior available in Appendix A, which is included in NCHRP Web- Only Document 78. 2.1.2 Type 2: Relationships Based on Specific Condition of Member in Its Ultimate 2.1.3 Type 3: Relationships Derived from Limit State Comprehensive Behavioral Model Design provisions may also be based on the condition of a The strength of this approach is that it is based on a com- member in its ultimate limit state. In this approach, there is prehensive behavioral model of the beam. This approach has one equilibrium diagram showing all of the forces that act on the potential to capture the true complexity of shear behav- a given section. This is a very powerful approach because it ior in which the angle of diagonal compression is calculated
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22 based on the calculated stiffness characteristics of the mem- these angles by considering slip deformations along crack ber, in which all mechanisms of resistance can contribute to interfaces. carrying shear, and in which failure by breakdown of one or Furthermore, the MCFT was derived from experiments on more mechanism of resistance can be considered. elements or panels in which there was a uniform distribution There are three principal shortcomings of this approach. of stress across the width of the test specimens. By contrast, First, there are the shortcomings of the behavioral model the LRFD Sectional Design Model is permitted by the LRFD itself. Second, the development of a hand-based design pro- specifications to be used for the design of end regions of cedure from a comprehensive behavioral model requires members for which there is a very non-uniform distribution many simplifications and can result in significantly reduced of stress and in the design of members that can have upper reliability of the model. Third, to fully understand the and lower flanges that are very stiff relative to the web and provisions requires an understanding of the underlying com- restrain the deformations of the web. These effects can lead prehensive behavioral model and that may be beyond the to (1) unconservative results because of the additional interests of most design engineers. stresses created by funneling the diagonal compressive The LRFD Sectional Design Model is an example of shear stresses into the supports or (2) conservative results because provisions that have been implemented in codes of practice of the restraint of the web deformations by the flanges. derived from a comprehensive model for behavior. This Determining internal stresses in an element corresponding design procedure was described in Section 1.1. The potential to a particular state of stress (vxy, fx, fy) by the MCFT is a mul- shortcomings of the MCFT and the effect of assumptions tistep and highly iterative process. By contrast, the comple- made in deriving the LRFD Sectional Design Model on the tion of a shear design by the LRFD Sectional Design Model effectiveness of these provisions are described below. is a comparatively simple hand-based procedure. Developing The MCFT is a smeared crack model for predicting this hand-based procedure from the MCFT required several the complete response of diagonally cracked concrete to in- assumptions. Predicting the full effect of these assumptions plane shear and membrane stresses as shown in Figure 6. is beyond the scope of this project but a few simple observa- Because the effect of cracking is smeared, it does not attempt tions follow: to model the development of individual discrete cracks. If the behavior of a member is dominated by the development of a 1. In a multilayer sectional analysis, such as conducted single discrete crack, then an approach based on fracture using Response 2000, the longitudinal strain varies over mechanics (50) may be more appropriate. It is also a rotating the depth of the member. When the MCFT is then used angle crack model that assumes that the direction of cracking to calculate the shear stress at each level, the distribu- will rotate as the orthotropic stiffness characteristics of the tion of shear stress over the depth of the member varies. element change over the loading history of the element. By contrast, in the LRFD Sectional Design Model the Research results suggest that this will only occur after very shear stress is assumed to be constant over the depth of significant changes in relative stiffness characteristics; little the member and only the calculated longitudinal strain to no crack rotation was observed in the girders tested as part at mid-depth, x, is used in calculating its value. If this of NCHRP Project 12-56. The evaluation of the angle of shear stress at mid-depth is similar to the average stress diagonal compression in the MCFT was made possible by over the depth of the member, as would be predicted by the assumption that the angle of diagonal compressive stress a multilayer analysis, then the effect of this assumption coincided with the angle of diagonal compressive strain. This is minimal. See Figure 18. If that is not the case, the has also been experimentally observed to be an approxima- effect can be significant. tion and the Disturbed Stress Field Model by Vecchio in 2. In the derivation of the LRFD Sectional Design 2000 (39) was developed to account for the difference in Model, the stress-strain relationship in concrete is dv v dv 2 Figure 18. Shear stress distribution.