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14 There is no mechanical reason to suggest that the concrete cracking was considered. The MCFT can predict the com- contribution to shear resistance at ultimate is equal to the plete response of an element subjected to shear and mem- diagonal cracking load, but experimental test data supported ber forces as described in Figure 6 and more fully explained the argument that the sum of the diagonal cracking strength in Appendix A (Appendix A is available on line as part of plus a shear reinforcement contribution calculated using a NCHRP Web-Only Document 78). 45-degree truss provided a reasonably conservative estimate Since the development of the MCFT, three other com- of shear capacity. Over time, additional expressions for pression field behavioral models developed worth noting the diagonal cracking strength were developed to account have been developed: the variable-angle softened truss for the influence of prestressing, flexure, and other axial model introduced by Belarbi and Hsu (3437), the fixed- loads. However, as noted in University of Illinois Bulletin angle softened truss model by Pang and Hsu (38), and the No. 493 (29), where the data that forms the basis for the disturbed stress field model by Vecchio (39). prestressed concrete shear design concepts of the Standard Specifications and ACI 318-05 (30) are reported, the equat- ing of the concrete contribution at ultimate to the shear at 1.2.3 Other Approaches and Design Provisions inclined cracking is a convenience justified by the simplicity of the result and not by a rational theoretical model. The MCFT provides a clear model for the flow of forces in both prestressed and non-prestressed (reinforced) concrete 1.2.2 Compression Field Approaches members and for calculating the angle of diagonal compres- for Modeling Shear Behavior sion and the concrete contribution based on the average ten- sile stress in the concrete. However other ways of looking at When the parallel chord truss model was developed, shear resistance remain. Mrsch (31, 32) argued in 1920 and 1922 that it was not Another approach for evaluating the angle of diagonal possible to calculate the angle of diagonal compression for compression is based on plasticity theory and an assumption there were four unknowns and only three equations (see that the diagonal compressive stress is limited to a fraction of Figure 12). This dilemma was overcome by Mitchell and the uniaxial compressive strength; 0.6f c is common. This Collins in the Compression Field Theory (33) through the model is used in some European design approaches. introduction of a compatibility relationship made possible Methods for calculating the concrete contribution to shear by the assumption that the direction of principal compres- resistance are far more varied because the concrete contribu- sive stress was equal to the direction of principal compres- tion at ultimate is really the sum of several mechanisms of sive strain. In addition, within the compression field theory, resistance as described in Figure 13. These mechanisms are the concept of compression softening was introduced. The shear in the uncracked compression zone, aggregate inter- principal tensile strain, 1, is considered to decrease the lock or interface shear transfer across cracks, dowel action, stiffness and strength of concrete in compression. In the and residual tensile stresses normal to cracks. In prestressed MCFT, the average tensile stress in the concrete after concrete members, such as bulb-tee girders, the bottom bulb M=0 jd . cos Av fv s . sin 0.5Nv f2 jd f2 M V Av 0.5Nv s Figure 12. Free body diagrams for development of shear relationship.