Unified approach to shear analysis and design

Abstract

The theory of reinforced concrete originated in the late 19th century with the development of the linear theory for bending. As the first rational theory in reinforced concrete behavior, it satisfied Navier's three principles of mechanics of materials: stress equilibrium; strain compatibility; and constitutive law of material. By the 1950s this linear bending theory was generalized to become the Bernoulli compatibility truss model which covered both the linear and non-linear theories for bending, axial load and their combination. For shear and torsion, however, a rational theory that satisfies Navier's principles took the entire 20th century to develop. By the 1990s the theory for shear and torsion contained four truss models: the equilibrium (plasticity) truss model; the Mohr compatibility truss model; the rotating-angle softened-truss model; and the fixed-angle softened-truss model. Together with the strut-and-tie model for local regions and the Bernoulli compatibility truss model for bending and axial load, a unified theory of six rational models for reinforced concrete was established.The two non-linear softened-truss models have capacities for predicting the behavior of reinforced concrete elements subjected to in-plane shear and normal stresses. These two models satisfy the two-dimensional stress equilibrium, Mohr's circular strain compatibility and the softened biaxial constitutive laws of concrete. As a result, they can be used to predict the strength as well as the load-deformation history of a membrane element. The rotating-angle softened-truss model is simple to use and is adequate for situation that does not require the ‘contribution of concrete’ ( V c ), while the fixed-angle softened-truss model is more complex but is capable of predicting V c . The rotating-angle softened-truss model has been used to analyze four types of structures that are subjected predominantly to shear: low-rise shear walls, framed shear walls, deep beams and shear transfer zones. In addition, both softened truss models are used to construct the concrete stiffness matrices used in the finite element analysis. These matrices contain only three moduli in the diagonal elements. The assumption that Poisson ratios are zero in the non-diagonal elements leads to undesirable limitations of applicability. To overcome this weakness, a general model of the stiffness matrix is proposed that includes two additional Poisson ratios. All five material properties in the general matrix are currently being established by new biaxial tests of panels using proportional loading and strain-control procedures.