An Enhanced Displacement-based Element to Account for Tension Shift Effects

Engineers and researchers often use nonlinear beam-column elements to simulate the response of reinforced concrete structures. Namely, distributed plasticity elements are arguably the most attractive due to the good compromise between accuracy and computational time. Formulation wise, three types of distributed plasticity beam elements can be distinguished: displacement-based, force-based and mixed formulations. The simplicity of numerical implementation renders the first class of elements particularly appealing from a practical point of view despite the fact that the classically employed shape functions yield exact solutions only for linear elastic problems and nodal loads. As a consequence, when material or geometrical nonlinearities are involved, equilibrium within the element is not strictly satisfied. Recent experimental tests on the inelastic behavior of bridge piers have shown that the curvature profile above the base section of a fixed structural member is different from the one simulated by a plane-section force-based beam formulation, which satisfies exact equilibrium and considers only the effect of the moment gradient. These tests confirmed in particular that the tension shift effects due to inclined cracks in concrete members are responsible for a curvature distribution that evolves in a bilinear shape along the member height during the inelastic phase of the response. This paper presents an enhanced displacement-based element strictly verifying axial equilibrium which ensures the axial force to remain constant within the element while maintaining the linearity of the curvature profile. This yields in two advantages over the classical displacement-based element: First, it yields a better prediction of the moment capacity as the axial force verifies equilibrium exactly, which is not the case in classical displacement-based elements where only an average equilibrium is guaranteed. Second, the combination between the axial force equilibrium and the linear curvature profile can be used to simulate the effects of tension shift in RC elements, both at the global and local level. The new element is first applied to an example column to illustrate its main features, and then used to simulate the response of a reinforced concrete bridge pier that was tested under cyclic loading. By comparing numerical to experimental results, it is shown that the proposed element predicts satisfactorily the global force-displacement response. Additionally, when compared to simulations using classical displacement-based or force-based beam elements, a greater accuracy can be obtained for the prediction of the evolution of curvatures and rebar strains over the height of the bridge pier for increasing ductility demands. The improved predictions come at the cost of a more involved state determination with respect to the classical displacement-based formulation.