Use of incompatible displacement modes in a finite element model to analyze the dynamic behavior of unreinforced masonry panels

A finite element model, where a non-conforming quadrilateral element is utilized, capable of analyzing the dynamic nonlinear behavior in a biaxial stress field of unreinforced masonry panels is presented. For the material, the linear elastic-plastic constitutive law is adopted. The formulation for the linear element and the extension for the linear elastic-plastic element are proposed. The solution is carried out by a direct step by step integration procedure in time domain, based on the Newmark method of the equilibrium equations, inclusive of inertial and damping actions, the latter evaluated using the Rayleigh hypothesis. The procedure was implemented in a computer program and verified by the analysis of an unreinforced masonry shear panel, the dynamic behavior of which was analyzed experimentally [1, 2]. The comparisons between the numeric results and laboratory test measurements show good agreement, proving the good performance of the non-conforming quadrilateral element also for time-dependent and markedly nonlinear analyses. In addition, the case of Parma Cathedral Bell-Tower subjected to a dynamic excitation available in literature, was analyzed using the proposed model. The same case was approached by a reliable finite element code [3], using quadratic serendipity elements and a more dense mesh than in the previous analysis. The results, in terms of kinematic parameters, stress and strain fields, etc. obtained by the two models, agree, proving that the use of a non-conforming quadrilateral element leads to analyses which are computationally economical and simple to use in input