A PLCP finite element crack model for the in-plane analysis of masonry

Masonry buildings represent the great part of the world’s historical heritage. During the past, these structures have proved their high vulnerability with respect to seismic actions and the structural safety of these constructions became a primary problem for both heritage preservation and life protection. The performance of masonry structures under seismic conditions is strongly influenced by cracking phenomena that must be taken into account to obtain a reliable evaluation of mechanical behavior. The present work presents a non-standard finite element (FE) model, coupled with a mathematical programming procedure, to simulate initiation, growth and propagation of cracks in masonry. The material is studied at the macro-level and modelled as a homogenous continuum by means of triangular elements. This hypothesis makes the method suitable also for other quasi-brittle material. The analysis is limited to the in-plane behavior. Tensile and shear cracks (i.e. sliding) are taken into account. The method distinguishes between micro and macro cracks. Micro-cracks develop in the initial phase of damage and are simulated by means of spread hardening plastic deformations, according to the classical theory of plasticity. Macro-cracks are represented instead as localized softening plastic deformations, which develop along the FE mesh edges. Cracks opening is detected at each node by a specific limit value of the tensile/shear force normal/tangential to a potential crack direction. When the limit value is reached, a cohesive crack starts to open, following a linear softening branch as proposed by Hillerborg. According to the Colonnetti approach, the non-linear response of the structure is obtained as the superposition of the linear-elastic responses to given external actions and plastic strains (representing the crack opening) conceived as unknown imposed strains. The unknowns of the problem are the nodal displacements of the elastic model (as in standard FE method) plus the inelastic non-compatible nodal displacements representing the crack openings named crack multipliers. The values of the crack multipliers are found by solving a parametric linear complementarity problem, assuming as parameter a load factor. The main advantage of using an LCP method is its ability to deal also with configurations in which instability and a multiplicity of solutions are possible (e.g. softening behavior). Different procedures and algorithms are developed to manage all the structural configurations characterized by a not positive semidefinite Hessian matrix because of the softening behavior of localized cracks (e.g. bifurcation point, inversion of load, loss of overall stability, etc.). Some numerical examples are presented to validate the method.