Numerical methods for the dynamic analysis of masonry structures

In this paper masonry is modelled as a non-linear elastic material, with zero tensile strength and infinite compressive strength. This constitutive equation is able to account for some of masonry's peculiarities, in particular its inability to withstand large tensile stresses. Assumptions underlying the model are that the infinitesimal strain is the sum on an elastic part and a inelastic part, and that the stress, negative semi-definite, depends linearly and isotropically on the former and is orthogonal to the latter, which is positive semi-definite. This equation, known as the equation of masonry-like or no-tension materials, has been implemented in the finite element code NOSA, with the purpose of studying the static behavior of masonry solids and modelling restoration and reinforcement operations on constructions of particular architectural interest. Regarding solution of the dynamics problem, it is necessary to directly integrate the equations of motion. In fact, due to the non-linearity of the adopted constitutive equation, the mode-superposition method is meaningless. Instead, we perform the integration with respect to the time of the system of ordinary differential equations obtained from discretising the structure into finite elements, by implementing the Newmark and the Hilber-Hughes-Taylor methods in NOSA. Moreover, the Newton-Raphson scheme, needed to solve the non-linear algebraic system obtained at each time step, has been adapted to the dynamic case. With the aim of evaluating the effectiveness of the Newmark and Hilber-Hughes-Taylor methods, some dynamic problems whose explicit solutions are known have been numerically solved. A well known, specific property of the longitudinal vibrations of finite and infinite beams made of a masonry-like material is that, due to the non-linearity of the constitutive equation, shock waves arise at the interface between the zone of positive strain and that in which the strain is negative, with a consequent loss of mechanical energy and progressive decay of the solution. Comparisons between the exact solutions and the corresponding approximate solutions obtained via the Newmark and Hilber-Hughes-Taylor methods show that in the examples considered here both numerical methods yield satisfactory results. However, these methods assume the smoothness of the velocity, while it is actually discontinuous in correspondence of the shock wave.