Effectiveness of the probability density evolution method for dynamic and reliability analyses of masonry structures

The main objective of this paper is to examine the possibility of using the probability density evolution theory (PDEM) to determine the evolution of the probability of some structural parameters, during dynamic processes of masonry buildings subjected to seismic actions. The study is mainly motivated by the computational burden that is required by the Monte Carlo method in the case of step-by-step dynamic analyses of structures with large size, complex geometry and a highly non-linear constitutive equation. The PDEM requires the deterministic solution of the dynamic system in a limited number of cases (much lower than that required by the Monte Carlo method), together with the numerical solution of a linear partial differential equation of the first order. First of all, the effectiveness of the method is verified in the case of a simple problem whose explicit solution is known, mainly to determine the most suitable numerical method for solving the differential equation. Then, the dynamic behavior of a masonry tower is analysed. The structure is modeled as a beam with a hollow rectangular section, made of a no-tension material with softening in compression. It is subjected to the action of a real earthquake. The Young's modulus of the material is assumed to be a random variable, and the probability density function of the displacement at the top of the tower is determined throughout the time-history. The results obtained at some time-steps are compared with those provided by the Monte Carlo method. Although the example examined is quite simple, the PDEM appears to be very promising to study more complex masonry structures.