A Numerical Approach for a Hemivariational between Adjacent Structures Inequality concerning the Seismic Interaction

A numerical treatment of a hemivariatinal inequality problem arising in dynamic structural mechanics is presented. This problem concerns the elastoplastic-fracturing unilateral contact between neighbouring civil engineering structures during earthquakes. By using concepts of the non-convex analysis, the problem formulation has as governing conditions a set of equations and inequalities, which is equivalent to a dynamic hemivariational inequality in the way introduced by P.D. Panagiotopoulos. The purpose here is to estimate numerically and to control actively the influence of the interaction effects on the seismic response of the adjacent structures. The latter can be obtained by adjusting the gap between the buildings and/or the contact material behaviour (hardening or softening) according to the optimal control theory in structural analysis. The numerical procedure is based on an incremental problem formulation and on a double discretization, in space by the finite element method and in time by the Houbolt method. The generally nonconvex constitutive contact laws are piece-wise linearized. So, in each time-step a nonconvex linear complementarity problem is solved with a reduced number of unknowns. Finally, the method is applied to a civil engineering example of adjacent buildings, and some conclusions useful for the praxis are discussed.