A variational approach for the numerical treatment of convex structural MDOF systems under stochastic loading

In this paper a stochastic variational principle is proposed for the numerical treatment of convex MDOF nonlinear systems under stochastic loading. Convex nonlinear systems are structural systems involving unilateral monotone contact zone (steel bolted T-stub connections, f.i.) and/or unilateral material properties (no-tensile material, f.i.). Unilateral contact problems of monotone nature, taking into account the “Non-smooth Mechanics” framework, can be formulated as constrained minimization problems of the potential energy of the structure, and are numerically treated by means of efficient convex optimisation algorithms. As a matter of fact detachment conditions holding on the unilateral contact boundary (simulating the possibility of partial separation of the adjacent nodes) can be mathematically described by subdifferential laws. This gives rise to a Linear Complementarity Problem (LCP) that completely describes mathematically the development of the detachment phenomena along the unilateral contact boundary. Actually, for this class of problems the response has to be treated with utmost care because explicit and conditionally stable integration algorithms cannot be employed. This highly non-linear behaviour has to be taken into account and coupled with the stochastic nature of the actions in the analysis procedures in order to achieve the relevant process of the stochastic structural response (e.g. displacement, strain field). In order to deal with the loading - and/or material properties - randomness in a natural and consistent manner, some versions of stochastic variational principle were proposed in literature. The characteristic equations of MDOF mechanical problems could be obtained based on those of the stochastic variational principle. The basic idea of the stochastic variational principle developed in this paper involves the expansion of all the random functions by Taylor series (about the averages of the random fields): Primary focus is placed on stochastic variational principle for convex non linear structural problem under stochastic dynamic excitation. Based on incremental theory of nonlinear analysis and subdifferential laws for the theoretical treatment of unilateral contact area, the corresponding formulation for MDOF is developed to use for solving nonlinear problems of random structural dynamics. Nonlinearities due to material could be included naturally. In addition, a numerical example is also utilized to test this method.