Stochastic response and reliability analyses of structures with random properties subject to stationary random excitation

A methodology to deal with the stochastic response and dynamic reliability of structures with random properties subject to random stationary excitation is developed to evaluate structural safety and reliability probabilistically. Uncertainties in the initial conditions, external forces and parameters of constitutive relations are modelled as random variables and random fields which are functions of space and/or time. It is considered that these uncertainties can be quantified only at the first and second order statistical level. A mean centered second-order perturbation approach is then derived to calculate the first and second order statistical moments of the stochastic response and dynamic reliability using this information.; The stochastic finite element method utilized is based on Galerkin finite element method where the discretization of the random response field of an element in space variable is performed using deterministic shape functions multiplied by random nodal processes. This gives the equations of motion of the nodal processes which are coupled stochastic differential equations in time. The nodal displacement and velocity response processes of the structural model will then become random vector processes due to both the random excitation and the uncertainty in the structural properties. The proposed solution method in time domain for the arising stochastic differential equations with random coefficients is a second order perturbation approach, grounded on the total probability theorem, and, can be compactly written. Moreover, the problem to be solved is independent of the dimension of the random variables involved. The frequency domain solutions, on the other hand, consider and observe convergence of modal expansions.; SFEM formulation is applied to structural engineering problems including the wave equation, linear elastic frames and geometrically nonlinear plane frames. Random eigenvalue problem is discussed. Nonlinear weighted integrals are derived for nonlinear plane frame analysis.; For reliability considerations, probability of failure of a system is considered to be defined in terms of first passage of assigned bounds of a set of response processes which may be selected as characteristic nodal point displacements or internal stresses and section forces. Reliability of the MDOF system is then approximated from a truncated modal expansion of these response processes. In the corresponding modal subspace, the limit state functions from all considered response processes map on hypersurfaces. This treatment leads to extreme decreases in system reliability calculations for MDOF structural systems with many failure elements, because of the very high rate of convergence of the modal expansions.