A spectral stochastic formulation for nonlinear framed structures

Abstract

The present paper proposes a stochastic formulation which enables the effective coupling of spectral stochastic finite elements with geometrically nonlinear analysis of framed structures. This is achieved by projecting the stochastic part of the incremental displacements, formulated in the framework of a Newton–Raphson solution scheme, to a polynomial chaos expansion basis. The nonlinear solution is consequently achieved by introducing these expansion coefficients in the iterative solution algorithm as unknowns to be evaluated. With this approach, an augmented system of nonlinear equations is produced and the values of the coefficients are obtained through convergence of the iterative algorithm. In a similar to the deterministic problems manner, the nonlinear load or displacement control algorithms are extended to their stochastic counterparts and a discussion on the appropriate choice of such algorithms depending on the problem at hand is presented. This work focuses on stochastic beam finite element systems in which uncertainty is considered in the system properties. The efficiency of the proposed methodology is demonstrated in benchmark problems with strong geometrically nonlinear behavior. In order to verify the validity of the proposed approach the results obtained are compared to those of Monte Carlo simulations and fair accuracy can be reported.