About the accuracy of a novel response surface method for the analysis of finite element modeled uncertain structures G. Falsone * , N. Impollonia Dipartimento di Costruzioni e Tecnologie Avanzate (DiCTA), Universita ` di Messina, Salita Sperone 31, 98166 S.Agata-MESSINA, Italy Abstract A study about the accuracy of a recently proposed method for the evaluation of the response of uncertain FE discretized structures is presented. The method can be considered a response surface technique employing ad hoc ratios of polynomials as performance functions, which have the favorable property to furnish the exact response for statically determinate structures. It is shown that when statically indeterminate structures are considered the results are still very accurate, in spite of the number of redundancies, especially if the performance functions include the cross terms. Namely, the explicit relationship between the derived and basic random variables obtained by the method is very close to the actual one. The method is very efficient from a computational point of view as a small number of sampling points are chosen and the evaluation of cross terms does not require extra analyses. The numerical examples evidence that the approach is always more accurate than the response surface method based on second order polynomial performance function. q 2003 Elsevier Ltd. All rights reserved. Keywords: Stochastic finite elements; Uncertain structures; Response surface method 1. Introduction The inclusion of the parameter uncertainty in the analysis of structures is essential for a more complete understanding of the structural behavior. Such uncertainty can arise because of the numerous assumptions made when modeling the geometry, material properties, constitutive laws, and boundary conditions of the structural members. In the literature, several techniques have been proposed to evaluate the response of structures with uncertain parameters (or stochastic structures). However, when reliability analyses are of concern only first or second order reliability methods (FORM or SORM) and response surface techniques appear to be the adequate tools. Finite element methods for linear and nonlinear struc- tures in conjunction with FORM or SORM have been successfully applied for structural reliability computations. The approach is effective for evaluating very small probabilities of failure and for small-scale problems. If these conditions are not fulfilled, both accuracy and efficiency are not assured. As regards accuracy, it should be mentioned that the linear or quadratic approximation of the limit state, assumed by FORM and SORM, respectively, may be very far from its exact shape. This common condition produces crude approximations if the probability of failure is not very small (i.e. for low reliability indices). The efficiency loss for large size structures with several uncertain parameters is due to the computational effort involved in the search of the design point. The search relies on the evaluation of the gradient of the system response with respect to the uncertain parameters, which may be carried out by direct differentiation, finite difference or perturbation procedures [1]. However, each of these techniques pos- sesses limitations which become crucial for complex structures (especially involving nonlinearity) to the point that in some cases the convergence to the design point is not guaranteed or the computational cost becomes too high. There are also cases in which the design point cannot be found. This may be due to the existence of multiple design points, i.e. the optimization algorithm finds a local minimum-distance point which is not the global minimum [2]. Difficulties may also arise with convergence of the usual iterative algorithm when one or more of the basic random variables are described by a discontinuous or by a truncated probability density function [3]. Moreover, although the finite element implementation of FORM and SORM in structural reliability analysis has been coherently coded [4], the algorithm development is cumbersome when nonlinear problems are addressed, so that the merging with commer- cial finite element programs is not straightforward. 0266-8920/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.probengmech.2003.11.005 Probabilistic Engineering Mechanics 19 (2004) 53–63 www.elsevier.com/locate/probengmech * Corresponding author. Tel.: þ 39-090-395322; fax: þ 39-090-395022. E-mail address: gfalsone@ingegneria.unime.it (G. Falsone).