Structural Safety and Reliability, Corotis et al. (eds), c  2001 Swets & Zeitlinger, ISBN 90 5809 197 X Stability analysis for imperfect systems with random loading York Schorling HOCHTIEF, Frankfurt, Germany Thomas Most & Christian Bucher Institute of Structural Mechanics, Bauhaus-Universität Weimar, Marienstrasse 15, D-99423 Weimar, Germany Keywords: stochastic stability, random vibration, non-linear vibration, random fields, stochastic fi nite elements method, probability theory, M onte C arlo simulation ABSTRACT: This paper shows an extension of research activities to stochastic dynamic stability problems. Both, the structure and the loading conditions are supposed to be random. The structures investigated are subjected to fluctuating loads in the vicinity of the static buckling load. The fluc- tuating components of the loading are modeled by stationary random processes, described by their power spectral density. As the loading is time dependent, the top Lyapunov exponent of the system has to be derived to determine the stability of the system. The top Lyapunov exponent can either be determined by nonlinear time integration of the system with accompanying stability analysis or by linear Itô analysis. The determination of the Lyapunov exponent by time integration shows one major difficulty: In statistical sense, the statement with respect to the Lyapunov exponent gets less precise as the structural response gets more critical. The Itô calculus principally represents an efficient linear analysis method to determine the second moment stability behavior of the structural system, suit- able for Finite Element analysis. Within this paper both approaches are performed and compared for geometrically perfect and imperfect systems. 1 INTRODUCTION This paper shows an extension of recent investigations in stochastic dynamic stability analysis. The authors considered geometrically imperfect structures with static loading (Schorling 1997), periodic loading conditions (Schorling and Bucher 1998, Schorling and Bucher 1999) and for random loading (Schorling, Bucher, and Purkert 1998). Geometrical uncertainties may be interpreted as randomly spatially distributed deviations from a perfect geometrical shape. Mathematically they are covered by point discretized random fields. The covariance matrix obtained can be diagonalized (Ghanem and Spanos 1991). The eigenvectors resulting from this transformation may be interpreted as orthogo- nal imperfection shapes which have probabilistic weights. Their influence on the dynamic stability behavior can be analyzed by standard methods of structural mechanics. The random loading is described by a scalar-valued random process. It is assumed to be stationary in time and normally distributed with a given mean value and power spectral density. It is represented in terms of a finite Fourier series with random coefficients (Rice 1948). Within this paper two analysis methods to determine the stability of the structure are presented and discussed. These methods base on different convergence criterions for asymptotic stability. The first method presented bases on the convergence criterion “stability with probability one (al- most sure stability)”. The stability of the structure is determined by analyzing the tangential equa- tions of motion of the structure, see e.g. Burmeister 1987, Eller 1988, Krätzig and Nawrotzki 1996. This procedure theoretically requires a time integration of the system with an accompanying stability 1