Structural Dynamics, EURODYN 2002, Grundmann & Schu¨ eller (eds.), c 2002 Swets & Zeitlinger, Lisse, ISBN 905809510 X Stochastic dynamic stability analysis of nonlinear structures Thomas Most & Christian Bucher Institute of Structural Mechanics, Bauhaus-Universit¨ at Weimar, Marienstrasse 15, D-99423 Weimar, Germany Summary This document presents an analysis method to investigate the dynamic stability behaviour of ran- dom geometrically imperfect systems loaded by random excitations. The analysis uses a time integration method to consider all nonlinearities of the structures. The method is based on the stability concept of Lya- punov. The obtained nonlinear numerical results are compared with the results of a well known linear analysis method, which is explained as well. The nonlinear and the linear method are applied on nonlinear multi-degree- of-freedom systems to compute the failure probability depending on the excitation intensity. 1 INTRODUCTION This paper gives an short overview of the investiga- tions in stochastic dynamic stability analysis. In pre- vious publications, geometrically imperfect structures with static loading (Schorling 1997), periodic load- ing conditions (Schorling and Bucher 1998, Schorling and Bucher 1999) and for random loading (Schor- ling, Bucher, and Purkert 1998, Schorling, Most, and Bucher 2001) were considered. These publications construe geometrical imperfections as randomly spa- tially distributed deviations from a perfect geome- try. Mathematically these imperfections can be mod- eled as random fields which are discretized by points equivalent to the nodes of the finite element model. The random field is characterized by the covariance matrix. The eigenvectors obtained by a diagonalisa- tion of this matrix (Ghanem and Spanos 1991) can be interpreted as orthogonal imperfection shapes with probabilistic weights. The stability behaviour of ge- ometrically imperfect systems can be analyzed sep- arately for each shape by using standard methods of structural mechanics. The random loading is assumed to be a scalar- valued white noise process, which can be discretized by using finite Fourier series with random coefficients (Rice 1948). This paper presents two stability analysis methods. These methods are base on different convergence cri- teria for asymptotic stability. By application to dif- ferent systems both methods are compared and dis- cussed. The first method is based on the convergence cri- terion ”stability with probability one”. To analyze the stability a time integration of the system with an ac- companying stability analysis until infinity is theoret- ically required, see e.g. Burmeister 1987, Eller 1988, Kr¨ atzig and Nawrotzki 1996. Principally all nonlin- earities of the system can be considered if the non- linear system matrices are computed time-step-wise. This time integration is the crucial numerical opera- tion. Implicit time integration methods of the New- mark type can fail due to ill-conditioned stiffness ma- trices in the vicinity of the stability border. An explicit time integration method is applied here which is lim- ited by a system-dependent critical time step. The second method is based on a convergence crite- rion of the stability expressed in term of second mo- ments (mean square stability). The Lyapunov expo- nents are derived by the Itˆ o calculus, see e.g. Soong and Grigoriu 1992. This method can be practically on linear systems. By considering only the first order terms of the asymptotic stiffness matrix series nonlin- ear systems can be linearized. The Itˆ o calculus does not require an extensive time integration procedure. Both methods are verified and applied to SDOF and MDOF systems by using the SL ang Software package (Bucher et al. 1995, Bucher and Schorling 1997). 2 METHOD OF ANALYSIS 2.1 Probabilistic Model 2.1.1 Random Imperfections To represent geometrical imperfections, which are in- terpreted as spatially fluctuating structural properties with respect to a perfect geometry, random fields with 1