Acta Appficandae Mathematicae 17: 269-286. 1989. 269 © 1989 Kluwer Academic Publishers. Printed #t the Netherlands. Buckling of Randomly Imperfect Beams W. DAY Department of Mathematics, Auburn University, AL 36849, U.S.A. A. J. KARWOWSKI Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, U.S.A. and G. C. PAPANICOLAOU Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A. (Received: 9 January 1989; accepted: 27 September 1989) Abstract. The qualitative behavior of buckled states of two different models of elastic beams is studied. It is assumed that random imperfections affect the governing nonlinear equations. It is shown that near the first critical value of the buckling load the stochastic bifurcation is described asymptotically by an algebraic equation whose coeffficients are Gaussian random variables. The corresponding asymptotic expansion for the displacement is to lowest order a Gaussian stochastic process. AMS subject classifications (1980). 58F14, 35C20, 35F20. Key words. Stochastic bifurcation, buckling, asymptotic expansion, nonlinear equation. I. Introduction In this paper we study the qualitative behavior of buckled states of two different models of elastic beams under compression by axial forces. In both examples, we assume that random imperfections affect the governing nonlinear equations. The imperfections are described by stationary random functions of the form f(x/e) with zero mean and with e a small parameter. This means that the stochastic perturb- ations are rapidly varying relative to the dimensions of the beam but are not necessarily small. Our object is to analyze the asymptotic bifurcation picture in a suitable scaling. We show that near the first critical value of the buckling load the stochastic bifurcation is described asymptotically by an algebraic equation of the second or third order whose coefficients are Gaussian random variables. The corresponding * Work supported by NSF Grant No. DCR81-14726. ** Work supported by NSF Grant No. DMS87-01895.
Buckling of Randomly Imperfect Beams
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