Hoff’s Problem in a Probabilistic Setting

Abstract

Hoff’s problem—that of investigation of the maximum load supported by an elastic column in a compression test—is considered in a probabilistic setting. The initial imperfections are assumed to be Gaussian random fields with given mean and autocorrelation functions, and the problem is solved by the Monte Carlo Method. The Fourier coefficients of the expansion for the initial imperfection function are simulated numerically; for each realization of the initial imperfection function, the maximum load supported by an elastic column in a compression test is found by the solution of a set of coupled nonlinear differential equations. For slightly imperfect columns, the closed solution is given in terms of Bessel and Lommel functions and turns out to compare well with the result of numerical integration. Results of the Monte Carlo solution are used in constructing the reliability function at a specified load. Reliability functions for different manufacturing processes (represented by different autocorrelation functions with equal variance) are calculated; design requirement suggests then that, other conditions being equal, the preference should be given to the manufacturing process resulting higher reliability.