M.Sc Thesis
M.Sc Student Nissenbaum Omer Exact Solutions of Average Buckling Loads for Stochastic Piecewise Homogeneous Columns Department of Mechanical Engineering Professor Emeritus Eli Altus

Abstract

An analytical exact solution of an average buckling load in a simply supported case is introduced for a stochastic two-region piecewise homogeneous morphology, in which the location between the two constant stiffness regions is uniformly distributed. Solution of the critical buckling load for a deterministic case (namely, a specific value of the stochastic parameter) is complex and can be achieved only by solving numerically the algebraic characteristic buckling equation which is transcendental, i.e. nonlinear, implicit, and has no simple analytical solution. However, it was found (actually, by accident) that the average bucking load for the whole ensemble is analytical and depends on the non-stochastic parameters of the problem in a simple form. It means that exact analytical average values can be found for a complex stochastic problem. Analytical analysis for the numerical results is performed. Two methods are utilized. One involves a Taylor expansion of the algebraic transcendental equation and the buckling load. The other is the Direct Functional Perturbation Method (DFPM), which is based on Functional (Fréchet) expansion of the differential buckling equation and the buckling load.  An evaluation for the buckling load is obtained for (a) the deterministic case, with comparison to the numerical results and the common energy based methods, and (b) the stochastic case, for a validation of the exact average. In addition, an analytical analysis is performed for the variance of the buckling load. The exact average solution is generalized to various parameters of the simply supported buckling problem, such as higher order modes of deflection and different stiffness ratios. It is then examined for different boundary conditions, both statically determinate and indeterminate, and for more complex morphologies which involve three regions of constant stiffness. The above analytical solutions may help in improving the accuracy of more complex stochastic buckling problems which have no exact solutions by extracting sub-ensembles with known exact average solutions. Example of such cases is presented.