Ph.D Thesis
Ph.D Student Totry Essam Buckling of Non-Uniform and Stochastically Heterogeneous Beams Department of Mechanical Engineering Professor Emeritus Eli Altus

Abstract

In this research, the statistical characteristics of the buckling load (P) of stochastically nonuniform beams are found analytically as a function of material morphology using the Function Perturbation Method (FPM). The material properties (stiffness or compliance) are considered as either deterministic or statistically homogeneous random fields. The statistical characteristics of P are obtained in three levels of accuracy according to different solution methods developed in this study.

1.   Minimum energy method

By assuming specific shape functions for the beam’s deflection, that satisfy both Kinematic and  kinetic boundary conditions, and following the well known minimum energy principle the functional characteristic equation for P is obtained. Since this equation is implicit for P, it is expanded into functional series near the property mean. Solving each perturbation order separately, an analytical solution is obtained for the statistical characteristics of P, having a direct correlation to material morphology.

2.   Optimized method (OFPM)

The OFPM is based on finding a new material property such that the second order perturbation term is minimized or vanishes. This requirement leads to a nonlinear differential equation. Solving this equation the new “ material property” which is morphology dependent is obtained. Expressing P using this new property, a much more accurate analytical solution for the whole region of the correlation length is obtained.

3.   Differential Functional perturbation method (DFPM)

Here the FPM is directly applied to the eigenvalue differential equation. This is done by expanding the eigenvalue problem into functional series which yields to a set of successive differential equations for the functional derivatives of both the eigenvalues and the eigenvectors (W). Successively solving each order, the functional derivatives of P and W are obtained.

The results of the above three methods are compared with the Stochastic Finite Elements Method and Monte Carlo results which shows the advantages of the FPM.