Ph.D Student | Totry Essam |
---|---|

Subject | Buckling of Non-Uniform and Stochastically Heterogeneous Beams |

Department | Department of Mechanical Engineering |

Supervisor | Professor Emeritus Eli Altus |

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.