# General methods for assessing critical loads

##### OBJECTIVE

To explain the energy methods for assessing critical loads for cases where it is not possible to get a closed-form solution to differential equilibrium equations.

##### SUMMARY

When certain assumptions are made about the nature of the deformation of an elastic system during the change of configuration associated with the buckling mode, the elastic system may be approximated by involving suitable and adjustable parameters which are determined in order that the neutral equilibrium conditions are fulfilled. Using this concept, practical approximate methods can be derived which are very useful to the design engineer. Some of the best known methods are presented in this lecture, i.e. the Rayleigh coefficient, the Rayleigh-Ritz method, and the Galerkin method. An outline of some numerical methods, such as the Euler finite difference method and the finite element method, is also given.

## Related content

### 1. INTRODUCTION

Critical stability loads may be determined using either of the following methods:

• by direct solution of the differential equilibrium equations with exact values of critical loads as a result
• by using approximate methods, which are, for the most part, based on energy methods, and which lead to approximate solutions of buckling problems

The solution of differential equilibrium equations to satisfy prescribed boundary conditions presents many difficulties and can only be achieved for simple buckling problems for structures with low degrees of freedom; such basic problems are presented and solved, in this way, in the Elastic instability modes lecture. This approach, however, will not be considered further in this lecture which instead concentrates on the alternative energy methods mentioned above. It should be noted that powerful iterative methods can also be used to solve stability problems.

When certain assumptions are made about the nature of the deformation of an elastic system during the change of configuration associated with neutral equilibrium (buckling mode), this elastic system may be approximated by one involving suitable and adjustable parameters or generalised coordinates which are to be determined in order that the neutral equilibrium conditions are fulfilled. This idea provides approximate methods which are very useful to the practical engineer. The best known methods are presented in this lecture, i.e. the Rayleigh coefficient, the Rayleigh-Ritz method, and the Galerkin method. An outline of some numerical methods, such as the Euler finite difference method and the finite element method, is also given.

If the adjustable parameters mentioned above are judiciously chosen and are of adequate number (in the case of an approximation of a continuous system), these approximate methods will give results very close to the exact solution, at the expense of an increased design effort.

#### Prerequisites

Concepts of stable and unstable elastic equilibrium

General criteria for elastic stability