10.1 INTRODUCTION

Among the most severe limitations of theoretical closed form solutions, presented in Chap. 4, are the idealized conditions, such as ideal elasticity, homogeneity and simple end-conditions, that are often assumed to facilitate the mathematical solution. These idealized conditions are rarely realized in natural phenomena, and their departure from real conditions is often considerable. The great advantage of numerical methods, such as the finite difference method to be described in the present chapter, or the numerical integration method, finite segment method and finite element method presented in Chapters 9, 11 and 12 lies in their ability to solve problems with conditions that are far too complicated for analytical methods. This enables us to introduce into the problem of flexural torsional stability some of the complexities like plasticity, residual stresses, initial deformations and general end-conditions and evaluate their effects on the result. Numerical methods have the additional advantage that they can be handled by electronic computers. The practical significance is that this makes it feasible to investigate a large number of variables such as loading conditions, end conditions, etc., and make parametric studies. The engineer can thus systematically analyze the effect of these possible conditions on the final result flexural-torsional stability of columns or lateral buckling of beams as the case may be.

For uniform members, generally encountered in civil engineering practice, the finite difference formulation to be described in this chapter is very powerful and economical in computer time. For more complicated structures and where local buckling...