Column Stability

Abstract

A central assumption of the theories for trusses, beams and frames developed in the previous chapters is that the equilibrium conditions have been formulated with reference to the original geometry, i.e. neglecting the fact that the structure via its deformation takes on a somewhat modified geometric configuration. In many cases the effect of the normal force on the stability of beam and frame structures can be analyzed by including small, but finite, displacements of the original configuration. In this chapter the theory is developed for a beam with a non-trivial normal force. The normal force is assumed to be given, or to be a parameter to be determined by the specific problem in question. The key point of the theory is, that when the beam is displaced in the transverse direction, the normal force acting in the beam is also displaced. When the normal force has a sufficient magnitude, this effect becomes important, and in the case of a compressive force the effective stiffness of the structure may be reduced, possibly leading to instability of the structure. A column design procedure based on a combination of material strength and the influence of initial imperfections is developed. The focus in this chapter is on the development of the basic principles in the context of the single structural element, while numerical methods for including the effect of reduced stiffness due to normal compressive forces are developed in Chapter 7.