TABLE OF CONTENTS In the previous chapter the question of finite deformations and non-linear material behaviour was discussed and methods were developed to allow the standard linear forms to be used in an iterative way to obtain solutions. In the present chapter we consider the more specialized problem of large displacements but with strains restricted to be small. Generally, we shall assume that 'small strain' stress-strain relations are adequate but for accurate determination of the displacements geometric non-linearity needs to be considered. Here, for instance, stresses arising from membrane action, usually neglected in plate flexure, may cause a considerable decrease of displacements as compared with the linear solution discussed in Chapters 11 and 12, even though displacements remain quite small. Conversely, it may be found that a load is reached where indeed a state may be attained where load-carrying capacity decreases with continuing deformation. This classic problem is that of structural stability and obviously has many practical implications. The applications of such an analysis are clearly of considerable importance in aerospace and automotive engineering applications, design of telescopes, wind loading on cooling towers, box girder bridges with thin diaphragms and other relatively 'slender' structures.
In this chapter we consider the above class of problems applied to beam, plate, and shell systems by examining the basic non-linear equilibrium equations. Such considerations lead also to the formulation of classical initial stability problems. These concepts are illustrated in detail by formulating the large deflection and initial stability problems for beams and flat plates.
