## ABSTRACT

Thus far, the development of the theory in Chapters 4 to 6 has been confined to line element systems (trusses, frames and grids). These are restricted classes of structure in which, for the purpose of simplicity, the assumption of beam bending, that is, plane sections remain plane in the deformed state, has been invoked, so that the generalized forces of axial load, moment and torque can be used to describe the load deformation characteristics of the member. This is essentially a strength-of-materials approach. In their immediate use (that is, for elementary small deflection analyses), the assumption appears to be a simplification; however, in the event of complications in the spatial and deflection geometry, it may have to be modified extensively, or the general theory of continuum mechanics may have to be resorted to. It was shown, for example, that if warping is included in the behaviour of cross sections of thin walled members (section 6.7), the concept of the bimoment as a generalized force becomes necessary. The present chapter develops the general theory for the approximate solution to elasticity problems involving either plane stress, plane strain, axisymmetry or general three-dimensional elasticity. The St Venant’s torsion problem is also discussed as a useful special class of restricted elasticity problem. The chapter concludes with a brief introduction to heat transfer because the topic is intimately connected with thermal stress analysis. This, in turn, introduces the concept of finite elements in space and time, that is, transient analysis. The basic mathematical preliminaries have been given in detail in Chapters 1 and 2, and in Chapter 3 the various approaches to the finite element method have been delineated. Herein, only the compatible displacement models will be studied. There will be, however, a parallel development of the conventional theory (using the contragredient principle and orthogonal interpolation functions (section 3.3), and the natural mode technique (section 3.4). Elements can be broadly classified (in so far as computations are concerned), into the categories of

those which can have their stiffness matrices calculated explicitly;

those which require numerical integration.