## ABSTRACT

Thus far it has been the intention to derive the fundamental relationships necessary for the analysis of continua via the finite element discretization procedure. For the continuum it will be shown that it is not necessary to invoke the analogy between element discretization and a physical element of the same shape, and that the finite element method may be looked upon as simply a piecewise application of Gauss’ divergence theorem to the region. However, in the case of line element structures such as frames and trusses the discretization process does coincide with the physical dimensions of the structure. The choice to begin the discussions with line element structures is thus made, not only because in the first instance of linear analysis the transformations are deceptively simple, leading to an easy understanding of the underlying principles, but, in addition, the theory presented herein embodies all the basic principles of matrix structural analysis. In this category of structures are found trusses, beams, grids, frames and cable nets. This last group needs special attention because of the necessity to undertake non-linear analysis of the node force-deflection relationships. The structures are interesting too, because low connectivity between nodes can lead to statical determinacy, for which member forces may be obtained without a knowledge of the deformation characteristics of the members. It is found, however, that even this simple class of structure is highly amenable to computer structural analysis. This inevitably leads to the possibilities of the introduction of computer methods as the very basis of structural mechanics. In this text, the emphasis will be on the stiffness analysis of structures based on the kinematic relationships between node displacements and member distortions. The reason for the choice of kinematic analysis in preference to static analysis is that the kinematic transformations of the form in eqn (2.122) are more readily determined than the force transformations in eqn (2.120). However, in this chapter a digression is made from the main theme of the book to examine statically determinate as well as statically indeterminate structures because of the usefulness of this class of structure in the study of elementary structural mechanics. It will be found that once the decision has been made to rely on the computer for the solution of the equilibrium equations it is expeditious to work always in terms of joints and joint equilibrium. This is in distinct contrast to hand methods of analysis, for example, in the analysis of determinate trusses where the equilibrium may be applied to either joints or sections depending on which is most advantageous for the analyst. Such arbitrary decisions are, of course, exceedingly difficult to incorporate into a general computer program.