ABSTRACT
By now, it should have become clear to the reader that in any finite element analysis we are not analyzing the actual physical problem, but a mathematical model of it. As a result, we introduce some simplifications, and hence some modeling errors. In reality all solids are three-dimensional. Fortunately, for many problems which are of practical interest, some simplifying assumptions can be made regarding the stress or strain distributions. For example, in Chapters 2 through 4 dealing with skeletal structures, line-type elements were used because of the predominance of the longitudinal stress. In Section 5.4.4, we have also seen that when the loading and/or geometry permit it, a solid can be analyzed as a plane stress or plane strain problem. There are also other simplifications for solids that posses a symmetry of revolution in both geometry and loading, and for flat solids loaded perpendicular to their plane. These will be dealt respectively in Chapters 10 and 11. However, unlike skeletal structures, whose discretization into an assembly of elements is relatively easy, the connecting joints naturally constitute the nodes, such an intuitive approach does not exist for a twoor three-dimensional continuum. There are no joints to be used as nodes or cleavage lines to be used as elements’ edges. Hence, the discretization becomes a process that requires an understanding of the physical problem at hand. It should be also added that the more physical details one tries to capture, the more complex the model becomes. In particular, the user has to decide on the choice of element type and size. These depend on the physical make-up of the body, the loading, and on how close to the actual behavior the user wants the results to be. He/she also has to decide whether the model can be simplified? And how could the results be checked? There are, of course, no definite answers to these questions. In this chapter dealing with plane problems, and in Chapters 10 and 11, we will formulate the finite element method, and in the process attempt to answer some of these questions. The user, however, is reminded that only practice makes perfect.
