Finite Element Method - The Basis (Volume 1)

4 Some preliminaries: the standard discrete system1.2 The structural element and the structural systemY 4X 4p3 4(2)y(1)(4)1 2x(3)5 6yp(1)2V 3U 33Nodes1xA typical element (1)Fig. 1.1 A typical structure built up from interconnected elements.To introduce the reader to the general concept of discrete systems we shall ®rstconsider a structural engineering example of linear elasticity.Figure 1.1 represents a two-dimensional structure assembled from individualcomponents and interconnected at the nodes numbered 1 to 6. The joints at thenodes, in this case, are pinned so that moments cannot be transmitted.As a starting point it will be assumed that by separate calculation, or for that matterfrom the results of an experiment, the characteristics of each element are preciselyknown. Thus, if a typical element labelled (1) and associated with nodes 1, 2, 3 isexamined, the forces acting at the nodes are uniquely de®ned by the displacementsof these nodes, the distributed loading acting on the element (p), and its initialstrain. The last may be due to temperature, shrinkage, or simply an initial `lack of®t'. The forces and the corresponding displacements are de®ned by appropriate components(U, V and u, v) in a common coordinate system.Listing the forces acting on all the nodes (three in the case illustrated) of the element(1) as a matrixy we have8>:q 1 1q 1 2q 1 39>=>;q 1 1 ˆU1V 1; etc: …1:1†y A limited knowledge of matrix algebra will be assumed throughout this book. This is necessary forreasonable conciseness and forms a convenient book-keeping form. For readers not familiar with the subjecta brief appendix (Appendix A) is included in which su cient principles of matrix algebra are given to followthe development intelligently. Matrices (and vectors) will be distinguished by bold print throughout.