Finite Element Method - The Basis (Volume 1)

8 Some preliminaries: the standard discrete systemThe element properties were assumed to follow a simple linear relationship. Inprinciple, similar relationships could be established for non-linear materials, butdiscussion of such problems will be held over at this stage.The calculation of the sti€ness coe cients of the bar which we have given here willbe found in many textbooks. Perhaps it is worthwhile mentioning here that the ®rstuse of bar assemblies for large structures was made as early as 1935 when Southwellproposed his classical relaxation method. 221.3 Assembly and analysis of a structureConsider again the hypothetical structure of Fig. 1.1. To obtain a complete solutionthe two conditions of(a) displacement compatibility and(b) equilibriumhave to be satis®ed throughout.Any system of nodal displacements a:8 9>< a 1 >=a ˆ.…1:7†>: >;listed now for the whole structure in which all the elements participate, automaticallysatis®es the ®rst condition.As the conditions of overall equilibrium have already been satis®ed within an element,all that is necessary is to establish equilibrium conditions at the nodes of thestructure. The resulting equations will contain the displacements as unknowns, andonce these have been solved the structural problem is determined. The internalforces in elements, or the stresses, can easily be found by using the characteristicsestablished a priori for each element by Eq. (1.4).Consider the structure to be loaded by external forces r:8 9>< r 1 >=r ˆ . …1:8†>: >;applied at the nodes in addition to the distributed loads applied to the individualelements. Again, any one of the forces r i must have the same number of componentsas that of the element reactions considered. In the example in question Xir i ˆ…1:9†Y ias the joints were assumed pinned, but at this stage the general case of an arbitrarynumber of components will be assumed.If now the equilibrium conditions of a typical node, i, are to be established, eachcomponent of r i has, in turn, to be equated to the sum of the component forcescontributed by the elements meeting at the node. Thus, considering all the forcea nr n