Approximate reanalysis based on the exact analytic expressions

Abstract

The first author has recently given the exact analytic expressions of the inverse of the stiffness matrix, the nodal displacements, and the stress resultants in linear elastic structures composed of prismatic elements. For structures of constant geometry, the expressions are explicit in terms of the unimodal stiffnesses of the components of the structures. However, the expressions are intractable in their exact form due to their inordinate length. It all has to do with the number of statically determinate substructures embedded in common engineering structures. This paper describes some preliminary results obtained from approximate analysis models for the internal forces using truncated expressions that are similar in form to the exact analytic ones. The approach is illustrated with numerical examples. HE classical methods of structural analysis methods are numerical techniques that compute the structural response to external loads for a given configuration of the structure and for given stiffnesses of its elements. Very often, engineers are faced with the need to reanalyze structures for the same geo- metrical configuration and for the same applied loads, but with modified stiffnesses. Such instances occur in structural design and, in particular, in optimal structural design where one usually tests many candidate stiffness distributions before an acceptable structure is found.1 Structural reanalysis is also present in the analysis of framed structures when the P-d effect is accounted for.2 Since the axial load in a member modifies its flexural rigidity, such structures are solved itera- tively. At a given iteration, the flexural stiffnesses of the members are computed, based on the axial loads of the previ- ous iteration. The procedure is repeated until convergence. When the stiffnesses of the individual members change, the analysis procedure must in principle be repeated with the modified parameters. There is no formula for the structural response in terms of the rigidities of its members. To alleviate the computational cost of multiple reanalysis, engineers have over the years developed reanalysis techniques that use results of previous analysis to predict the structural response when some or all of the rigidities are modified.3'4 Noteworthy is the reciprocal approximation,1 which expresses the nodal dis- placements as a linear Taylor polynomial in terms of the reciprocals of the member stiffnesses. Recently, the first author has written a series of papers 5'7 describing the analytic solution of the structural analysis prob- lem. For structures composed of uniform prismatic members of given geometry and assuming linear elastic behavior, the analytic expressions are exact equations for the inverse of the stiffness matrix, for the nodal displacements, and for the internal forces in the members, explicit in terms of the cross- sectional structural characteristics of the components of the structure. However, what seems to be Eldorado for the struc- tural designer—an exact formula for the structural response—