Analytic representation of member forces in linear elastic redundant trusses

The process of designing a truss of given geometry and material properties, where the design variables are the cross-sectional areas of the bars is hampered by the need to reanalyse the structure many times until an acceptable design is obtained. Currently, approximate explicit analysis models, based on truncated linear Taylor series expansions, are used to evaluate the structural response at the various candidate design points. Due to the approximate nature of the analysis model, the structure is designed iteratively until convergence of both the analysis equations and the design process. This paper presents for the first time the exact analytic expressions of the internal loads in a truss which is subjected to static loads. The stress resultants are the ratio of two multilinear polynomials in the element stillness. The number of terms of the polynomials is equal to the number of combinations of statically determinate stable structures which can be derived from the original structures. The coefficients of the polynomial expansions can be obtained from equilibrium considerations and from enforcing "global" compatibility of deformations. The expressions are explicit in both the external loads and the element stiffness. The applicability of the analytic equations hinges on the number of combinations of statically determinate stable substructures. In the case of small size structures, the present explicit equations circumvent the need for approximate reanalysis. In common engineering structures, the number of stable subsets is prohibitively large, which renders the analytic expressions intractable. The exact analytic expressions may, however, constitute a starting point for constructing approximate explicit analysis equations of improved quality.