# The explicit inverse of the stiffness matrix

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## Abstract

In two recent papers the author has given the exact analytic expression of the internal forces in linear elastic structures composed of uniform prismatic elements. It was shown that the member forces are the ratios of two multilinear homogeneous polynomials in the unimodal stiffnesses of the elements of the structure. The order of the polynomials is equal to the number of nodal degrees of freedom of the structure. The number of terms of each polynomial is equal to the number of statically determinate stable substructures which can be derived from the original structure. The coefficients of the polynomials can be computed by employing the equilibrium equations and by enforcing global compatibility of deformations. It was found empirically that the coefficients of the polynomial in the denominator were numerically equal to the square of the determinants of the statics matrices of the respective statically determinate substructures. As a consequence, the denominator became the sum of the stiffness matrices of the statically determinate substructures. This is in fact the Binet-Cauchy form of the determinant of the stiffness matrix of the structure. Bearing in mind that the inverse of the stiffness matrix can be expressed as the ratio of the adjoint matrix of the stiffness matrix divided by the determinant of the stiffness matrix it became clear that the expressions of the stress resultants stem from an explicit expression of the adjoint. The explicit expression of the adjoint of the stiffness matrix lies at the heart of this paper. It is shown that the adjoint is a congruent transformation of the (N-1) compound of the stiffness matrix, where N is the number of degrees of freedom of the structure. This cleared the way to use the Binct Cauchy theorem on the product of compound matrices to obtain an explicit expression for the adjoint and ipso facto, for the inverse of the stiffness matrix. Having now the displacements of the structure, the expression of the stress resultants, which was obtained independently, emerges in a very elegant manner. The member forces in a structure can be expressed as the weighted sum of the member forces in all its determinate substructures, when subjected to the applied loads. The weighting factors are the ratios of the determinants of (he stiffness matrices of the substructures, to the determinant of the stiffness matrix of the original structure. Both the explicit inverse of the stiffness matrix and the expression of the internal forces in the structure are, at present, of a theoretical nature. The number of terms involved in the polynomials is simply excessive for common engineering structures. However, ongoing reseach may yield more applicable expressions to be used, for instance, in (he field of automated design of structures. The theory is illustrated with the explicit analysis of a stayed mast.

Original language English 2101-2113 13 International Journal of Solids and Structures 29 16 https://doi.org/10.1016/0020-7683(92)90197-2 Published - 1992

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