A method to derive approximate explicit solutions for structural mechanics problems

The availability of explicit solutions, i.e. analytical relationships between the structural response and the design variables, allows a more direct and plain treatment of several structural problems. This paper is devoted to derive approximate explicit solutions in the framework of linear static analysis of finite element modeled structures with a given layout (fixed node positions). The proposed procedure is based on a factorization of the element stiffness matrix following the unimodal components concept, which allows a non-conventional assembly of the global stiffness matrix. The exact inversion of that matrix is a trivial task for the case of statically determinate structures, structures with few redundancies or few design variables. An approximate inverse of the stiffness matrix is herein derived for more general structural problems by resorting to the Sherman–Morrison–Woodbury formula.