The construction of free–free flexibility matrices as generalized stiffness inverses

Abstract

We present generalizations of the classical structural flexibility matrix. Direct or indirect computation of flexibilities as ‘influence coefficients’ has traditionally required pre-removal of rigid body modes by imposing appropriate support conditions. Here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free–free stiffness matrix. This entity is called a free–free flexibility matrix. It preserves exactly the rigid body modes. The definition is element independent. It only involves access to the stiffness generated by a standard finite element program as well as a separate geometric construction of the rigid body modes. With this information, the computation of the free–free flexibility can be done by solving linear equations and does not require the solution of an eigenvalue problem or performing a singular value decomposition. Flexibility expressions for symmetric and unsymmetric free–free stiffnesses are studied. For the unsymmetric case two flexibilities, one preserving the Penrose conditions and the other the spectral properties, are examined. The two versions coalesce for symmetric matrices.

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