A self-regularized approach for deriving the free–free flexibility and stiffness matrices

• Using the SVD, we make the singular matrix become the nonsingular bordered matrix.
• The constants and constraints of the bordered matrix can be explained.
• We derive the free–free flexibility/stiffness matrices from the bordered matrices.
• The equilibrium/compatibility of the unnatural force/displacement can be tested.

Motivated by Fichera’s idea for regularizing the rank-deficiency model, we derive the free–free flexibility matrices by inverting the bordered stiffness matrix. The singular stiffness matrix of a free–free structure is expanded to a bordered matrix by adding n slack variables, where n is the nullity of the singular stiffness matrix. Besides, the corresponding n constraints are accompanied to result in a nonsingular matrix. The constraints filter out the homogeneous solution for the regularized solution. By inverting the nonsingular matrix, we can obtain the free–free flexibility matrix from the submatrices. The value of the extra degree of freedom shows the role of no solution (nonzero case) or infinite solution (zero case) with respect to the loading vector. After constructing the bordered system, the equilibrium of the specified force and the compatibility of the specified displacement can be tested according the zero slack variable. Similarly, the free–free flexibility matrix is obtained from the free–free stiffness matrix. Finally, four examples, a rod with symmetric stiffness, a plane truss, a beam and a bar with unsymmetric stiffness, were demonstrated to see the validity of the present formulation.