The member stiffness determinant and its uses for the transcendental eigenproblems of structural engineering and other disciplines

Transcendental stiffness matrices are well established in vibration and buckling analysis, having been derived from exact analytical solutions of the differential equations for many structural members without recourse to finite–element discretization. Their assembly into the overall structural stiffness matrix gives a transcendental eigenproblem, solvable with certainty by the Wittrick–Williams (WW) algorithm, instead of the usual linear (algebraic) eigenproblem. This paper establishes a (normalized) member stiffness determinant, being the value of the stiffness matrix determinant if the member were modelled by infinitely many finite elements and its ends were clamped. It is derived for beams with uncoupled axial and Bernoulli–Euler flexural behaviour, by methods applicable to any member possessing transcendental stiffnesses. Multiplying the product of a structure's member stiffness determinants by the determinant of its transcendental overall stiffness matrix gives a determinant, which, unlike the determinant of the transcendental overall stiffness matrix itself, has no poles when plotted against the eigenparameter, leading to a more secure eigenvalue location. Numerical results demonstrate the resulting advantages. In addition, the member stiffness determinant can give exact determinants for members with various standard end conditions and also provides limits for finite–element solutions as the number of elements approaches infinity. Analogous advantages occur in all other disciplines where the WW algorithm is used, e.g. fluid vibrating in pipes, heat and mass diffusion and Sturm–Liouville problems