Defective eigensolutions and stiffness matrices

Abstract

The equilibrium configuration of a structural member is governed by a set of partial differential equations, which can be reduced to a set of ordinary differential equations depending on one spatial parameter alone by means of the Kantorovich method. The analytical solutions of the resulting boundary value problem are not straight forward when the associated eigenproblem is defective due to the insufficient number of eigenvectors. For the defective case, solution methods for uniform members are suggested. When the analytical solutions are used as shape functions, exact stiffness matrices are obtained. These matrices may be parametric to produce a dynamic stiffness matrix and stability matrix. The whole process from eigensolutions (for shape functions) to element matrix formulation is automated. The characteristic polynomial equation is first obtained by an analytical expansion method. The solution of the polynomial equation for the eigenvalues is standard. The ranks are checked and the generalized vectors are found. Finally, the element matrices are formed. The element matrices are free from all difficulties associated with the assumed shape function approach, e.g., rigid body modes, constant strains, spurious zero-energy modes, slow convergence, etc. A spatial helix is taken as an example.