Modal shapes and frequencies of the dynamic stiffness matrix of a beam-column

Abstract

The purpose of this thesis is to investigate the characteristics of the normal mode shapes the frequencies associated with the general stiffness matrix for a beam-column element, which is derived from the Bernoulli-Euler differential equation with the inclusion of the axial force and transverse inertial loading. The components of this general stiffness matrix contain hyperbolic and harmonic functions.

Three separate problems are analyzed. These include the statical beam-column bending problem, the dynamic beam problem in free vibration, and the dynamical beam-column problem in free vibration. In each case, the orthogonality conditions of the modal shapes and frequencies are established.

For each of the above three problems, the special case of rigid body motion is investigated. Each possess at most two rigid body modes, the remaining two families of modes shapes being associated with deformed geometry including, the resonant frequency of free vibration of the free-free beam, and the critical buckling load of a simply supported column, and the resonant frequency of a free-free beam-column.