Small displacements about equilibrium of a beam subjected to large static loads

Abstract

Small vibrations about the equilibrium configuration of a beam structure are treated. The beam is subjected to large static loads causing large displacements. The influence of these static loads on the small vibrations is investigated. In the stationary harmonic case, the symmetry of the dynamic stiffness matrix is discussed. A semifollower static moment is introduced to maintain the symmetry of the dynamic stiffness matrix and to create conservative subsystems. The beam is deformed only in bending and torsion, i.e., no extension of the geometric centerline is permitted and no transverse shear of the beam lamina is taken into account. The Euler-Bernoulli beam theory is applied to bending and the Saint-Venant theory to torsion. An Euler axis transformation is used for handling the large rotations in the nonlinear case. The material of the beam is assumed to be linearly elastic. A continuous model is made of the beam, i.e., the governing differential equations are derived and solved without recourse to a spatial discretization.