On coupled bending–torsional vibrations of beams with initial loads

The objective of this paper is to analyse the free vibration and mode shapes of straight beams where the coupling between the bending and torsion is induced by steady state lateral loads. The governing differential equations and boundary conditions for coupled vibrations of Euler–Bernoulli–Vlasov beams are performed by using the virtual work principle which includes the second order terms of finite beam rotations. Closed form solution is derived for the coupled frequencies and mode shapes of a symmetric beam with simply supported ends under uniform bending. A finite element model with seven degrees of freedoms per node is also presented. To illustrate the accuracy of this formulation, numerical solutions are presented and compared with available solutions.