Nonlinear Effects in Elastic Flexural-Torsional Vibrations of Beams of Arbitrary Cross Section

In this paper, a boundary element method is developed for the nonlinear flexural-torsional dynamic analysis of beams of arbitrary, simply or multiply connected, constant cross section, undergoing moderate large deflections and twisting rotations under general boundary conditions, taking into account the effects of rotary and torsional warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions as well as to twisting and/or axial loading. Four boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to the angle of twist and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique leads to a system of nonlinear coupled Differential-Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold-Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled to algebraic equations (DAE). The geometric, inertia, torsion and warping constants are evaluated employing the Boundary Element Method. The proposed model takes into account, both the Wagner's coefficients and the shortening effect. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy, the range of applications of the developed method as well as the influence of the nonlinear effects to the response of the beam. We also discussed here some recent patents relevant to nonlinear analysis techniques.