Warping shear stresses in nonlinear nonuniform torsional vibrations of bars by BEM

In this paper a boundary element method is developed for the evaluation of warping shear stresses of bars of an arbitrary doubly symmetric constant cross section undergoing nonuniform torsional vibrations taking into account the effect of geometrical nonlinearity. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are supported by the most general torsional and axial boundary conditions. The transverse displacement components are expressed so as to be valid for large twisting rotations (finite displacement — small strain theory), thus the arising governing differential equations and boundary conditions are in general nonlinear. Employing a variational approach, a coupled nonlinear initial boundary value problem with respect to the main unknown kinematical components and two boundary value problems with respect to the primary and secondary warping functions are formulated. The solution of the last two problems is performed by a pure BEM approach requiring exclusively boundary discretization of the bar’s cross section and leading to the evaluation of the warping shear stresses. The arising linear system of equations related to the secondary warping function is singular and a special technique is used to perform its regularization. The validity of the negligible axial inertia assumption is examined for the problem at hand.