Shear Deformation Effect in Nonlinear Analysis of Spatial Composite Beams in Variable Axial Loading by BEM

Abstract

In this paper a boundary element method is developed for the nonlinear analysis of composite beams of arbitrary doubly symmetric constant cross section, taking into account shear deformation effect. The composite beam consists of materials in contact each of which can surround a finite number of inclusions. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and numerically approximated employing a pure BEM approach, that is only boundary discretization is used. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows. i. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading. ii. The beam is supported by the most general linear boundary conditions including elastic support or restrain. iii. The analysis is not restricted to a linearized second - order one but is a nonlinear one arising from the fact that the axial force is nonlinearly coupled with the transverse deflections (additional terms are taken into account). iv. Shear deformation effect is taken into account. v. The shear deformation coefficients are evaluated using an energy approach, instead of Timoshenko’s and Cowper’s definitions, for which several authors have pointed out that one obtains unsatisfactory results or definitions given by other researchers, for which these factors take negative values. vi. The effect of the material’s Poisson ratio v is taken into account. vii. The proposed method employs a pure BEM approach (requiring only boundary discretization) resulting in line or parabolic elements instead of area elements of the FEM solutions (requiring the whole cross section to be discretized into triangular or quadrilateral area elements), while a small number of line elements are required to achieve high accuracy.