Beam theory for strongly orthotropic materials

Abstract

This paper presents a displacement-based model for orthotropic beams under plane linear elasticity based on the only kinematic assumption of transverse inextensibility. Any given axial and transverse loading as well as boundary conditions at the beam ends are considered. The solution is decomposed into the principal and the residual part (corresponding to the interior and the boundary problems) which are obtained by series expansions of polynomial functions and eigenfunctions, respectively. It is proved that the proposed one-dimensional theory gives both interior and boundary exact two-dimensional elasticity solutions for strongly orthotropic materials, i.e. for ratio between shear modulus and axial Young modulus approaching zero. For isotropic and orthotropic materials the accuracy of the beam model is also analysed and compared with that of other theories. In particular, the complementary energy error of the interior solution with respect to two-dimensional elasticity is evaluated, the asymptotic estimate of the characteristic decay length of end effects given in Choi and Horgan [J. Appl. Mech. ASME, 44, 424–430 (1977)] by two-dimensional analysis is reobtained and the range of validity of boundary solution is discussed. The numerical results presented agree very well with exact and finite element solutions even in the neighbourhood of clamped cross-sections, where the solution is mainly governed by the boundary problem.