Composites Engineering. Vol. 3, No. 12, pp. 1149-1164, 1993. Printed in Great Britain. G96-9526193 f6.00+ .OO 0 1993 Pergamon Press Ltd SAINT-VENANT ELASTICITY SOLUTIONS OF A TIP-LOADED ANISOTROPIC CANTILEVERED BEAM WITH AN ELLIPTICAL SECTION C. A. IE and J. B. KOSMATKA Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, CA 92093, U.S.A. (Received 19 January 1993; final version accepted 3 August 1993) Abstract-The Saint-Venant displacement distributions are developed, based upon the theory of elasticity, for a tip-loaded homogeneous cantilever beam having an elliptical cross-section and rectilinear anisotropy. These distributions are found by integrating the strain distributions, where the local in-plane deformation and out-of-plane warping of the cross-section are exactly deter- mined. A definition of the “anisotropic shear center” is presented by extending the classical definition for isotropic beams. The additional transverse beam displacement associated with shear deformation is determined for applied extension and flexure loads. Numerical results are presented, which show (1) the anisotropic shear center location is linearly dependent upon beam length and can be located outside the cross-section, (2) shear deformation can be negative for certain beam aspect ratios and material definitions, so that the inclusion of shear deformation can make the beam stiffer, as opposed to more flexible, and (3) the local in-plane cross-section deformations, out-of-plane cross-section warping, and the transverse shear stress distributions bear no resemblance to their isotropic counterparts. INTRODUCTION The elastic stress and displacement distributions of isotropic cantilever beams subjected to tip loads (i.e. extension, bending, torsion and flexure) have been exhaustively investigated by making use of Saint-Venant’s principle in the formulation of the boundary-value problem. Closed-form displacement and stress solutions exist for simple (elliptical) cross- sections; series solutions exist for slightly more complex (rectangular, triangular) cross- sections; and approximate solutions based upon the application of the Ritz method exist for arbitrary cross-sections. Detailed examples of these solutions can be found in many texts covering the theory of elasticity, for example, Sokolnikoff (1956) or Timoshenko and Goodier (1970). Conversely, the study of generally anisotropic cantilever beams subjected to tip loads has received far less attention. In Lekhnitskii’s monograph (1963), the stress and displace- ment distributions were formulated in terms of known quantities (geometric and material properties) and unknown functions which represented the local in-plane deformation and out-of-plane warping of the cross-section. A solution procedure for determining the stress distribution was presented based upon the use of Airy and Prandtl stress functions, where numerical examples include beams having an elliptical (closed-form), rectangular (series), or arbitrary cross-section (approximate). But no results were presented for the displace- ment distributions. Recently, Kosmatka and Dong (1991) developed a finite-element-based model for determining the complete displacement and stress distributions of a homogeneous prismatic anisotropic beam with an arbitrary cross-section, by solving the two-dimensional boundary problem in terms of the local in-plane deformation and out-of-plane warping of the cross-section. Numerical results were presented, which used these calculated displace- ment and stress distributions to study the beam behavior, determine important section constants, and show that the shear center location is linearly dependent upon beam length. In the current paper, we will develop the complete Saint-Ventant displacement and stress distributions, based upon the theory of elasticity, for a tip-loaded homogeneous cantilever beam having an elliptical cross-section and rectilinear anisotropy. The dis- placement distributions are found by integrating the strain distributions calculated by Lekhnitskii (1963), where the local in-plane deformation and out-of-plane warping of the 1149