Report on the activities of the international centre for mechanical sciences (CISM) in 1999

Abstract

Beamlike solutions for fully anisotropic elastic tubes of arbitrary closed cross section are derived following the exact beam theory introduced recently by Ladevèze and Simmonds [Comptes Rendus Acad. Sci. Paris 322 (1996) 455; Eur. J. Mech., A/Solids 17 (1998) 377]. Instead of using finite elements to compute the various operators that appear, here the linear shell theory of Koiter [A consistent first approximation in the general theory of thin elastic shells, The Theory of Thin Elastic Shells, Proc. IUTAM Sympos. Delft, Koiter, W.T. (Ed.), North-Holland, Amsterdam, 1959, p. 12] and Sanders [An improved first-approximation theory for thin shells, (1959) NASA Rept. No. 24] is used to exploit the relative thinness of the tube. Analytical, beamlike solutions (the analogues of Saint–Venant solutions in three-dimensional elasticity) are obtained which contain relative errors of O(h/R), where h is the shell thickness and R is some cross sectional radius. These errors are of the same order of magnitude as those contained unavoidably in the stress–strain relations of any first-approximation shell theory. In addition, beamlike stress–strain relations are obtained which express an overall bending strain vector and an overall extensional-shear strain vector in terms of the net traction and moment at any section. Numerical results are presented for tubes with elliptic cross sections. This work generalizes the analysis of Reissner and Tsai [J. Appl. Mech. 39 (1972) 148] by considering external surface loads and by allowing for overall transverse shearing forces in addition to a net axial force and complements the asymptotic analysis of Berdichevsky et al. [Comp. Eng. 2 (1992) 411] by allowing the tube to be of any length.