Theory of Thin Elastic Shells

Abstract

1. Introduction. The linear theory of thin elastic shells has received attention by numerous authors who have employed a variety of approximations in their work. Inasmuch as there is no difficulty in obtaining the stress differential equations of equilibrium and expressions for the components of strain, consistent with the assumptions for displacements, the works of these authors differ from one another essentially in the formulation of appropriate stress strain relations. With a few exceptions, the approximations introduced have been within the framework of the classical shell theory where, in addition to the smallness of (a) the thickness h in comparison with the least radius of curvature of the middle surface R, i.e., h/R « 1, (b) strains and displacements so that the quantities of second and higher order terms are neglected in comparison with the first order terms, it is also assumed that (c) the component of stress normal to the middle surface is small compared with other components of stress and it may, therefore, be neglected in the stress strain relations, and (d) plane cross sections normal to the undeformed middle surface remain normal to the deformed middle surface and suffer no extension. The last assumption implies neglect of the transverse shear deformation. Recent and notable contributions, where the effects of both transverse normal stress and shear deformation have been accounted for, are by Hildebrand, E. Reissner and Thomas [1], by Green and Zerna [2], and by E. Reissner [3] where references are made to previous works. From a practical point of view, reference [3], which is restricted to axisymmetric deformation of shells of revolution with elastically preferred directions along the normals to the middle surface, i.e., sandwich shells, contains results which represent some simplifications as compared with those given in [1] and [2]. The present paper is concerned with the formulation of suitable stress strain relations and the appropriate boundary conditions in the theory of small deformation of thin elastic isotropic shells of uniform thickness. The results, which include the effects of transverse normal stress, transverse shear deformation, as well as rotary inertia (dis-cussed separately in Sec. 4), are deduced by application of a recent variational theorem due to E. Reissner [4]. Since for the most part the underlying derivation for the stress strain relations is similar to that of reference [3], details of computation are omitted; it is felt, however, that the presentation of the final results will serve a useful purpose. 2. The coordinate system, notation, and preliminaries. Let & and £2 be the coordinates of a point on the middle surface of the shell and f be the distance measured along the outward normal to the middle surface. Further, let n be the unit normal vector at a point of the middle surface and ti and t2 (tx , t2 and n form a right-handed system) be the unit tangent vectors to the &-and ^-curves, respectively. Then the coordinate curves and £2 as lines of curvature (on the middle surface, f = 0) together