VI. On the extension and flexure of cylindrical and spherical thin elastic shells

Abstract

The various theories of thin elastic shells which have hitherto been proposed have been discussed by Mr. Love in a recent memoir, and it appears that most, it not all of them, depend upon the assumption that the three stresses which are usually denoted by R, S, T are zero; but, as I have recently pointed out, a very cursory examination of the subject is sufficient to show that this assumption cannot be rigorously true. It can, however, be proved that, when the external surfaces of a plane plate are not sub­jected to pressure or tangential stress, these stresses depend upon quantities propor­tional to the square of the thickness, and whenever this is the case they may be treated as zero in calculating the expression for the potential energy due to strain, because they give rise to terms proportional to the fifth power of the thickness, which may be neglected, since it is usually unnecessary to retain powers of the thickness higher than the cube. It will also, in the present paper, be shown by an indirect method that a similar proposition is true in the case of cylindrical and spherical shells, and, therefore, the fundamental hypothesis upon which Mr. Love has based his theory, although unsatisfactory as an assumption, leads to correct results. A general expression for the potential energy due to strain in curvilinear coordinates has also been obtained by Mr. Love, and the equations of motion and the boundary conditions have been, deduced therefrom by means of the Principle of Virtual Work, and if this expression and the equations to which it leads were correct, it would be unnecessary to propose a fresh theory of thin shells ; but although those portions of Mr. Love’s results which depend upon the thickness of the shell are undoubtedly correct, yet, for reasons which will be more fully stated hereafter, I am of opinion that the terms which depend upon the cube of thickness are not strictly accurate, inasmuch as he has omitted to take into account several terms of this order, both in the expression for the potential energy and elsewhere. His preliminary analysis is also of an exceedingly complicated character. Throughout the present paper the notation of Thomson and Tait’s “Natural Philosophy ” will be employed for stresses and elastic constants, but, for the purpose of facilitating comparison, Mr. Love's notation will be employed for strains and directions. It will also be convenient to denote the values of the various quantities involved, at a point P on the middle surface of the shell by unaccented letters; and the values of the same quantities at a point P' on the normal at P, whose distance from P is h ' by accented letters. The radius of the shell will also be a denoted by and its thickness by 2 h .