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

The usual theory of thin elastic shells is based upon the hypothesis that the three stresses R, S, T, may be treated as zero, where R is the normal traction perpendicular to the middle surface, and S and T are the two shearing stresses which tend to produce rotation about two lines of curvature of the middle surface. This hypothesis requires that these stresses should be at least of the order of the square of the thickness of the shell, for when this is the case they give rise to terms in the expression for the potential energy due to strain, which are proportional to the fifth power of the thickness, and which may be neglected, since it is usually unnecessary to retain powers of the thickness higher than the cube. It can be proved directly from the general equations of motion of an elastic solid, that this proposition is true in the case of a plane plate, provided the surfaces of the plate are not subjected to any pressures or tangential stresses, but there does not appear to be any simple method of establishing a similar proposition in the case of curved shells.