This paper is concerned with the application of the theory of thin shells to several problems for toroidal shells with elliptical cross section. These problems are as follows: ( a ) Closed shell subjected to uniform normal wall pressure. ( b ) Open shell subjected to end bending moments. ( c ) Combination of the results for the first and second problems in such a way as to obtain results for the stresses and deformations in Bourdon tubes. In all three problems the distribution of stresses is axially symmetric but only in the first problem are the displacements axially symmetric. The magnitude of stresses and deformations for given loads depends in all three problems on the magnitude of the two parameters bc/ah and b/c where b and c are the semiaxes of the elliptical section, a is the distance of the center of the section from the axis of revolution, and h is the thickness of the wall of the shell. For sufficiently small values of bc/ah trigonometric series solutions are obtained. For sufficiently large values of bc/ah asymptotic solutions are obtained. Numerical results are given for various quantities of practical interest as a function of bc/ah for the values 2, 1, 1/2, 1/4 of the semiaxes ratio b/c. It is suggested that the analysis be extended to still smaller values of b/c and to cross sections other than elliptical.

The determination of the stresses in cemented lap joints has become of practical importance because of the development of new methods which permit a strong bond to be established between wood, plastic, or metal sheets, or combinations of them by the use of cement adhesives. In this paper, the problem is divided into two parts, ( a ) determination of the loads at the edges of the joint; ( b ) determination of the stresses in the joint due to the applied loads. Solutions are obtained for two limiting cases, i.e., where the cement layer is so thin that its effect on the flexibility of the joint may be neglected; and where the joint flexibility is mainly due to that of the cement layer. In both cases expressions are obtained for the shearing stresses in the cement, and for the normal stresses in the cement in a direction perpendicular to the plane of the joint.

We investigate the effect of constitutive coupling of stretching, bending, and transverse shearing deformation on the deflection of an anisotropic cantilever beam with narrow rectangular cross-section. To this end, we have developed a hierarchy of beam models by applying a variational principle for displacements and transverse stresses to the associated plane stress problem.

Explicit solutions are obtained, in terms of modified Bessel functions, for the problems of transverse twisting and of tangential shearing of transversely shear-deformable shallow spherical shells with a small circular hole. The relevant stress concentration factors are calculated for the entire range of a rise-to-thickness ratio parameter and a transverse shear deformability parameter. The modification of known results obtained previously by shear deformable plate theory, and by shallow shell theory without consideration of transverse shear deformation effects, is delineated.

We obtain explicit solutions for the two problems considered, in terms of Poisson’s ratio ν and in terms of a parameter μ which is of the order of the ratio of hole radius a to plate thickness h, through application of a sixth-order theory of shear deformable plates, with this solution involving distinct edge zone and interior solution contributions when 1 < < μ. It is shown that in this range some relevant asymptotic Bessel function formulas furnish explicit examples concerning the distinction between first-order shear corrections and second-order (Timoshenko-type) shear corrections which have been established in a recent general analysis.

Equations for small finite displacements of shear-deformable plates are used to derive a one-dimensional theory of finite deformations of straight slender beams with one cross-sectional axis of symmetry. The equations of this beam theory are compared with the corresponding case of Kirchhoff’s equations, and with a generalization of Kirchhoff’s equations which accounts for the deformational effects of cross-sectional forces. Results of principal interest are: 1. The equilibrium equations are seven rather than six, in such a way as to account for cross-sectional warping. 2. In addition to the usual six force and moment components of beam theory, there are two further stress measures, (i) a differential plate bending moment, as in the corresponding linear theory, and (ii) a differential sheet bending moment which does not occur in linear theory. The general results are illustrated by the two specific problems of finite torsion of orthotropic beams, and of the buckling of an axially loaded cantilever, as a problem of bending-twisting instability caused by material anisotropy.

The equations of transverse bending of shear-deformable plates are used for the derivation of a system of one-dimensional equations for beams with unsymmetrical cross section, with account for warping stiffness, in addition to bending, shearing, and twisting stiffness. Significant results of the analysis include the observation that the rate of change of differential bending moment is given by the difference between torque contribution due to plate twisting moments and torque contribution due to plate shear stress resultants; a formula for shear center location which generalizes a result by Griffith and Taylor so as to account for transverse shear deformability and end-section warping restraint; a second-order compatibility equation for the differential bending moment; a contracted boundary condition of support for unsymmetrical cross-section beam theory in place of an explicit consideration of the warping deformation boundary layer; and construction of a problem where the effect of the conditions of support of the beam is such as to give noncoincident shear center and twist center locations.