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The paper presents a formulation of the two‐dimensional theory of shallow shells, including the effects of transverse shear deformation and of moments turning about the normal to the middle surface. The present formulation includes, as it must, Marguerre's theory. At the same time it is consistent with recent formulations of general linear shell theory, in particular in regard to the preservation of the static‐geometric duality. Various reductions of the equation of the theory are considered. Of particular significance and effectiveness among these are reductions for the special cases of (1) shells without moments about the middle surface normal, (where an earlier result of Naghdi is extended) (2) the shell without transverse shear deformability, (the static‐geometrical dual of case (1)). As an application of the general equations an explicit solution is obtained for the problem of stretching, twisting and bending of pretwisted rectangular plates.

In generalization of existing results of the linear theory of symmetrical bending and torsion of thin elastic shells of revolution and of the theory of pure bending and twisting of ring sector shells, this paper considers the differential equations of the theory of shells of revolution under the supposition of rotationally symmetric stress and strain, allowing for transverse shear deformation and for moments turning about the normals to the middle surface. In this way, making use of the simplified static‐geometric analogy which is associated with the introduction of the two mentioned effects, known results are rederived in a new way and some new results are obtained, in particular for the dislocation‐theory‐type problems of pure bending and twisting.

The side-force problem for a shallow helicoidal shell is shown to be the complete static-geometric analog of the pure bending problem for the same shell. The solution of the former in terms of elementary functions is obtained, without another set of independent calculations, simply by applying the rules of the static-geometric duality to that of the latter. The analogy also enables us to use the same computer program developed for the pure bending problem (without any modification) to generate numerical results for the side-force problem.

A direct time-domain method is used to analyze the titled problem. Special attention is paid to a system characterized by a general second-order equation with variable coefficients. The equation of flapping motion of a rigid rotor blade advancing in atmospheric turbulence belongs to this class. Steady-state mean-square response to ideal white noise and to exponentially correlated excitation is obtained by a perturbation series solution in a stiffness parameter. An upperbound of the same is derived. Explicit solution for the correlation matrix is obtained by the two-variable expansion method. Specialized to the rotor blade problem, the results have led to some new information concerning the blade behavior in a certain range of rotating speed. They have also served as useful check cases for computer programs developed for more general problems. Higher-order equations and their applications are also discussed.

The solution of the problem of pure bending of shallow helicoidal shells is obtained in terms of elementary functions. Numerical results for the direct and bending stresses as well as for the overall stiffness coefficient of the shell are given for various pitch-to-thickness ratios. An interesting aspect of the analysis is that the displacement state of the shell is multivalued in the polar angle θ while the stress state is unsymmetric but periodic in θ. Moreover, unlike the problem of axial torsion and extension where there is no edge effect, the stress state for the present problem exhibits both an interior state and an edge zone state.