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This paper presents the first phase in the analysis of the deformation of a shell of revolution when it is being crushed by a rigid wall. In this paper, the analysis accounts for finite deflections and rotations but assumes that the material remains linearly elastic. The load-deflection behavior is obtained for a hemispherical shell, and it shows that the shell cannot collapse with a flat contact region. The conditions for the buckling of the contact region are determined.

The authors have previously shown that a thin, complete spherical shell compressed between two parallel rigid plates deforms initially with the polar portion of the shell flattened against the plates and that at a critical deformation the flat region may buckle into an axisymmetric inward dimple. The present paper presents an analysis of the stresses and deflections produced during axisymmetric postbuckling and determines the deformation states at which the shell may buckle into a nonsymmetric shape. The analysis accounts for finite deflections and rotations, but assumes that the material remains linearly elastic throughout the deformation. An experiment shows that both the primary axisymmetric bifurcation point and the secondary nonsymmetric bifurcation point are stable for a shell with R/h ≃ 40.

The boundary-value problem of deformation of a rotationally symmetric shell is stated in terms of a new system of first-order ordinary differential equations which can be derived for any consistent linear bending theory of shells. The dependent variables contained in this system of equations are those quantities which appear in the natural boundary conditions on a rotationally symmetric edge of a shell of revolution. A numerical method of solution which combines the advantages of both the direct integration and the finite-difference approach is developed for the analysis of rotationally symmetric shells. This method eliminates the loss of accuracy encountered in the usual application of the direct integration approach to the analysis of shells. For the purpose of illustration, stresses and displacements of a pressurized torus are calculated and detailed numerical results are presented.

This paper is concerned with the vibration analysis of spherical shells, closed at one pole and open at the other, by means of the linear classical bending theory of shells. Frequency equations are derived ha terms of Legendre functions with complex indices, and for axisymmetric vibration the natural frequencies and mode shapes are deduced for opening angles ranging from a shallow to a closed spherical shell. It is found that for all opening angles the frequency spectrum consists of two coupled infinite sets of modes that can be labeled as bending (or flexural) and membrane modes. This distinction is made on the basis of the comparison of the strain energies due to bending and stretching of each mode. It is also found that the membrane modes are practically independent of thickness, whereas the bending modes vary with thickness. Previous analyses with the use of membrane theory have shown that one of two infinite sets of modes is spaced within a finite interval of the frequency spectrum. It is shown in this paper that this set of modes is a degenerate case of bending modes, and, if deduced by means of membrane theory, it is applicable only when the thickness of the shell is zero. When the bending theory is employed, then the frequency interval for this set of modes extends to infinity for every value of thickness that is greater than zero.