In this paper, the problem of determining the stresses and deformations in a thin, homogeneous, orthotropic shell of revolution under the action of axisymmetric loads is reduced to the solution of a single inhomogeneous second-order linear differential equation with a complex dependent variable. Asymptotic solutions are obtained which are uniformly valid in both the steep and shallow regions of the dome-shaped shell. The complementary, or “edge-effect,” solutions are expressed in terms of Thomson’s functions of a noninteger order. The order depends both on the shape of the meridian curve at the apex of the shell and on the ratio of the elastic moduli. The particular solution is found in terms of an appropriate linear combination of the Lommel’s function and Thomson’s functions. This particular solution is equivalent to the well-known “membrane” solution in the steep portion of the shell but in the shallow portion gives significant bending stresses. The particular and complementary solutions are used to investigate the behavior of orthotropic pressure vessels with rigid rings clamped to the edges.

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