Cylindrical shell (axial load / radial pressure / capped pressure) and spherical shell (internal/external pressure) — classical membrane theory of shells
This calculator gives exact closed-form results for the membrane stresses and deformations of thin-walled shells of revolution under uniform loading. It covers the most common cases in pressure vessel and piping engineering: a cylindrical shell under three distinct load types — uniform axial force p (per unit circumferential length), uniform radial pressure q with open ends, and uniform internal/external pressure q with capped ends — and a spherical shell under uniform internal or external pressure.
The equations come from the classical membrane theory of shells (no bending), which gives the two principal membrane stresses — meridional \sigma_1 and hoop \sigma_2 — together with the radial displacement \Delta R and axial/meridional displacement \Delta y. All results are exact within the membrane model: no approximation, no finite element required.
Sign convention: internal pressure q > 0, external pressure q < 0; tensile stresses are positive; outward radial displacement \Delta R > 0.
Model limits (first-order membrane theory for thin shells, R/t > 10): constant wall thickness, isotropic linear elastic, small deformations, axis-symmetric loading, remote from edges and supports. Results do not include edge bending stresses, discontinuity stresses, or through-thickness stress gradients. For R/t \le 10 or near edges and supports, use a bending-shell or finite-element analysis.
Covered cases of this table:
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