Influence of primary boundary conditions on the buckling of shallow cylindrical shells

Slender cylinders of intermediate length subjected to external pressure normally buckle into sinusoidal waves in the circumferential and meridional directions; the wave numbers are a function of the edge restraints, aspect ratio L/R and wall slenderness R/t. The influence of edge restraints diminishes with increasing L/R, and disappears in very long cylinders. All instability formulae purporting to evaluate the lowest eigenvalue require minimisation with respect to the circumferential wave number n; any 'exact' formulation requires the a priori evaluation of n. Therefore, in any investigation of the effect of boundary conditions on the buckling pressure q(cr), the evaluation of n is crucial. The effects of imperfections and consequent non-linear response are significant in this context, and shall form the subject of a future publication. The classical solution for the estimation of n involves 'standard' end conditions which consist only of radial restraints, and is derived from a minimisation, for example, of the Southwell equation with respect to n. For other sets of boundary conditions, the governing equations become almost intractable unless reduced theories, such as long wavelength theory, are used, and resort must be made to numerical methods such as Rayleigh-Ritz,finite differences or finite elements. To the authors' knowledge, a comprehensive theoretical or numerical study of the effect of boundary conditions on the buckling wavenumber and the critical pressure is not available in the literature. In this paper, a numerical study is presented of the influence of 11 separate sets of boundary conditions on the circumferential wavenumber n and the buckling pressure q(cr). Geometric ratios are considered in the practical range 0.5 less than or equal to L/R less than or equal to 5.0 and 300 less than or equal to R/t less than or equal to 3000. Finite element analysis is carried our using refined meshes of an element thoroughly tested against known solutions in this context. For each set of boundary conditions, simple expressions are derived for n and q(cr). It is found that, as the restraint of the cylinder edges increases, the meridional wavelength decreases, and the wavenumber n increases. The meridional wavenumber is not necessarily an integer value and therefore the concept of an effective length similar to that of columns is appropriate.