Unsymmetrical Wrinkling of Nonuniform Annular Plates and Spherical Caps Under Internal Pressure

Abstract

Unsymmetrical buckling of inhomogeneous annular plates and spherical shallow shells subjected to internal pressure is studied. The effect of material heterogeneity, shallowness and ratio of inner to outer radii on the buckling load is examined. The unsymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinate. A numerical method is employed to obtain the lowest load value, which leads to the appearance of waves in the circumferential direction. It is shown that if the elasticity modulus decreases away from the center of a plate, the critical pressure for unsymmetric buckling is sufficiently lower than for a plate with constant mechanical properties.

Appendix

The differential operators that appear in (1) are defined by

$$\begin{aligned} L(x,y)= & {} x''\left( \frac{y'}{r}+\frac{\ddot{y}}{r^2}\right) +y''\left( \frac{x'}{r}+\frac{\ddot{x}}{r^2}\right) -2\left( \frac{\dot{x}}{r}\right) '\left( \frac{\dot{y}}{r}\right) ', \\ L_1^{\pm }(y)= & {} 2 y'''+(2\pm \nu )\frac{y''}{r}+ 2\frac{\left( \ddot{y}\right) '}{r^2}-\frac{y'}{r^2}-3\frac{\ddot{y}}{r^3},\\ L_2^\pm (y)= & {} y''\pm \nu \left( \frac{y'}{r}+\frac{\ddot{y}}{r^2}\right) . \end{aligned}$$

The differential operators introduced in (13) are given by

$$\begin{aligned} \mathcal {L}_1 (g_1,w_n)= & {} g_1' L^{+}_{1n}(w_{n})+g_1''L_{2n}^{+}(w_n), \\ \mathcal {L}_2 (g_2,F_n)= & {} g_2' L^{-}_{1n}(F_{n})+g_2''L_{2n}^{-}(F_n), \\ \end{aligned}$$

where

$$\begin{aligned} L_1^{\pm }(y)= & {} 2 y'''+\frac{2\pm \nu }{r}y''- \frac{2n^2+1}{r^2}\ddot{y}+\frac{3n^2}{r^3}\ddot{y},\\ L_2^\pm (y)= & {} y''\pm \nu \left( \frac{y'}{r}-\frac{n^2}{r^2}\ddot{y}\right) . \end{aligned}$$