A WKB analysis of the buckling condition for a cylindrical shell of arbitrary thickness subjected to an external pressure

We consider the plane-strain buckling of a cylindrical shell of arbitrary thickness which is made of a Varga material and is subjected to an external hydrostatic pressure on its outer surface. The WKB method is used to solve the eigenvalue problem that results from the linear bifurcation analysis. We show that the circular cross-section buckles into a non-circular shape at a value of μ1 which depends on A1/A2 and a mode number, where A1 and A2 are the undeformed inner and outer radii, and μ1 is the ratio of the deformed inner radius to A1. In the large mode number limit, we find that the dependence of μ1 on A1/A2 has a boundary layer structure: it is constant over almost the entire region of 0 < A1/A2 < 1 and decreases sharply from this constant value to unity as A1/A2 tends to unity. Our asymptotic results for A1 − 1 = O(1) and A1 − 1 = O(1/n) are shown to agree with the numerical results obtained by using the compound matrix method.