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 mu(1) which depends on A(1)A(2) and a mode number, where A(1)and A(2) are the undeformed inner and outer radii, and mu(1) is the ratio of the deformed inner radius to A(1). In the large mode number limit, we find that the dependence of mu(1) on A(1)/A(2) has a boundary layer structure: it is constant over almost the entire region of 0 < A(1)/A(2) < 1 and decreases sharply from this constant value to unity as A(1)/A(2) tends to unity. Our asymptotic results for A(1) - 1 = O(1) and A(1) - 1 = O(1/n) are shown to agree with the numerical results obtained by using the compound matrix method.