Effect of the nonlinear pre-buckling state on the bifurcation point of conical shells

The effect of pre-buckling nonlinearity on the bifurcation point of a conical shell is examined on the basis of three shell theories: Donnell’s, Sanders’ and Timoshenko’s. The eigenvalue problem is solved iteratively about the nonlinear equilibrium state up to the bifurcation point. A new algorithm is presented for the real buckling behavior, dispensing with the need to cover the entire nonlinear pattern. This algorithm is very important for structures characterized by a softening process, in which the pre-buckling nonlinearity depresses the buckling level relative to the classical one.The procedure involves nonlinear partial differential equations, which are separated into two sets (using the perturbation technique) for the pre-buckling and buckling states, respectively and solved with the variable expanded in Fourier series in the circumferential direction, and by finite differences in the axial direction. A general computer code was developed and used in studying the effect of the pre-buckling nonlinearity on the buckling level, of the shell under axial compression, in the context of the three shell theories.