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.