Bifurcation Buckling as an Approximation of the Collapse Load for General Shells

Abstract

A computer program, STAGS, for nonlinear analysis of shells of general shape has been extended to include a branch for bifurcation buckling analysis. In the case of general shells, failure usually occurs by means of collapse at a limit point rather than through bifurcation. Therefore, the paper contains also a discussion of the practical applicability of the bifurcation buckling theory. Several example cases are presented in which results from a bifurcation buckling analysis are compared to results from a rigorous nonlinear analysis. It is emphasized by these examples that the buckling analysis may give results of little or no value if the shell geometry deteriorates appreciably (Brazier effect) or stresses are redistributed (statical indeterminance) in the subcritical load range. On the other hand, there are cases in which the less costly bifurcation analysis can be substituted for the rigorous collapse analysis. OMPLICATED shell structures for which the weight economy is of utmost importance, such as present space shuttle configurations, have resulted in an increase in the interest in two-dimension al shell analysis. Simultaneously remarkable improvements in numerical methods an4 in computer technology have greatly enhanced our capability to handle such problems. Hence, computer programs for the buckling analysis of shells of general shape qr loading are being offered to a public which to a large extent may be unaware of the limitations of the theory on which these programs are based. A computer program for buckling analysis of shells of general shape or loading based on the bifurcation theory must be used with some caution. A bifurcation point indicates a load level at and above which some new deformation mode is possible. Therefore, the bifurca- tion analysis is a rigorous solution of the problem only if the failure mode is orthogonal to the prebuckling deformation pattern. This, of course, is unlikely to be the case if the shape of the shell or its loading is of a general nature, and for such shells buckling or collapse will in most cases occur through the passing of a limit point (a maximum in a load displacement curve). However, the classical buckling theory (bifurcation from linear membrane solution) sometimes gives good approximations to the buckling load even for cases which are outside its scope. For an axially loaded cylinder with the edges restrained from radial displacements, the rigorous nonlinear analysis shows that the displacements approach infinity as the axial stress approaches the critical value The STAGS computer program has been under development for about three years. The program performs a nonlinear analysis of shells by use of a two-dimensional finite difference grid and an energy principle. Displacement and stress histories are computed corresponding to a given history of applied load, displacement or temperature. The theoretical background for STAGS is presented in Ref. 1 and its scope is discussed in more detail in Ref. 2. It applies to any shell for which a reference surface and a suitable set of gridlines (following shell boundaries) can be defined. The shell wall thickness can vary and material properties can vary with the surface coordinates and through the thickness. Cutouts in the shell wall and discrete stiffening can be included.