Conditions for symmetric, antisymmetric, and zero-stiffness bifurcation in view of imperfection sensitivity and insensitivity

In general, it is not clear how to design structures such that they are ab initio imperfection insensitive, i.e., without modifications of the original design after the diagnosis of imperfection sensitivity. Symmetry and antisymmetry of bifurcation paths, representing qualitative properties of a structure, not only simplify the postbuckling analysis, but also have an influence on the behavior of real, i.e., imperfect structures. The special case of a zero-stiffness postbuckling path incorporates both, symmetry and antisymmetry. Mathematical definitions of the three categories symmetric, antisymmetric, and zero-stiffness equilibrium paths are given. It is shown that symmetry as well as antisymmetry causes the vanishing of specific coefficients in an asymptotic series expansion, following from Koiter’s initial postbuckling analysis. Thereupon, methods of checking for the three categories of bifurcation behavior are discussed. Finally, the three categories are investigated in terms of necessary and/or sufficient conditions for imperfection insensitivity. For instance, a horizontal tangent of the postbuckling path at the bifurcation point is required for imperfection insensitivity. Four examples illustrate non-symmetric, symmetric, and antisymmetric bifurcation as well as zero-stiffness postbuckling behavior. For the first two examples, the approach of increasing the stiffness results in the conversion of the system from imperfection sensitivity into insensitivity. Remarkably, this happens without changing the prebuckling behavior and the buckling load of the structure. The third example demonstrates that imperfection sensitivity is an inevitable implication of antisymmetric bifurcation.