Imperfection sensitivity of hilltop branching points of systems with dihedral group symmetry

Imperfection sensitivity of a hilltop branching point occurring as a coincidence of a limit point and a double bifurcation point of a finite-dimensional, elastic, conservative system equivariant to the dihedral group is investigated. In the neighborhood of this point, the potential is expanded into a power series of independent state variables, loading parameter and imperfection magnitude. The form of the expansion is determined through exploitation of dihedral-group symmetry. For the perfect system, the hilltop branching point and bifurcated paths are shown to be all unstable. For an imperfect system, equilibrium paths in general break into a series of paths: including fundamental, complementary and aloof paths. The imperfection sensitivity laws for maximum (critical) points of loading on these paths are obtained as a novel finding of this paper. Critical points on the fundamental and complementary paths enjoy a piecewise linear law, which is less severe than a one-half or two-thirds power law for the double bifurcation point. By contrast, maximum points on aloof paths suffer more severe sensitivity. The hilltop branching point thus displays complex system of imperfection sensitivities. As numerical examples, imperfection sensitivity of simple structural models with the hilltop point is investigated to ensure the validity of the present formulation.