Analysis of bifurcation bending modes of arches and panels

This paper reports the results of numerical analysis of the bifurcation solutions of nonlinear boundary-value problems of plane bending of elastic arches and panels. Problems are formulated for a system of six nonlinear ordinary differential equations of the first order with independent fields of finite displacements and rotations. Two loading versions (by follower and conservative pressures) and two versions of boundary conditions (rigid clamping and pinning) are considered. In the case of clamped arches and panels, the set of solutions consists of symmetric and asymmetric bending modes which exist only for positive values of the load parameter. In the case of pinning, the set of solutions includes symmetric and asymmetrical modes which correspond to positive, negative, and zero values of the parameter. In both problems, the phase relations between the state parameter and the load parameter are bifurcated, ambiguous, have isolated branches, and admit a catastrophe – a finite jump from the fundamental equilibrium mode to a buckled mode.