Symmetric snap-through and equal potential energy load of non-uniform shallow arch under a concentrated load considering imperfection effect

Abstract

This paper deals with analytical investigation of non-uniform circular shallow arch under a concentrated load considering imperfection effect under pinned-pinned boundary conditions. After characterizing the non-uniformity by three piecewise constant stiffness segments, governing equilibrium equations including imperfection effect are derived via least potential energy principle and criteria for possible occurrence of symmetric snap-through are explicitly presented by using L'Hospital's rule after identifying two modified slenderness parameters. The existence of an equal potential energy load is rigorously shown in a straightforward manner and its parametric dependence on imperfection and non-uniformity is discussed. The relationship between equal potential energy load and buckle propagation load is studied by 3D FEA of buckle propagation of corresponding non-uniform shallow panel. Moreover, two limiting cases including rigid end case and rigid center case are analyzed by employing augmented potential energy with Lagrangian multipliers. An asymptotical solution shows that a finite modified slenderness (independent of imperfection) is sufficient to ensure the occurrence of symmetric snap-through when dimension of center segment is approaching to zero in rigid end case and this conclusion holds when there is some rotational restraint at ends. For rigid center case, the closed-form criterion considering imperfection for symmetric snap-through occurrence is presented. Equal potential energy load for two limiting cases is derived and analyzed in detail. This paper is expected to enhance the understanding of stability of imperfect non-uniform shallow arch.