Bistability of midpoint-fused arches with pinned-pinned boundary conditions

Abstract

Arranging multiple arches in a circular pattern and fusing them at their midpoint yields a three-dimensional configuration that we refer to as midpoint-fused arches (MFA). This study investigates the structural bistability of MFA, i.e., their ability to admit two distinct, force-free stable equilibrium states. Starting from an as-fabricated, stress-free configuration, MFA can invert into a stressed, toggled state reminiscent of an umbrella's ribs. We develop an analytical model for the response of a pinned-pinned MFA subjected to a concentrated mid-span load by minimizing the total potential energy. Individual arches are treated as spatially deforming, and kinematic compatibility relations are derived at the fusion point to couple their deformations. Various deformation symmetries are then exploited to simplify the problem. We demonstrate the model's utility by characterizing the force-displacement response of a two-arch MFA, identifying distinct deformation pathways and discussing the pathway transitions that occur during toggling. In particular, we show how the structure switches between symmetric and asymmetric deformation modes as it moves between stable configurations. The generality of the framework is further established through analysis of a three-arch MFA, which exhibits richer coupled deformation behaviour. Nonlinear finite-element simulations and table-top experiments corroborate the analytical predictions, showing close agreement in both equilibrium states and the associated transition responses.