Design of bistable arches by determining critical points in the force-displacement characteristic

The boundary conditions, as-fabricated shape, and height-to-depth ratio of an arch characterize its bistability. We investigate this by modeling planar arches with torsion and translational springs at the pin joints anchored to the ground. A pinned-pinned arch, a special case of the model, has superior bistable characteristics as compared to a fixed-fixed bistable arch, which is also a particular case of the model. Arches with revolute flexures at the ends retain bistable characteristics of the pinned-pinned arches while being amenable for easy fabrication. However, equilibrium equations for such arches become intractable for analytical solution unlike the extreme cases of fixed-fixed and pinned-pinned arches. Therefore, a semi-analytical method for analysis and shape-synthesis of bistable arches with general boundary conditions is developed in this work. This is done by numerically determining critical points in the force-displacement curve. These critical points correspond to switching and switch-back forces and travel between the two states thereby enabling synthesis for desired behavior. We present design and optimization examples of bistable arches with a variety of boundary conditions and as-fabricated shape without prestress. We also propose two approaches to design a new class of asymmetric bistable arches.