Statics of Shallow Bistable Arches

Abstract

Bistable arches have two force-free stable equilibrium configurations. They also show multimodality by switching between their stable states in multiple deformation pathways. These two attributes and their nonlinear force-displacement characteristic are desirable in a range of engineering applications. The analytical and semi-analytical methods developed in this work enable faster analysis than finite element analysis and also facilitate closed-form relationships for insightful design of arches. We show that arch profiles composed using the basis set of buckling mode shapes of the straight column with the corresponding boundary conditions exhibit bistability. We analyze such arches by expressing their deformed profiles also in the same basis set. We assume that the arches are slender and shallow to derive their potential energy comprising bending and compression strain energies as well as the work potential. We solve the equilibrium equations obtained by minimizing potential energy using a semi-analytical method for analytically intractable general boundary conditions. In this method, we obtain the critical points on the force-displacement curve corresponding to switching and switch back forces and travel of the mid-point of the arch. We use this method to analyze and optimize arches of varying as-fabricated stress-free shapes and boundary conditions. We obtain an analytical relationship between the arch-profiles in the force-free states of the arch by equating the force to zero in the aforementioned equilibrium equations for both fixed-fixed and pinned-pinned boundary conditions. This relationship is bilateral, i.e., in one form it can be a used for analysis and in another for design. We derive necessary and sufficient conditions as well as corollaries from the bilateral relationship pertaining to the shapes of bistable arches. Deformation pathways in bistable arches can also be three-dimensional . These spatial deformation pathways can help reduce the switching and switch-back forces and might also, at times, adversely affect bistability. We model spatial pathways by incorporating additional energy terms due to out-of-plane bending and torsion into analysis of planar arches. We use a geometric relation by St. Venant and Michell to capture the coupling amongst the in-plane and out-of-plane deformations and rotation of the cross-sections. Furthermore, we show that non-planar arches, i.e., spatial arches, can be bistable too. Our analysis is extended to spatial arches by modifying the geometric relation to consider the additional out-of-plane curvature. We also present design of two applications based on bistable arches: an RF-MEMS switch and a mechanical OR gate. RF-MEMS switch utilizes bimodality and a novel initially-retracting electrothermal actuator to realize ON and OFF states with only two electrical terminals. The mechanical OR gate uses the bilateral relationship to design arch-profiles that achieve the OR gate logic. Additionally, we present two studies on bistability in axisymmetric shallow thin shells. In the first study, we optimize the shell-profile for maximum travel by numerical and semi-analytical approaches and compare the results with the shell obtained by revolving the optimal arch for maximum travel. In the second study, we discuss the design of a passive universal gripper based on a bistable shell that can grasp objects of varying shape.