Snap-through behavior of non-uniform shallow arches or manually curved initially straight beams under a concentrated force

Abstract

This article presents a combined analytical, experimental and numerical investigation of non-uniform shallow circular arches and manually curved initial straight beams under lateral concentrated forces, with emphasis on non-uniformity. First, the power series method by Lyapunov artificial small parameter approach is introduced for analytically determining equilibrium paths for both shallow arches and curved beams in a similar manner. A key novelty lies in the precise estimation of truncation numbers even prior to iterations achieved through asymptotic results derived from Holder’s inequality. Detailed mathematical analysis is conducted on the monotonicity properties and asymptotic behaviors of the truncation number estimation function with respect to Holder’s exponent. The extreme case of non-uniform shallow circular arch under pinned BCs featuring rigid end segments and a small weak central segment is revisited asymptotically by identifying asymptotic dependence of deformation on smallness of central segment and intuitively interpreted by proposing a two-spring model. An experimental validation is provided for the previously theoretically identified unusual snap-through behavior of shallow arch under fixed-fixed boundary conditions with stiffer ends. Next, the snap-through behavior of manually curved initially straight beam with width non-uniformity is examined focusing on the loading position and the difference between width non-uniformity and thickness non-uniformity. Both experiments and theoretical calculations reveal a snap-through transition from load-induced (separating) mode to displacement-induced (non-separating) mode as the loading position shifts from the center to the ends. Strong agreement is observed between analytical solutions and experimental results. Finally, the averaged $1/I$ approach is validated for thickness-directional periodically fast-varying non-uniformity, while the least $I$ approach proves more accurate for width-directional non-uniformity. This paper intends to advance the understanding of how non-uniformity influences on snap-through behavior of shallow arch and curved beams under a single lateral concentrated force.