Locating dominant dynamics of curved structures: a refined classification of arch’s hardening/softening behaviors

Competing mechanisms due to mechanical and geometrical effects can lead to intricate nonlinear dynamics of curved structures like a shallow arch, e.g., slenderness ratio ξ (mechanical parameter), arch rise b (geometrical parameter). It is thus meaningful to correctly locate dominant dynamics in its parameter space. An auxiliary elasto-geometric parameter λ_a = ξb is newly introduced for shallow arches as inspired by a sagged cable, following which an asymptotic exploration of hardening/softening (H/S) dynamics of a shallow arch is presented by fully leveraging the so-called iso-λ_a curves located on the surface λ_a = ξb geometrically defined in three-dimensional parameter space (ξ–b–λ_a), i.e., a family of arch models associated with the same elasto-geometric parameter λ_a (but with different slenderness and rise). A critical iso-λ_a curve corresponding to separatrix of hardening and softening dynamics is identified. Away from the H/S transition curve, monotonic nonlinear dynamics (either hardening or softening) is found. In contrast, more subtle non-monotonic dynamics can occur in the vicinity of the H/S transition curve (separatrix), wherein a higher-order perturbation is required to capture relevant quintic mechanism besides the commonly treated cubic one. The current contribution leads to a full picture or classification of softening/hardening dynamics of shallow arches, and in particular, a closer look into subtle competing dynamics close to H/S transition.