Nonlinear Subharmonic Resonance Instability of an Arch-Type Structure Under a Vertical Base-Excitation

This study develops an analytical framework for investigating in-plane nonlinear subharmonic resonance in fixed–fixed circular arches under a vertical base-excitation, a phenomenon not adequately addressed in previous research. Based on Hamilton’s principle, the governing partial differential equation for in-plane nonlinear motion is first derived. The tangential displacement is then expressed as a modal superposition, and the system is reduced to a set of second-order ordinary differential equations via the Galerkin method. Using the method of multiple scales, the nonlinear 1/2-subharmonic resonance is solved, yielding closed-form, steady-state amplitude–phase relations and corresponding stability conditions. Validation against finite element simulations and Runge–Kutta analyses confirms the accuracy of the proposed approach. Dimensionless fundamental frequencies match finite element results exactly, with discrepancies in critical base-excitation below 2.5%. A close agreement is observed in both the amplitude–frequency and force–response curves with numerical predictions and Bolotin’s method, accurately capturing the characteristic hardening nonlinearity and three distinct dynamic regions spanning negligible vibration, stable resonance, and instability. Parametric studies further reveal key trends. Larger included angles intensify the vibration amplitude and promote saddle-node bifurcation, while narrowing stable operating regions. Higher slenderness ratios enhance structural flexibility and nonlinearity, shifting resonant peaks toward higher frequencies. Increased damping suppresses the response amplitude and raises the thresholds for vibration initiation and bifurcation.