In-plane dynamic instability of a shallow circular arch under a vertical-periodic uniformly distributed load along the arch axis

Abstract

This paper presents an analytical investigation on the in-plane dynamic instability of a shallow circular arch subjected to a vertical-periodic load uniformly distributed along the arch axis, which has not been previously reported in the literature hitherto. Equations of motion of the arch in the radial and tangential directions are derived by employing the Hamilton's principle. The trigonometric and hyperbolic functions are utilized as the mode shapes for the tangential and radial displacements. A method for determining critical instability regions bounded by the periodic solutions with the period T or 2T is presented for the antisymmetric and symmetric dynamic instability of the arch. Frequency sweep transient finite element analyses (FEA) are conducted to verify the analytical solution. Through comparisons, it is observed that analytical solutions agree well with the FEA results. It is shown that the bandwidth of instability regions bounded by the periodic solutions with a period 2T is significantly greater than that bounded by the periodic solutions with a period T. Effects of the rise-span ratio, damping ratio and static component of the periodic load on the dynamic instability are explored. It is found that the dynamic instability behavior of an arch is significantly influenced by the rise-span ratio and damping ratio and their increases lead to the decrease of the bandwidth of the dynamic instability region. It is also found that when the static component of the periodic load is considered, the dynamic instability region undergoes significant variations and moves towards to the lower excitation frequency owing to the decrease of vibration frequency of the arch. The bandwidth of the instability regions, however, increases with an increase of the static component. In addition, it is shown that dynamic instability of modal superposition may occur under a periodic load with a period between T and 2T and symmetric and antisymmetric instability modes are alternately changed during the dynamic instability of modal superposition.