Static and dynamic symmetric snap-through of non-uniform shallow arch under a pair of end moments considering critical slowing-down effect

This paper presents analytical analysis of static and dynamic symmetric snap-through of non-uniform shallow circular shallow arch under a pair of end moments with the same magnitude. The non-uniformity is characterized by dividing the arch into three piecewise constant-stiffness segments. Hamilton's principle is used to derive the governing differential equations by assuming negligible axial inertia. The snap-through load and snap-through criterion are analyzed in detail and by an index plot, the stiffer center case is shown to behave distinctly when stiffer end case is compared. The dynamic snap-through when the moment is slightly higher than the snap-through moment is analyzed analytically by a perturbation method, and a critical slowing effect is observed when the moment is approaching to the snap-through moment. Comparison with dynamic FEA shows a good agreement with the analytical result and analysis on theoretical finite time blow-up phenomenon reveals that when geometric parameters are corresponding to the critical snap-through condition, the initial quadratic phase's motion is slow and relatively blow-up phase's motion is fast. The analytical formulations have been extended to include two limiting cases including rigid end case and rigid center case by using the constrained Hamilton's principle by Lagrangian multipliers. Snap-through criteria analysis reveals closed-form criterion for rigid center case and an asymptotic result for rigid end case. This paper serves to enhance the knowledge on snap-through and critical slowing down for shallow arches with non-uniformity under end moments.