Dynamic buckling of classical/non-classical curved beams by nonlocal nonlinear finite element accounting for size dependent effect and using higher-order shear flexible model

This paper investigates the dynamic snap-through buckling of classical and non-classical curved beams subjected to a suddenly applied step load. The small scale effect prevalent in non-classical beams, viz, micro and nanobeam, is modeled using the nonlocal elasticity approach. The formulation accounts for moderately large deflection and rotation. The governing equilibrium equations are derived using the dynamic version of the principle of virtual work and are subsequently simplified in terms of the generalized displacements for the development of a nonlocal nonlinear finite element model. The spatial domain comprises of 3-noded higher-order curved beam elements based on shear flexible theory associated with sine function. The nonlinear governing equations are solved using the incremental stiffness matrices and by adopting direct time integration method. The critical dynamic buckling load is identified by the smallest load at which there is a sudden rise in the amplitude of vibration. The efficacy of model here is compared against the available analytical studies for the local and nonlocal beams. A detailed study is made to highlight the effects of the geometric parameter, initial condition, nonlocal parameter, load duration, and boundary conditions on the dynamic stability of both classical and non-classical curved beams. The nature and degree of participation of various eigen modes accountable for the dynamic snap-through behavior are examined a posteriori using the modal expansion approach. Some interesting observations made here are valuable for the optimal design of such structural members against fatigue and instability.