Nonlinear dynamic buckling of a viscoelastic model under impact loading

A simple nonlinear buckling analysis is applied to a one-degree-of-freedom arch under impact loading in which viscous damping may also be included. Such a loading consists of a falling body striking centrally the joint mass of the arch in such a way that a completely plastic impact can be postulated. When there is no damping the exact dynamic buckling load for such a kind of loading-associated with an unbounded motion can be established by using a static criterion (approach). More specifically, it was shown that the dynamic buckling load corresponds to that unstable equilibrium state where the total potential energy of the system is zero. Furthermore, it was proved that the second variation of the total potential energy at the foregoing unstable equilibrium state is negative definite. This implies that the curve loading versus displacement resulting by the vanishing of the total potential energy has always a maximum on the afore mentioned unstable state. It was also found that the system may become sensitive to initial conditions. If damping is included the foregoing static criterion yields lower bound buckling estimates. These findings were verified by employing a highly efficient approximate technique as well as the numerical scheme of Runge-Kutta for solving any nonlinear initial-value problem.