Unified analytical solution for dynamic elastic buckling of beams for various boundary conditions and loading rates

The problem of dynamic elastic buckling of Euler–Bernoulli beams subjected to axial loads is studied analytically. The dynamic axial loading is accomplished by a constant displacement rate of one end of the beam with respect to the other. The axial loading rates are considered slow enough to obviate the need to account for axial wave propagation effects. The dynamics of the beam is formulated in the modal domain considering the lowest static buckling mode. The dynamic stability of the beam is investigated as a response problem assuming an initial deformed geometry in terms of an eccentricity favouring the mode shape. The governing partial differential equation is condensed into a unified ordinary differential equation for various boundary conditions using a single dimensionless parameter in terms of beam geometry, material properties, loading rate, and appropriate coefficients corresponding to different boundary conditions. The results obtained for the dynamic response of beams for various values of the dimensionless parameter and initial eccentricity suggest that the solutions can be combined into a single unified analytical expression for the dynamic buckling load. It is shown that the corresponding dynamic response curves can also be collapsed into a single curve using the dimensionless peak load and the associated time parameter. The accuracy of the unified analytical expression for dynamic buckling is verified against exact solution of the ordinary differential equation considering three problems in terms of dimensional quantities for different boundary conditions and loading rates. Further the validity of the single mode dynamic buckling formulation is examined by comparing the results obtained for various boundary conditions with the numerical results from the dynamic response of a large number of degree of freedom finite element model of the beam without any restriction on the deformation mode shape. This unified solution has potential application in optimum design for dynamic collapse of truss type structures subjected to dynamic loads.