Buckling configurations and dynamic response of buckled Euler-Bernoulli beams with non-classical supports

Exact solutions of buckling configurations and vibration response of post-buckled configurations of beams with non-classical boundary conditions (e.g., elastically supported) are presented using the Euler-Bernoulli theory. The geometric nonlinearity arising from mid-plane stretching (i.e., the von Karman nonlinear strain) is considered in the formulation. The nonlinear equations are reduced to a single linear equation in terms of the transverse deflection by eliminating the axial displacement and incorporating the nonlinearity and the applied load into a constant. The resulting critical buckling loads and their associated mode shapes are obtained by solving the linearized buckling problem analytically. The buckling configurations are determined in terms of the applied axial load and the transverse deflection. The first buckled shape is the only stable equilibrium position for all boundary conditions considered. Then the pseudo-dynamic response of buckled beams is also determined analytically. Natural frequency versus buckling load and natural frequency versus amplitudes of buckling configurations are plotted for various non-classical boundary conditions.