Exact solution and dynamic buckling analysis of a beam-column system having the elliptic type loading

This paper presents a closed form solution to the dynamic stability problem of a beam-column system with hinged ends loaded by an axial periodically time-varying compressive force of an elliptic type, i.e., a₁cn²(t, k²) + a₂sn²(t, k²) + a₃dn²(t, k²). The solution to the governing equation is obtained in the form of Fourier sine series. The resulting ordinary differential equation is solved analytically. Finding the exact analytical solutions to the dynamic buckling problems is difficult. However, the availability of exact solutions can provide adequate understanding for the physical characteristics of the system. In this study, the frequency-response characteristics of the system, the effects of the static load, the driving forces, and the frequency ratio on the critical buckling load are also investigated.