In this paper, a mathematical model of large amplitude vibration of a uniform cantilever beam arising in the structural engineering is proposed. Two efficient and easy mathematical techniques called variational iteration method and He's variational approach are used to solve the governing differential equation of motion. To assess the accuracy of solutions, we compare the results with the Runge-Kutta 4th order. An excellent agreement of the approximate frequencies and periodic solutions with the numerical results and published results has been demonstrated. The results show that both methods can be easily extended to other nonlinear oscillations and it can be predicted that both methods can be found widely applicable in engineering and physics.

Seismic design loads for tunnels are characterized in terms of the deformations imposed on the structure by surrounding ground. The free-field ground deformations due to a seismic event are estimated, and the tunnel is designed to accommodate these deformations. Vertically propagating shear waves are the predominant form of earthquake loading that causes the ovaling deformations of circular tunnels to develop, resulting in a distortion of the cross sectional shape of the tunnel lining. In this paper, seismic behavior of circular tunnels has been investigated due to propagation of shear waves in the vertical direction using quasi-static analytical approaches as well as numerical methods. Analytical approaches are based on the closed-form solutions which compute the forces in the lining due to equivalent static ovaling deformations, while the numerical method carries out dynamic, nonlinear soil-structure interaction analysis. Based on comparisons made, the accuracy and reliability of the analytical solutions are evaluated and discussed. The results show that the axial forces determined using the analytical approaches are in acceptable agreement with numerical analysis results, while the computed bending moments are less comparable and show significant discrepancies. The differences between the analytical approaches are also investigated and addressed.