In this paper, nonlinear responses of a clamped-clamped buckled beam are investigated. Two efficient and easy mathematical techniques called He's Variational Approach and Laplace Iteration Method 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. 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.

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Herein, Reconstruction of Variational Iteration Method (RVIM) is used for computing solutions of the seventh-order Sawada-Kotera equation (sSK) and a Lax's seventh order KdV equations (LsKdV). The results are compared with the Adomian decomposition method (ADM) and the known analytical solutions. Results obtained expose effectiveness and capability of this method to solve the seven-order Sawada-Kotera (sSK) and a Lax's seven-order KdV (LsKdV) equations.

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.

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. The Structural engineering theory is based upon physical laws and empirical knowledge of the structural performance of different landscapes and materials. Many engineering structures can be modelled as a slender, flexible cantilever beam carrying a lumped mass with rotary inertia at an intermediate point along its span; hence they experience large-amplitude vibration (Wu, 2003;Herisanu and Marinca, 2010;Cveticanin and Kovacic, 2007;Hamdan and Shabaneh, 1997;Hamdan and Dado, 1997). Key wordsIn general, such problems are not amenable to exact treatment and approximate techniques must be resorted to. So, many new techniques have appeared in the open literature such as: variational iteration method (He, 2007a;Barari et al., 2011), homotopy perturbation method Saravi et al., 2013), homotopy analysis method (Liao, 2003;Khan et al., 2012)and some other methods Ghasempoor et al., 2102;Akbarzade and Khan, 2012;Alinia et al., 2011;Ganji, 2012;Torabi et al., 2012;

This paper presents the analytical simulation of string with large amplitudes using the Variational Iteration Method (VIM) and Hamiltonian Approach (HA). In order to verify the precision of the presented methods, current results were compared with He's Variational Approach and Runge-Kutta 4 th order. It has been found that these methods are well suited for a range of parameters and the approximate frequencies and periodic solutions show a good agreement with other techniques. 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.

In this paper, we extend variational iteration method (VIM) for deriving approximate analytical solution to seventh-order differential equations with specified initial conditions, also in this paper we applied a modified method to identification of Lagrange multiplier .By providing some examples, we illustrate the capability and reliability of the method.