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Scrutiny of non-linear differential equations Euler-Bernoulli beam with large rotational deviation by AGM

Frontiers of Mechanical Engineering 2014年第4期9卷 402-408页

作者： M. R. AKBARI M. NIMAFAR D. D. GANJI M, M. AKBARZADE Department of Civil Engineering and Chemical Engineering Universityof Tehran Tehran lran Department of Mechanical and Aerospace Engineering Politecnico diTorino Turin 24-10129 Italy Department of Mechanical Engineering Babol University of Technology Babol Iran Department of Mechanical Engineering Sari Branch Islamic AzadUniversity Sari Iran

The kinematic assumptions upon which the Euler-Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternative constitutive equations can allow for viscoelastic or plastic beam deformation. Euler-Bernoulli beam theory can also be extended to the analysis of curved beams, beam buckling, composite beams and geometrically nonlinear beam deflection. In this study, solving the nonlinear differential equation governing the calculation of the large rotation deviation of the beam （or column） has been discussed. Previously to calculate the rotational deviation of the beam, the assumption is made that the angular deviation of the beam is small. By considering the small slope in the linearization of the governing differential equation, the solving is easy. The result of this simplifica- tion in some cases will lead to an excessive error. In this paper nonlinear differential equations governing on this system are solved analytically by Akbari-Ganji＇s method （AGM）. Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration solving. The solution of the problem can be obtained very simply and easily. Furthermore, to enhance the accuracy of the results, the Taylor expansion is notneeded in most cases via AGM manner. Also, comparisons are made between AGM and numerical method （Runge- Kutta 4th）. The results reveal that this method is very effective and simple, and can be applied for other nonlinear problems.

关键词： AGM critical load of columns large deformations of beam nonlinear differential equation

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