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

Frontiers of Mechanical Engineering volume 9, pages 402–408 (2014)Cite this article

Abstract

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 simplification 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 not needed 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.

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Authors and Affiliations

  1. Department of Civil Engineering and Chemical Engineering, University of Tehran, Tehran, Iran

    M. R. Akbari

  2. Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Turin, 24-10129, Italy

    M. Nimafar

  3. Department of Mechanical Engineering, Babol University of Technology, Babol, Iran

    D. D. Ganji

  4. Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran

    M. M. Akbarzade

Authors
  1. M. R. Akbari
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  2. M. Nimafar
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  3. D. D. Ganji
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  4. M. M. Akbarzade
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Correspondence to M. R. Akbari.

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Akbari, M.R., Nimafar, M., Ganji, D.D. et al. Scrutiny of non-linear differential equations Euler-Bernoulli beam with large rotational deviation by AGM. Front. Mech. Eng. 9, 402–408 (2014). https://doi.org/10.1007/s11465-014-0316-8

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  • DOI: https://doi.org/10.1007/s11465-014-0316-8

Keywords

  • AGM
  • critical load of columns
  • large deformations of beam
  • nonlinear differential equation