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A homotopy analysis solution to large deformation of beams under static arbitrary distributed load

机译：静态任意分布荷载作用下梁大变形的同伦分析方法

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摘要

In this article, homotopy analysis method (HAM) is employed to investigate non-linear large deformation of Euler-Bernoulli beams subjected to an arbitrary distributed load. Constitutive equations of the problem are obtained. It is assumed that the length of the beam remains constant after applying external loads. Different auxiliary parameters and functions of the HAM and the extra auxiliary parameter, which is applied to initial guess of the solution, are employed to procure better convergence rate of the solution. The results of the solution are obtained for two different examples including constant cross sectional beam subjected to constant distributed load and periodic distributed load. Special base functions, orthogonal polynomials e.g. Chebyshev expansion, are employed as a tool to improve the convergence of the solution. The general solution, presented in this paper, can be used to attain the solution of the beam under arbitrary distributed load and flexural stiffness. Ultimately, it is shown that small deformation theory overestimates different quantities such as bending moment, shear force, etc. for large deflection of the beams in comparison with large deformation theory. Finally, it is concluded that solution of small deformation theory is far from reality for large deflection of straight Euler-Bernoulli beams.
机译：在本文中，使用同伦分析方法（HAM）研究承受任意分布载荷的Euler-Bernoulli梁的非线性大变形。得到了该问题的本构方程。假定在施加外部载​​荷之后，梁的长度保持恒定。 HAM的不同辅助参数和功能以及用于解决方案初始猜测的额外辅助参数可用于获得更好的解决方案收敛速度。该解决方案的结果是针对两个不同的示例获得的，包括承受恒定分布载荷的恒定横截面梁和周期性分布载荷。特殊基函数，正交多项式，例如Chebyshev扩展被用作改善解决方案收敛性的工具。本文提出的一般解可用于获得梁在任意分布载荷和挠曲刚度下的解。最终表明，与大变形理论相比，小变形理论高估了梁大挠度时的弯矩，剪切力等不同量。最后，得出结论，小变形理论的解决方案对于直线Euler-Bernoulli梁的大挠度远非现实。

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