The post-buckling of a clamped column made of nonlinear elastic material and subject to axial compression is investigated in this paper. The Ramberg−Osgood type constitutive relation is adopted and it is expanded by using Taylor series. Based on Euler−Bernoulli beam theory, the exact governing equations are expressed in terms of rotation angle. Because the equations contain strongly nonlinear terms that are functions of the rotation angle, analytical solutions are virtually impossible. In this respect, this paper is focused on presenting an alternative method to construct concise yet accurate analytical approximate solutions for post-buckling of the Ramberg−Osgood column that is related to large rotation amplitude. The improved harmonic balance method is used to solve the nonlinear governing equations which are simplified via the Maclaurin series expansion and orthogonal Chebyshev polynomials. In addition, numerical solutions by applying the shooting method on the governing equations are obtained for comparison. The second-order analytical approximate solutions presented in this paper show excellent accuracy by comparing with numerical solutions. The analytical approximate method and numerical result presented in this paper can be applied as design guidelines for designing engineering structures that sustain large deformation, such as slender nonlinear compression of aluminum alloy columns, rods or braces.

本文研究了由非线性弹性材料制成并受到轴向压缩的夹紧柱的后屈曲。采用 Ramberg-Osgood 型本构关系，并用泰勒级数对其进行扩展。基于欧拉-伯努利梁理论，精确的控制方程用旋转角表示。由于方程包含作为旋转角函数的强非线性项，因此解析解几乎是不可能的。在这方面，本文的重点是提出一种替代方法，为与大旋转幅度相关的 Ramberg-Osgood 柱的后屈曲构造简洁而准确的解析近似解。改进的谐波平衡法用于求解通过麦克劳林级数展开和正交切比雪夫多项式简化的非线性控制方程。此外，还得到了对控制方程应用打靶法的数值解进行比较。与数值解相比，本文提出的二阶解析近似解显示出优异的精度。本文提出的解析近似方法和数值结果可用作设计承受大变形的工程结构的设计指南，例如铝合金柱、杆或支撑的细长非线性压缩。得到对控制方程应用打靶法的数值解进行比较。与数值解相比，本文提出的二阶解析近似解显示出优异的精度。本文提出的解析近似方法和数值结果可用作设计承受大变形的工程结构的设计指南，例如铝合金柱、杆或支撑的细长非线性压缩。得到对控制方程应用打靶法的数值解进行比较。与数值解相比，本文提出的二阶解析近似解显示出优异的精度。本文提出的解析近似方法和数值结果可用作设计承受大变形的工程结构的设计指南，例如铝合金柱、杆或支撑的细长非线性压缩。