In this paper, a functionally graded material column that is simply supported at one end and clamped at the other is considered. The buckling mode is postulated as a high-order polynomial. Six novel closed-form solutions are found by the semi-inverse technique. These solutions can be used as benchmark problems with which numerous approximate solution techniques can be tested. Technical novelty consists in searching solutions via semi-inverse method, namely, by postulating the mode shape and searching for the variable flexural rigidity that matches the mode shape. The method is not universal in the sense that it does not develop method of finding the buckling loads for any, arbitrarily, axially graded columns; rather it furnishes closed-form solutions for flexural rigidity grading for columns that might possess the seventh-order polynomial mode shape. Still, this finding appears to be remarkable because it delivers the closed-form solution for the buckling loads.

在本文中，考虑了一端简单支撑而另一端夹紧的功能梯度材料柱。屈曲模式被假定为高阶多项式。通过半逆技术找到了六个新的封闭形式的解决方案。这些解决方案可以用作基准问题，可以测试许多近似解决方案技术。技术创新在于通过半逆方法搜索解决方案，即通过假设振型并寻找与振型匹配的可变抗弯刚度。该方法不是通用的，因为它没有开发出任何、任意、轴向分级的柱的屈曲载荷；相反，它为可能具有七阶多项式模态形状的柱的抗弯刚度分级提供了封闭形式的解决方案。尽管如此，这一发现似乎很了不起，因为它为屈曲载荷提供了封闭形式的解决方案。