Seventh-Order Polynomial Constituting the Exact Buckling Mode of a Functionally Graded Column

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.