Buckling and Vibration of Functionally Graded Material Columns Sharing Duncan's Mode Shape, and New Cases

In this study, the closed-form solution for the buckling of an inhomogeneous simply supported column that was uncovered by the noted British engineer Duncan in 1937, is first derived in a straightforward manner. It deals with buckling of a centrally compressed inhomogeneous column. It is also found that there are several other columns with variable axial functionally graded material (FGM) that share the same qualities as Duncan's column. It is then shown that the mode postulated by W.J. Duncan (1894–1970), FRS and the newly found modes, have a greater validity, namely the freely vibrating beam, albeit with different flexural rigidity than the centrally compressed one, may possess the same buckling mode. It is demonstrated also that there exists an inhomogeneous beam under axial compression whose vibration mode coincides with the buckling modes in the previous cases.