- مبلغ: ۸۶,۰۰۰ تومان
- مبلغ: ۹۱,۰۰۰ تومان
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 byW.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.
In this paper Duncan's  classic solution was derived systematically from the basic buckling equation of a member with axial FGM properties. Then, three other cases are derived using the same methodology: the first in known as Elishakoff's  shape another new first buckling mode, and a fourth one which is a second mode buckling case. Then the same method was extended to two other similar problems: vibrations and vibrations in the presence of axial loading, for the same four mode shapes. All three problems, although associated with different flexural rigidities of the corresponding beams share the same mode shapes. Explicit variation of the FGM properties for all the above cases was found analytically. The improvement in the magnitude of the buckling load can be utilized in all cases as compared to simply supported columns with constant properties. The improvement is small (less than 5% increase) and will be justified only in special cases where performance is the major consideration. It is remarkable that in all three problems, although associated with different flexural rigidities of the corresponding beams, they all share the same eigenfunction.