Buckling and Vibration of Functionally Graded Material Columns Sharing Duncan's Mode Shape, and New Cases Isaac Elishakoff a , Moshe Eisenberger b, ⁎, Axel Delmas c a Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, USA b Faculty of Civil and Environmental Engineering, Technion, Israel Institute of Technology, Technion City, Haifa 32000, Israel c Ecole Centrale Paris,, 92290 Châtenay-Malabry, France abstract article info Article history: Received 3 August 2015 Accepted 4 November 2015 Available online 14 November 2015 In this study, the closed-form solution for the buckling of an inhomogeneous simply supported column that was un- covered 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. © 2015 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved. 1. Introduction Duncan [3] devoted his study to efficacy of the Bubnov-Galerkin method. Inter alia, he communicated, without derivation, closed-form solution for buckling of inhomogeneous columns. As is known, the closed form solutions for inhomogeneous structures are extremely rare. Therefore, it is interesting to know how Duncan obtained his solution. Moreover, a pertinent question arises if there are other columns or beams for which Duncan's mode shape is valid, or if there are other similar examples. This study addresses above issues. It shows how one can derive Duncan's classic solution, and constructs analogous solutions for the vi- bration problems. Remarkably, it turns out, that there exists a vibrating column whose vibration mode coincides with Duncan's buckling mode. Note that monograph by Elishakoff [5] contains analysis for other candi- date mode shapes of beams in vibratory or buckling conditions. The present study is apparently the first one that addresses Duncan's mode shape directly. Duncan [3] proposed that the shape of the mode be taken as W ξ ðÞ¼ 7ξ-10ξ 3 þ 3ξ 5 ð1Þ and this shape satisfies the simple support conditions at the two ends. Later on, Elishakoff [5] suggested another mode W ξ ðÞ¼ ξ-2ξ 3 þ ξ 4 ð2Þ which has similar properties. These two case can be realized with spatial distribution of material properties that will be given below. The question that one may ask is if there exist other simple shapes that have similar properties. These new cases will yield different buckling loads and spatial distribution of the material properties. In the next section a general deri- vation is presented for the problem. Other recent studies of inhomoge- neous beams and columns include those of Akulenko and Nesterov [1], Caruntu [2], Ece, Ayadoğlu and Taskin [4], Sina and Navazi [10], Gilat, Caliò and Elishakoff [6], Huang and Li [7], Huang and Luo [8], Zarrinzadeh, Attarnejad and Shahba [11], and Maròti [9], among others. 2. Derivation of Duncan's solution and other new solutions Consider the governing differential equation for the buckling of centrally compressed inhomogeneous column simply supported at its two ends: D ξ ðÞ d 2 W dξ 2 þ P cr L 2 W ¼ 0: ð3Þ One can show that the function in Eq. (1), postulated by Duncan [3] satisfies the boundary conditions of the simple supports W 0 ð Þ¼ D ξ ðÞW ″ 0 ð Þ¼ W 1 ð Þ¼ D ξ ðÞW ″ 1 ð Þ¼ 0 ð4Þ where the prime denotes the differentiation with respect to ξ. We pose the following question: Is there an inhomogeneous column that has expression in Eq. (1) as its buckling mode? To answer this question, we observe that the second term in Eq. (1), namely, P cr L 2 w represents a Structures 5 (2016) 170–174 ⁎ Corresponding author. E-mail address: cvrmosh@technion.ac.il (M. Eisenberger). http://dx.doi.org/10.1016/j.istruc.2015.11.002 2352-0124/© 2015 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved. Contents lists available at ScienceDirect Structures journal homepage: www.elsevier.com/locate/structures