In this study, we investigate the buckling of a column made of functionally graded materials in axial direction. The column is simply supported at both ends. The mode shape is represented by a polynomial whose power is increased from four to nine; this power only restricted by the possibility of the hand- or symbolic-manipulation. It is shown that the solutions stemming from lower-order polynomial are all contained in the solution associated with higher-order polynomials. Numerous novel solutions are uncovered that could be used as benchmark solutions versus numerous approximate techniques available for eigenvalue problems.
EulerL. Sur la force des colonnes. Mem Acad Sci Belles-Lett Berlin1759; 13: 252–282 (in French).
2.
EngesserF. Über die Knickfestigkeit gerader Stäbe. Zeitschrift Architek Inginieure Hannover1899; 35: 455–462 (in German).
3.
DuncanNJ. Galerkin’s method in mechanics and differential equations. Aeronautical Research Committee reports and memoranda no. 1798, 3August1937. Fort Belvoir, VA: Defense Technical Information Center.
4.
ElishakoffI. Euler’s problem revisited: 222 years later. Meccanica2000; 35: 265–272.
5.
ElishakoffI. Inverse buckling problem for inhomogeneous columns. Int J Solids Struct2001; 38(3): 457–464.
CaliòIElishakoffI. Can harmonic functions constitute closed-form buckling modes of inhomogeneous columns?AIAA J2002; 40: 2532–2537.
8.
GilatRCaliòIElishakoffI. Inhomogeneous beams possessing an exponential mode shape. Mech Res Commun2010; 37(4): 417–426.
9.
ElishakoffIEisenbergerMDelmasA. Buckling and vibration of functionally graded material columns sharing Duncan’s mode shape, and new cases. Structures2016; 5: 170–174.
10.
ElishakoffIPentarasDGentiliniC. Mechanics of functionally graded material structures. Singapore: World Scientific, 2015.
11.
SureshSMortensenA. Fundamentals of functionally graded materials: processing and thermomechanical behaviour of graded metals and metal-ceramic composites. London: IOM Communications Ltd, 1998.
12.
ReddyJN. Analysis of functionally graded plates. Int J Numer Meth Eng2000; 47(1–3): 663–684.
13.
ŞimşekMReddyJN. A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory. Compos Struct2013; 101: 47–58.
14.
EisenbergerM. Buckling loads for variable cross-section members with variable axial forces. Int J Solids Struct1991; 27: 135–143.
15.
EisenbergerM. Buckling loads for variable cross section bars in a nonuniform thermal field. Mech Res Commun1992; 19: 259–266.
16.
AydoğduM. Semi-inverse method for vibration and buckling of axially functionally graded beams. J Reinf Plast Comp2008; 27(7): 683–691.
17.
LiQS. Exact solutions for the generalized Euler’s problem. J Appl Mech: T ASME2009; 76: 041015.
18.
MaalawiKY. Optimization of elastic columns using axial grading concept. Eng Struct2009; 31(12): 2922–2929.
19.
SinghKVLiG. Buckling of functionally graded and elastically restrained non-uniform columns. Compos Part B: Eng2009; 40: 393–403.
20.
CoskunSB. Determination of critical buckling loads for Euler columns of variable flexural stiffness with a continuous elastic restraint using homotopy perturbation method. Int J Nonlin Sci Num2009; 10: 191–197.
21.
DarbandiSMFirouz-AbadiRDHaddadpourH. Buckling of variable section columns under axial loading. J Eng Mech: ASCE2010; 136(4): 472–476.
22.
HuangYLiXF. Buckling analysis of non-uniform and axially graded columns with varying flexural rigidity. J Eng Mech: ASCE2011; 137: 73–81.
23.
HuangYLuoQZ. A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint. Comput Math Appl2011; 61: 2510–2517.
24.
BabilioE. Dynamics of an axially functionally graded beam under axial load. Eur Phys J: Spec Top2013; 222(7): 1519–1539.
BrangwynneCPMacKintoshFCKumarS, et al. Microtubules can bear enhanced compressive loads in living cells because of lateral reinforcement. J Cell Biol2006; 173(5): 733–741.
27.
IoakimidisNI. Application of quantifier elimination to inverse buckling problems. Acta Mech2017; 228(10): 3709–3724.
28.
KrutyiYS. Construction of the stability problem solution of a column with continuous variation of parameters. Math Method Phys Mech2016; 59(2): 142–151 (in Russian).
29.
BudianskyBHutchinsonJW. Buckling: progress and challenge. In: BesselingJFVan der HeijdenAMA (eds) Trends in solid mechanics 1979: proceedings of the symposium dedicated to the 65th birthday of W.T. Koiter. Delft: Delft University Press, 1979, pp. 93–116.
30.
ChakravertySPradhanKK. Vibration of functionally graded beams and plates. Amsterdam: Elsevier, 2016.
NguyenDK. Large displacement response of tapered cantilever beams made of axially functionally graded material. Compos Part B: Eng2013; 55: 298–305.
33.
ShahbaAAttarnejadRHajilarS. Free vibration and stability of axially functionally graded tapered Euler–Bernoulli beams. Shock Vib2011; 18(5): 683–896.
34.
ŞimşekMKocatürkTAkbaşŞD. Dynamic behavior of an axially functionally graded beam under action of a moving harmonic load. Compos Struct2012; 94(8): 2358–2364.
35.
ByskovE. Applicability of an asymptotic expansion for elastic buckling problems with mode interaction. AIAA J1979; 17(6): 630–633.
36.
FoxRLKapoorMP. Rates of change of eigenvalues and eigenvectors. AIAA J1968; 6(12): 2426–2429.
37.
VinogradovAM. Buckling of viscoelastic beam columns. AIAA J1987; 25(3): 479–483.
38.
BertCWJangSKStrizAG. Two new approximate methods for analyzing free vibration of structural components. AIAA J1988; 26(5): 612–618.
39.
LivneE. Accurate calculation of control-augmented structural eigenvalue sensitivities using reduced-order models. AIAA J1989; 27(7): 947–954.
40.
CanfieldRA. High-quality approximation of eigenvalues in structural optimization. AIAA J1990; 28(6): 1116–1122.
41.
ElishakoffIZazaNCurtinJ, et al. Apparently first closed-form solution for vibration of functionally graded rotating beams. AIAA J2014; 52(11): 2587–2593.
42.
StorchJElishakoffI. Buckling of axially graded columns: a fifth-order polynomial mode shape. AIAA J2018; 56(6): 2509–2513.
43.
ElishakoffIPadillaJMeraY, et al. Seventh-order polynomial constituting the exact buckling mode of a functionally graded column. AIAA J2021; 59(11): 4318–4325.
