The free vibration and buckling of functionally graded carbon nanotube reinforced composite beams with variable thickness resting on elastic foundations are investigated in the present paper. To account rotary inertia and transverse shear deformation effects, the Timoshenko beam theory is employed and governing equations are derived using Hamilton's principle. The obtained equations are solved using generalized differential quadrature method. Different carbon nanotube distributions through the thickness are considered, and the rule of mixture is used to describe the effective material properties of the functionally graded reinforced beams. The results are validated with available investigations, and the effects of boundary conditions, nanotube volume fraction and distribution, foundation and thickness ratio on both natural frequency and buckling load are studied. Finally, due to the weight optimization in aerospace and turbomachinery applications, the optimum beam shape and nanotube distribution are suggested to achieve the most capacity of bearing axial loads with fixed weight.
KeL-LYangJKitipornchaiS. Dynamic stability of functionally graded carbon nanotube-reinforced composite beams. Mech Adv Mater Struct2013; 20: 28–37.
6.
WattanasakulpongNUngbhakornV. Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation. Comput Mater Sci2013; 71: 201–208.
7.
RafieeMYangJKitipornchaiS. Large amplitude vibration of carbon nanotube reinforced functionally graded composite beams with piezoelectric layers. Compos Struct2013; 96: 716–725.
8.
RafieeMYangJKitipornchaiS. Thermal bifurcation buckling of piezoelectric carbon nanotube reinforced composite beams. Comput Math Appl2013; 66: 1147–1160.
9.
DeValveCPitchumaniR. Analysis of vibration damping in a rotating composite beam with embedded carbon nanotubes. Compos Struct2014; 110: 289–296.
10.
NareshBBabuAAEdwinPS, et al.Free vibration analysis of a carbon nanotube reinforced composite beam. Appl Mech Mater2014; 592: 2041–2045.
11.
LinFXiangY. Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories. Appl Math Modell2014; 38: 3741–3754.
12.
TagraraSBenachourABouiadjraMB, et al.On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams. Steel Compos Struct2015; 19: 1259–1277.
13.
WuHKitipornchaiSYangJ. Free vibration and buckling analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets. Int J Struct Stab Dyn2015; 15: 1540011.
14.
AlibeiglooALiewK. Elasticity solution of free vibration and bending behavior of functionally graded carbon nanotube-reinforced composite beam with thin piezoelectric layers using differential quadrature method. Int J Appl Mech2015; 7: 1550002.
15.
EbrahimiFFarazmandniaN. Thermo-mechanical vibration analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets based on a higher-order shear deformation beam theory. Mech Adv Mater Struct2017; 24: 820–829.
16.
HasanshahiBAzadiM. Active vibration control of a functionally graded carbon nanotube-reinforced composite beam subjected to follower force. Iran J Sci Technol, Trans Mech Eng2019; 43: 25–35.
17.
PouresmaeeliSFazelzadehS. Uncertain buckling and sensitivity analysis of functionally graded carbon nanotube-reinforced composite beam. Int J Appl Mech2017; 9: 1750071.
18.
Vo-DuyTHo-HuuVNguyen-ThoiT. Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method. Front Struct Civil Eng2019; 13: 324–336.
19.
RahmaniB. Adaptive fuzzy sliding mode control for vibration suppression of a rotating carbon nanotube-reinforced composite beam. J Vib Control2018; 24: 2447–2463.
20.
ShenasAGZiaeeSMalekzadehP. A unified higher-order beam theory for free vibration and buckling of FGCNT-reinforced microbeams embedded in elastic medium based on unifying stress–strain gradient framework. Iran J Sci Technol, Trans Mech Eng2019; 43: 469–492.
21.
TornabeneFFantuzziNViolaE, et al.Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory. Compos Struct2015; 119: 67–89.
22.
TornabeneFReddyJ. FGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foundations: a GDQ solution for static analysis with a posteriori stress and strain recovery. J Indian Inst Sci2013; 93: 635–688.
23.
TrabelsiSFrikhaAZghalS, et al.A modified FSDT-based four nodes finite shell element for thermal buckling analysis of functionally graded plates and cylindrical shells. Aerosp Sci2019; 178: 444–459.
24.
ZghalSFrikhaADammakF. Mechanical buckling analysis of functionally graded power-based and carbon nanotubes-reinforced composite plates and curved panels. Compos Part B: Eng2018; 150: 165–183.
25.
TornabeneFFantuzziNBacciocchiM, et al.Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Compos Part B: Eng2016; 89: 187–218.
26.
FrikhaAZghalSDammakF, et al.Dynamic analysis of functionally graded carbon nanotubes-reinforced plate and shell structures using a double directors finite shell element. Aerosp Sci2018; 78: 438–451.
27.
TornabeneFLiveraniACaligianaG. FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: a 2-D GDQ solution for free vibrations. Int J Mech Sci2011; 53: 446–470.
28.
TornabeneF. Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Comput Meth Appl Mech Eng2009; 198: 2911–2935.
29.
HuXXSakiyamaTMatsudaH, et al.Vibration analysis of twisted conical shells with tapered thickness. Int J Eng Sci2002; 40: 1579–1598.
30.
LeeJWLeeJY. An exact transfer matrix expression for bending vibration analysis of a rotating tapered beam. Appl Math Modell2018; 53: 167–188.
31.
SoltaniMAsgarianBMohriF. Elastic instability and free vibration analyses of tapered thin-walled beams by the power series method. J Constr Steel Res2014; 96: 106–126.
32.
BazehhourBGMousaviSMFarshidianfarA. Free vibration of high-speed rotating Timoshenko shaft with various boundary conditions: effect of centrifugally induced axial force. Arch Appl Mech2014; 84: 1691–1700.
33.
ReddyJN. Mechanics of laminated composite plates and shells: theory and analysis, Boca Raton: CRC Press, 2004.
34.
TornabeneFFantuzziNUbertiniF, et al.Strong formulation finite element method based on differential quadrature: a survey. Appl Mech Rev2015; 67: 020801.
TornabeneFViolaEInmanDJ. 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures. J Sound Vib2009; 328: 259–290.
37.
AbediokhchiJShakouriMKouchakzadehMA. Bending analysis of moderately thick functionally graded conical panels with various boundary conditions using GDQ method. Compos Struct2013; 103: 68–74.
38.
AbediokhchiJKouchakzadehMAShakouriM. Buckling analysis of cross-ply laminated conical panels using GDQ method. Compos B Eng2013; 55: 440–446.
39.
ShuCChewY. On the equivalence of generalized differential quadrature and highest order finite difference scheme. Comput Meth Appl Mech Eng1998; 155: 249–260.
40.
ShuCRichardsBE. Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. Int J Numer Methods Fluids1992; 15: 791–798.
41.
YanTKitipornchaiSYangJ, et al.Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load. Compos Struct2011; 93: 2992–3001.
42.
KeL-LYangJKitipornchaiS, et al.Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials. Mech Adv Mater Struct2009; 16: 488–502.
43.
BanerjeeJRSuHJacksonDR. Free vibration of rotating tapered beams using the dynamic stiffness method. J Sound Vib2006; 298: 1034–1054.
44.
Firouz-AbadiRDHaddadpourHNovinzadehAB. An asymptotic solution to transverse free vibrations of variable-section beams. J Sound Vib2007; 304: 530–540.
45.
TimoshenkoS. Theory of elastic stability, New York: McGraw-Hill, 1961.
46.
HanYElliottJ. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput Mater Sci2007; 39: 315–323.
