Vibration of sandwich beams reinforced by carbon nanotubes under a moving load

Nội dung text: Vibration of sandwich beams reinforced by carbon nanotubes under a moving load

1. Vietnam Journal of Science and Technology 59 (5) (2021) 662-676 doi:10.15625/2525-2518/59/5/15866 VIBRATION OF SANDWICH BEAMS REINFORCED BY CARBON NANOTUBES UNDER A MOVING LOAD Thi Thom Tran1, 2, Thi Hien Trinh1, Dinh Kien Nguyen1, 2, * 1Institute of Mechanics, VAST, 18 Hoang Quoc Viet, Cau Giay, Ha Noi, Viet Nam 2Graduate University of Science and Technology, VAST, 18 Hoang Quoc Viet, Cau, Giay, Ha Noi, Viet Nam *Email: thomtt0101@gmail.com Received: 29 January 2021; Accepted for publication: 9 September 2021 Abstract. This work studies the vibration of sandwich beams reinforced with carbon nanotubes (CNTs) under moving point loads. The cores of the beams are homogeneous while their two sides are made of carbon nanotube reinforced composites. The effective properties of two face sheets are estimated through a micromechanical model. A uniform distribution (UD) and four different types of functionally graded (FG) distributions, namely FG-X, FG- , FG-V, FG-O, are considered. Based on a third-order shear deformation theory, a finite element formulation is derived and used to investigate the vibration characteristics of the beams. The effects of carbon nanotube volume fraction, carbon nanotube distribution pattern and moving load velocity on beam vibration behavior are investigated and highlighted. The influence of layer thickness and span-to-height ratio on beam vibration is also examined and discussed. Keywords: Sandwich beam; carbon nanotube reinforcement, third-order shear deformation theory; moving load; vibration analysis. Classification numbers: 2.9.4, 5.4.2, 5.4.5. 1. INTRODUCTION Carbon nanotubes (CNTs) with high strength, high stiffness, high aspect ratio and low density are excellent reinforcement for composite materials. The analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams has drawn considerable attention from researchers in recent years. Ke et al. [1, 2] investigated the nonlinear free vibration and dynamic stability of FG nano beams reinforced by single-walled carbon nanotubes (SWCNTs) using Timoshenko theory. Their results show that an increase in CNT volume fraction leads to higher natural frequencies for both uniformly distributed CNT (UD-CNT) and FG-CNTRC beams. Yas and Heshmati [3] studied free and forced vibrations of an FG nanocomposite beam reinforced by randomly straight SWCNTs under a moving load. Free vibration and buckling analysis of FG-CNTRC Timoshenko beams resting on an elastic foundation are also described by Yas and Samadi [4]. Shen and Xiang [5] presented the nonlinear bending and the thermal
2. Vibration analysis of FG-CNTRC sandwich beams under moving load postbuckling analyses of CNTRC beams. The obtained results show that the CNT volume fraction has a significant influence on the load-deflection curves of the beam. Based on Timoshenko beam theory, Ansari et al. [6] studied forced vibration of nanocomposite beams reinforced by SWCNTs. The third-order shear deformation theory was adopted by Lin and Xiang [7] in determining vibration frequencies of UD- and FG-CNTRC beams with various boundary conditions. Nejati and Eslampanah [8] employed the two dimensional (2D) elasticity theory to obtain buckling loads and natural frequencies of cantilever FG-CNTRC beams under axial load. Based on the first-order shear deformation beam theory and von Kármán nonlinearity, Wu et al. [9] investigated nonlinear vibration of imperfect shear deformable FG-CNTRC beams. Nonlinear free vibration and post-buckling of FG-CNTRC beams resting on nonlinear foundation were studied by Shafiei and Setoodeh [10]. Recently, Mohseni and Shakouri [11] investigated the free vibration and buckling of FG-CNTRC beams with variable thickness resting on elastic foundations. Using FG-CNTRCs as facing composition in sandwich constructions to increase strength and stiffness, Wu and Kitipornchai [12] investigated free vibration and elastic buckling of sandwich beams with FG-CNTRC face sheets, giving a detail on the effects of CNT volume fraction, core-to-face sheet thickness ratio, slenderness ratio, and end supports on the free vibration and buckling behavior of sandwich beams. Ebrahimi and Farzamand Nia [13] proposed a higher-order shear deformation beam theory for free vibration analysis of FG-CNTRC sandwich beams in thermal environment. The effects of carbon nanotube volume fractions, slenderness ratio and core-to-face sheet thickness ratio on the vibration of the sandwich beams have been examined. The influence of material gradation on the vibration of beams carrying a moving load has been investigated in recent years [14, 15]. It has been shown that the variation of material properties in spatial directions has a significant influence on both free and forced vibrations of the beams. This topic is further explored in the present work by studying vibration of FG sandwich beams reinforced by CNTs. The core of the sandwich beams is homogeneous while its two face sheets are a FG-CNT reinforced material. The effective properties of the two face sheets are determined by an extended rule of mixture. Five types of CNT distribution, namely UD, FG-X, FG- , FG-V, FG-O, are considered. A third-order shear deformation finite element formulation is derived and employed to compute natural frequencies and investigate dynamic response of the beams. A parametric study is carried out to highlight the effects of carbon nanotube volume fraction, the type of carbon nanotube distribution, the beam geometry and moving load velocity on dynamic behavior of the sandwich beams. 2. FG-CNTRC SANDWICH BEAM Figure 1 shows a sandwich beam with FG-CNTRC face sheets subjected to a concentrated load F0 , moving from the left end to the right end of the beam at a constant speed v. In the figure, the Cartesian coordinate system (x,z) is chosen such that the x-axis lies on the beam mid- plane. The beam consists of three layers, a homogeneous core and two face sheets of CNTRC hh material. Denoting h ,,, h h h are the vertical coordinates of the bottom surface, two 022 1 2 3 interfaces between the layers, and the top surface. Five types of distribution of CNTs in the beam cross-section, as shown in Figure 2 and given in Table 1, namely the UD, FG-X, FG- , FG-V, FG-O, are considered. 663
3. Thi ThomTran, Thi Hien Trinh, Dinh Kien Nguyen z z F v 0 h3 = h/2 FG-CNTRC h2 Homogeneous h x y h1 FG-CNTRC h0 = -h/2 b Figure 1. Geometry and coordinate system of FG-CNTRC sandwich beams under moving load Homogeneous Homogeneous Homogeneous UD FG - X FG - O 0 Homogeneous Homogeneous FG - Ʌ FG - V Figure 2. Cross-section of sandwich beam with five types of CNT distribution Table 1. Volume fraction VCNT of CNTs in face sheets of sandwich beam Distribution type Bottom face sheet ()h01 z h Top face sheet ()h23 z h * * UD VCNT VCNT |h01 h 2 z | * |h23 h 2 z | * FG-X 2 VCNT 2 VCNT hh10 hh32 zh 0 * hz3 * FG-  2 VCNT 2 VCNT hh10 hh32 hz1 * zh 2 * FG-V 2 VCNT 2 VCNT hh10 hh32 | 2z h01 h | * | 2z h23 h | * FG-O 21 VCNT 21 VCNT hh10 hh32 * In Table 1,VCNT is the total CNT volume fraction in face sheets and it is the same for the w five types of the CNT distribution; defined by V * CN , CNT CNT m CNT m wwCN // CN 664
4. Vibration analysis of FG-CNTRC sandwich beams under moving load CNT m where wCN is the mass fraction of nanotube, and are the densities of carbon nanotube and matrix, respectively. The material properties of CNTRC layers are determined according to an extended rule of mixture as [12] CNT m 2 VVVVCNTmm3 CNT EVEVE11  1CNT 11 m ;; CNT m CNT m (1) EEEGGG22 22 12 12 CNT CNT CNT In Eq. (1), EE11, 22 and G12 are Young’s moduli and shear modulus, respectively, of mm the CNT; EG, and VVm 1 CNT are Young’s modulus, shear modulus and volume fraction of matrix phase, respectively; 1,,  2  3 are the CNT efficiency parameters. The Poisson’s ratio of the FG-CNTRC face sheets are determined as CNT m 12 12 VVECNT  12 m ;  21 22 ; (2) E11 CNT m where12 , are Poisson’s ratio of the CNT and matrix, respectively. The effective elastic and shear moduli of the kth layer are calculated as follows [12] ()()kkE11 E z ; G z G12 ( k 1,3) 1 12 21 (3) EEGG(2) cc; (2) in which EGcc, are the elastic and shear moduli of the core material. The effective mass density of the kth layer is defined as (k ) CNT m (2) c (z) VCNT Vm ( k 1,3); (4) where c is mass density of the core material. 3. MATHEMATICAL FORMULATION The Shi's third-order shear deformation theory [16] is adopted herewith to formulate the governing equations for the beam. This theory is derived from an elasticity formulation, rather than displacement hypothesis, which gives better results than the first-order and other simple higher order shear deformation theories. The displacements of a point in the beam in the x and z directions, u() x,, z t and w x,,, z t respectively, are given by zz5 3 uxztuxt(,,) (,) (5 w ) ( w );(,,) wxztwxt (,) (5) 043 0,xxh2 0, 0 where u0 x, t and w0 x, t are, respectively, the displacements in the x and z directions of a point on the x-axis;  is the cross-sectional rotation, and t is the time variable. In Eq. (5) and hereafter, the subscript comma is used to indicate the derivative with respect to the variable that follows. By using the transverse shear rotation 0 , defined as 0 w 0,x [17], the axial and transverse displacements in Eq. (5) can be rewritten as follows 665
5. Thi ThomTran, Thi Hien Trinh, Dinh Kien Nguyen 55z3 uxztuxtz(,,)(,);(,,)(,) 0  0 w 0,x 2 0 wxztwxt 0 (6) 43h Eq. (6) gives the axial strain  xx and shear strains  xz in the forms 5 5z3 1 1 2 xx u0, x z  0, x w 0, xx 22  0, x;5  xz z  0 (7) 4 3hh 4 The normal and shear stresses are given by linear elastic constitutive law as ()()()()k k k k xx E(); z  xx  xz G () z  xz ( k 1,2,3) (8) Using Eqs. (7) and (8) one can write the strain energy in the form 1 L U ()    dAdx xx xx xz xz 2 0 A L 2 12 5 5 10 A110,ux 2 A 120, u x  0, x w 0, xx A 22  0, x w 0, xx 2 A 340,0, u x  x (9) 2 4 4 3h 0 10 5 2522 1 1 1 2A440,x  0, x w 0, xxx 4 A 660, 25 B 11 2 B 22 4 B 440 dx 3h 4 9 h 16 2 h h where A = bh is the cross-section area; AAA11, 12 , 66 and BBB11,, 22 44 are the beam rigidities, defined as h3 AAAAAA,,,,, bEz ()1,,,,, zzzzzdz2 3 4 6 11 12 22 34 44 66 h0 3 hk b E(k ) z 1, z , z 2 , z 3 , z 4 , z 6 dz ; (10) k 1 hk 1 h/2 3 hk BBB, , bGz ( ) 1, zzdzb2 , 4 G (k ) z 1, zzdz 2 , 4 11 22 44  k 1 hh/2 k 1 The kinetic energy of the beam resulted from Eq. (5) has the form 1 L T ef(zw) u2 2 dAdx 2 0 A L 2 1 22 55 I11 u 0 w 0 2 I 12 u 0 0 w 0,x I 22 0 w 0,x (11) 2 44 0 10 10 5 25 2 2I340ud0 2 I 440   0 w 0,x 4 I 66  0 x 3hh 3 4 9h where the over dot denotes the derivative with respect to time variable; III11, 12 , 66 are the mass moments, defined as 666
6. Vibration analysis of FG-CNTRC sandwich beams under moving load h3 I,,,,, I I I I I b ()1,,,,, z z z2 z 3 z 4 z 6 dz 11 12 22 34 44 66 h 0 (12) 3 hk b (k )( z ) 1, z , z 2 , z 3 , z 4 , z 6 dz (k 1,2,3) k 1 hk 1 The potential of the moving load (V) is simply given by L V F w x, t x vt dx 00 (13) 0 where . is the Dirac delta function, and x is the abscissa, measured from the left end of the beam. 4. FG-CNTRC SANDWICH BEAM ELEMENT Consider a two-node beam element with length l. The element vector of nodal displacements (d) contains eight components as T d duw d d  (14) where TTT du u01 u 02 ;; d w w 01 w 0 x 1 w 02 w 0 x 2 d  01 02 (15) are, respectively, the vectors of values for uw00, and  0 at nodes 1 and 2. In the above equations and hereafter, a superscript ‘T’ denotes the transpose of a vector or a matrix. Linear functions are used to interpolate the axial displacement u0 and transverse shear rotation  0 , while cubic Hermite polynomials are used for the transverse displacement w0 as uw00 Ndu ;; Hdw  0 Nd (16) where N NN12and H HHHH1 2 3 4are the matrices of the linear and Hermite shape functions. Using the above interpolations, one can write the strain energy in Eq. (9) in the form 1 ne U dkT kd with k k k k k k (17) 2  uu uw u www   where ne is the total number of elements, and k is the element stiffness matrix. In the above equation, kuu, k uw , , k are the element stiffness matrices stemming from the axial, bending, shear deformations and their couplings with the following expressions 667
7. Thi ThomTran, Thi Hien Trinh, Dinh Kien Nguyen l l l 11 l kNNkNHk TTTTA dx; A dx ; 5 NN A NN A dx ; uu , x 11 , x uw , x 12 , xx u , x 12 , x2 , x 34 , x 0 0 0 43h 0 ll 11 k HTTTA H dx; k 5 H A N H A N dx ; ww , xx 22 , xx w , xx 22 , x2 , xx 44 , x (18) 00 43h l 1 1 1 1 1 1 k 25 NTTAA N N N NTTANNN B B B dx  ,x 22 , x24 , x 44 , x ,xx66 , 1124 22 44 0 16 12hh 9 16 2hh Similarly, the kinetic energy of the beam in Eq. (11) can be rewritten as 1 ne T dT m dwith m m m m m m m (19) 2  uu uw u ww w   where m is the element mass matrix. The expressions for the sub-matrices in the above equation are as follows l l l 11 mNNm TTTTI dx; NHm I dx ; 5 NNNN I I dx ; uu 11 uw 12 , x u 122 34 0 0 0 43h ll 11 m HTTTTI H H I H dx; m 5 H I N H I N dx ; (20) ww 11 , x 22 , x w , x 222 , x 44 00 43h l 1 1 1 m 25 NT I I I N dx  2224 44 66 0 16 2hh The discrete equation of motion for the dynamic analysis of the beam can be written in the form MD KD Fex (21) where M and K are, respectively, the global mass and stiffness matrices, obtained by assembling the matrices m and k over the elements; D and D are, respectively, the vectors of nodal displacements and accelerations; Fex is the vector of the nodal external force with the following form: ne Ffex  ex , (22) ex T wheref 0 0FHFHFHFH0 1 0 2 0 3 0 4 0 0 is the element nodal force vector. Except for ex the element under the load F0 , the element nodal force vector f is zero for all other elements, and the interpolation functions Hii 1, ,4 are evaluated at the current position of the force F0 . The system of Eq. (21) can be solved by the Newmark method. The average acceleration method which ensures the numerically unconditional stability is adopted herein. 5. NUMERICAL RESULTS AND DISCUSSION A FG-CNTRC sandwich beam with simply supported ends is considered in the numerical investigation in this section. Poly-methyl methacrylate (PMMA) with material properties as Em 2.5GPa,  m 1190 kg/ m3 , m 0.3 is chosen for the matrix phase; the armchair (10,10) SWCNTs with 668
8. Vibration analysis of FG-CNTRC sandwich beams under moving load CNT CNT CNT CNT 3 E11 5.6466TPa, EG22 7.08TPa, 12 1.9445TPa, 1400kg/m , and CNT  0.175are selected as reinforcements. The CNT efficiency parameters i are determined by matching Young's moduli EE11, 22 and shear modulus G12 of CNTRCs obtained from the rule of mixture given by Han and Elliott [18]. For example 1,  2 ,  3 0.137,1.022,0.715 for * * VCNT 0.12; 1,  2 ,  3 0.142,1.626,1.138 for VCNT 0.17; 1,  2 ,  3 0.141,1.585,1.109 * c c3 c for VCNT 0.28. Titanium alloy (Ti-6Al-4V) with E 113.8GPa,  4430 kg/m , 0.342 is chosen as the core material of the sandwich beam. The total thickness of the sandwich beam is 0.01 m. The ratio of homogeneous core thickness to face sheet thickness is defined by hhcf/ . The fundamental frequency parameter is defined as  LIA110/, 110 (23) where A110 and I110 are the values of A11 and I11 of a homogeneous beam made from pure core material, and  is the fundamental frequency. A uniform increment time step tT /200with T as the total time necessary for the load crossing the beam, is used for the Newmark procedure. For the convenience of discussion, the following dynamic magnification factor Dd is introduced w L/2, t Dd max , (24) wst 3 c where wst L F0 /48 E I is the static deflection of a pure Ti-6Al-4V beam under the load F 100 kN, acting at the mid-span. 0 5.1. Formulation verification Before investigating the vibration characteristics of the beam, the accuracy of the finite element model is firstly confirmed. Table 2. Comparison of frequency parameter of sandwich beams with hhcf/ 8, L/h=20 Distribution Source * * * VCNT 0.12 VCNT 0.17 VCNT 0.28 Wu and Kitipornchai [12] 0.1453 0.1588 0.1825 FG-V Present 0.1406 0.1545 0.1790 Wu and Kitipornchai [12] 0.1432 0.1560 0.1785 UD Present 0.1384 0.1517 0.1749 * Table 3. Comparison of frequency parameter of sandwich beams with L/h=20, VCNT 0.17 Distribution Source hhcf/8 hhcf/6 hhcf/4 Wu and Kitipornchai [12] 0.1588 0.1642 0.1743 FG-V Present 0.1545 0.1608 0.1720 Wu and Kitipornchai [12] 0.1560 0.1599 0.1668 UD Present 0.1517 0.1563 0.1643 669
9. Thi ThomTran, Thi Hien Trinh, Dinh Kien Nguyen * Table 4. Comparison of frequency parameter of sandwich beams with VCNT 0.17,hh c / f 8 Distribution Source L/h=10 L/h=20 L/h=30 Wu and Kitipornchai [12] 0.3124 0.1588 0.1062 FG-V Present 0.3119 0.1580 0.1056 Wu and Kitipornchai [12] 0.3070 0.1560 0.1043 UD Present 0.2995 0.1517 0.1014 Tables 2-4 compare the fundamental frequency parameter of the FG-CNTRC sandwich beam with the results of Wu and Kitipornchai [12], where the differential quadrature method has been employed. The frequency parameter is received for sandwich beam with two types of CNT distribution named as FG-V and UD. Very good agreement between the frequency parameter of the present work with that reported in [12] is obtained from Tables 2-4, regardless of the total * CNT volume fraction VCNT , ratio hhcf/ and aspect ratio L/h, noting that a Timoshenko beam theory is used to formulate governing equations in [12]. 5.2. Natural frequencies Table 5 lists fundamental frequency parameter of FG-CNTRC sandwich beam for five different types of CNT distribution. As seen from the Table, the frequency parameter  * increases with increasing the total CNTs volume fraction VCNT , especially more significantly increases for a smaller hhcf/ ratio. Table 5. The fundamental frequency parameter of sandwich beam for different hhcf/ and L/h ratios hhcf/8 hhcf/6 hhcf/4 L/h Type * VCNT 0.12 0.17 0.28 0.12 0.17 0.28 0.12 0.17 0.28 UD 0.0924 0.1013 0.1169 0.0937 0.1045 0.1230 0.0963 0.1098 0.1326 FG-X 0.0925 0.1014 0.1170 0.0938 0.1046 0.1231 0.0965 0.1101 0.1329 30 FG-O 0.0924 0.1013 0.1169 0.0936 0.1044 0.1228 0.0961 0.1096 0.1322 FG-V 0.0939 0.1033 0.1197 0.0960 0.1074 0.1271 0.1003 0.1150 0.1395 FG-  0.0910 0.0993 0.1045 0.0914 0.1014 0.1187 0.0921 0.1045 0.1253 UD 0.1384 0.1516 0.1749 0.1403 0.1563 0.1838 0.1441 0.1643 0.1982 FG-X 0.1384 0.1517 0.1750 0.1404 0.1565 0.1841 0.1444 0.1647 0.1987 20 FG-O 0.1383 0.1515 0.1748 0.1402 0.1561 0.1836 0.1438 0.1639 0.1976 FG-V 0.1406 0.1545 0.1790 0.1437 0.1608 0.1900 0.1501 0.1720 0.2085 FG- 0.1362 0.1487 0.1707 0.1368 0.1517 0.1775 0.1379 0.1517 0.1873 UD 0.2736 0.2994 0.3443 0.2774 0.3085 0.3616 0.2850 0.3242 0.3891 FG-X 0.2737 0.2995 0.3445 0.2777 0.3089 0.3621 0.2856 0.3250 0.3903 10 FG-O 0.2735 0.2992 0.3440 0.2771 0.3082 0.3611 0.2844 0.3234 0.3880 FG-V 0.2779 0.3051 0.3523 0.2841 0.3173 0.3736 0.2968 0.3392 0.4091 FG- 0.2692 0.2935 0.3360 0.2705 0.2995 0.3492 0.2727 0.3084 0.3680 UD 0.5243 0.5708 0.6500 0.5318 0.5878 0.6806 0.5465 0.6165 0.7282 5 FG-X 0.5246 0.5712 0.6506 0.5324 0.5886 0.6818 0.5478 0.6183 0.7308 FG-O 0.5240 0.5704 0.6495 0.5313 0.5871 0.6795 0.5451 0.6146 0.7256 670
10. Vibration analysis of FG-CNTRC sandwich beams under moving load FG-V 0.5325 0.5816 0.6650 0.5445 0.6042 0.7027 0.5686 0.6442 0.7640 FG-  0.5160 0.5597 0.6347 0.5188 0.5709 0.6577 0.5233 0.5871 0.6899 Of the five types of the CNT distribution, the FG-V sandwich beam has the highest frequency, while the FG-  sandwich beam gives the smallest result. It is easy to see that the results obtained for beam with UD, FG-X, FG-O distributions are very close together. Moreover, the decrease of the ratio hhcf/ leads to the increase in the frequency parameter, especially at smaller values of the aspect ratio L/h. This is because the sandwich beam gets higher stiffness corresponding to a smaller ratio hhcf/. Besides, the influence of the aspect ratio L/h on the frequency parameter  is also seen in the Table 5, where the frequency parameter increases with decreasing aspect ratio L/h. 5.3. Forced vibration Figure 3 shows the time histories for dimensionless mid-span deflection of the sandwich beam for three types of CNT distribution, namely FG-V, UD and FG- . The figure is plotted * with an aspect ratio Lh/ 20, a total CNTs volume fraction VCNT 0.17, two values of the ratio hc/ h f 4, h c / h f 8 and two values of the moving load velocity, v = 20 m/s, v = 100 m/s. Figure 3. Time histories for dimensionless mid-span transverse displacement for different types of * CNT distribution (L/h = 20,VCNT 0.17 ) 671
11. Thi ThomTran, Thi Hien Trinh, Dinh Kien Nguyen The difference of three types of CNT distribution is clearly seen from the figure, especially for the smaller values of hhcf/ (Figure 3a, c). The mid-span deflection of the sandwich beams corresponding to the distribution FG-V of CNT is the smallest, while that of the sandwich beam with FG-  type distribution is the highest. For both values of the hhcf/ ratio, the mid-span deflection increases with increasing moving load velocity. In addition, the sandwich beam is subjected to more vibration cycles when it is under load with lower moving velocities. This can be explained by the lower ratio of the moving load speed to the critical speed as in case of the isotropic beams [19]. As can be observed from Figure 3, the mid-span deflection of the FG-V beam insignificantly increases when increasing the hhcf/ ratio, regardless of the moving velocity. Table 6. Dynamic magnification factors for L/h = 20 and at different moving load velocities and hhcf/ ratios v (m/s) Type hhcf/8 hhcf/6 hhcf/4 * VCNT 0.12 0.17 0.28 0.12 0.17 0.28 0.12 0.17 0.28 20 UD 1.1778 0.9509 0.7275 1.1925 0.9419 0.6827 1.2102 0.9284 0.6216 FG-X 1.1766 0.9496 0.7264 1.1900 0.9401 0.6808 1.2040 0.9241 0.6177 FG-O 1.1790 0.9522 0.7284 1.1949 0.9436 0.6846 1.2162 0.9327 0.6256 FG-V 1.1367 0.9198 0.6929 1.1285 0.8958 0.6345 1.0993 0.8492 0.5669 FG-  1.2213 0.9969 0.7643 1.2622 0.9916 0.7363 1.3396 1.0171 0.7070 50 UD 1.2070 0.9905 0.7892 1.2138 0.9913 0.7506 1.2162 0.9907 0.6965 FG-X 1.2055 0.9898 0.7882 1.2107 0.9897 0.7487 1.2089 0.9868 0.6925 FG-O 1.2085 0.9912 0.7901 1.2167 0.9928 0.7523 1.2234 0.9946 0.7005 FG-V 1.1558 0.9649 0.7564 1.1357 0.9502 0.7036 1.1327 0.9173 0.6247 FG- 1.2621 1.0164 0.8239 1.3010 1.0335 0.8019 1.3753 1.0703 0.7810 100 UD 1.7810 1.4380 1.0152 1.8025 1.3967 0.9341 1.8286 1.3357 0.8311 FG-X 1.7791 1.4361 1.0134 1.7987 1.3932 0.9311 1.8194 1.3278 0.8251 FG-O 1.7828 1.4399 1.0168 1.8061 1.4002 0.9370 1.8378 1.3437 0.8370 FG-V 1.7180 1.3743 0.9576 1.7053 1.3052 0.8587 1.6619 1.1938 0.7281 FG- 1.8479 1.5068 1.0785 1.9093 1.4996 1.0211 2.0258 1.5083 0.9611 Table 7. Dynamic magnification factors for L/h = 5 and at different moving load velocities and hhcf/ ratios v (m/s) Type hhcf/8 hhcf/6 hhcf/4 0.12 0.17 0.28 0.12 0.17 0.28 0.12 0.17 0.28 20 UD 1.2313 1.0400 0.8053 1.2506 1.0266 0.7675 1.2844 1.0090 0.7259 FG-X 1.2297 1.0384 0.8036 1.2479 1.0239 0.7647 1.2775 1.0024 0.7205 FG-O 1.2328 1.0416 0.8068 1.2533 1.0295 0.7700 1.2911 1.0159 0.7313 672
12. Vibration analysis of FG-CNTRC sandwich beams under moving load FG-V 1.1933 1.0015 0.7680 1.1955 0.9721 0.7204 1.1842 0.9253 0.6593 FG-  1.2741 1.0839 0.8442 1.3166 1.0876 0.8097 1.3983 1.1137 0.8097 50 UD 1.2683 1.0724 0.8245 1.2914 1.0514 0.7873 1.3255 1.0224 0.7336 FG-X 1.2669 1.0707 0.8229 1.2886 1.0481 0.7842 1.3183 1.0171 0.7274 FG-O 1.2697 1.0741 0.8259 1.2940 1.0546 0.7902 1.3325 1.0276 0.7400 FG-V 1.2323 1.0292 0.7887 1.2339 0.9871 0.7350 1.2215 0.9464 0.6731 FG- 1.3052 1.1180 0.8616 1.3514 1.1216 0.8438 1.4395 1.1408 0.8285 100 UD 1.2662 1.1049 0.8656 1.2923 1.0982 0.8178 1.3409 1.0884 0.7540 FG-X 1.2644 1.1035 0.8638 1.2903 1.0955 0.8145 1.3356 1.0817 0.7472 FG-O 1.2679 1.1063 0.8673 1.2944 1.1009 0.8209 1.3462 1.0951 0.7607 FG-V 1.2338 1.0697 0.8238 1.2467 1.0446 0.7590 1.2568 0.9959 0.6741 FG- 1.3190 1.1416 0.9100 1.3599 1.1553 0.8830 1.4392 1.1908 0.8580 Figure 4. Variation of dynamic magnification factor with type of CNT distribution (L/h=20) Tables 6-7 present the dynamic magnification factor Dd of FG-CNTRC sandwich beam for five different types of CNT distribution and different moving load velocities. The aspect ratios L/h = 20 and L/h = 5 are respectively used in the computation in two tables. It is observed from the tables that the factor decreases with increasing the total CNTs volume fraction * VCNT , regardless of the CNT distribution. This decrease is observed more clearly when the moving load velocity is greater and the hhcf/ ratio is smaller. Similar to the frequency parameter, the factor obtained for three types of CNT distribution, UD, FG-X and FG-O, are 673
13. Thi ThomTran, Thi Hien Trinh, Dinh Kien Nguyen quite close together, while the factors Dd received from FG-V and FG-  distributions are the smallest and highest, respectively. In the Figure 4, the relation between the dynamic magnification factor and the moving load speed v is illustrated for FG-V, UD and FG-  sandwich beams. Two values of the total * * CNTs volume fraction VCNT 0.12, VCNT 0.28, and two values of the ratio * hc/ h f 4 and h c / h f 8 are chosen to plot the figure. Once again, we can see that when VCNT is higher and the hhcf/ ratio is smaller, the factor received from the beams corresponding to the three types of CNT distribution is markedly different. Moreover, for the beams with * VCNT 0.28, the factor significantly increases with increasing the hhcf/ ratio and this is clearly observed for all FG-V, UD and FG- sandwich beams. Especially, the dynamic magnification factor achieves the maximum value at a higher moving load speed for the * beam having a lower hhcf/ ratio and higherVCNT . 6. CONCLUSIONS Vibration of FG-CNTRC sandwich beams under a moving load with five different types of CNT distribution has been presented. The sandwich beam consists of three layers, a homogeneous core and two face sheets of CNTRC materials. The effective properties of two CNTRC faces are determined by the extended rule of mixture. Based on the third-order shear deformation theory, a finite element formulation has been derived and employed to investigate vibration characteristics of the beam. The obtained numerical results show that CNT distribution has an important influence on the vibration behavior of the beam. An increase of the total volume fraction of CNTs leads to an increase in the frequency parameter and a decrease in the dynamic magnification factor, and the effect of CNT volume fraction is more significant for the beam with a smaller hhcf/ ratio. This study also shows that among the five types of the CNT distribution, the sandwich beam with FG-V type of CNT distribution has the lowest dynamic magnification factor while its fundamental frequency is the highest. Noting that although the numerical investigation presented here is only for simply supported beams, the formulation derived in the present work can also be used for FG-CNTRC sandwich beam analysis with other boundary conditions. CRediT authorship contribution statement. Dr. Thi Thom Tran developed the computer code used in analysis and prepared the manuscript. Master Thi Hien Trinh performed the computations. Dr. Dinh Kien Nguyen set up the problem, checked the results and final manuscript. Declaration of competing interest. We declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. REFERENCES 1. Ke L. L., Yang J. and Kitipornchai S. - Nonlinear free vibration of functionally graded carbon nanotubereinforced composite beams, Compos. Struct. 92 (2010) 676-683. 2. Ke L. -L., Yang J. and Kitipornchai S. - Dynamic stability of functionally graded carbon nanotube reinforced composite beams, Mech. Adv. Mater. Struct. 20 (2013) 28-37. 674
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