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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (2) : 324-336     https://doi.org/10.1007/s11709-018-0466-6
RESEARCH ARTICLE |
Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method
T. VO-DUY1,2, V. HO-HUU1,2, T. NGUYEN-THOI1,2()
1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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Abstract

In the present study, the free vibration of laminated functionally graded carbon nanotube reinforced composite beams is analyzed. The laminated beam is made of perfectly bonded carbon nanotubes reinforced composite (CNTRC) layers. In each layer, single-walled carbon nanotubes are assumed to be uniformly distributed (UD) or functionally graded (FG) distributed along the thickness direction. Effective material properties of the two-phase composites, a mixture of carbon nanotubes (CNTs) and an isotropic polymer, are calculated using the extended rule of mixture. The first-order shear deformation theory is used to formulate a governing equation for predicting free vibration of laminated functionally graded carbon nanotubes reinforced composite (FG-CNTRC) beams. The governing equation is solved by the finite element method with various boundary conditions. Several numerical tests are performed to investigate the influence of the CNTs volume fractions, CNTs distributions, CNTs orientation angles, boundary conditions, length-to-thickness ratios and the numbers of layers on the frequencies of the laminated FG-CNTRC beams. Moreover, a laminated composite beam combined by various distribution types of CNTs is also studied.

Keywords free vibration analysis      laminated FG-CNTRC beam      finite element method      first-order shear deformation theory      composite material     
Corresponding Authors: T. NGUYEN-THOI   
Just Accepted Date: 30 January 2018   Online First Date: 29 March 2018    Issue Date: 12 March 2019
 Cite this article:   
T. VO-DUY,V. HO-HUU,T. NGUYEN-THOI. Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method[J]. Front. Struct. Civ. Eng., 2019, 13(2): 324-336.
 URL:  
http://journal.hep.com.cn/fsce/EN/10.1007/s11709-018-0466-6
http://journal.hep.com.cn/fsce/EN/Y2019/V13/I2/324
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T. VO-DUY
V. HO-HUU
T. NGUYEN-THOI
VCNT(z)=V CNT* UD
VCNT(z )=(1 +2zh)VCNT* FG-V
VCNT(z )=(1 2zh)VCNT* FG-Λ
VCNT(z )=4 |z| hVCNT* FG-X
VCNT(z )=2( 1 2|z|h )V CNT* FG-O
Tab.1  
Fig.1  Sketch of a laminated composite beam
parameters (unit) matrix fiber
Poisson’s coefficient nm = 0.3 v12CNT = 0.19
mass density (kg/m3) rm = 1190 rCNT = 17.2
Yong’s modulus (GPa) Em = 2.5 E 11CNT = 600, E22CNT = 600
shear modulus (GPa) G12CNT = 10
Tab.2  Material parameters of a FG-CNTRC beam
V CNT* BC mode FG-X UD FG-V FG-O
FEM (present) GDQM [2] FEM (present) GDQM [2] FEM (present) GDQM [2] FEM (present) GDQM [2]
0.12 CC 1 1.5953 1.6000 1.5052 1.5085 1.4046 1.4068 1.3166 1.3180
2 3.2568 3.2629 3.1317 3.1353 2.9980 2.9997 2.8763 2.8762
3 5.1517 5.1514 5.0022 4.9979 4.8433 4.8363 4.6940 4.6840
CH 1 1.3547 1.3577 1.2426 1.2444 1.1518 1.1529 1.0327 1.0331
2 3.1768 3.1817 3.0137 3.0159 2.8468 2.8472 2.6827 2.6814
3 5.1103 5.1092 4.9393 4.9342 4.7556 4.7474 4.5731 4.5619
HH 1 1.1139 1.1150 0.9748 0.9753 0.9451 0.9453 0.7529 0.7527
2 3.0780 3.0814 2.8722 2.8728 2.6436 2.6424 2.4588 2.4562
3 5.0713 5.0695 4.8765 4.8704 4.6768 4.6675 4.4445 4.4320
CF 1 0.4411 0.4416 0.3761 0.3764 0.3192 0.3193 0.2808 0.2809
2 1.8461 1.8497 1.6984 1.7006 1.5460 1.5473 1.4260 1.4266
3 3.8743 3.8777 3.6643 3.6648 3.4393 3.4380 3.2519 3.2489
0.17 CC 1 2.0409 2.0498 1.9083 1.9144 1.7677 1.7721 1.6471 1.6500
2 4.1962 4.2111 4.0088 4.0187 3.8242 3.8312 3.6527 3.6565
3 6.6638 6.6753 6.4310 6.4348 6.2143 6.2139 6.0025 5.9970
CH 1 1.7131 1.7188 1.5567 1.5602 1.4321 1.4344 1.2757 1.2769
2 4.0718 4.0843 3.8328 3.8402 3.6020 3.6064 3.3758 3.3772
3 6.5991 6.6094 6.3347 6.3370 6.0789 6.0765 5.8204 5.8126
HH 1 1.3808 1.3830 1.1989 1.1999 1.1601 1.1609 0.9155 0.9155
2 3.9201 3.9293 3.6235 3.6276 3.3075 3.3084 3.0591 3.0577
3 6.5365 6.5447 6.2367 6.2363 5.9550 5.9498 5.6247 5.6139
CF 1 0.5406 0.5413 0.4583 0.4587 0.3863 0.3866 0.3393 0.3394
2 2.3364 2.3437 2.1319 2.1365 1.9258 1.9287 1.7669 1.7685
3 4.9590 4.9706 4.6554 4.6614 4.3471 4.3500 4.0915 4.0913
Tab.3  Comparison of dimensionless frequency parameters, ω=ω L2 h ρmEm of FG-CNTRC beams for V CNT* = 0.12 and 0.17
BC mode FG-X FG-UD FG-Λ
FEM (present) GDQM [2] p-Ritz [34] FEM (present) GDQM [2] p-Ritz [34] FEM (present) GDQM [2] p-Ritz [34]
HH 1 1.6423 1.6493 1.6409 1.4362 1.4401 1.4348 1.3990 1.4027 1.3975
2 4.4443 4.4752 4.4333 4.1162 4.1362 4.1050 3.8487 3.8639 3.8370
3 7.2596 7.3068 7.2258 6.8940 6.9245 6.8595 6.7349 6.7618 6.6976
CF 1 0.6566 0.6586 0.6566 0.5601 0.5612 0.5600 0.4754 0.4761 0.4753
2 2.6797 2.6987 2.6763 2.4482 2.4614 2.4449 2.2578 2.2685 2.2543
3 5.5759 5.6150 5.5589 5.2175 5.2446 5.2005 4.9767 5.0007 4.9590
Tab.4  Comparison of dimensionless frequency parameters, ω=ω L2 h ρmEm of FG-CNTRC beams for VCNT * = 0.28
BC method frequency
15° 30° 45° 60° 75° 90°
CF FSDT [43] 0.9820 0.9249 0.7678 0.5551 0.3631 0.2723 0.2619
HSDT [50] 0.9832 0.9259 0.7683 0.5553 0.3631 0.2722 0.2618
present 0.9821 0.925 0.7679 0.5552 0.3632 0.2724 0.2619
HH FSDT [43] 2.6560 2.5105 2.1032 1.5368 1.0124 0.7611 0.7320
HSDT [50] 2.6563 2.5108 2.1033 1.5367 1.0121 0.7608 0.7317
present 2.6589 2.5133 2.1056 1.5386 1.0136 0.7620 0.7329
CH FSDT [43] 3.7305 3.5593 3.0573 2.3032 1.5511 1.1753 1.1312
present 3.7362 3.565 3.0625 2.3075 1.5541 1.1776 1.1335
CC FSDT [43] 4.8487 4.6635 4.0981 3.1843 2.1984 1.6815 1.6200
HSDT [50] 4.9116 4.7173 4.1307 3.1973 2.2019 1.6825 1.6205
present 4.8577 4.6725 4.1069 3.1922 2.2045 1.6862 1.6244
Tab.5  Comparison of dimensionless frequency parameter ω=ω L2 h ρE11 of four-layered angle-ply [q/–q/–q/q] beams
BC mode FG-X UD FG-V FG-O
CC 1 1.0433 0.9373 0.8185 0.7923
2 2.4475 2.2568 2.0305 1.9750
3 4.1450 3.8868 3.5747 3.4902
CH 1 0.8319 0.7506 0.6294 0.6443
2 2.2198 2.0285 1.7828 1.7613
3 3.9548 3.6757 3.3259 3.2623
HH 1 0.7117 0.6576 0.5191 0.5850
2 1.8890 1.6831 1.4651 1.4086
3 3.8043 3.5198 3.1156 3.1153
CF 1 0.1939 0.1679 0.1415 0.1361
2 1.0830 0.9614 0.8306 0.8011
3 2.6458 2.4040 2.1320 2.0635
Tab.6  First three dimensionless frequencies of cross-ply [0°/90°] FG-CNTRC beams
BC Mode FG-X UD FG-V FG-O
CC 1 1.5074 1.4961 1.4920 1.4885
2 3.1376 3.1195 3.1170 3.1123
3 5.0127 4.9876 4.9880 4.9824
CH 1 1.2436 1.2317 1.2269 1.2211
2 3.0184 2.9976 2.9917 2.9850
3 4.9491 4.9223 4.9205 4.9137
HH 1 0.9747 0.9621 0.9602 0.9486
2 2.8755 2.8516 2.8406 2.8330
3 4.8856 4.8567 4.8532 4.8443
CF 1 0.3758 0.3703 0.3663 0.3642
2 1.6999 1.6842 1.6760 1.6707
3 3.6698 3.6437 3.6350 3.6273
Tab.7  First three dimensionless frequencies of a cross-ply [0°/90°/0°] FG-CNTRC beams
BC mode frequency
15° 30° 45° 60° 75° 90°
CC 1 1.5147 1.4807 1.3610 1.1086 0.7332 0.4749 0.4454
2 3.1475 3.1019 2.9388 2.5600 1.8553 1.2603 1.1870
3 5.0244 4.9699 4.7711 4.2954 3.3204 2.3617 2.2349
CH 1 1.2525 1.2120 1.0789 0.8336 0.5235 0.3319 0.3107
2 3.0315 2.9713 2.7608 2.3090 1.5833 1.0442 0.9807
3 4.9628 4.8990 4.6644 4.1056 3.0377 2.0942 1.9753
HH 1 0.9851 0.9382 0.7976 0.5782 0.3452 0.2149 0.2009
2 2.8921 2.8156 2.5534 2.0319 1.3121 0.8413 0.7883
3 4.9015 4.8272 4.5520 3.8996 2.7437 1.8317 1.7223
CF 1 0.3806 0.3596 0.2992 0.2115 0.1241 0.0768 0.0718
2 1.7115 1.6589 1.4862 1.1628 0.7394 0.4711 0.4412
3 3.6864 3.6100 3.3476 2.8006 1.9308 1.2780 1.2008
Tab.8  First three dimensionless frequencies of four-layered angle-ply [q/–q/–q/q] FG-X beam
BC mode frequency
15° 30° 45° 60° 75° 90°
CC 1 1.5052 1.4710 1.3507 1.0985 0.7265 0.4729 0.4441
2 3.1317 3.0858 2.9216 2.5414 1.8402 1.2550 1.1836
3 5.0022 4.9471 4.7468 4.2687 3.2962 2.3521 2.2283
CH 1 1.2426 1.2021 1.0690 0.8250 0.5185 0.3305 0.3098
2 3.0137 2.9532 2.7417 2.2897 1.5695 1.0397 0.9778
3 4.9393 4.8749 4.6383 4.0771 3.0139 2.0855 1.9695
HH 1 0.9748 0.9282 0.7886 0.5715 0.3417 0.2139 0.2003
2 2.8722 2.7952 2.5323 2.0122 1.2998 0.8377 0.7859
3 4.8765 4.8015 4.5238 3.8692 2.7203 1.8239 1.7172
CF 1 0.3761 0.3553 0.2956 0.2090 0.1229 0.0765 0.0716
2 1.6984 1.6457 1.4731 1.1511 0.7324 0.4691 0.4399
3 3.6643 3.5874 3.3242 2.7773 1.9140 1.2726 1.1973
Tab.9  First three dimensionless frequencies of four-layered angle-ply [q/–q/–q/q] UD-CNTRC beam
BC mode frequency
15° 30° 45° 60° 75° 90°
CC 1 1.5024 1.4677 1.346 1.0928 0.7235 0.4733 0.4448
2 3.1309 3.0843 2.9177 2.5339 1.8345 1.2564 1.1855
3 5.0047 4.9487 4.7451 4.2618 3.2890 2.3550 2.2322
CH 1 1.2390 1.1981 1.0642 0.8203 0.5164 0.3308 0.3103
2 3.0100 2.9486 2.7348 2.2803 1.5638 1.0408 0.9794
3 4.9400 4.8743 4.6339 4.0670 3.0053 2.0879 1.9729
HH 1 0.9733 0.9266 0.7870 0.5703 0.3413 0.2142 0.2006
2 2.8641 2.7862 2.5209 2.0001 1.2939 0.8385 0.7872
3 4.8757 4.7992 4.5170 3.8566 2.7116 1.8259 1.7201
CF 1 0.3727 0.3520 0.2928 0.2071 0.1222 0.0766 0.0717
2 1.6921 1.6391 1.4656 1.1438 0.7291 0.4695 0.4406
3 3.6584 3.5807 3.3149 2.7655 1.9069 1.2739 1.1992
Tab.10  First three dimensionless frequencies of four-layered angle-ply [q/–q/–q/q] FG-V beam
BC mode frequency
15° 30° 45° 60° 75° 90°
CC 1 1.4997 1.4647 1.3425 1.0888 0.7199 0.4721 0.4442
2 3.1273 3.0804 2.9129 2.5273 1.8269 1.2534 1.1841
3 5.0004 4.9440 4.7392 4.2533 3.2775 2.3498 2.2297
CH 1 1.2344 1.1934 1.0594 0.8158 0.5133 0.3299 0.3098
2 3.0048 2.9431 2.7283 2.2724 1.5563 1.0381 0.9782
3 4.9347 4.8686 4.6267 4.0570 2.9935 2.0829 1.9705
HH 1 0.9639 0.9174 0.7785 0.5638 0.3381 0.2135 0.2003
2 2.8581 2.7799 2.5137 1.9924 1.2875 0.8362 0.7861
3 4.8688 4.7917 4.5075 3.8440 2.6988 1.8213 1.7179
CF 1 0.3710 0.3504 0.2913 0.2060 0.1215 0.0763 0.0716
2 1.6879 1.6348 1.4609 1.1390 0.7252 0.4683 0.4400
3 3.6524 3.5744 3.3076 2.7568 1.8982 1.2707 1.1977
Tab.11  First three dimensionless frequencies of four-layered angle-ply [q/–q/–q/q] FG-O beam
Fig.2  The first dimensionless frequency of angle-ply [q/–q/–q/q] FG-X beam with boundary condition of CC
Fig.3  The first dimensionless frequency of the angle-ply [45°/–45°/–45°/45°] FG-CNTRC beam with various distributions and boundary conditions for the case of VCNT * = 0.17
Fig.4  The first dimensionless frequency of angle-ply [45°/–45°/–45°/45°] FG-X beam with boundary condition of CC versus various length-to-thickness ratios
distribution BC mode 2 layers 3 layers 5 layers 10 layers
FG-X CC 1 1.5349 1.5202 1.5121 1.5086
2 3.1746 3.1549 3.1440 3.1392
3 5.0565 5.0332 5.0203 5.0146
CH 1 1.2772 1.2591 1.2493 1.2451
2 3.0675 3.0413 3.0269 3.0206
3 5.0002 4.9730 4.9580 4.9514
HH 1 1.0148 0.9930 0.9814 0.9764
2 2.9381 2.9046 2.8862 2.8782
3 4.9448 4.9133 4.8958 4.8882
CF 1 0.3942 0.3842 0.3789 0.3766
2 1.7437 1.7201 1.7074 1.7018
3 3.7325 3.6989 3.6805 3.6725
FG-V CC 1 1.4866 1.4984 1.5042 1.5066
2 3.1095 3.1256 3.1334 3.1366
3 4.9790 4.9983 5.0076 5.0114
CH 1 1.2243 1.2353 1.2407 1.2429
2 2.9832 3.0033 3.0131 3.0171
3 4.9107 4.9327 4.9433 4.9477
HH 1 0.9689 0.9722 0.9738 0.9745
2 2.8279 2.8551 2.8683 2.8737
3 4.8439 4.8677 4.8793 4.8840
CF 1 0.3630 0.3702 0.3739 0.3754
2 1.6675 1.6859 1.6949 1.6987
3 3.6224 3.6494 3.6626 3.6680
FG-O CC 1 1.4742 1.4934 1.5025 1.5062
2 3.0932 3.1189 3.1311 3.1360
3 4.9594 4.9903 5.0049 5.0108
CH 1 1.2044 1.2270 1.2378 1.2422
2 2.9598 2.9937 3.0098 3.0163
3 4.8867 4.9230 4.9400 4.9469
HH 1 0.9297 0.9553 0.9678 0.9730
2 2.8011 2.8441 2.8645 2.8728
3 4.8128 4.8552 4.8749 4.8830
CF 1 0.3558 0.3672 0.3728 0.3751
2 1.6490 1.6783 1.6923 1.6980
3 3.5955 3.6384 3.6588 3.6671
Tab.12  First three dimensionless frequencies of a beam with various numbers of layers
Fig.5  First dimensionless frequencies of the clamped-clamped FG-CNTRC beams with various numbers of layers for the case of VCNT * = 0.12
BC mode UD-X-UD X-UD-X V-X-V X-V-X Λ-X-V
CC 1 1.5105 1.5151 1.5079 1.5151 1.5688
2 3.1398 3.1470 3.1384 3.1480 3.2206
3 5.0129 5.0227 5.0135 5.0250 5.1102
CH 1 1.2484 1.2535 1.2453 1.2532 1.3201
2 3.0234 3.0319 3.0197 3.0323 3.1286
3 4.9511 4.9617 4.9503 4.9636 5.0624
HH 1 0.9812 0.9868 0.9798 0.9867 1.0685
2 2.8836 2.8936 2.8767 2.8930 3.0163
3 4.8894 4.9009 4.8875 4.9025 5.0165
CF 1 0.3789 0.3814 0.3762 0.3808 0.4193
2 1.7061 1.7127 1.7008 1.7121 1.8000
3 3.6764 3.6871 3.6710 3.6873 3.8114
Tab.13  First three dimensionless frequencies of laminated FG-CNTRC beams with various distributions of layers
1 C HSun, F Li, H MCheng, G QLu. Axial Young’s modulus prediction of single-walled carbon nanotube arrays with diameters from nanometer to meter scales. Applied Physics Letters, 2005, 87(19): 193101
https://doi.org/10.1063/1.2119409
2 M HYas, N Samadi. Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation. International Journal of Pressure Vessels and Piping, 2012, 98: 119–128 doi:10.1016/j.ijpvp.2012.07.012
3 SJedari Salami. Extended high order sandwich panel theory for bending analysis of sandwich beams with carbon nanotube reinforced face sheets. Physica E, Low-Dimensional Systems and Nanostructures, 2016, 76: 187–197
https://doi.org/10.1016/j.physe.2015.10.015
4 Z XLei, L W Zhang, K M Liew. Analysis of laminated CNT reinforced functionally graded plates using the element-free kp-Ritz method. Composites. Part B, Engineering, 2016, 84: 211–221
https://doi.org/10.1016/j.compositesb.2015.08.081
5 L WZhang, Z G Song, K M Liew. Optimal shape control of CNT reinforced functionally graded composite plates using piezoelectric patches. Composites. Part B, Engineering, 2016, 85: 140–149
https://doi.org/10.1016/j.compositesb.2015.09.044
6 HGhasemi, R Brighenti, XZhuang, JMuthu, TRabczuk. Optimization of fiber distribution in fiber reinforced composite by using NURBS functions. Computational Materials Science, 2014, 83: 463–473
https://doi.org/10.1016/j.commatsci.2013.11.032
7 MSilani, S Ziaei-Rad, HTalebi, TRabczuk. A semi-concurrent multiscale approach for modeling damage in nanocomposites. Theoretical and Applied Fracture Mechanics, 2014, 74: 30–38
https://doi.org/10.1016/j.tafmec.2014.06.009
8 HGhasemi, R Brighenti, XZhuang, JMuthu, TRabczuk. Optimal fiber content and distribution in fiber-reinforced solids using a reliability and NURBS based sequential optimization approach. Structural and Multidisciplinary Optimization, 2015, 51(1): 99–112
https://doi.org/10.1007/s00158-014-1114-y
9 K MHamdia, M A Msekh, M Silani, NVu-Bac, XZhuang, TNguyen-Thoi, TRabczuk. Uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling. Composite Structures, 2015, 133: 1177–1190
https://doi.org/10.1016/j.compstruct.2015.08.051
10 M AMsekh, M Silani, MJamshidian, PAreias, XZhuang, GZi, P He, TRabczuk. Predictions of J integral and tensile strength of clay/epoxy nanocomposites material using phase field model. Composites. Part B, Engineering, 2016, 93: 97–114
https://doi.org/10.1016/j.compositesb.2016.02.022
11 MSilani, H Talebi, A MHamouda, TRabczuk. Nonlocal damage modelling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15: 18–23
https://doi.org/10.1016/j.jocs.2015.11.007
12 NVu-Bac, R Rafiee, XZhuang, TLahmer, TRabczuk. Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites. Part B, Engineering, 2015, 68: 446–464
https://doi.org/10.1016/j.compositesb.2014.09.008
13 NVu-Bac, T Lahmer, YZhang, XZhuang, TRabczuk. Stochastic predictions of interfacial characteristic of polymeric nanocomposites (PNCs). Composites. Part B, Engineering, 2014, 59: 80–95
https://doi.org/10.1016/j.compositesb.2013.11.014
14 NVu-Bac, M Silani, TLahmer, XZhuang, TRabczuk. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535
https://doi.org/10.1016/j.commatsci.2014.04.066
15 HGhasemi, R Rafiee, XZhuang, JMuthu, TRabczuk. Uncertainties propagation in metamodel-based probabilistic optimization of CNT/polymer composite structure using stochastic multi-scale modeling. Computational Materials Science, 2014, 85: 295–305
https://doi.org/10.1016/j.commatsci.2014.01.020
16 H SShen. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Composite Structures, 2009, 91(1): 9–19
https://doi.org/10.1016/j.compstruct.2009.04.026
17 RAnsari, M Faghih Shojaei, VMohammadi, RGholami, FSadeghi. Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams. Composite Structures, 2014, 113: 316–327
https://doi.org/10.1016/j.compstruct.2014.03.015
18 LZhang, Z Lei, KLiew. Free vibration analysis of FG-CNT reinforced composite straight-sided quadrilateral plates resting on elastic foundations using the IMLS-Ritz method. Journal of Vibration and Control, 2017, 23(6): 1026–1043
https://doi.org/10.1177/1077546315587804
19 Z XLei, L W Zhang, K M Liew. Vibration of FG-CNT reinforced composite thick quadrilateral plates resting on Pasternak foundations. Engineering Analysis with Boundary Elements, 2016, 64: 1–11
https://doi.org/10.1016/j.enganabound.2015.11.014
20 MMirzaei, Y Kiani. Nonlinear free vibration of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets. Acta Mechanica, 2016, 227(7): 1869–1884
https://doi.org/10.1007/s00707-016-1593-6
21 YKiani. Free vibration of FG-CNT reinforced composite skew plates. Aerospace Science and Technology, 2016, 58: 178–188
https://doi.org/10.1016/j.ast.2016.08.018
22 HWu, S Kitipornchai, JYang. Free vibration and buckling analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets. International Journal of Structural Stability and Dynamics, 2015, 15(7): 1540011
https://doi.org/10.1142/S0219455415400118
23 H LWu, J Yang, SKitipornchai. Nonlinear vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections. Composites. Part B, Engineering, 2016, 90: 86–96
https://doi.org/10.1016/j.compositesb.2015.12.007
24 YKiani. Shear buckling of FG-CNT reinforced composite plates using Chebyshev-Ritz method. Composites. Part B, Engineering, 2016, 105: 176–187
https://doi.org/10.1016/j.compositesb.2016.09.001
25 MMirzaei, Y Kiani. Thermal buckling of temperature dependent FG-CNT reinforced composite plates. Meccanica, 2016, 51(9): 2185–2201
https://doi.org/10.1007/s11012-015-0348-0
26 YKiani. Thermal post-buckling of FG-CNT reinforced composite plates. Composite Structures, 2017, 159: 299–306
https://doi.org/10.1016/j.compstruct.2016.09.084
27 MRafiee, J Yang, SKitipornchai. Large amplitude vibration of carbon nanotube reinforced functionally graded composite beams with piezoelectric layers. Composite Structures, 2013, 96: 716–725
https://doi.org/10.1016/j.compstruct.2012.10.005
28 YKiani. Free vibration of functionally graded carbon nanotube reinforced composite plates integrated with piezoelectric layers. Computers & Mathematics with Applications (Oxford, England), 2016, 72(9): 2433–2449
https://doi.org/10.1016/j.camwa.2016.09.007
29 AAlibeigloo. Free vibration analysis of functionally graded carbon nanotube-reinforced composite cylindrical panel embedded in piezoelectric layers by using theory of elasticity. European Journal of Mechanics. A, Solids, 2014, 44: 104–115
https://doi.org/10.1016/j.euromechsol.2013.10.002
30 PMalekzadeh, M Shojaee. Buckling analysis of quadrilateral laminated plates with carbon nanotubes reinforced composite layers. Thin-walled Structures, 2013, 71: 108–118
https://doi.org/10.1016/j.tws.2013.05.008
31 PMalekzadeh, A R Zarei. Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers. Thin-walled Structures, 2014, 82: 221–232
https://doi.org/10.1016/j.tws.2014.04.016
32 Z XLei, L W Zhang, K M Liew. Free vibration analysis of laminated FG-CNT reinforced composite rectangular plates using the kp-Ritz method. Composite Structures, 2015, 127: 245–259
https://doi.org/10.1016/j.compstruct.2015.03.019
33 Z XLei, L W Zhang, K M Liew. Buckling analysis of CNT reinforced functionally graded laminated composite plates. Composite Structures, 2016, 152: 62–73
https://doi.org/10.1016/j.compstruct.2016.05.047
34 FLin, Y Xiang. Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories. Applied Mathematical Modelling, 2014, 38(15–16): 3741–3754
https://doi.org/10.1016/j.apm.2014.02.008
35 K MLiew, Z X Lei, L W Zhang. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Composite Structures, 2015, 120: 90–97
https://doi.org/10.1016/j.compstruct.2014.09.041
36 YQu, X Long, HLi, GMeng. A variational formulation for dynamic analysis of composite laminated beams based on a general higher-order shear deformation theory. Composite Structures, 2013, 102: 175–192
https://doi.org/10.1016/j.compstruct.2013.02.032
37 TVo-Duy, D Duong-Gia, VHo-Huu, H CVu-Do, TNguyen-Thoi. Multi-objective optimization of laminated composite beam structures using NSGA-II algorithm. Composite Structures, 2017, 168: 498–509
https://doi.org/10.1016/j.compstruct.2017.02.038
38 TVo-Duy, V Ho-Huu, T DDo-Thi, HDang-Trung, TNguyen-Thoi. A global numerical approach for lightweight design optimization of laminated composite plates subjected to frequency constraints. Composite Structures, 2017, 159: 646–655
https://doi.org/10.1016/j.compstruct.2016.09.059
39 VHo-Huu, T D Do-Thi, H Dang-Trung, TVo-Duy, TNguyen-Thoi. Optimization of laminated composite plates for maximizing buckling load using improved differential evolution and smoothed finite element method. Composite Structures, 2016, 146: 132–147
https://doi.org/10.1016/j.compstruct.2016.03.016
40 TVo-Duy, N Nguyen-Minh, HDang-Trung, ATran-Viet, TNguyen-Thoi. Damage assessment of laminated composite beam structures using damage locating vector (DLV) method. Frontiers of Structural and Civil Engineering, 2015, 9(4): 457–465
https://doi.org/10.1007/s11709-015-0303-0
41 DDinh-Cong, T Vo-Duy, NNguyen-Minh, VHo-Huu, TNguyen-Thoi. A two-stage assessment method using damage locating vector method and differential evolution algorithm for damage identification of cross-ply laminated composite beams. Advances in Structural Engineering, 2017, 20(12): 1807–1827
https://doi.org/10.1177/1369433217695620
42 TVo-Duy, V Ho-Huu, HDang-Trung, TNguyen-Thoi. A two-step approach for damage detection in laminated composite structures using modal strain energy method and an improved differential evolution algorithm. Composite Structures, 2016, 147: 42–53
https://doi.org/10.1016/j.compstruct.2016.03.027
43 KChandrashekhara, K Krishnamurthy, SRoy. Free vibration of composite beams including rotary inertia and shear deformation. Composite Structures, 1990, 14(4): 269–279
https://doi.org/10.1016/0263-8223(90)90010-C
44 A AKhdeir, J N Reddy. Free vibration of cross-ply laminated beams with arbitrary boundary conditions. International Journal of Engineering Science, 1994, 32(12): 1971–1980
https://doi.org/10.1016/0020-7225(94)90093-0
45 MKameswara Rao, Y MDesai, M RChitnis. Free vibrations of laminated beams using mixed theory. Composite Structures, 2001, 52(2): 149–160
https://doi.org/10.1016/S0263-8223(00)00162-8
46 G SRamtekkar, Y MDesai, A HShah. Natural vibrations of laminated composite beams by using mixed finite element modelling. Journal of Sound and Vibration, 2002, 257(4): 635–651
https://doi.org/10.1006/jsvi.2002.5072
47 MKisa. Free vibration analysis of a cantilever composite beam with multiple cracks. Composites Science and Technology, 2004, 64(9): 1391–1402
https://doi.org/10.1016/j.compscitech.2003.11.002
48 JLi, Q Huo, XLi, XKong, W Wu. Vibration analyses of laminated composite beams using refined higher-order shear deformation theory. International Journal of Mechanics and Materials in Design, 2014, 10(1): 43–52
https://doi.org/10.1007/s10999-013-9229-7
49 J LMantari, F G Canales. Free vibration and buckling of laminated beams via hybrid Ritz solution for various penalized boundary conditions. Composite Structures, 2016, 152: 306–315
https://doi.org/10.1016/j.compstruct.2016.05.037
50 T KNguyen, N D Nguyen, T P Vo, H T Thai. Trigonometric-series solution for analysis of laminated composite beams. Composite Structures, 2017, 160: 142–151
https://doi.org/10.1016/j.compstruct.2016.10.033
51 A SSayyad, Y M Ghugal, N S Naik. Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory. Curved and Layered Structures, 2015, 2(1): 279–289
https://doi.org/10.1515/cls-2015-0015
52 LJun, H Hongxing, SRongying. Dynamic finite element method for generally laminated composite beams. International Journal of Mechanical Sciences, 2008, 50(3): 466–480
https://doi.org/10.1016/j.ijmecsci.2007.09.014
53 GShi, K Y Lam. Finite element vibration analysis of composite beams based on higher-order beam theory. Journal of Sound and Vibration, 1999, 219(4): 707–721
https://doi.org/10.1006/jsvi.1998.1903
54 J NReddy, A Khdeir. Buckling and vibration of laminated composite plates using various plate theories. AIAA Journal, 1989, 27(12): 1808–1817
https://doi.org/10.2514/3.10338
55 SNatarajan, S Chakraborty, MThangavel, SBordas, TRabczuk. Size-dependent free flexural vibration behavior of functionally graded nanoplates. Computational Materials Science, 2012, 65: 74–80
https://doi.org/10.1016/j.commatsci.2012.06.031
56 FAmiri, D Millán, YShen, TRabczuk, MArroyo. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109
https://doi.org/10.1016/j.tafmec.2013.12.002
57 NNguyen-Thanh, K Zhou, XZhuang, PAreias, HNguyen-Xuan, YBazilevs, TRabczuk. Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 1157–1178
https://doi.org/10.1016/j.cma.2016.12.002
58 PAreias, T Rabczuk, M AMsekh. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 322–350
https://doi.org/10.1016/j.cma.2016.01.020
59 NNguyen-Thanh, J Kiendl, HNguyen-Xuan, RWüchner, K UBletzinger, YBazilevs, TRabczuk. Rotation free isogeometric thin shell analysis using PHT-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47–48): 3410–3424
https://doi.org/10.1016/j.cma.2011.08.014
60 TRabczuk, R Gracie, J HSong, TBelytschko. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71
61 PAreias, T Rabczuk. Finite strain fracture of plates and shells with configurational forces and edge rotations. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122
https://doi.org/10.1002/nme.4477
62 TChau-Dinh, G Zi, P SLee, TRabczuk, J HSong. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92–93: 242–256
https://doi.org/10.1016/j.compstruc.2011.10.021
63 NNguyen-Thanh, N Valizadeh, M NNguyen, HNguyen-Xuan, XZhuang, PAreias, GZi, Y Bazilevs, LDe Lorenzis, TRabczuk. An extended isogeometric thin shell analysis based on Kirchhoff-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291
https://doi.org/10.1016/j.cma.2014.08.025
64 TRabczuk, P M A Areias, T Belytschko. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548
https://doi.org/10.1002/nme.2013
65 PTan, N Nguyen-Thanh, KZhou. Extended isogeometric analysis based on Bézier extraction for an FGM plate by using the two-variable refined plate theory. Theoretical and Applied Fracture Mechanics, 2017, 89: 127–138
https://doi.org/10.1016/j.tafmec.2017.02.002
66 RKruse, N Nguyen-Thanh, LDe Lorenzis, T J RHughes. Isogeometric collocation for large deformation elasticity and frictional contact problems. Computer Methods in Applied Mechanics and Engineering, 2015, 296: 73–112
https://doi.org/10.1016/j.cma.2015.07.022
67 C HThai, H Nguyen-Xuan, NNguyen-Thanh, T HLe, TNguyen-Thoi, TRabczuk. Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach. International Journal for Numerical Methods in Engineering, 2012, 91(6): 571–603
https://doi.org/10.1002/nme.4282
68 JHuang, N Nguyen-Thanh, KZhou. Extended isogeometric analysis based on Bézier extraction for the buckling analysis of Mindlin-Reissner plates. Acta Mechanica, 2017, 228(9): 3077–3093
https://doi.org/10.1007/s00707-017-1861-0
69 NNguyen-Thanh, K Zhou. Extended isogeometric analysis based on PHT-splines for crack propagation near inclusions. International Journal for Numerical Methods in Engineering, 2017, 112(12): 1777–1800
https://doi.org/10.1002/nme.5581
70 O CZienkiewicz, R LTaylor, J ZZhu. The Finite Element Method: Its Basis and Fundamentals. 7th ed. Oxford: Butterworth-Heinemann, 2013
71 T J RHughes, J ACottrell, YBazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39–41): 4135–4195
https://doi.org/10.1016/j.cma.2004.10.008
72 O CZienkiewicz, R LTaylor, J MToo. Reduced integration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering, 1971, 3(2): 275–290
https://doi.org/10.1002/nme.1620030211
73 GPrathap, G R Bhashyam. Reduced integration and the shear-flexible beam element. International Journal for Numerical Methods in Engineering, 1982, 18(2): 195–210
https://doi.org/10.1002/nme.1620180205
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