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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Sun C H, Li F, Cheng H M, Lu G Q. 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
Yas M H, Samadi N. 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
Jedari Salami S. 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
Lei Z X, Zhang L W, Liew K M. Analysis of laminated CNT reinforced functionally graded plates using the element-free kp-Ritz method. Composites. Part B, Engineering, 2016, 84: 211–221
Zhang L W, Song Z G, Liew K M. Optimal shape control of CNT reinforced functionally graded composite plates using piezoelectric patches. Composites. Part B, Engineering, 2016, 85: 140–149
Ghasemi H, Brighenti R, Zhuang X, Muthu J, Rabczuk T. Optimization of fiber distribution in fiber reinforced composite by using NURBS functions. Computational Materials Science, 2014, 83: 463–473
Silani M, Ziaei-Rad S, Talebi H, Rabczuk T. A semi-concurrent multiscale approach for modeling damage in nanocomposites. Theoretical and Applied Fracture Mechanics, 2014, 74: 30–38
Ghasemi H, Brighenti R, Zhuang X, Muthu J, Rabczuk T. 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
Hamdia K M, Msekh M A, Silani M, Vu-Bac N, Zhuang X, Nguyen-Thoi T, Rabczuk T. Uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling. Composite Structures, 2015, 133: 1177–1190
Msekh M A, Silani M, Jamshidian M, Areias P, Zhuang X, Zi G, He P, Rabczuk T. Predictions of J integral and tensile strength of clay/ epoxy nanocomposites material using phase field model. Composites. Part B, Engineering, 2016, 93: 97–114
Silani M, Talebi H, Hamouda A M, Rabczuk T. Nonlocal damage modelling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15: 18–23
Vu-Bac N, Rafiee R, Zhuang X, Lahmer T, Rabczuk T. Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites. Part B, Engineering, 2015, 68: 446–464
Vu-Bac N, Lahmer T, Zhang Y, Zhuang X, Rabczuk T. Stochastic predictions of interfacial characteristic of polymeric nanocomposites (PNCs). Composites. Part B, Engineering, 2014, 59: 80–95
Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535
Ghasemi H, Rafiee R, Zhuang X, Muthu J, Rabczuk T. Uncertainties propagation in metamodel-based probabilistic optimization of CNT/ polymer composite structure using stochastic multi-scale modeling. Computational Materials Science, 2014, 85: 295–305
Shen H S. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Composite Structures, 2009, 91(1): 9–19
Ansari R, Faghih Shojaei M, Mohammadi V, Gholami R, Sadeghi F. Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams. Composite Structures, 2014, 113: 316–327
Zhang L, Lei Z, Liew K. 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
Lei Z X, Zhang L W, Liew K M. Vibration of FG-CNT reinforced composite thick quadrilateral plates resting on Pasternak foundations. Engineering Analysis with Boundary Elements, 2016, 64: 1–11
Mirzaei M, Kiani Y. Nonlinear free vibration of temperaturedependent sandwich beams with carbon nanotube-reinforced face sheets. Acta Mechanica, 2016, 227(7): 1869–1884
Kiani Y. Free vibration of FG-CNT reinforced composite skew plates. Aerospace Science and Technology, 2016, 58: 178–188
Wu H, Kitipornchai S, Yang J. Free vibration and buckling analysis of sandwich beams with functionally graded carbon nanotubereinforced composite face sheets. International Journal of Structural Stability and Dynamics, 2015, 15(7): 1540011
Wu H L, Yang J, Kitipornchai S. Nonlinear vibration of functionally graded carbon nanotube-reinforced composite beams with geometric imperfections. Composites. Part B, Engineering, 2016, 90: 86–96
Kiani Y. Shear buckling of FG-CNT reinforced composite plates using Chebyshev-Ritz method. Composites. Part B, Engineering, 2016, 105: 176–187
Mirzaei M, Kiani Y. Thermal buckling of temperature dependent FG-CNT reinforced composite plates. Meccanica, 2016, 51(9): 2185–2201
Kiani Y. Thermal post-buckling of FG-CNT reinforced composite plates. Composite Structures, 2017, 159: 299–306
Rafiee M, Yang J, Kitipornchai S. Large amplitude vibration of carbon nanotube reinforced functionally graded composite beams with piezoelectric layers. Composite Structures, 2013, 96: 716–725
Kiani Y. 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
Alibeigloo A. 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
Malekzadeh P, Shojaee M. Buckling analysis of quadrilateral laminated plates with carbon nanotubes reinforced composite layers. Thin-walled Structures, 2013, 71: 108–118
Malekzadeh P, Zarei A R. Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers. Thinwalled Structures, 2014, 82: 221–232
Lei Z X, Zhang L W, Liew K M. Free vibration analysis of laminated FG-CNT reinforced composite rectangular plates using the kp-Ritz method. Composite Structures, 2015, 127: 245–259
Lei Z X, Zhang L W, Liew K M. Buckling analysis of CNT reinforced functionally graded laminated composite plates. Composite Structures, 2016, 152: 62–73
Lin F, Xiang Y. 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
Liew K M, Lei Z X, Zhang L W. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Composite Structures, 2015, 120: 90–97
Qu Y, Long X, Li H, Meng G. 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
Vo-Duy T, Duong-Gia D, Ho-Huu V, Vu-Do H C, Nguyen-Thoi T. Multi-objective optimization of laminated composite beam structures using NSGA-II algorithm. Composite Structures, 2017, 168: 498–509
Vo-Duy T, Ho-Huu V, Do-Thi T D, Dang-Trung H, Nguyen-Thoi T. A global numerical approach for lightweight design optimization of laminated composite plates subjected to frequency constraints. Composite Structures, 2017, 159: 646–655
Ho-Huu V, Do-Thi T D, Dang-Trung H, Vo-Duy T, Nguyen-Thoi T. Optimization of laminated composite plates for maximizing buckling load using improved differential evolution and smoothed finite element method. Composite Structures, 2016, 146: 132–147
Vo-Duy T, Nguyen-Minh N, Dang-Trung H, Tran-Viet A, Nguyen-Thoi T. Damage assessment of laminated composite beam structures using damage locating vector (DLV) method. Frontiers of Structural and Civil Engineering, 2015, 9(4): 457–465
Dinh-Cong D, Vo-Duy T, Nguyen-Minh N, Ho-Huu V, Nguyen-Thoi T. 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
Vo-Duy T, Ho-Huu V, Dang-Trung H, Nguyen-Thoi T. 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
Chandrashekhara K, Krishnamurthy K, Roy S. Free vibration of composite beams including rotary inertia and shear deformation. Composite Structures, 1990, 14(4): 269–279
Khdeir A A, Reddy J N. Free vibration of cross-ply laminated beams with arbitrary boundary conditions. International Journal of Engineering Science, 1994, 32(12): 1971–1980
Kameswara Rao M, Desai Y M, Chitnis M R. Free vibrations of laminated beams using mixed theory. Composite Structures, 2001, 52(2): 149–160
Ramtekkar G S, Desai Y M, Shah A H. Natural vibrations of laminated composite beams by using mixed finite element modelling. Journal of Sound and Vibration, 2002, 257(4): 635–651
Kisa M. Free vibration analysis of a cantilever composite beam with multiple cracks. Composites Science and Technology, 2004, 64(9): 1391–1402
Li J, Huo Q, Li X, Kong X, Wu W. 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
Mantari J L, Canales F G. Free vibration and buckling of laminated beams via hybrid Ritz solution for various penalized boundary conditions. Composite Structures, 2016, 152: 306–315
Nguyen T K, Nguyen N D, Vo T P, Thai H T. Trigonometric-series solution for analysis of laminated composite beams. Composite Structures, 2017, 160: 142–151
Sayyad A S, Ghugal Y M, Naik N S. Bending analysis of laminated composite and sandwich beams according to refined trigonometric beam theory. Curved and Layered Structures, 2015, 2(1): 279–289
Jun L, Hongxing H, Rongying S. Dynamic finite element method for generally laminated composite beams. International Journal of Mechanical Sciences, 2008, 50(3): 466–480
Shi G, Lam K Y. Finite element vibration analysis of composite beams based on higher-order beam theory. Journal of Sound and Vibration, 1999, 219(4): 707–721
Reddy J N, Khdeir A. Buckling and vibration of laminated composite plates using various plate theories. AIAA Journal, 1989, 27(12): 1808–1817
Natarajan S, Chakraborty S, Thangavel M, Bordas S, Rabczuk T. Size-dependent free flexural vibration behavior of functionally graded nanoplates. Computational Materials Science, 2012, 65: 74–80
Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109
Nguyen-Thanh N, Zhou K, Zhuang X, Areias P, Nguyen-Xuan H, Bazilevs Y, Rabczuk T. 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
Areias P, Rabczuk T, Msekh M A. Phase-field analysis of finitestrain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 322–350
Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Wüchner R, Bletzinger K U, Bazilevs Y, Rabczuk T. Rotation free isogeometric thin shell analysis using PHT-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47–48): 3410–3424
Rabczuk T, Gracie R, Song J H, Belytschko T. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71
Areias P, Rabczuk T. 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
Chau-Dinh T, Zi G, Lee P S, Rabczuk T, Song J H. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92–93: 242–256
Nguyen-Thanh N, Valizadeh N, Nguyen M N, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchhoff-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291
Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548
Tan P, Nguyen-Thanh N, Zhou K. Extended isogeometric analysis based on Bézier extraction for an FGM plate by using the twovariable refined plate theory. Theoretical and Applied Fracture Mechanics, 2017, 89: 127–138
Kruse R, Nguyen-Thanh N, De Lorenzis L, Hughes T J R. Isogeometric collocation for large deformation elasticity and frictional contact problems. Computer Methods in Applied Mechanics and Engineering, 2015, 296: 73–112
Thai C H, Nguyen-Xuan H, Nguyen-Thanh N, Le T H, Nguyen-Thoi T, Rabczuk T. 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
Huang J, Nguyen-Thanh N, Zhou K. Extended isogeometric analysis based on Bézier extraction for the buckling analysis of Mindlin-Reissner plates. Acta Mechanica, 2017, 228(9): 3077–3093
Nguyen-Thanh N, Zhou K. Extended isogeometric analysis based on PHT-splines for crack propagation near inclusions. International Journal for Numerical Methods in Engineering, 2017, 112(12): 1777–1800
Zienkiewicz O C, Taylor R L, Zhu J Z. The Finite Element Method: Its Basis and Fundamentals. 7th ed. Oxford: Butterworth-Heinemann, 2013
Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39–41): 4135–4195
Zienkiewicz O C, Taylor R L, Too J M. Reduced integration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering, 1971, 3(2): 275–290
Prathap G, Bhashyam G R. Reduced integration and the shear-flexible beam element. International Journal for Numerical Methods in Engineering, 1982, 18(2): 195–210
Acknowledgements
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.08.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vo-Duy, T., Ho-Huu, V. & Nguyen-Thoi, T. Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method. Front. Struct. Civ. Eng. 13, 324–336 (2019). https://doi.org/10.1007/s11709-018-0466-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11709-018-0466-6