Advertisement

Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method

  • T. Vo-Duy
  • V. Ho-Huu
  • T. Nguyen-Thoi
Research Article
  • 46 Downloads

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2017.08.

References

  1. 1.
    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): 193101CrossRefGoogle Scholar
  2. 2.
    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: 119–128CrossRefGoogle Scholar
  3. 3.
    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: 187–197CrossRefGoogle Scholar
  4. 4.
    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: 211–221CrossRefGoogle Scholar
  5. 5.
    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: 140–149CrossRefGoogle Scholar
  6. 6.
    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: 463–473CrossRefGoogle Scholar
  7. 7.
    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: 30–38CrossRefGoogle Scholar
  8. 8.
    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): 99–112MathSciNetCrossRefGoogle Scholar
  9. 9.
    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: 1177–1190CrossRefGoogle Scholar
  10. 10.
    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: 97–114CrossRefGoogle Scholar
  11. 11.
    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: 18–23CrossRefGoogle Scholar
  12. 12.
    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: 446–464CrossRefGoogle Scholar
  13. 13.
    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: 80–95CrossRefGoogle Scholar
  14. 14.
    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: 520–535CrossRefGoogle Scholar
  15. 15.
    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: 295–305CrossRefGoogle Scholar
  16. 16.
    Shen H S. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Composite Structures, 2009, 91(1): 9–19CrossRefGoogle Scholar
  17. 17.
    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: 316–327CrossRefGoogle Scholar
  18. 18.
    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): 1026–1043MathSciNetCrossRefGoogle Scholar
  19. 19.
    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: 1–11MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mirzaei M, Kiani Y. Nonlinear free vibration of temperaturedependent sandwich beams with carbon nanotube-reinforced face sheets. Acta Mechanica, 2016, 227(7): 1869–1884MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kiani Y. Free vibration of FG-CNT reinforced composite skew plates. Aerospace Science and Technology, 2016, 58: 178–188CrossRefGoogle Scholar
  22. 22.
    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): 1540011MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    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: 86–96CrossRefGoogle Scholar
  24. 24.
    Kiani Y. Shear buckling of FG-CNT reinforced composite plates using Chebyshev-Ritz method. Composites. Part B, Engineering, 2016, 105: 176–187CrossRefGoogle Scholar
  25. 25.
    Mirzaei M, Kiani Y. Thermal buckling of temperature dependent FG-CNT reinforced composite plates. Meccanica, 2016, 51(9): 2185–2201MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kiani Y. Thermal post-buckling of FG-CNT reinforced composite plates. Composite Structures, 2017, 159: 299–306CrossRefGoogle Scholar
  27. 27.
    Rafiee M, Yang J, Kitipornchai S. Large amplitude vibration of carbon nanotube reinforced functionally graded composite beams with piezoelectric layers. Composite Structures, 2013, 96: 716–725CrossRefGoogle Scholar
  28. 28.
    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): 2433–2449MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    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: 104–115MathSciNetCrossRefGoogle Scholar
  30. 30.
    Malekzadeh P, Shojaee M. Buckling analysis of quadrilateral laminated plates with carbon nanotubes reinforced composite layers. Thin-walled Structures, 2013, 71: 108–118CrossRefGoogle Scholar
  31. 31.
    Malekzadeh P, Zarei A R. Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers. Thinwalled Structures, 2014, 82: 221–232Google Scholar
  32. 32.
    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: 245–259CrossRefGoogle Scholar
  33. 33.
    Lei Z X, Zhang L W, Liew K M. Buckling analysis of CNT reinforced functionally graded laminated composite plates. Composite Structures, 2016, 152: 62–73CrossRefGoogle Scholar
  34. 34.
    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): 3741–3754MathSciNetCrossRefGoogle Scholar
  35. 35.
    Liew K M, Lei Z X, Zhang L W. Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review. Composite Structures, 2015, 120: 90–97CrossRefGoogle Scholar
  36. 36.
    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: 175–192CrossRefGoogle Scholar
  37. 37.
    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: 498–509CrossRefGoogle Scholar
  38. 38.
    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: 646–655CrossRefGoogle Scholar
  39. 39.
    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: 132–147CrossRefGoogle Scholar
  40. 40.
    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): 457–465CrossRefGoogle Scholar
  41. 41.
    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): 1807–1827CrossRefGoogle Scholar
  42. 42.
    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: 42–53CrossRefGoogle Scholar
  43. 43.
    Chandrashekhara K, Krishnamurthy K, Roy S. Free vibration of composite beams including rotary inertia and shear deformation. Composite Structures, 1990, 14(4): 269–279CrossRefGoogle Scholar
  44. 44.
    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): 1971–1980MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Kameswara Rao M, Desai Y M, Chitnis M R. Free vibrations of laminated beams using mixed theory. Composite Structures, 2001, 52(2): 149–160CrossRefGoogle Scholar
  46. 46.
    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): 635–651CrossRefGoogle Scholar
  47. 47.
    Kisa M. Free vibration analysis of a cantilever composite beam with multiple cracks. Composites Science and Technology, 2004, 64(9): 1391–1402CrossRefGoogle Scholar
  48. 48.
    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): 43–52CrossRefGoogle Scholar
  49. 49.
    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: 306–315CrossRefGoogle Scholar
  50. 50.
    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: 142–151CrossRefGoogle Scholar
  51. 51.
    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): 279–289Google Scholar
  52. 52.
    Jun L, Hongxing H, Rongying S. Dynamic finite element method for generally laminated composite beams. International Journal of Mechanical Sciences, 2008, 50(3): 466–480CrossRefzbMATHGoogle Scholar
  53. 53.
    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): 707–721CrossRefGoogle Scholar
  54. 54.
    Reddy J N, Khdeir A. Buckling and vibration of laminated composite plates using various plate theories. AIAA Journal, 1989, 27(12): 1808–1817MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    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: 74–80CrossRefGoogle Scholar
  56. 56.
    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: 102–109CrossRefGoogle Scholar
  57. 57.
    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: 1157–1178MathSciNetCrossRefGoogle Scholar
  58. 58.
    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: 322–350MathSciNetCrossRefGoogle Scholar
  59. 59.
    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): 3410–3424MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    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): 48–71MathSciNetzbMATHGoogle Scholar
  61. 61.
    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): 1099–1122MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    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: 242–256CrossRefGoogle Scholar
  63. 63.
    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: 265–291MathSciNetCrossRefGoogle Scholar
  64. 64.
    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): 524–548MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    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: 127–138CrossRefGoogle Scholar
  66. 66.
    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: 73–112MathSciNetCrossRefGoogle Scholar
  67. 67.
    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): 571–603MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    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): 3077–3093MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    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): 1777–1800MathSciNetCrossRefGoogle Scholar
  70. 70.
    Zienkiewicz O C, Taylor R L, Zhu J Z. The Finite Element Method: Its Basis and Fundamentals. 7th ed. Oxford: Butterworth-Heinemann, 2013zbMATHGoogle Scholar
  71. 71.
    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): 4135–4195MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    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): 275–290CrossRefzbMATHGoogle Scholar
  73. 73.
    Prathap G, Bhashyam G R. Reduced integration and the shearflexible beam element. International Journal for Numerical Methods in Engineering, 1982, 18(2): 195–210CrossRefzbMATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Computational Mathematics and Engineering, Institute for Computational ScienceTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Civil EngineeringTon Duc Thang UniversityHo Chi Minh CityVietnam

Personalised recommendations