Computational Mechanics

, Volume 48, Issue 2, pp 175–193

# An efficient meshfree method for vibration analysis of laminated composite plates

• Tinh Quoc Bui
• Minh Ngoc Nguyen
• Chuanzeng Zhang
Original Paper

## Abstract

A detailed analysis of natural frequencies of laminated composite plates using the meshfree moving Kriging interpolation method is presented. The present formulation is based on the classical plate theory while the moving Kriging interpolation satisfying the delta property is employed to construct the shape functions. Since the advantage of the interpolation functions, the method is more convenient and no special techniques are needed in enforcing the essential boundary conditions. Numerical examples with different shapes of plates are presented and the achieved results are compared with reference solutions available in the literature. Several aspects of the model involving relevant parameters, fiber orientations, lay-up number, length-to-length, stiffness ratios, etc. affected on frequency are analyzed numerically in details. The convergence of the method on the natural frequency is also given. As a consequence, the applicability and the effectiveness of the present method for accurately computing natural frequencies of generally shaped laminates are demonstrated.

## Keywords

Vibration Meshfree method Moving Kriging interpolation Composite laminates

## References

1. 1.
Ochoa OO, Reddy JN (1992) Finite element analysis of composite laminates. Kluwer, DordrechtGoogle Scholar
2. 2.
Reddy JN (1985) A review of the literature on finite-element modeling of laminated composite plates. Shock Vib Digest 17: 3–8
3. 3.
Reddy JN, Averill RC (1991) Advances in the modeling of laminated plates. Comput Syst Eng 2: 541–555
4. 4.
Zhang YX, Yang CH (2009) Recent developments in finite element analysis for laminated composite plates. Compos Struct 88: 147–157
5. 5.
Hearmon R (1959) The frequency of flexural vibrations of rectangular orthotropic plates with clamped or simply supported edges. J Appl Mech 26: 537–542
6. 6.
Chow ST, Liew KM, Lam KY (1992) Transverse vibration of symmetrically laminated rectangular composite plates. Compos Struct 20: 213–226
7. 7.
Liew KM, Lim CW (1995) Vibratory characteristics of general laminates, I: symmetric trapezoids. J Sound Vib 183: 615–642
8. 8.
Leissa AW, Narita Y (1989) Vibration studies for simply supported symmetrically laminated rectangular plates. Compos Struct 12: 113–132
9. 9.
Liew KM (1996) Solving the vibration of thick symmetric laminates by Reissner/Mindlin plate theory and the p-Ritz method. J Sound Vib 198: 343–360
10. 10.
Liew KM, Lam KY, Chow ST (1989) Study on flexural vibration of triangular composite plates influenced by fibre orientation. Compos Struct 13: 123–132
11. 11.
Hung KC, Liew MK, Lim KM, Leong SL (1993) Boundary beam characteristics orthonormal polynomials in energy approach for vibration of symmetric laminates-I: classical boundary conditions. Compos Struct 26: 167–184
12. 12.
Venini P, Mariani C (1997) Free vibrations of uncertain composite plates via stochastic Ratleigh-Ritz approach. Comp Struct 64: 407–423
13. 13.
Hu XX, Sakiyama T, Lim CW, Xiong Y, Matsuda H, Morita C (2004) Vibration of angle-ply laminated plates with twist by Rayleigh–Ritz procedure. Comp Method Appl Mech Eng 193: 805–823
14. 14.
Wang YY, Lam KY, Liu GR (2000) Bending analysis of classical symmetric laminated composite plates by the strip element method. Mech Compos Mater Struct 7: 225–247Google Scholar
15. 15.
Liu GR, Lam KY (1994) Characterization of a horizontal crack in anisotropic laminated plates. Int J Solids Struct 31: 2965–2977
16. 16.
Secgin A, Sarigul AS (2008) Free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution (DSC) approach: algorithm and verification. J Sound Vib 315: 197–211
17. 17.
Ng CHW, Zhao YB, Wei GW (2004) Comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates. Comp Method Appl Mech Eng 193: 2483–2506
18. 18.
Bellman R, Kashef BG, Casti J (1972) Differential quarature: a technique for the rapid solution of nonlinear partial differential equations. J Comput Phys 10: 40–52
19. 19.
Lanhe W, Hua L, Daobin W (2005) Vibration analysis of generally laminated composite plates by the moving least square differential quadrature method. Compos Struct 68: 319–330
20. 20.
Bert CW, Malik M (1996) The differential quadrature method for irregular domains and application to plate vibration. Int J Mech Sci 38: 589–606
21. 21.
Zeng H, Bert CW (2001) A differential quadrature analysis of vibration for rectangular stiffened plates. J Sound Vib 241: 247–252
22. 22.
Liew KM, Huang YQ, Reddy JN (2003) Vibration analysis of symmetrically laminated plate based on the FSDT using the moving least squares differential quadrature method. Comp Method Appl Mech Eng 192: 2203–2222
23. 23.
Zhang JC, Ng TY, Liew KM (2003) Three-dimensional theory of elasticity for free vibration analysis of composite laminates via layerwise differential quadrature modeling. Int J Numer Method Eng 57: 1819–1844
24. 24.
Ferreira AJM (2003) A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates. Compos Struct 59: 385–392
25. 25.
Ferreira AJM, Roque CMC, Jorge RMN (2005) Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions. Comp Method Appl Mech Eng 194: 4265–4278
26. 26.
Roque CMC, Ferreira AJM, Jorge RMN (2007) A radial basis function approach for the free vibration analysis of functionally graded plates using a refined theory. J Sound Vib 300: 1048–1070
27. 27.
Roque CMC, Ferreira AJM (2009) New developments in the radial basis functions analysis of composite shells. Compos Struct 87: 141–150
28. 28.
Ferreira AJM (2005) Free vibration analysis of Timoshenko beams and Mindlin plates by radial basis functions. Int J Comp Meth 2: 15–31
29. 29.
Roque CMC, Ferreira AJM, Jorge RMN (2005) Modeling of composite and sandwich plates by a trigonometric layerwise deformation theory and radial basis functions. Compos Part B Eng 36: 559–572Google Scholar
30. 30.
Ferreira AJM, Fasshauer GE (2006) Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method. Comp Method Appl Mech Eng 196: 134–146
31. 31.
Ferreira AJM, Fasshauer GE (2007) Analysis of natural frequencies of composite plates by an RBF-pseudospectral method. Compos Struct 79: 202–210
32. 32.
Jiarang F, Jianqiao Y (1990) An exact solution for static and dynamics of laminated thick plates with orthotropic layers. Int J Solids Struct 26: 655–662
33. 33.
Srinivas S, Joga CV, Rao AK (1970) An exact analysis for vibration of simply supported homogeneous and laminated thick rectangular plates. J Sound Vib 12: 187–199
34. 34.
Srinivas S, Rao AK (1970) Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int J Solids Struct 6: 1463–1481
35. 35.
Vel SS, Batra RC (1999) Analytical solution for rectangular thick laminated plates subjected to arbitrary boundary conditions. AIAA J 37: 1646–1673
36. 36.
Pagano NJ (1970) Exact solutions for rectangular bidirectional composites and sandwich plates. J Compos Mater 4: 20–34Google Scholar
37. 37.
Leissa AW, Kang JH (2002) Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses. Int J Mech Sci 44: 1925–1945
38. 38.
Kang JH, Shim HJ (2004) Exact solutions for the free vibrations of rectangular plates having in-plane moments acting on two opposite simply supported edges. J Sound Vib 273: 933–948
39. 39.
Xing YF, Liu B (2009) New exact solutions for free vibrations of thin orthotropic rectangular plates. Compos Struct 89: 567–574
40. 40.
Belytschko T, Lu YY, Gu L (1994) Element free Galerkin method. Int J Numer Method Eng 37: 229–256
41. 41.
Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle method. Int J Numer Method Fluids 20: 1081–1106
42. 42.
Atluri SN, Zhu T (1998) A new meshless Petrov-Galerkin (MLPG) approach. Comp Mech 22: 117–127
43. 43.
Liu GR, Gu YT (2001) A point interpolation method for two-dimensional solids. Int J Numer Method Eng 50: 937–951
44. 44.
Li S, Liu WK (2004) Meshfree particle method. Springer, BerlinGoogle Scholar
45. 45.
Liu GR (2003) Meshfree methods: moving beyond the finite element method. CRC Press, Boca RatonGoogle Scholar
46. 46.
Wang J, Liew KM, Tan MJ, Rajendran S (2002) Analysis of rectangular laminated composite plates via FSDT meshless method. Int J Mech Sci 44: 1275–1293
47. 47.
Xiao JR, Gilhooley DF, Batra RC, Gillespie JW, Mccarthy MA (2008) Analysis of thick composite laminates using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless method. Compos Part B Eng 39: 414–427
48. 48.
Liew KM, Lim HK, Tan MK, He XQ (2002) Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method. Comp Mech 29: 486–497
49. 49.
Belinha J, Dinis LMJS (2006) Analysis of plates and laminates using the element-free Galerkin method. Comp Struct 84: 1547–1559
50. 50.
Amirani MC, Khalili SMR, Nemati N (2009) Free vibration analysis of sandwich beam with FG core using the element free Galerkin method. Compos Struct 90: 373–379
51. 51.
Liu GR, Zhao Z, Dai KY, Zhong Zh, Li GY, Han X (2008) Static and free vibration analysis of laminated composite plates using the conforming radial point interpolation method. Compos Sci Tech 68: 345–366Google Scholar
52. 52.
Xiang S, Wang KM, Ai YT, Sha YD, Shi H (2009) Analysis of isotropic, sandwich and laminated plates by a meshless method and various shear deformation theories. Compos Struct 91: 31–37
53. 53.
Dai KY, Liu GR, Lim MK, Chen XL (2004) A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates. J Sound Vib 269: 633–652
54. 54.
Chen XL, Liu GR, Lim SP (2003) An element free Galerkin method for the free vibration analysis of composite laminates of complicated shape. Compos Struct 59: 279–289
55. 55.
Belyschko T, Organ D, Krongauz Y (1995) A coupled finite element–element frer Galerkin method. Comp Mech 17: 186–195Google Scholar
56. 56.
Krysl P, Belyschko T (1995) Analysis of thin plates by the element free Galerkin method. Comp Mech 17: 26–35
57. 57.
Krysl P, Belyschko T (1996) Analysis of thin shells by the element free Galerkin method. Int J Solids Struct 33: 3057–3080
58. 58.
Donning B, Liu WK (1998) Meshless methods for shear- deformable beams and plates. Comp Method Appl Mech Eng 152: 47–72
59. 59.
Garcia O, Fancello EA, Barcellos CS, Duarte CA (2000) Hp-clouds in Mindlin’s thick plate model. Int J Numer Method Eng 47: 1381–1400
60. 60.
Liew KM, Huang YQ, Reddy JN (2003) Moving least squares differential quadrature method and its application to the analysis of shear deformable plates. Int J Numer Method Eng 56: 2331–2351
61. 61.
Liew KM, Huang YQ, Reddy JN (2004) Analysis of general shaped thin plates by the moving least-squares differential quadrature method. Finite Elem Anal Des 40: 1453–1474
62. 62.
Liu WK, Han W, Lu H, Li S, Cao J (2004) Reproducing kernel element method. Part I: Theoretical formulation. Comp Method Appl Mech Eng 193: 933–951
63. 63.
Li S, Lu H, Han W, Simkins DC Jr, Liu WK (2004) Reproducing kernel element method. Part II: Globally conforming I m/C n hierarchies. Comp Method Appl Mech Eng 193: 954–987
64. 64.
Simkins DC Jr, Li S, Lu H, Liu WK (2004) Reproducing kernel element method. Part IV: Globally compatible C n(n ≥ 1) triangular hierarchy. Comp Method Appl Mech Eng 1(193): 1013–1034
65. 65.
Wang D, Chen JS (2004) Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Comp Method Appl Mech Eng 193: 1065–1083
66. 66.
Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Method Eng 50: 435–466
67. 67.
Zhao X, Liu GR, Dai KY, Zhong ZH, Li GY, Han X (2009) A linearly conforming radial point interpolation method (LC-RPIM) for shells. Comp Mech 43: 403–413
68. 68.
Zhao X, Liu GR, Dai KY, Zhong ZH, Li GY, Han X (2009) Free-vibration analysis of shells via a linearly conforming radial point interpolation method (LC-RPIM). Finite Elem Anal Des 45: 917–924
69. 69.
Cui XY., Liu GR., Li GY., Zhang GY. (2011) Thin plate formulation without rotation DOFs based on radial point interpolation method. Int J Numer Method Eng 85: 958–986
70. 70.
Castellazzi G, Krysl P (2009) Displacement-based finite elements with nodal integration for Reissner-Mindlin plates. Int J Numer Method Eng 80: 135–162
71. 71.
Cui XY, Liu GR, Li GY, Zhang GY, Zheng G (2010) Analysis of plates and shells using an edge-based smoothed finite element method. Comp Mech 45: 141–156
72. 72.
Nguyen-Xuan H, Rabczuk T, Bordas S, Debongnie J (2008) A smoothed finite element method for plates. Comp Method Appl Mech Eng 197: 1184–1203
73. 73.
Lee PS, Bathe KJ (2004) Development of MITC isotropic triangular shell finite elements. Comput Struct 82: 945–962
74. 74.
Kim DN, Bathe KJ (2008) A 4-node 3D-shell element to model shell surface tractions and incompressible behavior. Comput Struct 86: 2027–2041
75. 75.
Kim DN, Bathe KJ (2009) A triangular six-node shell element. Comput Struct 87: 1451–1460
76. 76.
Lee PS, Bathe KJ (2010) The quadratic MITC plate and MITC shell elements in plate bending. Adv Eng Softw 41: 712–728
77. 77.
Chandrashekhar M, Ganguli R (2010) Nonlinear vibration analysis of composite laminated and sandwich plates with random material properties. Int J Mech Sci 52: 874–891
78. 78.
Gu L (2003) Moving Kriging interpolation and element free Galerkin method. Int J Numer Method Eng 56: 1–11
79. 79.
Tongsuk P, Kanok-Nukulchai W (2004) On the parametric refinement of moving Kriging interpolations for element free Galerkin method. In: Proceedings of Computational Mechanics WCCM VI in conjunction with APCOM’04, 5–10 Sep 2004, Beijing, ChinaGoogle Scholar
80. 80.
Tongsuk P, Kanok-Nukulchai W (2004) Further investigation of element free Galerkin method using moving Kriging interpolation. Int J Comp Meth 1: 1–21
81. 81.
Sayakoummane V, Kanok-Nukulchai W (2007) A meshless analysis of shells based on moving Kriging interpolation. Int J Comp Meth 4: 543–565
82. 82.
Bui QT, Nguyen NT, Nguyen-Dang H (2009) A moving Kriging interpolation-based meshless method for numerical simulation of Kirchhoff plate problems. Int J Numer Method Eng 77: 1371–1395
83. 83.
Bui QT., Nguyen NM., Zhang Ch. (2011) A moving Kriging interpolation-based element-free Galerkin method for structural dynamic analysis. Comp Method Appl Mech Eng 200: 1354–1366
84. 84.
Bui QT, Nguyen NM, Zhang Ch, Pham DAK (2010) An efficient meshfree method for analysis of two-dimensional piezoelectric structures. Smart Mater Struct (in review)Google Scholar
85. 85.
Reddy JN (1996) Mechanics of laminated composite plates and shells: theory and analysis. CRC Press, Boca RatonGoogle Scholar
86. 86.
Whitney JM (1987) Structural analysis of laminated anisotropic plates. Technomic Publishing Company Inc, Pennsylvania, USAGoogle Scholar
87. 87.
Liu Y, Hon YX, Liew KM (2006) A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems. Int J Numer Method Eng 66: 1153–1178
88. 88.
Cui XY, Liu GR, Li G (2011) A smoothed Hermite radial point interpolation method for thin plate analysis. Arch Appl Mech 81: 1–18
89. 89.
Abbassian F, Dawswell DJ, Knowles NC (1987) Free vibration benchmarks. Atkins Engineering Sciences, Glasgow, UKGoogle Scholar
90. 90.
Khosravifard A, Hermatiyan MR (2010) A new method for meshless integration in 2D and 3D Galerkin meshfree methods. Eng Anal Bound Elem 34: 30–40

## Authors and Affiliations

• Tinh Quoc Bui
• 1
Email author
• Minh Ngoc Nguyen
• 2
• Chuanzeng Zhang
• 1
1. 1.Department of Civil EngineeringUniversity of SiegenSiegenGermany
2. 2.Computational Engineering DepartmentRuhr University BochumBochumGermany