Skip to main content
SpringerLink
Log in

A novel 2-D six-parameter power-law distribution for three-dimensional dynamic analysis of thick multi-directional functionally graded rectangular plates resting on a two-parameter elastic foundation

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

This paper is motivated by the lack of studies in the technical literature concerning to the three-dimensional vibration analysis of bi-directional FG rectangular plates resting on two-parameter elastic foundations. The formulations are based on the three-dimensional elasticity theory. The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. This paper presents a novel 2-D six-parameter power-law distribution for ceramic volume fraction of 2-D FGM that gives designers a powerful tool for flexible designing of structures under multi-functional requirements. Various material profiles along the thickness and in the in-plane directions are illustrated using the 2-D power-law distribution. The effective material properties at a point are determined in terms of the local volume fractions and the material properties by the Mori-Tanaka scheme. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. Some new results for natural frequencies of the plates are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The interesting results indicate that a graded ceramic volume fraction in two directions has a higher capability to reduce the natural frequency than conventional 1-D FGM.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Koizumi M (1997) FGM activities in Japan. Composites, Part B, Eng 28:1–4

    Article  Google Scholar 

  2. Suresh S, Mortensen A (1998) Fundamentals of functionally graded materials. IOM Communications, London

    Google Scholar 

  3. Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG (1999) Functionally graded materials: design, processing and applications. Kluwer Academic, Dordrecht

    Book  Google Scholar 

  4. Qian LF, Batra RC (2005) Three-dimensional transient heat conduction in a functionally graded thick plate with a higher-order plate theory and a meshless local Petrov–Galerkin method. Comput Mech 35:214–226

    Article  MATH  Google Scholar 

  5. Yang J, Kitipornchai S, Liew KM (2004) Non-linear analysis of the thermo-electro-mechanical behaviour of shear deformable FGM plates with piezoelectric actuators. Int J Numer Methods Eng 59:1605–1632

    Article  MATH  Google Scholar 

  6. Naghdabadi R, Kordkheili SAH (2005) A finite element formulation for analysis of functionally graded plates and shells. Arch Appl Mech 74:375–386

    MATH  Google Scholar 

  7. Kordkheili SAH, Naghdabadi R (2007) Geometrically non-linear thermoelastic analysis of functionally graded shells using finite element method. Int J Numer Methods Eng 72:964–986

    Article  MATH  Google Scholar 

  8. Steinberg MA (1986) Materials for aerospace. Sci Am 255:59–64

    Article  Google Scholar 

  9. Xiang Y, Kitipornchai S, Liew KM (1996) Buckling and vibration of thick laminates on Pasternak foundations. J Eng Mech 122:54–63

    Article  Google Scholar 

  10. Xiang Y, Wang CM, Kitipornchai S (1994) Exact vibration solution for initially stressed Mindlin plates on Pasternak foundations. Int J Mech Sci 36:311–316

    Article  MATH  Google Scholar 

  11. Wang CM, Kitipornchai S, Xiang Y (1997) Relationships between buckling loads of Kirchhoff, Mindlin, and Reddy polygonal plates. J Eng Mech 123:1134–1137

    Article  Google Scholar 

  12. Gupta US, Lal R, Sagar R (1994) Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic Mindlin circular plates. Indian J Pure Appl Math 25:1317–1326

    MATH  Google Scholar 

  13. Ju F, Lee HPKH (1995) Free vibration of plates with stepped variations in thickness on non-homogeneous elastic foundations. J Sound Vib 183:533–545

    Article  ADS  MATH  Google Scholar 

  14. Gupta US, Lal R, Jain SK (1990) Effect of elastic foundation on axisymmetric vibrations of polar orthotropic circular plates of variable thickness. J Sound Vib 139:503–513

    Article  ADS  Google Scholar 

  15. Gupta US, Ansari AH (2002) Effect of elastic foundation on axisymmetric vibrations of polar orthotropic linearly tapered circular plates. J Sound Vib 254:411–426

    Article  ADS  Google Scholar 

  16. Laura PAA, Gutierrez RH (1991) Free vibrations of a solid circular plate of linearly varying thickness and attached to Winkler foundation. J Sound Vib 144:149–161

    Article  ADS  Google Scholar 

  17. Matsunaga H (2008) Free vibration and stability of functionally graded plates according to a 2D higher-order deformation theory. J Compos Struct 82:499–512

    Article  Google Scholar 

  18. Zhou D, Cheung YK, Lo SH, Au FTK (2004) Three-dimensional vibration analysis of rectangular thick plates on Pasternak foundation. Int J Numer Methods Eng 59:1313–1334

    Article  MATH  Google Scholar 

  19. Matsunaga H (2000) Vibration and stability of thick plates on elastic foundations. J Eng Mech 126:27–34

    Article  Google Scholar 

  20. Yas MH, Sobhani Aragh B (2010) Free vibration analysis of continuous grading fiber reinforced plates on elastic foundation. Int J Eng Sci 48:1881–1895

    Article  MATH  Google Scholar 

  21. Yas MH, Tahouneh V (2012) 3-D free vibration analysis of thick functionally graded annular plates on Pasternak elastic foundation via differential quadrature method (DQM). Acta Mech 223:43–62

    Article  MATH  MathSciNet  Google Scholar 

  22. Tahouneh V, Yas MH (2012) 3-D free vibration analysis of thick functionally graded annular sector plates on Pasternak elastic foundation via 2-D differential quadrature method. Acta Mech 223:1879–1897

    Article  MATH  MathSciNet  Google Scholar 

  23. Tahouneh V, Yas MH, Tourang H, Kabirian M (2012) Semi-analytical solution for three-dimensional vibration of thick continuous grading fiber reinforced (CGFR) annular plates on Pasternak elastic foundations with arbitrary boundary conditions on their circular edges. Meccanica. Online first. doi:10.1007/s11012-012-9669-4

    Google Scholar 

  24. Tahouneh V, Yas MH (2013) Semi-analytical solution for three-dimensional vibration analysis of thick multi-directional functionally graded annular sector plates under various boundary conditions. J Eng Mech, Online first. doi:10.1061/(ASCE)EM.1943-7889.0000653

  25. Liew KM, Han JB, Xiao ZM, Du H (1996) Differential quadrature method for Mindlin plates on Winkler foundations. Int J Mech Sci 38:405–421

    Article  MATH  Google Scholar 

  26. Zhou D, Lo SH, Au FTK, Cheung YK (2006) Three-dimensional free vibration of thick circular plates on Pasternak foundation. J Sound Vib 292:726–741

    Article  ADS  Google Scholar 

  27. Yang J, Shen HS (2001) Dynamic response of initially stressed functionally graded rectangular thin plates. J Compos Struct 54:497–508

    Article  Google Scholar 

  28. Cheng ZQ, Kitipornchai S (1999) Membrane analogy of buckling and vibration of inhomogeneous plates. J Eng Mech 125:1293–1297

    Article  Google Scholar 

  29. Batra RC, Jin J (2005) Natural frequencies of a functionally graded anisotropic rectangular plate. J Sound Vib 282:509–516

    Article  ADS  Google Scholar 

  30. Cheng ZQ, Batra RC (2000) Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates. J Sound Vib 229:879–895

    Article  ADS  MATH  Google Scholar 

  31. Malekzadeh P (2008) Three-dimensional free vibrations analysis of thick functionally graded plates on elastic foundations. J Compos Struct 89:367–373

    Article  Google Scholar 

  32. Bellman R, Casti J (1971) Differential quadrature and long term integration. J Math Anal Appl 34:235–238

    Article  MATH  MathSciNet  Google Scholar 

  33. Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49:1–28

    Article  ADS  Google Scholar 

  34. Malekzadeh P (2008) Differential quadrature large amplitude free vibration analysis of laminated skew plates based on FSDT. Compos Struct 83:189–200

    Article  Google Scholar 

  35. Shu C (2000) Differential quadrature and its application in engineering. Springer, Berlin

    Book  MATH  Google Scholar 

  36. Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 21:571–574

    Article  Google Scholar 

  37. Benveniste Y (1987) A new approach to the application of Mori-Tanaka’s theory of composite materials. Mech Mater 6:147–157

    Article  Google Scholar 

  38. Vel SS, Batra RC (2002) Exact solution for thermoelastic deformations of functionally graded thick rectangular plates. AIAA J 40:1421–1433

    Article  ADS  Google Scholar 

  39. Vel SS (2010) Exact elasticity solution for the vibration of functionally graded anisotropic cylindrical shells. Compos Struct 92:2712–2727

    Article  Google Scholar 

  40. Shu C, Wang CM (1999) Treatment of mixed and non-uniform boundary conditions in GDQ vibration analysis of rectangular plates. Eng Struct 21:125–134

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Shahre-Rey Branch, Islamic Azad University, Tehran, Iran

    V. Tahouneh

  2. Department of Mechanical Engineering, Tehran University, Tehran, Iran

    M. H. Naei

Authors
  1. V. Tahouneh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. H. Naei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Tahouneh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tahouneh, V., Naei, M.H. A novel 2-D six-parameter power-law distribution for three-dimensional dynamic analysis of thick multi-directional functionally graded rectangular plates resting on a two-parameter elastic foundation. Meccanica 49, 91–109 (2014). https://doi.org/10.1007/s11012-013-9776-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-013-9776-x

Keywords

Search

Navigation