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Transverse Vibration of Thick Triangular Plates Based on a Proposed Shear Deformation Theory

  • K. K. PradhanEmail author
  • S. Chakraverty
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)

Abstract

Natural frequencies of different thick triangular plates subject to classical boundary conditions are found based on a proposed shear deformation plate theory in this chapter. The stress distribution needs no shear correction factor in this proposed plate theory. The numerical formulation is performed by means of Rayleigh–Ritz method to obtain the generalized eigenvalue problem. The aim of this study is to find the effect of different physical and geometric parameters on natural frequencies. New results along with 3D mode shapes have been evaluated after the test of convergence and validation with the available results.

Keywords

Vibration Triangular plate Shear deformation theory Rayleigh–ritz method 3D mode shapes 

Notes

Acknowledgements

The first author is thankful for the funding provided by NPIU (TEQIP-III) against TEQIP-009582 and also Parala Maharaja Engineering College, Berhampur for permitting smooth progress in terms of official provisions.

References

  1. 1.
    Ansari R, Torabi J, Hassani R (2019) A comprehensive study on the free vibration of arbitrary shaped thick functionally graded CNT-reinforced composite plates. Eng Struct 181:653–669CrossRefGoogle Scholar
  2. 2.
    Aydogdu M (2009) A new shear deformation theory for laminated composite plates. Compos Struct 89:94–101CrossRefGoogle Scholar
  3. 3.
    Bhat RB (1987) Flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables. J Sound Vib 114(1):65–71CrossRefGoogle Scholar
  4. 4.
    Cheng ZQ, Batra RC (2000) Exact correspondece between eigenvalues of membranes and functionally graded simply supported polygonal plates. J Sound Vib 229(4):879–895CrossRefGoogle Scholar
  5. 5.
    Cheung YK, Zhou D (2002) Three-dimensional vibration analysis of clamped and completely free isosceles triangular plates. Int J Solids Struct 39:673–687CrossRefGoogle Scholar
  6. 6.
    Gorman DJ (1983) A highly accurate analytical solution for free vibraion analysis of simply supported right triangular plates. J Sound Vib 89(1):107–118CrossRefGoogle Scholar
  7. 7.
    Gorman DJ (1986) Free vibration analysis of right triangular plates with combinations of clamped-simply supported boundary conditions. J Sound Vib 106(3):419–431CrossRefGoogle Scholar
  8. 8.
    Gorman DJ (1989) Accurate free vibration analysis of right triangular plate with one free edge. J Sound Vib 131(1):115–125CrossRefGoogle Scholar
  9. 9.
    Hosseini-Hashemi S, Fadaee M, Taher HRD (2011) Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third order shear deformation plate theory. Appl Math Model 35:708–727MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kang SW, Lee JM (2001) Free vibration analysis of arbitrarily shaped plates with clamped edges using wave-type functions. J Sound Vib 242(1):9–26CrossRefGoogle Scholar
  11. 11.
    Lv X, Shi D (2018) Free vibration of arbitrary-shaped laminated triangular thin plates with elastic boundary conditions. Results Phys 11:523–533CrossRefGoogle Scholar
  12. 12.
    Mirza S, Bijlani M (1985) Vibration of triangular plates of variable thickness. Comput Struct 21:1129–1135CrossRefGoogle Scholar
  13. 13.
    Pradhan KK, Chakraverty S (2016) Natural frequencies of equilateral triangular plates under classical edge supports. In: Symposium on Statistical & Computational Modelling with Applications (SymSCMA—2016), Nov 2016, pp 30–34Google Scholar
  14. 14.
    Sakiyama T, Huang M (2000) Free-vibration analysis of right triangular plates with variable thickness. J Sound Vib 234(5):841–858CrossRefGoogle Scholar
  15. 15.
    Saliba HT (1990) Transverse free vibration of simply supported right triangular thin plates: a highly accurate simplified solution. J Sound Vib 139(2):289–297CrossRefGoogle Scholar
  16. 16.
    Shimpi RP, Patel HG (2006) Free vibrations of plates using two variable refined plate theory. J Sound Vib 296:979–999CrossRefGoogle Scholar
  17. 17.
    Shimpi RP, Patel HG, Arya H (2007) New first-order shear deformation plate theories. J Appl Mech 74:523–533CrossRefGoogle Scholar
  18. 18.
    Singh B, Chakraverty S (1992) Transverse vibration of triangular plates using characteristic orthogonal polynomials in two variables. Int J Mech Sci 34(12):947–955CrossRefGoogle Scholar
  19. 19.
    Singh B, Saxena V (1996) Transverse vibration of triangular plates with variable thickness. J Sound Vib 194(4):471–496CrossRefGoogle Scholar
  20. 20.
    Thai CH, Ferreira AJM, Bordas SPA, Rabczuk T, Nguyen-Xuan H (2014) Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory. Eur J Mech A Solids 43:89–108CrossRefGoogle Scholar
  21. 21.
    Wanji C, Cheung YK (1998) Refined triangular discrete Kirchoff plate element for thin plate bending, vibration and buckling analysis. Int J Numer Meth Eng 41:1507–1525CrossRefGoogle Scholar
  22. 22.
    Xiang S, Wang K, Ai Y, Sha Y, Shi H (2009) Analysis of isotropic, sandwich and laminated plates by a meshless method and various shear deformation theories. Compos Struct 91:31–37CrossRefGoogle Scholar
  23. 23.
    Zhong HZ (2000) Free vibration analysis of isosceles triangular Mindlin plates by the triangular differential quadrature method. J Sound Vib 237(4):697–708CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of Basic SciencePMEC BerhampurSitalapalliIndia
  2. 2.Department of MathematicsNIT RourkelaRourkelaIndia

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