A Study on the Structural Behaviour of AFG Non-uniform Plates on Elastic Foundation: Static and Free Vibration Analysis

• Hareram Lohar
• Anirban Mitra
• Sarmila Sahoo
Conference paper
Part of the Lecture Notes on Multidisciplinary Industrial Engineering book series (LNMUINEN)

Abstract

Plate on elastic foundation is an important realization of actual boundary condition of structures. The present paper studies the effect of the stiffness value of the elastic foundation on large deflection and free vibration of AFG non-uniform plate. Governing set of equation of the system is obtained by energy principle through variational method. Derived nonlinear equations are handled through utilizing an iterative method, which is direct substitution method. The effects of the elastic foundations are represented through load versus maximum deflection plot, deflected shape plot and backbone curves. The effects of the edged boundary conditions and non-uniformity of the plate shape are also highlighted.

Keywords

Large deflection Free vibration AFG plate Elastic foundation Minimum potential energy principle Hamilton principle Direct substitution method

Nomenclature

$$A_{0}$$

Cross-sectional area of the plate at root side

$$a$$

Length of the plate

$$b$$

$$d$$

Unknown coefficients

$$E_{0}$$

Elastic modulus of the plate material at root side

$$\left\{ f \right\}$$

$$I_{0}$$

Moment of inertia of the plate at root side

$$D$$

Flexural rigidity of plate

$$\left[ K \right]$$

Stiffness matrix

$$\left[ M \right]$$

Mass matrix

$$nf,nw,nv$$

Number of constituent functions for w, u and v, respectively

$$\upalpha_{i}$$

Set of orthogonal functions for u

$$\upbeta_{i}$$

Set of orthogonal functions for v

$$\phi_{i}$$

Set of orthogonal functions for w

$$K_{f}$$

Foundation stiffness

$$\omega_{1}$$

Fundamental linear frequency

$$nfg$$

Number of Gauss points

$$q$$

$$t_{0}$$

Thickness of the plate at root side

$$T$$

Kinetic energy of the system

$$u$$

Displacement field in x-axis

$$U$$

Strain energy stored in the system

$$v$$

Displacement field in y-axis

$$V$$

Potential energy of the external forces

$$w$$

Displacement field in z-axis

$$\delta$$

Variational operator

$$\mu$$

Poisson’s ratio

$$\rho_{0}$$

Density of the plate material at (ξ = 0)

$$\tau$$

Time coordinate

$$\xi , \eta$$

Normalized axial coordinates

$$\alpha$$

Taper parameter

$$\omega_{nf}$$

Nonlinear frequency

References

1. 1.
Jha, D.K., Kant, T., Singh, R.K.: A critical review of recent research on functionally graded plates. Compos. Struct. 96, 833–849 (2013)
2. 2.
Birman, V., Byrd, L.W.: Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev. 60, 195–216 (2007)
3. 3.
Kennedy, D., Cheng, R.K.H., Wei, S., AlcazarArevalo, F.J.: Equivalent layered models for functionally graded plates. Comput. Struct. 174, 113–121 (2016)
4. 4.
Singha, M.K., Prakash, T., Ganapathi, M.: Finite element analysis of functionally graded plates under transverse load. Finite Elem. Anal. Des. 47, 453–460 (2011)
5. 5.
Panyatong, M., Chinnaboon, B., Chucheepsakul, S.: Free vibration analysis of FG nanoplates embedded in elastic medium based on second-order shear deformation plate theory and nonlocal elasticity. Compos. Struct. 153, 428–441 (2016)
6. 6.
Benferhat, R., Daouadji, T.H., Mansour, M.S.: Free vibration analysis of FG plates resting on an elastic foundation and based on the neutral surface concept using higher-order shear deformation theory. C. R. Mecanique 344, 631–641 (2016)
7. 7.
Sharma, P., Parashar, S.K.: Free vibration analysis of shear-induced flexural vibration of FGPM annular plate using generalized differential quadrature method. Compos. Struct. 155, 213–222 (2016)
8. 8.
Abrate, S.: Functionally graded plates behave like homogeneous plates. Compos. Part B 39, 151–158 (2008)
9. 9.
Mohammadzadeh-Keleshteri, M., Asadi, H., Aghdam, M.M.: Geometrical nonlinear free vibration responses of FG-CNT reinforced composite annular sector plates integrated with piezoelectric layers. Compos. Struct. 171, 100–112 (2017)
10. 10.
Chi, S.H., Chung, Y.L.: Mechanical behavior of functionally graded material plates under transverse load—part I: analysis. Int. J. Solids Struct. 43, 3657–3674 (2006)
11. 11.
12. 12.
Uymaz, B., Aydogdu, M., Filiz, S.: Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method. Compos. Struct. 94, 1398–1405 (2012)
13. 13.
Xiang, T., Natarajan, S., Man, H., Song, C., Gao, W.: Free vibration and mechanical buckling of plates with in-plane material inhomogeneity—A three dimensional consistent approach. Compos. Struct. 118, 634–642 (2014)
14. 14.
Liu, D.Y., Wang, C.Y., Chen, W.Q.: Free vibration of FGM plates with in-plane material inhomogeneity. Compos. Struct. 92, 1047–1051 (2010)
15. 15.
Hussein, O.S., Mulani, S.B.: Optimization of in-plane functionally graded panels for buckling strength: Unstiffened, stiffened panels, and panels with cutouts. Thin-Walled Struct. 122, 173–181 (2018)
16. 16.
Kumar, S., Mitra, A., Roy, H.: Forced vibration response of axially functionally graded non-uniform plates considering geometric nonlinearity. Int. J. Mech. Sci. 128–129, 194–205 (2017)
17. 17.
Kumar, S., Mitra, A., Roy, H.: Forced vibration analysis of functionally graded plates with geometric nonlinearity. In: Proceedings of ASME 2015 Gas Turbine India Conference - GTINDIA2015, Hyderabad, India, pp. 1–8 (2015)Google Scholar
18. 18.
Kumar, S., Mitra, A., Roy, H.: Large amplitude free vibration analysis of axially functionally graded plates. In: Proceedings of ASME 2014 Gas Turbine India Conference - GTINDIA2014, New Delhi, India, pp. 1–8 (2014)Google Scholar
19. 19.
Taczała, M., Buczkowski, R., Kleiber, M.: Postbuckling analysis of functionally graded plates on an elastic foundation. Compos. Struct. 132, 842–847 (2015)
20. 20.
Ebrahimi, F., Jafari, A., Barati, M.R.: Vibration analysis of magneto-electro-elastic heterogeneous porous material plates resting on elastic foundations. Thin-Walled Struct. 119, 33–46 (2017)
21. 21.
Kutlu, A., Ugurlu, B., Omurtag, M.H.: A combined boundary-finite element procedure for dynamic analysis of plates with fluid and foundation interaction considering free surface effect. Ocean Eng. 145, 34–43 (2017)
22. 22.
Gupta, A., Talha, M., Chaudhari, V.K.: Natural frequency of functionally graded plates resting on elastic foundation using finite element method. Procedia Technol. 23, 163–170 (2016)
23. 23.
Mohammadzadeh, B., Noh, H.C.: Analytical method to investigate nonlinear dynamic responses of sandwich plates with FGM faces resting on elastic foundation considering blast loads. Compos. Struct. 174, 142–157 (2017)
24. 24.
Zamani, H.A., Aghdam, M.M., Sadighi, M.: Free vibration analysis of thick viscoelastic composite plates on visco-Pasternak foundation using higher-order theory. Compos. Struct. 182, 25–35 (2017)
25. 25.
Najafi, F., Shojaeefard, M.H., Googarchin, H.S.: Nonlinear low-velocity impact response of functionally graded plate with nonlinear three-parameter elastic foundation in thermal field. Compos. B 107, 123–140 (2016)
26. 26.
Wattanasakulpong, N., Chaikittiratana, A.: Exact solutions for static and dynamic analyses of carbon nanotube-reinforced composite plates with Pasternak elastic foundation. Appl. Math. Model. 39, 5459–5472 (2015)
27. 27.
Barati, M.R., Sadr, M.H., Zenkour, A.M.: Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic foundation. Int. J. Mech. Sci. 117, 309–320 (2016)
28. 28.
Shahsavari, D., Shahsavari, M., Li, L., Karami, B.: A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation. Aerosp. Sci. Technol. 72, 134–149 (2018)
29. 29.
Reddy, J.N.: Energy Principles & Variational Methods in Applied Mechanics, 2nd edn. Wiley, New Jersey, USA (2002)Google Scholar
30. 30.
Lohar, H., Mitra, A., Sahoo, S.: Geometric nonlinear free vibration of axially functionally graded non-uniform beams supported on elastic foundation. Curved Layer. Struct. 3, 223–239 (2016)Google Scholar
31. 31.
Mitra, A., Sahoo, P., Saha, K.N.: Nonlinear vibration analysis of simply supported stiffened plate by a variational method. Mech. Adv. Mater. Struct. 20, 373–396 (2013)
32. 32.
Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells, 2nd edn. McGraw-Hill Classic Textbook Reissue Series, New York, USA (1964)
33. 33.
Leissa, A.W.: The free vibration of rectangular plates. J. Sound Vib. 3, 257–293 (1973)
34. 34.
Saha, K.N., Misra, D., Pohit, G., Ghosal, S.: Large amplitude free vibration study of square plates under different boundary conditions through a static analysis. J. Sound Vib. 10, 1009–1028 (2004)