FE Analysis of FGM Plates on Arbitrarily Orthotropic Pasternak Foundations for Membrane Effects

Öz

In this study, the finite element analysis of sigmoid functionally graded material (S-FGM) plates resting on orthotropic Pasternak elastic foundations with different material angles is presented. For modelling, SAP2000 software package is used in which the required adjustments are made to obtain the expected behaviour of the plate-foundation system. The plate is modelled both using solid elements and layered shell elements by defining a number of solid elements and layers in the thickness direction having elastic properties equivalent to the properties of the S-FGM plate. The interaction between the plate and the foundation is treated by equalizing the vertical displacements of the plate and foundation nodal points. The orthotropic Pasternak foundation is modelled using plane strain elements with some adjustments to the elastic properties. The membrane effects of the simply supported S-FGM plate on Pasternak foundation are considered by assigning the edge boundaries of the system as pinned supports and these effects are excluded by converting all boundary nodes into roller supports except one of the corner nodes of the plate and the foundation due to the stability requirement. A number of verification examples are performed to demonstrate the convenience and robustness of the proposed model. This work can be easily extended to static and dynamic analyses of FGM plates with various geometries resting on arbitrarily orthotropic Pasternak elastic foundations for further studies.

Anahtar Kelimeler

• Functionally graded material plates
• membrane effects
• orthotropic Pasternak foundation
• material angle

Kaynakça

1. Suresh S., Mortenson A., Fundamentals of Functionally Graded Materials: Processing and Thermomechanical Behavior of Graded Metals and Metal-Ceramic Composites. IOM Communications Ltd, London, UK, 1998.
2. Koizumi M., FGM activities in Japan. Composites Part B: Eng 28(1-2), 1-4, 1997.
3. Tanigawa Y., Some basic thermoelastic problems for nonhomogeneous structural materials. Appl. Mech. Rev. 48(6), 287-300, 1995.
4. Suresh S., Mortensen A., Functionally graded metals and metalceramic composites 2: thermomechanical behaviour. Int. Mater. Rev. 42(3), 85-116, 1997.
5. Reddy J.N., Analysis of functionally graded plates. Int. J. Numer. Methods Eng., 47, 663-684, 2000.
6. Cheng Z.Q., Batra R.C. Deflection relationships between the homogenous Kirchhoff plate theory and different functionally graded plate theories. Archives of Mechanics 52, 143-158, 2000.
7. Bao G., Wang L., Multiple cracking in functionally graded ceramic/metal coatings. Int. J. Solids Struct., 32, 2853-2871, 1995.
8. Jin Z.H., Paulino G.H., Transient thermal stress analysis of an edge crack in a functionally graded material. Int. J. Fracture, 107, 73-98, 2001.

9. Delale F., Erdogan F., The crack problem for a nonhomogeneous plane. ASME J. Appl. Mech., 50, 609-614, 1983.
10. Erdogan F., Wu B.H., Crack problems in FGM layers under thermal stresses. J. Therm. Stress., 19, 237-265, 1996.
11. Chung Y.L., Chi S.H., The residual stress of functionally graded materials. J. Chin. Inst. Civ. Hydraul. Eng., 13, 1-9, 2001.
12. Chi S.H., Chung Y.L., Cracking in sigmoid functionally graded coating. J. Mech., 18, 41-53, 2002.
13. Chi S.H., Chung Y.L., Mechanical behavior of functionally graded material plates under transverse load-Part I: Analysis. Int. J. of Solids and Str., 43, 3657-3674, 2006a.
14. Chi S.H., Chung Y.L., Mechanical behavior of functionally graded material plates under transverse load-Part II: Numerical results. Int. J. of Solids and Str., 43, 3675-3691, 2006b.
15. Orakdöğen E., Küçükarslan S., Sofiyev A., Omurtag M.H., Finite element analysis of functionally graded plates for coupling effect of extension and bending. Meccanica, 45, 63-72, 2010.
16. Zenkour A.M., Generalized shear deformation theory for bending analysis of functionally graded plates. Appl. Math. Model., 30, 67-84, 2006.
17. Elishakoff I., Gentilini C., Viola E., Three-dimensional analysis of an all-round clamped plate made of functionally graded materials. Acta Mech., 180(1-4), 21-36, 2005.
18. Swaminathan K., Naveenkumar D.T., Zenkour A.M., Carrera E., Stress, vibration and buckling analyses of FGM plates-A state-of-the-art Review. Composite Structures, 120, 10-31, 2015.
19. Winkler E. Theory of Elasticity and Strength of Materials. Dominicus, Prague, 1867.
20. Pasternak P.L., On a new method of analysis of an elastic foundation by means of two foundation constants. Cosudarstrennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture, Moscow, USSR, 1-56, 1954.
21. Vallabhan C.V.G., Straughan W.T., Das Y.C., Refined Model of Analysis of Plates of Elastic Foundations. Journal of Engineering Mechanics, 117(12), 2830-2844, 1994.
22. Celik M., Saygun A., A method for the analysis of plates on a two-parameter foundation. Int. J. of Solids and Str., 36, 2891-2915, 1999.
23. Xiang Y., Wang C.M., Kitipornchai S., Exact vibration solution for initially stressed Mindlin plates on Pasternak foundation. Int. J. Mech. Sci., 36(4), 311-316, 1994.
24. Omurtag M.H., Ozutok A., Akoz A.Y., Free vibration analysis of Kirchhoff plates resting on elastic foundation by mixed finite element formulation based on Gateaux differential. Int. J. Numer. Methods Eng., 40(2), 295-317, 1997.
25. Matsunaga H. Vibration and stability of thick plates on elastic foundations. ASCE J. Eng. Mech., 126(1), 27-34, 2000.
26. Zhou D., Cheung Y.K., Lo S.H., Au F.T.K., Three-dimensional vibration analysis of rectangular thick plates on Pasternak foundations. Int. J. Numer. Methods. Eng., 59(10), 1313-1334, 2004.
27. Shen H.S., Postbuckling analysis of composite laminated plates on two-parameter elastic foundations. Int. J. Mech. Sci., 37(12), 1307-131, 1995.
28. Huang Z.Y., Lü C.F., Chen W.Q., Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations. Composite Structures, 85, 95-104, 2008.
29. Lee W.H., Han S.C., Park W.T., A refined higher order shear and normal deformation theory for E-,P-, and S-FGM plates on Pasternak elastic foundation. Composite Structures, 122, 330-342, 2015.
30. Tajeddini V., Ohadi A., Sadighi M., Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation. Int. J. Mech. Sci., 53(4), 300-308, 2011.
31. Baferani A.H., Saidi A.R., Ehteshami H., Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation. Composite Structures, 93(7), 1842-1853, 2011.
32. Malekzadeh P., Golbahar Haghighi M.R., Atashi M.M., Free vibration analysis of elastically supported functionally graded annular plates subjected to thermal environment. Meccanica, 46, 893-913, 2011.
33. Thai H.T., Choi D.H., A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation. Composites Part B, 43, 2335-2347, 2012.
34. Mansouri M.H., Shariyat M., Differential quadrature thermal buckling analysis of general quadrilateral orthotropic auxetic FGM plates on elastic foundations. Thin-Walled Structures, 112, 194-207, 2017.
35. Hamarat M.A., Dynamic analysis of structures resting on two parameter elastic foundation. MSc Dissertation, Istanbul Technical University, Istanbul, 2012.
36. Hamarat M.A., Çalık-Karaköse Ü.H., Orakdöğen E., Seismic Analysis of Structures Resting on Two Parameter Elastic Foundation. 15th World Conference on Earthquake Engineering. Lisbon, Portugal, September, 2012.
37. Kutlu A., Analysis of free vibrations of Mindlin plates resting on arbitrarily orthotropic Pasternak foundation with mixed finite elements. MSc Dissertation, Istanbul Technical University, Istanbul, 2007.
38. Kutlu A., Omurtag M.H., Large deflection bending analysis of elliptic plates on orthotropic elastic foundation with mixed finite element method, International Journal of Mechanical Sciences, 65, 64–74, 2012.
39. Kutlu A., Ugurlu B., Omurtag M.H., Ergin A., Dynamic response of Mindlin plates resting on arbitrarily orthotropic Pasternak foundation and partially in contact with fluid. Ocean Engineering, 42,112-125, 2012.
40. Kolahchi R., Safari M., Esmailpour M., Dynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium. Composite Structures, 150, 255-265, 2016.
41. Heydari M. M., Bidgoli A. H., Golshani H. R., Beygipoor G., Davoodi, A., Nonlinear bending analysis of functionally graded CNT-reinforced composite Mindlin polymeric temperature-dependent plate resting on orthotropic elastomeric medium using GDQM. Nonlinear Dynamics, 79(2), 1425- 1441, 2015.
42. Arani A. G., Cheraghbak A., Kolahchi R., Dynamic buckling of FGM viscoelastic nanoplates resting on orthotropic elastic medium based on sinusoidal shear deformation theory. Structural engineering and mechanics: An international journal, 60(3), 489-505, 2016.
43. SAP2000, v18., Integrated Finite Elements Analysis and Design of Structures, Computers and Structures. p. Inc, Berkeley, CA, 2016.
44. Elmacı Ş., Spectral analysis and determination of free vibration characteristics of quadratic plates resting on arbitrarily orthotropic two parameter elastic foundation. MSc Dissertation, Istanbul Technical University, Istanbul, 2019.
45. Aykılıç B., Spectral analysis of circular and elliptic plates resting on arbitrarily orthotropic Pasternak type foundation. MSc Dissertation, Istanbul Technical University, Istanbul, 2019.
46. Lam K.Y., Wang C.M., He X.Q., Canonical exact solution for Levy-plates on two parameter foundation using Green’s functions. Eng. Struct., 22(4),364-378, 2000.