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Forced vibration analysis of Mindlin plates resting on Winkler foundation

Yaprak Itır Özdemir



The purpose of this paper is to study shear locking-free parametric earthquake analysis of thick and thin plates resting on Winkler foundation using Mindlin’s theory, to determine the effects of the thickness/span ratio, the aspect ratio and the boundary conditions on the linear responses of thick and thin plates subjected to earthquake excitations. In the analysis, finite element method is used for spatial integration and the Newmark-β method is used for the time integration. Finite element formulation of the equations of the thick plate theory is derived by using higher order displacement shape functions. A computer program using finite element method is coded in C++ to analyze the plates clamped or simply supported along all four edges. In the analysis, 8-noded finite element is used. Graphs are presented that should help engineers in the design of thick plates subjected to earthquake excitations. It is concluded that 8-noded finite element can be effectively used in the earthquake analysis of thick plates. It is also concluded that, in general, the changes in the thickness/span ratio are more effective on the maximum responses considered in this study than the changes in the aspect ratio.


earthquake analysis; thick plate; Mindlin’s theory; 8-noded finite element; Winkler foundation

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Ayvaz Y, Daloğlu A, Doğangün A (1998). Application of a modified Vlasov model to earthquake analysis of the plates resting on elastic foundations. Journal of Sound & Vibration, 212(3), 499-509.

Ayvaz Y, Oguzhan CB (2008). Free vibration analysis of plates resting on elastic foundations using modified Vlasov model. Structural Engi-neering & Mechanics, 28(6), 635-658.

Bathe KJ (1996). Finite Element Procedures. Prentice Hall, Upper Saddle River, New Jersey.

Benferhat R, Daouadji TH, Mansour MS, Hadji L (2016). Effect of porosi-ty on the bending and free vibration response of functionally graded plates resting on Winkler-Pasternak foundations. Earthquakes & Structures, 10(6), 1429-1449.

Cook RD, Malkus, DS, Michael EP (1989). Concepts and Applications of Finite Element Analysis. John Wiley & Sons, Inc., Canada.

Grice RM, Pinnington RJ (2002). Analysis of the flexural vibration of a thin-plate box using a combination of finite element analysis and an-alytical impedances. Journal of Sound & Vibration, 249(3), 499-527.

Hetenyi M (1950). A general solution for the bending of beams on an elastic foundation of arbitrary continuity. Journal of Applied. Physics, 21, 55-58.

Kutlu A, Uğurlu B, Omurtag MH (2012). Dynamic response of Mindlin plates resting on arbitrarily orthotropic Pasternak foundation and partially in contact with fluid. Ocean Engineering, 42, 112-125.

Leissa AW (1973). The free vibration of rectangular plates. Journal of Sound & Vibration, 31(3), 257-294.

Lok TS, Cheng QH (2001). Free and forced vibration of simply supported, orthotropic sandwich panel. Computers & Structures, 79(3), 301-312.

Mindlin RD (1951). Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics, 18, 31-38.

Omurtag MH, Kadıoğlu F (1998). Free vibration analysis of orthotropic plates resting on Pasternak foundation by mixed finite element for-mulation. Computers & Structures, 67, 253-265.

Özdemir YI (2007). Parametric Analysis of Thick Plates Subjected to Earthquake Excitations by Using Mindlin’s Theory. Ph.D. thesis, Ka-radeniz Technical University, Trabzon.

Özdemir YI (2012). Development of a higher order finite element on a Winkler foundation. Finite Element. Analysis and Design, 48, 1400-1408.

Özdemir YI, Bekiroğlu S, Ayvaz Y (2007). Shear locking-free analysis of thick plates using Mindlin’s theory. Structural Engineering & Me-chanics, 27(3), 311-331.

Özgan K, Daloglu AT (2012). Free vibration analysis of thick plates on elastic foundations using modified Vlasov model with higher order finite elements. International Journal of Engineering & Materials Sciences, 19, 279-291.

Pasternak PL (1954). New method of calculation for flexible substruc-tures on two-parameter elastic foundation. Gasudarstvennoe Iz-datelstoo. Literatury po Stroitelstvu I Architekture, 1-56, Moscow.

Providakis CP, Beskos DE (1989a). Free and forced vibrations of plates by boundary elements. Computer Method Applied Mechanics, 74, 231-250.

Providakis CP, Beskos DE (1989b). Free and forced vibrations of plates by boundary and interior elements. International Journal of Numer-ical Methods in Engineering, 28, 1977-1994.

Qiu J, Feng ZC (2000). Parameter dependence of the impact dynamics of thin plates. Computers & Structures, 75(5), 491-506.

Senjanovic I, Tomic M, Hadzic N, Vladimir N (2017). Dynamic finite element formulations for moderately thick plate vibrations based on the modified Mindlin theory. Engineering Structures, 136, 100-113.

Sheikholeslami SA, Saidi AR. (2013). Vibration analysis of functionally graded rectangular plates resting on elastic foundation using higher-order shear and normal deformable plate theory. Computers & Struc-tures, 106, 350-361.

Si WJ, Lam KY, Gang SW (2005). Vibration analysis of rectangular plates with one or more guided edges via bicubic B-spline method. Shock & Vibration, 12(5), 363-376.

Tahouneh V (2014). Free vibration analysis of thick CGFR annular sector plates resting on elastic foundations. Structural Engineering & Mechanics, 50(6), 773-796.

Tedesco JW, McDougal WG, Ross CA (1999). Structural Dynamics. Addison Wesley Longman Inc., California.

Timoshenko S, Woinowsky-Krieger S (1959). Theory of Plates and Shells. Second edition, McGraw-Hill, New York.

Ugural AC (1981). Stresses in Plates and Shells. McGraw-Hill, New York.

Vlasov VZ, Leont’ev NN (1989). Beam, plates and shells on elastic foundations. GIFML, Moscow.

Weaver W, Johnston PR (1984). Finite Elements for Structural Analysis. Prentice Hall, Inc., Englewood Cliffs, New Jersey.

Winkler E (1867). Theory of Elasticity and Strength. Dominicus Pague, Czechoslovakia.

Wu LH (2012). Free vibration of arbitrary quadrilateral thick plates with internal colums and uniform elastic edge supports by pb-2 Ritz method. Structural Engineering & Mechanics, 44(3), 267-288.

Zamani HA, Aghdam MM, Sadighi M (2017). Free vibration analysis of thick viscoelastic composite plates on visco-Pasternak foundation using higher-order theory. Computers & Structures, 182, 25-35.

Zienkiewich OC, Taylor RL, Too JM (1971). Reduced integration tech-nique in general analysis of plates and shells. International Journal of Numerical Methods in Engineering, 3, 275-290.

Zienkiewich OC, Taylor RL, Too JM (1989). The Finite Element Method. Fourth ed., McGraw-Hill, New York.


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