Buckling and Free Vibration Analysis of Fiber Metal-laminated Plates Resting on Partial Elastic Foundation

Document Type: Research Paper


1 University Complex of Materials and Manufacturing Technology, Malek Ashtar University of Technology, Lavizan, Tehran, Iran

2 Department of Mechanical Engineering, Malek Ashtar University of Technology, Lavizan, Tehran, Iran


This research presents, buckling and free vibration analysis of fiber metal-laminated (FML) plates on a total and partial elastic foundation using the generalized differential quadrature method (GDQM). The partial foundation consists of multi-section Winkler and Pasternak type elastic foundation. Taking into consideration the first-order shear deformation theory (FSDT), FML plate is modeled and its equations of motion and boundary conditions are derived using Hamilton's principle. The formulations include Heaviside function effects due to the nonhomogeneous foundation. The novelty of this study is considering the effects of partial foundation and in-plane loading, in addition to considering the various boundary conditions of FML plate. A computer program is written using the present formulation for calculating the natural frequencies and buckling loadings of composite plates without contacting with elastic foundation and composite plates resting on partial foundations. The validation is done by comparison of continuous element model with available results in the literature. The results show that the constant of total or partial spring, elastic foundation parameter, thickness ratio, frequency mode number and boundary conditions play an important role on the critical buckling load and natural frequency of the FML plate resting on partial foundation under in-plane force.


Main Subjects

[1] Winkler, E., Die Lehre von der Elasticitaet und Festigkeit: mit besonderer Rücksicht auf ihre Anwendung in der Technik für polytechnische Schulen, Bauakademien, Ingenieue, Maschinenbauer, Architecten, etc. Dominicus, 1867.

[2] Pasternak, P., On a new method of analysis of an elastic foundation by means of two foundation constants. Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture, Moscow, 1954.

[3] Timoshenko, S., Theory of Elastic Stability 2e. Tata McGraw-Hill Education, 1970.

[4] Vlasov, V.Z., Beams, plates and shells on elastic foundations. Israel Program for Scientific Translations, Jerusalem, 1966.

[5] Seide, P., Compressive buckling of a long simply supported plate on an elastic foundation, Journal of the Aeronautical Sciences, 25(6), 1958, 382-384.

[6] Cheung, Y., Zinkiewicz, O., Plates and tanks on elastic foundations—an application of finite element method, International Journal of Solids and Structures, 1(4), 1965, 451-461.

[7] Cheung, Y., Nag, D., Plates and beams on elastic foundations–linear and non-linear behaviour, Geotechnique, 18(2), 1968, 250-260.

[8] Akbarov, S.D., KocatÜrk, T., On the bending problems of anisotropic (orthotropic) plates resting on elastic foundations that react in compression only, International Journal of Solids and Structures, 34(28), 1997, 3673-3689.

[9] Wekezer, J.W., Chilton, D.S., Plates on Elastic Foundation, Journal of Structural Engineering, 116(11), 1990, 3236-3241.

[10] Henwood, D., Whiteman, J., Yettram, A., Fourier series solution for a rectangular thick plate with free edges on an elastic foundation, International Journal for Numerical Methods in Engineering, 18(12), 1982, 1801-1820.

[11] Katsikadelis, J., Armenakas, A., Plates on elastic foundation by BIE method, Journal of Engineering Mechanics, 110(7), 1984, 1086-1105.

[12] Puttonen, J., Varpasuo, P., Boundary element analysis of a plate on elastic foundations, International Journal for Numerical Methods in Engineering, 23(2), 1986, 287-303.

[13] Prakash, B.G., Flexure of Clamped Rectangular Plates Resting on Winkler Foundations, Journal of Structural Mechanics, 7(2), 1979, 131-142.

[14] Xiang, Y., Wang, C.M., Kitipornchai, S., Exact vibration solution for initially stressed Mindlin plates on Pasternak foundations, International Journal of Mechanical Sciences, 36(4), 1994, 311-316.

[15] Mindlin, R., Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics, 18, 1951, 31.

[16] Xiang, Y., Kitipornchai, S., Liew, K.M., Buckling and Vibration of Thick Laminates on Pasternak Foundations, Journal of Engineering Mechanics, 122(1), 1996, 54-63.

[17] Matsunaga, H., Vibration and Stability of Thick Plates on Elastic Foundations, Journal of Engineering Mechanics, 126(1), 2000, 27-34.

[18] Civalek, Ö., Free vibration analysis of symmetrically laminated composite plates with first-order shear deformation theory (FSDT) by discrete singular convolution method, Finite Elements in Analysis and Design, 44(12), 2008, 725-731.

[19] Civalek, Ö., Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods, Applied Mathematical Modelling, 31(3), 2007, 606-624.

[20] Akavci, S., Buckling and free vibration analysis of symmetric and antisymmetric laminated composite plates on an elastic foundation, Journal of Reinforced Plastics and Composites, 26(18), 2007, 1907-1919.

[21] Zenkour, A.M., The refined sinusoidal theory for FGM plates on elastic foundations, International Journal of Mechanical Sciences, 51(11), 2009, 869-880.

[22] Malekzadeh, P., Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations, Composite Structures, 89(3), 2009, 367-373.

[23] Malekzadeh, K., Khalili, S.M.R., Abbaspour, P., Vibration of non-ideal simply supported laminated plate on an elastic foundation subjected to in-plane stresse, Composite Structures, 92(6), 2010, 1478-1484.

[24] Dehghan, M., Baradaran, G.H., Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method. Applied Mathematics and Computation, 218(6), 2011, 2772-2784.

[25] Sobhy, M., Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions, Composite Structures, 99, 2013, 76-87.

[26] Dehghany, M., Farajpour, A., Free vibration of simply supported rectangular plates on Pasternak foundation: An exact and three-dimensional solution, Engineering Solid Mechanics, 2(1), 2014, 29-42.

[27] Thinh, T.I., Nguyen, M.C., Ninh, D.G., Dynamic stiffness formulation for vibration analysis of thick composite plates resting on non-homogenous foundations, Composite Structures, 108, 2014, 684-695.

[28] Mantari, J.L., Free vibration of advanced composite plates resting on elastic foundations based on refined non-polynomial theory, Meccanica, 50(9), 2015, 2369-2390.

[29] Gupta, A., Talha, M., Seemann, W., Free vibration and flexural response of functionally graded plates resting on Winkler–Pasternak elastic foundations using non-polynomial higher order shear and normal deformation theory, Mechanics of Advanced Materials and Structures, 25(6), 2018, 523-538.

[30] Moradi-Dastjerdi, R., Momeni-Khabisi, H., Vibrational behavior of sandwich plates with functionally graded wavy carbon nanotube-reinforced face sheets resting on Pasternak elastic foundation, Journal of Vibration and Control, 24(11), 2018, 2327-2343.

[31] Mansouri, M.H., Shariyat, M., Differential quadrature thermal buckling analysis of general quadrilateral orthotropic auxetic FGM plates on elastic foundations, Thin-Walled Structures, 112, 2017, 194-207.

[32] Reddy, J.N., Mechanics of laminated composite plates and shells: theory and analysis. CRC press, 2004.

[33] Jones, R.M., Mechanics of composite materials, 193, Scripta Book Company Washington, DC, 1975.

[34] Tauchert, T.R., Energy principles in structural mechanics. McGraw-Hill Companies, 1974.

[35] Bracewell, R., Heaviside's Unit Step Functio, The Fourier Transform and Its Applications, 2000, 61-65.

[36] Bellman, R., Casti, J., Differential quadrature and long-term integration, Journal of Mathematical Analysis and Applications, 34(2), 1971, 235-238.

[37] Bellman, R., Kashef, B., Casti, J., Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics, 10(1), 1972, 40-52.

[38] Shu, C., Differential quadrature and its application in engineering, Springer Science & Business Media, 2012.

[39] Shu, C., Richards, R.E., Application of generalized differential quadrature to solve two‐dimensional incompressible Navier‐Stokes equations, International Journal for Numerical Methods in Fluids, 15(7), 1992, 791-798.

[40] Khdeir, A., Free vibration and buckling of symmetric cross-ply laminated plates by an exact method, Journal of Sound and Vibration, 126(3), 1988, 447-461.

[41] Aiello, M.A., Ombres, L., Buckling and vibrations of unsymmetric laminates resting on elastic foundations under inplane and shear forces, Composite Structures, 44(1), 1999, 31-41.