Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theory

Document Type: Research Paper

Authors

1 SRES College of Engineering, Kopargaon, Maharashtra, India.

2 Head of Department of Applied Mechanics, Govt. College of EngineeringKarad, Shivaji University, Maharashtra-415124, India

Abstract

A trigonometric plate theory is assessed for the static bending analysis of plates resting on Winkler elastic foundation. The theory considers the effects of transverse shear and normal strains. The theory accounts for realistic variation of the transverse shear stress through the thickness and satisfies the traction free conditions at the top and bottom surfaces of the plate without using shear correction factors. The governing equations of equilibrium and the associated boundary conditions of the theory are obtained using the principle of virtual work. A closed-form solution is obtained using double trigonometric series. The numerical results are obtained for flexure of simply supported plates subjected to various static loadings. The displacements and stresses are obtained for three different values of foundation modulus. The numerical results are also generated using higher order shear deformation theory of Reddy, first order shear deformation theory of Mindlin, and classical plate theory for the comparison of the present results.

Keywords

Main Subjects

[1] Kirchhoff, G.R., Uber das gleichgewicht und die bewegung einer elastischen Scheibe, Journal for Pure and Applied Mathematics, 40, 1850, 51-88.

[2] Mindlin, R.D., Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics, 18, 1951, 31-38.

[3] Reddy, J.N., A simple higher order theory for laminated composite plates, Journal of Applied Mechanics, 51, 1984, 745-752.

[4] Matsunaga, H., Vibration and stability of thick plates on elastic foundations, Journal of Engineering Mechanics, 126(1), 2000, 27–34.

[5] Huang, M.H., Thambiratnam, D.P., Analysis of plate resting on elastic supports and elastic foundation by finite strip method, Computers and Structures,79(29-30), 2001, 2547-2557.

[6] Chen, W.Q., Bian, Z.G., A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation, Applied Mathematical Modelling,28(10), 2004, 877–890.

[7] Atmane, H.A., Tounsi, A., Mechab, I., Bedia, E.A.A., Free vibration analysis of functionally graded plates resting on Winkler–Pasternak elastic foundations using a new shear deformation theory, International Journal of Mechanics and Materials in Design, 6(2), 2010, 113-121.

[8] Thai, H.T., Park, M., Choi, D.H., A simple refined theory for bending, buckling, and vibration of thick plates resting on elastic foundation, International Journal of Mechanical Science, 73, 2013, 40–52.

[9] Zenkour, A.M., Bending of orthotropic plates resting on Pasternak's foundations by mixed shear deformation theory, Acta Mechanica Sinica,27(6), 2011, 956–962.

[10] Zenkour, A.M., Allam, M.N.M., Shaker, M.O., Radwan, A.H., On the simple and mixed first-order theories for plates resting on elastic foundations, Acta Mechanica,220(1-4), 2011, 33–46.

[11] Sayyad, A.S., Flexure of thick orthotropic plates by exponential shear deformation theory, Latin American Journal of Solids and Structures,10, 2013, 473-490.

[12] Akbas, S.D., Vibration and static analysis of functionally graded porous plates,Journal of Applied and Computational Mechanics, 3(3), 2017, 199-207.

[13] Akbas, S.D., Stability of a non-homogenous porous plate by using generalized differantial quadrature method, International Journal of Engineering & Applied Sciences, 9(2), 2017, 147-155.

[14] Akbas, S.D., Static analysis of a nano plate by using generalized differential quadrature method,International Journal of Engineering & Applied Sciences, 8(2), 2016, 30-39.

[15] Civalek, O., Analysis of thick rectangular plates with symmetric cross-ply laminates based on first-order shear deformation theory, Journal of Composite Materials, 42(26), 2008, 2853-2867.

[16] Gurses, M., Civalek, O., Korkmaz, A.K., Ersoy, H., Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first-order shear deformation theory, International Journal for Numerical Methods in Engineering, 79, 2009, 290–313.

[17] Sayyad, A.S., Ghugal, Y.M., On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results, Composite Structures, 129, 2015, 177–201.

[18] Sayyad, A.S., Ghugal, Y.M., Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature, Composite Structures, 171, 2017, 486–504.

[19] Ghugal, Y.M., Sayyad, A.S., A static flexure of thick isotropic plates using trigonometric shear deformation theory, Journal of Solid Mechanics, 2(1), pp. 79-90, 2010.

[20] Ghugal, Y.M., Sayyad, A.S., Free vibration of thick isotropic plates using trigonometric shear deformation theory, Journal of Solid Mechanics, 3(2), 2011, 172-182.

[21] Ghugal, Y.M., Sayyad, A.S., Static flexure of thick orthotropic plates using trigonometric shear deformation theory, Journal of Structural Engineering, 39(5), 2013, 512-521.

[22] Ghugal, Y.M., Sayyad, A.S., Free vibration of thick orthotropic plates using trigonometric shear deformation theory, Latin American Journal of Solids and Structures, 8, 2011, 229-243.

[23] Timoshenko, S.P., Goodier, J.M., Theory of Elasticity, McGraw-Hill, Singapore, 1970.