Abstract
This article investigates the bending response of an orthotropic rectangular plate resting on two-parameter elastic foundations. Analytical solutions for deflection and stresses are developed by means of the simple and mixed first-order shear deformation plate theories. The present mixed plate theory accounts for variable transverse shear stress distributions through the thickness and does not require a shear correction factor. The governing equations that include the interaction between the plate and the foundations are obtained. Numerical results are presented to demonstrate the behavior of the system. The results are compared with those obtained in the literature using three-dimensional elasticity theory or higher-order shear deformation plate theory to check the accuracy of the simple and mixed first-order shear deformation theories.
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Akavci S.S., Yerli H.R., Dogan A.: The first order shear deformation theory for symmetrically laminated composite plates on elastic foundation. Arab. J. Sci. Eng. 32, 341–348 (2007)
Jaiswal O.R., Iyengar R.N.: Dynamic response of a beam on elastic foundation of finite depth under a moving force. Acta Mech. 96, 67–83 (1993)
Chudinovich I., Constanda C.: Integral representations of the solutions for a bending plate on an elastic foundation. Acta Mech. 139, 33–42 (2000)
Tsiatas G.C.: Nonlinear analysis of non-uniform beams on nonlinear elastic foundation. Acta Mech. 209, 141–152 (2010)
Dumir P.C.: Nonlinear dynamic response of isotropic thin rectangular plates on elastic foundations. Acta Mech. 71, 233–244 (2003)
Chien R.D., Chen C.S.: Nonlinear vibration of laminated plates on an elastic foundation. Thin-Walled Struct. 44, 852–860 (2006)
Celep Z., Güler K.: Axisymmetric forced vibrations of an elastic free circular plate on a tensionless two parameter foundation. J. Sound Vib. 301, 495–509 (2007)
Liew K.M., Han J.B., Xiao Z.M., Du H.: Differential quadrature method for Mindlin plates on Winkler foundations. Int. J. Mech. Sci. 38, 405–421 (1996)
Eratll N., Akoz A.Y.: The mixed finite element formulation for the thick plates on elastic foundations. Comput. Struct. 65, 515–529 (1997)
Han J.B., Liew K.M.: Numerical differential quadrature method for Reissner/Mindlin plates on two-parameter foundations. Int. J. Mech. Sci. 39, 977–989 (1997)
Omurtage M.H., Kadioglu F.: Free vibration analysis of orthotropic plates resting on Pasternak foundation by mixed finite element formulation. Comput. Struct. 67, 253–265 (1998)
Chen W.Q., Lü C.F., Bian Z.G.: A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. Appl. Math. Modell. 28, 877–890 (2004)
Singh B.N., Lal A., Kumar R.: Post buckling response of laminated composite plate on elastic foundation with random. Commun. Nonlin. Sci. Numer. Simul. 14, 284–300 (2007)
Zenkour A.M., Allam M.N.M., Sobhy M.: Bending analysis of FG viscoelastic sandwich beams with elastic cores resting on Pasternak’s elastic foundations. Acta Mech. 212, 233–252 (2010)
Reissner E.: On the theory of bending of elastic plates. J. Math. Phys. 23, 184–191 (1944)
Reissner E.: The effect of transverse shear deformation on the bending of elastic plates. ASME J. Appl. Mech. 12, 69–77 (1945)
Mindlin R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. ASME J. Appl. Mech. 18, 31–38 (1951)
Fares M.E., Allam M.N.M., Zenkour A.M.: Hamilton’s mixed variational formula for dynamical problems of anisotropic elastic bodies. SM Arch. 14, 103–114 (1989)
Zenkour A.M.: Maupertuis-Lagrange mixed variational formula for laminated composite structures with a refined higher-order beam theory. Int. J. Non-linear Mech. 32, 989–1001 (1997)
Fares M.E., Zenkour A.M.: Mixed variational formula for the thermal bending of laminated plates. J. Thermal Stresses 22, 347–365 (1999)
Zenkour A.M.: Natural vibration analysis of symmetrical cross-ply laminated plates using a mixed variational formulation. Eur. J. Mech. A/Solids 19, 469–485 (2000)
Zenkour A.M.: Buckling and free vibration of elastic plates using simple and mixed shear deformation theories. Acta Mech. 146, 183–197 (2001)
Zenkour A.M., Mashat D.S.: Bending analysis of a ceramic-metal arched bridge using a mixed first-order theory. Meccanica 44, 721–731 (2009)
Buczkowski R., Torbacki W.: Finite element modeling of thick plates on two-parameter elastic foundation. Int. J. Numer. Anal. Meth. Geomech. 25, 1409–1427 (2001)
Timoshenko S.P., Woinowsky-Krieger W.: Theory of Plates and Shells. McGraw-Hill, New-York (1970)
Lam K.Y., Wang C.M., He X.Q.: Canonical exact solution for Levyplates on two parameter foundation using Green’s functions. Eng. Struct. 22, 364–378 (2000)
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Zenkour, A.M., Allam, M.N.M., Shaker, M.O. et al. On the simple and mixed first-order theories for plates resting on elastic foundations. Acta Mech 220, 33–46 (2011). https://doi.org/10.1007/s00707-011-0453-7
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DOI: https://doi.org/10.1007/s00707-011-0453-7