Abstract
This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation. Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space. The operator matrix of the equation set is proven to be a Hamilton operator matrix. Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition. There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions, with opposite sides simply supported and opposite sides clamped. Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given. Analytical solutions using two examples are presented to show the use of the new methods described in this paper. To verify the accuracy and convergence, a fully simply supported plate that is fully and simply supported under uniformly distributed load is used to compare the classical Navier method, the Levy method and the new method. Results show that the new technique has good accuracy and better convergence speed than other methods, especially in relation to internal forces. A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods, with solutions compared to those produced by the Galerkin method.
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References
Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. Mc-Graw-Hill, New York (1970)
Selvadurai, A.P.S.: Elastic Analysis of Soil-Foundation Interaction. Elsevier, Amsterdam (1979)
Kong, J., Cheung, Y.K.: A generalized spline finite strip for the analysis of plates. Thin. Wall Struct. 22, 181–202 (1995)
Cheung, Y.K., Zienkiewicz, O.C.: Plates and tanks on elastic foundation-an application of the finite element method. International Journal of Solids and Structures 1, 451–461 (1965)
Sadecka, L.: A Finite/infinite element analysis of thick plate on a layered foundation. Computers and Structures 76, 603–610 (2000)
Silva, A.R.D., Silveira, R.A.M., Goncalves, P.B.: Numerical methods for analysis of plates on tensionless elastic foundations. International Journal of Solids and Structures 38, 2083–2100 (2001)
Sladek, J., Sladek, V., Zhang, C., et al.: Analysis of orthotropic thick plates by meshless local Petrov-Galerkin (MLPG) method. International Journal for Numerical Methods in Engineering 67, 1830–1850 (2006)
Zhong, W.X.: A New Systematic Methodology for Theory of Elasticity. Dalian University of Technology Press, Dalian (1995) (in Chinese)
Yao, W.A., Zhong, W.X., Lim, C.W.: Symplectic Elasticity. World Scientific, Singapore (2009)
Yao, W.A., Xu, C.: A restudy of the paradox on an elastic wedge based on the Hamiltonian system. ASME Journal of Applied Mechanics 68, 678–681 (2001)
Zhong, W.X., Yao, W.A.: New solution system for plate bending. Computational Mechanics in Structural Engineering 31, 17–30 (1999)
Lim, C.W., Yao, W.A., Cui, S.: Benchmark symplectic solutions for bending of corner-supported rectangular thin plates. The IES Journal Part A: Civil & Structural Engineering 1, 106–115 (2008)
Lim, C.W., Cui, S., Yao, W.A.: On new symplectic elasticity approach for exact bending solutions of rectangular thin plates with two opposite sides simply supported. International Journal of Solids and Structures 44, 5396–5411 (2007)
Yao, W.A., Su, B., Zhong, W.X.: Hamiltonian system for orthotropic plate bending based on analogy theory. Science in China (series E) 44, 258–264 (2001)
Yao, W.A. Sui, Y.F.: Symplectic solution system for Reissner plate bending. Applied Mathematics and Mechanics 25, 178–185 (2003)
Long, Y.L., Wen, X.L., Xie, C.F.: An implementation of a root finding algorithm for transcendental functions in a complex plane. Journal on Numerical Methods and Computer Applications 15, 88–92 (1994) (in Chinese)
Mbakogu, F.C., Pavlovic, M.N.: Bending of clamped orthotropic rectangular plates: a variational symbolic solution. Computers and Structures 77, 117–128 (2000)
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The project was supported by the National Natural Science Foundation of China (10772039 and 10632030) and the National Basic Research Program of China (973 Program) (2010CB832704).
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Yao, WA., Hu, XF. & Xiao, F. Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation. Acta Mech Sin 27, 929–937 (2011). https://doi.org/10.1007/s10409-011-0532-y
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DOI: https://doi.org/10.1007/s10409-011-0532-y