## 留言板

 引用本文: 李锐, 田宇, 郑新然, 王博. 求解弹性地基上自由矩形中厚板弯曲问题的辛-叠加方法[J]. 应用数学和力学, 2018, 39(8): 875-891.
LI Rui, TIAN Yu, ZHENG Xinran, WANG Bo. A Symplectic Superposition Method for Bending Problems of Free-Edge Rectangular Thick Plates Resting on Elastic Foundations[J]. Applied Mathematics and Mechanics, 2018, 39(8): 875-891. doi: 10.21656/1000-0887.390186
 Citation: LI Rui, TIAN Yu, ZHENG Xinran, WANG Bo. A Symplectic Superposition Method for Bending Problems of Free-Edge Rectangular Thick Plates Resting on Elastic Foundations[J]. Applied Mathematics and Mechanics, 2018, 39(8): 875-891.

• 中图分类号: O302

## A Symplectic Superposition Method for Bending Problems of Free-Edge Rectangular Thick Plates Resting on Elastic Foundations

Funds: The National Basic Research Program of China(973 Program)（2014CB049000）；The National Natural Science Foundation of China（11302038）
• 摘要: 基于近年来提出的辛-叠加方法，解析求解了弹性地基上自由矩形中厚板的弯曲问题.首先将原问题拆分为3类子问题，在Hamilton体系下，运用辛几何方法推导出子问题对应的弹性地基上对边滑支矩形板弯曲问题的辛解析解；以此为基础，通过叠加法思想，求出弹性地基上四边自由矩形中厚板的弯曲解.与半逆法等传统解析方法相比，辛-叠加方法兼备了辛方法理性和叠加法规律性的优点，在求解过程中不需要预先假定解的形式，而是由弹性力学基本方程出发，经过逐步严格推导获得解析解，因而大大拓展了可求解问题的范围，成为一种求解以矩形板问题为代表的弹性力学高阶偏微分方程复杂边值问题的有效解析方法.
•  [1] MINDLIN R D. Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates[J]. Journal of Applied Mechanics,1951,18(1): 31-38. [2] LOK T S, CHENG Q H. Bending and forced vibration response of a clamped orthotropic thick plate and sandwich panel[J]. Journal of Sound & Vibration,2001,245(1): 63-78. [3] HENWOOD D J, WHITEMAN J R, YETTRAM A L. Finite difference solution of a system of first-order partial differential equations[J]. International Journal for Numerical Methods in Engineering,1981,17(9): 1385-1395. [4] BUCZKOWSKI R, TORBACKI W. Finite element modelling of thick plates on two-parameter elastic foundation[J]. International Journal for Numerical and Analytical Methods in Geomechanics,2001,25(14): 1409-1427. [5] SHEN P, HE P. Bending analysis of rectangular moderately thick plates using spline finite element method[J].Computers & Structures,1995,54(6): 1023-1029. [6] PEREIRA W L A, KARAM V J, CARRER J A M, et al. A dynamic formulation for the analysis of thick elastic plates by the boundary element method[J]. Engineering Analysis With Boundary Elements,2012,36(7): 1138-1150. [7] LIEW K M, HAN J B. Bending solution for thick plates with quadrature boundary[J]. Journal of Engineering Mechanics,1998,124(1): 9-17. [8] LIU F L, LIEW K M. Differential cubature method for static solutions of arbitrarily shaped thick plates[J].International Journal of Solids and Structures,1998,35(28/29): 3655-3674. [9] CIVALEK . Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method[J]. International Journal of Mechanical Sciences,2007,49(6): 752-765. [10] FERREIRA A J M, CASTRO L M S, BERTOLUZZA S. Analysis of plates on Winkler foundation by wavelet collocation[J]. Meccanica,2011,46(4): 865-873. [11] L C F, LIM C W, CHEN W Q. Semi-analytical analysis for multi-directional functionally graded plates: 3-D elasticity solutions[J]. International Journal for Numerical Methods in Engineering,2010,79(1): 25-44. [12] YAO W, ZHONG W, LIM C W. Symplectic Elasticity[M]. Singapore: World Scientific, 2009. [13] LI R, ZHONG Y, LI M. Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method[J]. Proceedings of the Royal Society A: Mathematical Physical & Engineering Sciences,2013,469(2153): 20681. [14] LI R, NI X Q, CHEN G G. Symplectic superposition method for benchmark flexure solutions for rectangular thick plates[J]. Journal of Engineering Mechanics,2015,141(2): 04014119. [15] LI R, WANG B, LI G. Benchmark bending solutions of rectangular thin plates point-supported at two adjacent corners[J]. Applied Mathematics Letters,2015,40: 53-58. [16] LI R, WANG P, TIAN Y, et al. A unified analytic solution approach to static bending and free vibration problems of rectangular thin plates[J]. Scientific Reports,2015,5: 17054. [17] WANG B, LI P, LI R. Symplectic superposition method for new analytic buckling solutions of rectangular thin plates[J]. International Journal of Mechanical Science s, 2016,119: 432-441. [18] LI R, TIANY, ZHENGX, et al. New analytic bending solutions of rectangular thin plates with a corner point-supported and its adjacent corner free[J]. European Journal of Mechanics: A/Solids,2017,66: 103-113. [19] LI R, WANG P, ZHENG X, et al. New benchmark solutions for free vibration of clamped rectangular thick plates and their variants[J]. Applied Mathematics Letters,2017,78: 88-94. [20] LI R, WANG P, WANG B, et al. New analytic free vibration solutions of rectangular thick plates with a free corner by the symplectic superposition method[J]. Journal of Vibration and Acoustics,2018,140(3): 031016. [21] LI R, ZHENG X, WANG H, et al. New analytic buckling solutions of rectangular thin plates with all edges free[J]. International Journal of Mechanical Sciences,2018,144: 67-73.

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##### 出版历程
• 收稿日期:  2018-06-28
• 刊出日期:  2018-08-15

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