| Citation: |
CHENG Lin-yan, TAN Zhi-ming. ANALYTIC SOLUTIONS TO THE BENDING PROBLEM OF RECTANGULAR THIN PLATES SIMPLY SUPPORTED ON FOUR SIDES ON A WINKLER FOUNDATION WITH HORIZONTAL FRICTION[J]. Engineering Mechanics. DOI: 10.6052/j.issn.1000-4750.2024.05.S040
|
Assuming that the horizontal friction between the plate and the foundation is proportional to the horizontal displacement difference between them, the differential equations for a rectangular plate resting on an elastic foundation with horizontal friction were established according to the force equilibrium and the constitutive relationship, and the analytical solution in series form was obtained for the four-side simply-supported plate. The effects of the horizontal friction on the deflection and bending moment of plates were analyzed by taking the concentrated load and the square uniform load acting at the center of the plate as examples. The results indicate that the horizontal friction between the plate and the foundation has a reduction effect on the deflection and bending moment of the rectangular plate. When the horizontal friction parameter is greater than 0.01, the influence of the horizontal friction on the deflection and bending moment may exceed 2% and should be considered. The size of the plate and the size of the load affect the reduction effect of the horizontal friction, which should be considered carefully according to the actual working conditions. When a square uniform load is applied at the center of a square plate with a relative length of 5, the horizontal friction can cause the maximum deflection of the plate (i.e., the deflection at the center of the load) to decrease by about 60 %, and the maximum section bending moment (i.e.,the bending moment at the center of the load) to decrease by about 70 %.
| [1] |
黄义, 何芳社. 弹性地基上的梁、板、壳[M]. 北京: 科学出版社, 2005: 61 − 95.
HUANG Yi, HE Fangshe. Beams, plates, and shells on elastic foundations [M]. Beijing: Science Press, 2005: 61 − 95. (in Chinese)
|
| [2] |
WANG Y H, THAM L G, CHEUNG Y K. Beams and plates on elastic foundations: A review [J]. Progress in Structural Engineering and Materials, 2005, 7(4): 174 − 182. doi: 10.1002/pse.202
|
| [3] |
BAI E, ZHANG C L, CHEN A, et al. Analytical solution of the bending problem of free orthotropic rectangular thin plate on two-parameter elastic foundation [J]. ZAMM - Journal of Applied Mathematics and Mechanics, 2021, 101(10): e202000358. doi: 10.1002/zamm.202000358
|
| [4] |
ZHENG Y F, XU L L, CHEN C P. Nonlinear bending analysis of magnetoelectroelastic rectangular plates using higher order shear deformation theory [J]. Journal of Mechanical Science and Technology, 2021, 35(3): 1099 − 1108. doi: 10.1007/s12206-021-0223-y
|
| [5] |
JAIN R, AZAM M S, SINGH P P. Bending analysis of functionally graded plates resting on elastic foundation: A Rayleigh-Ritz approach and ANN method [J]. Mechanics of Advanced Materials and Structures, 2024, 31(29): 11484 − 11502. doi: 10.1080/15376494.2024.2307474
|
| [6] |
LIU Fan, SONG Lina, JIANG Maosheng, et al. Generalized finite difference method for solving the bending problem of variable thickness thin plate [J]. Engineering Analysis with Boundary Elements, 2022, 139: 69 − 76. doi: 10.1016/j.enganabound.2022.03.008
|
| [7] |
谈至明, 周玉民, 刘少文, 等. 不等尺寸双层混凝土路面结构力学模型研究[J]. 工程力学, 2010, 27(3): 132 − 137.
TAN Zhiming, ZHOU Yumin, LIU Shaowen, et al. Mechanistic model of double-layered concrete pavement structures with unequal planar dimensions [J]. Engineering Mechanics, 2010, 27(3): 132 − 137. (in Chinese)
|
| [8] |
许金余, 吴彰春, 冷培义. 弹性地基板的无界边界元──有限元耦合计算法[J]. 工程力学, 1994, 11(3): 137 − 143.
XU Jinyu, WU Zhangchun, LENG Peiyi. A calculation method of coupling Infinite BE and FE for plate based on elastic half space [J]. Engineering Mechanics, 1994, 11(3): 137 − 143. (in Chinese)
|
| [9] |
NAVIER C L M H. Extrait des recherches sur la flexion des plans élastiques [J]. Bulletin des Sciences, par la Société Philomathique de Paris, 1823: 92 − 102.
|
| [10] |
LEVY M. Mémoire sur la théorie des plaques élastiques planes [J]. Journal de Mathématiques Pures et Appliquées, 1877, 3: 219 − 306.
|
| [11] |
石小平, 姚祖康. Пастернак基础上四边自由矩形厚板的解[J]. 同济大学学报, 1989, 17(2): 173 − 184.
SHI Xiaoping, YAO Zukang. The solution of a rectangular thick plate with free edges on a Пастернак foundation [J]. Journal of Tongji University, 1989, 17(2): 173 − 184. (in Chinese)
|
| [12] |
王克林, 黄义. 弹性地基上四边自由的矩形厚板[J]. 固体力学学报, 1986(1): 37 − 49.
WANG Kelin, HUANG Yi. Thick rectangular plates with four free edges on elastic foundations [J]. Acta Mechanica Solida Sinica, 1986(1): 37 − 49. (in Chinese)
|
| [13] |
黄炎. 弹性地基上的矩形板弯曲问题的解析解法[J]. 国防科技大学学报, 1992, 14(2): 47 − 52.
HUANG Yan. Analytical method for solving bending problem of rectangular plates on elastic foundations [J]. Journal of National University of Defense Technology, 1992, 14(2): 47 − 52. (in Chinese)
|
| [14] |
谈至明. 铺面力学[M]. 2版. 北京: 人民交通出版社, 2021: 132 − 179.
TAN Zhiming. Pavement mechanics [M]. 2nd ed. Beijing: China Communications Press, 2021: 132 − 179. (in Chinese)
|
| [15] |
赵明华, 郑玥, 刘猛, 等. 考虑纵横耦合变形的土工格室加筋体变形分析[J]. 湖南大学学报(自然科学版), 2019, 46(9): 89 − 99.
ZHAO Minghua, ZHENG Yue, LIU Meng, et al. Deformation analysis of geocell-reinforcement considering vertical-horizontal coupling distortion [J]. Journal of Hunan University (Natural Sciences), 2019, 46(9): 89 − 99. (in Chinese)
|
| [16] |
谈至明, 成林燕. 考虑地基水平摩阻的Winkler地基上板模型解析[J]. 力学季刊, 2023, 44(1): 101 − 112.
TAN Zhiming, CHENG Linyan. Model and solution of a plate on Winkler foundation with horizontal friction [J]. Chinese Quarterly of Mechanics, 2023, 44(1): 101 − 112. (in Chinese)
|
| [17] |
ZHANG Y T, ROESLER J R, HUANG Z Y. A method for evaluating CRCP performance based on edge-loaded FWD test [J]. Materials and Structures, 2020, 53(2): 46. doi: 10.1617/s11527-020-01481-0
|
| [18] |
TARR S M, OKAMOTO P A, SHEEHAN M J, et al. Bond interaction between concrete pavement and lean concrete base [J]. Transportation Research Record, 1999, 1668(1): 9 − 16. doi: 10.3141/1668-02
|
Supported by: Beijing Renhe Information Technology Co.,Ltd. 
DownLoad: