直角坐标系下层状黏弹性地基Biot固结解析刚度矩阵解

寇磊; 白云

doi:10.3879/j.issn.1000-0887.2016.01.007

直角坐标系下层状黏弹性地基Biot固结解析刚度矩阵解

doi: 10.3879/j.issn.1000-0887.2016.01.007

寇磊¹,
白云²

¹郑州大学水利与环境学院, 郑州 450002;²同济大学地下建筑与工程系, 上海 200092

基金项目: 教育部长江学者和创新团队发展计划(IRT1029)

详细信息

作者简介:
寇磊（1983—），男，博士（通讯作者. E-mail: klyhe@163.com）.

中图分类号: TU433
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- 被引次数: 0
出版历程
- 收稿日期: 2015-10-14
- 修回日期: 2015-11-28
- 刊出日期: 2016-01-16

Analytical Stiffness Matrixes for Biot Consolidation of Multilayered Viscoelastic Foundations in the Cartesian Coordinate System

KOU Lei¹,
BAI Yun²

¹School of Water Conservancy & Environment, Zhengzhou University,Zhengzhou 450002, P.R.China;²Department of Geotechnical Engineering, Tongji University, Shanghai 200092, P.R.China

摘要

摘要: 基于直角坐标系下Biot固结的基本控制方程，并考虑软土土骨架的黏弹性特性，通过Fourier-Laplace积分变换、解耦变换、微分方程组理论和矩阵理论，推导了黏弹性地基Biot固结三维空间问题和平面应变问题在积分变换域的解析解，进而得到对应问题的单元刚度矩阵. 然后根据对号入座原则组装得到层状黏弹性地基Biot固结对应问题的总体刚度矩阵.通过求解总体刚度矩阵形成的线性代数方程，得到层状黏弹性地基Biot固结对应问题在积分变换域内的解答.最后应用Fourier-Laplace逆变换得到其物理域内的解.对比求解黏弹性Biot固结问题退化的弹性Biot固结问题与已有解答，验证了刚度矩阵计算方法的正确性，为层状黏弹性地基Biot固结问题提供了理论基础.
- 直角坐标系 /
- 层状黏弹性地基 /
- Biot固结 /
- 解析刚度矩阵法
Abstract: Based on the basic viscoelastic equations of Biot consolidation in the Cartesian coordinate system and in view of the viscoelasticity of soft soil skeleton, the analytical solutions to 3D problems and plane strain problems of Biot consolidation in the integral transform domain were obtained through the FourierLaplace transform and the decoupling transform according to the differential equation theory and the matrix theory, and in turn the corresponding element stiffness matrixes were derived. The global stiffness matrixes for Biot consolidation 3D problems and plane strain problems of multilayered viscoelastic foundations were assembled with the matrix matching method, and the solutions to the corresponding problems of multilayered viscoelastic foundations in the transform domain were obtained in the solution of the algebraic equations for the global stiffness matrixes. The solutions in the physical domain were acquired through the inverse FourierLaplace transform. The validity of the proposed method was examined in the comparison of the present results of 2 examples, where viscoelastic Biot consolidation was reduced to elastic Biot consolidation, with the previous reference solutions. The analytical stiffness matrixes provide a theoretical base for Biot consolidation of multilayered viscoelastic foundations.
- Cartesian coordinate system /
- multilayered viscoelastic foundation /
- Biot consolidation /
- analytical stiffness matrix method

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参考文献(18)

[1]	白冰, 周健. 饱和软粘土及其在冲击荷载作用后的次固结变形[J]. 武汉交通科技大学学报, 1999,23(6): 584-588.(BAI Bing, ZHOU Jian. The secondary consolidation deformation of soft clay after impact loading[J]. Journal of Wuhan Transportation University,1999,23(6): 584-588.(in Chinese))
[2]	Leorueil S. Stress-strain-strain rate relation for the compressibility of sensitive natural clays[J]. Géotechnique,1985,35(2): 159-180.
[3]	Leomeil S, Kabbaj M, Bouchard R, Tavenas F. Discussion on stress-strain-strain rate relation for the compressibility of sensitive natural clays[J]. Géotechnique,1986,36(2): 288-290.
[4]	Biot M A. General theory of three-dimensional consolidation[J]. Journal of Applied Physics,1941,12(2): 155-164.
[5]	艾智勇, 王全胜. 有限土层轴对称Biot 固结的一个新的解析解[J]. 应用数学和力学, 2008,29(12): 1472-1478.(AI Zhi-yong, WANG Quan-sheng. A new analytical solution for axisymmetric Biot’s consolidation of a finite soil layer[J]. Applied Mathematics and Mechanics,2008,29(12): 1472-1478.(in Chinese))
[6]	艾智勇, 吴超. 渗透各向异性土层的平面应变固结[J]. 应用数学和力学, 2009,30(11): 1349-1356.(AI Zhi-yong, WU Chao. Plane strain consolidation of soil layer with an isotropic permeability[J]. Applied Mathematics and Mechanics,2009,30(11): 1349-1356.(in Chinese))
[7]	Ai Z Y, Cheng Z Y. Transfer matrix solutions to plane strain and three-dimensional Biot’s consolidation of multilayered soils[J]. Mechanics of Materials,2009,41(3): 244-251.
[8]	艾智勇, 王全胜. 多层横观各向同性地基轴对称固结的传递矩阵解[J]. 岩土力学, 2009,30(4): 921-925.(AI Zhi-yong, WANG Quan-sheng. Transfer matrix solutions for axisymmetric consolidation of multilayered transversely isotropic soils[J]. Chinese Journal of Rock and Soil Mechanics,2009,30(4): 921-925.(in Chinese))
[9]	AI Zhi-yong, CAO Guo-jun, CHENG Yi-chong. Analytical layer-element solutions of Biot’s consolidation with anisotropic permeability and incompressible fluid and solid constituents[J]. Applied Mathematical Modelling,2012,36(10): 4817-4829.
[10]	Booker J R, Small J C. Finite layer analysis of consolidation Ⅰ[J]. International Journal for Numerical and Analytical Methods in Geomechanics,1982,6(2): 151-171.
[11]	Booker J R, Small J C. Finite layer analysis of consolidation Ⅱ[J]. International Journal for Numerical and Analytical Methods in Geomechanics,1982,6(2): 173-194.
[12]	顾尧章, 金波. 轴对称荷载作用下多层地基的Biot 固结及其变形[J]. 工程力学, 1992,9(3): 81-94.(GU Yao-zhang, JIN Bo. Biot consolidation of multilayer subsoils under 3D-axisymmetric load[J]. Engineering Mechanics,1992,9(3): 81-94.(in Chinese))
[13]	方诗圣, 王建国, 王秀喜. 层状饱和土Biot 固结问题的状态空间法[J]. 力学学报, 2003,35(2): 206-211.(FANG Shi-sheng, WANG Jian-guo, WANG Xiu-xi. Thestate space method of the Biot consolidation problem for multilayered porous media[J]. Chinese Journal of Theoretical and Applied Mechanics,2003,35(2): 206-211.(in Chinese))
[14]	宰金珉, 梅国雄. 有限层法求解三维比奥固结问题[J]. 岩土工程学报, 2002,24(1): 31-33.(ZAI Jin-min, MEI Guo-xiong. Finite layer analysis of three dimensional Biot consolidation[J]. Chinese Journal of Geotechnical Engineering,2002,24(1): 31-33.(in Chinese))
[15]	陈胜立, 张建民. 横观各向同性饱和地基轴对称Biot 固结问题的解析解[J]. 岩土工程学报, 2002,24(1): 26-30.(CHEN Sheng-li, ZHANG Jian-min. An analysis of axisymmetric consolidation for transversely isotropic saturated soils[J]. Chinese Journal of Geotechnical Engineering, 2002,24(1): 26-30.(in Chinese))
[16]	寇磊, 白云. 直角坐标系黏弹性层状地基荷载作用下的解析刚度矩阵解[J]. 中南大学学报(自然科学版), 2014,45(7): 2346-2352.(KOU Lei , BAI Yun. Analytical stiffness matrix of viscoelastic layered foundation under loading in Cartesian coordinate system[J]. Journal of Central South University (Science and Technology),2014,45(7): 2346-2352.(in Chinese))
[17]	Booker J R, Small J C. A Fourier-Laplace transform finite element method for the analysis of contaminant transport in porous media[J]. International Journal for Numerical and Analytical Methods in Geomechanics,1996,23(14):1763-1796.
[18]	Talbot A. The accurate numerical inversion of Laplace transforms[J]. IMA Journal of Applied Mathematics,1979,23(1): 97-120.