倾斜层状地基的精准边界元法及其应用

肖莎; 岳中琦

doi:10.11779/CJGE20240005

倾斜层状地基的精准边界元法及其应用 English Version

肖莎^1,,
岳中琦^{2, 3}

1.
北京工业大学城市与工程安全减灾教育部重点实验室, 北京 100124
2.
哈尔滨工业大学理学院, 广东深圳 518055
3.
香港大学土木工程系, 香港 999077

基金项目:

国家自然科学基金项目 42207182

北京市自然科学基金项目 8242006

详细信息

作者简介:
肖莎（1994—），女，博士，助理研究员，主要从事非均匀岩土介质弹性力学和边界元法等方面的研究。E-mail: xiaosha@bjut.edu.cn

中图分类号: TU43;O34
计量
- 文章访问数: 0
- HTML全文浏览量: 0
- PDF下载量: 0
出版历程
- 收稿日期: 2024-01-01
- 网络出版日期: 2024-09-23
- 刊出日期: 2025-04-30

Precise boundary element method for non-horizontally layered foundations and its applications

XIAO Sha^1,,
YUE Zhongqi^{2, 3}

1.
Key Laboratory of Urban Security and Disaster Engineering of China Ministry of Education, Beijing University of Technology, Beijing 100124, China
2.
School of Science, Harbin Institute of Technology, Shenzhen 518055, China
3.
Department of Civil Engineering, The University of Hong Kong, Hong Kong 999077, China

摘要

摘要: 发展了一种新型边界元法，精准计算基础荷载作用下倾斜层状地基附加应力和沉降。该边界元法采用最新发展的横观各向同性层状材料基本解，8结点等参单元离散加载域及其附近区域的边界，6结点无穷单元离散远场边界。引入结点位于单元内部的不连续单元，消除边界积分方程自由项系数的求解困难，给出离散边界积分方程中非奇异和奇异积分的数值计算方法。数值验证表明发展的数值方法具有很高的计算精度和效率。算例分析详细揭示了层状岩土材料各向异性和倾斜分层对弹性场的影响。
- 倾斜层状地基 /
- 横观各向同性 /
- 边界元法 /
- 附加应力和沉降
Abstract: A new boundary element method (BEM) is developed for accurately calculating the additional stresses and settlements in non-horizontally layered foundations. The proposed BEM utilizes the fundamental solutions for multilayered solids with transverse isotropy (TI) under point-concentrated loads. The eight-noded isoparametric elements are used to discretize a core region around the traction area, whereas the six-noded infinite elements are used to discretize a far-field region beyond the traction area. To avoid calculating the coefficient of the free term for the source point at the strike line between the boundary and the material interface, the discontinuous isoparametric and infinite elements with nodes located within elements are positioned near the strike lines. Numerical methods for non-singular and various singular integrals in the discretized boundary integral equations are developed. The numerical verification shows that the proposed BEM has very high accuracy and computational efficiency. The numerical examples illustrate the effects of anisotropy and non-horizontal layered structures of the foundations on the elastic fields.
- non-horizontally layered foundation /
- transverse isotropy /
- BEM /
- additional stress and settlement

HTML全文

图 1 边界荷载作用下横观各向同性倾斜分层地基

Figure 1. Non-horizontally layered and TI foundations under action of boundary tractions

下载: 全尺寸图片幻灯片

倾斜层状地基边界面的分区（ $S = {S_F} + {S_I}$ ）

Subdomains of boundaries on non-horizontally layered foundations ( $S = {S_F} + {S_I}$ )

下载: 全尺寸图片幻灯片

图 3 两种8结点等参单元

Figure 3. Two types of 8-noded isoparametric elements

下载: 全尺寸图片幻灯片

图 4 3种类型6结点无穷单元

Figure 4. Three types of 6-noded infinite elements

下载: 全尺寸图片幻灯片

图 5 二维单元的离散

Figure 5. Discretization of a two-dimensional element

下载: 全尺寸图片幻灯片

图 6 两层铺路模型

Figure 6. A two-layered pavement model

下载: 全尺寸图片幻灯片

图 7 由3种岩石组成的倾斜层状地基

Figure 7. Non-horizontally layered model with three different rocks

下载: 全尺寸图片幻灯片

图 8 水平边界加载区域及临近区域的离散

Figure 8. Discretization of a loading area and its neighboring area

下载: 全尺寸图片幻灯片

位移沿水平方向的变化 $(y = 0.5{\text{ m, }}z = 1{\text{ m}})$

Variation of displacements along horizontal direction $(y = 0.5{\text{ m, }}z = 1{\text{ m}})$

下载: 全尺寸图片幻灯片

正应力沿水平方向的变化 $(y = 0.5{\text{ m, }}z = 1{\text{ m}})$

Variation of normal stresses along horizontal direction $(y = 0.5{\text{ m, }}z = 1{\text{ m}})$

下载: 全尺寸图片幻灯片

切应力沿水平方向的变化 $(y = 0.5{\text{ m, }}z = 1{\text{ m}})$

Variation of shear stresses along horizontal direction $(y = 0.5{\text{ m, }}z = 1{\text{ m}})$

下载: 全尺寸图片幻灯片

位移沿深度方向的变化 $(x = 0{\text{ m, }}y = 1{\text{ m}})$

Variation of displacements along depth direction $(x = 0{\text{ m, }}y = 1{\text{ m}})$

下载: 全尺寸图片幻灯片

正应力沿深度方向的变化 ${\text{(}}x = 0{\text{ m, }}y = 1{\text{ m)}}$

Variation of normal stresses along depth direction ${\text{(}}x = 0{\text{ m, }}y = 1{\text{ m)}}$

下载: 全尺寸图片幻灯片

切应力沿深度方向的变化 $(x = 0{\text{ m, }}y = 1{\text{ m}})$

Variation of shear stresses along depth direction $(x = 0{\text{ m, }}y = 1{\text{ m}})$

下载: 全尺寸图片幻灯片

表 1 层状铺路模型表面竖向位移

Table 1 Vertical displacements on boundary for pavement model 单位: mm

距离r	Cai等^[11]	数值解	相对误差/%
0	0.49	0.4975	1.5
200	0.43	0.4402	2.4
600	0.31	0.3052	1.5
900	0.24	0.2352	2.0
1200	0.19	0.1857	2.2

下载: 导出CSV

表 2 横观各向同性岩石的弹性参数

Table 2 Elastic parameters of TI rocks

岩石类型	${E_{x'}}/{E_{z'}}$	${v_{x'y'}}/{v_{x'z'}}$	${\mu _{x'y'}}/{\mu _{x'z'}}$
1：各向同性	1.0	1.0	1.0
2：横观各向同性	3.0	1.0	1.0
3：横观各向同性	1.0	1.0	0.8
4：横观各向同性	2.0	1.0	1.0

下载: 导出CSV

表 3 不同倾角条件下倾斜层状地基界面的正应力间断值

Table 3 Jumps of normal stresses across interfaces under different inclination angles 单位: MPa

$\theta$ /(°)	${\sigma _{xx}}$		${\sigma _{yy}}$		${\sigma _{zz}}$
$\theta$ /(°)	界面1	界面2	界面1	界面2	界面1	界面2
60	8.3999	11.3703	4.1384	2.6056	29.3390	8.7858
90	0.0000	0.0000	6.5755	1.2066	26.9099	17.8794
120	14.7349	2.4015	7.3428	8.6723	13.9174	20.0039

下载: 导出CSV

表 4 不同倾角条件下倾斜层状地基界面的切应力间断值

Table 4 Jumps of shear stresses across interfaces under different inclination angles 单位: MPa

$\theta$ /(°)	${\sigma _{xx}}$		${\sigma _{yy}}$		${\sigma _{zz}}$
$\theta$ /(°)	界面1	界面2	界面1	界面2	界面1	界面2
60	8.3999	11.3703	3.0046	0.0046	2.2134	0.7746
90	0.0000	0.0000	0.0000	0.0000	1.9836	1.9660
120	14.7349	2.4015	2.2408	3.4545	0.7841	2.1519

下载: 导出CSV

参考文献(25)

[1]	李广信, 张丙印, 于玉贞. 土力学[M]. 3版. 北京: 清华大学出版社, 2022. LI Guangxin, ZHANG Bingyin, YU Yuzhen. Soil Mechanics[M]. 3rd ed. Beijing: Tsinghua University Press, 2022. (in Chinese)
[2]	BURMISTER D M. The general theory of stresses and displacements in layered soil systems. Ⅲ[J]. Journal of Applied Physics, 1945, 16(5): 296-302. doi: 10.1063/1.1707590
[3]	CHEN W T. Computation of stresses and displacements in a layered elastic medium[J]. International Journal of Engineering Science, 1971, 9(9): 775-800. doi: 10.1016/0020-7225(71)90072-3
[4]	XIAO H T, YUE Z Q. Elastic fields in two joined transversely isotropic media of infinite extent as a result of rectangular loading[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2013, 37(3): 247-277. doi: 10.1002/nag.1098
[5]	朱桂春, 史培新, 王占生. 层状横观各向同性地基变形问题的一个近似解[J]. 岩土工程学报, 2020, 42(4): 790-796. doi: 10.11779/CJGE202004024 ZHU Guichun, SHI Peixin, WANG Zhansheng. Approximate solution for deformation problems of transversely isotropic multi-layered soils[J]. Chinese Journal of Geotechnical Engineering, 2020, 42(4): 790-796. (in Chinese) doi: 10.11779/CJGE202004024
[6]	时刚, 高广运, 冯世进. 饱和层状地基的薄层法基本解及其旁轴边界[J]. 岩土工程学报, 2010, 32(5): 664-671. http://cge.nhri.cn/article/id/13399 SHI Gang, GAO Guangyun, FENG Shijin. Basic solution of saturated layered ground by thin layered method and its paraxial boundary[J]. Chinese Journal of Geotechnical Engineering, 2010, 32(5): 664-671. (in Chinese) http://cge.nhri.cn/article/id/13399
[7]	PAN E N. Green's functions for geophysics: a review[J]. Reports on Progress in Physics, 2019, 82(10): 106801. doi: 10.1088/1361-6633/ab1877
[8]	OZAWA Y, MAINA J, MATSUI K. Influence of cross-anisotropy material behavior on back-calculation analysis of multi-layered systems[C]// 6th International conference on road and airfield pavement technology. Sapporo, 2008.
[9]	WANG C D, YE Z Q, RUAN Z W. Displacement and stress distributions under a uniform inclined rectangular load on a cross-anisotropic geomaterial[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2009, 33(6): 709-748. doi: 10.1002/nag.738
[10]	艾智勇, 杨轲舒. 横观各向同性层状地基上弹性矩形板的参数研究[J]. 岩土工程学报, 2016, 38(8): 1442-1446. doi: 10.11779/CJGE201608011 AI Zhiyong, YANG Keshu. Parametric study on an elastic rectangle plate on transversely isotropic multi-layered soils[J]. Chinese Journal of Geotechnical Engineering, 2016, 38(8): 1442-1446. (in Chinese) doi: 10.11779/CJGE201608011
[11]	CAI Y C, SANGGHALEH A, PAN E N. Effect of anisotropic base/interlayer on the mechanistic responses of layered pavements[J]. Computers and Geotechnics, 2015, 65: 250-257. doi: 10.1016/j.compgeo.2014.12.014
[12]	POULOS H G. Stresses and displacements in an elastic layer underlain by a rough rigid base[J]. Géotechnique, 1967, 17(4): 378-410. doi: 10.1680/geot.1967.17.4.378
[13]	CHOW Y K. Vertical deformation of rigid foundations of arbitrary shape on layered soil media[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1987, 11(1): 1-15. doi: 10.1002/nag.1610110102
[14]	林皋, 韩泽军, 李建波. 层状地基任意形状刚性基础动力响应求解[J]. 力学学报, 2012, 44(6): 1016-1027. LIN Gao, HAN Zejun, LI Jianbo. Solution of the dynamic response of rigid foundation of arbitrary shape on multi-layered soil[J]. Chinese Journal of Theoretical and Applied Mechanics, 2012, 44(6): 1016-1027. (in Chinese)
[15]	巴振宁, 胡黎明, 梁建文. 层状横观各向同性地基上刚性条形基础动力刚度系数[J]. 土木工程学报, 2017, 50(9): 67-81. BA Zhenning, HU Liming, LIANG Jianwen. Dynamic impedance functions of rigid strip foundation on a multi-layered transversely isotropic half-space[J]. China Civil Engineering Journal, 2017, 50(9): 67-81. (in Chinese)
[16]	YUE Z Q, YIN J H. Layered elastic model for analysis of Cone Penetration Testing[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1999, 23(8): 829-843. doi: 10.1002/(SICI)1096-9853(199907)23:8<829::AID-NAG16>3.0.CO;2-X
[17]	黄茂松, 钟锐, 任青. 层状地基中沉箱加桩复合基础的水平-摇摆振动[J]. 岩土工程学报, 2012, 34(5): 790-797. http://cge.nhri.cn/article/id/14565 HUANG Maosong, ZHONG Rui, REN Qing. Lateral vibration of caisson-pile composite foundation in layered soils[J]. Chinese Journal of Geotechnical Engineering, 2012, 34(5): 790-797. (in Chinese) http://cge.nhri.cn/article/id/14565
[18]	王志新, 王波, 李昊, 等. 飞机荷载引起的层状地基附加应力及对下穿隧道的影响范围研究[J]. 土木工程学报, 2020, 53(增刊1): 258-264. WANG Zhixin, WANG Bo, LI Hao, et al. The additional stress of layered foundation caused by aircraft load and its influence scope on the underpass tunnel[J]. China Civil Engineering Journal, 2020, 53(S1): 258-264. (in Chinese)
[19]	SELVADURAI A P S. The analytical method in geomechanics[J]. Applied Mechanics Reviews, 2007, 60(3): 87-106. doi: 10.1115/1.2730845
[20]	XIAO S, YUE Z Q. Elastic response of transversely isotropic and non-homogeneous geomaterials under circular ring concentrated and axisymmetric distributed loads[J]. Engineering Analysis with Boundary Elements, 2024, 158: 385-404. doi: 10.1016/j.enganabound.2023.10.025
[21]	XIAO S, DU X L, YUE Z Q. Axisymmetric elastic field in layered non-homogeneous and transversely isotropic geo-materials due to surface traction[J]. Computers and Geotechnics, 2023, 155: 105226. doi: 10.1016/j.compgeo.2022.105226
[22]	ALMEIDA PEREIRA O J B, PARREIRA P. Direct evaluation of Cauchy-principal-value integrals in boundary elements for infinite and semi-infinite three-dimensional domains[J]. Engineering Analysis with Boundary Elements, 1994, 13(4): 313-320. doi: 10.1016/0955-7997(94)90025-6
[23]	MOSER W, DUENSER C, BEER G. Mapped infinite elements for three-dimensional multi-region boundary element analysis[J]. International Journal for Numerical Methods in Engineering, 2004, 61(3): 317-328. doi: 10.1002/nme.1073
[24]	XIAO S, YUE Z, XIAO H. Boundary element analysis of elastic fields in non-horizontally layered halfspace whose horizontal boundary subject to tractions[J]. Engineering Analysis with Boundary Elements, 2018, 95: 105-123. doi: 10.1016/j.enganabound.2018.06.020
[25]	XIAO S, YUE Z Q. Complete solutions for elastic fields induced by point load vector in functionally graded material model with transverse isotropy[J]. Applied Mathematics and Mechanics, 2023, 44(3): 411-430. doi: 10.1007/s10483-023-2958-8