桩侧摩擦力在均质地基中引起的竖向位移计算公式

江小兵; 朱沈阳; 张浩博

doi:10.20265/j.cnki.issn.1007-2993.2023-0646

桩侧摩擦力在均质地基中引起的竖向位移计算公式

doi: 10.20265/j.cnki.issn.1007-2993.2023-0646

1.
中国有色金属工业西安勘察设计研究院有限公司，陕西西安　710054
2.
陕西省建设工程质量安全监督总站，陕西西安　710016
3.
西安理工大学，陕西西安　710048

详细信息

作者简介:
江小兵，男，1979年生，硕士，高级工程师，主要从事岩土工程勘察与检测工作。E-mail：155075214@qq.com

中图分类号: TU470
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- 被引次数: 0
出版历程
- 收稿日期: 2023-07-17
- 修回日期: 2023-11-05
- 录用日期: 2023-12-25
- 刊出日期: 2025-02-21

Calculation formula of vertical displacement caused by side friction of pile in homogeneous foundation soils

1.
Xi’an Investigation Institute of China National Nonferrous Metals Industry, Xi’an 710054, Shaanxi, China
2.
Shaanxi Province Construction Engineering Quality and Safety Supervision Station, Xi’an 710016, Shaanxi, China
3.
Xi’an University of Technology, Xi’an 710048, Shaanxi, China

摘要

摘要: 根据半无限体内受集中力作用的Mindlin位移解，通过线积分完整地推导出均质地基中桩侧摩阻均匀分布、桩侧摩阻随深度呈线性增加分布以及桩侧摩阻随深度呈任意形态分布情况下引起的地基竖向位移计算公式，通过有限元计算程序和具体工程算例验证了所推导公式的正确性和可行性。文中所推导公式对于纯摩擦单桩沉降的估算、桩土体受荷前期弹性变形阶段的计算、弹塑性变形及有限元计算的对比验算，均具有重要的作用。西安部分地区地层主要由压缩性中等偏低的黄土和古土壤组成，土质较均匀，运用本文计算方法可以取得较为可靠的计算结果，具有实际应用价值。
- Mindlin解 /
- 桩侧摩擦力分布形式 /
- 竖向位移 /
- 公式推导 /
- 有限元计算程序验证 /
- 公式应用
Abstract: According to Mindlin’s displacement solution regarding a semi-infinite mass subjected to a point load, the calculation formulas for vertical displacement caused by skin friction in piles under uniformly distributed loads have been derived thoroughly. This includes scenarios with linearly increasing distribution loads along the length of the pile and arbitrary distribution loads within uniform foundations. Through concrete project examples, it is confirmed that these formulas are both accurate and practical. The formula presented plays a crucial role in calculating the settlement of pure friction single piles, the elastic deformation in the early stage of pile and soil under load, and the comparison of elastic-plastic deformation and finite element calculation. The strata in some areas of Xi’an are mainly composed of medium and low compressible loess and paleosol, and the soil quality is relatively uniform. The results obtained by this calculation method are reliable and have practical application value.
- Mindlin’s solution /
- the distribution form of skin friction of pile /
- vertical displacement /
- derivation of formula /
- verification of finite element calculation program /
- application of formula

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图 1 h深度处集中荷载P引起的任意点M的竖向位移^[11]

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图 2 均匀竖向线荷载作用引起的地基竖向位移

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图 3 桩侧摩阻均匀分布引起的地基竖向位移计算

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图 4 桩侧摩阻随深度呈线性增加分布引起的竖向位移计算

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图 5 桩侧摩阻随深度呈任意形态分布引起的竖向位移计算

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图 6 有限元计算模型及桩侧施加侧摩阻力情况

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图 7 土样弹性模量试验曲线

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图 8 Q-s 曲线图

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表 1 模型1桩中心点下各点竖向位移

距桩顶深度/m	S₁/ mm	S₁′/ mm	差异率/ %	S₂/ mm	S₂′/ mm	差异率/ %
0	34.0	33.6	1.2	19.4	19.2	1.0
2	35.0	34.6	1.1	20.9	20.7	1.0
4	33.8	33.4	1.2	22.8	22.6	0.9
6	32.8	32.5	0.9	25.0	24.7	1.2
8	31.8	31.5	0.9	27.3	27.1	0.7
10	30.9	30.6	1.0	29.6	29.4	0.7
15	27.8	27.5	1.0	33.7	33.3	1.2
20	19.4	19.1	1.5	25.3	25.0	1.2
22	14.6	14.3	2.1	17.7	17.5	1.1
26	10.3	10.0	3.0	11.6	11.2	3.4
30	8.3	7.9	3.6	9.0	8.6	4.4
35	6.8	6.3	7.4	7.2	6.7	6.9
40	5.8	5.2	10.3	6.0	5.4	10.0
50	4.5	3.6	20.0	4.6	3.7	19.6
注：S₁，S₂分别为代入式（3）、式（8）计算位移；S₁′，S₂′分别为桩侧均布荷载、递增荷载作用下有限元计算位移。

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表 2 模型2桩中心点下各点竖向位移

距桩顶深度/m	S₃/ mm	S₃′/ mm	差异率/ %	S₄/ mm	S₄′/ mm	差异率/ %
0	29.8	29.2	2.0	12.8	12.6	1.6
2	31.0	30.5	1.6	14.4	14.1	2.1
4	30.4	29.9	1.6	16.2	15.8	2.5
6	30.0	29.4	2.0	18.1	17.7	2.2
10	29.2	28.7	1.7	22.2	21.8	1.8
15	28.2	27.7	1.8	27.4	26.9	1.9
20	27.0	26.5	1.9	32.1	31.5	2.0
25	25.0	24.4	2.4	35.0	34.3	2.0
30	16.7	16.2	3.0	24.3	23.6	2.9
32	11.1	11.0	0.9	14.3	13.7	4.2
34	9.1	8.8	3.3	11.1	10.5	5.4
36	8.0	7.5	6.3	9.3	8.7	6.5
40	6.6	5.9	9.1	7.4	6.6	10.8
45	5.5	4.7	14.5	6.0	5.1	16.3
50	4.7	3.8	19.1	5.1	4.1	19.6
备注：S₃，S₄分别为代入式（3）、式（8）计算位移；S₃′，S₄′分别为桩侧均布荷载、递增荷载作用下有限元计算位移。

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表 3 西安地区典型钻孔灌注桩试桩工程算例

编号	工程名称	成孔工艺	桩顶荷载/kN	平均侧摩阻/kPa	桩径/m	桩长/m	计算桩底竖向位移/mm	计算桩身压缩量/mm	计算总沉降量/mm	实测试桩平均沉降量/mm
1	华山机械厂	锅锥成孔	4400	77.0	0.70	26.0	2.63	4.95	7.58	6.52
2	景天佳苑住宅楼	锅锥成孔	3690	63.1	0.60	31.0	1.96	6.74	8.70	4.14
3	西勘商住楼	锅锥成孔	5900	81.3	0.70	33.0	2.89	8.43	11.32	9.77
4	旺座时代A区1^#、2^#楼	锅锥成孔	8250	107.2	0.70	35.0	3.85	10.42	14.27	12.46
5	豪盛时代C区	锅锥成孔	8200	113.0	0.70	33.0	4.02	10.65	14.67	13.25
6	长安区土地综合开发公司1^#、2^#楼	锅锥成孔	5000	82.9	0.60	32.0	2.58	8.32	10.90	10.94
7	西关正街商住楼B座	锅锥成孔	8400	97.9	0.70	39.0	3.59	12.90	16.49	16.83
8	世家星城三期93^#楼	锅锥成孔	5000	75.9	0.60	35.0	2.40	9.38	11.78	8.38
9	乾唐雁月1^#楼	锅锥成孔	7000	88.4	0.70	36.0	3.19	9.92	13.11	12.60
10	融发·沁园西区20^#、 21^#住宅楼	锅锥成孔	2300	49.0	0.60	25.0	1.46	3.28	4.74	3.60
11	陕西省公务员培训楼	锅锥成孔	5000	83.0	0.60	32.0	2.58	8.57	11.15	8.37
12	金源都市公寓	锅锥成孔	3520	62.0	0.60	30.0	1.92	5.84	7.76	6.94
13	广丰大厦	锅锥成孔	11000	125.0	0.80	35.0	5.02	11.61	16.63	16.04
14	西安市卫生学校高层住宅楼	锅锥成孔	7200	93.5	0.70	35.0	3.36	9.63	12.99	13.11
15	元丰怡家住宅小区9^#、11^#楼	锅锥成孔	6000	106.0	0.60	30.0	3.27	9.95	13.22	12.34
16	长丰园Ⅱ区2^#高层住宅楼	旋挖成孔	6600	113.0	0.60	31.0	3.50	11.31	14.81	13.82
17	西安现代花城A、B座	旋挖成孔	7200	93.5	0.70	36.0	3.29	10.86	14.15	14.04
18	西安小寨住宅小区1^#住宅楼	旋挖成孔	6800	81.4	0.70	38.0	2.97	10.17	13.14	10.86
19	旺座曲江四期H座	旋挖成孔	7600	106.1	0.60	38.0	3.40	14.59	17.99	20.12
20	旺座曲江四期Ｎ座	旋挖成孔	8400	109.8	0.60	40.6	3.56	17.23	20.79	21.19
注：按式（3）计算时，土的弹性模量统一取80 MPa，土的泊松比取0.38，桩的弹性模量根据砼的强度等级C30～C55取值30～36 GPa。

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参考文献(15)

[1]	袁聚云, 赵锡宏. 竖向均布荷载作用在地基内部时的土中应力公式[J]. 上海力学,1995,16(3):213-222. (YUAN J Y, ZHAO X H. Formulas for the calculation of stresses in soil subjecting to distributed loading beneath the surface of the ground[J]. Shanghai Journal of Mechanics,1995,16(3):213-222. (in Chinese) YUAN J Y, ZHAO X H. Formulas for the calculation of stresses in soil subjecting to distributed loading beneath the surface of the ground[J]. Shanghai Journal of Mechanics, 1995, 16（3）: 213-222. (in Chinese)
[2]	周罡, 林荫. 用Mindlin应力解求单桩沉降的方法[J]. 地下空间,2001,21(3):173-177. (ZHOU G, LIN Y. Method for calculation of single pile settlement by Mindlin stress solution[J]. Underground Space,2001,21(3):173-177. (in Chinese) ZHOU G, LIN Y. Method for calculation of single pile settlement by Mindlin stress solution[J]. Underground Space, 2001, 21（3）: 173-177. (in Chinese)
[3]	王士杰, 张梅, 张吉占. Mindlin应力解的应用理论研究[J]. 工程力学,2001,18(6):141-148. (WANG S J, ZHANG M, ZHANG J Z. On Mindlin stress formulas[J]. Engineering Mechanics,2001,18(6):141-148. (in Chinese) doi: 10.3969/j.issn.1000-4750.2001.06.020 WANG S J, ZHANG M, ZHANG J Z. On Mindlin stress formulas[J]. Engineering Mechanics, 2001, 18（6）: 141-148. (in Chinese) doi: 10.3969/j.issn.1000-4750.2001.06.020
[4]	中华人民共和国住房和城乡建设部. 建筑地基基础设计规范: GB 50007–2011[S]. 北京: 中国计划出版社, 2012. (Ministry of Housing and Urban-Rural Development of the People's Republic of China. Code for design of building foundation: GB 50007–2011[S]. Beijing: China Planning Press, 2012. (in Chinese) Ministry of Housing and Urban-Rural Development of the People's Republic of China. Code for design of building foundation: GB 50007–2011[S]. Beijing: China Planning Press, 2012. (in Chinese)
[5]	GEDDES J D. Stresses in foundation soils due to vertical subsurface loading[J]. Géotechnique,1966,16(3):231-255.
[6]	中华人民共和国建设部. 建筑桩基技术规范: JGJ 94–2008[S]. 北京: 中国建筑工业出版社, 2008. (Ministry of Housing and Urban-Rural Development of the People's Republic of China. Technical code for building pile foundations: JGJ 94–2008[S]. Beijing: China Architecture & Building Press, 2008. (in Chinese) Ministry of Housing and Urban-Rural Development of the People's Republic of China. Technical code for building pile foundations: JGJ 94–2008[S]. Beijing: China Architecture & Building Press, 2008. (in Chinese)
[7]	杨敏, 王树娟, 王伯钧, 等. 使用Geddes应力系数公式求解单桩沉降[J]. 同济大学学报,1997,25(4):379-385. (YANG M, WANG S J, WANG B J, et al. On analysis of single pile by using Geddes’ stress formula[J]. Journal of Tongji University,1997,25(4):379-385. (in Chinese) YANG M, WANG S J, WANG B J, et al. On analysis of single pile by using Geddes’ stress formula[J]. Journal of Tongji University, 1997, 25（4）: 379-385. (in Chinese)
[8]	吕亚茹, 刘汉龙, 王新泉, 等. 地基附加应力和桩基沉降计算的修正Geddes应力解[J]. 四川大学学报(工程科学版),2013,45(2):68-74. (LYU Y R, LIU H L, WANG X Q, et al. A modified Geddes solution for estimating induced stress coefficient of subsoil and settlement of pile foundation[J]. Journal of Sichuan University (Engineering Science Edition),2013,45(2):68-74. (in Chinese) LYU Y R, LIU H L, WANG X Q, et al. A modified Geddes solution for estimating induced stress coefficient of subsoil and settlement of pile foundation[J]. Journal of Sichuan University (Engineering Science Edition), 2013, 45（2）: 68-74. (in Chinese)
[9]	毕平均. 基于Geddes应力解的桩筏基础沉降计算[J]. 地下空间与工程学报,2012,8(5):1048-1051,1058. (BI P J. Settlement calculation of the piled raft foundations based on Geddes solution[J]. Chinese Journal of Underground Space and Engineering,2012,8(5):1048-1051,1058. (in Chinese) BI P J. Settlement calculation of the piled raft foundations based on Geddes solution[J]. Chinese Journal of Underground Space and Engineering, 2012, 8（5）: 1048-1051,1058. (in Chinese)
[10]	POULOS H G, DAVIS E H. Elastic solutions for soil and rock mechanics[M]. New York: Wiley, 1974.
[11]	POULOS H G, DAVIS E H. 岩土力学弹性解[M]. 孙幼兰, 译. 徐州: 中国矿业大学出版社, 1990. (POULOS H G, DAVIS E H. The rock and soil mechanics elastic solution[M]. SUN Y L, trans. Nanjing: China University of Mining and Technology Publishing House, 1990. (in Chinese) POULOS H G, DAVIS E H. The rock and soil mechanics elastic solution[M]. SUN Y L, trans. Nanjing: China University of Mining and Technology Publishing House, 1990. (in Chinese)
[12]	薛定宇, 陈阳泉. 高等应用数学问题的Matlab求解[M]. 北京: 清华大学出版社, 2004. (XUE D Y, CHEN Y Q. The Solution of higher applied mathematics problems by Matlab[M]. Beijing: Tsinghua University Press, 2004. (in Chinese) XUE D Y, CHEN Y Q. The Solution of higher applied mathematics problems by Matlab[M]. Beijing: Tsinghua University Press, 2004. (in Chinese)
[13]	中华人民共和国水利部. 土工试验规程: SL 237–1999[S]. 北京: 中国水利水电出版社, 1999. (Ministry of Water Resources of the People’s Republic of China. Specification of soil test: SL 237–1999[S]. Beijing: China Water & Power Press, 1999. (in Chinese) Ministry of Water Resources of the People’s Republic of China. Specification of soil test: SL 237–1999[S]. Beijing: China Water & Power Press, 1999. (in Chinese)
[14]	贾堤, 石峰, 郑刚, 等. 深基坑工程数值模拟土体弹性模量取值的探讨[J]. 岩土工程学报,2008,30(S):155-158. (JIA D, SHI F, ZHENG G, et al. Elastic modulus of soil used in numerical simulation of deep foundation pits[J]. Chinese Journal of Geotechnical Engineering,2008,30(S):155-158. (in Chinese) JIA D, SHI F, ZHENG G, et al. Elastic modulus of soil used in numerical simulation of deep foundation pits[J]. Chinese Journal of Geotechnical Engineering, 2008, 30（S）: 155-158. (in Chinese)
[15]	中华人民共和国住房和城乡建设部. 建筑基桩检测技术规范: JGJ 106–2014[S]. 北京: 中国建筑工业出版社, 2014. (Ministry of Housing and Urban-Rural Development of the People's Republic of China. Technical code for testing of building foundation piles: JGJ 106–2014[S]. Beijing: China Architecture & Building Press, 2014. (in Chinese) Ministry of Housing and Urban-Rural Development of the People's Republic of China. Technical code for testing of building foundation piles: JGJ 106–2014[S]. Beijing: China Architecture & Building Press, 2014. (in Chinese)