基于Mindlin-Geddes解的桩基沉降计算改进方法

王小波; 李建光; 王笃礼

doi:10.20265/j.cnki.issn.1007-2993.2025-0281

基于Mindlin-Geddes解的桩基沉降计算改进方法

doi: 10.20265/j.cnki.issn.1007-2993.2025-0281

1.
中航勘察设计研究院有限公司，北京　100098
2.
上海勘测设计研究院有限公司，上海　200434

详细信息

作者简介:
王小波，男，1978年生，博士，研究员，主要从事岩土工程咨询、设计及数值仿真研究工作。E-mail：jack_tsinghua@126.com

中图分类号: TU473.1
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- 文章访问数: 73
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- 被引次数: 0
出版历程
- 收稿日期: 2025-06-25
- 修回日期: 2025-08-03
- 录用日期: 2025-08-25
- 刊出日期: 2026-08-08

An improved method of pile foundation settlement calculation based on Mindlin-Geddes solution

1.
AVIC Institute of Geotechnical Engineering Co., Ltd., Beijing 100098, China
2.
Shanghai Investigation Design and Reserarch Institute Co., Ltd., Shanghai 200434, China

摘要

摘要: 目前桩基沉降计算多采用Mindlin-Geddes应力解法，为达到精度要求，需对地层进行细致划分，导致该方法计算过程繁琐、效率低下。为提升桩基沉降计算效率，通过解析桩端及桩侧阻力引起的附加应力场，结合Mindlin-Geddes应力解与分层总和法，推导了平均竖向应力影响系数的积分表达式，并改进了桩基沉降计算公式。同时，编制出不同阻力分布模式下的平均竖向应力影响系数的计算表格。无量纲应力面积算例和工程案例分析计算表明，平均竖向应力影响系数方法与规范方法计算结果吻合，精度满足要求，且不需要对每层土再进行细分计算，有效提升了桩基沉降计算效率与精度，为桩基工程设计提供了新的计算方法与技术手段。
- Mindlin-Geddes应力解 /
- 平均竖向应力影响系数 /
- 积分表达式 /
- 阻力分布模式
Abstract: Currently, the calculation of pile foundation settlement mostly relies on the Mindlin-Geddes stress solution. To meet accuracy requirements, this method demands a detailed subdivision of soil strata, resulting in a cumbersome and inefficient computational process. To improve computational efficiency, by combining the Mindlin-Geddes stress solution with the layered summation method, the integral expressions of the average vertical stress influence coefficient were derived, and the pile foundation settlement calculation formulas were improved. Meanwhile, calculation tables of the average vertical stress influence coefficients under different resistance distribution patterns were compiled. Through the calculation and analysis of a dimensionless stress area example and an engineering case, the results show that the average vertical stress influence coefficient method agrees well with the calculation results of the standard method, the accuracy meets the requirements. Moreover, it does not need to further subdivide each soil layer for calculation, effectively improves the efficiency and accuracy of pile foundation settlement calculation, and provides new calculation methods and technical means for the design of pile foundation projects.
- Mindlin-Geddes stress solution /
- average vertical stress influence coefficient /
- integral expression /
- resistance distribution pattern

HTML全文

图 1 Mindlin解计算示意图

Figure 1. Schematic diagram of Mindlin solution calculation

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图 2 桩端阻力和桩侧阻力分布示意图

Figure 2. Schematic diagram of the distribution of pile tip resistance and pile shaft resistance

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图 3 桩基沉降计算示意图

Figure 3. Schematic diagram of pile foundation settlement calculation

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图 4 桩基平面布置图

Figure 4. Plan layout of pile foundation

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表 1 岩土体计算参数

Table 1. Calculation parameters of rock and soil mass

岩土编号	岩土名称	重度/（kN∙m⁻³）	压缩模量/MPa
⑨	圆砾、卵石	20.20	40.00
⑨₁	细中砂	20.20	32.00
⑩	黏土	18.86	14.25
⑩₁	粉质黏土、重粉质黏土	19.75	15.63
⑩₃	黏质粉土、砂质粉土	20.40	23.80
⑪	细中砂	20.50	35.00
⑫	粉质黏土、重粉质黏土	20.13	16.12
⑫₃	黏土	19.16	15.79
⑬	粉细砂	20.80	40.00

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表 2 本文方法与规范方法计算结果对比

Table 2. Comparison of calculation results between the method in this paper and the specification method

布桩方式	本文方法（平均竖向应力影响系数法）		规范方法（竖向应力影响系数法）
布桩方式	桩沉降s_p/mm	总沉降s/mm	桩沉降s_p/mm	总沉降s/mm
方式一	48.9	52.1	49.4	52.6
方式二	37.0	39.7	37.3	39.9

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参考文献(20)

[1]	MINDLIN R D. Force at a point in the interior of a semi-infinite solid[J]. Physics, 1936, 7(5): 195-202. doi: 10.1063/1.1745385
[2]	GEDDES J D. Stresses in foundation soils due to vertical subsurface loading[J]. Géotechnique, 1966, 16(3): 231-255. doi: 10.1680/geot.1966.16.3.231
[3]	POULOS H G, DAVIS E H. Elastic solutions for soil and rock mechanics[M]. New York: Wiley, 1974.
[4]	RANDOLPH M F, WROTH C P. Analysis of deformation of vertically loaded piles[J]. Journal of the Geotechnical Engineering Division, 1978, 104(12): 1465-1488. doi: 10.1061/ajgeb6.0000729
[5]	徐志英. 以明特林(Mindlin)公式为根据的地基中垂直应力的計算公式[J]. 土木工程学报, 1957, 4(4): 485-497. (XU Z Y. Calculation of vertical pressures in soil beneath loaded areas based on Mindlin's formula[J]. Chinese Journal of Civil Engineering, 1957, 4(4): 485-497. (in Chinese) XU Z Y. Calculation of vertical pressures in soil beneath loaded areas based on Mindlin's formula[J]. Chinese Journal of Civil Engineering, 1957, 4（4）: 485-497. (in Chinese)
[6]	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) doi: 10.3969/j.issn.1673-0836.2001.03.003 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) doi: 10.3969/j.issn.1673-0836.2001.03.003
[7]	邱明兵, 刘金砺, 秋仁东, 等. 基于Mindlin解的单桩竖向附加应力系数[J]. 土木工程学报, 2014, 47(3): 130-137. (QIU M B, LIU J L, QIU R D, et al. Study on vertical additional stress coefficient for the single piles based on Mindlin's stress solution theory[J]. China Civil Engineering Journal, 2014, 47(3): 130-137. (in Chinese) doi: 10.15951/j.tmgcxb.2014.03.014 QIU M B, LIU J L, QIU R D, et al. Study on vertical additional stress coefficient for the single piles based on Mindlin's stress solution theory[J]. China Civil Engineering Journal, 2014, 47（3）: 130-137. (in Chinese) doi: 10.15951/j.tmgcxb.2014.03.014
[8]	姚梅红. Mindlin桩基沉降竖向应力影响系数的Excel算法[J]. 福建建筑, 2012(12): 103-105. (YAO M H. Mindlin influence coefficient of vertical stress in pile foundation settlement algorithm of Excel[J]. Fujian Architecture & Construction, 2012(12): 103-105. (in Chinese) YAO M H. Mindlin influence coefficient of vertical stress in pile foundation settlement algorithm of Excel[J]. Fujian Architecture & Construction, 2012（12）: 103-105. (in Chinese)
[9]	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)
[10]	中华人民共和国住房和城乡建设部. 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)
[11]	林智勇, 戴自航, 苏美选. 基于Mindlin位移解考虑桩径影响的桩基沉降计算[J]. 福州大学学报(自然科学版), 2009, 37(4): 588-592. (LIN Z Y, DAI Z H, SU M X. Settlement calculation of pile foundations based on Mindlin’s displacement formula considering influence of pile diameters[J]. Journal of Fuzhou University (Natural Science Edition), 2009, 37(4): 588-592. (in Chinese) LIN Z Y, DAI Z H, SU M X. Settlement calculation of pile foundations based on Mindlin’s displacement formula considering influence of pile diameters[J]. Journal of Fuzhou University (Natural Science Edition), 2009, 37（4）: 588-592. (in Chinese)
[12]	王涛. 基于Mindlin公式考虑桩径影响的应力计算方法[J]. 建筑结构, 2006, 36(10): 46-49. (WANG T. New stress calculation method for Mindlin formula considering impact of piles diameters[J]. Building Structure, 2006, 36(10): 46-49. (in Chinese) WANG T. New stress calculation method for Mindlin formula considering impact of piles diameters[J]. Building Structure, 2006, 36（10）: 46-49. (in Chinese)
[13]	王涛, 高文生, 刘金砺. 桩基变刚度调平设计的实施方法研究[J]. 岩土工程学报, 2010, 32(4): 531-537. (WANG T, GAO W S, LIU J L. Study on implemention method for optimization design of pile foundation stiffness to reduce differential settlement[J]. Chinese Journal of Geotechnical Engineering, 2010, 32(4): 531-537. (in Chinese) WANG T, GAO W S, LIU J L. Study on implemention method for optimization design of pile foundation stiffness to reduce differential settlement[J]. Chinese Journal of Geotechnical Engineering, 2010, 32（4）: 531-537. (in Chinese)
[14]	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)
[15]	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) WANG S J, ZHANG M, ZHANG J Z. On Mindlin stress formulas[J]. Engineering Mechanics, 2001, 18（6）: 141-148. (in Chinese)
[16]	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)
[17]	江小兵, 朱沈阳, 张浩博. 桩侧摩擦力在均质地基中引起的竖向位移计算公式[J]. 岩土工程技术, 2025, 39(1): 6-13. (JIANG X B, ZHU S Y, ZHANG H B. Calculation formula of vertical displacement caused by side friction of pile in homogeneous foundation soils[J]. Geotechnical Engineering Technique, 2025, 39(1): 6-13. (in Chinese) JIANG X B, ZHU S Y, ZHANG H B. Calculation formula of vertical displacement caused by side friction of pile in homogeneous foundation soils[J]. Geotechnical Engineering Technique, 2025, 39（1）: 6-13. (in Chinese)
[18]	刘金砺, 秋仁东, 邱明兵, 等. 不同条件下桩侧阻力端阻力性状及侧阻力分布概化与应用[J]. 岩土工程学报, 2014, 36(11): 1953-1970. (LIU J L, QIU R D, QIU M B, et al. Behaviors of shaft resistance and tip resistance of piles under different conditions and conceptualization and application of distribution of shaft resistance[J]. Chinese Journal of Geotechnical Engineering, 2014, 36(11): 1953-1970. (in Chinese) doi: 10.11779/CJGE201411001 LIU J L, QIU R D, QIU M B, et al. Behaviors of shaft resistance and tip resistance of piles under different conditions and conceptualization and application of distribution of shaft resistance[J]. Chinese Journal of Geotechnical Engineering, 2014, 36（11）: 1953-1970. (in Chinese) doi: 10.11779/CJGE201411001
[19]	王涛, 刘金砺, 王旭. 基于桩侧阻概化模式的基桩均化附加应力系数研究[J]. 岩土工程学报, 2018, 40(4): 665-672. (WANG T, LIU J L, WANG X. Homogenized additional stress coefficient of foundation piles based on generalized mode of pile shaft resistance[J]. Chinese Journal of Geotechnical Engineering, 2018, 40(4): 665-672. (in Chinese) doi: 10.11779/CJGE201804010 WANG T, LIU J L, WANG X. Homogenized additional stress coefficient of foundation piles based on generalized mode of pile shaft resistance[J]. Chinese Journal of Geotechnical Engineering, 2018, 40（4）: 665-672. (in Chinese) doi: 10.11779/CJGE201804010
[20]	王笃礼, 王小波, 李建光. 考虑侧阻分布特征的桩–土–筏板共同作用研究[J]. 建筑结构, 2021, 51(S2): 1661-1667. (WANG D L, WANG X B, LI J G. Research on pile-soil-raft interaction considering shaft resistance distribution characteristics[J]. Building Structure, 2021, 51(S2): 1661-1667. (in Chinese) WANG D L, WANG X B, LI J G. Research on pile-soil-raft interaction considering shaft resistance distribution characteristics[J]. Building Structure, 2021, 51（S2）: 1661-1667. (in Chinese)