开发和利用地下空间是当今城市发展的必然趋势，随之而来的施工扰动给周围环境带来了严重的影响。城市地铁隧道的开挖会影响上覆地层中既有管线的稳定性，如何确保管线的正常安全使用已成为当前地铁隧道施工的关键技术课题，对地铁隧道上覆管线变形规律的研究不仅具有重要的理论意义，而且对地铁隧道的安全快速施工具有重要的参考价值。工程实际中，受施工扰动影响，软岩地铁隧道上覆岩土体强度参数将产生不同程度的劣化，引起的地层移动将带动管线产生变形。现有关于地铁隧道对上覆管线变形的研究，主要集中在单一地层变形对管线的影响方面，尚缺乏对低岩跨比条件下地铁隧道开挖扰动对其上覆地层及管线变形影响规律的分析研究。本文以南宁某低岩跨比地铁区间隧道为工程背景，采用理论分析、数值模拟和室内试验相结合的研究方法对该隧道上覆岩土体及其管线在不同扰动程度下的变形规律进行了系统研究，主要结论如下：

（1）地铁隧道开挖后，其上覆岩层会受到不同程度的扰动损伤。在不同岩跨比条件下，岩层呈现出不同的破断塌落形态。当上覆岩层满足低岩跨比条件时，隧道开挖后岩层在地层压力作用下的塌落范围将穿越土岩分界面延伸至表土层，进而引发上覆表土层的沉降变形。低岩跨比条件下，隧道上覆岩层塌落形态呈“倒漏斗形”分布，上覆表土层下沉形态呈“漏斗形”分布。

（2）基于隧道上覆岩层的“倒漏斗形”与上覆表土层的“漏斗形”移动分布规律，构建了低岩跨比软岩地铁隧道上覆岩土层的“类双曲线”移动模型，提出了类双曲移动边界的判别条件和地表沉降槽宽度的求解方法。结果表明：地表沉降槽宽度随隧道上覆表土层厚度的增加而增大，随上覆岩层厚度的增加而逐渐减小，随上覆岩土层塑性变形范围的增加而增大。

（3）考虑施工间隙和注浆充填量对隧道围岩塑性变形范围的影响，建立了隧道围岩变形力学分析模型，得到围岩塑性区位移与地层损失参数的函数关系。考虑开挖扰动对地层损失的影响，基于霍克-布朗准则提出了开挖扰动下隧道上覆岩土层损失率的计算公式。考虑管土脱空效应，采用双层弹簧弹性地基梁模型，推导了隧道开挖扰动下上覆管线竖向挠曲位移、弯矩、剪力及转角的计算公式。结果表明：开挖扰动、管线埋深及管隧夹角均为隧道上覆管线变形的关键控制因素，管线转角、弯矩及剪力均与开挖扰动、管线埋深及管隧夹角呈正相关，管线竖向挠曲位移随扰动系数和管线埋深的增加而增大，随管隧夹角的增加而减小。

（4）采用理论分析与数值模拟相结合的方式，基于有限元软件ABAQUS建立了管隧垂直以及管隧平行两种工况下的三维数值计算模型，分析了开挖扰动、管线埋深、管隧夹角以及管材等因素对隧道上覆管线变形的影响规律。结果表明：开挖扰动和管线埋深对管线的竖向挠曲位移影响显著，管隧夹角对管线的变形范围影响显著，管材对管线的水平位移影响显著。

（5）运用本文管线变形理论对南宁某低岩跨比软岩地铁区间隧道上覆地层及管线的变形进行了分析计算，将计算结果和的现场监测数据进行了对比分析，验证了本文管线变形理论的工程适用性。基于分析结果对管线的安全风险进行评估，提出了低岩跨比工况条件下地铁隧道上覆管线的变形加固措施，并在现场取得了良好的管线变形控制效果。

The development and utilization of underground space is an inevitable trend in today's urban development, and the accompanying construction disturbances have brought serious impacts on the surrounding environment. The excavation of urban subway tunnel will affect the stability of existing pipelines in the overlying strata. How to ensure the normal and safe use of pipelines has become a key technical topic in the current construction of subway tunnel. The research on the deformation law of the overlying pipelines of subway tunnel not only has important theoretical significance, but also has important reference value for the safe and rapid construction of subway tunnel. In engineering practice, affected by construction disturbance, the strength parameters of the overlying rock and soil mass of the soft rock subway tunnel will deteriorate to varying degrees, and the resulting ground movement will drive the pipeline to deform. The existing research on the deformation of subway tunnel to the overlying pipeline mainly focuses on the influence of single stratum deformation on the pipeline, and there is still a lack of analysis and research on the influence of subway tunnel excavation disturbance on the overlying stratum and pipeline deformation under the condition of low rock­thickness span ratio. This article takes a low rock­thickness span ratio subway section tunnel in Nanning as the engineering background, and uses a combination of theoretical analysis, numerical simulation, and indoor experiments to systematically study the deformation laws of the overlying rock and soil and its pipelines under different degrees of disturbance. The main conclusions are as follows:

(1) After the excavation of subway tunnel, the overlying strata will be disturbed and damaged to varying degrees. Under different rock span ratios, rock layers exhibit different forms of fracture and collapse. When the overlying rock layer meets the condition of low rock­thickness span ratio, the collapse range of the rock layer under the action of formation pressure after tunnel excavation will extend through the soil rock interface to the topsoil layer, thereby causing settlement deformation of the overlying topsoil layer. Under low rock­thickness span ratio conditions, the collapse of the overlying rock strata in the tunnel is distributed in an inverted funnel shape, while the subsidence of the overlying topsoil layer is distributed in a funnel shape.

(2) Based on the "inverted funnel" movement distribution law of the overlying strata and the "funnel" movement distribution law of the overlying topsoil of the tunnel, the "quasi hyperbola" movement model of the overlying strata and soil of the subway tunnel in soft rock with low rock­thickness span ratio is constructed. The results show that the width of the surface settlement trough increases with the increase of the thickness of the overlying soil layer in the tunnel, gradually decreases with the increase of the thickness of the overlying rock layer, and increases with the increase of the plastic deformation range of the overlying rock layer.

(3) Considering the influence of construction gap and grouting filling amount on the plastic deformation range of tunnel surrounding rock, a mechanical analysis model for tunnel surrounding rock deformation was established, and the functional relationship between the displacement of the plastic zone of the surrounding rock and the parameters of strata loss was obtained. Considering the impact of excavation disturbance on geological losses, a formula for calculating the loss rate of overlying rock and soil layers in tunnels under excavation disturbance is proposed based on the Hoek-Brown criterion. Considering the void effect of pipe soil, a double layer spring elastic foundation beam model was used to derive the calculation formulas for the vertical deflection displacement, bending moment, shear force, and rotation angle of the overlying pipeline under tunnel excavation disturbance. The results show that excavation disturbance, pipeline burial depth, and pipe tunnel angle are key control factors for the deformation of the overlying pipeline in the tunnel. The pipeline angle, bending moment, and shear force are positively correlated with excavation disturbance, pipeline burial depth, and pipe tunnel angle. The vertical deflection displacement of the pipeline increases with the increase of disturbance coefficient and pipeline burial depth, and decreases with the increase of pipe tunnel angle.

(4) By combining theoretical analysis and numerical simulation, a three-dimensional numerical calculation model was established based on the finite element software ABAQUS under two working conditions of vertical and parallel pipe tunnels. The influence of excavation disturbance, pipeline burial depth, pipe tunnel angle, and pipe material on the deformation of the overlying pipe line in the tunnel was analyzed. The results show that excavation disturbance and pipeline burial depth have a significant impact on the vertical deflection displacement of the pipeline, the angle between the pipeline and the tunnel has a significant impact on the deformation range of the pipeline, and the pipe material has a significant impact on the horizontal displacement of the pipeline.

(5) The deformation theory of pipelines in this article was applied to analyze and calculate the deformation of the overlying strata and pipelines of a low rock­thickness to span ratio soft rock subway section tunnel in Nanning. The calculation results were compared with on-site monitoring data, verifying the engineering applicability of the pipeline deformation theory in this paper. Based on the analysis results, the safety risk of the pipeline is assessed, and the deformation reinforcement measures for the overlying pipeline of subway tunnel under the condition of low rock­thickness span ratio are proposed, and good deformation control effect of the pipeline is achieved on the site.

[1]王 安. 2022年城市轨道交通运营数据速报[EB/OL]. 中国政府网，2023-1-20.

[2]方玉树，郑颖人. 重庆市轻轨一号线工程埋置深度的工程地质研究[J]. 工程地质学报，1994，(1)：73-81.

[3]张先锋. 青岛地铁暗挖车站埋深的探讨[C]//中国土木工程学会隧道及地下工程学会第九届年会论文集：《铁道工程学报》编辑部，1996：328-331.

[4]张先锋. 对硬岩地层地铁车站结构设计的认识与思考[J]. 岩石力学与工程学报，2003，(3)：476-480.

[5]郑颖人，徐 浩，王 成，等. 隧洞破坏机理及深浅埋分界标准[J]. 浙江大学学报(工学版)，2010，44(10)：1851-1856，1875.

[6]郑颖人，朱合华，方正昌. 地下工程围岩稳定分析与设计理论[M]. 北京：人民交通出版社，2012.

[7]王旭东，周生华，迟建平，等. 上软下硬地层浅埋暗挖隧道覆跨比研究[J]. 地下空间与工程学报，2011，7(4)：700-705.

[8]郑建国. 土岩组合地层大跨度浅埋暗挖车站施工环境效应研究[D]. 青岛：中国海洋大学，2011.

[9]Wang X D, Yuan Y, WU X F, et al. Study on method to determining the rational depth of metro station in composed of soil and weathered rock stratum[C]//Proceedings of the 6th European Congress on Computational Methods in AppliedSciences and Engineering Vienna, 2012.

[10]张 佩，路德春，杜修力，等. 深埋隧道与浅埋隧道划分方法研究[J]. 岩土工程学报，2013，35(S2)：422-427.

[11]胡智民. 土岩组合地层浅埋隧道埋深确定方法研究[J]. 隧道建设，2015，35(04)：322-327.

[12]张自光，仇文革. 上软下硬地层暗挖地铁隧道合理岩跨比探讨及实例分析[J]. 现代隧道技术，2015，52(06)：28-35+42.

[13]刘 俊，刘新荣，赖 勇，等. 不同覆跨比下浅埋软弱隧道的破坏模式[J]. 中南大学学报(自然科学版)，2016，47(05)：1744-1751.

[14]刘运思，牟天光，郭 磊，等. 不同覆跨比下洞桩法导洞开挖引发地表变形规律研究[J]. 公路交通科技，2020，37(12)：100-107.

[15]张坤鹏. 地铁隧道施工影响下土岩复合地层渐进破坏模式及演化规律[D]. 青岛：青岛理工大学，2021.

[16]张俊儒，刘雨萌，燕 波. 机械化作业条件下考虑覆跨比修正的大断面隧道稳定性评价方法[J]. 中国铁道科学，2022，43(01)：92-100.

[17]李 创，陈德刚，周 明，等. 堆载作用下土岩复合地层隧道坍塌特征及演变规律研究[J]. 青岛理工大学学报，2023，44(01)：1-10.

[18]ATTEWELL P B, YEATES J, SELBY A R. Soil movements induced by tunnelling and their effects on pipelines and structures[M]. London: Blackie and Son Ltd, 1986.

[19]MARTOS F. Concerning an approximate equation of the subsidence trough and its time factors[C]//International strata control congress. Leipzig, 1958: 191-205.

[20]PECK R B. Deep excavations and tunneling in soft ground[C]//Proceedings 7th ICSMFE, State of the Art Volume. Mexico City, 1969: 225-290.

[21]CLOUGH G W, SCHMIDT B. Design and performance of excavations and tunnels in softclay[C]//Soft Clay Engineering. New York: Elsevier, 1981: 569-634.

[22]O'REILLY M P, NEW B M. Settlements above tunnels in the United Kingdom-their magnitude and prediction[C]//Proceedings Tunneling 82 Symposium. London, 1982: 173-181.

[23]LEACH G. Pipeline response to tunneling[Z]. 1985.

[24]MAIR R J, TAYLOR R N, Bracegirdle A. Subsurface settlement profiles above tunnels in clays[J]. Geotechnique, 1993, 43(2): 361-362.

[25]MOH Z C, JU D H, HWANG R N. Ground movements around tunnels in soft ground[C]//Proceedings Intemational Symposium on Geotechnical Aspects of Underground Construction in Soft Ground. London: Balkema AA, 1996: 725-730.

[26]韩 煊. 隧道施工引起地层位移及建筑物变形预测的实用方法研究[D]. 西安：西安理工大学，2007.

[27]韩 煊，李 宁，J.R.Standing. Peck公式在我国隧道施工地面变形预测中的适用性分析[J]. 岩土力学，2007(01)：23-28.

[28]SAGASETA C. Analysis of undraind soil deformation due to ground loss[J]. Geotechnique, 1987, 37(3): 301-320.

[29]VERRUIJT A, BOOKER J R. Surface settlements due to deformation of a tunnel in an elastic half plane[J]. Geotechnique, 1996, 46(4): 753-756.

[30]LOGANATHAN N, POULOS H G. Analytical prediction for tunneling-induced ground movements in clays[J]. Journal of Geotechnical and geoenvironmental engineering, 1998, 124(9): 846-856.

[31]CELESTINO T B, GOMES R, BORTOLUCCI A A.Errors in ground distortions due to settlement trough adjustment[J]. Tunnelling and underground space technology, 2000, 15(1): 97-100.

[32]VORSTER T E B, KLAR A, SOGA K, et al. Estimating the Effects of Tunneling on Existing Pipelines[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2005, 131(11): 1399-1410.

[33]刘建航，侯学渊. 盾构法隧道[M]. 北京：中国铁道出版社，1991.

[34]MAIR R J. Centrifugal modelling of tunnel construction in soft clay[D]. Cambridge: University of Cambridge, 1978.

[35]FINNO R J, MOLNAR K M, ROSSOw E C. Analysis of effects of deep braced excavations on adjacent buried utilities[R]. US Department of Transportation, 2003.

[36]李兴高，王 霆. 刚性管线纵向应变计算及安全评价[J]. 岩土力学，2008，29(12)：3299-3302.

[37]李兴高，王 霆. 柔性管线安全评价的简便方法[J]. 岩土力学，2008，29(07)：1861-1864.

[38]ZHANG C R, YU J, HUANG M S. Effects of tunnelling on existing pipelines in layered soils[J]. Computers and Geotechnics, 2003, 30: 12-25.

[39]KLAR A, VORSTER T, SOGA K, et al. Soil-pipe-tunnel interaction: comparison between winkler and elastic continuum solutions[R]. UK: Department of Engineering, University of Cambridge, 2004.

[40]KLAR A, MARSHALL A M, SOGA K, et al. Tunneling effects on jointed pipelines[J]. Canadian Geotechnical Journal, 2008, 45(1): 131-139.

[41]张坤勇，王 宇，艾英钵. 任意荷载下管土相互作用解答[J]. 岩土工程学报，2010，32(08)：1189-1193.

[42]WANG Y, WANG Q, ZHANG K Y. An Analytical Model for Pipe-Soil-Tunneling Interaction[J]. Procedia Engineering, 2011, 14: 3127-3135.

[43]张陈蓉，俞 剑，黄茂松. 隧道开挖对邻近非连续接口地埋管线的影响分析[J]. 岩土工程学报，2013，35(06)：1018-1026.

[44]张 桓，张子新. 盾构隧道开挖引起既有管线的竖向变形[J]. 同济大学学报(自然科学版)，2013，41(08)：1172-1178.

[45]WANG Y, MOORE I D. Simplified design equations for joints in buried flexible pipesbased on Hetenyi solutions[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2013, 140(3): 4013020.

[46]YU J, ZHANG CR, HUANG M S. Soil-pipe interaction due to tunnelling: Assessment of Winkler modulus for underground pipelines[J]. Computers and Geotechnics, 2013, 50: 17-28.

[47]杨成永，张彦斌，张伦政. 匀质地下管线大变形的控制微分方程及其近似解[J]. 中国铁道科学，2014，35(01)：42-46.

[48]杨成永，程 霖，余 乐，等. 匀质地下管线沉降的傅里叶级数解[J]. 中国铁道科学，2017，38(02)：71-76.

[49]WANG Y, MOORE I D. Simplified design model for rigid pipe joints based on the two-pipe approximation[J]. Canadian Geotechnical Journal, 2014, 52(5): 626-637.

[50]KLAR A, ELKAYAM I, MARSHALL A M. Design Oriented Linear-Equivalent Approach for Evaluating the Effect of Tunneling on Pipelines[J]. Journal of Geotechnical & Geoenvironmental Engineering, 2016, 142(1): 4015062.

[51]程 霖，杨成永，李延川，等. 带接头地下管线变形的傅里叶级数解[J]. 湖南大学学报(自然科学版)，2018，45(11)：149-156.

[52]林存刚，黄茂松. 基于Pasternak地基的盾构隧道开挖非连续地下管线的挠曲[J]. 岩土工程学报，2019，41(07)：1200-1207.

[53]王雨，王凯旋. 地铁隧道开挖下刚性接口管线安全控制研究[J]. 中国安全科学学报，2021，31(08)：97-103.

[54]KLAR A, MARSHALL A M. Shell versus beam representation of pipes in the evaluation of tunneling effects on pipelines[J]. Tunnelling and Underground Space Technology, 2008, 23(4): 431-437.

[55]MARSHALL A M, ELKAYAM I, KLAR A, et al. Centrifuge and discrete element modelling of tunnelling effects on pipelines[C]//Proceedings of the 7th International Conference on Physical Modelling in Geotechnics. Zurich, 2010: 633-637.

[56]WANG Y, SHI J W, NG C W W. Numerical modeling of tunneling effect on buriedpipelines[J]. Canadian Geotechnical Journal, 2011, 48(7): 1125-1137.

[57]SHI J W, WANG Y, NG C W W. Buried pipeline responses to ground displacements induced by adjacent static pipe bursting[J]. Canadian Geotechnical Journal, 2012, 50(5): 481-492.

[58]BALKAYA M, MOOREI D, SAGLAMER A. Study of non-uniform bedding due to voids under jointed PVC water distribution pipes[J]. Geotextiles and Geomembranes, 2012, 34: 39-50.

[59]张陈蓉，俞 剑，黄茂松. 基坑开挖对邻近地下管线影响的变形控制标准[J]. 岩土力学，2012，33(07)：2027-2034.

[60]王 霆，罗富荣，刘维宁，等. 地铁车站洞桩法施工引起的地表沉降和邻近柔性接头管线变形研究[J]. 土木工程学报，2012，45(02)：155-161.

[61]高永涛，于咏妍，吴顺川. 通车荷载下不同覆土厚度的地铁隧道开挖对管线安全性的影响[J]. 岩石力学与工程学报，2013，32(S2)：3557-3564.

[62]高丙丽. 地铁隧道暗挖施工对既有管线的变形影响规律及其控制技术[J]. 现代隧道技术，2014，51(04)：96-101.

[63]WHAM B P, ARGYROU C, OROURKE T D. Jointed pipeline response to tunneling-induced ground deformation[J]. Canadian Geotechnical Journal, 2016, 53(11): 1794-1806.

[64]SHI J W, WANG Y, NG C W W. Numerical parametric study of tunneling-induced joint rotation angle in jointed pipelines[J]. Canadian Geotechnical Journal, 2016, 53(12): 2058-2071.

[65]邵 羽. 盾构双隧道施工对临近地埋管线的影响研究[D]. 南宁：广西大学，2017.

[66]SHAO Y, DUAN ZB, LIU Y, et al. Estimating the effects of tunnelling on preexisting jointed pipelines[J]. Advances in Civil Engineering, 2019: 1-12.

[67]杨成永，董 烨，程 霖，等. 穿越施工中管线与土层共同变形规律分析[J]. 铁道工程学报，2020，37(01)：93-100.

[68]SINGHAI A C. Behavior of jointed ductile iron pipelines[J]. Journal of Transportation Engineering, 1984, 110(2): 235-250.

[69]TRAUTMANN C H, O ROURFCE T D, KULHAWY F H. Uplift force-displacement response of buried pipe[J]. Journal of Geotechnical Engineering, 1985, 111(9): 1061-1076.

[70]KUSAKABE O, TAKAGI N, KIMURA T, et al. Centrifuge model tests on the influence of axisymmetric excavation on buried pipes[C]//Ground movements and structures, Proceedings of the 3rd International conference. London: Pentech Press, 1985: 133-128.

[71]EL HMADI K, O ROURKE M J. Soil springs for buried pipeline axial motion[J]. Journal of Geotechnical Engineering, 1988, 114(11): 1335-1339.

[72]VORSTER TEB, MAIR R J, SOGA K, et al. Centrifuge modelling of the effect of tunnelling on buried pipelines: mechanisms observed[C]//Proceedings of the 5th International Symposium on Geotechnical Aspects of Underground Construction in Soft Ground. A msterdam, 2005: 327-333.

[73]MARSHALL A M, KLAR A, MAIR R J. Tunneling beneath buried pipes: view of soil strain and its effect on pipeline behavior[J]. Jourmal of Geotechnical & Geoenvironmental Engineering, 2010, 136(12): 1664-1672.

[74]张伦政. 地下管线与土层相互作用数值模拟与离心模型试验研究[D]. 北京：北京交通大学，2014.

[75]WHAM B P, O'ROURKE T D. Jointed pipeline response to large ground deformation[J]. Journal of Pipeline Systems Engineering & Practice, 2016, 7(1): 4015009.

[76]SHI J W, WANG Y, NG C W. Three-dimensional centrifuge modeling of ground and pipeline response to tunnel excavation[J]. Journal of Geotechnical & Geoenvironmental Engineering, 2016, 142(11): 4016054.

[77]朱叶艇，张 桓，张子新，等. 盾构隧道推进对邻近地下管线影响的物理模型试验研究[J]. 岩土力学，2016，37(S2：151-160.

[78]MA S K, SHAO Y, LIU Y, et al. Responses of pipeline to side-by-side twin tunnelling at different depths: 3D centrifuge tests and numerical modelling[J]. Tunnelling & Underground Space Technology, 2017, 66: 157-173.

[79]马少坤，刘 莹，邵 羽，等. 盾构双隧道不同开挖顺序及不同布置形式对管线的影响研究[J]. 岩土工程学报，2018，40(04)：689-697.

[80]程 霖，杨成永，马文辉，等. 地铁隧道开挖引起的管线变形计算与试验研究[J]. 华中科技大学学报(自然科学版)，2022，50(04)：7-13.

[81]TB10003-2016. 铁路隧道设计规范[S]. 北京：中国铁道出版社，2016.

[82]孙运江. 采动覆岩破断移动机理及“类双曲线”模型研究[D]. 中国矿业大学(北京)，2019.

[83]左建平，陈忠辉，王怀文，等. 深部煤矿采动诱发断层活动规律的研究[J]. 煤炭学报，2009，34(3)：305-309.

[84]Griffith A A. The phenomena of rupture and flow in solids [J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1921, 221: 163-98.

[85]Griffith A A. The theory of rupture [C]. Proceedings of the 1st International Congress for Applied Mechanics. Delft, Netherlands, 1924: 54-63.

[86]阿特金森B. K. 岩石断裂力学[M]. 北京：地震出版社，1992.

[87]Zuo Jianping, Liu Huihai, Li Hongtao. A theoretical derivation of the Hoek-Brown failure criterion for rock materials [J]. Journal of Rock Mechanicsand Geotechnical Engineering, 2015, 7: 361-366.

[88]李洪涛，左建平，李 辉. Hoek-Brown强度准则的断裂力学理论研究[J]. 岩土工程学报，2004，26(2)：212-215.

[89]沈明荣. 岩体力学[M]. 上海：同济大学出版社，1999.

[90]李阿康. 基于复变函数方法的浅埋圆形隧道围岩变形与应力理论解研究[D]. 安徽建筑大学，2018.

[91]黎春林，缪林昌. 盾构隧道施工土体扰动范围研究[J]. 岩土力学，2016，37(03)：759-766.

[92]HOEK E, BROWN E T. Underground excavations in rock [A]. London: Institution of Mining and Metallurgy, 1980: 510-530.

[93]HOEK E, MARINOS P. A brief history of the development of the Hoek-Brown failure criterion[J]. Soils and Rocks, 2007, (2): 1-8.

[94]卓 莉，何江达，谢红强，等. 基于Hoek-Brown准则确定岩石材料强度参数的新方法[J]. 岩石力学与工程学报，2015，34(S1)：2773-2782.

[95]朱才辉，李 宁. 隧道施工诱发地表沉降估算方法及其规律分析[J]. 岩土力学，2016，37(S2)：533-542.

[96]严长征. 盾构隧道近距离共同作用机理及施工技术研究[D]. 上海：同济大学，2007.

[97]KLAR A, VORSTER T, SOGA K, et al. Soil-pipe interaction due to tunnelling: comparison between winkler andelastic continuum solutions[J]. Geotechnique, 2005, 55(6): 461-466.

[98]程尧舜. 弹性力学基础[M]. 上海：同济大学出版社，2009.

[99]吴贤国，丁保军，张立茂，等. 基于贝叶斯网络的地铁施工风险管理研究[J]. 中国安全科学学报，2014，24(01)：84-89.