随着城市地下轨道交通的快速发展，管片在盾构隧道建设中的应用十分广泛。以地铁隧道为代表的城市地下轨道交通隧道，通常承受不均匀地层荷载作用。因此，合理设计管片结构形式、充分发挥结构能力成为了需要重点研究的课题。本文研究地层荷载与管片厚度间的关系，提出适用于不均匀地层荷载的盾构隧道变刚度管片，给出变刚度管片设计建议，推导变刚度管片力学模型，分析变刚度管片的可行性与稳定性。论文主要研究成果如下：

(1) 考虑不均匀地层荷载，提出一种变刚度管片设计理念，管片外轮廓保持圆形以方便施工，内轮廓形状是与地层荷载有关的函数，基于等应力设计方法，给出变刚度管片厚度的设计方法，分别设计椭圆形、单抛物线形和双抛物线形三种管片内轮廓形式，对比三种变刚度管片的优缺点及内部空间利用率，选择椭圆形内轮廓的外圆内椭管片作为变刚度管片的主要形式。

(2) 借鉴阶梯折算法的思想，推导变刚度管片力学模型，对变刚度管片的力学特性与变形特性展开研究，得到变刚度管片结构的内力分布规律与变形规律，研究结构参数与地层参数变化对变刚度管片力学特性的影响，结合变刚度管片力学特性与安全要求，给出变刚度管片结构最优尺寸组合建议。

(3) 通过数值模拟方法，补充验证变刚度管片理论分析结论。搭建变刚度管片模型试验平台，通过模型试验，分析变刚度管片的内力特性和变形特性，并与理论分析结果进行对比验证。结果表明，变刚度管片内力分布规律的试验结果与理论分析结果一致，弯矩方面，理论计算值大于试验值；轴力方面，试验值大于理论计算值。变刚度管片位移的试验结果与理论计算结果一致，位移试验值大于理论计算值。

(4) 选取国内不同地区的工程实例，针对其地质条件设计对应的变刚度管片，根据现行规范要求，分析不同地质条件下变刚度管片的稳定性与经济性，并与等刚度管片进行对比，确定变刚度管片在今后类似工程中的可行性。

研究结果对于盾构隧道管片的合理设计具有重要的现实意义，同时为类似工程提供理论上的指导和工程上的借鉴。

With the rapid development of urban underground rail transit, shield tunnel linings are widely used in shield tunnel construction. Urban underground rail transit tunnels represented by subway tunnels usually bear uneven ground.Urban underground rail transit tunnels, represented by subway tunnels, are usually subjected to uneven ground loading. Therefore, rationally designing the shield tunnel linings and giving full play to the structural capability has become a key research topic.the reasonable design of the shield tunnel linings structure form and the full play of the structural capacity become the subject of key research. In this paper, we study the relationship between ground load and shield tunnel linings thickness, propose the variable stiffness shield tunnel linings for shield tunnels with uneven ground load, suggest the design of variable stiffness shield tunnel linings form, derives the mechanical model of variable stiffness shield tunnel linings, and analyzes the variable stiffness shield tunnel linings  feasibility and stability of variable stiffness shield tunnel linings. The main research results of the paper are as follows.

(1) Considering the uneven formation load, a variable stiffness shield tunnel lining structure is proposed. The outer contour of the shield tunnel lining remains circular to facilitate construction, and the inner contour shape is a function related to the formation load. Based on the equal stress design method, the variable stiffness shield tunnel lining is given. The design method of the thickness of the shield tunnel lining is to design three types of inner contours of the shield tunnel lining, ellipse, single parabola and double parabola respectively, compare the advantages and disadvantages of the three variable stiffness shield tunnel linings and the utilization rate of internal space, and finally choose the outer contour of the elliptical inner profile. The inner elliptical shield tunnel lining is the main form of the variable stiffness shield tunnel lining.Study the connection between the ground load and the thickness of the shield tunnel linings, propose a variable stiffness shield tunnel linings structure, the outer profile of this type of shield tunnel linings keeps round to facilitate the construction, the shape of the inner profile is a function related to the ground load, give the design method of the thickness of the variable stiffness shield tunnel linings based on the equal stress design method, design three types of inner profile forms of the shield tunnel linings, elliptical, single parabolic and double parabolic, compare the three types of variable stiffness shield tunnel linings The advantages and disadvantages of the three types of variable stiffness shield tunnel linings and the utilization rate of internal space are compared, and finally the elliptical inner profile of the outer circle and inner ellipse shield tunnel linings is selected as the main form of variable stiffness shield tunnel linings.

(2) Drawing on the idea of the step fold method, deduce the mechanical model of the variable stiffness shield tunnel lining, study the mechanical properties and deformation characteristics of the variable stiffness shield tunnel lining, and obtain the internal force distribution law and deformation law of the variable stiffness shield tunnel lining structure. The influence of the change of formation parameters on the mechanical properties of the variable stiffness shield tunnel lining, combined with the mechanical properties and safety requirements of the variable stiffness shield tunnel lining, the optimal size combination of the variable stiffness shield tunnel lining structure is proposed.The mechanical model of the variable stiffness shield tunnel linings is derived by the idea of the step-folding method, and the mechanical and deformation characteristics of the variable stiffness shield tunnel linings are studied. The internal force distribution and deformation characteristics of the variable stiffness shield tunnel linings structure are analyzed, and the influence of the changes of the structural parameters and the ground parameters on the mechanical characteristics of the variable stiffness is studied.

(3) The theoretical analysis conclusion of the variable stiffness shield tunnel lining is supplemented and verified by the numerical simulation method. A variable stiffness shield tunnel lining model test platform is built, and through the model test, the internal force characteristics and deformation characteristics of the variable stiffness shield tunnel lining are analyzed, and the results are compared and verified with the theoretical analysis results. The results show that the experimental results of the distribution law of the internal force of the variable stiffness shield tunnel lining are consistent with the theoretical analysis results. In terms of bending moment, the theoretical calculation value is larger than the experimental value; in terms of axial force, the experimental value is larger than the theoretical calculation value. The experimental results of the variable stiffness shield tunnel lining displacement are consistent with the theoretical calculation results, and the displacement experimental value is greater than the theoretical calculation value.To verify the correctness of the theoretical analysis of the variable stiffness shield tunnel linings by numerical simulation. A model test platform is built to analyze the internal force characteristics and deformation characteristics of the variable stiffness shield tunnel linings through the model test, and the results are compared with the theoretical calculations to verify. The results show that the internal force distribution pattern of the variable stiffness shield tunnel linings obtained from the test is consistent with the theoretical calculation, and the theoretical calculation value is larger than the test value in terms of bending moment, and the test value is larger than the theoretical calculation value in terms of axial force. The test results of deformation of the variable stiffness shield tunnel linings are consistent with the theoretical calculation results. The test value of displacement is greater than the theoretical calculation value.

(4) Select engineering examples from different regions in China, design corresponding variable stiffness shield tunnel linings according to their geological conditions, and analyze the stability and economy of variable stiffness shield tunnel linings under different geological conditions according to the requirements of current specifications, and compare them with equal stiffness shield tunnel linings. Determine the feasibility of variable stiffness shield tunnel linings in similar projects in the future. The project examples in different regions of China are selected, and the corresponding variable stiffness shield tunnel linings are designed for their geological conditions. The stability and economy of variable stiffness shield tunnel linings under different geological conditions are analyzed according to the current code requirements, and compared with equal stiffness shield tunnel linings to determine the feasibility of variable stiffness shield tunnel linings in similar projects in the future.

The results of the study are of great practical significance for the rational design of shield tunnel shield tunnel linings, and provide theoretical guidance and engineering reference for similar projects.

[1] 中国城市轨道交通协会. 中国城市轨道交通年度报告2020[J]. 都市快轨交通, 2021(04):14-22.

[2] 刘建航, 候学渊. 盾构法隧道[M]. 北京:中国铁道出版社,1991:152-325.

[3] 于书甄, 杜漠远. 隧道施工[M]. 北京:人民交通出版社,1999:27-89.

[4] 张凤祥, 朱合华, 傅明德. 盾构隧道[M]. 北京:人民交通出版社,2004:108-224.

[5] 鲍绥意. 盾构技术理论与实践[M]. 北京:中国建筑工业出版社,2012:46-82.

[6] 朱世友. 国内地铁盾构区间隧道管片结构设计的现状与发展[J]. 现代隧道技术,2002(06):23-28.

[7] 朱伟, 胡如军, 钟小春. 几种盾构隧道管片设计方法的比较[J]. 地下空间,2003(04):352-356+453.

[8] Hewet,B.M. Shield and Compressed Air Tunnelling[J]. McGraw-Hill Book Company, Inc.N.Y.,1922:7-13.

[9] Schmidt，B. Tunnel Lining Design—Do the TheoriesWork[J], Proceedings of Austraial-New Zealand Geome-chanics Conference Perth Australia, 1984:8-10.

[10]高渠清. 论地下结构物计算理论的演变及研究方向[J]. 刘启山、高渠清隧道及地下工程论文集, 北京:中国铁道出版社,1996:8-12.

[11]关宝树. 隧道力学概论[M]. 四川:西南交通大学出版社,1993:62-148.

[12]Bull,A., Strew in the Linings of Shield-Driven Tunnels[J]. Trans ASCE, Nov 1994.

[13]Engelbreth K.. Tunnel Stress Analysis[J]. Geotechnidue,1961(3):32-38.

[14]刘铁雄. 日本隧道标准规范(盾构篇)及解释[M]. 四川:西南交通大学出版社,1993:46-82.

[15]Morgan,H.D. A Contribution to the Analysis of Stressin a Circular Tunnel[J]. Geotechnique, Vol.11(1),1961:24-27.

[16]T. D.奥罗克主编. 隧道衬砌设计指南[M]. 北京:中国铁道出版社,1987:12-86.

[17]久保结城. シールドセグメントの應力に對する繼手剛性の影響[J]. 土木學會論文報告集,1968(150):34-40.

[18]Peck, R. B.. State of the Art of Soft Ground Tunnelling[J]. Proceedings of RapidExcavation and Tunnelling Conference. Chicago, Vol.1,1972:40-44.

[19]Dar, S. M.，Stress Analysis of Hollow Cylindrical Inclusions[J]. Journal of GeotechnicalEngineering Division, ASCE. Vol.100, No.GT2, Feb,1974:24-30.

[20]Muir Wood, A. M. The Circular Tunnel in Elastic Ground[J], Geotechnique, Vol.25(1),1975:16-22.

[21]Curbs, D. J. Discussion of Muir Wood, A. M. The Circular Tunnel in Elastic Ground[J]. Geotechnique.Vol.26(2),1976:28-32.

[22]村上博智, 小泉淳. シールドセグメントりングの耐荷械構ついて[J]. 土木學會論文報告集,1978(150):103-115.

[23]侯学渊. 地下圆形结构弹塑性理论[J]. 同济大学学报,1982(4):50-61.

[24]半谷. 二次覆工を有するシールドトンネル覆工の力學的特性に關する研究[J]. 鐵道技術研究所報告,1985(13):25-33.

[25]堀地紀行. トンネル軸方向の剛性を考慮したセグメントりングの解析法[J]. トンネルと地下,1986,17(8):609-616.

[26]张厚美, 张正林, 王建华. 盾构隧道装配式管片接头三维有限元分析[J]. 上海交通大学学报,2003(04):566-569.

[27]张厚美, 连烈坤, 过迟. 盾构隧洞双层衬砌接头相互作用模型[J]. 岩石力学与工程学报,2003(01):70-74.

[28]张厚美, 傅德明, 过迟. 盾构隧道管片接头荷载试验研究[J]. 现代隧道技术,2002(06):28-33+41.

[29]张厚美, 叶均良, 过迟. 盾构隧道管片接头抗弯刚度的经验公式[J]. 现代隧道技术,2002(02):12-16+52.

[30]何川, 张建刚, 杨征. 武汉长江隧道管片衬砌结构力学特征模型试验研究[J]. 土木工程学报,2008(12):85-90.

[31]陈馈, 洪开荣, 吴学松. 盾构施工技术[M]. 人民交通出版社,2009:54-123.

[32]杨新安, 丁春林, 徐前卫. 城市隧道工程[M]. 同济大学出版社,2015:14-36.

[33]何川, 张建刚, 杨征. 大断面水下盾构隧道结构力学特性[M]. 科学出版社,2010:56-143.

[34]朱合华, 庄晓莹, 张雪健. 盾构管片接头破坏的弹塑性—损伤三维有限元模型研究[J]. 岩土工程学报,2015,37(10):1826-1834.

[35]朱合华, 黄伯麒, 李晓军, 等. 盾构衬砌管片接头内力—变形统一模型及试验分析[J]. 岩土工程学报,2014,36(12):2153-2160.

[36]彭益成, 丁文其, 朱合华, 等. 盾构隧道衬砌结构的壳—接头模型研究[J]. 岩土工程学报,2013,35(10):1823-1829.

[37]朱合华, 崔茂玉, 杨金松. 盾构衬砌管片的设计模型与荷载分布的研究[J]. 岩土工程学报,2000(02):190-194.

[38]朱合华, 陶履彬. 盾构隧道衬砌结构受力分析的梁—弹簧系统模型[J]. 岩土力学,1998(02):26-32.

[39]苏宗贤, 何川. 盾构隧道管片衬砌内力分析的壳—弹簧—接触模型及其应用[J]. 工程力学,2007(10):131-136.

[40]朱伟, 黄正荣, 梁精华. 盾构衬砌管片的壳—弹簧设计模型研究[J]. 岩土工程学报,2006(08):940-947.

[41]胡志平, 罗丽娟, 蔡志勇. 盾构隧道管片衬砌的平板壳—弹性铰—地基系统模型[J].岩土力学,2005(09):1403-1408.

[42]Kappers C, Grubl F, Ostermeier B. Structural Analyses of Segmental Lining-Coupled Beam and Spring Analyses Versus 3D-FEM Calculations with Shell Elements[J]. World Tunnel Congress & 32nd ITA General Assembly,2006:75-83.

[43]晏启祥, 王春艳, 郑代靖, 等. 大断面水下盾构隧道管片设计参数及其统计分析[J]. 铁道标准设计,2016,60(02):99-104+118.

[44]陈炜韬, 姜志毅, 董宇苍, 等. 大直径双护盾TBM公路隧道管片分块设计探讨[J]. 铁道科学与工程学报,2018,15(01):170-177.

[45]陈炜韬, 傅支黔, 马建新. 大直径双护盾TBM隧道管片厚度设计研究[J]. 铁道建筑,2017,57(10):60-62.

[46]丁军霞, 冯卫星, 杜学玲. 盾构隧道管片衬砌内力及变形的影响因素分析[J]. 石家庄铁道学院学报,2004(03):75-79.

[47]赵国旭, 何川. 盾构隧道管片设计的主要影响因素分析[J]. 铁道建筑,2003(12):25-29.

[48]黄式浩, 狄宏规, 王友文, 姚琦钰. 管片厚度对大直径盾构隧道受力及变形的影响[J]. 华东交通大学学报,2020,37(01):15-22.

[49]黄钟晖, 廖少明, 刘国彬. 上海软土盾构法隧道管片厚度的优化[J]. 岩土力学,2000(04):326-330.

[50]黄钟晖, 廖少明, 刘国彬. 管片厚度对隧道受力及使用性能的影响[J]. 建筑技术,2000(07):471-472.

[51]杨友彬. 跨座式单轨盾构隧道管片厚度研究[J]. 现代城市轨道交通,2019(09):75-79.

[52]Xin Huang, Yeting Zhu,Zixin Zhang, et al. Mechanical behaviour of segmental lining of a sub-rectangular shield tunnel under self-weight[J]. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research,2018(74):103-121.

[53]Xian Liu, Zhen Liu,Yuhang Ye, et al. Mechanical behavior of quasi-rectangular segmental tunnel linings: Further insights from full-scale ring tests[J]. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research,2018(79):97-112.

[54]Xian Liu, Yuhang Ye,Zhen Liu, et al. Mechanical behavior of Quasi-rectangular segmental tunnel linings: First results from full-scale ring tests[J]. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research,2018(71):83-92.

[55]朱叶艇, 朱雁飞, 张子新, 等. 异形盾构管片原型试验混凝土裂缝宽度预测与可视化[J]. 上海交通大学学报,2019,53(10):1259-1268.

[56]嵇中. 超大矩形断面隧道衬砌结构优化设计研究[J]. 中国市政工程,2018(06):82-85+106-107.

[57]郭艳华, 李岳, 马新亮. 公路隧道变截面衬砌外轮廓的确定[C]. 中国交通土建工程学术论文集（2006）中国土木工程学会,2006:634-637.

[58]唐志成, 何川, 林刚. 地铁盾构隧道管片结构力学行为模型试验研究[J]. 岩土工程学报,2005(01):85-89.

[59]吴圣智, 王明年, 于丽, 等. TBM隧道回填层—管片受力特征模型试验研究[J]. 岩土力学,2018,39(11):3976-3982+4009.

[60]胡雄玉, 杨清浩, 何川, 等. 层状围岩中管片衬砌受力特征的模型试验研究[J]. 岩土工程学报,2018,40(10):1773-1781.

[61]叶飞, 杨鹏博, 毛家骅, 等. 基于模型试验的盾构隧道纵向刚度分析[J]. 岩土工程学报,2015,37(01):83-90.

[62]杨钊, 罗会武. 地铁盾构管片接头抗弯刚度模型试验与数值分析[J]. 地下空间与工程学报,2018,14(06):1542-1548.

[63]梁禹, 苏文辉, 方理刚, 等.大直径江底盾构隧道衬砌结构受力现场测试与分析[J]. 隧道建设,2014,34(07):637-641.

[64]叶飞, 何川, 王士民. 盾构隧道施工期衬砌管片受力特性及其影响分析[J]. 岩土力学,2011,32(06):1801-1807+1812.

[65]刘四进, 封坤, 何川, 等. 大断面盾构隧道管片接头抗弯力学模型研究[J]. 工程力学,2015,32(12):215-224.

[66]《中国公路学报》编辑部. 中国隧道工程学术研究综述2015[J]. 中国公路学报,2015,28(05):1-65.

[67]封坤. 大断面水下盾构隧道管片衬砌结构的力学行为研究[D].西南交通大学,2012.

[68]苏宗贤. 超大断面水下盾构隧道原型结构试验及结构分析模型研究[D]. 成都:西南交通大学,2008.

[69]高云龙. 地铁水下盾构隧道管片衬砌结构力学行为的现场试验研究[D]. 西南交通大学, 2012.

[70]刘怀恒. 支护荷载反演及安全度预测[J]. 西安矿业学院学报,1991(04):1-8.

[71]黄宏伟, 刘怀恒. 地下工程结构的可靠度分析[J]. 西安矿业学院学报,1991(03):17-25.

[72]刘怀恒, 熊顺成. 隧洞衬砌变形监控及安全预测[J]. 岩石力学与工程学报,1990(02):91-99.

[73]刘怀恒. 地下工程位移反分析—原理、应用及发展[J]. 西安矿业学院学报,1988(03):1-11.

[74]钟小春, 朱伟. 盾构衬砌管片土压力反分析研究[J]. 岩土力学,2006(10):1743-1748.

[75]何涛. 地下结构随机荷载反演与可靠性分析研究[D]. 同济大学,2007.

[76]张中生, 陈子荫, 蒋斌松. 地下结构荷载反算中解的稳定性研究[J]. 中国矿业,1999(05):82-87.

[77]张中生. 地下结构荷载反算中的测线网优化布置[J]. 岩石力学与工程学报,2000(02):219-224.

[78]张中生, 陈子荫, 朱维申. 地下结构荷载的广义反演方法[J]. 土木工程学报,2001(02):38-42.

[79]关宝树. 拼装衬砌上的土压力分布与变形[J]. 隧道译丛,1978(1):18-20.

[80]陈伟, 彭振斌, 唐孟雄. 盾构管片工作性能试验研究[J]. 岩石力学与工程学报,2004,23 (6):959-963.

[81]周济民, 何川, 肖明清, 等. 狮子洋水下盾构隧道衬砌结构受力的现场测试与计算分析[J]. 铁道学报,2012,34(07):115-121.

[82]梁禹, 苏文辉, 方理刚, 等.大直径江底盾构隧道衬砌结构受力现场测试与分析[J]. 隧道建设,2014,34(07):637-641.

[83]张君禄, 段峰虎, 廖文来, 等. 湛江湾跨海盾构隧道管片现场监测试验研究[J]. 岩石力学与工程学报,2014,33 (S1):2878-2884.

[84]叶冠林, 王吉云, 王建华, 等. 超大断面盾构隧道管片施工荷载现场监测研究[J]. 现代隧道技术,2010,47(05):85-89.

[85]王建, 叶宇航, 刘加福, 等. 软土地基大直径地铁盾构隧道运营期衬砌结构受力特性现场测试研究[J]. 隧道建设,2017,37(07):781-787.

[86]唐孟雄, 陈如桂, 陈伟. 广州地铁盾构隧道施工中管片受力监测与分析[J]. 土木工程学报,2009,42(03):118-124.

[87]张厚美, 张良辉, 马广州. 盾构隧道围岩压力的现场监测试验研究[J]. 隧道建设,2006(S2):8-11 +46.

[88]叶飞, 何川, 王士民. 盾构隧道施工期衬砌管片受力特性及其影响分析[J]. 岩土力学,2011,32(06):1801一1807+1812.

[89]叶开沅. 非均匀变厚度弹性体力学的若干问题的一般解，非均匀变厚度梁的弯曲、稳定性和自由振动问题[J]. 兰州大学学报,力学专号, 1979(1):133-157.

[90]何川, 封坤, 方勇. 盾构法修建地铁隧道的技术现状与展望[J]. 西南交通大学学报，2015,50(1):97-109.

[91]吴林. 盾构隧道双层衬砌计算模型与力学特性研究[D]. 西南交通大学,2011.

[92]周济民. 水下盾构法隧道双层衬砌结构力学特性[D]. 西南交通大学, 2012.

[93]封坤. 大断面水下盾构隧道管片衬砌结构的力学行为研究[D]. 西南交通大学,2012.

[94]钟小春. 盾构隧道管片土压力的研究[D]. 南京:河海大学,2005.

[95]Renato E.B，Kael R. Tunnel design and construction inextremely difficult ground conditions[J]. Tunnel,1998(8):23-31.

[96]Marta D. Tunnel Complex unloaded by a deep excavation[J]. Computers and Geotechnics,2001,28(3):469-493.

[97]Redendiz D，Romo M P. Settlement upon soft ground tunnelling[J]. Theoretical Solution,1981:65-74.

[98]李围, 何川. 超大断面越江盾构隧道结构设计与力学分析[J]. 中国公路学报,2007,20(3):76-80.

[99]李围,何川. 盾构隧道通用管片力学行为与控制拼装方式研究[J]. 铁道学报,2007,29(2):77-82.

[100]赵国旭, 何川. 隧道管片设计优化分析[J]. 中国铁道科学,2003,24(6):61-66.

[101]朱伟, 黄正荣, 梁精华. 盾构衬砌管片的壳—弹簧设计模型研究[J]. 岩土工程学报,2006,28(8):940-947.

[102]苏宗贤, 何川. 荷载—结构模式的壳—弹簧—接触计算模型[J]. 西南交通大学学报,2007 42 (3):288-292.

[103]吴兰婷. 盾构隧道管片接头力学行为的有限元分析[D]. 成都:西南交通大学,2005.

[104]邱培, 李守巨, 上官子昌. 盾构隧道管片内力特性分析和管片配筋设计[J]. 辽宁工程技术大学学报(自然科学版),2016,35(10):1111-1116.

[105]叶开沅, 汤任基, 甄继庆. 非均匀变截面弹性圆环在任意载荷下的弯曲问题[J]. 应用数学和力学,1981(01):1-12.

[106]叶开沅, 纪振义. 非均匀变截面梁的稳定和自由振动的阶梯折算法[J]. 兰州大学学报,1990(02):40-46.

[107]张光富. 应用阶梯函数计算任意变截面曲梁的位移[J]. 西安矿业学院学报,1982(01):82-90.

[108]刘礼华, 周剑波, 雷仲庄. 变截面圆环稳定计算的有限解析法[J]. 武汉水利电力大学学报,1995(05):575-578.

[109]董长军, 刘世忠, 冀伟, 等. 变曲率曲梁的单元等效节点荷载分析[J]. 铁道工程学报,2018,35(09):31-34.

[110]马希龄. 对称变截面平面曲杆的单元刚度矩阵[J]. 力学与实践,2000(02):47-49.

[111]朱先奎, 刘光廷. 阶梯形变截面圆拱和圆环稳定计算的有效方法[J]. 力学与实践,1997(01):26-28.

[112]朱先奎, 刘光廷 .变截面圆拱和圆环稳定计算的阶梯折算法[J]. 建筑结构,1999(01):52-56.

[113]苏海东, 周朝, 颉志强. 基于精确几何的曲梁分析新方法[J]. 长江科学院院报,2018,35(04):151-157+166.

[114]于学馥. 轴变论与围岩变形破坏的基本规律[J]. 铀矿冶,1982(01):8-17+7.

[115]于学馥, 乔端. 轴变论和围岩稳定轴比三规律[J].有色金属,1981(03):8-15.

[116]杨雪强, 何世秀, 高桐. 对巷道开孔卸载参数的研究[J].岩土力学,1995(01):46-53.

[117]郑建伟, 鞠文君, 孙晓冬, 等. 巷道变形差异化响应特征及支护设计分析[J]. 采矿与岩层控制工程学报,2021(03):1-9.

[118]小夫. 轴变论—关于地压问题的一个理论(上)[J]. 有色金属,1959(16):9-18.

[119]小夫. 轴变论—关于地压问题的几个理论(下)[J]. 有色金属,1959(17):1-9.

[120]闫振雄, 王培涛. 空心包体应变计地应力计算方法的探讨[J]. 岩石力学与工程学报,2018,37(S1):3568-3574.

[121]王新志, 赵永刚, 踞旭, 等. 变厚度圆薄板非对称非线性弯曲问题[J]. 应用数学和力学,2005(04):386-393.

[122]王新志, 韩明君, 赵永刚, 等. 变厚度扁锥壳的非线性固有频率[J]. 应用数学和力学,2005(03):253-258.

[123]叶开沅, 童晓华, 纪振义. 非均匀变厚度梁的动力响应的一般解[J]. 应用数学和力学,1992(09):753-764.

[124]叶开沅, 刘平. 在常温度场中非均匀变厚度高速旋转圆盘的等强度设计[J].应用数学和力学,1986(09):769-778.

[125]叶开沅, 刘平. 非均匀变厚度圆盘的定常热传导[J]. 应用数学和力学,1984(05):619-624.

[126]叶开沅, 丁延岭. 非均匀变厚度连续梁的弯曲[J]. 兰州大学学报,1982(04):30-41.

[127]叶开沅, 丁延岭. 非均匀变厚度连续梁的稳定性和自由振动[J]. 兰州大学学报,1982(02):42-74.

[128]叶开沅. 变厚度弹性圆薄板问题[J]. 物理学报,1955(03):207-218.

[129]何玲, 杨观赐, 章杰. CAD/CAE技术应用[M]. 南京大学出版社,2019:18-75.

[130]费康, 彭劼. ABAQUS岩土工程实例详解[M]. 人民邮电出版社:CAE分析大系,2017:242-285.

[131]齐威. ABAQUS6.14超级学习手册[M]. 人民邮电出版社,2016:2-5.

[132]法国达索系统. Abaqus关键词参考指南[M]. 南京东南大学出版社,2015:1-375.

[133]王凯, 赵辉, 周凯歌, 等. 南昌地铁3号线盾构隧道穿湖段管片受力特征研究[J]. 现代隧道技术,2019,56(S2):400-408.

[134]王立新. 太原地铁2号线盾构隧道管片规格研究[J]. 铁道工程学报,2016,33(08):111-115.

[135]张冬梅, 邹伟彪, 闫静雅. 软土盾构隧道横向大变形侧向注浆控制机理研究[J]. 岩土工程学报,2014,36(12):2203-2212.

[136]张颖, 许鹏, 尹亮亮, 等. 沈阳地铁盾构法施工的三维数值模拟研究[J]. 地质灾害与环境保护,2015,26(01):108-112.