分层土中盾构隧道掌子面极限推力研究

分层土中盾构隧道掌子面极限推力研究 Research on ultimate thrust of shield tunnel face in layered soil corresp first-author 傅鹤林，博士，教授，从事隧道与地下工程研究；E-mail：fu.h.l@csu.edu.cn 傅鹤林，博士，教授，从事隧道与地下工程研究；E-mail：fu.h.l@csu.edu.cn fu.h.l@csu.edu.cn 中南大学 土木工程学院，湖南 长沙，410075 中南大学 土木工程学院，湖南 长沙，410075 School of Civil Engineering, Central South University, Changsha 410075, China School of Civil Engineering, Central South University, Changsha 410075, China 陈灿华 在盾构隧道开挖过程中，掌子面受力十分复杂，盾构机对掌子面的推力对维持掌子面前方土体的稳定起着关键性作用，为此，基于筒仓理论和极限平衡法，改进传统楔形体模型，考虑盾构隧道掌子面处土体分层对掌子面极限推力的影响，建立折线型滑动模型，推导掌子面极限推力的计算公式。依据实际工程建立相应的数值模拟模型，并对计算公式中各参数敏感性进行分析。研究结果表明：考虑掌子面土体分层时极限推力的数值模拟解与由本文所得公式计算的理论解误差很小，验证了本文所述模型和计算方法的可靠性与准确性；掌子面极限推力随隧道埋深和直径增加而增大；分层土中引起极限推力变化的主要土体是上层软弱土；盾构直径越大，土体黏聚力和内摩擦角对极限推力的影响越大，且内摩擦角对极限推力的影响更显著；当盾构直径较小时，土体黏聚力和内摩擦角对极限推力的影响较小。 During the excavation of a shield tunnel, the force on the tunnel face is very complicated, and the thrust of the shield machine on the face plays a key role in maintaining the stability of the soil in front of the tunnel. Based on the silo theory and limit equilibrium method, the traditional wedge-shaped model was improved, and the influence of the layered soil at the face on the ultimate thrust was considered. A broken-line sliding model was established, and the calculation formula for the ultimate thrust of the face was deduced. The corresponding numerical simulation model was established based on the actual project. The results indicate that when considering the stratification of the palm soil mass, the numerical simulation solution of the ultimate thrust force is very close to the theoretical solution obtained by the caculation formula in this paper, which verifies the reliability and accuracy of the model and calculation method described in this paper. The numerical solution is very different from the calculation result when the soil layering at the face is not considered, indicating that the soil layering at the face has great influence on the ultimate thrust of the face, which cannot be ignored in the calculation and proves the rationality of the model in this paper. The ultimate thrust of the face increases with the increase of the buried depth and diameter of the tunnel, the main soil mass that causes the ultimate thrust change in the layered soil is the upper soft soil. The larger the shield diameter, the greater the influence of soil cohesion and internal friction angle on the ultimate thrust, and the more significant the influence of internal friction angle. When the shield diameter is smaller, the two influences on the ultimate thrust are smaller. 分层土 极限推力 筒仓理论 极限平衡法 折线型滑动模型 layered soil ultimate thrust silo theory limit equilibrium method broken-line sliding mode 在盾构隧道开挖过程中，掌子面推力对保持盾构机前方土体平衡起着极其重要的作用，推力过大和过小都会对上方建构筑物产生严重的不良影响，因此，研究盾构隧道掌子面极限推力具有十分重要的理论价值和经济价值。 研究盾构掌子面极限推力的理论分析法包括极限分析上限法、极限平衡法和筒仓理论等。刘克奇等 [ 1 1 ] 基于滑移线理论和极限分析上限定理，利用空间离散技术提出了一种盾构施工掌子面三维滑移破裂模型；李得建等 [ 2 2 ] 基于非线性破坏准则和极限分析上限法，推导了隧道掌子面被动三维支护力的上限计算公式；张道兵等 [ 3 3 ] 采用考虑孔隙水压力效应的极限分析上限法，推导了掌子面极限支护力的表达式；吕玺琳等 [ 4 4 ] 基于极限分析法及极限平衡法给出了掌子面的最小支护力的计算公式和影响系数；王秀英等 [ 5 5 ] 基于极限平衡理论，采用水平条分法推导了掌子面极限支护力计算公式；HORN [ 6 6 ] 基于筒仓理论和极限平衡法建立了三维楔形体-筒仓破坏模型，为后续研究打下了基础；JANCSECZ等 [ 7 7 ] 对该模型进行了改进并对盾构隧道开挖面最小极限支护力进行了研究；BROERE [ 8 8 ] 将三维楔形体模型应用于复合地层掌子面稳定性分析，但模型的适用性有待加强；王国富等 [ 9 9 - 10 10 9-10 ] 基于筒仓理论，分别推导了曲线盾构隧道开挖面极限支护力与突变地质界面下开挖面极限支护力的计算公式；傅鹤林等 [ 11 11 ] 以筒仓理论及极限平衡法为基础，在砂土地层盾构下穿既有线情况下，对掌子面极限支护力进行计算，并给出了计算公式。 但现有研究中普遍没有重视掌子面处土体分层情况，只是简单地将掌子面前方土体视为均匀分布，此类简化方法对于小直径的盾构隧道工程可能适用，但对于大直径和超大直径的盾构隧道工程容易造成较大的误差。为此，本文作者基于筒仓理论和极限平衡法，对传统楔形体模型进行改进，建立考虑盾构隧道掌子面土体分层的折线型滑动模型，推导相应的盾构隧道掌子面前方存在多层土体时的极限推力计算公式，并按实际工程建立的数值模拟进行验证，证明建立的折线型滑动模型及推导的理论计算公式的正确性。与其他模型相比，本文提供的计算公式更简便，计算难度显著降低。对盾构隧道掌子面极限推力的敏感性进行分析，以期完善盾构隧道中掌子面极限推力计算理论。 1 掌子面极限推力公式推导 1 考虑掌子面处土体分层时，土体滑动面呈折线型滑动，破坏土体可分为 n 层，为方便叙述，令 n =3，计算模型如 图1 图1 所示。当破坏土体为多层即 n > 3时，均可按照 n =3的方法推导。将折线形破坏模型分成3部分，即由2个棱台及1个楔形体构成。其中，下方折线型滑动部分为隧道掌子面失稳时的下滑土体，上部分棱柱体为楔形体下滑时所带动的掌子面上方下沉土体。分析折线型破坏模型各个部分的受力情况，上、中、下3部分的受力如 图2 图2 所示 [ 12 12 - 13 13 12-13 ] 。 (a) 盾构机前方土体示意图；(b) 折线型破坏模型 <math id="M1"><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679278&type= 3.30200005 3.38666677 —第 i 层高度， <math id="M2"><msub><mrow><mi>H</mi></mrow><mrow><mn 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679326&type= 3.13266683 3.38666677 —第 i 层倾斜滑动面上的支撑力； <math id="M5"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679284&type= 2.79399991 4.06400013 —模型上部总荷载(kN)； <math id="M6"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679287&type= 2.79399991 4.06400013 和 <math id="M7"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679290&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679301&type= 2.79399991 4.06400013 —第二、三层土体之间的相互作用力； <math id="M8"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679292&type= 2.70933342 3.38666677 —竖直滑动面上的剪力； <math id="M9"><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679310&type= 2.96333337 3.38666677 —第 i 层倾斜滑动面上的剪力； <math id="M10"><msub><mrow><mi>Q</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679307&type= 3.21733332 4.06400013 — <math id="M11"><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math> 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mathvariant="normal">n</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679370&type= 36.57600021 4.48733330 式中： <math id="M22"><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679361&type= 14.90133286 4.48733330 [ 14 14 ] ； P 为盾构机作用在掌子面土体的推力。联立式( 1 1 )~( 6 6 )，得到掌子面盾构推力 P 的表达式： <math display="block" id="M23"><mi>P</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679363&type= 54.35599899 8.21266651 式中： <math id="M24"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679365&type= 3.64066648 3.72533321 、 <math id="M25"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679367&type= 3.64066648 3.72533321 和 <math id="M26"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679374&type= 3.64066648 3.72533321 为简化计算公式定义的参数。 <math display="block" id="M27"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679375&type= 31.07266808 4.48733330 <math display="block" id="M28"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679377&type= 55.71067047 3.72533321 <math display="block" id="M29"><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679386&type= 57.99666977 4.48733330 将掌子面推力转化为作用在掌子面上的均布荷载，得到掌子面极限推力的应力 p k ： <math display="block" id="M30"><msub><mrow><mi>p</mi></mrow><mrow><mi mathvariant="normal">k</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>P</mi></mrow><mrow><mi mathvariant="normal">π</mi><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679381&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679389&type= 13.63133430 7.78933382 式( 7 7 )~( 10 10 )中参数求解如下 [ 15 15 ] ： <math display="block" id="M31"><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>R</mi><mo stretchy="false">(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679382&type= 69.59600067 4.48733330 <math display="block" id="M32"><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>R</mi><mo stretchy="false">(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679399&type= 51.39266586 4.48733330 <math display="block" id="M33"><msub><mrow><mi>G</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msubsup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>R</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679401&type= 26.58533096 4.82600021 <math display="block" id="M34"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>-</mo><mn mathvariant="normal">2</mn><mi>c</mi><mi>η</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">2</mn><mi>η</mi><mi>K</mi><mi>h</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>φ</mi></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn><mi>η</mi><mi>K</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>φ</mi></mrow></mfrac><mi mathvariant="normal">π</mi><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679403&type= 52.49333191 8.55133343 <math display="block" id="M35"><mi>η</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mi>R</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>R</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><mi>θ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679391&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679405&type= 21.59000015 7.28133297 <math display="block" id="M36"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679393&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679392&type= 68.07199860 7.19666624 <math display="block" id="M37"><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679394&type= 49.44533539 7.28133297 <math display="block" id="M38"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679406&type= 53.17066193 3.72533321 <math display="block" id="M39"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">]</mo><mo>⋅</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679408&type= 51.81600189 4.48733330 <math display="block" id="M40"><mo stretchy="false">[</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679396&type= 56.81133270 7.19666624 <math display="block" id="M41"><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">[</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679413&type= 59.43600082 7.19666624 <math display="block" id="M42"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">3</mn><msubsup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mn mathvariant="normal">3</mn><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679415&type= 44.70399857 7.28133297 <math display="block" id="M43"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679417&type= 36.83000183 3.72533321 <math display="block" id="M44"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">]</mo><mn mathvariant="normal">2</mn><mi>R</mi><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679419&type= 76.79267120 4.48733330 <math display="block" id="M45"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn 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mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>+</mo><mi>R</mi><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679423&type= 75.01467133 4.82600021 <math display="block" id="M47"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">[</mo><mn mathvariant="normal">8</mn><msup><mrow><mi>R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679425&type= 43.68799973 7.28133297 式中： h 为隧道的埋深； K 为隧道上覆土的侧压力系数， <math id="M48"><mi>K</mi><mo>=</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679410&type= 18.03400040 3.38666677 [ 16 16 ] ； K i 为掌子面第 i 层土体的侧压力系数； c 为隧道上覆土的黏聚力； c i 为掌子面第 i 层土体的黏聚力； <math id="M49"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679412&type= 1.52400005 3.38666677 为隧道上覆土体的重度； <math id="M50"><msub><mrow><mi>γ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679429&type= 2.28600001 4.48733330 为掌子面第 i 层土体的重度。 考虑掌子面处于地下水面线以下以及土体分层时，水的存在会改变土体的参数以及模型中的部分参数如内摩擦角和黏聚力，此时，需对参数进行调整。计算掌子面的极限推力时，还需要考虑孔隙水压力。在不忽略渗流时，还需要考虑渗透力对掌子面推力的影响，可根据高健等 [ 17 17 ] 所使用的方法进行计算。 2 数值模拟对比验证 1 某隧道工程位于沿海城市，地质条件复杂，采用直径为15 m的盾构机开挖。在开挖面处土体分为3层，具体土体参数见 表1 表1 。为了对该工程的顺利开挖提供指导并验证本文公式的合理性，采用数值模拟对开挖过程的扰动效应进行研究。 材料 泊松比 <math id="M51"><mi>ν</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679431&type= 1.52400005 2.37066650 重度 <math id="M52"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679433&type= 1.35466671 3.04800010 /(kN·m -3 ) 内摩擦角 <math id="M53"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679435&type= 1.86266661 3.04800010 /(°) 黏聚力 c /kPa 土层1 0.32 17.8 10 8 土层2 0.28 18.0 20 15 土层3 0.26 18.5 30 30 管片 0.20 25.0 — — 在研究过程中，采用有限差分法构建数值计算模型，如 图3 图3 所示。模型长×宽×高为80 m× 50 m×50 m，隧道埋深为12.5 m，盾构机直径为 15 m，管片厚度为0.5 m。土层在掌子面分为3层，计算参数见 表1 表1 。模型上表面为自由面，约束其余表面的法向位移。土体采用Mohr-Coulomb本构材料，管片采用线弹性单元。 数值模拟步骤如下：1) 建立计算模型，计算初始地应力，使模型达到初始平衡；2) 开挖隧道，及时加上管片支护，并在开挖面上施加与原始地层侧向静止土压力相等的推力，定义该推力与原始地层侧向静止土压力的比值为推力压力比；3) 令推力压力比从1.0逐渐减小，计算达到平衡并记录掌子面前方土体最大位移；4) 根据实际数值模拟结果和参考已有研究 [ 11 11 , 15 15 ] ，当上述位移超过洞径的0.3%即45 mm时，则认为掌子面前方土体发生破坏并停止计算，此时，对应的推力即为掌子面极限推力的数值解 [ 18 18 ] 。 提取数值模拟结果中土体的竖向位移云图，如 图4 图4 所示。从 图4 图4 可以看到：当盾构推力 p k = 240 kPa时，掌子面以及掌子面前方土体没有发生位移，说明在该推力作用下，掌子面处于稳定状态，隧道沉降变形主要出现在管片附近，这是土压力作用在管片上所致；随着推力不断减小，掌子面处土体开始产生位移，首先滑动的是第一层土，随后第二层土和第三层土也开始滑动，但第三层土滑动程度比第一及第二层土的小。当掌子面发生破坏时，滑动体呈现楔形体和棱柱体形状，说明采用筒仓理论及极限平衡分析法构建的折线型滑动模型是可靠的。 p k /kPa：(a) 240；(b) 220；(c) 200；(d) 180；(e) 160；(f) 140；(g) 120；(h) 100；(i) 80；(j) 64；(k) 60；(l) 40 对施加在掌子面的推力进行记录，得到推力压力比与掌子面土体最大位移关系曲线，如 图5 图5 所示。从 图5 图5 可以看出：当推力压力比在1.0~0.9之间时( p k =220~240 kPa)，推力压力比减小引起掌子面处土体的最大位移变化极小；当推力压力比在0.9~0.3之间时( p k =70~220 kPa)，推力压力比减小使得掌子面处土体最大位移明显增大，两者呈线性相关；当推力压力比小于0.2时( p k =40 kPa)，模型计算达不到收敛状态，此时，掌子面前方土体已经发生主动破坏；随着推力压力比不断减小，掌子面处最大位移呈现不敏感变化、敏感变化以及不收敛变化3个阶段，这与文献[ 13 13 ]的研究结果一致。根据 图5 图5 中曲线以及前述判定破坏条件，当最大位移等于45 mm时，极限推力压力比为0.267，对应施加在掌子面处的极限推力的数值解 p k =64 kPa。 将 表1 表1 中的参数代入 式(7) 式(7) 至 式(21) 式(21) 中，最终计算得到施加在掌子面的极限推力的理论解 p k = 67.3 kPa，数值解与理论解相对误差为5.2%，可见数值解与理论解相对误差较小，验证了本文所得的模型和理论计算公式的正确性。 不考虑掌子面处土体分层，忽略 式(7) 式(7) 至 式(21) 式(21) 中第一和第二层土的相关参数，将 表1 表1 中3层土体参数的加权平均值作为 式(7) 式(7) 至 式(21) 式(21) 中第三层土的参数进行计算，得到不考虑掌子面土体分层的极限推力 p k =80.6 kPa；比考虑掌子面土体分层时的数值解64.0 kPa和理论解67.3 kPa分别大25.9%和19.8%，结果偏差较大，说明掌子面土体分层对极限推力的影响不可忽略。按照本文得出的计算公式能够有效缩小掌子面极限推力的取值范围，在实际施工中更有参考价值，而且本文中推导的理论计算公式较其他模型的计算公式更简便，不需要进行大量积分和迭代计算，大大降低了计算难度。 3 敏感性因素分析 1 由推导的计算公式可以看出，掌子面处极限推力主要受隧道埋深( h )、分层土体厚度( H 1 、 H 2 及 H 3 )、盾构直径( R ) 、 黏聚力( c )和内摩擦角( <math id="M54"><mi>φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679458&type= 2.11666679 3.38666677 )的影响，下面对这些因素的敏感性进行分析 [ 19 19 - 20 20 19-20 ] ，以便为实际工程提供参考。 在保持3层土总厚度不变的条件下，改变第一层、第二层和第三层土厚度及改变隧道埋深，掌子面处的极限推力的变化情况如 图6 图6 所示。从 图6 图6 可以看到：随着隧道埋深 h 增加，掌子面处的极限推力随之增加，且增加的幅度有所下降，这说明随着隧道埋深 h 增加，上部土压力荷载逐渐减小，塌落拱逐渐形成；分别增加3层土的厚度，都会使掌子面极限推力随之增大；第一层土厚度 H 1 对推力的影响最大，第二层土厚度 H 2 对推力的影响次之，第三层土厚度 H 3 则对推力影响最小。由此可见：当掌子面处为多层土体时，第一层软弱土是引起掌子面极限推力变化的主要土体。 (a) 第一层；(b) 第二层；(c) 第三层 图7 图7 所示为3层土体的不同黏聚力比及不同直径对掌子面极限推力的影响。从 图7 图7 可以看到：随着盾构机直径增加，掌子面极限推力也随着增加，这是由于盾构机直径增加，将增大楔形滑动体的体积，从而需要在掌子面处施加更大的推力才能阻止土体滑动；掌子面极限推力随着3层土的黏聚力比增加而增加，导致掌子面极限推力变化的土层主要是第一层软弱土层，其黏聚力越小，第一层土越容易发生滑动，且第一层土相比第二层及第三层土也更容易发生滑动；在盾构机直径较小时，3层土体的黏聚力之比对掌子面极限推力的影响极小，盾构机直径越大，3层土体的黏聚力之比对掌子面极限推力的影响越大。 图8 图8 所示为3层土体的不同内摩擦角之比时掌子面极限推力的变化情况。从 图8 图8 可以看到：随着第一层土、第二层土与第三层土的内摩擦角之比不断增大，掌子面极限推力也随之增加，且比黏聚力的影响更大。这说明在折线型滑动模型滑动过程中，土体的内摩擦角对掌子面极限推力起决定作用，而黏聚力对极限推力的影响较小；当盾构直径较小时，土层的不均匀性对掌子面的极限推力影响较小。 4 结论 1 1) 基于筒仓理论和极限平衡法，改进传统楔形体模型，考虑到盾构隧道掌子面处土体分层对掌子面极限推力造成较大影响，建立了折线型滑动模型，推导了掌子面处土体分层时的极限推力计算公式。 2) 通过数值模拟验证了模型和计算公式的正确性和可靠性。数值模拟结果表明，考虑掌子面土体分层时极限推力的数值解与推导的极限推力计算公式得到的理论解相对误差为5.2%；而不考虑掌子面土体分层时，数值解与通过计算公式得到的理论解相对误差达到25.9%。因此，在实际工程中，当掌子面存在多层土体时，采用本文所得计算公式计算掌子面极限推力更合理。本文计算模型更加简洁，采用常规的力学平衡方程求解能得到合理的计算结果，不需要进行大量积分和迭代计算，显著降低了计算难度。 3) 掌子面极限推力随隧道埋深和直径增加而增大。分层土中引起极限推力变化的主要土体是上层软弱土。盾构直径越大，土体黏聚力和内摩擦角对极限推力的影响越大，且内摩擦角对极限推力的影响更显著；当盾构直径较小时，黏聚力和内摩擦角对极限推力的影响较小。 参考文献 刘克奇 , 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Journal of Central South University(Science and Technology), 2023, 54(4): 1370-1378. 国家自然科学基金资助项目(51538009)；贵州省科技计划项目([2020]2Y035) Project(51538009) supported by the National Natural Science Foundation of China; Project([2020]2Y035) supported by the Science and Technology Plan Program of Guizhou Province 2022-06-10 2022-08-25 2023-04-26 2023-05-22 37691273 中南大学学报(自然科学版) 4 54