Bipolar coordinate solving method of ground displacement caused by shallow tunnel excavation
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摘要: 基于双极坐标系和Mohr-Coulomb准则，考虑剪胀特性，在Jeffery和Massinas半无限空间圆形隧道围岩应力解的基础上联立平衡方程，推导了浅埋隧道施工拱顶方向的地层位移弹塑性解，并通过Peck公式、Park公式、Loganathan-Poulos公式和实测数据进行验证，揭示了地层变形机理和既有研究之间的联系，给出了考虑地层参数和施工因素影响的经验参数(地层损失率和间隙参数)定量取值方法。研究结果表明：弹塑性解的假定条件更少，与Peck公式、Park公式的差值在2%以内，与Loganathan-Poulos公式的差值为9.5%；双极坐标求解法从弹塑性分析的角度进一步解释了地层变形机理、Peck公式和各类修正弹性公式，即浅埋隧道开挖也会引起地层弹性和塑性变形，以Peck公式为代表的经验公式法在计算地层变形时，地层损失率在不同地区和施工控制条件下的取值分别对应着不同地层参数(黏聚力、内摩擦角、泊松比、重度和弹性模量)和施工边界条件(埋深、开挖半径、支护力)下的地层弹塑性变形；以Park公式、Loganathan-Poulos公式为代表的各类修正弹性解也可近似看作通过断面椭圆化和下沉等手段的修正来抵消理想弹性解与弹塑性解之间的差值。可见，弹塑性解与既有公式结合能更好地指导现场施工。Abstract: Based on the bipolar coordinate system and Mohr-Coulomb criterion, the elastoplastic solution of ground displacement caused by the shallow tunnel excavation in vault direction was derived by considering the characteristic of dilatancy and combining the surrounding rock stress solution of Jeffery and Massinas semi-infinite spatial circular tunnel with the equilibrium equations. The new solution was further verified by the Peck formula, Park formula, Loganathan-Poulos formula, and measured data. The relationships between the ground deformation mechanism and existing solutions were revealed. The quantitative methods for determining the empirical parameters (ground loss rate and gap parameter) subjected to ground parameters and construction factors were provided. Research results show that fewer hypotheses are needed by the elastoplastic solution. An difference of less than 2% is presented in comparison with the Peck formula and Park formula, and an difference of 9.5% is shown in comparison with the Loganathan-Poulos formula. The ground deformation mechanism, Peck formula, and various modified elastic formulas are further elucidated from the perspective of elastoplastic analysis by the bipolar coordinate solving method. In other words, both the elastic and plastic deformations can be caused by the shallow tunnel excavation. When the ground deformation is calculated by the empirical formula methods, typified by the Peck formula, the values of ground loss rate in diverse regions and construction conditions correspond to the ground elastoplastic deformations under different ground parameters (cohesion, internal friction angle, Poisson's ratio, weight, and elastic modulus) and construction boundary conditions (buried depth, excavation radius, and support force). Various modified elastic solutions, typified by the Park formula and Loganathan-Poulos formula, can also be deemed as the modifications offsetting the discrepancy between the ideal elastic solution and the elastoplastic solution by the approximate means of the cross-section ovalization and sedimentation. Therefore, the field construction can be better guided by combining the elastoplastic solution with the existing formula.
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图 2 双极坐标系中的半无限空间隧道模型
Figure 2. Semi-infinite space tunnel model in bipolar coordinate system
图 4 不同隧道支护力下的最大地表沉降对比
Figure 4. Comparison of maximum ground settlements under different tunnel support forces
图 5 弹性、塑性和剪胀工况下隧道拱顶沉降
Figure 5. Tunnel vault settlements under elastic, plastic and dilatant conditions
图 9 黄土地层中地表最大沉降和间隙参数的关系
Figure 9. Relationship between maximum surface settlement and ground gap parameter in loess strata
表 1 采用不同公式计算的地表最大沉降结果
Table 1. Maximum ground settlements calculated by different formulas
公式名称 地表沉降/cm 与本文弹塑性解的相对误差/% 本文弹塑性解 2.64 理想弹性解 1.75 33.7 Peck公式 2.61 1.1 Loganathan-Poulos公式 2.39 9.5 Park公式 2.69 1.9 
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表 2 地层损失率统计及计算值对比
Table 2. Comparison of ground loss ratios between statistic and calculation
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表 3 中国部分地区地铁施工间隙参数的经验取值
Table 3. Empirical values of ground gap parameter in subway construction in some areas of China
地区 样本数 地层特征 g/mm 北京 81 砂土、黏性土互层 4.2~146.5 广州 31 黏性土、砂土、风化岩 8.9~97.6 深圳 27 淤泥质土、粉质黏土、风化岩 17.0~341.0 武汉 37 黏性土、软土、砂土 15.2~164.1 南京 25 淤泥质黏土、粉质黏土 2.0~187.4 西安 54 黄土 4.6~134.1
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