工程科学与技术   2021, Vol. 53 Issue (2): 95-102

Upper Bound Limit Analysis of Limit Support Pressure for Shield Excavation Face in Composite Ground
DAI Zhonghai, HU Zaiqiang
Inst. of Geotechnical Eng., Xi’an Univ. of Technol., Xi’an 710048, China
Abstract: With the further development and utilization of underground space, the geological conditions faced by shield tunneling are more complex and diverse. When the shield tunnels in the composite strata, due to the differences in mechanical properties between the strata, it is easy to cause instability and damage due to improper determination of the supporting force of the excavation surface, which will bring negative impacts to the projects. It is difficult to determine the support pressure of the excavation face due to the stiffness differences between the upper and lower strata, when the shield tunnel passes through the composite strata. Based on the combined logarithmic helix failure model utilized as the instability model of the shield excavation face in the composite strata, the upper limit expression of the limit support force of shield excavation face in the composite strata was derived by using the upper bound theorem of limit analysis. The optimal solution of the ultimate support force was obtained by linear optimization technology, and the rationality of the combined logarithmic spiral failure mode was verified by comparing with the existing literatures. At the same time, the influence of different parameters on the ultimate support force and the critical failure mode were discussed, and the schematic diagram of the critical sliding surface was presented. The results showed that when the internal friction angle and cohesion of the upper stratum were fixed, the ultimate support force decreased with the increase of the internal friction angle and cohesion of the lower stratum. With the increase of the diameter of the tunnel, the ultimate support force increased significantly. The depth coefficient was defined as the ratio of the depth of the upper stratum to the diameter of the tunnel. The smaller the depth coefficient was, the more sensitive the ultimate support force was to the variations of the shear strength parameters of the stratum. But for a single stratum, the change of depth coefficient has no effect on the ultimate support force. The conclusions had a certain guiding significance for the practical engineering.
Key words: shield tunnel    composite strata    upper bound limit analysis method    stability of excavation face

1 计算原理和计算模型 1.1 计算原理

 $\int_S {{{{T}}_i}} {{v}}_i^*{\rm{d}}S + \int_V {{{{F}}_i}} {{v}}_i^*{\rm{d}}V = \int_V {{{\sigma}} _{ij}^*} \dot {{\varepsilon}} _{ij}^*{\rm{d}}V$ (1)

1.2 计算模型

 图1 盾构开挖面破坏模型 Fig. 1 Failure mechanism of the shield excavation face

 $\left\{ \begin{array}{l} {r_1}\left( \theta \right) = {r_{OA}}{{\rm{e}}^{\left( {\theta - {\theta _A}} \right)\tan\; {\varphi _1}}}, \\ {r_{\rm{2}}}\left( \theta \right) = {r_{OE}}{{\rm{e}}^{\left( {{\theta _E} - \theta } \right)\tan \;{\varphi _1}}}, \\ {r_{\rm{3}}}\left( \theta \right) = {r_{OB}}{{\rm{e}}^{\left( {{\theta _B} - \theta } \right)\tan \;{\varphi _2}}} \end{array} \right.$ (2)

2 外功率与内能耗散率 2.1 外功率

 图2 重力功率计算示意图 Fig. 2 Schematic diagram for calculating power of the gravity

 \begin{aligned}[b] {\rm{d}}{W_{OEF}} =& \gamma {\rm{d}}A \cdot {v_G}\sin\; \theta = {\rm{ }}\gamma \cdot \frac{1}{2}{r_2^2}{\rm{d}}\theta \cdot \frac{2}{3}\omega {r_2}\sin\; \theta = \\ &\frac{{\rm{1}}}{3}\gamma \omega {r_2^3}\sin\; \theta {\rm{d}}\theta \end{aligned} (3)

 \begin{aligned}[b] {W_{OEF}} =& \int_{{\theta _E}}^{{\theta _F}} {\frac{2}{3}\omega {r_2}\gamma } \sin\; \theta \frac{1}{2}{r_2}^2{\rm{d}}\theta = \\ &\frac{1}{3}\gamma \omega \int_{{\theta _E}}^{{\theta _F}} {{r_2}^{\rm{3}}} \sin \;\theta {\rm{d}}\theta \end{aligned} (4)

 \begin{aligned}[b] {W_\gamma } =& \frac{1}{3}\gamma \omega \int_{{\theta _E}}^{{\theta _F}} {{r_2}^{\rm{3}}} \sin \;\theta {\rm{d}}\theta {\rm{ + }}\frac{1}{3}\gamma \omega \int_{{\theta _B}}^{{\theta _E}} {{r_3^3}} \sin\; \theta {\rm{d}}\theta - \\ &\frac{1}{3}\gamma \omega \int_{{\theta _A}}^{{\theta _F}} {{r_{\rm{1}}}^{\rm{3}}} \sin\; \theta {\rm{d}}\theta - \gamma \omega \frac{2}{3}{r_{OA}}\sin\; {\theta _A}\frac{1}{2}D{r_{OA}} \cdot \\ &\sin\; {\theta _A} = \gamma \omega r_{OA}^3\left( {{f_1} + {f_2} - {f_3} - {f_4}} \right) \end{aligned} (5)

 \begin{aligned}[b] {f_1} =& \frac{{{{\sin }^3}{\theta _A}}}{{{{\sin }^3}{\theta _B}}}\frac{{{{\rm{e}}^{3\left( {{\theta _B} - {\theta _E}} \right)\tan \;{\varphi _2}}}}}{{{\rm{3}}\left( {{\rm{1 + 9ta}}{{\rm{n}}^2}{\varphi _1}} \right)}}\left[ {\cos \;{\theta _E} + 3\sin \;{\theta _E}\tan \;{\varphi _1}} \right. - \\ &{{\rm{e}}^{3\left( {{\theta _E} - {\theta _F}} \right)\tan\; {\varphi _1}}}\left. {\left. {\left( {\cos\; {\theta _F} + } \right.3\sin \;{\theta _F}\tan \;{\varphi _1}} \right)} \right]\\[-10pt] \end{aligned} (6)
 \begin{aligned}[b] {f_{\rm{2}}} =& \frac{{{{\sin }^3}{\theta _A}}}{{{{\sin }^3}{\theta _B}}}\frac{{\rm{1}}}{{{\rm{3}}\left( {{\rm{1 + 9ta}}{{\rm{n}}^2}{\varphi _2}} \right)}}\left[ {\left( {\cos\; {\theta _B} + 3\sin \;{\theta _B}\tan \;{\varphi _2}} \right) - } \right. \\ &\left. {{{\rm{e}}^{3\left( {{\theta _B} - {\theta _E}} \right)\tan \;{\varphi _2}}}\left( {\cos \;{\theta _E} + 3\sin\; {\theta _E}\tan \;{\varphi _2}} \right)} \right]\\[-12pt] \end{aligned} (7)
 \begin{aligned}[b] {f_{\rm{3}}} =& \frac{{\rm{1}}}{{{\rm{3}}\left( {{\rm{1 + 9ta}}{{\rm{n}}^2}{\varphi _1}} \right)}}\left[ {{{\rm{e}}^{3\left( {{\theta _F} - {\theta _A}} \right)\tan\; {\varphi _1}}}( {3\tan \;{\varphi _1}\sin \;{\theta _F} -} } \right. \\ &\;\;\;\;\;\;\;\cos\; {\theta _F} )\left. { - \left( {3\tan\; {\varphi _1}\sin \;{\theta _A} - \cos\; {\theta _A}} \right)} \right] \end{aligned} (8)
 ${f_{\rm{4}}} = \frac{{\sin \left( {{\theta _A} - {\theta _B}} \right){{\sin }^2}{\theta _A}}}{{{\rm{3}}\sin \;{\theta _B}}}$ (9)

 ${r_{OA}}\sin \;{\theta _A} = {r_{OB}}\sin \;{\theta _B}$ (10)
 $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;D\sin \;{\theta _A} = {r_{OB}}\sin \left( {{\theta _A} - {\theta _B}} \right)$ (11)
 $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{r_{OB}}\cos\; {\theta _B} - {r_{OE}}\cos\; {\theta _E} = \left( {1 - n} \right)D$ (12)

 ${r_{OE}} = {r_{OB}}{{\rm{e}}^{\left( {{\theta _B} - {\theta _E}} \right)\tan \;{\varphi _{\rm{2}}}}}$ (13)

 $\cos \;{\theta _B} - \cos\; {\theta _E}{{\rm{e}}^{\left( {{\theta _B} - {\theta _E}} \right)\tan\; {\varphi _2}}} = \left( {1 - n} \right)\frac{{\sin \left( {{\theta _A} - {\theta _B}} \right)}}{{\sin\; {\theta _A}}}$ (14)

 $\left\{ \begin{array}{l} {r_{OF}} = {r_{OA}}{{\rm{e}}^{\left( {{\theta _F} - {\theta _A}} \right)\tan \;{\varphi _1}}}, \\ {r_{OF}} = {r_{OE}}{{\rm{e}}^{\left( {{\theta _E} - {\theta _F}} \right)\tan \;{\varphi _1}}} \end{array} \right.$ (15)

 \begin{aligned}[b] {\theta _F} =& \frac{{\rm{1}}}{{{\rm{2}}\tan\; {\varphi _1}}}\left[ {{\theta _A}\tan\; {\varphi _1} + {\theta _B}\tan\; {\varphi _2} + {\theta _E}} \right. \cdot \\ &\left. {\left. {\left( {\tan\; {\varphi _1} - } \right.\tan\; {\varphi _2}} \right) + \ln \left( {{{\sin\; {\theta _A}} / {\sin \;{\theta _B}}}} \right)} \right] \end{aligned} (16)

 图3 极限支护力功率计算示意图 Fig. 3 Schematic diagram of limit supporting force power calculation

 $OP = \frac{{{r_{OA}}\sin \;{\theta _A}}}{{\sin\; \theta }}$ (17)

P点处的速度大小为：

 $v = \frac{{{r_{OA}}\sin\; {\theta _A}}}{{\sin \;\theta }}\omega$ (18)

 \begin{aligned}[b] {W_T} =& - \int_{{\theta _B}}^{{\theta _A}} {{\sigma _T}{r_{OA}}\frac{{\sin\; {\theta _A}}}{{\sin\; \theta }}\cos\; \theta \omega r} \frac{{{\rm{d}}\theta }}{{\sin\; \theta }} = \\ & - \frac{{\rm{1}}}{{\rm{2}}}\omega {\sigma _T}r_{OA}^2\left( {\frac{{{{\sin }^2}{\theta _A}}}{{{{\sin }^{\rm{2}}}{\theta _B}}} - 1} \right) \end{aligned} (19)

 ${W}_{{\text{总}}}={W}_{\gamma }+{W}_{T}$ (20)
2.2 内能耗散率

 ${D_{AF}} = \int_{{\theta _A}}^{{\theta _F}} {{c_1}} \cdot \omega {r_1}\cos\; {\varphi _1} \cdot \frac{{{r_1}{\rm{d}}\theta }}{{\cos \;{\varphi _1}}} = \frac{{{c_1}\omega r_{OA}^2}}{{2\tan\; {\varphi _1}}}{g_1}$ (21)
 ${D_{FE}} = \int_{{\theta _E}}^{{\theta _F}} {{c_1}} \cdot \omega {r_2}\cos\; {\varphi _1} \cdot \frac{{{r_2}{\rm{d}}\theta }}{{\cos\; {\varphi _1}}} = \frac{{{c_1}\omega r_{OA}^2}}{{2\tan\; {\varphi _1}}}{g_2}$ (22)
 ${D_{BE}} = \int_{{\theta _B}}^{{\theta _E}} {{c_2}} \cdot \omega {r_3}\cos\; {\varphi _2} \cdot \frac{{{r_3}{\rm{d}}\theta }}{{\cos \;{\varphi _2}}} = \frac{{{c_2}\omega r_{OA}^2}}{{2\tan \;{\varphi _2}}}{g_3}$ (23)

 ${g_1}{\rm{ = }}{{\rm{e}}^{2\left( {{\theta _F} - {\theta _A}} \right)\tan\; {\varphi _1}}} - 1$ (24)
 ${g_2}{\rm{ = }}\frac{{{{\sin }^2}{\theta _A}}}{{{{\sin }^2}{\theta _B}}}{{\rm{e}}^{2\left( {{\theta _B} - {\theta _E}} \right)\tan\; {\varphi _2}}}\left[ {1 - {{\rm{e}}^{2\left( {{\theta _E} - {\theta _F}} \right)\tan\; {\varphi _1}}}} \right]$ (25)
 ${g_3}{\rm{ = }}\frac{{{{\sin }^2}{\theta _A}}}{{{{\sin }^2}{\theta _B}}}\left[ {1 - {{\rm{e}}^{2\left( {{\theta _B} - {\theta _E}} \right)\tan\; {\varphi _2}}}} \right]$ (26)

 ${D}_{{\text{总}}}={D}_{AF}+{D}_{FE}+{D}_{BE}$ (27)
2.3 极限支护力的求解

 \begin{aligned}[b] {\sigma _T}{\rm{ = }}&\frac{{{{\sin }^2}{\theta _B}}}{{{{\sin }^2}{\theta _A} - {{\sin }^2}{\theta _B}}}\left[ {{\rm{2}}\gamma D\frac{{\sin \;{\theta _B}}}{{\sin \left( {{\theta _A} - {\theta _B}} \right)}}} \right.\left( {{f_1} + {f_2}} \right. - \\ &\left. {\left. {{f_3} - {f_4}} \right) - \frac{{{c_1}}}{{\tan\; {\varphi _1}}}\left( {{g_1} + {g_2}} \right) - \frac{{{c_2}}}{{\tan\; {\varphi _2}}}{g_3}} \right] \end{aligned} (28)

 $\left\{ \begin{array}{l} 0 < {\theta _B} < {\theta _A} < {\text{π} / 2}, \\ {\theta _A} < {\theta _E} < {\text{π}} , \\ {\theta _B} < {\theta _E} < {\theta _F} \end{array} \right.$ (29)
3 验　证

Cheng等[21]利用随机极限分析法研究了复合地层盾构开挖面稳定性。将本文计算结果与其数据对比，如表1所示。其他参数取值如下： ${\varphi _1} = {\varphi _2} = {30^ {\circ} }$ $\gamma = 20\;{\rm{kN/}}{{\rm{m}}^{\rm{3}}}$ $n = 0.5$ $D = 10$ m，c1c2分别为上部地层和下部地层的黏聚力大小。

4 参数分析

4.1 地层强度参数

 图4 内摩擦角对不同深度系数下极限支护力的影响 Fig. 4 Effect of the internal friction angle on limit support pressure with different depth coefficient

 图5 黏聚力对不同深度系数下极限支护力的影响 Fig. 5 Effect of the cohesion on limit support pressure with different depth coefficient

4.2 隧道直径及土体重度

 图6 隧道直径对极限支护力的影响 Fig. 6 Effect of the tunnel diameter on limit support pressure

 图7 土体重度对极限支护力的影响 Fig. 7 Effect of the soil unit weight on limit support pressure

4.3 深度系数

 图8 不同内摩擦角时深度系数的影响 Fig. 8 Effect of the depth coefficient with different internal friction angle

 图9 不同黏聚力时深度系数的影响 Fig. 9 Effect of the depth coefficient with different cohesion

4.4 临界滑动面示意图

 图10 临界滑动面示意图 Fig. 10 Schematic diagram of the critical failure surface

5 结　论

1）当上部地层内摩擦角一定，随着下部地层内摩擦角的增大，极限支护力显著降低，且深度系数越小，这一趋势越明显。当上部地层黏聚力一定时，随着下部地层黏聚力的增大，极限支护力呈线性减小趋势。同样地，深度系数越小，下降幅度越大。

2）随着隧道直径的增大，极限支护力显著增大；随着土体重度的增加，极限支护力逐渐增大。

3）当上部地层内摩擦角不变时，随着下部地层内摩擦角的增大，破坏块体的体积减小，下部地层破坏面向隧道面移动，即较大的摩擦角产生较小的破坏范围。当上部地层黏聚力不变，下部地层黏聚力增大，破坏面稍向隧道面移动，但变化并不明显。

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