工程地质学报  2017, Vol. 25 Issue (6): 1633-1639   (1477 KB)
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OPTIMAL CONTROL OF TBM EXCAVATION RATE CONSIDERING NEIGHBORING SOIL VOID
LIU Zhen, LI Jieming, YANG Xu, LU Yiqi, ZHOU Cuiying
School of Civil Engineering, Sun Yatsen University, Guangzhou 510275
Abstract: Tunneling with neighboring soil void is a particular challenge for underground railway construction in karst terrain. To reduce the disturbance to neighboring soil voids generated by the TBM and to secure the TBM from sudden settlement, axis deviation and ground collapse, it's necessary to conduct TBM tunneling parameters optimization. Aiming at the existing circumstances that TBM tunneling parameters are mostly determined by engineering experiences, a mechanical model based on Mindlin solution is built and an optimal control problem about excavation rate is proposed using the excavation rate regarded as a controlled variable, and the variation of energy density is regarded as the performance index. The problem is then solved using Gradient method. This optimization method has been introduced to the construction of a TBM tunnel in Line 9, Guangzhou Railway System. The results indicate that the model proposed in this study can effectively reflect the disturbance at the top of neighboring soil voids caused by bulkhead thrust in TBM tunneling. The optimization based on optimal excavation rate control problem agrees with engineering experiences.
Key words: Karst    Soil void    TBM tunneling    Driving speed    Optimal control

0 引言

1 穿越邻近土洞的掘进速度最优控制

 图 1 盾构穿越邻近土洞模型示意图 Fig. 1 TBM excavation with neighboring soil void

① 土体为均质弹性半无限体，盾构在土体中沿水平直线掘进；②盾构正面推力近似为作用于圆形开挖面上的均布荷载，不考虑盾壳摩擦和同步注浆的影响。同时，土洞状态按照普氏理论来考虑。

1.1 系统状态方程

 $\dot x = u\left( t \right)$ (1)

1.2 允许控制集

 ${U_{ad}} = \left\{ {u\left( t \right)|{v_{{\rm{min}}}} \le u\left( t \right) \le {v_{{\rm{max}}}}} \right\}$ (2)

1.3 初终端条件

 $x({t_0}) = {x_0} - {R_d}$ (3)

 $x({t_1}) = {x_0} + {R_d}$ (4)

1.4 性能指标函数

 $J\left( u \right) = \int_{{t_0}}^{{t_1}} {\frac{{{\rm{d}}{w_0}}}{{{\rm{d}}t}}{\sigma _0}{\rm{d}}t}$ (5)

 \begin{align} &{{w}_{0}}\left( x \right)=\iint\limits_{\Omega }{\frac{P}{{\rm{ }}\!\!\pi\!\!{\rm{ }}{{r}^{2}}}\frac{(x{\rm{ }}-{\rm{ }}{{x}_{0}})}{16{\rm{ }}\!\!\pi\!\!{\rm{ }}G\left( 1{\rm{ }}-{\rm{ }}\mu \right)}\left[ \frac{{\rm{ }}{{z}_{0}}~-{\rm{ }}z}{R_{1}^{3}}+\frac{\left( 3-4\mu \right)({{z}_{0}}-z)}{R_{2}^{3}} \right.} \\ &\quad \quad \left. -\frac{6{{z}_{0}}z({{z}_{0}}z)}{R_{2}^{5}}+\frac{4\left( 1-\mu \right)\left( 1-2\mu \right)}{{{R}_{2}}({{R}_{2}}+{{z}_{0}}+z){\rm{ }}} \right]{\rm{d}}s \\ \end{align} (6)

 \begin{align} & {{\sigma }_{0}}\left( x \right)=\iint\limits_{\Omega }{\frac{P}{\rm{ }\!\!\pi\!\!\rm{ }{{r}^{2}}}\frac{(x\rm{ }-\rm{ }{{x}_{0}})}{8\rm{ }\!\!\pi\!\!\rm{ }\left( 1\rm{ }-\rm{ }\mu \right)}\left[ \frac{\rm{ }1+2\mu }{R_{1}^{3}}-\frac{\left( 1-2\mu \right)}{R_{2}^{3}} \right.} \\ & \quad \quad -\frac{3{{({{z}_{0}}-z)}^{2}}}{R_{1}^{5}}-\frac{3\left( 3-4\mu \right){{({{z}_{0}}+z)}^{2}}}{R_{2}^{5}} \\ & \quad \quad \left. +\frac{6c}{R_{2}^{5}}\left( z\rm{ + }\left( 1-2\mu \right)({{z}_{0}}~+\rm{ }z)\rm{ }+\frac{5{{z}_{0}}{{({{z}_{0}}~+\rm{ }z)}^{2}}}{R_{2}^{5}} \right) \right]\rm{d}s~ \\ \end{align} (7)

 $\Omega = {\rm{ }}\left\{ {\left( {x,m,n} \right)\left| {\sqrt {{{\left( {m - y} \right)}^2} + {\rm{ }}{{\left( {n - z} \right)}^2} \le {\rm{ }}{r^2}} } \right.} \right\}$ (8)

R1R2分别有如下定义：

 ${R_1} = \sqrt {{{({x_0} - x)}^2} + {{({y_0} - y)}^2} + {{({z_0} - z)}^2}}$ (9)

 ${R_2} = {\rm{ }}\sqrt {{{({x_0} - x)}^2} + {{({y_0} - y)}^2} + {{({z_0} + z)}^2}}$ (10)

 $\dot x = {b_1}{{\rm{e}}^{\frac{P}{{{b_2}}}}} + {b_3}$ (11)

 $\left\{ {\begin{array}{*{20}{l}} {\dot x = u\left( t \right)}\\ {x({t_0}) = {x_0} - {R_d},x({t_1}) = {x_0} + {R_d}} \end{array}} \right.$ (12)

2 掘进速度最优控制问题求解
2.1 位移/应力函数的数值积分

 \begin{align} &{{I}_{w}}\left( x \right)=\iint\limits_{\Omega }{\frac{{{z}_{0}}-z}{R_{1}^{3}}+\frac{\left( 3-4\mu \right)({{z}_{0}}-z)}{R_{2}^{3}}-\frac{6{{z}_{0}}z({{z}_{0}}+z)}{R_{2}^{5}}} \\ &\quad \quad +\frac{\rm{ }4\left( 1-\mu \right)\left( 1-2\mu \right)}{{{R}_{2}}({{R}_{2}}+{{z}_{0}}+z)~}{\rm{d}}s \\ \end{align} (13)

 图 2 被积函数数值积分示意图 Fig. 2 Numerical integration

 图 3 Iw-x关系曲线 Fig. 3 Iw-x curve

2.2 引入罚函数

 $J\left( u \right) = \int_{{t_0}}^{{t_1}} {\frac{{{\rm{d}}{w_0}}}{{{\rm{d}}t}}} {\sigma _0}{\rm{d}}t + N{\left[ {x\left( {{t_1}} \right) - \left( {{x_0} + {R_d}} \right)} \right]^2}$ (14)

2.3 梯度法求解

 $H = - \frac{{{\rm{d}}{w_0}}}{{{\rm{d}}t}}{\sigma _0} + \lambda u$ (15)

 $\mathop {{\rm{max}}}\limits_{u \in {U_{ad}}} H({x^*}\left( t \right),u,\lambda \left( t \right)) = H({x^*}\left( t \right),{u^*}\left( t \right),\lambda \left( t \right))$ (16)

 $\dot \lambda = - \frac{{\partial H}}{{\partial x}}$ (17)

 $H({x^*}\left( t \right),{u^*}\left( t \right),\lambda \left( t \right)) \equiv H\left| {_{t_1^*}} \right. = 0$ (18)

(1) 选取罚因子N>0，设置终端状态容差ε>0；

(2) 选取初始控制函数u0(t)，设置容许误差δ>0，令K=0；

(3) 用uK(t)代替u(t)，根据状态方程(式(1))求得xK(t)；

(4) 根据λK(t1)=$2N\left[ {x\left( {{t_1}} \right) - \left( {{x_0} + {R_d}} \right)} \right]$和式(17)，反向积分求得λK(t)；

(5) 计算梯度

 $h({u_K}\left( t \right)) = - \frac{{\partial H}}{{\partial u}}{\rm{ }}({x_K}\left( t \right),{u_K}\left( t \right),{\lambda _K}\left( t \right))$ (19)

(6) 修正控制函数：

 \begin{align} & {{u}_{K+1}}\left(t \right)=~~ \\ & \quad \left\{ \begin{array}{*{35}{l}} {{v}_{\rm{max}}}, & {{u}_{K}}+a_{K}^{*}\cdot h\left({{u}_{K}}\left(t \right) \right)\ge {{v}_{\rm{max}}} \\ \rm{ }{{u}_{K}}+a_{K}^{*}\cdot h\left({{u}_{K}}\left(t \right) \right), & {{v}_{\rm{min}}}＜{{u}_{K}}+a_{K}^{*}\cdot h\left({{u}_{K}}\left(t \right) \right)＜{{v}_{\rm{max}}} \\ {{v}_{\rm{min}}}, & ~{{u}_{K}}+a_{K}^{*}\cdot h\left({{u}_{K}}\left(t \right) \right)\le {{v}_{\rm{min}}} \\ \end{array} \right. \\ \end{align} (20)

(7) 计算误差，若

 $\left| {\frac{{J({u_{K + 1}}) - J({u_K})}}{{J({u_K})}}} \right| ＜ \delta$ (21)

(8) 令K=K+1，转至步骤(3)；

(9) 计算终端状态误差，若

 $\left| {{x_K}({t_1}) - ({x_0} + {R_d})} \right| ＜ \varepsilon$ (22)

 \begin{align} & J({{u}_{K}}+a_{K}^{*}\cdot h({{u}_{K}}\left( t \right)))= \\ & \quad \quad \quad \underset{{{a}_{k}}>0}{\mathop{\text{min}}}\,J({{u}_{K}}+{{a}_{K}}\cdot h({{u}_{K}}\left( t \right))) \\ \end{align} (22)

3 工程实例

 图 4 花都广场—马鞍山公园区间部分地质剖面图(朱小藻，2013) Fig. 4 Geological profile of Hua-Ma section(Zhu, 2013)

 图 5 盾构掘进速度最优控制及相应位移 Fig. 5 Optimal control of excavation rate

4 结论

(1) 根据弹性力学Mindlin解建立的盾构在含土洞地层中掘进的力学模型，能求出盾构正面推力使土洞顶部产生的竖向位移和应力，有效反映了盾构掘进对土洞的扰动。

(2) 在掘进模型的基础上，提出了盾构掘进速度最优控制问题，该问题以盾构掘进速度为控制变量，以掘进引起的土洞洞顶能量密度变化量为性能指标，控制约束和初终端条件则结合工程实际设定，再通过梯度法对其进行数值求解，能对穿越土洞时盾构的掘进速度控制策略进行优化。

(3) 以广州地铁九号线花—马区间盾构穿越邻近土洞问题为例，求得掘进速度的最优控制为以较高速度掘进一段时间后，改以较低速度匀速穿越，与工程经验符合，具有一定合理性。该方法为穿越邻近土洞时优化盾构掘进参数、降低土洞所受扰动提供了新的思路，具有一定参考意义。

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