Method and application of deformation control of excavations in soft ground
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摘要: 基坑变形控制是软土地区基坑工程的核心内容，不仅与自身工程安全密切相关，更涉及到对周边环境的影响。随着城市地上、地下各类建（构）筑物越来越密集，基坑工程施工产生的变形对环境影响的控制愈加成为基坑工程的焦点问题。首先，从基坑施工全过程控制的视角，分析了基坑施工全过程各阶段的变形特征、机理以及对环境的影响。进而，将基坑变形及其对环境影响的控制划分为“基于基坑支护体系的变形控制”和“基于邻近基坑保护对象的变形控制”两类方法。针对基于邻近基坑保护对象的变形控制，提出了不是基于对基坑支护体系，而是直接着眼于保护对象的变形主动控制理论，通过对关键区域土体的应力和变形的控制，实现对保护对象的测控一体化靶向控制。此外，提出了基坑无支撑支护理论并发展了一系列软弱土地区基坑绿色无支撑支护技术，实现了在较大的深度条件下也可进行坑无支撑支护设计。通过“基坑施工全过程控制”“基坑变形主动控制理论”“基坑无支撑支护控制体系”的变形控制理论及工程应用，努力推动基坑工程变形控制向“高效、智能、绿色、低碳”方向发展。Abstract: The main task of excavations in soft ground is the deformation control, which is closely rated to their safety and environmental impact. With the increase of the buildings and structures in the urban areas, the construction-induced deformation has become the focus of the excavations. The characteristics, mechanism and environmental impact of the deformation caused by each excavation phase are analyzed in a view of the whole-process control. Furthermore, the control methods for the deformation and environmental impact of the excavations are classified into two types, i.e., the control based on the retaining system of the excavations and that based on the protected objects adjacent to them. For the latter type, the active control theory is proposed focusing on the deformation of the protected objects instead of the retaining system. This active targeting technology integrated with the measurement and control for the protected objects is realized by controlling the stress and deformation of the key zone. Finally, the strut-free retaining theory is proposed and a series of strut-free retaining technologies are developed for the excavations in soft ground. The design of strut-free retaining for the excavations with relatively large depth can be realized using these technologies. The theories and applications of the whole-process control, the active control and the strut-free retaining system promote the deformation control of the excavations towards the efficient, intelligent, green and low-carbon aim.
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Keywords:
- excavation /
- deformation control /
- whole process /
- strut-free retaining /
- active control
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0. 引言
地下洞室对弹性波的散射问题在地震工程、地下结构抗爆、地球物理勘探等领域有着广泛的应用。近年来，许多研究人员基于波函数展开法[1-2]对单相介质中衬砌结构的动力响应进行了探讨。然而，土和岩石具有多孔性和多相性，单相弹性连续介质无法准确描述实际情况。因此，波在两相介质中的传播和动力响应问题受到广泛关注。应用Biot饱和多孔介质波动理论，周香莲等[3]求解了全空间圆柱形衬砌洞室对平面波散射的解析解。Jiang等[4]引入大圆弧假设，解决了平面P波入射下饱和多孔弹性半空间中圆形衬砌洞室周围的散射问题。为了更符合实际工程，张海等[5]使用辅助函数配合波函数展开法和傅立叶级数展开方法，进一步研究了SH波入射下含直边界半圆形衬砌洞室的动力响应。Ding等[6]利用波函数展开法，给出了平面P波入射下饱和多孔弹性介质中复合衬砌动应力集中的解析解。Hasheminejad等[7]解析分析P波和SV波作用下饱和多孔弹性介质中变壁厚内衬的圆柱形洞室的动应力集中问题。Liu等[8]基于平面复变理论得到了多孔弹性半空间中衬砌洞室对P1和SV波散射的解析解。徐长节等[9]，Xu等[10]基于非局部弹性理论分析了孔径效应和孔隙动力学效应对饱和土中圆柱形衬砌动力响应的影响。Ding等[11]基于非局部Biot理论得到了平面P波入射下饱和多孔介质中浅埋复合衬砌洞室动力响应的解析解。
综上，饱和多孔介质中衬砌洞室的动力响应已经得到了广泛而充分的研究。可实际上，土是一种三相介质，即使地下水位以下的土，也可能存在不完全饱和的情况。Li等[12]总结了众多非饱和孔隙弹性模型以及非饱和多孔介质波动特性的研究，表明由于气相的存在，非饱和介质的动态响应在许多方面与传统饱和多孔介质的动态反应显著不同。然而，目前有关非饱和多孔介质中衬砌洞室对地震波散射问题的研究还较为匮乏。Li等[13]研究了非饱和多孔介质中圆形洞室在内部荷载作用下的动力响应。Tan等[14]研究了无限非饱和多孔弹性介质中不完全接触的衬砌洞室对平面P波的散射问题。以考虑混合物理论的非饱和多孔介质动力理论为基础，采用波函数展开法和分离变量法，分别推导了平面P波和SV波入射时衬砌洞室周围位移应力的解析解。分析了不同入射频率下饱和度对衬砌洞室周围动应力集中系数的影响。
1. 分析模型
1.1 数学模型和基本方程
为了简化研究模型，把非饱和多孔介质中圆柱形衬砌洞室，视为无限非饱和多孔介质全空间中一个无限长度的衬砌洞室（图 1）。
衬砌洞室的内径为a，外径为b。本研究基于Wei等[15]提出的非饱和多孔介质动力学理论。控制方程表示为
nS0ρS0¨uS=(MSS+G)∇∇⋅uS+G∇⋅∇uS+MSW∇∇⋅uW+MSN∇∇⋅uN+ˆμW(˙uW−˙uS)+ˆμN(˙uN−˙uS)，nW0ρW0¨uW=MSW∇∇⋅uS+MWW∇∇⋅uW+MWN∇∇⋅uN−ˆμW(˙uW−˙uS)，nN0ρN0u ¨N=MSN∇∇⋅uS+MWN∇∇⋅uW+MNN∇∇⋅uN−ˆμN(˙uN−˙uS)。} (1) 式中：上角标S，W，N分别为非饱和介质中的固体骨架部分、液相部分和气相部分；G为非饱和多孔介质的剪切模量；nα0,ρα0,uα分别为各相的初始体积分数、初始密度和初始位移；MSS，MWW，MNN，MSW，MSN，MWN为与体积模量、剪切模量及模量间耦合相关的弹性系数；ˆμf为渗透率相关的系数，f=W,N。Wei等[15]给出了相关参数的表达式。采用Van等[16]提出的土水特征曲线V-G模型。
非饱和多孔介质中的本构关系表示为
σS=(MSS∇⋅uS+MSW∇⋅uW+MSN∇⋅uN)I+2G[∇uS+(∇uS)T]，σw=(MSW∇⋅uS+MWW∇⋅uW+MWN∇⋅uN)I ，σN=(MSN∇⋅uS+MWN∇⋅uW+MNN∇⋅uN)I 。} (2) 式中：σα为各相的应力张量，α=S,W,N；I为单位张量矩阵。
衬砌介质被视为连续、各向同性的单相弹性介质，衬砌介质位移为uL，衬砌介质的拉梅常数为λL，μL，衬砌密度为ρL。
1.2 动力学方程的求解
根据亥姆霍兹(Helmholtz)矢量分解原理，非饱和多孔介质方程中的位移可以表示为两部分：
uα = ∇φα + ∇×Ψα(α=S,W,N)。 (3) 式中：φα,Ψα分别为非饱和多孔介质中α相的标量势函数和矢量势函数。
本文考虑简谐波入射，势函数可表示为φ= φ(x,y,z)e−iωt。将式（3）代入方程（1）可得
(∇2−k2j)φSj=0，φWj=δWjφSj，φNj=δNjφSj(j=1,2,3)，} (4a) (∇2−k24)ΨS=0，ΨW=ζWΨS，ΨN=ζNΨS。} (4b) 从式（4a），（4b）中可以看出非饱和多孔介质中有3种压缩波1种剪切波，ki为3种压缩波波数, i=1,2,3，k4为剪切波的波数，δWj，δNj分别为液相、气相压缩波参与系数，j=1,2,3; ζW，ζN分别为液相、气相剪切波参与系数。详细表达式参见文献[13]。
将式（4a），（4b）中固相波函数转换到圆柱坐标系下并使用分离变量法求解，可得固体骨架的标量势函数和矢量势函数的通解为
φSj(r,θ)=∞∑n=0An,jKn(kjr)cosnθ (j=1,2,3)， (5) ΨS(r,θ)=∞∑n=0BnKn(k4r)sinnθ。 (6) 式中：An,j，Bn为待定系数；Kn为第二类修正Bessel函数，下标n代表阶数。
同理可以得到衬砌介质中标量势函数和矢量势函数的通解：
φL(r,θ)=∞∑n=0(A(1)nIn(kL1r)+A(2)nKn(kL1r))einθ， (7) ΨL(r,θ)=∞∑n=0(B(1)nIn(kL2r)+B(2)nKn(kL2r))einθ。 (8) 式中：φL，ΨL分别表示衬砌介质中的标量势函数和矢量势函数；A(1)n，A(2)n，B(1)n和B(2)n为待定系数，分别表示P和SV波的振幅；In为第一类虚宗量Bessel函数，下标n代表阶数。
2. 波场分析与求解
2.1 波场分析
假设具有圆频率的平面谐波P1或SV波沿x轴正方向传播，用贝塞尔-傅立叶级数表示为
入射P1波：φi=φ0∞∑n=0εninJn(k0r)cos nθe−iωt， (9) 入射SV波：Ψi=Ψ0∞∑n=0εninJn(k0r)cos nθe−iωt。 (10) 式中：k0=ω/cp(入射P1波)或者k0=ω/cs(入射SV波)表示入射波的波数，φ0(Ψ0)表示入射波振幅，Jn(.)表示第一类n阶贝塞尔函数，若n=0则εn=1，若n⩾则 {\varepsilon _n} = 2 且 i = \sqrt { - 1} ，t表示时间， {\text{e}^{ - \text{i}\omega t}} 为时间因子将从后续表达式中省略。
入射平面波进入非饱和多孔介质，由于衬砌洞室的存在，会在非饱和多孔介质中产生散射波，在衬砌介质中产生透射波和内表面的反射波。以P1波入射为例，非饱和多孔介质中的波场可表示为
\left. \begin{array}{l}{\varphi }^{\text{S}}={\varphi }_{0}{\displaystyle \sum _{n=0}^{\infty }{\varepsilon }_{n}}{i}^{n}{\text{J}}_{n}({k}_{\text{0}}r)\text{cos }n\theta \text{+}{\displaystyle \sum _{n=0}^{\infty }{A}_{n,j}{\text{K}}_{n}}({k}_{j}r)\cos n\theta \text{，}\\ {\varphi }^{\rm{W}}={\delta }_{1}^{\rm{W}}{\varphi }_{0}{\displaystyle \sum _{n=0}^{\infty }{\varepsilon }_{n}}{i}^{n}{\text{J}}_{n}({k}_{\text{0}}r)\text{cos }n\theta \text{+}\\ \;\;\;\;\;\;\;\;{\displaystyle \sum _{j=1}^{3}{\delta }_{j}^{\rm{W}}{\displaystyle \sum _{n=0}^{\infty }{A}_{n,j}{\text{K}}_{n}}({k}_{j}r)\cos n\theta }\text{，}\\ {\varphi }^{\rm{N}}={\delta }_{1}^{\rm{N}}{\varphi }_{0}{\displaystyle \sum _{n=0}^{\infty }{\varepsilon }_{n}}{i}^{n}{\text{J}}_{n}({k}_{\text{0}}r)\text{cos }n\theta \text{+}\\ \;\;\;\;\;\;\;\;{\displaystyle \sum _{j=1}^{3}{\delta }_{j}^{\rm{N}}{\displaystyle \sum _{n=0}^{\infty }{A}_{n,j}{\text{K}}_{n}}({k}_{j}r)\cos n\theta \text{ }}\text{，}\end{array} \right\} (11a) \left. \begin{array}{l}{\varPsi }^{\rm{S}}={\displaystyle \sum _{n=0}^{\infty }{B}_{n}{\text{K}}_{n}}({k}_{4}r)\sin n\theta \text{，}\\ {\varPsi }^{\rm{W}}={\zeta }^{\rm{W}}{\varPsi }^{\rm{S}}\text{，}\\ {\varPsi }^{\rm{N}}={\zeta }^{\rm{N}}{\varPsi }^{\rm{S}}。\end{array} \right\} (11b) 衬砌介质中的总波场可表示为
\left. \begin{array}{l}{\varphi }_\text{L}={\displaystyle \sum _{n=0}^{\infty }\left({A}_{n}^{{}^{(1)}}{\text{I}}_{n}({k}_{\text{L}_{1}}r)+{A}_{n}^{{}^{(2)}}{\text{K}}_{n}({k}_{\text{L}_{1}}r)\right)}\cos n\theta \text{，}\\ {\varPsi }_\text{L}={\displaystyle \sum _{n=0}^{\infty }\left({B}_{n}^{{}^{(1)}}{\text{I}}_{n}({k}_{\text{L}_{2}}r)+{B}_{n}^{{}^{(2)}}{\text{K}}_{n}({k}_{\text{L}_{2}}r)\right)}\sin n\theta 。\end{array} \right\} (12) 2.2 边界条件
当 r = a 时，在衬砌介质中：
\left. \begin{array}{l}{\sigma }_{rr}^{\text{L}}=0\text{，}\\ {\sigma }_{r\theta }^{\text{L}}=0。\end{array} \right\} (13) 当 r = b 时，非饱和多孔介质与衬砌介质满足连续变形条件与界面不透水边界条件：
\left. \begin{array}{l}{u}_{r}{}^{\rm L}={u}_{r}{}^{\rm{S}}={u}_{r}{}^{\rm{W}}={u}_{r}{}^{\rm{N}}\text{，}\\ {u}_{\theta }{}^{\rm L}={u}_{\theta }{}^{\rm{S}}\text{，}\\ {\sigma }_{r\theta }{}^{\text{L}}={\sigma }_{r\theta }{}^{\rm{S}}\text{，}\\ {\sigma }_{rr}{}^{\text{L}}={\sigma }_{rr}{}^{\rm{S}}+{\sigma }^{\rm{W}}+{\sigma }^{\rm{N}}。\end{array} \right\} (14) 式中： u_{^r}^\alpha ， u_\theta ^\alpha 分别为非饱和多孔介质各相位移法向和切向分量， \alpha = \text{S,W,N} ； {u_r}^{\rm L} ， {u_\theta }^{\rm L} 分别为衬砌的法向、切向位移分量； {\sigma _{rr}}^{\rm S} ， {\sigma _{r\theta }}^{\rm S} 分别为非饱和多孔介质固相法向、切向应力分量； {\sigma ^{\rm W}} ， {\sigma ^{\rm N}} 分别为非饱和多孔介质的孔隙水压力和孔隙气压力； {\sigma _{rr}}^{\text{L}} ， {\sigma _{r\theta }}^{\text{L}} 分别为衬砌法向和切向应力分量。
2.3 求解
将非饱和多孔介质和衬砌介质中总波场势函数的式（11），（12），代入应力和位移的式（2）中，根据边界条件，可得到波场中未知幅值系数的方程：
\left[\begin{array}{llllllll} G_1 & G_2 & G_3 & G_4 & G_5 & G_6 & G_7 & G_8 \\ H_1 & H_2 & H_3 & H_4 & H_5 & H_6 & H_7 & H_8 \\ T_1 & T_2 & T_3 & T_4 & 0 & 0 & 0 & 0 \\ U_1 & U_2 & U_3 & U_4 & 0 & 0 & 0 & 0 \\ V_1 & V_2 & V_3 & V_5 & -\left.D_1\right|_{r=b} & -\left.D_2\right|_{r=b} & -\left.D_3\right|_{r=b} & -\left.D_4\right|_{r=b} \\ Y_1 & Y_2 & Y_3 & Y_5 & -\left.F_1\right|_{r=b} & -\left.F_2\right|_{r=b} & -\left.F_3\right|_{r=b} & -\left.F_4\right|_{r=b} \\ 0 & 0 & 0 & 0 & \left.D_1\right|_{r=a} & \left.D_2\right|_{r=a} & \left.D_3\right|_{r=a} & \left.D_4\right|_{r=a} \\ 0 & 0 & 0 & 0 & \left.F_1\right|_{r=a} & \left.F_2\right|_{r=a} & \left.F_3\right|_{r=a} & \left.F_4\right|_{r=a} \end{array}\right]\left[\begin{array}{l} A_{n, 1} \\ A_{n, 2} \\ A_{n, 3} \\ B_n \\ A_n^{(1)} \\ A_n^{(2)} \\ B_n^{(1)} \\ B_n^{(2)} \end{array}\right]=\left[\begin{array}{l} G_9 \\ H_9 \\ T_5 \\ U_5 \\ -V_4 \\ -Y_4 \\ 0 \\ 0 \end{array}\right] . (15) 系数 {G}_{i}，{H}_{i}，{D}_{i}，{F}_{i}，{T}_{i}，{U}_{i}，{V}_{i}和{Y}_{i} 的表达式由上述公式计算得出，表达式略。通过求解式（15），得到未知波幅系数，以确定非饱和多孔介质和衬砌中的波场，从而获得其中位移和应力的解析解答。
3. 验证和算例
3.1 退化验证
将衬砌外内半径之比取1.000001，其结果将会退化为非饱和多孔介质中无衬砌洞室的情况。与文献[13]计算结果对比，取参数（n0=0.34，Sr=0.6），图 2给出了P波入射下衬砌洞室与空腔洞室动应力集中因子对比结果。从图 2中可以看出，当衬砌很薄时，衬砌洞室退化后的结果与洞室的计算结果一致。
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图 2 非饱和多孔介质中衬砌外内半径之比为1.000001时衬砌洞室与空腔洞室集中因子对比结果Figure 2. Comparison results of concentration factors between cavity and lining cavity with 1.000001 of ratio of outer radius to inner radius in unsaturated porous media3.2 算例分析
工程抗震研究的重点是衬砌的动应力集中因子(dynamic stress concentration factor，简称DSCF)，本文中考虑介质中环向应力与入射波产生的环向应力的比值，表达式为 {\sigma _{\theta \theta }}^{\rm S}/{\sigma _0} ，其中
\begin{align} {\sigma _{\theta \theta }}^{\rm S} =& \left( {{M_\text{SS}}{\mathit{\nabla }^2}{\varphi ^{\rm S}} + {M_\text{SW}}{\mathit{\nabla }^2}{\varphi ^{\rm W}} + {M_{\rm{SN}}}{\mathit{\nabla }^2}{\varphi ^{\rm N}}} \right) + \\ &\frac{{2G}}{r}\left( {\frac{{\partial {\varphi ^{\rm S}}}}{{\partial r}} + \frac{1}{r}\frac{{{\partial ^2}{\varphi ^{\rm S}}}}{{\partial {\theta ^2}}}} \right) - 2G\left[ {\frac{\partial }{{\partial r}}\left( {\frac{1}{r}\frac{{\partial {\varPsi ^{\rm S}}}}{{\partial \theta }}} \right)} \right] \text{，} \end{align} (16) \begin{array}{c} {\sigma _0} = [2G + {M_\text{SS}} + {M_\text{SW}} + {M_\text{SN}} + \delta _1^{\rm W}({M_\text{WW}} + {M_\text{SW}} + {M_\text{WN}}) + \\ {\delta }_{1}^{\rm{N}}\left({M}_\text{SN}+{M}_\text{WN}+{M}_\text{NN}\right)]{k}_{1}^{2}\text{ (P}波入射\text{)} \text{，} \end{array} (17a) {\sigma _0} = Gk_4^2\;\;\;\;\;\; (\text{SV}波入射) 。 (17b) 从应力表达式中可以看到动剪切模量是一个关键因素，在以往的研究中很少将饱和度作为动剪切模量的考量，实际饱和度的变化对土中动剪切模量影响很大[17]。采用下式计算非饱和土动剪切模量[17]：
G = {G_0} + \frac{{2050}}{\alpha }\ln (\sqrt {{{({S_e})}^{ - 2}} - 1} + {({S_e})^{ - 1}})\tan \phi ' 。 (18) 式中： {G_0} 为土体饱和时动剪切模量； \phi ' 为土体饱和时内摩擦角。
本文分别分析P波和SV波入射时，不同频率下饱和度对衬砌洞室周围动应力集中因子的影响。为避免孔洞尺寸对分析结果的影响采用无量纲频率 \eta = a \times {k_0} ，式中a为衬砌内半径，k0为入射波波数。衬砌介质的参数取 {\lambda _{\rm{L}}} = 3.5×107 kPa， {\mu _{\rm{L}}} = 3.5×107 kPa， {\rho _{\rm{L}}} = 2500{\text{ }}\rm kg/{m^3} 。非饱和多孔介质的材料参数及土水特征曲线参数 \alpha ， d ， {S_{r\rm{W}}}{\text{ }} ， {S_{r\rm{N}}}{\text{ }} ，如表 1所示[13, 17]。
表 1 非饱和多孔介质材料参数Table 1. Material parameters of unsaturated porous medium孔隙率 {n_0} \rho _0^{\rm S}/(kg·m-3) \rho _0^{\rm W}/(kg·m-3) \rho _0^{\rm N}/(kg·m-3) {K_\rm{S}}/\rm kPa 0.36 2650 1000 1.1 3.6×107 {K_\rm{W}}/\rm kPa {K_\rm{N}}/\rm kPa {\nu ^{\rm W}} /(\rm Pa \cdot s) {\nu ^{\rm N}} /(\rm Pa \cdot s) {G_0}/\rm kPa 2×106 110 1.0×10-3 1.8×10-5 9.23×107 \widehat K{\text{/}} kPa \kappa /(\rm m \cdot {\rm s^{ - 1}} ) {\alpha _\text{B}} {S_{r\rm{W}}}{\text{ }} {S_{r\rm{N}}}{\text{ }} 2.0 \times {10^5} 2.5 \times {10^{ - 12}} 1 0.05 1 \upsilon d \alpha \phi ' /(°) 0.3 2 2.0 \times {10^{ - 5}} 20 取衬砌外内半径之比为 \vartheta = 1.1 ，分别计算 \eta 为0.25，1.0，2.0情况下饱和度为Sr为0.2，0.6，0.8，1.0时衬砌外壁动应力集中因子分布情况。图 3，4分别给出了P波和SV波入射时衬砌洞室周边动力集中因子分布曲线。从图中可以看出，P波和SV波入射时衬砌周边的应力均对称分布在入射波两侧，饱和介质与非饱和介质中动应力分布有明显差异。P波入射时，饱和度变化对动应力分布形状影响较大，饱和时动应力分布形状与非饱和状态时有显著不同；SV波入射时，饱和度变化对动应力分布形状影响较小，饱和时动应力大于非饱和状态。P波入射下，当 \eta =0.25时，Sr < 1.0动应力集中因子的最大值随饱和度的增加逐渐增大，在Sr为0.2，0.6时分布形式较一致呈“8”字形，最大值出现在80°，280°；在Sr=0.8时图像分布形式变为3个凸起，最大值出现在70°，180°和290°。Sr=1.0时动应力集中因子呈椭圆形沿洞中均匀分布，最大值角度出现在80°，280°。当 \eta =1.0时，Sr < 1.0动应力集中因子最大值随饱和度的增加逐渐增大，动应力集中因子分布呈“豌豆状”，随饱和度增加逐渐饱满。Sr=1.0时动应力集中因子呈椭圆形分布在坐标系的中上部，最大值出现在0°，360°。当 \eta =2.0时，动应力集中因子最大值随饱和度的增加逐渐增大，曲线集中分布在坐标系的上半部分，随饱和度增加最大值出现位置逐渐向0°坐标轴靠拢。P波入射时，最小值位置出现在入射面和入射背面，基本不受饱和度和入射频率的影响。
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图 3 P波入射时考虑饱和度对剪切模量的影响下衬砌洞室周边动力集中因子分布曲线Figure 3. Curves of dynamic stress concentration factors around lining cavity considering influences of saturation on shear modulus under incidence of P-waves![]()
图 4 SV波入射时考虑饱和度对剪切模量的影响下衬砌洞室周边动力集中因子分布曲线Figure 4. Curves of dynamic stress concentration factors around lining cavity considering influences of saturation on shear modulus under SV-waves由于P波与SV波入射的振动形式不同，衬砌洞室周边动应力集中因子分布不同，在SV波入射时动应力集中因子呈蝶状分布。对比图 3，4可以发现不同相对频率下，饱和时动应力集中因子最大值要大于入射 {P_{}} 波时的动应力集中因子。在图 4中可以发现，SV波入射时，饱和与非饱和情况存在明显差异，相比于Sr=1.0，Sr < 1.0时环向应力数值更小。在Sr < 1.0时饱和度对动应力集中因子影响较小，但随相对频率变化动应力集中因子变化明显，整体上随着入射频率的增加衬砌周边动应力逐渐增加，由于Sr=1.0与Sr < 1.0增幅不一致，在相对频率为2.0时饱和与非饱和动应力集中因子最接近。Sr < 1.0时最大值出现角度不变为40°，320°；Sr=1时最大值出现角度变化为50°，310°；30°，330°；20°，340°。 SV 波入射时最小值出现在轴线上。
4. 结论
推导了平面P波和SV波入射下非饱和多孔介质中内嵌圆柱形衬砌洞室的动力响应解析解。考虑了饱和度对剪切模量的影响，求解并分析了饱和度和频率对圆柱形衬砌洞室动应力集中因子的影响，得到3点结论。
（1）饱和度对动应力集中因子的影响不容忽视，无论P波还是SV波入射，饱和时动应力分布与非饱和时有显著不同。且P波入射时，饱和度对动应力集中因子的分布形式影响较大，随着饱和度增加，动应力因子最大值出现角度逐渐向0°偏移，动应力集中因子最大值逐渐增加；SV波入射下，饱和度小于1时，饱和度对动应力集中因子影响较小。
（2）由于P波与SV波的振动形式不同，衬砌洞室周边动应力集中因子分布不同。P波入射时，动应力集中因子分布更多呈椭圆形或“豌豆”状；SV波入射时，动应力集中因子分布呈蝶状。
（3）入射频率主要影响动应力集中因子的大小。随着入射频率的增加，P波和SV波入射下洞室周边动应力集中因子最大值均逐渐增加。
致谢: 感谢土力学和岩土工程界各位同行的信任，让笔者有幸成为今年黄文熙讲座的主讲人。笔者自1989年师从于顾晓鲁教授，开始了基坑工程领域的学习，建立了对岩土工程浓厚的兴趣，并持续至今，积累了一些粗浅的认识和工程经验。感谢团队刁钰副教授、程雪松副教授、周海祚副教授、张天奇副研究员、雷华阳教授、刘畅副教授，以及笔者的学生杜一鸣博士、李志伟博士、曾超峰副教授、魏少伟博士、刘景锦博士等对本文提供的巨大帮助！感谢笔者的博士生苏奕铭、黄建友、栗晴瀚、何晓佩、焦陈磊、甘伟等，他们对本文也提供了很多具体帮助。感谢深圳市工勘岩土集团有限公司雷斌先生为本文提供了三级支护工程图片。 -
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图 3 某工程中群孔效应引发的周边建筑物沉降
Figure 3. Settlements of adjacent buildings induced by group borehole effects
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图 5 单孔和群孔成孔引起地表沉降
Figure 5. Ground surface settlements induced by single and group borehole
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图 6 多孔合并前后地表曲线对比
Figure 6. Comparison of ground surface settlements with and without simplification of group borehole
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图 10 部分空孔回填控制群孔效应影响
Figure 10. Filling of partial boreholes to control group borehole effects
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图 12 基坑预降水引起地下连续墙变形
Figure 12. Wall deflections induced by pre-dewatering of excavations
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图 13 基坑预降水引起地下连续墙变形和建筑物沉降
Figure 13. Wall deflections induced by pre-dewatering of excavations
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图 14 某大面积基坑降水井及监测点平面布置
Figure 14. Plan of dewatering wells and field monitoring paints
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图 15 某大面积基坑预降水过程中围护结构变形情况
Figure 15. Wall deflections induced by pre-dewatering of excavations
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图 16 考虑预降水4个效应的变形计算模型
Figure 16. Deformation prediction model considering 4 effects of pre-excavation dewatering
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图 17 承压层抽水引发土体变形发展规律
Figure 17. Prediction model for deformation considering 4 effects of pre-dewatering of excavations
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图 18 基坑水平支撑平面和支护桩侧移变形监测点
Figure 18. Plan of structs and monitoring points of lateral displacements of retaining piles
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图 20 不同变形模式下坑外地表土体位移对比
Figure 20. Comparison of ground surface deformations under different lateral deformation modes of retaining wall
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图 21 围护结构不同变形模式下坑外深层土体沉降对比
Figure 21. Comparison of ground deformations behind retaining wall under different lateral deformation models of retaining wall
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图 24 纵墙墙体拉应变最大值变化曲线
Figure 24. Variation of maximum tensile strain of longitudinal wall
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图 25 纵墙墙体拉应变最大值随角度变化曲线
Figure 25. Relationship between maximum tensile strain of longitudinal wall and arbitrary angle
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图 26 围护结构为内凸型模式时坑外不同位置处隧道变形
Figure 26. Deformations of tunnels at different locations caused by convex deformation of retaining structures
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图 27 围护结构不同变形模式下隧道变形影响区
Figure 27. Influenced zones determined by different profiles of deflection of retaining structures
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图 28 某实际工程隧道渗漏引发的沉降和错台
Figure 28. Settlements and dislocation of tunnel segments induced by leakage of water and soils
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图 31 隧道底部不同距离两点渗漏模拟试验
Figure 31. Results of two-leakage-point tests with different spacings
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图 32 不同砂土–黏土界面位置时隧道底部渗漏试验
Figure 32. Large-scale model tests considering position of sand-clay interface relative to tunnel
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图 36 多种保护措施下左线隧道水平位移对比
Figure 36. Comparison of tunnel displacements with different types of protection measures
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图 39 一期、二期基坑开挖时地铁结构Y4测点的水平位移
Figure 39. Horizontal displacements of metro structures at Y4 during 1st and 2nd stages of excavation
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图 40 主动控制的关键区域土体
Figure 40. Key soil zone to control deformation of structures to be protected
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图 41 袖阀管注浆对土体水平变形影响的试验布置图
Figure 41. Field tests on effect of TAM grouting on lateral displacement of soils
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图 42 注浆量及注浆距离对土体侧向变形的影响
Figure 42. Effects of grouting volume and distance on lateral displacement of soils
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图 43 超孔压及A点土体位移随时间发展曲线
Figure 43. Development of excess pore water pressure and horizontal displacement at point A with time
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图 44 袖阀管注浆对隧道位移控制现场试验
Figure 44. Field tests on effects of TAM grouting on control of deformation of tunnels
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图 45 隧道水平位移、水平收敛及随时间的变化规律
Figure 45. Development of horizontal displacement and convergence of tunnels with time due to TAM grouting
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图 47 注浆引起的隧道水平位移增量和水平收敛增量
Figure 47. Increments and convergence increments of horizontal displacement of tunnels caused by TAM grouting
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图 48 第一次注浆前后地铁隧道结构的水平位移
Figure 48. Horizontal displacements of metro structures before and after 1st TAM grouting
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图 49 工况4中4次注浆前后隧道的水平位移
Figure 49. Horizontal displacements of tunnel before and after 4 times of TAM grouting for case 4
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图 51 珠海成层土中袖阀管注浆引起土体水平位移
Figure 51. Horizontal displacements due to TAM grouting in stratified soils in Zhuhai
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图 53 珠海成层土中囊体扩张引起土体水平位移
Figure 53. Horizontal displacements due to capsule grouting in stratified soils in Zhuhai
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图 54 天津成层土中囊体扩张引起土体水平位移
Figure 54. Horizontal displacements due to capsule grouting in stratified soils in Tianjin
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图 55 囊体扩张对桩侧向变形控制现场试验
Figure 55. Field tests on lateral deformation of piles due to capsule expansion
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图 56 隧道与基坑关系及试验平、剖面布置图
Figure 56. Plan and profile of field test tunnel and excavation
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图 58 囊体膨胀主动控制前后隧道Z1测点水平位移
Figure 58. Increments of lateral displacement of tunnel at Z1
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图 60 基坑外各含水层典型观测井水位变化曲线
Figure 60. Variation of water level in aquifers during and after dewatering
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图 61 第Ⅰ微承压含水层回灌时各含水层水位变化曲线
Figure 61. Variation of water level in aquifer due to recharge of artesian aquifer Ⅰ
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图 62 基坑内开始抽水后对第Ⅰ微承压含水层回灌时各含水层水位变化曲线
Figure 62. Variation of water-level in aquifer due to recharge of artesian aquifer Ⅰ after commencement of dewatering inside diaphragm
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图 64 考虑反压土作用的悬臂支护分析模型
Figure 64. Analysis model for cantilever retaining piles considering effects of earth berm
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图 69 多级支护3种破坏模式与多级支护宽度关系
Figure 69. Failure modes of multi-level retaining excavations with respect to width
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图 71 砂土中竖直桩、倾斜桩、斜直组合支护桩模型试验
Figure 71. Model tests on inclined retaining piles in sand
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图 72 不同支护结构直桩桩身变形
Figure 72. Wall deformation of vertical wall for different retaining structures
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图 74 内撑式和无支撑支护结构变形对比
Figure 74. Comparison of deformations of strut-free inclined retaining piles and braced vertical retaining piles
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图 75 倾斜桩无支撑支护结构形式
Figure 75. Comparison of bending moments of strut-free inclined retaining piles and braced vertical retaining piles
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图 77 倾斜桩稳定破坏模式与倾斜角的关系
Figure 77. Variation of failure mode of inclined retaining piles with respect to angle of inclination
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图 81 7种试验工况极限挖深对比
Figure 81. Ultimate depths of excavations with different types of retaining structures
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图 86 不同支护形式的坑内土体隆起
Figure 86. Uplifts of soils in excavations with different strut forms
表 1 土层物理和力学指标
Table 1 Physical and mechanical parameters of soils
层号 土层 层厚/m \gamma
/(kN·m-3)w
/%e \varphi
/(°)c
/kPa① 人工填土 3.66 17.5 — — 10.0 8.0 ②1 淤泥质砂土 7.97 20.0 17.9 0.549 22.6 — ②2 淤泥 8.80 15.2 77.8 2.079 1.5 2.1 ②3 黏土 3.86 18.0 32.3 0.977 17.1 21.4 ②4 淤泥质土 12.16 16.4 53.7 4.670 6.6 7.5 ②5 粗砂 8.00 20.2 15.2 0.504 29.1 — 
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