﻿ 黏土中考虑土体卸荷效应的后注浆压密模型
 哈尔滨工业大学学报  2020, Vol. 52 Issue (11): 71-79  DOI: 10.11918/201902121 0

### 引用本文

WU Yue, ZHAO Chunfeng, WANG Youbao, FEI Yi. Compaction grouting model in clay considering unloading effect[J]. Journal of Harbin Institute of Technology, 2020, 52(11): 71-79. DOI: 10.11918/201902121.

### 文章历史

1. 岩土与地下工程教育部重点实验室(同济大学)，上海 200092;
2. 同济大学 地下建筑与工程系，上海 200092

WU Yue1,2, ZHAO Chunfeng1,2, WANG Youbao1,2, FEI Yi1,2
1. Key Laboratory of Geotechnical and Underground Engineering(Tongji University), Ministry of Education, Shanghai 200092, China;
2. Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China
Keywords: post-grouting    spherical cavity expansion    compaction grouting    pressure distribution    ultimate grouting pressure

1 压密注浆模型 1.1 基本假定

1) 忽略浆液在土体交界面处的渗透效应，认为注浆过程中仅存在浆液对土体的压密效应;

2) 忽略浆液沿结构物表面的扩散运动，注浆体在土体中呈球形扩散，压密注浆过程相当于在土体中扩张一个半径为Ru的球形浆体，如图 1所示，p为注浆压力，R0为球形浆体的初始半径，Rp为土体塑性区最大半径;

 图 1 压密注浆模型 Fig. 1 Model of compaction grouting

3) 土体在初始状态下为均质各向同性体;

4) 土体在受到浆体扩张挤压后发生弹塑性变形，且变形在应力施加后立刻发生;

5) 浆液和土颗粒不可压缩;

6) 忽略重力对土体压缩的影响.

1.2 理论推导

 $\frac{{{\rm{d}}{\sigma _{\rm{r}}}}}{{{\rm{d}}r}} + 2\frac{{{\sigma _{\rm{r}}} - {\sigma _\theta }}}{r} = 0.$ (1)

 $\left\{ {\begin{array}{*{20}{l}} {{\varepsilon _{\rm{r}}} = - \frac{{{\rm{d}}{u_{\rm{r}}}}}{{{\rm{d}}r}},}\\ {{\varepsilon _\theta } = - \frac{{{u_{\rm{r}}}}}{r}.} \end{array}} \right.$ (2)

 $\left\{ {\begin{array}{*{20}{l}} {{\varepsilon _{\rm{r}}} = \frac{{{\sigma _{\rm{r}}} - 2v{\sigma _\theta }}}{E},}\\ {{\varepsilon _\theta } = \frac{{(1 - v){\sigma _\theta } - v{\sigma _{\rm{r}}}}}{E}.} \end{array}} \right.$ (3)

 ${\sigma _r}{|_{r = {R_{\rm{u}}}}} = p,{\sigma _{\rm{r}}}{|_{r \to \infty }} = {p_0}.$ (4)

 $\left\{ {\begin{array}{*{20}{l}} {{\sigma _{\rm{r}}} = {p_0} + (p - {p_0}) \cdot {{\left( {\frac{{{R_{\rm{u}}}}}{r}} \right)}^3},}\\ {{\sigma _\theta } = {p_0} - \frac{{(p - {p_0})}}{2} \cdot {{\left( {\frac{{{R_{\rm{u}}}}}{r}} \right)}^3},}\\ {{u_r} = \frac{{(1 + v)(p - {p_0})r}}{{2E}} \cdot {{\left( {\frac{{{R_{\rm{u}}}}}{r}} \right)}^3},} \end{array}} \right.$ (5)
1.3 弹塑性应力分析

 $F = ({\sigma _{\rm{r}}} - {\sigma _\theta }) - ({\sigma _{\rm{r}}} + {\sigma _\theta }){\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi - 2c{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi = 0.$ (6)

 ${{\sigma _{\rm{r}}} - \alpha {\sigma _\theta } = Y,}$ (7)

 ${\alpha = \frac{{1 + {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{1 - {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }},Y = \frac{{2c{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }}{{1 - {\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi }},}$ (8)

p=py时，浆体边缘土体首先出现屈服，根据屈服准则式(6)和式(5)得土体进入塑性状态的临界压力为

 ${p_y} = \frac{{2[Y + (\alpha - 1){p_0}]}}{{2 + \alpha }} + {p_0}.$ (9)

 $\left\{ {\begin{array}{*{20}{l}} {{\sigma _{\rm{r}}} = {p_0} + ({p_y} - {p_0}) \cdot {{\left( {\frac{{{R_{\rm{p}}}}}{r}} \right)}^3},}\\ {{\sigma _\theta } = {p_0} - \frac{{({p_y} - {p_0})}}{2} \cdot {{\left( {\frac{{{R_{\rm{p}}}}}{r}} \right)}^3},}\\ {{u_{\rm{r}}} = \frac{{(1 + v)({p_y} - {p_0})r}}{{2E}} \cdot {{\left( {\frac{{{R_{\rm{p}}}}}{r}} \right)}^3}.} \end{array}} \right.$ (10)

 $\left\{ {\begin{array}{*{20}{l}} {{\sigma _{\rm{r}}} = \frac{Y}{{1 - \alpha }} + A \cdot {r^{\frac{{2(1 - \alpha )}}{\alpha }}},}\\ {{\sigma _\theta } = \frac{Y}{{1 - \alpha }} + \frac{A}{\alpha } \cdot {r^{\frac{{2(1 - \alpha )}}{\alpha }}}.} \end{array}} \right.$ (11)

 $A = \left( {p - \frac{Y}{{1 - \alpha }}} \right) \cdot R_{\rm{u}}^{\frac{{2(\alpha - 1)}}{\alpha }} = \left( {{p_y} - \frac{Y}{{1 - \alpha }}} \right) \cdot R_{\rm{p}}^{\frac{{2(\alpha - 1)}}{\alpha }}.$ (12)

 $\left\{ {\begin{array}{*{20}{l}} {{\sigma _{\rm{r}}} = \frac{Y}{{1 - \alpha }} + \left( {{p_y} - \frac{Y}{{1 - \alpha }}} \right) \cdot {{\left( {\frac{{{R_{\rm{p}}}}}{r}} \right)}^{\frac{{2(\alpha - 1)}}{\alpha }}},}\\ {{\sigma _\theta } = \frac{Y}{{1 - \alpha }} + \frac{1}{\alpha } \cdot \left( {{p_y} - \frac{Y}{{1 - \alpha }}} \right) \cdot {{\left( {\frac{{{R_{\rm{p}}}}}{r}} \right)}^{\frac{{2(\alpha - 1)}}{\alpha }}}.} \end{array}} \right.$ (13)

 $\beta = \frac{{{R_{\rm{p}}}}}{{{R_{\rm{u}}}}} = {\left[ {\frac{{(\alpha - 1)p + Y}}{{(\alpha - 1){p_y} + Y}}} \right]^{\frac{\alpha }{{2(\alpha - 1)}}}}.$ (14)
1.4 弹塑性位移分析

r=Rp代入式(10)可得弹塑性边界位移为

 ${u_{{\rm{rp}}}} = \frac{{(1 + v)({p_y} - {p_0})}}{{2E}}{R_{\rm{p}}} = \delta {R_{\rm{p}}}.$ (15)

 $\chi \frac{{4\pi }}{3}(R_{\rm{u}}^3 - R_0^3) = \chi \frac{{4\pi }}{3}({r^3} - r_0^3) + \varDelta .$ (16)

 ${R_{\rm{u}}^3 - R_0^3 = R_{\rm{p}}^3 - {{({R_{\rm{p}}} - {u_{{\rm{rp}}}})}^3},}$ (17)
 ${1 - {{\left( {\frac{{{R_0}}}{{{R_{\rm{u}}}}}} \right)}^3} = {{\left( {\frac{{{R_{\rm{p}}}}}{{{R_{\rm{u}}}}}} \right)}^3}({\delta ^3} - 3{\delta ^2} + 3\delta ).}$ (18)

 $\xi = \frac{{{R_{\rm{u}}}}}{{{R_0}}} = \frac{1}{{\sqrt[3]{{1 - {{\left( {\frac{{{R_{\rm{p}}}}}{{{R_{\rm{u}}}}}} \right)}^3}({\delta ^3} - 3{\delta ^2} + 3\delta )}}}}.$ (19)

 $\zeta = \frac{{{R_{\rm{p}}}}}{{{R_0}}} = \beta \cdot \xi .$ (20)

 ${u_{\rm{r}}} = r - {r_0}.$ (21)

 ${r_0} = r - {u_{\rm{r}}}.$ (22)

 $\chi \frac{4}{3}\pi {({R_{\rm{p}}} - {u_{{r_{\rm{p}}}}})^3} - \chi \frac{4}{3}\pi r_0^3 = \chi \frac{4}{3}\pi R_{\rm{p}}^3 - \chi \frac{4}{3}\pi {r^3}.$ (23)

 ${u_{\rm{r}}} = r - \sqrt[3]{{{r^3} - 3R_{\rm{p}}^2{u_{{r_{\rm{p}}}}} + 3{R_{\rm{p}}}u_{{r_{\rm{p}}}}^2 - u_{{r_{\rm{p}}}}^3}}.$ (24)
1.5 极限注浆压力

 $\begin{array}{l} p = \frac{{[(\alpha - 1){p_y} + Y]{{\left[ {1 - {{\left( {\frac{{{R_0}}}{{{R_u}}}} \right)}^3}} \right]}^{\frac{{2(\alpha - 1)}}{{3\alpha }}}}}}{{\alpha - 1}} \times \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\left( {\frac{1}{{{\delta ^3} - 3{\delta ^2} + 3\delta }}} \right)^{\frac{{2(\alpha - 1)}}{{3\alpha }}}} - \frac{Y}{{\alpha - 1}}. \end{array}$ (25)

 $\begin{array}{*{20}{c}} {{p_{\rm{u}}} = \mathop {{\rm{lim}}}\limits_{{R_{\rm{u}}} \to \infty } p = \frac{{[(\alpha - 1){p_y} + Y]}}{{\alpha - 1}} \times }\\ {{{\left( {\frac{1}{{{\delta ^3} - 3{\delta ^2} + 3\delta }}} \right)}^{\frac{{2(\alpha - 1)}}{{3\alpha }}}} - \frac{Y}{{\alpha - 1}}.} \end{array}$ (26)
1.6 孔周土体弹塑性应力场和位移场

 ${\sigma _{\rm{r}}} = \left\{ {\begin{array}{*{20}{l}} {{p_0} + ({p_y} - {p_0}) \cdot {{\left( {\frac{{{R_{\rm{p}}}}}{r}} \right)}^3},r \ge {R_{\rm{p}}},}\\ {\frac{Y}{{1 - \alpha }} + \left( {{p_y} - \frac{Y}{{1 - \alpha }}} \right) \cdot {{\left( {\frac{{{R_{\rm{p}}}}}{r}} \right)}^{\frac{{2(\alpha - 1)}}{\alpha }}},{R_{\rm{u}}} \le r < {R_{\rm{p}}}.} \end{array}} \right.$ (27)
 ${\sigma _\theta }{\sigma _\theta } = \left\{ {\begin{array}{*{20}{l}} {{p_0} - \frac{{({p_y} - {p_0})}}{2} \cdot {{\left( {\frac{{{R_{\rm{p}}}}}{r}} \right)}^3},r \ge {R_{\rm{p}}},}\\ {\frac{Y}{{1 - \alpha }} + \frac{1}{\alpha } \cdot \left( {{p_y} - \frac{Y}{{1 - \alpha }}} \right) \cdot {{\left( {\frac{{{R_{\rm{p}}}}}{r}} \right)}^{\frac{{2(\alpha - 1)}}{\alpha }}},{R_{\rm{u}}} \le r < {R_{\rm{p}}}.} \end{array}} \right.$ (28)
 ${u_{\rm{r}}} = \left\{ {\begin{array}{*{20}{l}} {\frac{{(1 + v)({p_y} - {p_0})r}}{{2E}} \cdot {{\left( {\frac{{{R_{\rm{p}}}}}{r}} \right)}^3},r \ge {R_{\rm{p}}},}\\ {r - \sqrt[3]{{{r^3} - 3R_{\rm{p}}^2{u_{{\rm{rp}}}} + 3{R_{\rm{p}}}u_{{\rm{rp}}}^2 - u_{{\rm{rp}}}^3}},{R_{\rm{u}}} \le r < {R_{\rm{p}}}.} \end{array}} \right.$ (29)
2 土体卸荷效应影响

 ${E_i} = {C_0}\eta {s_{\rm{u}}}.$ (30)

 图 2 估计初始切线模量方法[18] Fig. 2 Method for estimating the initial tangent elastic modulus[18]

 ${R_{{\rm{OC}}}} = \frac{{{P_{\rm{c}}}}}{{{P_{{\rm{ul}}}}}}.$ (31)

 $\varsigma = \frac{{{P_{\rm{c}}} - {P_{{\rm{ul}}}}}}{{{P_{\rm{c}}}}}.$ (32)

 ${R_{{\rm{OC}}}} = \frac{{{P_{\rm{c}}}}}{{{P_{{\rm{ul}}}}}} = \frac{1}{{1 - \varsigma }}.$ (33)

3 参数确定与模型适用性的讨论 3.1 参数确定

3.2模型适用性

4 实例分析

4.1 不同卸荷程度下球形浆体和土体塑性区扩张率

 图 3 不同卸荷比下浆体扩张率和土体塑性区扩张率与注浆压力关系 Fig. 3 Relation between expansion rates and grouting pressure under different unloading ratios

4.2 不同卸荷程度下塑性区半径与浆体扩散半径关系

 图 4 塑性区半径与浆体扩散半径关系 Fig. 4 Relation between plastic zone radius Rp and grout diffusion radius Ru
4.3 不同卸荷程度下径向和环向应力沿径向分布关系

 图 5 注浆压力370 kPa不同卸荷比下土体径向和环向应力沿径向分布 Fig. 5 Stress distributions along radial direction with grouting pressure of 370 kPa under different unloading ratios

 图 6 同一注浆量不同卸荷比下土体径向和环向应力沿径向分布 Fig. 6 Stress distributions along radial direction with same grouting volume under different unloading ratios

 ${\sigma _\theta } = {p_0} = \frac{{({p_y} - {p_0})}}{2}.$ (34)

4.4 不同卸荷程度下径向位移沿径向分布关系

4.5 不同注浆量下径向、环向应力和径向位移沿径向分布关系

 图 9 卸荷比0.5不同注浆量下土体径向和环向应力沿径向分布 Fig. 9 Stress distributions along radial direction under unloading ratio of 0.5 with different grouting volumes