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 同济大学学报(自然科学版)  2017, Vol. 45 Issue (9): 1290-1297.  DOI: 10.11908/j.issn.0253-374x.2017.09.006 0

引用本文

GAO Guangyun, NIE Chunxiao, ZHANG Haiqiu, LEI Dan. Radial Consolidation Solution of Plastic Wick Drain Combined Vacuum Preloading[J]. Journal of Tongji University (Natural Science), 2017, 45(9): 1290-1297. DOI: 10.11908/j.issn.0253-374x.2017.09.006.

文章历史

1. 同济大学 土木工程学院, 上海 200092;
2. 同济大学 岩土及地下工程教育部重点实验室, 上海 200092;
3. Center for Geomechanics and Railway Engineering, Faculty of Engineering, University of Wollongong, Australia

GAO Guangyun1,2, NIE Chunxiao1,2, ZHANG Haiqiu3, LEI Dan1,2
1. College of Civil Engineering, Tongji University, Shanghai 200092, China;
2. Key Laboratory of Geotechnical and Underground Engineering of the Minisitry of Education, Tongji University, Shanghai 200092, China;
3. Center for Geomechanics and Railway Engineering, Faculty of Engineering, University of Wollongong, Australia
Abstract: A system of radial consolidation by combining the plastic wick drain with the vacuum preloading is one of the most popular way for soft ground improvement, which can not only reduce the height of surcharge preloading on dam, but also reduce the lateral displacement of soil. The consolidation solution of plastic wick drains and vacuum preloading based on the nonlinear relationship can well forecast and analyze this kind of ground improvement. In the process of analytical solution, the relationship between the void ratio and the mean effective stress, as well as the horizontal permeability coefficient under one-dimensional consolidation is considered to be semi logarithmic linear, rather than previously thought one-dimensional linear. Because of disturbing on soil around PVD in smear zone, permeability around PVD in the zone distributes in parabolic form. Besides, the effect of vacuum preloading on consolidation is taken into consideration. This analytical solution is verified through degradation method and compared with test results. Finally, the effects of κ, Ck/Cc are analyzed on consolidation. The results indicate that consolidation rate increases with the increasing of Ck/Cc and decreasing of κ.

1 径向固结基本方程

 图 1 塑料排水板平面布置图 Fig.1 Layout of plastic wick drain
 图 2 塑料排水板工作原理图 Fig.2 Working principle of plastic wick drain

 ${k_{\rm{s}}}\left( r \right) = {k_0}\left( {\kappa - 1} \right)\left( {A - B + C\frac{r}{{{r_{\rm{w}}}}}} \right)\left( {A + B - C\frac{r}{{{r_{\rm{w}}}}}} \right)$ (1)

HANSBO[8]的解析解被广泛认可，据此进行径向排水固结方程求解.塑料排水板的排水速率为

 ${q_1} = 2\pi rv\left( {{\rm{d}}z} \right)$ (2)

 ${q_2} = \pi \left( {r_e^2 - {r^2}} \right)\frac{{\partial \varepsilon }}{{\partial t}}\left( {{\rm{d}}z} \right)$ (3)

 $2\pi rv = \pi \left( {r_e^2 - {r^2}} \right)\left( {\frac{{\partial \varepsilon }}{{\partial t}}} \right)$ (4)

 $\frac{{\partial \varepsilon }}{{\partial t}} = - {m_{\rm{v}}}\frac{{\partial \bar u}}{{\partial t}}$ (5)

 $v' = {k_{\rm{s}}}\frac{1}{{{\gamma _{\rm{w}}}}}\frac{{\partial {u_{\rm{s}}}}}{{\partial r}};{r_{\rm{w}}} \le r \le {r_{\rm{s}}}$ (6)
 $v = {k_{\rm{h}}}\frac{1}{{{\gamma _{\rm{w}}}}}\frac{{\partial u}}{{\partial r}};{r_{\rm{s}}} \le r \le {r_{\rm{e}}}$ (7)

 $\begin{array}{*{20}{c}} {\frac{{\partial {u_{\rm{s}}}}}{{\partial r}} = - \frac{\kappa }{{\kappa - 1}}\frac{{\partial \bar u}}{{\partial t}}\frac{1}{{2{c_{\rm{h}}}}} \cdot }\\ {\frac{1}{{\left( {A - B + {C_{\rm{r}}}/{r_{\rm{w}}}} \right)\left( {A + B - {C_{\rm{r}}}/{r_{\rm{w}}}} \right)}}\left( {\frac{{r_{\rm{e}}^2 - {r^2}}}{r}} \right)} \end{array}$ (8)
 $\frac{{\partial u}}{{\partial r}} = - \frac{{\partial \bar u}}{{\partial t}}\frac{1}{{2{c_{\rm{h}}}}}\left( {\frac{{r_{\rm{e}}^2 - {r^2}}}{r}} \right)$ (9)

 $\begin{array}{l} {u_{\rm{s}}} = - \frac{\kappa }{{\kappa - 1}}\frac{{\partial \bar u}}{{\partial t}}\frac{1}{{2{c_{\rm{h}}}}}\frac{1}{{2A}} \cdot \\ \left( \begin{array}{l} D\ln \left( {r/{r_{\rm{w}}}} \right) + E\ln \left( {\frac{{A - B + C\frac{r}{{{r_{\rm{w}}}}}}}{{A - B + C}}} \right) + \\ F\ln \left( {\frac{{A + B - C\frac{r}{{{r_{\rm{w}}}}}}}{{A + B - C}}} \right) \end{array} \right) + {\left( {{u_{\rm{s}}}} \right)_{r = {r_{\rm{w}}}}} \end{array}$ (10)
 $u = - \frac{{\partial \bar u}}{{\partial t}}\frac{1}{{2{c_{\rm{h}}}}}\left( {r_{\rm{e}}^2\ln \left( {r/{r_{\rm{w}}}} \right) - \frac{{{r^2}}}{2} + G} \right) + {\left( {{u_{\rm{s}}}} \right)_{r = {r_{\rm{w}}}}}$ (11)

 $D = \frac{{2Ar_{\rm{e}}^2}}{{{A^2} - {B^2}}};$
 $E = \frac{{\left( {{A^2} + 2AB + {B^2}} \right)r_{\rm{w}}^2 - {C^2}r_{\rm{e}}^2}}{{{C^2}\left( {A - B} \right)}};$
 $F = \frac{{\left( {{A^2} + 2AB + {B^2}} \right)r_{\rm{w}}^2 - {C^2}r_{\rm{e}}^2}}{{{C^2}\left( {A + B} \right)}};$
 $\begin{array}{l} G = - \left( {r_{\rm{e}}^2\ln s - r_{\rm{w}}^2\frac{{{s^2}}}{2}} \right) + \\ \frac{\kappa }{{\kappa - 1}}\frac{1}{{2A}}\left[ \begin{array}{l} D\ln s + E\ln \left( {\frac{{A - B + Cs}}{{A - B + C}}} \right) + \\ F\ln \left( {\frac{{A + B - Cs}}{{A + B - C}}} \right) \end{array} \right]; \end{array}$

 $\begin{array}{l} \pi \left( {r_{\rm{e}}^2 - r_{\rm{w}}^2} \right)\bar ul = \\ 2\pi \int_0^l {\int_{{r_{\rm{w}}}}^{{r_{\rm{s}}}} {r{u_{\rm{s}}}\left( {r,z} \right){\rm{d}}r{\rm{d}}z} } + 2\pi \int_0^l {\int_{{r_{\rm{s}}}}^{{r_{\rm{e}}}} {ru\left( {r,z} \right){\rm{d}}r{\rm{d}}z} } \end{array}$ (12)

 $\bar u = \frac{2}{{\left( {{N^2} - 1} \right)l}}\left[ {\int_0^l {\int_1^s {y{u_{\rm{s}}}\left( y \right){\rm{d}}y{\rm{d}}z} } + \int_0^l {\int_s^N {yu\left( y \right){\rm{d}}r{\rm{d}}z} } } \right]$ (13)

 $\bar u = - \frac{{\partial \bar u}}{{\partial t}}\frac{1}{{2{c_{\rm{h}}}}}\alpha - {p_0}\frac{{1 + {k_l}}}{2}$ (14)

 $\alpha = \frac{{2r_{\rm{w}}^2}}{{{r_{\rm{e}}} - r_{\rm{w}}^2}}\left[ {\left( {\frac{\kappa }{{\kappa - 1}}\frac{1}{{2A}}} \right)\left( {Da + Eb + Fc} \right) + d} \right]$ (15)

 $a = \frac{1}{2}\left[ {{s^2}\left( {\ln s - \frac{1}{2}} \right)} \right] + \frac{1}{4}$
 $\begin{array}{*{20}{c}} {b = \ln \left( {\frac{{A - B + Cs}}{{A - B + C}}} \right)\left( {\frac{{{s^2}}}{2} - \frac{{{{\left( {A - B} \right)}^2}}}{{2{C^2}}}} \right) - }\\ {\frac{{{s^2}}}{4} + \frac{{\left( {s - 1} \right)\left( {A - B} \right)}}{{2C}} + \frac{1}{4}} \end{array}$
 $\begin{array}{*{20}{c}} {c = \ln \left( {\frac{{A + B - {C_{\rm{s}}}}}{{A + B - C}}} \right)\left( {\frac{{{s^2}}}{2} - \frac{{{{\left( {A + B} \right)}^2}}}{{2{C^2}}}} \right) - }\\ {\frac{{{s^2}}}{4} - \frac{{\left( {s - 1} \right)\left( {A + B} \right)}}{{2C}} + \frac{1}{4}} \end{array}$
 $\begin{array}{l} d = \frac{1}{8}\left( {4{N^4}\ln N - 3{N^4} + {s^4} + 2{N^2}{s^2} - } \right.\\ \;\;\;\;\;\;\left. {4{N^2}{s^2}\ln s} \right) + E\left( {\frac{{{N^2} - {s^2}}}{2}} \right) \end{array}$

CHU等[14]进行了两组现场试验，试验土样为两层，第一层土厚4~5 m，为软粘土，第二层土厚10~16 m，为海相粘土.试验时通过排水板施加真空预压，发现随着深度增加，真空负压有近似线性减小、孔压近似线性增加的规律.据此原理，INDRARATNA等[15]认为真空预压随深度的分布如图 3所示.可知土体在排水边界处的孔压为

 图 3 模型分析中真空度的分布情况 Fig.3 Distribution of vacuum pressure in analytical model
 ${\left( {{u_{\rm{s}}}} \right)_{r = {r_{\rm{w}}}}} = - {p_0}\left[ {1 - \left( {1 - {k_1}} \right)\frac{z}{l}} \right]$ (16)

2 土体的本构方程

 图 4 有效应力、渗透系数与孔隙比的经验对数线性关系 Fig.4 Empirical linear relationship between effective stress, permeability coefficient and void ratio
 $\begin{array}{l} e = {e_0} - {C_{\rm{c}}}\log \left( {\frac{{\sigma '}}{{{{\sigma '}_0}}}} \right)\\ e = {e_0} + {C_{\rm{k}}}\log \left( {\frac{k}{{{k_0}}}} \right) \end{array}$ (17)

 $W = \frac{{\bar u}}{{\Delta \sigma}}\;\;\left( {0 \le W \le 1} \right)$ (18)

 $\sigma ' = {{\sigma '}_0} + \Delta \sigma - W\Delta \sigma$ (19)

 ${m_{\rm{v}}} = \frac{1}{{1 + {e_0}}}\frac{{\partial e}}{{\partial \sigma '}}$ (20)

 ${m_{\rm{v}}} = \frac{{ - 0.434{C_{\rm{c}}}}}{{\sigma '\left( {1 + {e_0}} \right)}}$ (21)

 $\frac{{{m_{v0}}}}{{{m_{\rm{v}}}}} = \frac{{\sigma '}}{{{{\sigma '}_0}}}$ (22)

 $\frac{k}{{{k_0}}} = {\left( {\frac{{\sigma '}}{{{{\sigma '}_0}}}} \right)^{ - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}}}$ (23)

 ${c_{\rm{h}}} = \frac{{{k_{\rm{h}}}}}{{{m_{\rm{v}}}{\gamma _{\rm{w}}}}}$ (24)

 $\frac{{{c_{\rm{h}}}}}{{{c_{{\rm{h0}}}}}} = {\left( {1 + \frac{{\Delta \sigma }}{{{{\sigma '}_0}}} - \frac{{W\Delta \sigma }}{{{{\sigma '}_0}}}} \right)^{1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}}}$ (25)

3 方程的求解

 $\partial T = - \frac{{\partial W}}{{W + P}}\frac{{{c_{{\rm{h0}}}}}}{{{c_{\rm{h}}}}}$ (26)

Y=W+P，得

 $\begin{array}{*{20}{c}} {\frac{1}{{W + P}}{{\left( {1 + \frac{{\Delta \sigma }}{{{{\sigma '}_0}}} - W\frac{{\Delta \sigma }}{{{{\sigma '}_0}}}} \right)}^{ - \left( {1 - \frac{{{C_{\rm{r}}}}}{{{C_{\rm{k}}}}}} \right)}} = }\\ {\frac{1}{Y}\left( {1 + \frac{{\Delta \sigma }}{{{{\sigma '}_0}}} + P\frac{{\Delta \sigma }}{{{{\sigma '}_0}}} - Y\frac{{\Delta \sigma }}{{{{\sigma '}_0}}}} \right)} \end{array}$ (27)

 $\begin{array}{*{20}{c}} {\frac{{{c_{\rm{h}}}}}{{Y{c_{{\rm{h0}}}}}} = {{\left( {1 + \frac{{\Delta \sigma }}{{{{\sigma '}_0}}} + P\frac{{\Delta \sigma }}{{{{\sigma '}_0}}}} \right)}^{ - \left( {1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}} \right)}} \cdot }\\ {\sum\limits_{j = 0}^\infty {\frac{{{{\left\{ {1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}} \right\}}_j}}}{{j!}}{{\left( {1 + P + \frac{{{{\sigma '}_0}}}{{\Delta \sigma }}} \right)}^{ - j}}} {{\left( Y \right)}^{j - 1}}} \end{array}$ (28)
 $\begin{array}{*{20}{c}} {T\left( Y \right) = - {{\left( {1 + \frac{{\Delta \sigma }}{{{{\sigma '}_0}}} + P\frac{{\Delta \sigma }}{{{{\sigma '}_0}}}} \right)}^{ - \left( {1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}} \right)}} \cdot }\\ {\left[ \begin{array}{l} \ln \left( {\frac{Y}{{1 + P}}} \right) + \left( {1 - \frac{{{C_{\rm{c}}}}}{{{C_{\rm{k}}}}}} \right){\left( {1 + \frac{{{{\sigma '}_0}}}{{\Delta \sigma }} + P} \right)^{ - 1}} \cdot \\ \left( {Y - \left( {1 + P} \right)} \right) + \sum\limits_{j = 2}^\infty {\frac{{{{\left\{ {1 - \frac{{{C_{\rm{r}}}}}{{{C_{\rm{k}}}}}} \right\}}_j}}}{{j!\left( {j - n + 1} \right)}} \cdot } \\ {\left( {1 + \frac{{{{\sigma '}_0}}}{{\Delta \sigma }}} \right)^{ - j}}\left( {{Y^j} - {{\left( {1 + P} \right)}^j}} \right) \end{array} \right]} \end{array}$ (29)

 $U = \frac{{1 - W}}{{1 - {W_\infty }}} \times 100$ (30)

4 退化验证

 图 5 N和kl不同取值下与INDRARATNA(2005) 解析解退化验证 Fig.5 Degradation analysis of the obtained solutions with different values of N and kl

 图 6 N、k和s不同取值下与WALKER(2006) 解析解退化验证 Fig.6 Degradation analysis of the obtained solutions under different values of N, k and s

WALKER解[17]可考虑孔隙比与有效应力和渗透系数的非线性关系，但无法体现WALKER解[12]中涂抹区渗透系数抛物线分布特性，本文对此不足进行改进.与WALKER解[17]对比验时，退化涂抹区渗透系数抛物线分布影响这一特性，仅考虑Ck/Cc的影响.对比结果如图 7所示.为了能够证明本文解的广泛适用性，分别取N为9和19，Ck/Cc分别取为1.55与2.93.退化涂抹区渗透系数后的结果与Walker解[17]结果吻合性非常好，说明本文解析解具有可以考虑孔隙比与有效应力和渗透系数的非线性关系对固结的影响这一优点.

 图 7 N和Ck/Cc不同取值下与Walker(2012) 解析解退化验证 Fig.7 Degradation analysis of the obtained solutions with different values of N and Ck/Cc
5 试验验证

 ${U_{{\rm{hs}}}} = \frac{\rho }{{{\rho _\infty }}}$ (31)

 $\rho = \frac{{H{C_{\rm{c}}}}}{{1 + {e_0}}}\log \left[ {1 + \frac{{\Delta \sigma \left( {1 - W} \right)}}{{{{\sigma '}_0}}}} \right]$ (32)

 ${U_{{\rm{hs}}}} = \frac{{\log \left[ {1 + \frac{{\Delta \sigma }}{{{{\sigma '}_0}\left( {1 - W} \right)}}} \right]}}{{\log \left[ {1 + \left( {\frac{{\Delta \sigma }}{{{{\sigma '}_0}}}} \right)} \right]}}$ (33)

 图 8 试验固结与解析解固结随时间变化图 Fig.8 Consolidation rate comparison of test and analytical solution
6 模型的应用与参数分析

 图 9 k与Cc/Ck不同取值对固结速率的影响 Fig.9 Variation of consolidation rate with different k and Cc/Ck values

7 结论

(1) 本文推导的解析解能够综合考虑真空预压、涂抹区水平渗透系数的抛物线分布、孔隙比与水平渗透系数和有效应力的非线性关系对固结的影响.

(2) 通过退化，本文的解析解分别与INDRARATNA解、WALKER解进行了对比验证，证明了该解析解的正确性.

(3) 本文解析解与试验结果对比非常吻合，证明了该解析解的正确性与实用性，可用于塑料排水板与真空预压相结合的软土地基加固计算和预测.

(4) Ck/Cc值越大，固结需要时间越短，因此，Ck/Cc增大有促使固结加快的作用.若施工对涂抹区造成扰动，κ增大，增加固结所需时间，对工程不利.