工程科学与技术   2018, Vol. 50 Issue (2): 32-41
半透水边界砂井地基的真空预压固结解
胡亚元     
浙江大学 滨海和城市岩土工程研究中心,浙江 杭州 310058
基金项目: 国家自然科学基金资助项目(51178419)
摘要: 针对砂井地基顶部存在垫层和底部存在下卧层的现状,采用半透水边界反映其对砂井地基固结的影响。假定地基变形符合等应变条件,考虑水平向渗透系数沿半径发生变化,建立上下边界为半透水边界时真空预压砂井地基的固结理论,通过分离变量法获得固结度和沉降的级数解析解答,分析底部透水系数、顶部透水系数、井阻、渗透系数沿半径分布特性对砂井地基固结度和沉降的影响规律。研究表明:当底部半透水时,最终负孔压和有效应力呈沿地基深度衰减的变化规律。当其他条件相同时,底部透水系数越大的地基固结度越大,但最终沉降和径向固结时间因子相同时的固结沉降却越小。当底部边界不透水时,最终沉降不会随着顶部透水系数改变而改变,但当底部边界半透水或完全透水时,最终沉降随着顶部透水系数增大而增大;顶部透水系数越大的地基固结度越大,径向固结时间因子相同时的固结沉降也越大。均匀井阻的大小不会影响负压最终竖向分布特性,也不会改变最终沉降,但会明显降低砂井地基固结度,延长固结完成时间。渗透系数径向分布模式会对砂井地基的固结度产生影响;其中不考虑涂抹作用的工况1计算的固结度最大,假定涂抹区渗透系数恒定的工况2计算的固结度最小,假定涂抹区内渗透系数向原状土渗透系数线性过渡的工况3计算的固结度介于两者之间。半透水边界对真空预压砂井地基固结过程的影响规律与堆载预压砂井地基相比存在明显不同。
关键词: 砂井地基    真空预压    半透水边界    固结沉降    固结解析理论    
Consolidation Solution for Sand-drained Ground with Impeded Boundaries Under Vacuum Preloading
HU Yayuan     
Research Center of Coastal and Urban Geotechnical Eng.,Zhejiang Univ.,Hangzhou 310058,China
Abstract: Since the cushion generally exists on the top and a substratum below the bottom of sand-drained ground,impeded boundaries were used to describe its effect on consolidation process of sand-drained ground.Assuming that ground deformation satisfies the condition of equal vertical strain and considering the radial variation of horizontal permeability coefficient,consolidation theory was established for sand-drained ground with top and bottom impeded boundaries under vacuum preloading.The series analytical solutions of consolidation degree and settlement were achieved by the separated variable method.The effect of the pervious coefficient of top boundary,the pervious coefficient of bottom boundary,the well resistance and the radial distribution feature of permeability coefficient on the consolidation degree and settlement of sand-drained ground were analyzed.The researches show that the final negative pore pressure and the effective stress exhibit a decrease with increasing depth of ground.When the other conditions are same,the ground with big bottom pervious coefficient has big consolidation degree,while it has small final settlement and the consolidation settlement at same radial consolidation time factor.When the bottom boundary of ground is impervious,the final settlement of ground hardlychange with the top pervious coefficient,but when the bottom boundary of ground is impeded or completely permeable,the final settlement increases with top pervious coefficient.The ground with big top pervious coefficient has big consolidation degree and consolidation settlement at the same radial consolidation time factor.Resistance of sand drain neither affect the final vertical distribution features of negative pressure,nor change the magnitude of the final settlement,while the resistance of sand drain can significantly decreases the consolidation degree of sand-drained ground and extends the finish time of consolidation.The radial distribution mode of permeability coefficient affects the consolidation degree of sand-drained ground.The consolidation degree of mode 1 is the biggest,mode 2 is the smallest,and mode 3 falls between mode 1 and mode 2.The effect laws of impeded boundary on the consolidation process of sand-drained ground under vacuum preloading are obviously different from those under surcharged preloading.
Key words: sand-drained ground    vacuum preloading    impeded boundary    consolidation settlement    analytical theory of consolidation    

采用真空排水预压法加固地基是地基处理的常用方法之一,近年来在软土工程中得到越来越多地应用[110]。为理论指导真空排水预压法加固地基的工程实践,许多岩土专家采用解析法对真空条件下砂井地基的固结过程进行深入研究。董志良[1]借鉴谢康和等[11]提出的砂井固结理论,获得了砂井地基存在负压边界时真空预压解析解。Indraratna等[2]考虑了竖井排水体内真空度沿深度方向的线性衰减,借鉴Hansbo[12]砂井固结理论,获得了只考虑砂井地基径向渗透条件下的固结解析解。彭劼[3]获得了真空荷载随时间变化时砂井地基的固结解析解答;郭彪等[7]获得了考虑水平向渗透系数随径向变化及真空度随深度变化的砂井地基固结度公式。在上述理论分析中,均把砂井地基顶部处理成完全透水边界,把下卧层处理成水头梯度恒定边界(完全不透水[1,4]或固定值[2,6])。但在砂井地基实际工程中,砂垫层(或横向排水体)透水性如不理想或下卧层具有一定透水性时,应将其考虑为半透水边界。有关半透水边界条件下堆载预压砂井地基的固结课题目前已取得较为丰富的研究成果[1318]。李西斌等[13]研究了循坏荷载下半透水边界对1维固结特性的影响;王瑞春等[14]研究了半透水边界对黏弹性砂井地基固结特性的影响;孙举[15]、刘加才[16]和李赞[17]等研究了半透水边界对双层复合地基、双层竖井地基和砂墙地基固结特性的影响;侯丰等[18]采用二元介质损伤模型分析结构性土的堆载预压固结特性。但对于真空预压砂井地基,由于普遍认为半透水边界对真空预压固结过程的影响与堆载预压相同,因此目前还未引起岩土工程界的重视,少有文献报道该方面的研究成果。实际上,由于真空预压引起的超孔隙水压力与堆载预压相比存在正负差别,砂井边界透水特性对真空预压的影响与堆载预压相比存在较大的不同,因此有必要研究半透水边界下真空预压的独特固结规律和沉降特性[4]。为实现这一目的,假定地基符合等应变条件和水平向渗透系数随径向变化,运用偏微分方程中的数学物理方法获得上下边界为半透水边界时真空预压砂井地基的固结解析解答,首次阐明真空预压中造成真空度沿深度衰减的力学根源在于下边界的透水性。

1 等应变条件下的固结解析解 1.1 固结方程

砂井地基单井固结计算简图如图1所示。

图1 砂井地基单井固结计算简图 Fig. 1 Analytical sketch of consolidation for unit cell in sand-drained ground

图1中, $r$ ${\textit{z}}$ 为径向及水平向坐标, $t$ 为时间, $H$ 为软土层层厚, ${r_{{w}}}$ 为砂井半径, ${d_{{w}}}$ 为砂井直径, ${r_{{e}}}$ 为砂井影响区半径, ${d_{{e}}}$ 为砂井影响区直径, ${\gamma _{{w}}}$ 为水容重, $ - {u_0}$ 为作用在砂井地基上边界的负孔隙压力。理论求解砂井地基固结问题所做的基本假定为:

1)等应变假定成立。地基只有竖向变形而无侧向变形,同一深度平面上任意一点的竖向变形相等。

2)有效应力原理成立。饱和土的应力–应变关系符合线弹性本构模型,地基压缩模量为 ${E_{{s}}}$

3)达西定理成立。垂直向渗透系数 ${k_{{v}}}$ 和砂井井料的渗透系数 ${k_{{w}}}$ 为常数,水平向渗透系数沿径向呈 ${k_{{h}}}f(r)$ 变化。其中: ${k_{{h}}}$ 为原状土的水平向渗透系数; $f(r)$ 为涂抹系数,反映砂井施工对水平向渗透系数的影响,为随 $r$ 变化的函数。

4)土区的径竖向固结可分别单独考虑。计算竖向固结时 ${k_{{h}}}$ 取0,计算径向固结时 ${k_{{v}}}$ 取0。综合考虑径竖向的总固结度按Carrillo建议的方法计算。

5)假定砂井地基的上下边界为半透水边界,土区顶部边界条件为:

${\left. {\frac{{\partial {u_{\textit{z}}}}}{{\partial {\textit{z}}}}} \right|_{{\textit{z}} = H}} = - \frac{{{R_{{U}}}}}{H}({u_{\textit{z}}} + {u_0})$ (1)

底部边界条件为:

${\left. {\frac{{\partial {u_{\textit{z}}}}}{{\partial {\textit{z}}}}} \right|_{{\textit{z}} = 0}} = \frac{{{R_{{L}}}}}{H}{u_{\textit{z}}}$ (2)

式中: ${u_{\textit{z}}}$ 为土区只考虑竖向固结时的孔压; ${R_{{U}}}$ ${R_{{L}}}$ 分别为土区顶部和底部透水系数,当 ${R_{{U}}} = {R_{{L}}} = \infty $ 时为透水边界,当 ${R_{{L}}} = 0$ 时底部为不透水边界。

砂井区域的边界条件为:

${\left. {\frac{{\partial {u_{ w}}}}{{\partial {\textit{z}}}}} \right|_{{\textit{z}} = H}} = - \frac{{{R_{{{wU}}}}}}{H}({u_{{w}}} + {u_0})$ (3)
${\left. {\frac{{\partial {u_{ w}}}}{{\partial {\textit{z}}}}} \right|_{{\textit{z}} = 0}} = - \frac{{{R_{{{wL}}}}}}{H}{u_{{w}}} $ (4)

式中: ${u_{{w}}}$ 为砂井排水体内的孔压; ${R_{{{wU}}}}$ ${R_{{{wL}}}}$ 分别为砂井顶部和底部的透水系数,当 ${R_{{{wU}}}} = {R_{{{wL}}}} = \infty $ 时为透水边界,当 ${R_{{{wL}}}} = 0$ 时底部为不透水边界。

6)假定在 $r = {r_{{e}}}$ 处沿径向为不透水边界,根据基本假定4)有:

${\left. {\frac{{\partial {u_{r}}}}{{\partial {r}}}} \right|_{r = {r_{{e}}}}} = 0$ (5)

式中, ${u_r}$ 为土区只考虑径向固结的孔压。

7)假定砂井中孔压沿径向的变化可忽略不计,任一深度 ${\textit{z}}$ 处砂井中向上水流的增量等于土中渗入砂井的流量,由此推导出砂井渗流连续方程为[11]

$\frac{{\partial {}^2{u_{{w}}}}}{{\partial {{\textit{z}}^2}}} = - \frac{{2{k_{{h}}}f({r_{{w}}})}}{{{k_{{w}}}{r_{{w}}}}}{\left. {\frac{{\partial {u_r}}}{{\partial r}}} \right|_{r = {r_{{w}}}}}$ (6)

8)假定土区和砂井中初始孔压为0。根据基本假定4)有:

${\left. {{u_{\textit{z}}}} \right|_{t = 0}} = 0$ (7)
${\left. {{u_{r}}} \right|_{t = 0}} = 0$ (8)

根据假定1)~8),单独考虑竖向固结的固结方程为:

$\frac{{\partial {\varepsilon _{\upsilon 1}}}}{{\partial t}} = - \frac{1}{{{E_{{s}}}}}\frac{{\partial {u_{\textit{z}}}}}{{\partial t}}$ (9)
$ - \frac{{{k_{{v}}}}}{{{\gamma _{{w}}}}}\frac{{{\partial ^2}{u_{\textit{z}}}}}{{\partial {{\textit{z}}^2}}} = \frac{{\partial {\varepsilon _{\upsilon 1}}}}{{\partial t}}$ (10)

式中, ${\varepsilon _{\upsilon 1}}$ 为仅考虑竖向固结时土体的体积应变, ${E_{{s}}}$ 为地基压缩模量。单独考虑径向固结的固结方程为[11]

$\frac{{\partial {\varepsilon _{\upsilon 2}}}}{{\partial t}} = - \frac{1}{{{E_{{s}}}}}\frac{{\partial {{\bar u}_r}}}{{\partial t}}$ (11)
$ - \frac{1}{r}\frac{\partial }{{\partial r}}[\frac{{{k_{{h}}}f(r)}}{{{\gamma _{{w}}}}}r\frac{{\partial {u_r}}}{{\partial r}}] = \frac{{\partial {\varepsilon _{\upsilon 2}}}}{{\partial t}}$ (12)

式中: ${\varepsilon _{\upsilon 2}}$ 为仅考虑径向固结时土体的体积应变; ${\bar u_{ r}}$ 为径向平均孔压,其公式为:

${\bar u_{r}} = \frac{1}{{{\text{π}} (r_{{e}}^2 - r_{{w}}^2)}}\int_{{r_{{w}}}}^{{r_{{{ e}}}}} {2{\text{π}} r{u_{r}}{{d }}r} $ (13)
1.2 方程求解 1.2.1 竖向固结方程的求解

由于式(1)不满足齐次边界方程,为使边界条件齐次化,令

${{\textit{z}}^*} = H - {\textit{z}}$ (14)
$u_{\textit{z}}^* = {u_{\textit{z}}} + ({\alpha _{\textit{z}}} - {\beta _{\textit{z}}}\frac{{{{\textit{z}}^*}}}{H}){u_0}$ (15)

式中:

${\alpha _{\textit{z}}} = \frac{{(1 + {R_{{L}}}){R_{{U}}}}}{{(1 + {R_{{L}}}){R_{{U}}} + {R_{{L}}}}}$ (16)
${\beta _{\textit{z}}} = \frac{{{R_{{L}}}{R_{{U}}}}}{{({R_{{L}}} + 1){R_{{U}}} + {R_{{L}}}}}$ (17)

根据式(9)、(10)、(14)、(15)得竖向固结方程为:

$\frac{{{k_{ v}}{E_{{s}}}}}{{{\gamma _{ w}}}}\frac{{{\partial ^2}u_{\textit{z}}^*}}{{\partial {{\textit{z}}^*}^2}} = \frac{{\partial u_{\textit{z}}^*}}{{\partial t}}$ (18)

根据式(1)、(2)、(14)、(15)得竖向固结的边界条件为:

${\left. {\frac{{\partial u_{\textit{z}}^*}}{{\partial {{\textit{z}}^*}}}} \right|_{{{\textit{z}}^*} = 0}} = \frac{{{R_{{U}}}}}{H}u_{\textit{z}}^*$ (19)
${\left. {\frac{{\partial u_{\textit{z}}^*}}{{\partial {{\textit{z}}^*}}}} \right|_{{{\textit{z}}^*} = H}} = - \frac{{{R_{{L}}}}}{H}u_{\textit{z}}^*$ (20)

根据式(7)和(15)得竖向固结的初始条件为:

${\left. {u_{\textit{z}}^*} \right|_{t = 0}} = ({\alpha _{\textit{z}}} - \frac{{{\beta _{\textit{z}}}}}{H}{{\textit{z}}^*}){u_0}$ (21)

对式(18)~(21)采用分离变量法求解得:

$u_{\textit{z}}^* = {u_0}\sum\limits_{m = 1}^\infty {\frac{{{a_{{\textit{z}}m}}}}{{{G_{{\textit{z}}m}}}}[\frac{{{\lambda _{{\textit{z}}m}}}}{{{R_{{U}}}}}\cos (\frac{{{\lambda _{{\textit{z}}m}}}}{H}{{\textit{z}}^*}) + \sin } (\frac{{{\lambda _{{\textit{z}}m}}}}{H}{{\textit{z}}^*})] \cdot {{ e}^{ - {T_{{v}}}\lambda _{{\textit{z}}m}^2}}$ (22)

式中, ${T_{{v}}} = {c_{\textit{z}}}t/{H^2}$ 为竖向固结时间因子; $\lambda _{{\textit{z}}m}^{} > 0$ 为特征值,按从小到大的顺序由方程(23)确定:

$\tan \; \lambda _{{\textit{z}}m}^{} = \frac{{\lambda _{{\textit{z}}m}^{}({R_{{U}}} + {R_{{L}}})}}{{\lambda _{{\textit{z}}m}^2 - {R_{{U}}}{R_{{L}}}}}$ (23)

${a_{{\textit{z}}m}}$ 按式(24)计算:

$\begin{aligned} {a_{{\textit{z}}m}} = & {\alpha _{\textit{z}}}[\frac{1}{{{R_{{U}}}}}\sin\; {\lambda _{{\textit{z}}m}} + \frac{{1 - \cos\; {\lambda _{{\textit{z}}m}}}}{{{\lambda _{{\textit{z}}m}}}}] - {\beta _{\textit{z}}}[ - \frac{1}{{{\lambda _{{\textit{z}}m}}{R_{{U}}}}} + \\& (\frac{1}{{{R_{{U}}}}} + \frac{1}{{\lambda _{{\textit{z}}m}^2}})\sin\; {\lambda _{{\textit{z}}m}} + (\frac{1}{{{R_{{U}}}}} - 1)\frac{{\cos\; {\lambda _{{\textit{z}}m}}}}{{{\lambda _{{\textit{z}}m}}}}]\end{aligned}$ (24)

${G_{{\textit{z}}m}}$ 按式(25)计算:

$\begin{aligned}{G_{{\textit{z}}m}} = & \frac{1}{2}[{(\frac{{{\lambda _{{\textit{z}}m}}}}{{{R_{{U}}}}})^2} + 1 + \frac{1}{{{R_{{U}}}}} + \frac{1}{2}(\frac{{{\lambda _{{\textit{z}}m}}}}{{R_{{U}}^2}} - \frac{1}{{{\lambda _{{\textit{z}}m}}}})\sin (2{\lambda _{{\textit{z}}m}}) - \\& \frac{1}{{{R_{{U}}}}}\cos (2{\lambda _{{\textit{z}}m}})]\end{aligned}$ (25)

由于总应力保持不变,根据有效应力原理、式(14)~(15)和(22)可得 $t$ 时刻竖向固结引起的地基沉降 ${S\!_{{{v}}{\textit{z}}}}(t)$ 为:

$\begin{aligned}{S\!_{{{v}}{\textit{z}}}}(t) = & - \int_0^H {\frac{{{u_{\textit{z}}}({\textit{z}},t)}}{{{E_{{s}}}}}{ d}{\textit{z}}} = \frac{{H{u_0}}}{{{E_{{s}}}}}({\alpha _{\textit{z}}} - \frac{{{\beta _{\textit{z}}}}}{2}) - \\& \frac{{H{u_0}}}{{{E_{{s}}}}}\sum\limits_{m = 1}^\infty {\frac{{{A_{{\textit{z}}m}}}}{{{G_{{\textit{z}}m}}}}[\frac{{\sin\; {\lambda _{{\textit{z}}m}}}}{{{R_{{U}}}}} + \frac{{1 - \cos\; {\lambda _{{\textit{z}}m}}}}{{{\lambda _{{\textit{z}}m}}}}]{{ e}^{ - {T_{{v}}}\lambda _{{\textit{z}}m}^2}}} \end{aligned}$ (26)

竖向固结引起的地基最终沉降量 ${S\!_{{{v}}{\textit{z}}}}(\infty )$ 为:

${S\!_{{{v}}{\textit{z}}}}(\infty ) = \frac{{H{u_0}}}{{{E_{{s}}}}}({\alpha _{\textit{z}}} - \frac{{{\beta _{\textit{z}}}}}{2})$ (27)

故按沉降定义的平均固结度由式(26)~(27)得:

$\begin{aligned}[b]{{\overline U}_{\textit{z}}}(t) = & \frac{{{S\!_{{{v}}{\textit{z}}}}(t)}}{{{S\!_{{{v}}{\textit{z}}}}(\infty )}} = 1 - \frac{2}{{2{\alpha _{\textit{z}}} - {\beta _{\textit{z}}}}}\sum\limits_{m = 1}^\infty {\frac{{{A_{{\textit{z}}m}}}}{{{G_{{\textit{z}}m}}}}[\frac{{\sin \;{\lambda _{{\textit{z}}m}}}}{{{R_{{U}}}}}} + \\& \frac{{1 - \cos {\lambda _{{\textit{z}}m}}}}{{{\lambda _{{\textit{z}}m}}}}]{{ e}^{ - {T_{ v}}\lambda _{{\textit{z}}m}^2}}\end{aligned}$ (28)

${U_{\textit{z}}} > 30\% $ 时,式(28)可只取第1个特征值 $\lambda _{{\textit{z}}1}^{}$ 所对应的特解作为近似值,由此得:

${{\overline U}_{\textit{z}}}(t) = 1 - \frac{{{A_{{\textit{z}}1}}}}{{{G_{{\textit{z}}1}}}}\left(\frac{{\sin\; {\lambda _{{\textit{z}}1}}}}{{{R_{{U}}}}} + \frac{{1 - \cos \;{\lambda _{{\textit{z}}1}}}}{{{\lambda _{{\textit{z}}1}}}}\right)\frac{{2{{ e}^{ - {T_{ v}}\lambda _{{\textit{z}}1}^2}}}}{{2{\alpha _{\textit{z}}} - {\beta _{\textit{z}}}}}$ (29)
1.2.2 径向固结方程的求解

由于式(3)不满足齐次边界方程,为使边界条件齐次化,令

$u_{{w}}^* = {u_{{w}}} + ({\alpha _{{w}}} - {\beta _{{w}}}\frac{{{{\textit{z}}^*}}}{H}){u_0}$ (30)
$u_{r}^* = {u_{r}} + ({\alpha _{{w}}} - {\beta _{{w}}}\frac{{{{\textit{z}}^*}}}{H}){u_0}$ (31)

其中:

${\alpha _{{w}}} = \frac{{(1 + {R_{{{wL}}}}){R_{{{wU}}}}}}{{(1 + {R_{{{wL}}}}){R_{{{wU}}}} + {R_{{{wL}}}}}}$ (32)
${\beta _{{w}}} = \frac{{{R_{{{wL}}}}{R_{{{wU}}}}}}{{({R_{{{wL}}}} + 1){R_{{{wU}}}} + {R_{{{wL}}}}}}$ (33)

利用式(14)、(30)、(31),式(6)、(11)、(12)分别变为:

$\frac{{\partial {}^2u_{{w}}^*}}{{\partial {{\textit{z}}^*}^2}} = - \frac{{2{k_{{h}}}f({r_{{w}}})}}{{{r_{{w}}}{k_{{w}}}}}{\left. {\frac{{\partial u_{r}^*}}{{\partial r}}} \right|_{r = {r_{{w}}}}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$ (34)
$\frac{{\partial {\varepsilon _{\upsilon 2}}}}{{\partial t}} = - \frac{1}{{{E_{{s}}}}}\frac{{\partial \bar u_{r}^*}}{{\partial t}}$ (35)
$\quad \quad\quad\quad\quad\quad - \frac{1}{r}\frac{\partial }{{\partial r}}[\frac{{{k_{{h}}}f(r)}}{{{\gamma _{{w}}}}}r\frac{{\partial u_{r}^*}}{{\partial r}}] = \frac{{\partial {\varepsilon _{\upsilon 2}}}}{{\partial t}}$ (36)

式中, $\bar u_{r}^*$ $u_{r}^*$ 的径向平均值,有:

$\bar u_{r}^* = \frac{1}{{{\text{π}} (r_{{e}}^2 - r_{{w}}^2)}}\int_{\displaystyle{r_{{w}}}}^{\displaystyle{r_{{e}}}} {2{\text{π}} ru_{r}^*{ d}r} $ (37)

利用式(14)、(30)、(31),式(3)、(4)、(8)分别变为:

${\left. {\frac{{\partial u_{{w}}^*}}{{\partial {{\textit{z}}^*}}}} \right|_{{{\textit{z}}^*} = 0}} = \frac{{{R_{{{wU}}}}}}{H}u_{{w}}^*$ (38)
${\left. {\frac{{\partial u_{{w}}^*}}{{\partial {{\textit{z}}^*}}}} \right|_{{{\textit{z}}^*} = H}} = - \frac{{{R_{{{wL}}}}}}{H}u_{{w}}^*$ (39)
${\left. {u_r^*} \right|_{t = 0}} = ({\alpha _{{w}}} - \frac{{{\beta _{{w}}}}}{H}{{\textit{z}}^*}){u_0}$ (40)

$\xi = \frac{{8{c_{{h}}}}}{{d_{{e}}^2{F_{{a}}}}} ,\; \rho = \frac{{2{k_{{h}}}({n^2} - 1)}}{{{k_{{w}}}r_{{e}}^2{F_{{a}}}}}$ (41)

式中,

${c_{{h}}} = \frac{{{k_{{h}}}{E_{{s}}}}}{{{\gamma _{{w}}}}} , \;n = \frac{{{r_{{e}}}}}{{{r_{{w}}}}}$ (42)
${F_{{a}}} = \frac{2}{{r_{{e}}^2 - r_w^2}}\int_{{r_{{w}}}}^{{r_{{e}}}} {r[\int_{{r_{{w}}}}^r {\frac{{\operatorname{d} x}}{{xf\left( x \right)}}} } - \frac{1}{{r_{{e}}^2}}\int_{{r_{{w}}}}^r {\frac{{x\operatorname{d} x}}{{f\left( x \right)}}]} { d}r$ (43)

采用分离变量法对式(34)~(36)求解,并利用式(38)~(43)得:

$\begin{aligned}[b] {u_{{w}}^*} = & {{u_0}\sum\limits_{m = 1}^\infty {\frac{{{A_{{{w}}m}}}}{{{G_{{{w}}m}}}}\frac{{\xi - {\eta _m}}}{\xi } \times } }\\& {[\frac{{{\lambda _{{{w}}m}}}}{{{R_{{{wU}}}}}}\cos (\frac{{{\lambda _{{{w}}m}}}}{H}{\textit{z}^*}) + \sin (\frac{{{\lambda _{{{w}}m}}}}{H}{\textit{z}^*})]{{ e}^{ - {\eta _m}t}}}\end{aligned}$ (44)
$\bar u_{r}^* = {u_0}\sum\limits_{m = 1}^\infty {\frac{{{A_{{{w}}m}}}}{{{G_{{{w}}m}}}}[\frac{{{\lambda _{{{w}}m}}}}{{{R_{{{wU}}}}}}\cos (\frac{{{\lambda _{{{w}}m}}}}{H}{{\textit{z}}^*}) + \sin (\frac{{{\lambda _{{{w}}m}}}}{H}{{\textit{z}}^*})]} {{ e}^{ - {\eta _m}t}}$ (45)

式中, $\lambda _{{{w}}m}^{} > 0$ 为特征值,按从小到大的顺序由方程(46)确定:

$\tan \;\lambda _{{{w}}m}^{} = \frac{{\lambda _{{{w}}m}^{}({R_{{{wU}}}} + {R_{{{wL}}}})}}{{\lambda _{{{w}}m}^2 - {R_{{{wU}}}}{R_{{{wL}}}}}}$ (46)

${\eta _m}$ 的计算式为:

${\eta _m} = \frac{{\xi \lambda _{{{w}}m}^2}}{{{\rho ^2}{H^2} + \lambda _{{{w}}m}^2}} = \frac{{8{c_{{h}}}}}{{{d}_{{e}}^2}}/\left( {{F_{{a}}} + \frac{8}{{\lambda _{{{w}}m}^2}}\frac{{{n^2} - 1}}{{{n^2}}}{R_{{J}}}} \right)$ (47)

式中, ${R_{{J}}} = ({k_{{h}}}/{k_{{w}}}){(H/{d_{{w}}})^2}$ 为井阻因子。

${A_{{{w}}m}}$ 按式(48)计算:

$\begin{aligned}& {A_{{{w}}m}} = {\alpha _{{w}}}[\frac{1}{{{R_{{{wU}}}}}}\sin\; {\lambda _{{{w}}m}}+ \frac{{1 - \cos \;{\lambda _{{{w}}m}}}}{{{\lambda _{{{w}}m}}}}] + {\beta _{{w}}}[\frac{1}{{{\lambda _{{{w}}m}}{R_{{{wU}}}}}} - \\& \quad \quad (\frac{1}{{{R_{{{wU}}}}}} + \frac{1}{{\lambda _{{{w}}m}^2}})\sin \;{\lambda _{wm}} - (\frac{1}{{{R_{{{wU}}}}}} - 1)\frac{{\cos\; {\lambda _{{{w}}m}}}}{{{\lambda _{{{w}}m}}}}]\end{aligned}$ (48)

${G_{{{w}}m}}$ 按式(49)计算:

$\begin{aligned}{G_{{{w}}m}} = & \frac{1}{2}[\frac{1}{2}(\frac{{{\lambda _{{{w}}m}}}}{{R_{{{wU}}}^2}} - \frac{1}{{{\lambda _{{{w}}m}}}})\sin (2{\lambda _{{{w}}m}}) - \frac{1}{{{R_{{{wU}}}}}}\cos (2{\lambda _{{{w}}m}})+\\ & {(\frac{{{\lambda _{{{w}}m}}}}{{{R_{{{wU}}}}}})^2} + 1 + \frac{1}{{{R_{{{wU}}}}}}]\end{aligned}$ (49)

由于总应力保持不变,根据有效应力原理及式(14)、(31)、(45)可得 $t$ 时刻径向固结引起的地基沉降量 ${S\!_{{{v}}{r}}}(t)$ 为:

$\begin{aligned}{S\!_{{{v}}{r}}}(t) = & - \int_0^H {\frac{{{{\bar u}_{r}}({\textit{z}},t)}}{{{E_{{s}}}}}{ d}{\textit{z}}} = \frac{{H{u_0}}}{{{E_{{s}}}}}({\alpha _{{w}}} - \frac{{{\beta _{{w}}}}}{2}) - \\ & \frac{{H{u_0}}}{{{E_{{s}}}}}\sum\limits_{m = 1}^\infty {\frac{{{A_{{{w}}m}}}}{{{G_{{{w}}m}}}}[\frac{{\sin \;{\lambda _{{{w}}m}}}}{{{R_{{{wU}}}}}} + \frac{{1 - \cos \;{\lambda _{{{w}}m}}}}{{{\lambda _{{{w}}m}}}}]{{ e}^{ - {\eta _m}t}}} \end{aligned}$ (50)

径向固结引起的地基最终沉降量 ${S\!_{{{v}}{r}}}(\infty )$ 为:

${S\!_{{{v}}{r}}}(\infty ) = \frac{{H{u_0}}}{{{E_{{s}}}}}({\alpha _{{w}}} - \frac{{{\beta _{{w}}}}}{2})$ (51)

故径向固结引起的按沉降定义的平均固结度为:

$\begin{aligned}{{\overline U}_r}(t) = & \frac{{{S\!_{{{v}}{r}}}(t)}}{{{S\!_{{{v}}{r}}}(\infty )}} = 1 - \frac{2}{{2{\alpha _{{w}}} - {\beta _{{w}}}}}\sum\limits_{m = 1}^\infty {\frac{{{A_{{{w}}m}}}}{{{G_{{{w}}m}}}}[\frac{{\sin \;{\lambda _{{{w}}m}}}}{{{R_{{{wU}}}}}} + } \\& \frac{{1 - \cos \;{\lambda _{{{w}}m}}}}{{{\lambda _{{{w}}m}}}}]{{ e}^{ - {\eta _m}t}}\end{aligned}$ (52)

如果把式(44)中第1个特征值 $\lambda _{{{w}}1}^{}$ 所对应的特解作为 $u_{{w}}^*$ 的解,并近似地按式(53)确定特解系数 ${A_{{{w}}1}}/{G_{{{w}}1}}$

$\frac{1}{H}\int_0^H {{{\left. {\bar u_r^*} \right|}_{t = 0}}{ d}{\textit{z}}} = {u_0}({\alpha _{{w}}} - \frac{{{\beta _{{w}}}}}{2})$ (53)

则可得到近似解为:

${\overline U_r}(t) = 1 - {{ e}^{\scriptstyle - 8{T_{{h}}}/\left( {{F_{{a}}} + \textstyle\frac{8}{{\lambda _{{{w}}1}^2}}\textstyle\frac{{{n^2} - 1}}{{{n^2}}}{\scriptstyle R_{{J}}}} \right)}}$ (54)

式中, ${T_{{h}}} = {c_{{h}}}t/{{d_{{e}}^2}}$ 为径向固结时间因子。

1.3 Carrillo法的必要条件

根据式(14)、(15)、(22)和有效应力原理可知,竖向固结引起的最终孔压和有效应力为:

${\left. {{{\left. { - u'({\textit{z}})} \right|}_{t = \infty }} = \sigma '({\textit{z}})} \right|_{t = \infty }} = ({\alpha _{\textit{z}}} - {\beta _{\textit{z}}}\frac{{H - {\textit{z}}}}{H}){u_0}$ (55)

根据式(16)~(17)可知, ${\alpha _{\textit{z}}}$ ${\beta _{\textit{z}}}$ ${R_{{U}}}$ ${R_{{L}}}$ 有关,故在真空预压法中,边界的透水特性会影响竖向固结引起的最终孔压和有效应力值。在堆载预压法中,边界透水特性不会对竖向固结引起的最终孔压和有效应力产生影响[15]。这说明半透水边界对真空预压竖向固结效果的影响规律与堆载预压相比具有较大不同。

同理,根据式(14)、(31)、(45)和有效应力原理可知,径向固结引起的最终孔压和有效应力为:

${\left. { - u'({\textit{z}})} \right|_{t = \infty }} = {\left. {\sigma '({\textit{z}})} \right|_{t = \infty }} = ({\alpha _{{w}}} - {\beta _{{w}}}\frac{{H - {\textit{z}}}}{H}){u_0}$ (56)

根据式(32)、(33)可知, ${\alpha _{{w}}}$ ${\beta _{{w}}}$ ${R_{{{wU}}}}$ ${R_{{{wL}}}}$ 值有关,故在真空预压法中,边界的透水特性同样会影响径向固结引起的最终孔压和有效应力值。在堆载预压法中,半透水边界的透水特性不会对径向固结引起的最终孔压和有效应力产生影响[1416]。这说明半透水边界的透水特性对真空预压径向固结效果的影响规律与堆载预压相比也存在较大的不同。

从式(55)和(56)可知,若 ${\alpha _{\textit{z}}}$ ${\alpha _{{w}}}$ ${\beta _{\textit{z}}}$ ${\beta _{{w}}}$ 不各自相等,则竖向渗透固结引起的最终有效应力不等于径向渗透固结引起的最终有效应力。这种情况下竖向固结和径向固结会产生渗流耦合,无法采用Carrillo法进行砂井总固结度计算,甚至难以采用解析法对其进行求解。为便于工程应用,在土力学中常常抓住关键因素而忽略次要因素以简化理论分析。借鉴该研究思路解决这一难题。砂井地基沉降主要由径向固结完成,为能够应用Carrillo法进行砂井总固结度计算,按 ${\alpha _{{w}}}$ ${\beta _{{w}}}$ ${\alpha _{\textit{z}}}$ ${\beta _{\textit{z}}}$ 取值,即假定

${\alpha _{\textit{z}}} = {\alpha _{{w}}}, {\beta _{\textit{z}}} = {\beta _{{w}}}$ (57)

在此条件下有:

${R_{{{wU}}}} = {R_{{U}}}, {R_{{{wL}}}} = {R_{{L}}}$ (58)

根据式(55)~(57)、(27)、(51)可知竖向固结引起的最终有效应力分布形状和地基最终沉降与径向固结引起的相同,借鉴文献[1]和[11]研究成果,砂井地基总固结度可采用Carrillo方法计算:

$\overline U(t) = 1 - [1 - {\overline U_{r}}(t)][1 - {\overline U_{\textit{z}}}(t)]$ (59)

$\sigma {'_\infty }$ 为固结增加的最终有效应力,图2给出了 ${R_{{{wU}}}} = {R_{{U}}} = \infty $ 时, ${R_{{{wL}}}}$ $ {R_{{L}}}$ )对 $\sigma {'_\infty }/{u_0}$ 沿深度分布特性的影响;图3给出了 ${R_{{{wL}}}} = {R_{{L}}} = 1$ 时, ${R_{{{wU}}}}$ $ {R_{{U}}}$ )对 $\sigma {'_\infty }/{u_0}$ 沿深度分布特性的影响。

图2 ${{ R}_{{\bf{wU}}}} \!=\! {{ R}_{\bf{U}}} \!=\! { \infty} $ ${{ R}_{{\bf{wL}}}}$ $ {{ R}_{\bf{L}}}$ )对 $ {\sigma {'_{ \infty}}/{ u_{ 0}}}$ 随深度 ${{z}}$ 分布的影响 Fig. 2 Effect of ${{ R}_{{\bf{wL}}}}$ ( $ {{ R}_{\bf{L}}}$ ) on the distribution of $ {\sigma {'_{ \infty}}/{ u_{ 0}}}$ along with depth ${{z}}$ at ${{ R}_{{\bf{wU}}}} = {{ R}_{\bf{U}}} = { \infty} $

图3 ${{ R}_{{\bf{wL}}}} = {{ R}_{\bf{L}}} = { 1}$ ${{ R}_{{\bf{wU}}}}$ $ {{ R}_{\bf{U}}}$ )对 $\sigma {'_{ \infty}}/{{ u}_{ 0}}$ 随深度 ${{z}}$ 分布的影响 Fig. 3 Effect of ${{ R}_{{\bf{wU}}}}$ ( ${{ R}_{\bf{U}}}$ ) on the distribution of ${\sigma} {'_{ \infty}}/{{ u}_{ 0}}$ along with depth ${{z}}$ at ${{ R}_{{\bf{wL}}}} = {{ R}_{\bf{L}}} = { 1}$

造成真空预压有效应力随深度衰减的原因是真空预压造成砂井加固土体区域内的孔压小于砂井底部土体中的孔压。在半透水条件下,砂井底部土体的孔隙水在负孔压作用下流入砂井地基中,同时砂井上部在抽真空作用下孔隙水从砂井土体中流出。当土体真空预压固结引起的负孔压分布使得砂井范围内任意深度砂井的上部排水量等于下部流入量时,土体的孔隙比保持不变,因而土体不再发生沉降变形;相应地,负孔压和有效应力随深度的分布特性也不再发生变化,至此真空预压固结结束。这说明只要真空泵施加的真空度保持恒定,当底部为半透水边界时,由于预压期间(包括固结完成时)砂井地基孔压始终低于底部四周土体,孔隙水将不断地从砂井底部流入砂井,因此真空预压的沉降稳定条件是砂井范围内任意深度孔隙水的上部排水量等于下部流入量。堆载预压的稳定条件是超孔隙压力趋近于0;相应地,砂井的排水量也趋近于0。由于真空预压的稳定条件与堆载预压的稳定条件不同,因此在半透水边界砂井地基中,真空预压固结增加的土体有效应力会随深度发生衰减变化,堆载预压固结增加的土体有效应力则不会出现这种情况。

1.4 理论解的退化

${R_{{{wU}}}} \!=\! {R_{{U}}} \!=\! \infty $ ${R_{{{wL}}}} \!=\! {R_{{L}}} \!=\! 0$ 时,此时有 ${\alpha _{\textit{z}}} \!=\! {\alpha _{{w}}} \!=\! 1$ ${\beta _{\textit{z}}} = {\beta _{{w}}} = 0$ ${\lambda _{{\textit{z}}m}} = {\lambda _{{{w}}m}} = (m + 0.5){\text{π}} $ ,竖向固结的解由式(24)、(25)、(28)得:

${\overline U_{\textit{z}}}(t) = 1 - \frac{8}{{{{\text{π}} ^2}}}\sum\limits_{m = 1}^\infty {\frac{1}{{{{(2m + 1)}^2}}}} {{ e}^{ - {{\text{π}} ^2}{T_{ v}}{{(2m + 1)}^2}}}$ (60)

径向固结的解由式(48)、(49)、(52)得:

${\overline U_r}(t) = 1 - \sum\limits_{m = 1}^\infty {\frac{2}{{{{[(m + 0.5){\text{π}} ]}^2}}}{{ e}^{ - {\eta _m}t}}} $ (61)

式(60)、(61)与负压下董志良[1]的解答相同。

2 透水系数对固结度与沉降的影响分析

实际工程中砂井地基固结主要由径向渗透完成,因此本文数值分析忽略竖向固结的影响,只分析径向渗透引起的砂井固结特性。根据竖向排水体施工工况,砂井地基固结理论中常用的涂抹系数 $f\left( r \right)$ 分布模式有以下几种:

工况1:不考虑砂井的涂抹作用,有:

$f\left( r \right) = 1,\; {r_{{w}}} \le r \le {r_{{e}}}$ (62)

根据式(43)可得 ${F_{{a}}}$ 为:

${F_{{a}}} = \frac{{{n^2}}}{{{n^2} - 1}}\ln n - \frac{{3{n^2} - 1}}{{4{n^2}}}$ (63)

工况2:砂井涂抹区渗透系数为常数 ${k_{{s}}}$ ,令 $\delta = {k_{{s}}}/{k_{{h}}}$ ,此时 $f\left( r \right)$ 分布模式为:

$f\left( r \right) = \left\{ {\begin{aligned}& {\delta, \;{r_{{w}}} \le r \le {r_{{s}}}};\\& {1,\;{r_{{s}}} \le r \le {r_{{e}}}}\end{aligned}} \right.$ (64)

式中, ${r_{{s}}}$ 为涂抹区半径,令 $s = {r_{{s}}}/{r_{{w}}}$ ,根据式(43)得:

${F_{{a}}} = \frac{{{n^2}}}{{{n^2} - 1}}[\ln n + (\frac{1}{\delta } - 1)(\ln s + \frac{{1 - {s^2}}}{{{n^2}}} + \frac{{{s^4} - 1}}{{4{n^4}}})] - \frac{{3{n^2} - 1}}{{4{n^2}}}$ (65)

工况3:砂井涂抹区渗透系数为线性分布,有:

$f\left( r \right) = \left\{ {\begin{array}{*{20}{c}}\!\!\!\!\!{\displaystyle\frac{{(r{{ - }}{r_{{w}}}) + \delta ({r_{{s}}}{{ - }}r)}}{{{r_{{s}}}{{ - }}{r_{{w}}}}},}& {{r_{{w}}} \le r \le r;}\\\!\!\!\!\!{1,}& {{r_{{s}}} \le r \le {r_{{e}}}}\end{array}} \right.$ (66)

根据式(43)可得:

$\begin{aligned}{F_{{a}}} = & \frac{{{n^2}}}{{{n^2} - 1}} [\frac{{s - 1}}{{\delta s - 1}}\ln (\delta s) - \frac{{{{(s - 1)}^2}}}{{{n^2}(1 - \delta )}} +\\[6pt]& \frac{{2(s - 1)(\delta s - 1)}}{{{n^2}(1 - \delta ){}^2}}\ln \frac{1}{\delta } - \frac{{2{s^3} - 3{s^2} + 1}}{{3{n^4}(1 - \delta )}}(s - 1) - \\[6pt]& \frac{{(s - 1)(\delta s - 1)}}{{{n^4}(1 - \delta ){}^2}}\!\![\frac{{{s^2} - 1}}{2} \!\!-\!\! \frac{{(s - 1)(\delta s - 1)}}{{1 - \delta }} \!\!+\!\! \frac{{{{(\delta s - 1)}^2}}}{{{{(1 - \delta )}^2}}}\ln \frac{1}{\delta }] - \\[6pt]&\frac{{({n^2} - {s^2}){{(1 - s)}^2}}}{{{n^4}(1 - \delta )}} + \ln \frac{n}{s} - \frac{3}{4} + \frac{{4{n^2}{s^2} - {s^4}}}{{4{n^4}}}]\quad\quad\;\;\;\; (\;67\;)\end{aligned}$ (67)
2.1 顶部透水系数恒定、底部半透水情形

为反映边界透水特性对砂井地基沉降的影响,把 ${\overline S\!_{{{v}}{r}}}(t) = {S\!_{{ v}r}}(t)/({u_0}H/{E_{{s}}})$ 称为无量纲地基沉降。

假定 $f\left( r \right)$ 按工况2(式(64))分布, $n = {r_{{e}}}/{r_{{w}}} = 15$ $s \!=\! {r_{{s}}}/{r_{{w}}} \!=\! 2$ $\delta \!=\! {k_{{s}}}/{k_{{h}}} \!=\! 0.25$ ${R_{{J}}} = ({k_{{h}}}/{k_{{w}}}){(H/{d_{{w}}})^2} \!=\! 1 \sim 4$ ,顶部(上边界)完全透水( ${R_{{{wU}}}} = \infty $ ),根据式(50)、(52)计算,可得底部透水系数 ${R_{{{wL}}}}$ 变化对砂井地基平均径向固结度 ${\overline U_{r}}$ 和无量纲地基沉降 ${\bar S\!_{{{v}}{r}}}(t)$ 的影响规律,如图47所示。

图4 ${{ R}_{\bf{J}}} = { 1}$ ${{ R}_{{\bf{wL}}}}$ 对固结度的影响 Fig. 4 Effect of ${{ R}_{{\bf{wL}}}}$ on consolidation degree at ${{ R}_{\bf{J}}} = { 1}$

图5 ${{ R}_{\bf{J}}} = { 1}$ ${{ R}_{{\bf{wL}}}}$ 对无量纲沉降的影响 Fig. 5 Effect of ${{ R}_{{\bf{wL}}}}$ on dimensionless settlement at ${{ R}_{\bf{J}}} = { 1}$

图6 ${{ R}_{\bf{J}}} = { 4}$ ${{ R}_{{\bf{wL}}}}$ 对固结度的影响 Fig. 6 Effect of ${{ R}_{{\bf{wL}}}}$ on consolidation degree at ${{ R}_{\bf{J}}} = { 4}$

图7 ${{ R}_{\bf{J}}} = { 4}$ ${{ R}_{{\bf{wL}}}}$ 对无量纲沉降的影响 Fig. 7 Effect of ${{ R}_{{\bf{wL}}}}$ on dimensionless settlement at ${{ R}_{\bf{J}}} = { 4}$

图47可知:随着砂井底部透水系数 ${R_{{{wL}}}}$ 逐渐增大,真空预压的固结度也逐渐增大; ${R_{{J}}}$ 越大,底部透水系数 ${R_{{{wL}}}}$ 对固结度的影响越明显。但其沉降却随底部透水系数的增加明显减少。造成这一现象的原因是某一时刻的地基固结沉降是固结度与最终沉降之间的乘积。从式(32)、(33)、(51)可以看出,当 ${R_{{{wU}}}} = \infty $ 时, ${\alpha _{{w}}} = 1$ ,随着底部透水系数增大, ${\beta _{{w}}}$ 值从0增加到1;相应地,无量纲最终沉降从1减少到0.5。由于透水系数对最终沉降的影响比对固结度的影响大,因此在相同固结时间因子下地基沉降呈现随着底部透水系数增加而逐渐减小的规律。在堆载预压法中,底部透水条件的变化不会对堆载预压最终沉降产生影响;由于底部透水系数越大固结度越大,故在相同固结时间因子下堆载预压沉降呈随底部透水系数增加而增加的规律[1416]。故底部透水条件对真空预压的影响与堆载预压相比两者有较大的差别。

2.2 底部透水系数恒定顶部半透水情形

假定 $f\left( r \right)$ 按工况2(式(64))分布, $n = 15$ $s = 2$ $\delta = 0.25$ ${R_{{J}}} = 4$ ,底部不透水( ${R_{{{wL}}}} = 0$ ),根据式(50)分析计算,可得顶部透水系数 ${R_{{{wU}}}}$ 变化对无量纲地基沉降 ${\overline S_{{{v}}{r}}}(t)$ 的影响规律,如图8所示。

图8 底部不透水和 ${{ R}_{\bf{J}}} = { 4}$ ${{ R}_{{\bf{wU}}}}$ 对无量纲沉降的影响 Fig. 8 Effect of ${{ R}_{{\bf{wU}}}}$ on dimensionless settlement for impermeable bottom and ${{ R}_{\bf{J}}} = { 4}$

图8可知:当 ${R_{{{wL}}}} = 0$ ,即底部不透水时,顶部透水系数 ${R_{{{wU}}}}$ 不会对地基的最终沉降产生影响;顶部透水系数 ${R_{{{wU}}}}$ 越大,固结度越大。因此,在相同固结时间因子下,地基沉降呈随顶部透水系数增加而增加的变化规律。

假定 $f\left( r \right)$ 按工况2分布: $n = 15$ $s = 2$ $\delta = 0.25$ ${R_{{J}}} = 1 \sim 4$ ;底部半透水,底部透水系数为 ${R_{{{wL}}}} = 1$ 。根据式(52)、(50)分析计算可得顶部透水系数 ${R_{{{wU}}}}$ 变化对砂井地基固结度 ${\overline U_r}$ 和无量纲沉降 ${\overline S_{{{{ v}}}{r}}}(t)$ 的影响规律,如图910所示。

图9 ${{ R}_{{\bf{wL}}}} = { 1}$ ${{ R}_{\bf{J}}} = { 1}$ ${{ R}_{{\bf{wU}}}}$ 对固结度的影响 Fig. 9 Effect of ${{ R}_{{\bf{wU}}}}$ on consolidation degree at ${{ R}_{\bf{wL}}} = { 1}$ and ${{ R}_{\bf{J}}} = { 1}$

图10 ${{ R}_{\bf{wL}}} { =} { 1}$ ${{ R}_{\bf{J}}} = { 1}$ ${{ R}_{{\bf{wU}}}}$ 对无量纲沉降的影响 Fig. 10 Effect of ${{ R}_{{\bf{wU}}}}$ on dimensionless settlement at ${{ R}_{\bf{wL}}} = { 1}$ and ${{ R}_{\bf{J}}} { =} { 1}$

图910可知:当 ${R_{{{wL}}}} = 1$ ,即底部半透水时,随着顶部透水系数 ${R_{{{wU}}}}$ 逐渐增大,真空预压的固结度均逐渐增大。同时从式(32)、(33)、(51)可以看出,当 ${R_{{{wL}}}} = 1$ 时,随着 ${R_{{{wU}}}}$ 从1增大到 $\infty $ 时, ${\alpha _{{w}}} = 2{\beta _{{w}}}$ 从2/3增加到1;相应地,无量纲最终沉降 ${\widetilde S_{{{{ v}}}{r}}}(t)$ 从0.5增加到0.75。由于顶部的透水系数越大,最终沉降也越大,因此图10呈现相同固结时间因子下地基沉降随上部透水系数增加而逐渐增大的变化规律。

2.3 渗透系数不同分布对固结度和沉降的影响

假定 $n \!=\! 15$ $s \!=\! 2$ $\delta \!=\! 0.25$ ${R_{{J}}} \!=\! 1$ ${R_{{{wU}}}} \!=\! 10$ ${R_{{{wL}}}} \!=\! 1$ ,根据式(52)、(63)、(65)和(67)分析计算,可得涂抹区渗透系数沿径向不同分布模式对径向固结度 ${\overline U_{r}}$ 的影响规律,如图11所示。

图11 渗透系数分布对固结度影响 Fig. 11 Effect of distribution of permeability coefficient on consolidation degree

图11可知:当其他条件相同时,由于工况1不折减涂抹区渗透系数,根据工况1渗透系数分布模式计算的固结度最大;工况2按涂抹区受施工干扰后的最小渗透系数进行砂井固结度计算,根据其计算的固结度最小;工况3考虑了涂抹区从最小渗透系数向原状土渗透系数的过渡变化段,根据其计算的固结度介于两者之间。渗透系数沿径向分布模式不会影响真空预压的地基最终沉降,所以渗透系数分布模式对固结沉降的影响规律与固结度相同。

3 算 例

浙江舟山煤炭中转码头采用真空预压分区块处理老旧海堤间淤泥质软土地基[8]。根据工程地质报告,取地基压缩模量 ${E_{{s}}}$ =1.52 MPa,竖直向渗透系数 ${k_{{v}}}$ =3.04 $ \times $ 10–9 m/s,水平向渗透系数 ${k_{{h}}}$ =3.68 $ \times $ 10–9 m/s。采用真空预压结合塑料排水板加固软土地基,加固深度为 $H$ =25 m。塑料排水板截面尺寸为100 mm $ \times $ 6 mm,折合成砂井半径为 ${r_{{w}}}$ =0.033 8 m;间距为1.2 m,正方形布置,折合成砂井影响半径为 ${r_{{e}}}$ =0.677 cm,井径比 $n$ =20。考虑砂井施工引起的涂抹作用,涂抹系数 $f\left( r \right)$ 按工况2中的式(64)分布,涂抹区半径 ${r_{{s}}}$ =0.080 1 m, $s$ =2.37,涂抹区的渗透系数为 ${k_{{s}}}$ =0.92 $ \times $ 10–9 m/s, $\delta $ =0.25。塑料排水板砂井的渗透系数为 ${k_{{w}}}$ =1.2 $ \times $ 10–4 m/s,井阻因子 ${R_{{J}}} = ({k_{ h}}/{k_{ w}}){(H/{d_{ w}})^2} = 5.03$

图12按文献[1]与本文理论计算了不同深度的孔压值,并与实测值进行对比分析。

图12可以看出:两种理论在固结前期与实测值均存在不同程度的偏差,整体来说都偏大;固结与前期抽真空压力不稳有关。与不考虑底部透水条件的文献[1]相比,本文理论获得的孔压计算值更接近实测数据。

图12 两种不同理论的超孔隙水压力计算值与实测值比较 Fig. 12 Comparison between the calculated excess pore pressure values by two different theories and measured data

图13比较了文献[1]和本文理论计算的沉降值与实测值,两者在固结前期与实测沉降均比较接近;但随着固结时间的增长,文献[1]的预测值逐渐偏离实测值,而本文理论的预测值与实测结果依然较吻合。

图13 两种不同理论的沉降计算值与实测值比较 Fig. 13 Comparison between the calculated settlement values by two different theories and measured data

4 结 论

考虑水平向渗透系数随半径发生变化的分布特征,建立上下边界为半透水边界时真空预压砂井地基的固结解析理论。由于在实际工程中一般忽略竖直向固结而只考虑水平向固结,因此着重数值分析了水平向渗透引起的固结度和固结沉降发展规律,得出如下结论:

1)当顶部透水系数恒定时,砂井地基的固结度随底部透水系数的增加而增大,但径向固结时间因子相同时地基固结沉降随底部透水系数的增大而减小。砂井地基的最终沉降也随底部透水系数的增大而减小。

2)当底部不透水时,砂井地基的固结度随顶部透水系数的增加而增大,径向固结时间因子相同时的地基固结沉降也随顶部透水系数的增大而增大,但最终沉降不随顶部透水系数的变化而变化;不管顶部透水和土层渗透系数如何变化,最终负孔压和最终有效应力不会随深度发生衰减。当底部为半透水边界且透水系数恒定时,砂井地基的固结度随顶部透水系数的增加而增大,径向固结时间因子相同时的地基固结沉降和地基的最终沉降也随顶部透水系数的增大而增大。

3)数值分析了渗透系数不同分布模式对固结度的影响规律,结果表明:涂抹区的渗透系数向原状土的渗透系数靠拢得越快,土层的固结度越大。

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