﻿ Logistic模型在软黏土一维固结水力梯度预测中的应用 Research on Logistic Model of Hydraulic Gradient in One-Dimensional Consolidation of Saturated Clay

Vol.08 No.02(2018), Article ID:25326,13 pages
10.12677/APF.2018.82004

Research on Logistic Model of Hydraulic Gradient in One-Dimensional Consolidation of Saturated Clay

Jinzhu Li1, Lu Lan1,2, Xinyu Xie1,2

1Ningbo Institute of Technology, Zhejiang University, Ningbo Zhejiang

2Research Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou Zhejiang

Received: May 21st, 2018; accepted: May 31st, 2018; published: Jun. 7th, 2018

ABSTRACT

During the consolidation of saturated clay, the drainage rate is positively related to hydraulic gradient. The formula and changeable curve with time of hydraulic gradient are acquired based on Terzaghi’s one-dimensional consolidation, and the “S” type curve is explained in the paper. Logistic function is used instead of complex formula of hydraulic gradient through nonlinear fitting. The relationships between function parameters and soil consolidation parameters are defined to establish Logistic model. One-dimensional consolidation of saturated clay in instantaneous and linear loading conditions is simulated with FEM software, thus comparing changeable curves of hydraulic gradient in two conditions which fits well with Logistic function. The results indicate that Logistic model can not only quantitatively describe variation of hydraulic gradient in Terzaghi’s one-dimensional consolidation, but also other consolidation conditions such as linear loading.

Keywords:Saturated Clay, One-Dimensional Consolidation, Hydraulic Gradient, Logistic Model

Logistic模型在软黏土一维固结水力梯度预测中的应用

1浙江大学宁波理工学院，浙江 宁波

2浙江大学滨海和城市岩土工程研究中心，浙江 杭州

1. 引言

2. 水力梯度曲线的一般描述

$i=\frac{2}{{\gamma }_{w}}\frac{p}{H}\sum _{n=0}^{\infty }\mathrm{cos}\frac{Mz}{H}{\text{e}}^{-{M}^{2}{T}_{v}}$ (1)

${T}_{v}=\frac{{c}_{v}t}{{H}^{2}}$ (2)

${c}_{v}=\frac{k{E}_{s}}{{\gamma }_{w}}$ (3)

3. 水力梯度的Logistic模型

3.1. Logistic模型

$i=-\frac{{A}_{1}-{A}_{2}}{1+{\left(t/{t}_{0}\right)}^{\delta }}+{A}_{2}$ (4)

Logistic函数和水力梯度两阶段曲线具备有界性、单调递增性及存在反弯点等共性。公式(4)对t进行

Figure 1. Variation of hydraulic gradient with time

${i}^{\prime }=-\frac{\left({A}_{1}-{A}_{2}\right)\delta {\left(t/{t}_{0}\right)}^{\delta -1}}{{\left[1+{\left(t/{t}_{0}\right)}^{\delta }\right]}^{2}{t}_{0}}$ (5)

${i}^{″}=\frac{\left({A}_{1}-{A}_{2}\right)\delta {\left(t/{t}_{0}\right)}^{\delta -2}}{{\left[1+{\left(t/{t}_{0}\right)}^{\delta }\right]}^{3}{t}_{0}{}^{2}}\left[1-\delta +\left(1+\delta \right){\left(\frac{t}{{t}_{0}}\right)}^{\delta }\right]$ (6)

${t}_{1}={\left[1/\left(\lambda -1\right)-1\right]}^{1/\delta }{t}_{0}$ (7)

${t}_{2,3}={\left[\left(\delta -1\right)/\left(\delta +1\right)\right]}^{1/\delta }{t}_{0}$ (8)

Figure 2. First derivative of Logistic function

Figure 3. Second derivative of Logistic function

3.2. 算例分析

${i}_{\mathrm{max}}=\frac{\alpha p}{{\gamma }_{w}z}$ (9)

Figure 4. Schematic of physical model

Figure 5. Variation of hydraulic gradient i with time

Figure 6. Variation of hydraulic gradient imax with z

Table 1. Logistic model and parameters upward

Table 2. Logistic model and parameters downward

${t}_{0}=\frac{\beta {z}^{2}}{{c}_{v}}$ (10)

${A}_{1}$${A}_{2}$${t}_{0}$$\delta$ 参数值代入式(4)可得上升阶段水力梯度i的Logistic公式，见式(11)。

$i=\frac{\alpha p}{{\gamma }_{w}z}-\frac{\frac{\alpha p}{{\gamma }_{w}z}}{1+{\left(\frac{{c}_{v}t}{\beta {z}^{2}}\right)}^{\delta }}$ (11)

(12)

$\delta =A+B\frac{z}{H}$ (13)

$i=\frac{\frac{\alpha p}{{\gamma }_{w}z}}{1+{\left(\frac{{c}_{v}t}{{\beta }^{\prime }zH}\right)}^{A+B\frac{z}{H}}}$ (14)

$i=\frac{\alpha p}{{\gamma }_{w}z}-\frac{\frac{\alpha p}{{\gamma }_{w}z}}{1+{\left(\frac{{c}_{v}t}{\beta {z}^{2}}\right)}^{\delta }},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t<{\left[1/\left(\lambda -1\right)-1\right]}^{1/\delta }{t}_{0}$ (15)

$i=\frac{\frac{\alpha p}{{\gamma }_{w}z}}{1+{\left(\frac{{c}_{v}t}{{\beta }^{\prime }zH}\right)}^{A+B\frac{z}{H}}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge {\left[1/\left(\lambda -1\right)-1\right]}^{1/\delta }{t}_{0}$ (16)

3.3. 数值模拟

3.3.1. 瞬时加载结果分析

Figure 7. Distribution of pore pressure (t = 10,000 d)

Figure 8. Distribution of settlement (t = 10,000 d)

Figure 9. Distribution of settlement (t = 10,000 d)

Figure 10. Distribution of settlement (t = 10,000 d)

$i=\left\{\begin{array}{l}6.6-\frac{6.64}{1+{\left(t/45\right)}^{2.84}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\le 153\text{\hspace{0.17em}}\text{d}\\ \frac{6.64}{1+{\left(t/984.375\right)}^{1.696}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>153\text{\hspace{0.17em}}\text{d}\end{array}$ (17)

3.3.2. 线性加载结果分析

$i=\left\{\begin{array}{l}6.51-\frac{6.51}{1+{\left(t/62\right)}^{3.49}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\le 189\text{\hspace{0.17em}}\text{d}\\ \frac{6.51}{1+{\left(t/1160\right)}^{1.75}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>189\text{\hspace{0.17em}}\text{d}\end{array}$ (18)

Figure 11. Comparison of ABAQUS numerical solution and formula (15)

Figure 12. Variation of linear load with time

4. 结论

1) 水力梯度在固结过程中随时间先增大后减小，其变化曲线分为上升和下降两阶段的S型曲线，且

Figure 13. Distribution of pore pressure after consolidation

Figure 14. i-t curve of ABAQUS numerical solution

2) 水力梯度的Logistic模型中，无量纲参数α、β等取值应视具体情况考虑。本文以Terzaghi一维固结解析解为基础，虽存在一定的拟合误差，但能较好反映固结过程中水力梯度的变化规律，预测固结排水速率，分析固结过程机理。

Research on Logistic Model of Hydraulic Gradient in One-Dimensional Consolidation of Saturated Clay[J]. 渗流力学进展, 2018, 08(02): 22-34. https://doi.org/10.12677/APF.2018.82004

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