Stochastic Medium Model for the Settlement Calculation of Prefabricated Vertical Drains of Soft Soil Foundations in the Coastal Area of South China

Abstract

The prefabricated vertical drain (PVD) is an essential means to mitigate the settlement of soft soil foundations in coastal areas of South China. The commonly used elastoplastic analytical method cannot directly reflect the interaction of different PVDs and the resulting displacement of soft soil. At the same time, these elastoplastic analysis and numerical simulation methods are greatly influenced by the values adopted for rock and soil material parameters. In this paper, we present a stochastic medium model that can directly reflect the interaction of different PVDs and the resulting displacement of soft soil. It is not affected by the characteristics of rock and soil themselves and can also reflect the actual deformation process of soft soil. According to engineering practice, the settlement curves of soft soil foundations in coastal areas of South China with PVDs exhibited distinct normal distribution characteristics, which was consistent with the description of settlement by the stochastic medium model. Hence, based on the stochastic medium model, this paper analyzed the settlement mechanism of PVDs and established a stochastic medium model for the settlement calculation of PVDs. A function for the soft soil foundation in the coastal area of South China cross-section settlement curve was presented by back analysis of the PVD model. We chose the stochastic medium model based on this methodology to explore the interaction between different PVDs. The above models were then applied to an expressway in South China. Comparing actual settlement monitoring values to calculated values obtained with the PVD model, the error between the two models was less than 15%. This research provides a new settlement calculation method of PVDs in soft soil foundations in the coastal area of South China and a new basis for designing soft soil foundations.

1. Introduction

Soft soil foundations in the coastal area of South China are prone to uneven settlement due to seepage. PVD preloading is an important method to address the problem of uneven settlement of soft soil foundations in the coastal area of South China. The PVD plays an essential role in solving the seepage problem in soft soil foundation engineering [1,2,3,4,5]. Construction projects involving soft soil foundations in the coastal area of South China are increasing [6,7,8,9,10,11,12,13]. The PVD has large-scale application prospects in the future. The basic function of the PVD is to facilitate vertical drainage by its advance incorporation into a soft soil foundation in the coastal area of South China. The principle of the PVD is to shorten the drainage distance of soft soil, which is associated with excess pore pressure in soft soil under surcharge loading. If excess pore pressure occurs, pore water flows from areas of high excess pore pressure to regions of low excess pore pressure. PVDs accelerate consolidation and soft soil foundation in the settlement of the coastal area of South China.

Research on settlement calculation of PVDs in soft soil foundations in the coastal area of South China has mainly focused on analytical calculations and numerical simulations. Concerning the analytical calculation approach, Barron et al. [14] deduced an analytical solution for the settlement of a single PVD by assuming ideal drainage conditions of a soft soil foundation in the coastal area of South China, and Hird et al. [15] extended the settlement calculation of PVDs to plane strain conditions. To explore a more accurate analytical solution, Hansbo [16] considered the perimeter of the PVD in the model, and Lu et al. [17] considered the area of the PVD in the model. More and more PVD models are being used in the engineering field [18,19]. In practical engineering, soft soil foundation settlement in the coastal area of South China is usually uneven, with the assumption that plane strain conditions are inadequate. Indraratna and Redana [20] were accounting for smear and well resistance in plane strain analysis of PVDs, but the influence of the PVD itself was not considered. Hansbo [21] considered the smear and well resistance associated with PVDs and accounted for change in displacement due to the installation of PVDs, but the whole derivation process assumed that the drainage amount was the same in the consolidation process of the same volume of soft soil foundation in the coastal area of South China. The change in displacement in the process of soft soil foundation in the coastal area of South China reinforcement was ignored, and the settlement of a single PVD was only analytically calculated.

In terms of the numerical simulation approach, the cross section of a PVD has often been considered as a rectangle whose width is larger than its thickness [22]. Ngo et al. [23] proposed a numerical method suitable for the settlement calculation of soft soil foundations in the coastal area of South China by changing the size of the PVD cross section. In past research, a cylindrical model has been used to simulate PVDs [24,25]. Huang et al. [26] established a settlement equation by constructing a cylindrical model with an elliptical cross section. However, the adoption of an elliptical cross section makes appropriate meshing of PVDs difficult in finite element analysis. To develop a more accurate PVD model, Abuel-Naga et al. [27] and Atkinson et al. [28] further improved the model proposed by Hansbo. However, these two equivalent methods considering the shape of the PVD itself did not consider the associated displacement change in soft soil foundation consolidation in the coastal area of South China.

PVDs are often used for soft soil foundation in the coastal area of South China treatment in practical engineering. It has been fully proven that a PVD can be applied to solve the settlement problem of soft soil foundations in the coastal area of South China. Many scholars have proposed analytic functions to calculate the settlement of soft soil foundations in the coastal area of South China with PVDs. However, the aforementioned analytical calculation methods only considered the influence of a single PVD on the settlement calculation. In addition, the analytical calculation methods mainly focused on making the analytical results more accurate and the PVD model more representative of the actual soft soil foundation in the coastal area of South China.

Soft soil foundation in the coastal area of South China settlement curves exhibit the characteristics of a normal distribution, which is consistent with the description of settlement by the stochastic medium theory [29]. The commonly used elastoplastic analytical method cannot directly reflect the interaction of different PVDs and the movement law of soft soil element. At the same time, these elastoplastic analysis methods and numerical simulation methods are greatly influenced by test parameters of rock and soil properties. However, the stochastic medium model can directly reflect the interaction of different PVDs and the movement law of soft soil elements. It is not affected by test parameters of rock and soil properties and can also reflect the actual deformation process of soft soil. The stochastic medium model is calculated according to the measured values, so the stochastic medium model can consider creep. The elastoviscoplaticity approach indirectly reflects creep by simulating the constitutive model. It is difficult to apply the elastoviscoplaticity approach to soft soil foundation with PVD, but the stochastic medium method can. The radial orientation of the stochastic medium model changes from the center of a single drainage plate to the center of a group of drainage plates. In this paper, we propose a stochastic medium model for the settlement calculation of PVDs in soft soil foundations in the coastal area of South China by combining a settlement calculation model of a single PVD drain with the stochastic medium model. A function for the soft soil foundation in the coastal area of South China cross-section settlement curve is derived, and in this way, the interaction between multiple PVDs is effectively explored. The method was applied in the analysis of embankment construction for an expressway to verify its rationality and reliability.

2. Model Development

For this paper we focused on deep soft soil without considering any hard layer below. Considering the low permeability of soft soil foundations in the coastal area of South China and the construction of PVDs, the settlement curve of soft soil foundation in the coastal area of South China cross-sections was analyzed, and a stochastic medium model for the settlement calculation of a single PVD was established. The drainage area of the single PVD was defined as a cylinder in the settlement model with the PVD as the axis, and the influence radius of the PVD with respect to settlement was regarded as a fixed value. The stochastic medium model for the settlement calculation of PVDs was established by superposition of multiple PVDs. A function for the soft soil foundation in the coastal area of South China cross-section settlement curve was obtained by back analysis of the PVD model. The specific model construction method is shown in Figure 1.

In engineering, settlement may be accelerated by the addition of PVDs to a soft soil foundation in the coastal area of South China. The settlement curve of the soft soil foundation in the coastal area of South China follows a normal distribution, which is consistent with the description of settlement by the stochastic medium theory [30,31,32]. Therefore, the soft soil foundation in the coastal area of South China may be regarded as a stochastic medium in the study of its settlement.

In view of the settlement of a soft soil foundation in the coastal area of South China caused by a PVD, the stochastic medium model states that after the formation is drained, the groundwater level underneath the PVD assumes the shape of a funnel whose axis aligns with that of the PVD. As shown in Figure 2a, h is the groundwater level before the PVD is laid, ρ(η) is the influence radius of the PVD at depth η after it has been laid, θ is the angle in the horizontal direction with the center line of the PVD as the axis, and H is the maximum height of the groundwater level after the PVD has been laid. With the use of a single PVD, the soil is assumed to be horizontally isotropic.

A cylindrical coordinate system was established, as shown in Figure 2b. The coordinates of the soft soil at a point are ($\mathit{\epsilon }$,$\mathit{\theta },\mathit{\eta })$, and an infinitesimal wedge of soil has dimensions ($\mathit{d}\mathit{\epsilon }$,$\mathit{d}\mathit{\theta },\mathit{d}\mathit{\eta })$. The drainage funnel equation is denoted as $\mathit{f}\left(\mathit{\rho }\right)$, and the compressive vertical deformation of the soil wedge ($\mathit{d}\mathit{\epsilon }$,$\mathit{d}\mathit{\theta },\mathit{d}\mathit{\eta })$ is$\mathit{dS}$. We then have

$$\mathit{dS}=-\frac{\Delta \mathit{e}}{{1+\mathit{e}}_0}\mathit{dZ}=\frac{{\mathit{\alpha }}_{\mathit{v}}\Delta \mathit{p}}{{1+\mathit{e}}_0}\mathit{dZ}=\frac{{\mathit{\alpha }}_{\mathit{v}}{(\mathit{Z}-\mathit{H})\mathit{\gamma }}_{\mathit{w}}}{{1+\mathit{e}}_0}\mathit{dZ}$$

(1)

where $\mathit{C}\left(\mathit{Z}\right)=\frac{{\mathit{\alpha }}_{\mathit{\nu }}{(\mathit{Z}-\mathit{H})\mathit{\gamma }}_{\mathit{w}}}{{1+\mathit{e}}_0}$ is the settlement function of the soil at depth $\mathit{Z}\mathrm{,} {\mathit{\alpha }}_{\mathit{\nu }}$ is the compressibility, ${\mathit{e}}_0$ is the initial void ratio of the soft soil, and ${\mathit{\gamma }}_{\mathit{w}}$ is the unit weight of the soft soil. According to the stochastic medium theory [33], the settlement (${\mathit{W}}_{\mathit{e}}$) caused by soft soil compression located at a distance of $\mathit{\rho }$ away from the central axis of the PVD is as follows:

$${\mathit{W}}_{\mathit{e}}\left(\mathit{\rho }\right)=\frac{1}{{\mathit{\gamma }}^2\left(\mathit{\eta }\right)}\exp \left[-\frac{\mathit{\pi }}{{\mathit{\gamma }}^2\left(\mathit{\eta }\right)}\left({\mathit{\rho }}^2{+\mathit{\epsilon }}^2-2\mathit{\rho }\mathit{\epsilon }\cos \mathit{\theta }\right)\right]\mathit{\epsilon }\mathit{d}\mathit{\epsilon }\mathit{d}\mathit{\theta }\mathit{d}\mathit{\eta }$$

(2)

where $\mathit{\gamma }\left(\mathit{\eta }\right)=\frac{\mathit{\eta }}{\mathit{tan}\mathit{\beta }}$, $\mathrm{and} \mathit{\beta }$ is the major influence angle of the soil, which affects the shape of the surface settlement trough. The effective stress increment of soft soil is $\Delta {\mathit{p}=(\mathit{Z}-\mathit{H})\mathit{\gamma }}_{\mathit{w}}$. The compressibility is ${\mathit{\alpha }}_{\mathit{\nu }}=\Delta \mathit{e}/\Delta \mathit{p}$. On the horizontal plane of $\mathit{\eta }$, the surface settlement is caused by soft soil consolidation due to drainage at ρ away from the central axis of the PVD.${ \mathit{W}}_{\mathit{e}}(\mathit{\rho },\mathit{Z})$ is expressed as follows:

$${\mathit{W}}_{\mathit{e}}\left(\mathit{\rho },\mathit{Z}\right)={{\displaystyle \int }}_0^{2\mathit{\pi }}{{\displaystyle \int }}_0^{\mathit{\gamma }\left(\mathit{\eta }\right)}{\mathit{W}}_{\mathit{e}}\left(\mathit{\rho }\right)\mathit{C}\left(\mathit{\eta }\right)\mathit{d}\mathit{\eta }\mathit{d}\mathit{\theta }$$

(3)

In this work, we studied the layout of PVDs according to equilateral triangles, as shown in Figure 3. It is necessary to consider the composite effect of several PVDs. For simplicity, the influence range of each PVD was assumed to be a circle. For the regular triangle drainage board, its main influence range is equivalent to the adjacent regular hexagon distribution drainage board, and the influence range of three drainage plates is equivalent to a circle whose area is equal to the polygon area. Therefore, the circular superposition area of the three drainage plates is equal to the area of the adjacent regular hexagon. The radius S₂ of the equivalent circle is calculated as follows:

$${\mathit{S}}_2=\sqrt{\frac{\sqrt{3}}{2\mathit{\pi }}}{\mathit{S}}_1$$

(4)

where S₁ is the distance between the center lines of two PVDs.

A single PVD within a group of PVDs forms a drainage funnel, as shown with a solid line in Figure 2c, and its main part is the cylinder surface. Thus, the influence radius ρ(η) of a PVD drain among several PVDs is a constant value R₀, which is not equal to S₂.

$$\mathit{\rho }({\mathit{\eta })=\mathit{R}}_0$$

(5)

In the process of determining the influence radius of the single PVD within multiple PVDs, the problem of the interaction between multiple PVDs in soft soil foundations in the coastal area of South China is solved.

Substituting Equation (5) into Equation (3), the following is obtained:

$${\mathit{W}}_{\left(\mathit{R}\right)}={{\displaystyle \int }}_{\mathit{h}}^{\mathit{H}+\mathit{h}}{{\displaystyle \int }}_0^{2\mathit{\pi }}\frac{{2\mathit{R}}_0\mathit{tan}\mathit{\beta }}{\mathit{\eta }}·\exp \left\{-\frac{{\mathit{\pi }\mathit{tan}}^2\mathit{\beta }}{{\mathit{\eta }}^2}\left[{\left({\mathit{R}-\mathit{R}}_0\right)}^2{+2\mathit{RR}}_{\mathit{0}}\left(1-\cos \mathit{\theta }\right)\right]\right\}{\mathit{C}}_{\left(\mathit{Z}\right)}\mathit{d}\mathit{\theta }\mathit{d}\mathit{\eta }$$

(6)

where R is the distance between the measured settlement point and the center line of the soft soil foundation in the coastal area of South China. Based on the stochastic medium model, the settlement of soft soil foundation in the coastal area of South China is normally distributed [34]. To calculate the settlement of soft soil foundations in the coastal area of South China more accurately, Equation (6) is further optimized. Combined with the shape characteristics of the PVD, the drainage range caused by the PVD is regarded as a cylindrical drainage area, so the stochastic medium model and PVD are comprehensively combined. A single PVD model based on the stochastic medium model is established by Gaussian integration of Equation (6).

${\mathit{C}}_{\mathit{z}}=\frac{{\mathit{\alpha }}_{\mathit{v}}\mathit{Z}\mathit{\gamma }}{{1+\mathit{e}}_0}$ is substituted into Equation (6), ${\mathit{\theta }}^{\prime }=\frac{\mathit{\theta }-\mathit{\pi }}{\mathit{\pi }}$ is defined, with ${\mathit{\eta }}^{\prime }=\frac{\mathit{\eta }-\mathit{h}/2}{\mathit{h}/2}$, and these equations are inserted into the Gauss-type integral:

$${\mathit{W}}_{\left(\mathit{R}\right)}={{\displaystyle \int }}_{-1}^1{{\displaystyle \int }}_{-1}^1\frac{{\mathit{\pi }HR}_0{\mathit{\alpha }}_{\mathit{v}}{\mathit{\gamma }}_{\mathit{w}}\mathit{tan}\mathit{\beta }}{2\left({1+\mathit{e}}_0\right)}·\exp \left[-\frac{{4\mathit{\pi }\mathit{tan}}^2\mathit{\beta }}{{\left({\mathit{H}\mathit{\eta }}^{\prime }+\mathit{H}+2\mathit{h}\right)}^2}\left({\mathit{R}}^2{+\mathit{R}}_0^2+2\left|\mathit{R}\right|{\mathit{R}}_0{\cos \mathit{\pi }\mathit{\theta }}^{\prime }\right)\right]{\mathit{d}\mathit{\theta }}^{\prime }{\mathit{d}\mathit{\eta }}^{\prime }$$

(7)

The unknown quantity in the exponential part of Equation (7) cannot be integrated analytically. According to the Gauss–Legendre integration method, the Legendre polynomial is ${\mathit{L}}_{\mathit{n}}\left(\mathit{x}\right)=\frac{1}{2^{\mathit{n}}\mathit{n}!}\frac{{\mathit{d}}^{\mathit{n}}}{{\mathrm{dx}}^{\mathit{n}}}{\left[\right(\mathit{x}}^2{-1)}^{\mathit{n}}]$, where ${\mathit{L}}_0\left(\mathit{x}\right)=1$, which constitutes an orthogonal system in the interval of $[$−1, 1$]$. The n base points of the Gauss quadrature equation correspond to the n zeros of the Legendre polynomials of degree n.

Based on the properties of Legendre polynomials, the following is obtained:

$$\left({1-\mathit{x}}^2\right){\mathit{L}}_{\mathit{n}}^{\prime }\left(\mathit{x}\right)=\mathit{n}\left[{\mathit{L}}_{\mathit{n}-1}\left(\mathit{x}\right){-\mathrm{xL}}_{\mathit{n}}\left(\mathit{x}\right)\right]$$

(8)

The weight coefficients corresponding to the quadrature function are:

$${\mathit{A}}_{\mathit{k}}=\frac{{2(1-\mathit{x}}_{\mathit{k}}^2)}{{{[\mathrm{nL}}_{\mathit{n}-1}{(\mathit{x}}_{\mathit{k}}\left)\right]}^2} \mathit{k}=1, 2,\dots ,\mathit{n}$$

(9)

The Gaussian-type integral ${{\displaystyle \int }}_{-1}^1\mathit{f}\left(\mathit{x}\right)\mathit{dx}$ can be expanded to form ${\displaystyle \sum }_{\mathit{k}=1}^{\mathit{n}}\mathit{f}\left(\mathit{x}\right){\mathit{A}}_{\mathit{k}}$, and the error is as follows:

$$\mathit{E}\left(\mathit{f}\right)=\frac{2^{2\mathit{n}+1}{\left(\mathit{n}!\right)}^4}{\left(2\mathit{n}+1\right){\left[\left(2\mathit{n}\right)!\right]}^3}{\mathit{f}}^{\left(2\mathit{n}\right)}\left(\mathit{\xi }\right) -1<\mathit{\xi }<1$$

(10)

In regard to the double integral of $I={{\displaystyle \int }}_{-1}^1{{\displaystyle \int }}_{-1}^1{\mathit{f}}_1\left({\mathit{t}}_1{,\mathit{t}}_2\right){\mathit{dt}}_1{\mathit{dt}}_2$.

$$I={{\displaystyle \int }}_{-1}^1{\mathit{\psi }(\mathit{t}}_2{)\mathit{dt}}_2={\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{A}}_{\mathit{j}}\mathit{\psi }\left({\mathit{\lambda }}_{2\mathit{j}}\right)={\sum }_{\mathit{i}=1}^{\mathit{m}}{\sum }_{\mathit{j}=1}^{\mathit{n}}{\mathit{A}}_{\mathit{i}}{\mathit{A}}_{\mathit{j}}{\mathit{f}(\mathit{\lambda }}_{1\mathit{i}}{,\mathit{\lambda }}_{2\mathit{j}})$$

(11)

where m and n are the number of base points and ${\mathit{A}}_{\mathit{i}}$ and ${\mathit{A}}_{\mathit{j}}$ are the weighted coefficients corresponding to t₁ and t₂, respectively. For the same integral, the more base points of the integral there are, the greater the number of calculations and the longer the computation. A certain number of base points should be selected according to the required accuracy. In this paper, the Gauss–Legendre quadrature method with five base points is employed to solve the quadratic integral in the model.

The integrand is taken as:

$${\mathit{f}}_{0(\mathit{R},\mathit{\beta })}=\frac{{\mathit{\pi }\mathit{HR}}_0{\mathit{\alpha }}_{\mathit{v}}{\mathit{\gamma }}_{\mathit{w}}\mathit{tan}\mathit{\beta }}{{1+\mathit{e}}_0}·\mathit{exp}\left[-\frac{{4\mathit{\pi }\mathit{tan}}^2\mathit{\beta }}{{\left({\mathit{H}\mathit{\eta }}^{\prime }+\mathit{H}+2\mathit{h}\right)}^2}\left({\mathit{R}}^2{+\mathit{R}}_0^2+2\left|\mathit{R}\right|{\mathit{R}}_0\cos \mathit{\pi }\mathit{\theta }\right)\right]$$

(12)

The Gauss-type integral is expanded into five base points:

$$\mathit{f}({\mathit{R},\mathit{\beta })=\mathrm{\Lambda }[\mathit{A}}^{\left(5\right)}]\mathrm{\Lambda }$$

(13)

where $\mathrm{\Lambda }$ is an array with elements ${(\mathit{x}}_1{,\mathit{x}}_2{,\mathit{x}}_3{,\mathit{x}}_4{,\mathit{x}}_5)$ and

$${\mathit{A}}^{\left(5\right)}{=\mathit{A}}_{ij}=\mathit{f}\left({\mathit{x}}_{\mathit{i}}{,\mathit{x}}_{\mathit{j}},\mathit{R},\mathit{\beta }\right), 1\le \mathit{i}, \mathit{j}\le 5$$

(14)

With the use of mathematical software, Equation (12) is regarded as function of θ, η, R, and β. The Gauss–Legendre quadrature coefficients are listed in

Table 1. Gauss–Legendre quadrature basis points and coefficients.

n	${\mathbf{x}}_{\mathbf{k}}$	${\mathbf{A}}_{\mathbf{k}}$
2	±0.5774	1.0000
3	0	0.8889
	±0.7746	0.55565
4	±0.3400	0.6521
	±0.8611	0.3479
5	0	0.56898
	±0.5385	0.4786
	±0.9062	0.2369

.

Equation (13) is a stochastic medium model based on the settlement calculation of a single PVD in a soft soil foundation in the coastal area of South China. In the process of solving the single PVD model based on a stochastic medium by the Gauss–Legendre integral, the seepage problem in the consolidation process of a soft soil foundation in the coastal area of South China is solved.

A soft soil foundation in the coastal area of South China with a PVD is affected by horizontal seepage, which is associated with a force that arises due to the loading of the foundation. Soft soil foundations in the coastal area of South China are often treated with PVDs. Here, unique models are used, PVDs are combined into a single PVD of equivalent volume, and the overall settlement curve of the soft soil foundation in the coastal area of South China cross-section is obtained. The loading regime was the different surcharge height on soft soil. Different surcharge heights correspond to different pressure conditions. The magnitude of the stress associated with the central PVD will depend on surcharge loads. Based on this, the cross-section settlement function of the soft soil foundation in the coastal area of South China is assumed to be a normal distribution with the maximum value at the center line.

The specific calculation method is as follows:

The equivalent radius S₂ of each drain is calculated with Equation (4) and substituted into Equation (14) to obtain a stochastic medium model for the settlement calculation of PVDs in soft soil foundations in the coastal area of South China.

$$F(\mathit{R},\mathit{\beta })={\sum }_{\mathit{i}=0}^{(\mathit{n}-1)/2}\mathit{f}(\mathit{R},\mathit{\beta })$$

(15)

where $F(\mathit{R},\mathit{\beta })$ is the function of the settlement curve of the soft soil foundation in the coastal area of South China cross-section, n is the number of measured settlement points, and R is the distance between the measured settlement point and the center line of the soft soil foundation in the coastal area of South China cross-section. Calculation of consolidation settlement based on the stochastic medium model relies on the assumption that only vertical compression occurs in soft soil during consolidation. This reflects the characteristics of soft soil: soft soil is prone to vertical drainage under load.

For n measured settlement values $\mathit{F}\left(\mathit{R},\mathit{\beta }\right)$, a residual function $\Delta \mathit{f}\left(\mathit{\beta }\right)$ is obtained by the least-squares method.

$$\Delta \mathit{f}\left(\mathit{\beta }\right)={\sum }_{\mathit{j}=1}^{\mathit{n}}{[\mathit{F}\left(\mathit{R},\mathit{\beta }\right)-\mathit{F}\left(\mathit{R}\right)]}^2$$

(16)

The minimum value of $\mathit{\beta }$ in $\Delta \mathit{f}\left(\mathit{\beta }\right)$ is the optimal value of $\mathit{\beta }$. The total settlement curve of the soft soil foundation in the coastal area of South China cross-section is obtained by substituting $\mathit{\beta }$ into Equation (16). $\mathit{F}\left(\mathit{R}\right)$ is the settlement value of the measured settlement point. Different $\mathit{\beta }$ values correspond to different surcharge loads and can be varied according to different loading conditions. This method can be applied in soft soil under surcharge loads; surcharge can accelerate the consolidation of soft soil. The parameters controlling settlement are the parameters of measured settlement points. They come from field measurements and are calibrated by multiple measurements in the field.

The sensitivity analysis of settlement is carried out using the method of controlling variables, that is, changing one parameter and fixing the other parameters, to determine the sensitivity degree of each parameter and the change law with the parameters, involving six parameters: the maximum height of the groundwater level after the PVD has been laid H, compressibility${ \mathit{\alpha }}_{\mathit{\nu }}$, void ratio e, the unit weight of the soft soil${ \mathit{\gamma }}_{\mathit{w}}$, the number of measured settlement points n, and the distance between the measured settlement point and the center line of the soft soil foundation$\mathit{R}$. Since the initial values of different model parameters have certain influence on the sensitivity analysis, it is proposed to select the data in line with the actual project as the initial values and let them change the amplitude to ± 50%. The results are shown in

Table 2. Sensitivity analysis of parameter change to settlement.

Change Rate of Each Parameter (%)	H (m)		e		n		$\mathit{R}$ (m)		${\mathit{\alpha }}_{\mathit{\nu }}$ $\left(\mathbf{MPa}{-}^1\right)$		${\mathit{\gamma }}_{\mathit{w}}$ $(\mathbf{kN}/{\mathbf{m}}^3)$
	f (m)	∆f (%)	f (m)	∆f (%)	f (m)	∆f (%)	f (m)	∆f (%)	f (m)	∆f (%)	f (m)	∆f (%)
+50	2.35	+10.6	2.25	+7.1	2.39	+13.8	2.42	+15.2	2.22	+5.7	2.3	+9.5
+20	2.27	+8	2.21	+5.2	2.34	+11.4	2.37	+12.9	2.19	+4.3	2.26	+7.6
0	2.1	0	2.1	0	2.1	0	2.1	0	2.1	0	2.1	0
−20	1.96	−6.6	2.02	−3.8	−1.89	10	1.84	12.6	−2.04	4.2	1.96	−6.7
−50	1.88	−10.4	1.95	−0.71	−1.85	11.9	1.79	14.8	−1.97	6.2	1.91	−9

, where f represents the settlement and ∆f represents the settlement change rate.

3. Case Study

3.1. Site Description or Field Data

The proposed model was applied to the construction of embankments for an expressway to verify its reliability. The proposed model was tested against field data in order to verify its robustness and applicability. The expressway is located on a deltaic alluvial plain, and the terrain is flat. The subsurface profile is divided into four layers, the first layer is silt, the second layer is silt, the third layer is muddy loam, and the fourth layer is muddy silt. The soft soil in South China has strong nonlinear creep characteristics, and the nonlinear characteristics gradually appear through the increase of deviator stress level. The relationship between pore pressure increment and stress increment is not linear.

The length of the PVDs on this project was 30.3 m. Settlement of the soft soil foundation in the coastal area of South China was calculated for cumulative surcharge heights of 0.541 m, 0.942 m, 3.928 m, and 6.059 m. Samples were collected via drilling at the left, middle, and right of the K41 + 277 section of the expressway. The geotechnical parameters are listed in

Table 3. Geotechnical parameters of K41+277 section of the expressway.

Drilling Position	Moisture Content (%)	Specific Gravity	Wet Density (g/cm3)	Dry Density (g/cm3)	Degree of Saturation (%)	Void Ratio	Liquid Limit (%)	Plastic Limit (%)	Plasticity Index	Liquidity Index	Classification and Name of the Soil Samples
K41+277 (left)	54.2	2.73	1.91	1.24	100.0	1.204	26.2	14.1	12.1	3.31	silty soil
K41+277 (middle)	25.0	2.72	1.89	1.51	85.1	0.799	27.7	14.7	13.0	0.79	mild clay
K41+277 (right)	29.2	2.72	1.95	1.51	99.0	0.802	27.3	13.5	13.8	1.14	silty soil

, and photographs of the samples are shown in Figure 4. The test was conducted according to GB/T 36197-2018/ISO 10381-2. The sampling standard was 1m in length and 200mm in diameter. After loading the soft soil, we obtained samples of the soft soil to carry out a consolidation test. The deformation value of the consolidation test corresponds to the change of consolidation parameters caused by surcharge load. The consolidation test sample itself has been consolidated in the early stage. The settlement after surcharge is generated on the basis of early consolidation.

After land levelling, a sand cushion was laid first, and PVDs were then laid. It was observed that the PVDs were equidistantly distributed in a triangular pattern as per Figure 3, at a spacing of S₁ = 1.2–1.5 m. The site layout is shown in Figure 5.

After the PVDs were laid, on-site monitoring instruments were installed (Figure 6). Field settlement monitoring relied on layered settlement pipes and inclinometer tubes. The inclinometer tube monitors the horizontal settlement of soft soil, and the layered settlement tube mainly monitors the vertical settlement of soft soil; the whole construction cycle is under monitoring, as shown in Figure 6. We obtained settlement data through on-site monitoring.

Three settlement measurement points were selected in the field, as indicated in

Table 4. Measured values of the settlement points for the expressway.

Measured Settlement Points	R	β
$n_1$	0	−1.2
$n_2$	16	−1.08
$n_3$	24	−0.9

. Calculations were carried out with the model proposed in this paper.

3.2. Model Application

The average geotechnical parameters of the left, middle, and right of the K41+277 section were taken as the initial values for the model. The calculation steps were as follows:

(1)

It was supposed that the water table is located at the ground surface (h = 0).

(2)

Initial values of H = 3.0 m, ${\mathit{\alpha }}_{\mathit{\nu }}$ = 1.5 × 10⁻⁶ MPa⁻¹, e = 1.5, and ${\mathit{\gamma }}_{\mathit{w}}$= 9.8 kN/m³ were adopted and determined into Equation (7).

(3)

Function f(R, β) was expanded according to Equations (13) and (14), and single PVD model function f(θ, η, R, β) was obtained.

(4)

The function of the soft soil foundation in the coastal area of South China cross-section settlement curve F(R, β) was superimposed according to Equation (15).

The residual function $\mathit{\Delta }\mathit{f}\left(\mathit{\beta }\right)$ was formulated as follows:

$$\mathit{\Delta }\mathit{f}\left(\mathit{\beta }\right)={\left[\mathit{F}\left(0,\mathit{\beta }\right)-\left(-1.2\right)\right]}^2+{\left[\mathit{F}\left(16,\mathit{\beta }\right)-\left(-1.08\right)\right]}^2+{\left[\mathit{F}\left(24,\mathit{\beta }\right)-\left(-0.9\right)\right]}^2$$

(17)

A curve of △f(β) with β in the interval of [0.11, 1.55] was obtained with mathematical software, as shown in Figure 7.

Δf(β) was always greater than zero, and it had an extreme value in the interval where β was meaningful. According to Figure 7, the value of β where Δf(β) is minimum is 0.443. Substituting this value into Equations (7) and (16), a function of the soft soil foundation in the coastal area of South China cross-section settlement curve and the settlement curve of the soft soil foundation in the coastal area of South China cross-section were obtained, as shown in Figure 8.

Layered sand blowing and surcharge loading were implemented during construction. The void ratio and compressibility of the soft soil changed with the construction process of surcharge loading→consolidation→surcharge loading.

Table 5. Results of K41+277 consolidation tests for the expressway.

	e				α (MPa−1)
Load	50 kPa	100 kPa	200 kPa	400 kPa	50 kPa	100 kPa	200 kPa	400 kPa
K41+277 (left)	1.525	1.319	1.130	0.961	6.849	4.123	1.893	0.845
K41+277 (middle)	1.162	1.093	1.006	0.912	2.905	1.382	0.862	0.473
K41+277 (right)	1.737	1.551	1.273	1.057	2.732	3.718	2.778	1.081

shows the results of consolidation tests where the soft soil parameters changed with the load increments of 50 kPa, 100 kPa, 200 kPa, and 400 kPa. The compressibility calculation is shown in Equation (18).

$$\mathit{\alpha }=1000·(\frac{e_1-e_2}{p_2-p_1})$$

(18)

1000 is the unit conversion factor, p is the pressure, e is the void ratio.

Selecting the average values of the void ratio and compressibility in the left, middle, and right boreholes of section K41+277 under equal load action, as listed in Table 4, the least-squares method was applied to fit quadratic curves, as shown in Figure 9.

When the applied load is 50–400 kPa, the void ratio and compressibility of the soft soil decreases with an increase in pressure. The pressures corresponding to the four surcharge loads applied in the field were 10.16 kPa, 17.69 kPa, 73.77 kPa, and 113.79 kPa. The corresponding void ratio and compressibility values were obtained according to fitting function in Figure 9, and are listed in

Table 6. Calculation results of the pressure, void ratio, and compressibility coefficient for the K41+277 section of the expressway.

Surcharge Height (m)	p (kPa)	e	$\mathit{\alpha }$ (MPa−1)
0.541	10.16	1.590	4.941
0.942	17.69	1.566	4.775
3.928	73.77	1.401	3.645
6.059	113.79	1.301	2.954

.

3.3. Analysis of Results

The measured settlement values under the four on-site surcharge loads were compared to the settlement values calculated with the PVD model. The settlement values were 98, 288, 167, and 46 days after the application of the first, second, third, and fourth surcharge loads, respectively. The β values corresponding to the four surcharge heights of 0.541 m, 0.942 m, 3.928 m, and 6.059m were calculated with Equations (15) and (16), and were determined as 0.00611, 0.08773, 0.19351, and 0.27170, respectively. The calculated and measured settlement values at each surcharge load level are shown in Figure 10 and Figure 11, respectively.

The ratio of the difference between the maximum measured and calculated settlements to the maximum measured settlement was adopted as the relative error, which was calculated for each of the surcharge loads as shown in

Table 7. Comparison of the calculated and measured settlement values.

Surcharge Load Number	Height of the Surcharge Load (mm)	Maximum Measured Settlement (mm)	Maximum Calculated Settlement (mm)	Difference (mm)	Relative Error (%)
1	541	140	123	17	12.1
2	942	1355	1207	148	10.9
3	3928	2017	1900	117	5.8
4	6059	2101	2035	66	3.1

.

The following observations were made of the results:

(1)

The model curve was almost a straight line in the interval of −24-24 m when the surcharge height was 0.541 m.

(2)

The maximum settlement values under the first, second, third, and fourth surcharge loads obtained according to the PVD model were 0.123 m, 1.207 m, 1.900 m, and 2.035 m, respectively, and the relative errors compared to the measured settlements of 17 mm, 1355 mm, 2017 mm, and 2101 mm were 12.1%, 10.9%, 5.8%, and 3.1%, respectively. In the range of 50-400 kPa, the higher the surcharge height was, the greater the surcharge load was. The greater the surcharge load was, the higher the accuracy of the PVDs model was. This is because the settlement of soft soil is higher at the initial stage of loading. At the same time, in the initial stage of loading, settlement caused by mechanical load and construction load accounted for a large proportion of the total settlement. Therefore, the larger the stacking height was, the smaller the calculation error of the PVD model was.

(3)

The PVD spacing for the expressway in southern China was 1.2-1.5m, the influence range of a single PVD was 15 m [35,36], the measured settlement curve of the soft soil foundation in the coastal area of South China cross-section and the settlement curve obtained with the PVD model looked similar to quadratic functions that were convex upward, and the maximum value occurs at the center line of the cross-section. The settlement is highest at the center line of the foundation because the horizontal confinement of the soil is the greatest at this point.

(4)

The settlement value of the PVD model was always slightly smaller than the measured settlement value [36]. The PVD model only calculated the settlement of the soft soil itself, but the mechanical load and construction load on site would also produce settlement.

4. Conclusions

(1)

In this paper, the stochastic medium model is combined with the PVD. The stochastic medium model for the settlement calculation of PVDs was established by the superposition of multiple PVDs. A function for the soft soil foundation in the coastal area of South China cross-section settlement curve was obtained. The problem of interaction between multiple PVDs in the settlement calculation was explored in a way closer to the reality.

(2)

This article included an application to the soft soil foundation in the coastal area of South China treatment of an expressway. Comparing actual monitoring data to calculation results of the stochastic medium PVD model, the relative error between the predicted and actual measured values of the soft soil foundation in the coastal area of South China cross-section was less than 15%. When the preloading strength was between 50–400 kPa, the greater the surcharge load was, the smaller the error between the measured and predicted maximum settlements was.

(3)

The model proposed in this paper could be applied to calculate the settlement of soft soil foundations in the coastal area of South China under surcharge preloading and could be extended to three-dimensional settlement calculations of soft soil foundations in the coastal area of South China. The optimal surcharge height could be determined by progressively increasing the surcharge height.