# Analytical Solution for Consolidation Behaviors of Combined Composite Foundation Reinforced with Penetrated PCCSs and Floating DM Columns

Authors: Chaozhe Zhang; Dingwen Zhang; Songyu Liu; Jianyong Han; Chen Jiang; Yue Zhao
Journal: Geofluids
 (2023)

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## Abstract

As the composite pile, the precast concrete piles reinforced with cement-treated soil (PCCS) is formed by driving the precast cement (PC) pile into the deep mixing (DM) column, which has been successfully and widely utilized to support buildings and embankments over soft soil. To increase the pile spacing and give full play to the economic merits of the PCCS, a reinforcement scheme, which involves the combined use of rigid piles and flexible columns, employing penetrated PCCSs and floating DM columns is proposed and utilized for soft soil ground treatment. However, there is a lack of feasible method for consolidation behaviors of this combined composite foundation (CCF) reinforced with penetrated PCCSs and floating DM columns under flexible loads. This paper developed an analytical solution to predict the average consolidation degree of this CCF based on a cylinder consolidation model and double-layer ground consolidation theory. The excess pore pressure and average consolidation degree were calculated by considering the composite pile penetration into the cushion. The analytical method agrees well with results obtained by numerical analysis. Additionally, a parametric study was conducted systematically to analyze the effect of key influence factors on the average consolidation degree of this CCF. The results indicate that the consolidation rate of this CCF can be much faster than that of the natural ground. The consolidation rate strongly depends on the compressive modulus and area replacement ratio of PCCSs. The increasing inner core-outer core modulus ratio and the inner core-subsoil modulus ratio increase the consolidation rate of this CCF. In addition, the consolidation rate increases with the gravel cushion-subsoil modulus ratio, while it decreases with the loading period.


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## Body

## 1. Introduction

The composite ground has been widely used in ground improvement due to its convenience and efficiency [1–7]. A variety of pile techniques have been developed to meet various engineering needs, such as stone columns, deep mixing (DM) columns, T-shaped DM columns, jet grouted columns, and precast cement (PC) piles [8–11]. To give full play to merits of different pile techniques, the combined use of multiple pile techniques for soft soil ground improvement was developed [12–15]. Because the combined composite foundation (CCF) can take advantage of multiple performances of every pile technique, it is becoming more well-liked everywhere, but notably in China. [16, 17]. In this CCF system, the piles have different lengths and diameters and use various materials. The common type of the CCF is the composite ground reinforced by penetrated rigid piles (long piles) and floating flexible or semirigid piles (short piles). In general, the penetrated rigid piles can control the settlements by transferring loads to the deep firm stratum or bearing stratum via piles. The floating columns can improve the bearing capacity of shallow soft soil layers [13].Similarly, the combination concept of two types of piles also applies to the single pile to form the composite pile, which can exploit both merits. The precast concrete pile reinforced with cement-treated soil (PCCS) is a novel type of composite pile, which is created by driving the PC pile, termed inner core, into the DM column, termed outer core [18–21]. In order to increase the pile spacing and give full play to the economic merits of the PCCS, a novel CCF reinforced by penetrated PCCSs and floating DM columns is proposed and utilized to improve soft soils.In recent years, the several types of combined composite foundations with reinforcing elements have been proposed, including the PC piles united with lime columns, steel pipe piles united with sand columns, lime columns united with sand columns [22] as well as cement fly-ash gravel (CFG) piles united with lime-soil columns [23]. A comparison of CCFs reinforced by different multiple vertical reinforcement elements was carried out to reveal the bearing mechanism and performance. According to the deformation consistency of pile-soil and the bilinear elastoplastic model of the soil, Zheng et al. [24] proposed the analytical solution for the CCF under elastic and plastic circumstances to calculate the composite modulus accurately and built a design method for the multicolumn composite foundation with DM columns united with stone columns [25].The bearing performances of the PCCS were investigated by means of experimental, theoretical, and numerical analysis under vertical and lateral loads [26–31]. The inner core is installed to bear external loads, while the outer core can increase the shaft friction of the inner core and effectively transfer the inner core’s axial force to the surrounding soil [18]. The PCCS can reduce costs by nearly 30% under the same bearing capacity provided by the PCCS and the piles with the same geometric dimensioning [32]. Wang et al. [20] developed a practical analytical approach for predicting the bearing capacity of the PCCS considering the pile-soil interaction and nonhomogeneity of subsoil shear. Wang et al. [19] proposed a modified p-y curve model to calculate the lateral bearing capacity of the PCCS based on field tests of the PCCSs and indicated that cemented soil could reduce the bending moment and lateral deflections. Based on tested results, Wang et al. [33] found that the stiffened deep mixing (SDM) column-supported embankment could significantly reduce the postconstruction settlement by 27.3%-68.3 compared to the conventional DM column-reinforced composite foundation. Ye et al. [26] established a series of finite element (FE) models to analyze the bearing mechanism of the SDM column-supported embankment.In addition, several methodologies consist of the field-scale tests, numerical analysis as well as analytical solutions for studying the consolidation behaviors of the composite foundations reinforced by different vertical elements. Because of the accuracy and convenience for end-users, analytical solutions are more popular than field measurement and numerical methods [16, 34, 35]. Considerable achievements on the consolidation of the permeable column- and impermeable pile-reinforced composite foundations which adopted Biot’s consolidation theory have been obtained [13, 36, 37]. As for the granular column-improved composite grounds under various time-dependent loading, many analytical solutions have been proposed considering well resistance, smear effect, consolidation of stone columns, and several types of drained boundary [38–42]. The analytical methods for consolidation behaviors of impermeable pile-improved soft grounds also have been developed considering floating and penetrated piles, piercing into the gravel cushion, a partial drained boundary as well as the varying vertical stress increment with depth in subsoils [2, 3, 9]. In addition, the consolidation behaviors of the CCF improved by cohesive piles with prefabricated vertical drains have been investigated. Lu et al. [16] presented two analytical models of the composite foundation improved with impervious columns and vertical drains under different distribution modes. However, there are limited research on the consolidation of the CCF with multiple types of piles. Abusharar et al. [43] studied the consolidation of a CCF with CFG piles united with lime columns under embankment load and found that this CCF can increase the consolidation rate and embankment stability. Yang et al. [13] proposed a theoretical method of the CCF improved by impervious long piles and impervious short DM columns. Since, the CCF reinforced by penetrated PCCSs and floating DM columns is a novel technique, whose consolidation behaviors need further research.The main purpose of this work is to propose an analytical solution for consolidation of the CCF reinforced with penetrated PCCSs and floating DM columns, which is simplified as a double-layered consolidation foundation based on a quasiequal-strain assumption [44]. The analytical solution of the average consolidation degree of this CCF is derived by the boundary conditions of vertical additional stress increment linearly with depth in subsoils. Moreover, the proposed analytical method is validated by numerical analysis. A sensitive analysis is conducted to describe the key influence factors on the consolidation behaviors of this CCF.


## 2. Mathematical Model

### 2.1. Model Description and Basic Assumptions

The penetrated PCCSs as the long piles and floating DM columns as the short columns can be arranged in four types of distribution modes, including triangular, intensive triangular, rectangular, and intensive rectangular distributions, as shown in Figure1. The primary step in analyzing the consolidation behaviors is partitioning this multipile composite ground into the square unit cells. The unit cell is adopted as the representative of the CCP and converted into a cylindrical unit cell using based on equivalent hypothesize of the sectional area [16, 17]. In general, the unit cell can be adopted in two patterns: (1) Form A is a PCCS at the center with N DM columns dispersed separately at its boundary and (2) Form B is a DM column at the center with N PCCSs distributed separately at its perimeter. The cross-sectional area of each PCCS distributed independently at its boundary within the unit cell makes up the value of N. With respect to triangular and rectangular distribution patterns, selecting any pile at the center for the consolidation path in the subsoil is constant, since the PCCS and DM columns can be regarded as impervious piles. However, for the intensive rectangular distribution patterns, it is necessary to select the PCCS located at the center. Otherwise, it makes the problem more complex when other DM columns are distributed at the circumferential boundary. Therefore, Form B is adopted in this paper to analyze the cases of rectangular and intensive rectangular distributions.Figure 1
Layout of this CCF reinforced with penetrated PCCSs and floating DM columns.In the current study, the penetrated PCCS with a long inner core as the long pile and the floating DM column as the short column is installed in a rectangular mode with identical center-to-center spacing ofSp. This penetrated PCCS penetrates the whole soft soil layers seated on an impermeable bearing layer in this composite ground, and one PCCS at the center and eight DM columns are distributed at the boundary, where the value of N is 3. Figure 2 presents the cylinder model with an equal cross-sectional area transferred from the unit cell. Its simplified model comprises two improved areas according to the cross-section of DM column, which are named Region A and Region B with the permeability coefficient (kv1 and kv2), compressive modulus (Es1 and Es2), consolidation coefficient (cv1 and cv2), and length (hl and h2), respectively. rdm and rpc are the radius of the PC pile and the DM column, respectively. Edm and Epc are the compressive modulus of the concrete and the cemented soil, respectively. rn is the radius of the unit cell, and re is the outside radius of the simplified model of the CCF, which can be determined as follows:
(1)re2=rn2−Nrdm2,π4rn2=Sp2.Figure 2
Schematic diagram of consolidation model; (a) plan view; (b) elevation plan.
(a)(b)The reinforced gravel cushion laying on the piles and surrounding soil is subjected to a ramp load,pt. Water flow is prohibited across the bottom and lateral surfaces of the model due to the underlying impermeable layer, whilst is allowed across the ground surface to analyze the single drainage of the CCF.The following assumptions during the analysis in this paper are given below:(i)
Darcy’s law is obeyed for the seepage of the water in the surrounding soil.kv1, kv2, Es1, and Es2 are assumed to remain constant. The subsoils are regarded as fully saturated state(ii)
PC piles and DM columns are assumed to be impervious, where no flow and the consolidation occur at the PC pile-DM column interface and DM column-soil interface(iii)
There is no interface slip between the PC pile-cemented soil of the PCCS(iv)
It is considered that the vertical strains of the DM column and its surrounding soil are equal at any depth(v)
A quasiequal strain assumption is used between the PCCS and its surrounding soil due to the pile top pierced into the gravel cushion to generate the relatively large penetration,δ1 [45]. The deformation of the surrounding soil is relatively close to the total of the pile shaft compression and the pile head penetration value. Since the pile tip is located on a firm soil layer, the pile tip piercing into the underlying soil layer is not taken into account. Therefore, the volumetric strain, εv, can be obtained as follows:(2)εv=σ¯pcEpc+δ1h1=σ¯dm1Edm+δ1h1=σ¯dm2Edm=σ¯s1−u¯1Es1,(3)δ1=π1−μc2pacrp2Ec,where σ¯pc, σ¯dm1, and σ¯dm2 are the vertical stress of inner core and outer core of the PCCS as well as the DM column, respectively. σ¯s1 and u¯s1 are the vertical stress and the excess pore pressure of the soil around piles in Region A, respectively. pac, μc, and Ec are the vertical stress of the pile top, the Poisson’s ratio, and the compressive modulus of the cushion, respectively.Substituting Equation (3) into Equation (2) yields
(4)εv=σ¯pcEepc=σ¯dm1Eedm=σ¯dm2Edm=σ¯s1−u¯1Es1,where Eepc and Eedm are the equivalent compressive moduli of PC pile and DM column of the PCCS:
(5)Eepc=2h1EcEpc2h1Ec+π1−μc2Epcrsp,Eedm=2h1EcEdm2h1Ec+π1−μc2Edmrdm.


### 2.2. Governing Equation

#### 2.2.1. Consolidation Equation for the Subsoil in Region A

In the axisymmetric consolidation model, the average vertical stress in Region A within the CCF is transferred from the cushion to the DM columns, the PCCSs, and the soil around piles.σt,z is the average total stress in this CCF induced by the external load, pt. The equilibrium equation at any depth in Region A can be written as follows:
(6)1−m1−m2σ¯s1+m11−ρσ¯dm1+ρσ¯pc+m2σ¯dm2=σt,z,where m1 and m2 are the area replacement ratios of the PCCS and DM column, respectively, and ρ is the ratio between the inner and outer cores’ cross-sectional areas.From Equation (4) and Equation (6), the vertical volumetric strain of the PCCSs, DM columns, and the subsoil at any depth, εv, can be expressed as follows:
(7)εv=σt,z−1−m1−m2u¯s1Ecomp1,(8)Ecomp1=m11−ρEedm+ρEepc+m2Edm+1−m1−m2Es1,where Ecomp1 is the composite compressive modulus of this CCF in Region A.From Equation (7), the time derivative of εv of this CCF can be obtained as follows:
(9)∂εv∂t=−1−m1−m2Ecomp1∂u¯s1∂t+1Ecomp1∂σt,z∂t.The vertical flow of the pore water of the surrounding soil is permitted in the impervious column reinforced ground; the consolidation equation of the subsoil in Region A can be written as follows:(10)∂εv∂t=−kv1γw∂2u¯s1∂z2.By substituting Equation (9) into Equation (10), the governing differential equation for the consolidation for the subsoil in Region A can be derived:
(11)∂u¯s1∂t=kv1Ecomp11−m1−m2γw∂2u¯s1∂z2+11−m1−m2∂σt,z∂t.A simplified version of Equation (11) can be rearranged as follows:
(12)∂u¯s1∂t=cv1e∂2u¯s1∂z2+11−m1−m2∂σt,z∂t,cv1e=Ecomplkv11−m1−m2γw=1+m11−ρEedm+ρEepc+m2Edm21−m1−m2Es1cv1,where cv1e is defined as the equivalent vertical consolidation coefficient of the subsoil in Region A.


#### 2.2.2. Consolidation Equation for the Subsoil in Region B

From the equilibrium equation of the PCCS and the soil around the inner core in Region B, the following relation can be obtained:(13)1−m1ρσ¯s2+m1ρσ¯pc=σt,z,where σ¯s2 is the vertical stress of d the soil around the inner core in Region B.Similarly,εv for the inner core of the PCCS and its surrounding soil in Region B, can be written as follows:
(14)εv=σ¯pcEpc=σ¯s2−u¯s2Es2,where u¯s2 is the average excess pore pressure of the soil around the inner core in Region B.Combining Equation (13) and Equation (14) yeilds
(15)εv=σt,z−1−m1ρu¯s2Ecomp2,(16)Ecomp2=1−m1ρEs2+m1ρEpc,where Ecomp2 is the composite compressive modulus of this CCF in Region B.It can be derived by taking the time derivative ofεv that
(17)∂εv∂t=−1−m1ρEcomp2∂u¯s2∂t+1Ecomp2∂σt,z∂t.Similar to the derivation of Equation (10), the consolidation equation of this CCF in Region B can be obtained:
(18)∂εv∂t=−kv2γw∂2u¯s2∂z2.From Equation (17) and Equation (18), the governing equation for the consolidation of the subsoil in Region B can be derived:
(19)∂u¯s2∂t=cv2e∂2u¯s2∂z2+11−m1ρ∂σt,z∂t,where cv2e is the equivalent vertical consolidation coefficient of the subsoil in Region B, considering the effect of the inner core of the PCCS:
(20)cv2e=kv2Ecomp21−m1ργw=1+m1ρEpc1−m1ρEs2cv2.


## 2.1. Model Description and Basic Assumptions

The penetrated PCCSs as the long piles and floating DM columns as the short columns can be arranged in four types of distribution modes, including triangular, intensive triangular, rectangular, and intensive rectangular distributions, as shown in Figure1. The primary step in analyzing the consolidation behaviors is partitioning this multipile composite ground into the square unit cells. The unit cell is adopted as the representative of the CCP and converted into a cylindrical unit cell using based on equivalent hypothesize of the sectional area [16, 17]. In general, the unit cell can be adopted in two patterns: (1) Form A is a PCCS at the center with N DM columns dispersed separately at its boundary and (2) Form B is a DM column at the center with N PCCSs distributed separately at its perimeter. The cross-sectional area of each PCCS distributed independently at its boundary within the unit cell makes up the value of N. With respect to triangular and rectangular distribution patterns, selecting any pile at the center for the consolidation path in the subsoil is constant, since the PCCS and DM columns can be regarded as impervious piles. However, for the intensive rectangular distribution patterns, it is necessary to select the PCCS located at the center. Otherwise, it makes the problem more complex when other DM columns are distributed at the circumferential boundary. Therefore, Form B is adopted in this paper to analyze the cases of rectangular and intensive rectangular distributions.Figure 1
Layout of this CCF reinforced with penetrated PCCSs and floating DM columns.In the current study, the penetrated PCCS with a long inner core as the long pile and the floating DM column as the short column is installed in a rectangular mode with identical center-to-center spacing ofSp. This penetrated PCCS penetrates the whole soft soil layers seated on an impermeable bearing layer in this composite ground, and one PCCS at the center and eight DM columns are distributed at the boundary, where the value of N is 3. Figure 2 presents the cylinder model with an equal cross-sectional area transferred from the unit cell. Its simplified model comprises two improved areas according to the cross-section of DM column, which are named Region A and Region B with the permeability coefficient (kv1 and kv2), compressive modulus (Es1 and Es2), consolidation coefficient (cv1 and cv2), and length (hl and h2), respectively. rdm and rpc are the radius of the PC pile and the DM column, respectively. Edm and Epc are the compressive modulus of the concrete and the cemented soil, respectively. rn is the radius of the unit cell, and re is the outside radius of the simplified model of the CCF, which can be determined as follows:
(1)re2=rn2−Nrdm2,π4rn2=Sp2.Figure 2
Schematic diagram of consolidation model; (a) plan view; (b) elevation plan.
(a)(b)The reinforced gravel cushion laying on the piles and surrounding soil is subjected to a ramp load,pt. Water flow is prohibited across the bottom and lateral surfaces of the model due to the underlying impermeable layer, whilst is allowed across the ground surface to analyze the single drainage of the CCF.The following assumptions during the analysis in this paper are given below:(i)
Darcy’s law is obeyed for the seepage of the water in the surrounding soil.kv1, kv2, Es1, and Es2 are assumed to remain constant. The subsoils are regarded as fully saturated state(ii)
PC piles and DM columns are assumed to be impervious, where no flow and the consolidation occur at the PC pile-DM column interface and DM column-soil interface(iii)
There is no interface slip between the PC pile-cemented soil of the PCCS(iv)
It is considered that the vertical strains of the DM column and its surrounding soil are equal at any depth(v)
A quasiequal strain assumption is used between the PCCS and its surrounding soil due to the pile top pierced into the gravel cushion to generate the relatively large penetration,δ1 [45]. The deformation of the surrounding soil is relatively close to the total of the pile shaft compression and the pile head penetration value. Since the pile tip is located on a firm soil layer, the pile tip piercing into the underlying soil layer is not taken into account. Therefore, the volumetric strain, εv, can be obtained as follows:(2)εv=σ¯pcEpc+δ1h1=σ¯dm1Edm+δ1h1=σ¯dm2Edm=σ¯s1−u¯1Es1,(3)δ1=π1−μc2pacrp2Ec,where σ¯pc, σ¯dm1, and σ¯dm2 are the vertical stress of inner core and outer core of the PCCS as well as the DM column, respectively. σ¯s1 and u¯s1 are the vertical stress and the excess pore pressure of the soil around piles in Region A, respectively. pac, μc, and Ec are the vertical stress of the pile top, the Poisson’s ratio, and the compressive modulus of the cushion, respectively.Substituting Equation (3) into Equation (2) yields
(4)εv=σ¯pcEepc=σ¯dm1Eedm=σ¯dm2Edm=σ¯s1−u¯1Es1,where Eepc and Eedm are the equivalent compressive moduli of PC pile and DM column of the PCCS:
(5)Eepc=2h1EcEpc2h1Ec+π1−μc2Epcrsp,Eedm=2h1EcEdm2h1Ec+π1−μc2Edmrdm.


## 2.2. Governing Equation

### 2.2.1. Consolidation Equation for the Subsoil in Region A

In the axisymmetric consolidation model, the average vertical stress in Region A within the CCF is transferred from the cushion to the DM columns, the PCCSs, and the soil around piles.σt,z is the average total stress in this CCF induced by the external load, pt. The equilibrium equation at any depth in Region A can be written as follows:
(6)1−m1−m2σ¯s1+m11−ρσ¯dm1+ρσ¯pc+m2σ¯dm2=σt,z,where m1 and m2 are the area replacement ratios of the PCCS and DM column, respectively, and ρ is the ratio between the inner and outer cores’ cross-sectional areas.From Equation (4) and Equation (6), the vertical volumetric strain of the PCCSs, DM columns, and the subsoil at any depth, εv, can be expressed as follows:
(7)εv=σt,z−1−m1−m2u¯s1Ecomp1,(8)Ecomp1=m11−ρEedm+ρEepc+m2Edm+1−m1−m2Es1,where Ecomp1 is the composite compressive modulus of this CCF in Region A.From Equation (7), the time derivative of εv of this CCF can be obtained as follows:
(9)∂εv∂t=−1−m1−m2Ecomp1∂u¯s1∂t+1Ecomp1∂σt,z∂t.The vertical flow of the pore water of the surrounding soil is permitted in the impervious column reinforced ground; the consolidation equation of the subsoil in Region A can be written as follows:(10)∂εv∂t=−kv1γw∂2u¯s1∂z2.By substituting Equation (9) into Equation (10), the governing differential equation for the consolidation for the subsoil in Region A can be derived:
(11)∂u¯s1∂t=kv1Ecomp11−m1−m2γw∂2u¯s1∂z2+11−m1−m2∂σt,z∂t.A simplified version of Equation (11) can be rearranged as follows:
(12)∂u¯s1∂t=cv1e∂2u¯s1∂z2+11−m1−m2∂σt,z∂t,cv1e=Ecomplkv11−m1−m2γw=1+m11−ρEedm+ρEepc+m2Edm21−m1−m2Es1cv1,where cv1e is defined as the equivalent vertical consolidation coefficient of the subsoil in Region A.


### 2.2.2. Consolidation Equation for the Subsoil in Region B

From the equilibrium equation of the PCCS and the soil around the inner core in Region B, the following relation can be obtained:(13)1−m1ρσ¯s2+m1ρσ¯pc=σt,z,where σ¯s2 is the vertical stress of d the soil around the inner core in Region B.Similarly,εv for the inner core of the PCCS and its surrounding soil in Region B, can be written as follows:
(14)εv=σ¯pcEpc=σ¯s2−u¯s2Es2,where u¯s2 is the average excess pore pressure of the soil around the inner core in Region B.Combining Equation (13) and Equation (14) yeilds
(15)εv=σt,z−1−m1ρu¯s2Ecomp2,(16)Ecomp2=1−m1ρEs2+m1ρEpc,where Ecomp2 is the composite compressive modulus of this CCF in Region B.It can be derived by taking the time derivative ofεv that
(17)∂εv∂t=−1−m1ρEcomp2∂u¯s2∂t+1Ecomp2∂σt,z∂t.Similar to the derivation of Equation (10), the consolidation equation of this CCF in Region B can be obtained:
(18)∂εv∂t=−kv2γw∂2u¯s2∂z2.From Equation (17) and Equation (18), the governing equation for the consolidation of the subsoil in Region B can be derived:
(19)∂u¯s2∂t=cv2e∂2u¯s2∂z2+11−m1ρ∂σt,z∂t,where cv2e is the equivalent vertical consolidation coefficient of the subsoil in Region B, considering the effect of the inner core of the PCCS:
(20)cv2e=kv2Ecomp21−m1ργw=1+m1ρEpc1−m1ρEs2cv2.


## 2.2.1. Consolidation Equation for the Subsoil in Region A

In the axisymmetric consolidation model, the average vertical stress in Region A within the CCF is transferred from the cushion to the DM columns, the PCCSs, and the soil around piles.σt,z is the average total stress in this CCF induced by the external load, pt. The equilibrium equation at any depth in Region A can be written as follows:
(6)1−m1−m2σ¯s1+m11−ρσ¯dm1+ρσ¯pc+m2σ¯dm2=σt,z,where m1 and m2 are the area replacement ratios of the PCCS and DM column, respectively, and ρ is the ratio between the inner and outer cores’ cross-sectional areas.From Equation (4) and Equation (6), the vertical volumetric strain of the PCCSs, DM columns, and the subsoil at any depth, εv, can be expressed as follows:
(7)εv=σt,z−1−m1−m2u¯s1Ecomp1,(8)Ecomp1=m11−ρEedm+ρEepc+m2Edm+1−m1−m2Es1,where Ecomp1 is the composite compressive modulus of this CCF in Region A.From Equation (7), the time derivative of εv of this CCF can be obtained as follows:
(9)∂εv∂t=−1−m1−m2Ecomp1∂u¯s1∂t+1Ecomp1∂σt,z∂t.The vertical flow of the pore water of the surrounding soil is permitted in the impervious column reinforced ground; the consolidation equation of the subsoil in Region A can be written as follows:(10)∂εv∂t=−kv1γw∂2u¯s1∂z2.By substituting Equation (9) into Equation (10), the governing differential equation for the consolidation for the subsoil in Region A can be derived:
(11)∂u¯s1∂t=kv1Ecomp11−m1−m2γw∂2u¯s1∂z2+11−m1−m2∂σt,z∂t.A simplified version of Equation (11) can be rearranged as follows:
(12)∂u¯s1∂t=cv1e∂2u¯s1∂z2+11−m1−m2∂σt,z∂t,cv1e=Ecomplkv11−m1−m2γw=1+m11−ρEedm+ρEepc+m2Edm21−m1−m2Es1cv1,where cv1e is defined as the equivalent vertical consolidation coefficient of the subsoil in Region A.


## 2.2.2. Consolidation Equation for the Subsoil in Region B

From the equilibrium equation of the PCCS and the soil around the inner core in Region B, the following relation can be obtained:(13)1−m1ρσ¯s2+m1ρσ¯pc=σt,z,where σ¯s2 is the vertical stress of d the soil around the inner core in Region B.Similarly,εv for the inner core of the PCCS and its surrounding soil in Region B, can be written as follows:
(14)εv=σ¯pcEpc=σ¯s2−u¯s2Es2,where u¯s2 is the average excess pore pressure of the soil around the inner core in Region B.Combining Equation (13) and Equation (14) yeilds
(15)εv=σt,z−1−m1ρu¯s2Ecomp2,(16)Ecomp2=1−m1ρEs2+m1ρEpc,where Ecomp2 is the composite compressive modulus of this CCF in Region B.It can be derived by taking the time derivative ofεv that
(17)∂εv∂t=−1−m1ρEcomp2∂u¯s2∂t+1Ecomp2∂σt,z∂t.Similar to the derivation of Equation (10), the consolidation equation of this CCF in Region B can be obtained:
(18)∂εv∂t=−kv2γw∂2u¯s2∂z2.From Equation (17) and Equation (18), the governing equation for the consolidation of the subsoil in Region B can be derived:
(19)∂u¯s2∂t=cv2e∂2u¯s2∂z2+11−m1ρ∂σt,z∂t,where cv2e is the equivalent vertical consolidation coefficient of the subsoil in Region B, considering the effect of the inner core of the PCCS:
(20)cv2e=kv2Ecomp21−m1ργw=1+m1ρEpc1−m1ρEs2cv2.


## 3. Analytical Solutions

Equation (21) with cv1e and cv2e is similar with the governing equations provided by Xie [44] in format. Based on the above derivation, this CCF reinforced with penetrated PCCSs and floating DM columns can be regarded as a double-layered ground.
(21)∂u¯s1∂t=cv1e∂2u¯s1∂z2+11−m1−m2∂σt,z∂t,∂u¯s2∂t=cv2e∂2u¯s2∂z2+11−m1ρ∂σt,z∂t.


### 3.1. Boundary Conditions

Since this CCF with the top previous surface and the bottom impervious surface, the boundary conditions of this consolidation model are as follows:(22)u¯s1t,z=0,z=0,∂u¯s2t,z∂z=0,z=h.Equation (23) can be derived by expressing Equation (6) and Equation (13) based on the principle of effective stress.
(23)1−m1−m2σ¯s1−u¯s1+m11−ρσ¯dm1+ρσ¯pc+m2σ¯dm2=1−m1ρσ¯s2−u¯s2+m1ρσ¯pc.According to Equations (6), (13), and (23), the continuity conditions of the boundary pore pressure can be obtained as follows:
(24)1−m1−m2u¯s1=1−m1ρu¯s2.Based on the continuous condition of seepage on the Region A-Region B interface, the following relation can be obtained:(25)1−m1−m2kv1∂u¯s1∂z=1−m1ρkv2∂u¯s2∂z.From Equations (7) and (15), the excess pore water pressure at initial time can be obtained:
(26)t=0:u¯s1=σt,z1−m1−m2,u¯s2=σt,z1−m1ρ.As shown in Figure3, this CCF is subjected to a ramp load, which increases with time until tu, and then remains a constant value, pu. The additional stress distribution of this CCF resulting from pu assumed to vary linearly along the depth as shown in Figure 4 can be expressed as follows:
(27)σt,z=σ0h−zthtu,t<tu,σ0h−zh,t≥tu,where σ0 is the vertical stress of the ground surface.Figure 3
Relationship between load and time.Figure 4
Relationship between of the additional stress and depth.


### 3.2. Derivation of Pore Pressure and Consolidation Degree under Ramp Loading

In this paper,a, b, and c as the dimensionless parameters are adopted to simplify the derivation as follows:
(28)a=kv2kv1,b=Ecomp11−m1ρEcomp21−m1−m2,c=h2h1.Based on the consolidation model of double-layered ground under ramp loading, the governing equations of the composite ground can be solved:(29)u¯s1=∑m=1∞sinλmzH1e−βmtCmTmt,(30)u¯s2=∑m=1∞Amcosμλmh−zh1e−βmtCmTmt,where Am, Cm, λm, and βm are unknown coefficients and Tmt is the pending function united with the external load.According to the boundary condition of the initial stress and water flow, the above-mentioned parameters can be obtained as follows:(31)Am=sinλmcosμcλm,(32)βm=cv1eλm2h12,(33)Cm=2λm1+bcAm2,(34)μ=cv1ecv2e,(35)absinλmsinμcλm=cosλmcosμcλm.The eigenvaluesλm are the root of the eigen-Equation (35). From Equation (29), Equation (30), and Equation (21), the pending function of Tmt can be calculated by
(36)Tm=11−m1−m2σutu∫0teβmtdt,0<z≤h1,Tm=11−m1ρσutu∫0teβmtdt,h1<z≤h.Substituting Equations (31)–(35) to Equation (29) and Equation (30), the average excess pore water pressure of this CCF reinforced with penetrated PCCSs and floating DM columns in Region A (u¯1) and in Region B (u¯2) can be derived during 0≤t<tu.
(37)u¯s1=σu1−m1−m2∑m=1∞Cmλm2Tvtusinλmzh11−e−λm2Tv1,u¯s2=σu1−m1ρ∑m=1∞AmCmλm2Tvtucosμλmh−zh11−e−λm2Tv1,where Tvtu is the time factor when t equals to tu and Tv1 is the time function, which can be obtained as
(38)Tvtu=cv1etuh12,Tv1=cv1eth12.Furthermore, the functions ofu¯1 of this CCF for t>tc can be derived as
(39)u¯s1=σu1−m1−m2∑m=1∞Cmλm2Tvtusinλmzh1e−λm2Tv1eλm2Tvtu−1,u¯s2=σu1−m1ρ∑m=1∞AmCmλm2Tvtucosμλmh−zh1e−λm2Tv1eλm2Tvtu−1.In this study, the average consolidation degree of this CCF concerning the settlement,Us, can be written as follows:
(40)Ust=1/Es1∫0h1σ¯s1−u¯s1dz+1/Es2∫h1hσ¯s2−u¯s2dz1/Es1∫0h1σ¯s1uzdz+1/Es2∫h1hσ¯s2uzdz,where σ¯s1u and σ¯s2u are the final stress of the soil in Region A and Region B, respectively.Combining Equation (4) and Equation (6), the vertical stress of the surrounding soil in Region A can be obtained:
(41)σ¯s1=σt,z+m11−ρYdm+m1ρYpc+m2Ydm2u¯s11−m1−m2+m11−ρYdm+ρYpc+m2Ydm2.From Equation (41), σ¯s1u can be derived:
(42)σ¯s1u=σt,z1−m1−m2+m11−ρYdm+ρYpc+m2Ydm2,whereYdm and Ypc are the dimensionless parameters, which can be expressed by
(43)Ydm=EedmEs1;Ydm2=EdmEs1;Ypc=EepcEs1.Similarly, combining Equation (4) and Equation (13), the average and final stresses of the soil in Region B can be obtained:
(44)σ¯s2=σt,z+m1ρYpc2u¯s21−m1ρ+m1ρYpc2,(45)σ¯s2u=σt,z1−m1ρ+m1ρYpc2,where Ypc2=Epc/Es2.Substituting Equations (37)–(39) and Equations (41)–(45) into Equation (40), the overall average consolidation degree of this CCF during 0≤t<tu can be obtained:
(46)Ut=tut−2hh1Tvtcmv1e∑m=1∞Cm1−e−λm2Tv1/λm3cosλm−1−mv2e∑m=1∞AmCm1−e−λm2Tv1/μλm3sinμλmh−h1/h1mv1eh12−2h1h+mv2eh−h12.Similarly,Ut of the CCF during t≥tu can be derived as
(47)Ut=1+2hh1Tvtcmv1e∑m=1∞Cme−λm2Tv1eλm2Tvtc−1/λm3cosλm−1−mv2e∑m=1∞AmCme−λm2Tv1eλm2Tvtc−1/μλm3sinμλmh−h1/h1mv1eh12−2h1h+mv2eh−h12,mv1e=1Es1e=11−m1−m2+m11−ρYdm+ρYpc+m2Ydm2Es1,mv2e=1Es2e=11−m1ρ+m1ρYpc2Es2,where mv1e and mv2e are the volume compressibility equivalent coefficients of the surrounding soil in Region A and Region B, respectively.


### 3.3. Model Calibration

Numerical analysis using Plaxis 3D (2016) was carried out to verify the accuracy of the current analytical solution for this CCF. A cubic model (height=20m and pilespacingofPCCSs=4m) is adopted for numerical simulation, as shown in Figure 5. The penetrated PCCS as the long pile consisted of the outer core with a length of 12 m and the inner core with a length of 20 m. The floating DM column is regarded as the short pile with a length of 12 m. The diameter of the outer core and the DM column is 800 mm, while the diameter of the inner core is 400 mm. m1 and m2 are 0.15 and 0.09, respectively. A 0.3 m thickness gravel cushion is seated on the surface of this CCF. After the completion of initial geostatic stress balance, a ramp uniform external load pu=100kPa is applied linearly on the gravel cushion within 60 days (tu=60d), followed by consolidation of 1000 days.Figure 5
Finite element model.Mohr-Coulomb (MC) model is adopted to express the elastic-perfectly plastic behaviors of the DM column, subsoil, and gravel cushion, while concrete is regarded as isotropic linear-elastic material. The Young’s modulus,E, of every material can be obtained from using the equation E=Es1+v1+2v/1−v, where Es and v are the constrained modulus and Poisson’s ratio of the material, respectively. Table 1 presents the primary material parameters. A 10-node tetrahedral cell is used to model the pile, DC column, soil, and gravel cushion.Table 1
Parameters in the numerical simulation.
MaterialUnit weight,γ (kN/m3)Compression modulus,Es (MPa)Effective cohesion,c′ (kPa)Effective friction angle,φ′ (°)vPermeability coefficient,kv (m/s)Soft soil1834180.41×10−8Gravel cushion19300320.3—Cemented soil20150500360.3—Concrete2420000——0.20—Regarding the drainage boundary condition, the lateral and bottom surfaces are regarded as impermeable surfaces, while the water flow is permitted through the ground surface. The friction coefficients of DM column-soil and inner-outer cores of the PCCS are 0.1 (Brinkgreve et al., 2008) and 0.4 [28, 29], respectively.Figure6 presents the comparison of Ut of this CCF obtained from the current analytical method and FE model. It indicates that the consolidation rate obtained from numerical simulation agrees well with that predicted by the current analytical method. However, the theoretical values are slightly bigger than the numerical ones for t>tu. The main reason is that the nonlinear variation of the additional stress of subsoils in the numerical analysis is different from that assumed to vary linearly along with the depth in the analytical method.Figure 6
Comparison of the current analytical solution and numerical analysis.


## 3.1. Boundary Conditions

Since this CCF with the top previous surface and the bottom impervious surface, the boundary conditions of this consolidation model are as follows:(22)u¯s1t,z=0,z=0,∂u¯s2t,z∂z=0,z=h.Equation (23) can be derived by expressing Equation (6) and Equation (13) based on the principle of effective stress.
(23)1−m1−m2σ¯s1−u¯s1+m11−ρσ¯dm1+ρσ¯pc+m2σ¯dm2=1−m1ρσ¯s2−u¯s2+m1ρσ¯pc.According to Equations (6), (13), and (23), the continuity conditions of the boundary pore pressure can be obtained as follows:
(24)1−m1−m2u¯s1=1−m1ρu¯s2.Based on the continuous condition of seepage on the Region A-Region B interface, the following relation can be obtained:(25)1−m1−m2kv1∂u¯s1∂z=1−m1ρkv2∂u¯s2∂z.From Equations (7) and (15), the excess pore water pressure at initial time can be obtained:
(26)t=0:u¯s1=σt,z1−m1−m2,u¯s2=σt,z1−m1ρ.As shown in Figure3, this CCF is subjected to a ramp load, which increases with time until tu, and then remains a constant value, pu. The additional stress distribution of this CCF resulting from pu assumed to vary linearly along the depth as shown in Figure 4 can be expressed as follows:
(27)σt,z=σ0h−zthtu,t<tu,σ0h−zh,t≥tu,where σ0 is the vertical stress of the ground surface.Figure 3
Relationship between load and time.Figure 4
Relationship between of the additional stress and depth.


## 3.2. Derivation of Pore Pressure and Consolidation Degree under Ramp Loading

In this paper,a, b, and c as the dimensionless parameters are adopted to simplify the derivation as follows:
(28)a=kv2kv1,b=Ecomp11−m1ρEcomp21−m1−m2,c=h2h1.Based on the consolidation model of double-layered ground under ramp loading, the governing equations of the composite ground can be solved:(29)u¯s1=∑m=1∞sinλmzH1e−βmtCmTmt,(30)u¯s2=∑m=1∞Amcosμλmh−zh1e−βmtCmTmt,where Am, Cm, λm, and βm are unknown coefficients and Tmt is the pending function united with the external load.According to the boundary condition of the initial stress and water flow, the above-mentioned parameters can be obtained as follows:(31)Am=sinλmcosμcλm,(32)βm=cv1eλm2h12,(33)Cm=2λm1+bcAm2,(34)μ=cv1ecv2e,(35)absinλmsinμcλm=cosλmcosμcλm.The eigenvaluesλm are the root of the eigen-Equation (35). From Equation (29), Equation (30), and Equation (21), the pending function of Tmt can be calculated by
(36)Tm=11−m1−m2σutu∫0teβmtdt,0<z≤h1,Tm=11−m1ρσutu∫0teβmtdt,h1<z≤h.Substituting Equations (31)–(35) to Equation (29) and Equation (30), the average excess pore water pressure of this CCF reinforced with penetrated PCCSs and floating DM columns in Region A (u¯1) and in Region B (u¯2) can be derived during 0≤t<tu.
(37)u¯s1=σu1−m1−m2∑m=1∞Cmλm2Tvtusinλmzh11−e−λm2Tv1,u¯s2=σu1−m1ρ∑m=1∞AmCmλm2Tvtucosμλmh−zh11−e−λm2Tv1,where Tvtu is the time factor when t equals to tu and Tv1 is the time function, which can be obtained as
(38)Tvtu=cv1etuh12,Tv1=cv1eth12.Furthermore, the functions ofu¯1 of this CCF for t>tc can be derived as
(39)u¯s1=σu1−m1−m2∑m=1∞Cmλm2Tvtusinλmzh1e−λm2Tv1eλm2Tvtu−1,u¯s2=σu1−m1ρ∑m=1∞AmCmλm2Tvtucosμλmh−zh1e−λm2Tv1eλm2Tvtu−1.In this study, the average consolidation degree of this CCF concerning the settlement,Us, can be written as follows:
(40)Ust=1/Es1∫0h1σ¯s1−u¯s1dz+1/Es2∫h1hσ¯s2−u¯s2dz1/Es1∫0h1σ¯s1uzdz+1/Es2∫h1hσ¯s2uzdz,where σ¯s1u and σ¯s2u are the final stress of the soil in Region A and Region B, respectively.Combining Equation (4) and Equation (6), the vertical stress of the surrounding soil in Region A can be obtained:
(41)σ¯s1=σt,z+m11−ρYdm+m1ρYpc+m2Ydm2u¯s11−m1−m2+m11−ρYdm+ρYpc+m2Ydm2.From Equation (41), σ¯s1u can be derived:
(42)σ¯s1u=σt,z1−m1−m2+m11−ρYdm+ρYpc+m2Ydm2,whereYdm and Ypc are the dimensionless parameters, which can be expressed by
(43)Ydm=EedmEs1;Ydm2=EdmEs1;Ypc=EepcEs1.Similarly, combining Equation (4) and Equation (13), the average and final stresses of the soil in Region B can be obtained:
(44)σ¯s2=σt,z+m1ρYpc2u¯s21−m1ρ+m1ρYpc2,(45)σ¯s2u=σt,z1−m1ρ+m1ρYpc2,where Ypc2=Epc/Es2.Substituting Equations (37)–(39) and Equations (41)–(45) into Equation (40), the overall average consolidation degree of this CCF during 0≤t<tu can be obtained:
(46)Ut=tut−2hh1Tvtcmv1e∑m=1∞Cm1−e−λm2Tv1/λm3cosλm−1−mv2e∑m=1∞AmCm1−e−λm2Tv1/μλm3sinμλmh−h1/h1mv1eh12−2h1h+mv2eh−h12.Similarly,Ut of the CCF during t≥tu can be derived as
(47)Ut=1+2hh1Tvtcmv1e∑m=1∞Cme−λm2Tv1eλm2Tvtc−1/λm3cosλm−1−mv2e∑m=1∞AmCme−λm2Tv1eλm2Tvtc−1/μλm3sinμλmh−h1/h1mv1eh12−2h1h+mv2eh−h12,mv1e=1Es1e=11−m1−m2+m11−ρYdm+ρYpc+m2Ydm2Es1,mv2e=1Es2e=11−m1ρ+m1ρYpc2Es2,where mv1e and mv2e are the volume compressibility equivalent coefficients of the surrounding soil in Region A and Region B, respectively.


## 3.3. Model Calibration

Numerical analysis using Plaxis 3D (2016) was carried out to verify the accuracy of the current analytical solution for this CCF. A cubic model (height=20m and pilespacingofPCCSs=4m) is adopted for numerical simulation, as shown in Figure 5. The penetrated PCCS as the long pile consisted of the outer core with a length of 12 m and the inner core with a length of 20 m. The floating DM column is regarded as the short pile with a length of 12 m. The diameter of the outer core and the DM column is 800 mm, while the diameter of the inner core is 400 mm. m1 and m2 are 0.15 and 0.09, respectively. A 0.3 m thickness gravel cushion is seated on the surface of this CCF. After the completion of initial geostatic stress balance, a ramp uniform external load pu=100kPa is applied linearly on the gravel cushion within 60 days (tu=60d), followed by consolidation of 1000 days.Figure 5
Finite element model.Mohr-Coulomb (MC) model is adopted to express the elastic-perfectly plastic behaviors of the DM column, subsoil, and gravel cushion, while concrete is regarded as isotropic linear-elastic material. The Young’s modulus,E, of every material can be obtained from using the equation E=Es1+v1+2v/1−v, where Es and v are the constrained modulus and Poisson’s ratio of the material, respectively. Table 1 presents the primary material parameters. A 10-node tetrahedral cell is used to model the pile, DC column, soil, and gravel cushion.Table 1
Parameters in the numerical simulation.
MaterialUnit weight,γ (kN/m3)Compression modulus,Es (MPa)Effective cohesion,c′ (kPa)Effective friction angle,φ′ (°)vPermeability coefficient,kv (m/s)Soft soil1834180.41×10−8Gravel cushion19300320.3—Cemented soil20150500360.3—Concrete2420000——0.20—Regarding the drainage boundary condition, the lateral and bottom surfaces are regarded as impermeable surfaces, while the water flow is permitted through the ground surface. The friction coefficients of DM column-soil and inner-outer cores of the PCCS are 0.1 (Brinkgreve et al., 2008) and 0.4 [28, 29], respectively.Figure6 presents the comparison of Ut of this CCF obtained from the current analytical method and FE model. It indicates that the consolidation rate obtained from numerical simulation agrees well with that predicted by the current analytical method. However, the theoretical values are slightly bigger than the numerical ones for t>tu. The main reason is that the nonlinear variation of the additional stress of subsoils in the numerical analysis is different from that assumed to vary linearly along with the depth in the analytical method.Figure 6
Comparison of the current analytical solution and numerical analysis.


## 4. Parametric Study of This CCF

A parametrical study is conducted to study the consolidation behaviors of the CCF reinforced with penetrated PCCSs and floating DM columns. In this analysis, the real time is replaced by a dimensionless time factorTv=cv1t/h. The effect of seven key influence factors on consolidation behaviors of this CCF is analyzed. All the parameters of the FE model are adopted for the baseline unit cell.


### 4.1. Effect ofm1

Figure7 depicts the average degree of consolidation, Us versus time factor Tv for m1. It can be seen from Figure 7 that the consolidation rate increases with m1 for the constant m2. This appears to be only effective up to a certain value of m1, resulting from that the larger m1, the smaller the reinforcement zone of the pile and the shorter the pore water seepage path of the soil around piles.Figure 7
Effect ofm1 on the consolidation rate.


### 4.2. Effect ofm2

The average degree of consolidationUs versus time factor Tv for varying m2 in Figure 8 considers the two cases that (Ldm/Lpc) are 0.3 and 0.6. The consolidation rate slightly increases with m2 varying from 0.09 to 0.18. Compared with m1, DM columns have inapparent influence on the consolidation rate of this CCF with penetrated PCCSs and floating DM columns. Besides, the effect on consolidation rate of the CCF with Ldm/Lpc of 0.6 is relatively significant than that of the CCF with Ldm/Lpc of 0.3, which is attributed to the floating effect of the DM columns.Figure 8
Effect ofm2 on the consolidation rate.
(a)
Lpc/Ldm=0.3(b)
Lpc/Ldm=0.6


### 4.3. Effect ofEpc/Edm

Figure9 presents the effect of inner core-outer core modulus ratio (Epc/Edm) on consolidation behavior of CCFs. Edm includes 0, 150 MPa, 200 MPa, 300 MPa, and 600 MPa, keeping Epc as 30 GPa. When the value of Edm is assigned to zero, the CCF reinforced with penetrated PCCSs and DM columns degenerated to the CCF improved by PC piles united with DM columns. Figure 9 indicates that the average consolidation rate increases with Epc/Edm and the equivalent consolidation coefficient increases with Edm. Moreover, compared with the traditional the CCF improved by PC piles united with DM columns, the consolidation rate of the CCF improved by penetrated PCCSs increases obviously, implying that the outer core can accelerate the consolidation rate of this CCF to some extent.Figure 9
Effect ofEpc/Edm on the consolidation rate.


### 4.4. Effect ofEpc/Es

Figure10 illustrates the variations of Us and Tv for the different the inner core-subsoil modulus ratio (Epc/Es). Epc consists of 3 GPa, 3 GPa, 18 GPa, and 30 GPa, and Es adopted the constant value of 3 MPa. The corresponding Epc/Es includes 1500, 3000, 6000, 30000, and 60000. It can be seen that the average consolidation rate increases as Epc/Es increases with a decreasing growth rate. The reason is that the larger the consolidation coefficient, the higher the consolidation rate. Besides, the equivalent consolidation coefficient increasing with Epc/Es affects the consolidation rate.Figure 10
Effect ofEpc/Es on the consolidation rate.


### 4.5. Effect ofEc/Es

The PCCS piercing value and the composite compression modulus of Region A are both influenced by the stiffness of the cushion.Us and Tv for the different the cushion-subsoil modulus ratio (Ec/Es) are shown in Figure 11, where Ec consists of 0, 3 MPa, 15 MPa, and 30 MPa keeping Es constant as 3 MPa. The corresponding Ec/Es includes 1, 5, 10, and 15. The case with Ec/Es of 1 indicates that no cushion was arranged on the surface of the CCF. It can be seen that the average degree of consolidation increases slightly with Ec/Es and the growth rate decreases with Ec/Es. The effect of Ec/Es on the consolidation rate is less obvious than that of Epc/Es on the consolidation rate because of the smaller range of Ec/Es. The consolidation rate for the CCF untreated with gravel cushion (Ec/Es=1) is lower than that for the CCF with the cushion. Without the gravel cushion, the soil around the piles needs to bear more loads and generates larger excess pore pressure, which takes longer time to dissipate.Figure 11
Effect ofEc/Es on the consolidation rate.


### 4.6. Effect ofkv1/kv2

Figure12 presents the effect of the permeability coefficient ratio of the surrounding soil in Region A and Region B, (kv1/kv2) on consolidation characteristic, where kv1 consists of 10-9 m/s, 10-8 m/s, 5×10−8m/s, and 10-7 m/s, keeping kv2 constant as 10-8 m/s. The corresponding kv1/kv2 comprises 0.5, 1, 5, and 10. The consolidation rate increases as kv1 increases with a decreasing growth rate. Since the CCF reinforced with penetrated PCCSs and floating DM columns is simplified as double-layer ground whose permeability is reflected by the value of kv1/kv2, the larger kv1/kv2 indicates the higher permeability of the subsoil to accelerate the consolidation of the surrounding soil.Figure 12
Effect ofkv1/kv2 on the consolidation rate.


### 4.7. Effect oftu

The effect of the loading time (tu) on the consolidation of this CCF is investigated by varying tu from 60 d to 8 d, 30 d, 80 d, and 0 d in Figure 13. The case of tu as 0 d can be regarded as instant loading. It can be seen from Figure 13 that the average consolidation rate of the CCF decreases with tu. Since the additional stress increment increases with time and remains constant until tu, the longer the loading period, the longer the time it takes for excess pore pressure to accumulate and dissipate.Figure 13
Influence ofTvtc on the average degree of consolidation.


## 4.1. Effect ofm1

Figure7 depicts the average degree of consolidation, Us versus time factor Tv for m1. It can be seen from Figure 7 that the consolidation rate increases with m1 for the constant m2. This appears to be only effective up to a certain value of m1, resulting from that the larger m1, the smaller the reinforcement zone of the pile and the shorter the pore water seepage path of the soil around piles.Figure 7
Effect ofm1 on the consolidation rate.


## 4.2. Effect ofm2

The average degree of consolidationUs versus time factor Tv for varying m2 in Figure 8 considers the two cases that (Ldm/Lpc) are 0.3 and 0.6. The consolidation rate slightly increases with m2 varying from 0.09 to 0.18. Compared with m1, DM columns have inapparent influence on the consolidation rate of this CCF with penetrated PCCSs and floating DM columns. Besides, the effect on consolidation rate of the CCF with Ldm/Lpc of 0.6 is relatively significant than that of the CCF with Ldm/Lpc of 0.3, which is attributed to the floating effect of the DM columns.Figure 8
Effect ofm2 on the consolidation rate.
(a)
Lpc/Ldm=0.3(b)
Lpc/Ldm=0.6


## 4.3. Effect ofEpc/Edm

Figure9 presents the effect of inner core-outer core modulus ratio (Epc/Edm) on consolidation behavior of CCFs. Edm includes 0, 150 MPa, 200 MPa, 300 MPa, and 600 MPa, keeping Epc as 30 GPa. When the value of Edm is assigned to zero, the CCF reinforced with penetrated PCCSs and DM columns degenerated to the CCF improved by PC piles united with DM columns. Figure 9 indicates that the average consolidation rate increases with Epc/Edm and the equivalent consolidation coefficient increases with Edm. Moreover, compared with the traditional the CCF improved by PC piles united with DM columns, the consolidation rate of the CCF improved by penetrated PCCSs increases obviously, implying that the outer core can accelerate the consolidation rate of this CCF to some extent.Figure 9
Effect ofEpc/Edm on the consolidation rate.


## 4.4. Effect ofEpc/Es

Figure10 illustrates the variations of Us and Tv for the different the inner core-subsoil modulus ratio (Epc/Es). Epc consists of 3 GPa, 3 GPa, 18 GPa, and 30 GPa, and Es adopted the constant value of 3 MPa. The corresponding Epc/Es includes 1500, 3000, 6000, 30000, and 60000. It can be seen that the average consolidation rate increases as Epc/Es increases with a decreasing growth rate. The reason is that the larger the consolidation coefficient, the higher the consolidation rate. Besides, the equivalent consolidation coefficient increasing with Epc/Es affects the consolidation rate.Figure 10
Effect ofEpc/Es on the consolidation rate.


## 4.5. Effect ofEc/Es

The PCCS piercing value and the composite compression modulus of Region A are both influenced by the stiffness of the cushion.Us and Tv for the different the cushion-subsoil modulus ratio (Ec/Es) are shown in Figure 11, where Ec consists of 0, 3 MPa, 15 MPa, and 30 MPa keeping Es constant as 3 MPa. The corresponding Ec/Es includes 1, 5, 10, and 15. The case with Ec/Es of 1 indicates that no cushion was arranged on the surface of the CCF. It can be seen that the average degree of consolidation increases slightly with Ec/Es and the growth rate decreases with Ec/Es. The effect of Ec/Es on the consolidation rate is less obvious than that of Epc/Es on the consolidation rate because of the smaller range of Ec/Es. The consolidation rate for the CCF untreated with gravel cushion (Ec/Es=1) is lower than that for the CCF with the cushion. Without the gravel cushion, the soil around the piles needs to bear more loads and generates larger excess pore pressure, which takes longer time to dissipate.Figure 11
Effect ofEc/Es on the consolidation rate.


## 4.6. Effect ofkv1/kv2

Figure12 presents the effect of the permeability coefficient ratio of the surrounding soil in Region A and Region B, (kv1/kv2) on consolidation characteristic, where kv1 consists of 10-9 m/s, 10-8 m/s, 5×10−8m/s, and 10-7 m/s, keeping kv2 constant as 10-8 m/s. The corresponding kv1/kv2 comprises 0.5, 1, 5, and 10. The consolidation rate increases as kv1 increases with a decreasing growth rate. Since the CCF reinforced with penetrated PCCSs and floating DM columns is simplified as double-layer ground whose permeability is reflected by the value of kv1/kv2, the larger kv1/kv2 indicates the higher permeability of the subsoil to accelerate the consolidation of the surrounding soil.Figure 12
Effect ofkv1/kv2 on the consolidation rate.


## 4.7. Effect oftu

The effect of the loading time (tu) on the consolidation of this CCF is investigated by varying tu from 60 d to 8 d, 30 d, 80 d, and 0 d in Figure 13. The case of tu as 0 d can be regarded as instant loading. It can be seen from Figure 13 that the average consolidation rate of the CCF decreases with tu. Since the additional stress increment increases with time and remains constant until tu, the longer the loading period, the longer the time it takes for excess pore pressure to accumulate and dissipate.Figure 13
Influence ofTvtc on the average degree of consolidation.


## 5. Conclusion and Limitation

In order to give full play to the merits of different pile techniques, the CCF reinforced with penetrated PCCSs and floating DM columns is proposed. Based on an idealized double-layered consolidation model and modified equal-strain assumptions, an analytical method for the consolidation of this CCF is derived. Besides, the average consolidation degree predicted from this analytical solution is verified by that obtained from numerical analysis. Finally, a parametric study is carried out to analyze the effect of key factors on consolidation of this CCF. The following conclusions can be drawn:(1)
The equivalent compressive moduli ofEcomp1/1−m1−m2 and Ecomp2/1−m1 of the surrounding soil in Region A and Region B, respectively, are larger than that of the untreated-soil ground. Therefore, the consolidation rate of this CCF can be much faster than that of the natural ground(2)
The theoretical values of consolidation rate are slightly bigger than the numerical ones fort>tu. The reason is that the nonlinear variation of the additional stress of subsoils assumed in the numerical simulation is different from that assumed to vary linearly along with the depth in the analytical method(3)
The average consolidation rate of this CCF is mainly affected byEpc and m1 because of the PCCSs penetrating the soft soils. m2 has limited effect on the consolidation rate of this CCF induced by the floating effect of the DM columns(4)
The average consolidation rate is improved slightly withEpc/Es and Ec/Es in this CCF. The average consolidation rate decreases with tu. The consolidation rate increases as kv1/kv2 increases with a decreasing growth rate. Besides, the consolidation rate of this CCF with the gravel cushion is higher than that of the CCF without the gravel cushion. The consolidation rate of the CCF reinforced with penetrated PCCSs and DM columns increases obviously compared with the traditional CCF improved by PC piles united with DM columns(5)
The additional stress of subsoils in the analytical method was simplified as the linear variation along with the depth, which may be different from that in situ. The further research will be conducted to improve the current method



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*Source: 7371850-2023-02-18.xml*