# Analytical Solution for Consolidation of Vertical Drains-Impervious Piles with Multiple Composite Foundations under Nonuniform Initial Pore Pressure and Time-Dependent Well Resistance

Authors: Yuguo Zhang; Yamin Zhao
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

For the vertical drains-impervious pile multiple composite foundation consolidation problem, the effects of nonuniform distribution of initial pore pressure, variation of well resistance with time, and combined radial and vertical seepage in the soil are considered, and a consolidation control equation based on the assumption of equal strain is established based on the axisymmetric model. The analytical solution method is used to give an analytical solution for the consolidation of multiple composite foundations with vertical drains-impervious piles with trapezoidal, rectangular, positive triangle, and inverted triangular distribution of initial pore pressure. The reasonableness of the solution of this study is verified by degradation analysis. According to the solution of the study, the consolidation properties of the multicomposite foundation were analyzed. The results show that the initial pore pressure is faster to consolidate with trapezoidal distribution than with rectangular distribution, the slowest to consolidate with positive triangular distribution, and the fastest to consolidate with inverted triangular distribution, and the larger the initial pore pressure value at the base of the soil layer, the slower the consolidation; the consolidation rate of the foundation all decreases with the increase of the well resistance factor, and the change of the well resistance mainly has an important influence on the later consolidation of the foundation; the vertical permeability coefficient of the soil has a greater influence on the consolidation of the foundation at the initial stage when the foundation thickness is small, and the vertical permeability coefficient of the soil has greater influence on the consolidation of the foundation at the later stage when the foundation thickness is large; the consolidation rate of foundation increases with the increase of replacement rate of impervious pile, compression modulus, and vertical permeability coefficient of soil. Compared with the single pile foundation, the multicomposite foundation can significantly improve the consolidation rate of foundation.


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## Body

## 1. Introduction

Composite foundation technology has a wide application in the field of airports, roads, and terminals because of its good economy and applicability [1–3]. A large number of engineering applications have promoted the development of technology and theory, and the types of composite foundations have changed from the early single pile type to the joint use of multiple pile types today. According to the permeability performance of piles, composite foundations can be divided into drainage pile composite foundations and impervious pile composite foundations. The permeability of impervious piles is less than that of soft clay soils, and the permeability of soft clay soils is usually much less than that of vertical drains, and composite foundations usually solve the problems of poor drainage capacity and insufficient bearing capacity of foundations. The vertical drain has strong drainage capacity, which can promote the discharge of water in the foundation; impervious piles have good bearing capacity, which can improve the bearing capacity of foundation. The combination of the two can give full play to their respective advantages, in order to reduce postconstruction settlement, accelerate foundation consolidation, and improve bearing capacity. The theory of composite foundation consolidation is an important part of composite foundation consolidation, and currently, the established analytical solutions for composite foundation consolidation are able to take into account the variation of total stress with time and depth [4, 5], variable load effects [6, 7], variation of soil permeability coefficient [8], and the influence of various factors such as considering semipermeable drainage boundary conditions [9, 10]. There are more studies about drainage consolidation in sand wells and the theory is more mature, and the multivariate composite foundation consolidation theory draws on the research methods of sand well foundation consolidation theory.In the research of sand well consolidation theory, Carrillo [11] first established the expression of the relationship between the average consolidation degree of unidirectional seepage consolidation and the total average consolidation degree of multidirectional seepage, which provided a theoretical basis for simplifying multidimensional consolidation calculations, and thus, the research of sand well consolidation theory was rapidly developed. Barron [12] obtained analytical solutions for the consolidation of ideal sand well foundations based on the Terzaghi consolidation theory, solved based on the equal strain assumption and the free strain assumption, respectively. Yoshikuni and Nakanodo [13] derived the consolidation equations for free strain in vertical drain foundations based on the study of Barron [12] and obtained solutions that can integrate combined radial and vertical seepage. Hansbo et al. [14] gave a more practical solution for consolidation considering the well resistance effect and smearing effect. Basack and Nimbalkar [15] developed and validated a finite-difference numerical model based on the assumption of free strain to calculate the consolidation settlement of road foundations and concluded that the free strain ratio equal strain change is close to the actual deformation of the foundation, which is more difficult to solve, but Barron [12] showed that there is no significant difference between the two results. Xie and Zeng [16] derived an analytical solution for radial consolidation of sand well foundations based on the assumption of equal strain. Tang and Onitsuka [17] investigated the consolidation of double-layered sand well foundations considering that the soils are stratified in nature. Wang and Jiao [18] discarded the assumption of equal flow around the well and developed a computational model that considers both radial and vertical seepage within the well and obtained the analytical solution for consolidation in both cases of perforated and unperforated soft clay layers. Conte and Troncone [19] considered the variation of external loads with time and derived an analytical solution for radial consolidation of sand well foundations under variable loads. However, the above studies either ignored the well resistance effect of sand wells or assumed that the permeability coefficient of sand wells remained constant when considering the well resistance effect. Zhang et al. [20] considered that the actual drainage boundary conditions are usually incomplete drainage boundaries, and in order to make up for the deficiencies of the boundary conditions in the conventional sand well consolidation theory under instantaneous constant loading, first, the solution of the three-dimensional consolidation equations for sand well foundations was obtained by introducing continuous drainage boundary conditions and using the characteristic function method to obtain the analytical solution for the consolidation of sand well foundations considering the smearing and well resistance effects.The site test studies showed that the permeability coefficient of the sand well foundation will gradually decrease during the consolidation process, which will reduce the consolidation rate of the sand well foundation [21–23], so the change of well resistance with time during the consolidation process should be considered. Deng et al. [24] derived an analytical solution for radial consolidation of sand well foundations under time-dependent variation of well resistance, considering that the permeability coefficient of sand wells decreases gradually during the consolidation process. Nguyen and Kim [25] studied the large deformation consolidation of sand well foundations by considering the variation of sand well permeability coefficient with time with the law of exponential function. Kim et al. [26] consolidation behavior of soft soil with vertical drains considering well resistance and smear effect under cyclic loadings was investigated. Using the variables separation method, a series of analytical solutions were derived to calculate the excess pore water pressure and the average degree of consolidation of the soil subjected to various cyclic loadings including trapezoidal cyclic loading, rectangular cyclic loading, triangular cyclic loading, and Haversine cyclic loading. The correctness of the proposed solutions was verified through degenerating into the existing solutions. Consolidation has an important influence on soil strength and building stability [27], and these studies on the theory of sand well foundations have been relatively mature, and with the development of composite foundation technology, sand well foundations have laid a solid theoretical foundation for the combined composite foundations with the joint use of multiple pile types in recent years.With the development of composite foundation technology, the combined composite foundation with multiple pile types used jointly is gradually popularized in engineering, and its consolidation characteristics have been studied with certain results, and scholars have also studied the consolidation theory of multiple composite foundations composed of different materials. Wu et al. [28] proposed the reinforcement method of prestressed pipe pile combined with plastic drainage board to strengthen soft clay foundation, and the reinforced soft clay foundation and the dissipation law of pore water pressure were analyzed by engineering examples. Chen et al. [29] studied the consolidation problem of composite foundations with drained powder-jet piles, and through a simplified analysis of the foundation model, with the drainage slab as the center, the dissipation law of excess pore water pressure is infiltration from the surrounding to the center, assuming that the powder-jet piles are impermeable; considering that the drainage slab has the effect of bearing the load and satisfying the assumption of equal strain, the consolidation expression of the foundation is obtained using the axisymmetric consolidation theory. The compression modulus of the drainage slab and the coating area is the same as that of the soil. Liu et al. [30] considered the effects of variation of total pile-soil stress, compression and pore water pressure of vertical drainage body, drainage performance of pile body, and distance between pile and drainage body on consolidation of composite foundation. The analytical solutions of pore pressure, consolidation degree, and pile-soil stress ratio of composite foundation under embankment considering pile drainage performance, pile-soil load transfer, compressibility of drainage body, and well resistance are derived. It is considered that pore pressure exists in both drainage piles and weakly drained piles in composite foundation and assumed that pore pressure varies proportionally along the depth, which is not in accordance with the actual situation. Ye et al. [31, 32] investigated a combined composite foundation consolidation problem of cemented soil piles combined with plastic drainage boards. The pore water pressure was modeled as percolation from the center to the periphery, the piles were assumed to be permeable piles, the bearing and deformation of the drainage boards were not considered, equivalent vertical and radial consolidation coefficients were established, and the expressions for the consolidation of composite foundations under transient loading were derived. Qin et al. [33] assumed that the flow rate of pore water seeping into the sand well body along the radial direction is equal to the amount of water discharged from the sand well body, considered the smearing effect of the sand well, and derived an analytical solution for the consolidation of composite foundations with sand well combined with cemented soil mixing piles. Yu et al. [34, 35] simplified the composite foundation with impermeable long piles and permeable short piles under combined seepage by considering the volume compression of permeable piles based on the axisymmetric consolidation model, equated the soil within the permeable piles as the interpile soil with larger permeability coefficient, and formed a double-layer impermeable pile composite foundation with different permeability coefficients in the reinforcement zone with impermeable long piles, so as to obtain the analytical solution of the composite foundation with permeable piles combined with impermeable piles. Liu et al. [36] gave a simplified method for calculating the settlement of a composite foundation consolidation used plum blossom pile layout scheme and based on an axisymmetric consolidation model with a combination of powder-jetting piles and drainage board. Yang et al. [37] reported that based on the equal vertical strain assumption, the governing equations for average excess pore water pressures within the surrounding soil of the composite ground with long rigid piles and impervious short piles are derived under ramp loading. The corresponding analytical solutions are developed based on the one-dimensional consolidation theory of a double-layer ground. Prakash and Krishnamoorthy [38] investigated the effect of lime and CFG pile composite foundations on the factor of safety of embankments at different time intervals during the consolidation of foundations. The above multicomposite foundation composed of different materials incorporates the advantages of each pile type, which can save the project cost and also provides a guide for the actual project.In the analysis of the previous related soft foundation treatment methods, based on the actual engineering, the problem of consolidation for multicomposite foundations is proposed, and the study of the consolidation theory of multicomposite foundations is the basis for predicting settlement changes with time; however, in actual engineering, the initial pore pressure caused by external loads in the foundation is often nonuniformly distributed along the depth [39, 40]. While considering the variation of well resistance with time, the vertical seepage of the soil is almost always not considered, and the vertical seepage of the soil affects the distribution pattern of the superstatic pore pressure during the consolidation of the foundation; if the vertical seepage of the soil is neglected, it is not only inconsistent with the actual consolidation process of the multicomposite foundation but may also significantly underestimate the consolidation rate of the multicomposite foundation [41, 42]. Therefore, when considering the consolidation of multifaceted composite foundations when the well resistance varies with time, the effect of vertical seepage of the soil on the consolidation process of the foundation should be considered. Based on the above research, the research on the multifaceted composite foundation is still in the development stage, and the existing research results on the consolidation theory of vertical drain-impervious pile multiple composite foundation are less; therefore, the research on vertical drain-impervious pile multicomposite foundation under the consideration of initial pore pressure nonuniformity and well resistance change with time can provide a theoretical reference for engineering construction.Based on the assumption that the permeability coefficient of sand wells decreases exponentially as a function of time as proposed by Deng et al. [24], this study derives an analytical solution for the consolidation of multifaceted composite foundations with vertical and radial combined seepage in the soil under a trapezoidal distribution of the initial pore pressure and gives the analytical solution for the consolidation of multicomposite foundations with rectangular, positive triangular, and inverted triangular distributions of the initial pore pressure. The reasonableness of the solution is verified by comparing the degenerate solution with the existing analytical solution. The consolidation properties of the vertical drain-impervious pile multiple composite foundations were analyzed by preparing a calculation program and plotting the relevant consolidation curves.


## 2. Establishment of Consolidation Control Equation

### 2.1. Calculation Sketch and Basic Assumptions

Figure1 shows a computational model for a combined vertical drain-impervious column multicomposite foundation consolidation problem. Multicomposite foundations are mostly arranged in rectangles and triangles. In this study, both vertical drains and impervious columns are arranged in rectangles, and one unit is taken as the research object, with the vertical drain being the center of the model and the impervious column located at the outer boundary of the model, where H is the thickness of the soft clay layer, q is the instantaneous applied external load, rw, re, and rn are the radius of the vertical drain, the radius of the zone of influence, and the radius of the entire unit foundation, respectively, kh and kv are the horizontal and vertical permeability coefficients of the foundation soil, respectively, kw is the vertical permeability coefficient of the vertical drain, uw and us are the pore pressures within the vertical drain and soil at any point, respectively, Ep and Es are the compression modulus of the impervious column and natural soil, respectively, and r and z are the radial and vertical coordinates, respectively, assuming that the drainage conditions of the multicomposite foundation are permeable at the top and impermeable at the bottom.Figure 1
Vertical drain-impervious column multicomposite foundation consolidation model. (a) Layout plan. (b) Cutaway diagram.
(a)(b)In the derivation of this study, the following assumptions are made about the model.(1)
The equal strain condition holds, i.e., there is no lateral deformation in the foundation and the vertical deformation at any point at the same depth is equal.(2)
The load is applied instantaneously, and the resulting initial pore pressure is nonuniformly distributed along the depth direction.(3)
Ignore the radial seepage in the vertical drains(4)
Water in the soil has both radial and vertical seepage, and the seepage obeys Darcy’s law.(5)
Considering the well resistance effect of the vertical drain and considering that the permeability coefficient of the vertical drain decreases gradually during the consolidation process, according to the relevant experimental and theoretical studies carried out in the literature [21–24], an exponential function can be used to describe the variation of the permeability coefficient of the vertical drain with the consolidation time t, i.e.,(1)kw=kw0e−At.(6)
The flow around the well is continuously equal, i.e., the flow of water from the soil into the vertical drains at any depth is equal to the increment of water flowing upwards in the well, expressed as follows.(2)2πrdzkhγw∂us∂r|r=rw=−πrw2dzkw0e−Atγw∂2u¯w∂z2.In the above equation,kw0 is the initial permeability coefficient of the vertical drain; A is the test constant greater than 0; and γw is the water weight.Figure2 shows the four cases of initial pore pressure distribution along the depth, where PT and PB are the initial pore pressure values at the top and bottom of the soil layer, respectively. When PT=PB=P0, as shown in Figure 2(a), the initial pore pressure is evenly distributed. When PT=0, as shown in Figure 2(b), the initial pore pressure is distributed in a positive triangle along the depth. When PB=0, as shown in Figure 2(c), the initial pore pressure is distributed in an inverted triangle. When PT≠PB≠0, as shown in Figure 2(d), the initial pore pressure is distributed in a trapezoid shape.Figure 2
Nonuniform distribution of initial excess pore water pressure along depth. (a) Rectangular. (b) Positive triangle. (c) Inverted triangle. (d) Trapezoidal.
(a)(b)(c)(d)


### 2.2. Solving Conditions and Control Equation

The external load is supported by the vertical drains, soil, and impervious columns, which according to the vertical equilibrium condition gives(3)πrw2σ¯w+πre2−rw2σ¯s+πrn2−re2σ¯p=πrn2q,(4)σ¯w−uwEw=σ¯s−u¯sEs=σ¯pEp=εv,where σ¯w, σ¯s, and σ¯p are the average stresses in the vertical drain, soil, and impervious columns, respectively, in a multicomposite foundation, Ew is the modulus of compression of the vertical drain, and u¯s is the average pore water pressure at any depth in the soil.The expressions for the average pore water pressure in the foundation soil at any depth and the average pore water pressure across the foundation at any depth are(5)u¯s=1πre2−rw2∫rwre2πrusdr,(6)u¯=1πrn2∫0re2πrusrdr=New2−1Nnw2u¯s,where New=re/rw, Nnw=rn/rw, where εv and εz are the volumetric and vertical strains in the foundation, respectively, and are equal when only vertical deformation is present.From equations (3), (4), and (6), the partial derivative of the vertical strain εz with respect to time is(7)∂εz∂t=−1Ecom∂u¯∂t,where u¯ is the average pore water pressure across the foundation at any depth, and Ecom is the composite modulus of elasticity, which can be expressed as(8)Ecom=Ew+New2−1Es+Nnw2−New2EpNnw2.Referring to the study of Xie and Zeng [16] in the literature, the basic consolidation equation for the axisymmetric case can be obtained according to the assumption of equal strain as follows:(9)∂εv∂t=−khγw1r∂∂rr∂us∂r−kvγw∂2u¯s∂z2rw≤r≤re.To simplify the calculation, a functionwz,t, which is independent of r, is introduced and can be expressed as(10)wz,t=kvγw∂2u¯s∂z2+∂εv∂t.Substituting (10) into (9) gives(11)wz,t=−khγw1r∂∂rr∂us∂rrw≤r≤re.Boundary conditions:(12)r=rw,us=uw=u¯w,(13)r=re,∂us∂r=0,(14)z=0,u=0,(15)z=H,∂u∂z=0.Initial conditions:(16)t=0,u¯z=u0z=PT+PB−PTZH,where u0 is the initial pore pressure of the foundation, and u is the pore pressure at any point of the foundation.Integrating (11) with respect to r and using the boundary condition (13) yields(17)∂us∂r=wz,tγw2khrn2r−r.Integrating (17) with respect to r and using the boundary condition (12) yields(18)us=z,tγw2khre2lnrrw−r2−rw22+uw.The combination of equation (5) and equation (18) yields(19)u¯s=wz,tγwre2Fa2kh+u¯w,where Fa=lnNew+4N4ew−1/4N4ew−3/4N2ew/N2ew−1Substituting (17) and (1) into (2) gives(20)kw0e−Atγw∂2u¯w∂z2=−wz,tNew2−1.Then, from (6),(7), (10), and (20), the following can be derived:(21)Nnw2kvγw∂2u¯∂z2=New2−1Ecom∂u¯∂t−kw0e−Atγw∂2u¯w∂z2.Taking the partial derivative of (19) twice with respect to z and using (6) gives(22)Nnw2New2−1∂2u¯∂z2=∂2u¯w∂z2+γwre2Fa2kh−1Ecom∂3u¯∂z2∂t+Nnw2New2−1kvγw∂4u¯∂z4.Combining (21) and (22) and eliminating u¯w yields the control equation as(23)Be−At∂4u¯∂z4−Ce−At∂3u¯∂z2∂t−De−At+E∂2u¯∂z2+∂u¯∂t=0,where B=Nnw2/New2−12re2Fa/2Ecomkv/γwkw0/kh; C=1/New2−1re2Fa/2kw0/kh; D=Nnw2/New2−12Ecomkw0/γw; E=Nnw2/New2−1Ecomkv/γw, where B, C, D, and E are all constants.


## 2.1. Calculation Sketch and Basic Assumptions

Figure1 shows a computational model for a combined vertical drain-impervious column multicomposite foundation consolidation problem. Multicomposite foundations are mostly arranged in rectangles and triangles. In this study, both vertical drains and impervious columns are arranged in rectangles, and one unit is taken as the research object, with the vertical drain being the center of the model and the impervious column located at the outer boundary of the model, where H is the thickness of the soft clay layer, q is the instantaneous applied external load, rw, re, and rn are the radius of the vertical drain, the radius of the zone of influence, and the radius of the entire unit foundation, respectively, kh and kv are the horizontal and vertical permeability coefficients of the foundation soil, respectively, kw is the vertical permeability coefficient of the vertical drain, uw and us are the pore pressures within the vertical drain and soil at any point, respectively, Ep and Es are the compression modulus of the impervious column and natural soil, respectively, and r and z are the radial and vertical coordinates, respectively, assuming that the drainage conditions of the multicomposite foundation are permeable at the top and impermeable at the bottom.Figure 1
Vertical drain-impervious column multicomposite foundation consolidation model. (a) Layout plan. (b) Cutaway diagram.
(a)(b)In the derivation of this study, the following assumptions are made about the model.(1)
The equal strain condition holds, i.e., there is no lateral deformation in the foundation and the vertical deformation at any point at the same depth is equal.(2)
The load is applied instantaneously, and the resulting initial pore pressure is nonuniformly distributed along the depth direction.(3)
Ignore the radial seepage in the vertical drains(4)
Water in the soil has both radial and vertical seepage, and the seepage obeys Darcy’s law.(5)
Considering the well resistance effect of the vertical drain and considering that the permeability coefficient of the vertical drain decreases gradually during the consolidation process, according to the relevant experimental and theoretical studies carried out in the literature [21–24], an exponential function can be used to describe the variation of the permeability coefficient of the vertical drain with the consolidation time t, i.e.,(1)kw=kw0e−At.(6)
The flow around the well is continuously equal, i.e., the flow of water from the soil into the vertical drains at any depth is equal to the increment of water flowing upwards in the well, expressed as follows.(2)2πrdzkhγw∂us∂r|r=rw=−πrw2dzkw0e−Atγw∂2u¯w∂z2.In the above equation,kw0 is the initial permeability coefficient of the vertical drain; A is the test constant greater than 0; and γw is the water weight.Figure2 shows the four cases of initial pore pressure distribution along the depth, where PT and PB are the initial pore pressure values at the top and bottom of the soil layer, respectively. When PT=PB=P0, as shown in Figure 2(a), the initial pore pressure is evenly distributed. When PT=0, as shown in Figure 2(b), the initial pore pressure is distributed in a positive triangle along the depth. When PB=0, as shown in Figure 2(c), the initial pore pressure is distributed in an inverted triangle. When PT≠PB≠0, as shown in Figure 2(d), the initial pore pressure is distributed in a trapezoid shape.Figure 2
Nonuniform distribution of initial excess pore water pressure along depth. (a) Rectangular. (b) Positive triangle. (c) Inverted triangle. (d) Trapezoidal.
(a)(b)(c)(d)


## 2.2. Solving Conditions and Control Equation

The external load is supported by the vertical drains, soil, and impervious columns, which according to the vertical equilibrium condition gives(3)πrw2σ¯w+πre2−rw2σ¯s+πrn2−re2σ¯p=πrn2q,(4)σ¯w−uwEw=σ¯s−u¯sEs=σ¯pEp=εv,where σ¯w, σ¯s, and σ¯p are the average stresses in the vertical drain, soil, and impervious columns, respectively, in a multicomposite foundation, Ew is the modulus of compression of the vertical drain, and u¯s is the average pore water pressure at any depth in the soil.The expressions for the average pore water pressure in the foundation soil at any depth and the average pore water pressure across the foundation at any depth are(5)u¯s=1πre2−rw2∫rwre2πrusdr,(6)u¯=1πrn2∫0re2πrusrdr=New2−1Nnw2u¯s,where New=re/rw, Nnw=rn/rw, where εv and εz are the volumetric and vertical strains in the foundation, respectively, and are equal when only vertical deformation is present.From equations (3), (4), and (6), the partial derivative of the vertical strain εz with respect to time is(7)∂εz∂t=−1Ecom∂u¯∂t,where u¯ is the average pore water pressure across the foundation at any depth, and Ecom is the composite modulus of elasticity, which can be expressed as(8)Ecom=Ew+New2−1Es+Nnw2−New2EpNnw2.Referring to the study of Xie and Zeng [16] in the literature, the basic consolidation equation for the axisymmetric case can be obtained according to the assumption of equal strain as follows:(9)∂εv∂t=−khγw1r∂∂rr∂us∂r−kvγw∂2u¯s∂z2rw≤r≤re.To simplify the calculation, a functionwz,t, which is independent of r, is introduced and can be expressed as(10)wz,t=kvγw∂2u¯s∂z2+∂εv∂t.Substituting (10) into (9) gives(11)wz,t=−khγw1r∂∂rr∂us∂rrw≤r≤re.Boundary conditions:(12)r=rw,us=uw=u¯w,(13)r=re,∂us∂r=0,(14)z=0,u=0,(15)z=H,∂u∂z=0.Initial conditions:(16)t=0,u¯z=u0z=PT+PB−PTZH,where u0 is the initial pore pressure of the foundation, and u is the pore pressure at any point of the foundation.Integrating (11) with respect to r and using the boundary condition (13) yields(17)∂us∂r=wz,tγw2khrn2r−r.Integrating (17) with respect to r and using the boundary condition (12) yields(18)us=z,tγw2khre2lnrrw−r2−rw22+uw.The combination of equation (5) and equation (18) yields(19)u¯s=wz,tγwre2Fa2kh+u¯w,where Fa=lnNew+4N4ew−1/4N4ew−3/4N2ew/N2ew−1Substituting (17) and (1) into (2) gives(20)kw0e−Atγw∂2u¯w∂z2=−wz,tNew2−1.Then, from (6),(7), (10), and (20), the following can be derived:(21)Nnw2kvγw∂2u¯∂z2=New2−1Ecom∂u¯∂t−kw0e−Atγw∂2u¯w∂z2.Taking the partial derivative of (19) twice with respect to z and using (6) gives(22)Nnw2New2−1∂2u¯∂z2=∂2u¯w∂z2+γwre2Fa2kh−1Ecom∂3u¯∂z2∂t+Nnw2New2−1kvγw∂4u¯∂z4.Combining (21) and (22) and eliminating u¯w yields the control equation as(23)Be−At∂4u¯∂z4−Ce−At∂3u¯∂z2∂t−De−At+E∂2u¯∂z2+∂u¯∂t=0,where B=Nnw2/New2−12re2Fa/2Ecomkv/γwkw0/kh; C=1/New2−1re2Fa/2kw0/kh; D=Nnw2/New2−12Ecomkw0/γw; E=Nnw2/New2−1Ecomkv/γw, where B, C, D, and E are all constants.


## 3. Solution of Average Pore Pressure and Average Consolidation of Composite Foundation

According to Lu et al. [43], it is assumed that the solution of (23) takes the form(24)u¯=∑m=0∞TmtsinMHz,where M=2m+1π/2, m=0,1,2⋯Substituting (24) into (23), the following can be obtained:(25)Tm′t+BM4/H4+DM2/H2e−At+EM2/H21+Ce−AtM2/H2Tmt=0.(25) is a variable coefficient first-order partial differential equation; for any m, the function Tmt is a function related to time t only. From the expressions for B, C, and E in (23), the solution to (25) gives the expression for Tmt as(26)Tmt=GmImt,(27)Imt=exp−EM2tH2+DAClnH2+CM2e−AtH2+CM2,where Gm is the constant coefficient associated with m.Substituting (26) and (27) into equation (24) gives(28)u¯=∑m=0∞GmImt⋅sinMzH.Substituting the initial condition (16) into equation (28) and using the trigonometric orthogonality, the following can be obtained:(29)Gm=∫0Hu0zsinMz/Hdz∫0Hsin2Mz/Hdz=2MpT+−1mpB−pTM.Substituting (29) into (28) gives the average pore pressure over the entire foundation at any depth as(30)u¯=∑m=0∞2MpT+−1mpB−pTMImt⋅sinMzH.Thus, the total average consolidation of a multiple composite foundations as defined by stress can be derived as(31)Ut=1−∑m=0∞4ImtM2pB+pTpT+−1mpB−pTM.In summary, (30) and (31) are the solutions for the consolidation of a vertical drain-impervious column multicomposite foundation when a trapezoidal distribution of the initial pore pressure is considered. Based on the four distributions of the initial pore pressure in Figure 2, further expressions for the average pore pressure and average consolidation for each case are given.Case 1.
AtPT=PB=P0, the initial pore pressure is rectangularly distributed, as shown in Figure 2(a).
Degenerate (30) and (31) to the instantaneously loaded initial pore pressure homogeneous vertical drain and impervious column multicomposite foundation consolidation solution, i.e.,(32)u¯=p0∑m=0∞2MImt⋅sinMzH,(33)Ut=1−∑m=0∞2M2Imt.Case 2.
AtPT=0, the initial pore pressure has a positive triangular distribution, as shown in Figure 2(b).
According to (30) and (31), the following can be obtained:(34)u¯=pB∑m=0∞−1m2M2Imt⋅sinMzH,(35)Ut=1−∑m=0∞−1m4M3Imt.Case 3.
AtPB=0, the initial pore pressure has an inverted triangular distribution, as shown in Figure 2(c).
According to (30) and (31), the following can be obtained:(36)u¯=pT∑m=0∞2M1−−1m1MImt⋅sinMzH,(37)Ut=1−∑m=0∞4M21−−1m1MImt.
At this point, all the analytical solutions for multicomposite foundation consolidation of combined vertical drains-impervious columns with trapezoidal, rectangular, positive triangle, and inverted triangular distribution of initial pore pressure under this model have been given.


## 4. Degenerations of the Obtained Solution

For the consolidation analytic solution obtained in this study, its reasonableness needs to be verified and analyzed, and the reasonableness of the solution can be effectively verified by using the degeneracy analysis of the solution. According to the degradation methods in the literature [26, 44], the degradation analysis is performed on the consolidation analytical solutions obtained in this study. The above three special cases of pore pressure and consolidation degree are degenerated from (30) and (31), where (32) and (33) are the solutions for consolidation when the initial pore pressure is homogeneous and can be degenerated again, and the steps and methods of degeneration are as follows.If so thatNew=Nnw, i.e., there are no impervious piles in the foundation. Since there are no more impervious piles, the other parameters need to be degraded in the same way, where Ecom=Es, and New=n.(1)
When the vertical seepage of the soil is not considered, i.e., whenkv=0, equation (33) can be degraded to(38)U=1−∑m=0∞2M21+CM2/H2exp−At1+CM2/H2D/AC.Combining the expressions forC and D in equation (23), equation (38) is in perfect agreement with the analytical solution for consolidation of sand well foundations when the smear zone is not considered by Deng et al. [24]. At this point, the multifaceted composite foundation consolidation solution degenerates into a single sand well foundation consolidation solution.(2)
When the radial seepage of the soil is not considered, i.e.,kh=0, and the sand well foundation consolidation problem becomes a one-dimensional consolidated natural soil foundation, equation (38) can be degraded to(39)U=1−∑m=0∞2M2exp−cvM2tH2.(39) is the analytical solution of the one-dimensional consolidation of Terzaghi natural foundation.Through the degradation study of the analytical solution of this study, it can be seen that the degraded solution of this study is completely consistent with the existing analytical solution in different cases, which verifies the correctness of the solution of this study to a certain extent. The analytical solution of consolidation in this study takes into account the nonuniform distribution of initial pore pressure, the variation of well resistance with time, and the combined radial and vertical seepage of the soil and is therefore more applicable to engineering.


## 5. Analysis of Consolidation Properties

Based on the above analysis, the solution for the consolidation of vertical drain-impervious pile multiple composite foundation with nonhomogeneous initial pore pressure and time variation of well resistance has been derived, and the analytical solution for consolidation can reflect the factors affecting the consolidation rate of this multiple composite foundation more comprehensively. To facilitate calculation and plotting, the functionIm is dimensionless, where the time factor Th=cht/4re2, where ch is the radial consolidation factor of the soil, ch=khEs/γw. Let At=αTh and the well resistance factor α=4Are2/ch, then the function ImTh can be expressed as(40)ImTh=exp−M2LTh+DC4re2αchlnH2+CM2exp−αThH2+CM2.By preparing a computational program for the solution of this study and plotting the relevant consolidation curves using the method of control variables, a parametric study of the consolidation properties of multivariate composite foundations was carried out using the consolidation analytical solution derived in this study.Figure3 shows the consolidation curves for different distributions of initial pore pressure. The initial pore pressure distribution conditions have a significant effect on the consolidation of multicomposite foundations, with a trapezoidal distribution consolidating faster than a rectangular distribution, a positive triangular distribution consolidating the slowest, and an inverted triangular distribution consolidating the fastest. The results show that the effect of considering the nonuniform distribution of the initial pore pressure along the depth is not negligible.Figure 3
Consolidation curves for different distributions of initial pore pressure.Figures4 and 5 show the curves of the effect of the well resistance factor α on the consolidation of vertical drain-impervious pile multiple composite foundations when only radial seepage is considered in the soil and when radial-vertical seepage is considered. The consolidation rate of the foundation decreases with the increase of the well resistance factor, regardless of whether the vertical seepage of the soil is considered or not, and the effect of the change of the well resistance on the consolidation of the foundation is mainly in the late stage of consolidation. As can be seen from Figure 4, when only radial seepage is considered, the larger the well resistance factor, the earlier the consolidation process stabilizes, and the smaller the average consolidation degree after stabilization. As can be seen from Figure 5, when considering radial-vertical seepage, the foundation can continue to consolidate through vertical seepage at a later stage of consolidation, thus maximizing the average consolidation of the foundation.Figure 4
Influence ofα on average consolidation degree when only radial flow is considered.Figure 5
Influence ofα on average consolidation degree when radial-vertical flow is considered.The curve of the effect of the pile-soil compression modulus ratio on the consolidation of the foundation is shown in Figure6. As the pile-soil modulus ratio increases, the consolidation rate of the foundation is accelerated. In engineering, impervious piles can be CFG piles, cement mix piles, or even concrete pipe piles, and the stiffness varies greatly, and the increase of stiffness of impervious piles has a significant effect on the consolidation rate of composite foundations. When the compressive modulus of impervious piles gradually becomes larger, the stress of impervious piles also becomes larger. In the design of composite foundation, pile-soil share the upper load; when the stress concentration effect of impervious pile is more and more obvious, it makes the load acting on pore pressure dissipation smaller and smaller, so the consolidation speed of vertical drain-impervious pile multiple composite foundation is accelerated with the increase of impervious pile compression modulus.Figure 6
Influence of pile-soil compression modulus ratio on foundation consolidation curve.Figure7 shows the influence of the initial pore pressure values at the top and bottom of the soil layer on the consolidation process. When PB/PT=1 and PB/PT=0, the initial pore pressure is uniformly distributed along the depth and the initial pore pressure is distributed in an inverted triangle along the depth, respectively, as shown in the figure; when the initial pore pressure value at the top of the soil layer is constant, the larger the initial pore pressure value at the bottom of the soil layer and the slower the foundation consolidation.Figure 7
Curve of the effect ofPB/PT variation on the consolidation process.Figure8 shows the effect of the impervious pile replacement rate on the consolidation of the multiple composite foundations. As the radius of the entire foundation decreases, this leads to a decrease in Nnw. The smaller the replacement rate of the impervious pile, the slower the rate of consolidation of the whole foundation because the modulus of elasticity of the impervious piles is much larger than the modulus of elasticity of the vertical drain and the soil, the smaller the replacement rate of the impervious piles, the smaller the average modulus of elasticity of the whole foundation, the smaller the external load borne by the impervious piles themselves, and the slower the rate of consolidation of the foundation accordingly. When Nnw=11, the replacement rate of impervious piles is zero, i.e., there are no impervious piles in the composite foundation. The multiple composite foundation can significantly increase the consolidation rate of the foundation compared to a single pile type foundation.Figure 8
Effect of impervious pile replacement rate on foundation consolidation.Figures9–11 show the curves of the effect of the vertical permeability coefficient kv of the soil on the consolidation of the vertical drain-impervious pile multiple composite foundation for different foundation thicknesses H. As shown in Figures 9–11, the vertical percolation of the soil can accelerate the consolidation rate of the foundation for different foundation thicknesses H. Moreover, the consolidation rate of the foundation increases as the vertical percolation coefficient kv of the soil increases, but for different foundation thicknesses H, the effect of the vertical percolation coefficient kv of the soil on the consolidation properties of the foundation is significantly different. When the foundation thickness H is small (e.g., H/re=6.5), the vertical permeability coefficient kv of the soil has a greater influence on the initial consolidation properties of the foundation and the average consolidation curves at a later stage coincide almost exactly. When H/re=16.2, the soil vertical permeability coefficient kv has some influence on the initial and later consolidation properties of the foundation, but its influence is limited. When the foundation thickness H is large (e.g., H/re=26), the vertical permeability coefficient kv of the soil has a greater influence on the consolidation properties of the foundation at a later stage, and the average consolidation curves at the beginning coincide almost exactly.Figure 9
Influence ofkv on average consolidation degree when H/re=6.5.Figure 10
Influence ofkv on average consolidation degree when H/re=16.2.Figure 11
Influence ofkv on average consolidation degree when H/re=26.Figure12 shows the comparison between the present solution and the existing solution under transient loading. Under the same parameter conditions, when the smearing effect is not considered, the consolidation speed from fast to slow is Chen et al.’ [29] multicomposite foundation solution, this study’s solution, Lu et al.’ [45] multicomposite foundation solution, and Deng et al.’ [24] sand well foundation solution in order. The reason for the faster consolidation in the solution of Chen et al. [29] is that the radial and vertical seepage in the vertical drain is not considered, and its drainage capacity is considered infinite in the calculation process, i.e., there is no effect of well resistance in the vertical drain. In contrast, the solution in this study considers that there is vertical well resistance in the vertical drain, and the vertical permeability coefficient in the vertical drain gradually decreases during consolidation. The solution of Lu et al. [45] considers both radial and vertical seepage in the pile, i.e., the pile has well resistance in both radial and vertical directions, so the consolidation rate is slower than that in this study. The solution of Deng et al. [24] is a single vertical drain foundation without impervious piles and without considering the vertical seepage of the soil, so the consolidation rate is the slowest.Figure 12
Comparison of the solution in this study with the existing solution.


## 6. Conclusion

This study establishes a vertical drain-impervious pile multivariate composite foundation consolidation model and gives the analytical solution of consolidation considering the nonuniform initial pore pressure, the change of well resistance with time, and the combined radial and vertical seepage of soil. By comparing the degenerate solution with the existing analytical solution, the reasonableness of the analytical solution is verified. The consolidation characteristics of the multifaceted composite foundation are systematically investigated through parametric analysis. The main conclusions are as follows.(1)
The nonuniform distribution of initial pore pressure along the depth has a significant effect on the consolidation of multifaceted composite foundations, with a trapezoidal distribution of initial pore pressure consolidating faster than a rectangular distribution, a positive triangular distribution consolidating the slowest, and an inverted triangular distribution consolidating the fastest. The larger the initial pore pressure value at the base of the soil layer, the slower the consolidation.(2)
Whether or not the vertical seepage in the soil is considered, the consolidation rate of the multicomposite foundation decreases with the increasing well resistance factor. The change in well resistance mainly has an important influence on the consolidation properties of the foundation at a later stage.(3)
When the foundation thicknessH is small, the vertical permeability coefficient of the soil has a greater influence on the initial consolidation properties of the foundation, and when the foundation thickness H is large, the vertical permeability coefficient of the soil has a greater influence on the later consolidation properties of the foundation.(4)
The effect of impervious piles on the consolidation rate of multifaceted composite foundations is mainly reflected in the stress concentration resulting from having a higher compression modulus, and the larger the compression modulus of impervious piles, the faster the foundation consolidation.(5)
The consolidation rate of multiple composite foundations increases with the increase in the replacement rate of impervious piles and vertical permeability coefficient of the soil. Compared with the single pile type foundation, the multicomposite foundation can significantly increase the consolidation rate of the foundation.



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*Source: 2181792-2022-05-11.xml*