# A Three-Section-Settlement Calculation Method for Composite Foundation Reinforced by Geogrid-Encased Stone Columns

Authors: Binhui Ma; Zhiyong Hu; Zhuo Li; Kai Cai; Minghua Zhao; Chengbin He; Qiunan Chen; Bingchu Chen; Xiaocheng Huang
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

The analysis of the bearing characteristics and deformation mechanism of composite foundation reinforced with geogrid-encased stone columns is presented in order to obtain its settlement calculation method. The settlement of composite foundation is divided into three sections which are the reinforced section, unreinforced section, and underlying stratum. Based on Hooke’s law of space problem and the thoughts of the layer-wise summation method, the relative slip displacement between pile and soil of reinforced section without plastic zone is analyzed. The settlement of reinforced section is calculated by the layered iteration method based on the pile element model. The compatibility of vertical and radial deformations of unreinforced section is analyzed based on the pile-soil element model. The settlement of underlying stratum is still calculated by the layer-wise summation method. Finally, two engineering examples are analyzed and the results show that the settlement calculated by the presented method is close to the measured one. The method overcomes the defect that the calculated results by the other existing methods are more dangerous and it is more feasible and can be applied in engineering practice.


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## Body

## 1. Introduction

Stone column has been widely used in soft clay improving engineering as a common reinforcement [1]. The bearing capacity of stone column mainly depends on the lateral restraint provided by the surrounding soil [2]. When the lateral restraint force is insufficient, the top part of the stone column body is prone to swelling, resulting in the reduction of its bearing capacity and the increase of settlement, which fails to achieve the expected effect of reinforcement [3]. In engineering practice, the treatment method of covering the geogrid encasement into stone column in the depth range of 2∼3 times of pile diameter is adopted to make use of the hoop effect of the geogrid encasement on the pile body to increase its lateral restraint force and limit the swelling deformation of the pile body, so as to improve the bearing capacity and reduce the settlement. This new type can be called composite foundation by geogrid-encased stone columns. Because of the effect of the geogrid encasement, the bearing capacity and deformation characteristics of the stone column have changed, so it is not appropriate to simply apply the existing settlement calculation method of the composite foundation to analyze the settlement of the composite foundation reinforced by geogrid-encased stone columns. Therefore, how to combine the characteristics of geogrid-encased stone columns and deeply discuss the settlement calculation method of that has an important theoretical significance for engineering design.At present, the research methods of geogrid-encased stone columns are mostly laboratory tests, numerical simulation and field tests, etc. The research mainly focuses on the analysis of bearing capacity and stability. The bearing capacity of geogrid-encased stone columns is early analyzed by Zhou and Zhang [4] with the limit equilibrium theory. Chen et al. [5–7], respectively, derived the bearing capacity calculation formula of the top-reinforced stone column based on the expansion theory of circular hole, the slider balance method, and the upper limit analysis method. Based on the progress of limit equilibrium theory and considering the failure mode of geogrid-reinforced crushed stone pile, Zhao et al. [8,9] deduced the calculation formula of ultimate bearing capacity of single reinforced crushed stone pile. However, the research on settlement calculation of composite foundation reinforced by geogrid-encased stone columns is relatively few. Cao and Yang [10] divided geogrid-encased stone column into reinforced section, unreinforced plastic section, and unreinforced elastic section according to the relative slip between column and surrounding soil, established corresponding settlement analysis model according to the deformation characteristics of different sections, and obtained the settlement calculation method of geogrid-encased stone column composite foundation under flexible foundation. Wu and Chen [11] also divided the reinforcement into three sections and derived settlement formulas for each section based on elastic mechanics and pile-soil coordinated deformation. According to the observation and conclusion of numerical simulation results, Tan et al. [12] established the analytical solution of the ultimate bearing capacity of geogrid-encased stone column in soft clay. Esameldin and Zalihe [13] studied the bearing capacity of geogrid-encased stone column in different soft soils through laboratory experiments.Although these research studies on the bearing deformation mechanism of the geogrid-encased stone column are fine, they are still with some limitations and shortcomings. For example, the friction of geogrid-encased stone column caused by relative slip of column and soil is too complicated to be considered. The lateral deformation of column decreases rapidly with depth, and there is not necessarily a plastic zone, so the assumption that there are three sections is not necessarily valid. In addition, from the calculation results, the theoretical value is less than the measured value, and the calculation is biased to the danger. Therefore, on the basis of previous research, the composite foundation reinforced by geogrid-encased stone column is divided into three sections, which are the reinforced section, unreinforced section, and underlying stratum. Based on Hooke’s law of space problem and the thoughts of the layer-wise summation method, the deformation of stone column is calculated based on the pile element by the layer-iteration method, considering the relative slip of pile and soil without plastic zone. The pile-soil unit model is introduced for analysis according to the coordination of the vertical and radial deformations of pile and soil in the unreinforced section. The settlement of the underlying stratum is calculated by the layer-wise summation method, and then the settlement calculation method of the composite foundation is proposed.


## 2. Deformation Mechanism Analysis

Generally speaking, the settlement of composite foundation reinforced by geogrid-encased stone columns consists of cushion, reinforcement area, and underlying layer [1] (Figure 1).Figure 1
Composition of composite foundation reinforced by geogrid-encased stone columns.Because the thickness of the cushion is generally smaller and the compaction degree is higher, the settlement deformation is basically completed in the construction stage, so it can be ignored. Therefore, the settlementS of geogrid-encased stone column composite foundation can be regarded as the sum of the compression amount S1 in the reinforcement area and the compression amount S2 in the underlying strata, namely,(1)S=S1+S2.According to the existing research results [9–11], the reinforcement area can be divided into reinforced section and unreinforced section according to whether the pile is wrapped by geogrid encasement, so S1 can be expressed as(2)S1=S11+S12.In the formula,S11 and S12 are the compression amount of the stiffened segment and the nonstiffened segment, respectively. Thus, equation (1) can be rewritten as(3)S=S11+S12+S2.Formula (3) is the settlement calculation method of composite foundation reinforced by geogrid-encased stone column that is commonly used at present, in which the compression deformation of the underlying strata S2 is solved by the traditional layered summation method. It can be seen from this that the key to the settlement calculation of composite foundation lies in the determination of compression deformation S11 and S12 in the reinforcement area.


### 2.1. Reinforced Section

As the pile body is wrapped by the grille sleeve, the integrity and deformation stiffness of the reinforced section are greatly improved, and a relatively clear pile-soil contact surface is formed by means of the grille. The deformation mechanism is similar to that of a flexible pile or semirigid pile. However, the pile body is composed of loose materials and has no cohesive force. Under the action of load, the stress concentration of the pile body produces ineligible swelling deformation. When the composite foundation bears a large load, there is a relative displacement or displacement trend at the pile-soil interface of the reinforced section, which leads to the existence of pile lateral friction at the pile-soil interface. In this paper, the pile element is used and the layered iterative method is used to calculate.


### 2.2. Unreinforced Section

Without the lateral restraint of geogrid, the stiffness of unreinforced section of stone column decreases obviously. There are vertical and radial deformations existing in the unreinforced section under the vertical load, especially the radial deformation is extremely obvious because of the sudden change of the stiffness of stone column at the junction of the reinforced and unreinforced section, which is also the reason for the bulging failure of the geogrid-encased stone columns [14]. Without geogrid encasement, stone columns are embedded into the surrounding soil, which makes the boundary between columns and soil very fuzzy, and the relative displacement small to be neglected. In addition, due to stress diffusion, the additional stress of the pile decreases sharply with depth, and its lateral deformation decreases correspondingly. Therefore, the bulging deformation of the whole unreinforced section is not uniform. However, considering the continuous and coordinated deformation characteristics of the pile-soil, the unreinforced section is analyzed by pile-soil element to simplify the calculation.Therefore, the settlement analysis method of composite foundation reinforced by geogrid-encased stone column established in this paper must fully reflect the deformation mechanics mechanism of the reinforced section and the unreinforced section. Therefore, the settlement deformation of reinforced segment and unreinforced segment will be analyzed, respectively, on this basis.


## 2.1. Reinforced Section

As the pile body is wrapped by the grille sleeve, the integrity and deformation stiffness of the reinforced section are greatly improved, and a relatively clear pile-soil contact surface is formed by means of the grille. The deformation mechanism is similar to that of a flexible pile or semirigid pile. However, the pile body is composed of loose materials and has no cohesive force. Under the action of load, the stress concentration of the pile body produces ineligible swelling deformation. When the composite foundation bears a large load, there is a relative displacement or displacement trend at the pile-soil interface of the reinforced section, which leads to the existence of pile lateral friction at the pile-soil interface. In this paper, the pile element is used and the layered iterative method is used to calculate.


## 2.2. Unreinforced Section

Without the lateral restraint of geogrid, the stiffness of unreinforced section of stone column decreases obviously. There are vertical and radial deformations existing in the unreinforced section under the vertical load, especially the radial deformation is extremely obvious because of the sudden change of the stiffness of stone column at the junction of the reinforced and unreinforced section, which is also the reason for the bulging failure of the geogrid-encased stone columns [14]. Without geogrid encasement, stone columns are embedded into the surrounding soil, which makes the boundary between columns and soil very fuzzy, and the relative displacement small to be neglected. In addition, due to stress diffusion, the additional stress of the pile decreases sharply with depth, and its lateral deformation decreases correspondingly. Therefore, the bulging deformation of the whole unreinforced section is not uniform. However, considering the continuous and coordinated deformation characteristics of the pile-soil, the unreinforced section is analyzed by pile-soil element to simplify the calculation.Therefore, the settlement analysis method of composite foundation reinforced by geogrid-encased stone column established in this paper must fully reflect the deformation mechanics mechanism of the reinforced section and the unreinforced section. Therefore, the settlement deformation of reinforced segment and unreinforced segment will be analyzed, respectively, on this basis.


## 3. Compression Deformation Analysis

### 3.1. The Basic Assumptions

In this paper, the research is mainly limited to the reinforced stone column formed by the two-way high-strength geogrid at the depth of 2∼3 times of the pile diameter below the top of the stone column, and to simplify the calculation, the following basic assumptions are made:(a)
The geogrid encasement has sufficient strength and stiffness and will not be plastic deformation or tension due to pile deformation(b)
The mesh hole of the geogrid encasement is smaller than the particle size of gravel, and the gravel particles are not allowed to bulge out from the sleeve(c)
Under normal working conditions, the pile body and the soil around the pile can be linear elastomer(d)
Only the influence of additional stress on the compression deformation of pile body and soil around pile is considered, not the influence of dead weight(e)
Effect of pile-group is not considered in the composite foundation(f)
The microelement body of the selected segment is very small, and the volume change of the pile and soil element body can be ignoredThe pile body and the unit body composed of soil around the pile within its influence range were taken as research objects for analysis, as shown in Figure2 where a is the radius of pile body and b is the radius of pile and soil unit body, which can be obtained from the distance of pile body and the way of pile layout:(4)de=cgsd=2b,where de is the diameter of the element body; sd is the center distance between piles; and cg is the influence coefficient of pile layout (for plum-shaped or equilateral triangular pile, cg = 1.05; for square pile, cg = 1.13). Under the above conditions, the compression deformation analysis method of reinforced section and unreinforced section of composite foundation is discussed.Figure 2
The unit of pile and soil.


### 3.2. Compression Deformation Analysis of Reinforced Section

In view of the characteristics and deformation characteristics of stone column, this paper adopts the iterative method of subcascade generation for compression deformation analysis. Now, the reinforced section is divided intoM microsegment units, the length of the reinforced section is l0, and then the thickness of the i microsegment unit ΔHi is(5)ΔHi=l0M.The microsegment unit of pile bodyi-th of the stiffened section was taken for analysis, as shown in Figure 3. According to the basic assumption, the pile deformation is uniform and the volume remains the same before and after deformation. At the same time, it can be obtained by ignoring the high-order trace:(6)aΔhi=2ΔHiΔri,where Δhi and Δri are the vertical and radial deformations of the i-th pile microsegment element, respectively.Figure 3
Deformation of thei-th microelement of pile.Based on the coordinated and continuous deformation of pile and soil, it can be known that the swelling amount of pile body should be equal to the shrinkage amount of soil around the pile; that is, the mean circumjacent stressσ¯θpi of the microsegment element body is equal to the mean radial stress σ¯rpi.According to Hookie’s law in spatial problems, it can be concluded that(7)εzpi=ΔhiΔHi=1Epσ¯zpi−μpσ¯rpi+σ¯θpi=1Epσ¯zpi−2μpσ¯rpi,(8)εrpi=Δria=1Epσ¯rpi−μpσ¯zpi+σ¯θpi=1Ep1−μpσ¯rpi−μpσ¯zpi,where εrpi and εzpi are the radial and vertical strains of the i-th microsegment element of pile body, respectively; Ep is the compression modulus of pile body; μp is Poisson’s ratio of the pile; and σ¯zpiis the average vertical additional stress of the i-th pile microsegment element, and the average value of the additional stress at the bottom and top of the element can be taken as(9)σ¯zpi=σzpi+σzpi−12.From equations (7) and (8), the vertical and radial deformation of the i-th pile element microsegment element can be obtained as(10)Δhi=ΔHiEpσ¯zpi−2μpσ¯rpi,(11)Δri=aEp1−μpσ¯rpi−μpσ¯zpi.Substituting equation (9) into equations (10) and (11), the additional stress σzpi on the bottom of the pile microsegment unit is(12)σzpi=AiΔhi+BiΔri−σzpi−1,where Ai and Bi are calculating parameters, Ai=2Ep/ΔHi1+2μp2/1−μp−2μp2, Bi=4Epμp/1−μp−2μp2a.Take thei-th pile microsegment unit for force analysis, as shown in Figure 4, and from the vertical static balance of the pile, we can get(13)πa+Δri−12σzpi−1+2πa+ΔriΔHi−Δhiτi=πa+Δri2σzpi.Figure 4
Stress analysis of thei-th microelement of pile.In the formula,τi is the average frictional force on the pile side of the i-th pile microsection unit. From equation (6), we can see that Δri is a function of Δhi, and we can make Δri = f1(Δhi); substituting Δri for equation (12), we can see that σzpi is a function of Δhi, so that σzpi = f2(Δhi); equation (13) can obtain another functional relationship between σzpi and Δhi, and σzpi = f3(Δhi).Simultaneously, equations (12) and (13) can be used to obtain a univariate quadratic equation about Δhi; that is,(14)F1iΔhi2+F2iΔhi+F3i=0,where F1i, F2i, and F3i are the calculation parameters of the i-th pile microsegment unit, F1i=aτi/ΔHi+a2σzpi/4ΔHi2, F2i=aτi+a2σzpi/ΔHi−a2Ai−a3Bi/2ΔHi, F3i=2a2σzpi−2aτiΔHi.Δhi can be obtained from equation (14):(15)Δhi=−F2i+F2i2−4F1iF3i2F1i,τi=ca+k0pszitanφa,where ca is the cohesion between piles and soil; φa is the friction angle at the pile-soil interface; pszi is the vertical additional stress of soil around the pile at the depth zi in the middle of the microsegment element of the i-th pile; and k0 is the static earth pressure coefficient of soil around the pile. Because the soil around the pile has a certain shear strength, it is impossible to transfer excessive shear stress, so the pile lateral friction resistance is related to the soil’s shear strength and is τ finite value. It can be considered that the pile lateral friction resistance does not exceed the soil’s shear strength τ, and the pile lateral friction resistance of the whole reinforced section is(16)τz=ca+k0psztanφa≤τ.


### 3.3. Compression Deformation Analysis of Unreinforced Section

Similar to the above method, the unreinforced segment is divided intoN pile-soil microsegment units, and then the thickness of the j-th pile-soil microsegment unit ΔHj is(17)ΔHj=l1N.From the above analysis, it can be seen that the impact of vertical deformation and radial deformation should be taken into account in calculating the compression deformation of nonstiffened segment as well as that of stiffened segment, but the relative slip of pile and soil should be ignored in nonstiffened segment and analyzed according to the pile-soil element.Therefore, the compression deformation of nonstiffened segment can be analyzed by using the compression deformation method of stiffened segment. Thej-th pile and soil element was taken for force analysis, as shown in Figure 5.Figure 5
Stress analysis of thej-th pile-soil microelement in the unreinforced section.Considering the symmetry of pile-soil microsegment element and the stress coordination at the boundary of pile-soil element, the vertical stress at the boundary of pile-soil microsegment elementτj = 0, and then the vertical static equilibrium of pile-soil microsegment element can be obtained as follows:(18)πa+Δrj−12σzpj−1+πb2−a+Δrj−12σzsj−1=πa+Δrj2σzpj+πb2−a+Δrj2σzsj,where Δr(j−i) and Δrj are, respectively, the radial deformations of the (j − 1)th and j-th micropile soil elements; σzp(j−i) and σzpj are the (j − 1)th and j-th micropile soil element vertical stress at the bottom of the element pile: σzs(j−1) and σzsj are the vertical stress at the bottom of the (j − 1)th and j-th microelement soil element, and σzs(jj − i) = σzp(jj−i)/n(j−i), σzsj = σzpj/nj, where n(j−i) and nj are the pile-soil stress ratios at the bottom of the (j − 1)th and j-th pile-soil units, respectively.According to the reinforced section of compression deformation calculation, it is a cubic equation of one variable Δhj as follows:(19)λ3j⋅Δhj3+λ2j⋅Δhj2+λ1jΔhj=λj,where λj, λ1j, λ2j, and λ3j are parameters, λj=nja+Δrj−12+b2σzpj−1+njb2−a+Δrj−12σzsj−1, λ1j=b2+nj−1a2F1j+F2ja/2ΔZj−nj−1a2σzpj−1/ΔZj, λ2j=nj−1a2/ΔZjF1j+λ2ja/2ΔZj−σzpj−1/4ΔZj, λ3j=nj−1a2/4ΔZj2F1j+F2ja/2ΔZj.An analytical solution can be obtained from formula (19); that is,(20)λj=nja+Δrj−12+b2σzpj−1+njb2−a+Δrj−12σzsj−1,λ1j=b2+nj−1a2F1j+F2ja2ΔZj−nj−1a2σzpj−1ΔZj,λ2j=nj−1a2ΔZjF1j+λ2ja2ΔZj−σzpj−14ΔZj,λ3j=nj−1a24ΔZj2F1j+F2ja2ΔZj,Δhj=−qj2+qj22+pj333+−qj2−qj22+pj333−λ2j3λ3j,where qj and pj are parameters, qj=λ2j2λ2j2−9λ1jλ3j−27λjλ3j2/27λ3j3, pj=3λ1jλ3j−λ2j2/3λ3j2.


### 3.4. Calculation of Compressive Deformation in Reinforced Section

From the above analysis, it is known that the compression deformation amountS11 of the reinforced section in the reinforced area is S11=∑i=1MΔhi, and the compression deformation amount S12 of the unreinforced section in the reinforced area is S12=∑j=1NΔhj. The calculation formula of the compression deformation amount of the entire reinforced area is S1:(21)S1=S11+S12=∑i=1MΔhi+∑j=1NΔhj.


### 3.5. Determination of Related Calculating Parameters

The abovementioned analysis method for the settlement of the reinforced stone column reinforcement zone has been established. From equations (15) and (20), it can be known that the compression deformation S1 in the reinforced section is related to the following parameters:(22)S1=Fa,b,ΔHi,ΔHj,Δri,Δrj,σzpi,σzsi,σzpj,σzsj,nj,where a, b, ΔHi, and ΔHj are geometric parameters for composite foundation, which can be determined by the engineering condition and the calculation precision; Δri, σzpi, σzsi, Δrj, σzpj, σzsj, and nj are stress-strain parameters, which need to be calculated and determined. The determination process of parameters is as follows.Step 1.
Determining Δri and Δrj. In the reinforced section, Δri is the radial deformation of the top of the (i + 1)-th pile microsegment unit (that is the bottom of the i-th pile microsegment unit). Since the compression deformation analysis of the pile is recursive from top to bottom, the i-th element is calculated from the (i − 1)-th element, and Δri can be obtained by analogy from the radial initial deformation Δr0. Since the first microsegment element of the pile body is close to the bottom of the foundation or cushion, which has a greater constraint on the pile, Δr0 = 0 is assumed in order to simplify the calculation. Δr(j−1) can also be obtained by using the same method in the unreinforced section, where the first microelement of the pile body is the radial deformation of the interface between the reinforced section and the unreinforced section. The radial deformation of the top surface of the first pile-soil element is equal to the one of the bottom surfaces of the adjacent pile-soil element in the unreinforced section, that is, Δrj|j=0 = Δri|i=M.Step 2.
Determiningσzp and σzs. In the reinforced section, σzp(i−1) is the vertical additional stress of the top surface of the i-th pile-soil unit, and σzs(i-1) is the vertical additional stress of the surrounding soil. Likewise, Δr(i−1), σzp(i−1) and σzs(i−1) can be determined by the initial condition, that is, σzp0 and σzs0 of the first pile-soil unit, which are calculated as follows:(23)σzs0=ps=p1+mn−1,(24)σzp0=pp=nps,where pp and ps are the vertical additional stresses on the top of the pile and surrounding soil; p is the vertical pressure on the top of the composite foundation; m is the area replacement rate; and n is the stress ratio of the pile and soil, which can be calculated by the following formula (14) when there is no measured value:(25)n=Ep2kpEslnb/a+kskp,where Ep is the compression modulus of the pile, Es is the compression modulus of the soil, kp is the coefficient of lateral pressure of pile, and ks is the coefficient of lateral pressure of the soil around the pile.
In the unreinforced section, considering the continuity of stress, the vertical additional stress of the pile and soil at the top of the unreinforced section is equal to the one at the bottom of the reinforced section, that is,(26)σzpj|j=0=σzpi|i=M,σzsj|j=0=σzsi|i=M.Step 3.
Determining the pile-soil stress rationj of the element. Considering that the stone column is divided into many units and the thickness is small, the pile-soil stress ratio varies little in a pile-soil unit. To simplify the calculation, it can be assumed that the pile-soil stress ratio of the unit remains unchanged, that is,(27)nj=σzpjσzsj≈σzpj−1σzsj−1≈σzp0σzs0.


## 3.1. The Basic Assumptions

In this paper, the research is mainly limited to the reinforced stone column formed by the two-way high-strength geogrid at the depth of 2∼3 times of the pile diameter below the top of the stone column, and to simplify the calculation, the following basic assumptions are made:(a)
The geogrid encasement has sufficient strength and stiffness and will not be plastic deformation or tension due to pile deformation(b)
The mesh hole of the geogrid encasement is smaller than the particle size of gravel, and the gravel particles are not allowed to bulge out from the sleeve(c)
Under normal working conditions, the pile body and the soil around the pile can be linear elastomer(d)
Only the influence of additional stress on the compression deformation of pile body and soil around pile is considered, not the influence of dead weight(e)
Effect of pile-group is not considered in the composite foundation(f)
The microelement body of the selected segment is very small, and the volume change of the pile and soil element body can be ignoredThe pile body and the unit body composed of soil around the pile within its influence range were taken as research objects for analysis, as shown in Figure2 where a is the radius of pile body and b is the radius of pile and soil unit body, which can be obtained from the distance of pile body and the way of pile layout:(4)de=cgsd=2b,where de is the diameter of the element body; sd is the center distance between piles; and cg is the influence coefficient of pile layout (for plum-shaped or equilateral triangular pile, cg = 1.05; for square pile, cg = 1.13). Under the above conditions, the compression deformation analysis method of reinforced section and unreinforced section of composite foundation is discussed.Figure 2
The unit of pile and soil.


## 3.2. Compression Deformation Analysis of Reinforced Section

In view of the characteristics and deformation characteristics of stone column, this paper adopts the iterative method of subcascade generation for compression deformation analysis. Now, the reinforced section is divided intoM microsegment units, the length of the reinforced section is l0, and then the thickness of the i microsegment unit ΔHi is(5)ΔHi=l0M.The microsegment unit of pile bodyi-th of the stiffened section was taken for analysis, as shown in Figure 3. According to the basic assumption, the pile deformation is uniform and the volume remains the same before and after deformation. At the same time, it can be obtained by ignoring the high-order trace:(6)aΔhi=2ΔHiΔri,where Δhi and Δri are the vertical and radial deformations of the i-th pile microsegment element, respectively.Figure 3
Deformation of thei-th microelement of pile.Based on the coordinated and continuous deformation of pile and soil, it can be known that the swelling amount of pile body should be equal to the shrinkage amount of soil around the pile; that is, the mean circumjacent stressσ¯θpi of the microsegment element body is equal to the mean radial stress σ¯rpi.According to Hookie’s law in spatial problems, it can be concluded that(7)εzpi=ΔhiΔHi=1Epσ¯zpi−μpσ¯rpi+σ¯θpi=1Epσ¯zpi−2μpσ¯rpi,(8)εrpi=Δria=1Epσ¯rpi−μpσ¯zpi+σ¯θpi=1Ep1−μpσ¯rpi−μpσ¯zpi,where εrpi and εzpi are the radial and vertical strains of the i-th microsegment element of pile body, respectively; Ep is the compression modulus of pile body; μp is Poisson’s ratio of the pile; and σ¯zpiis the average vertical additional stress of the i-th pile microsegment element, and the average value of the additional stress at the bottom and top of the element can be taken as(9)σ¯zpi=σzpi+σzpi−12.From equations (7) and (8), the vertical and radial deformation of the i-th pile element microsegment element can be obtained as(10)Δhi=ΔHiEpσ¯zpi−2μpσ¯rpi,(11)Δri=aEp1−μpσ¯rpi−μpσ¯zpi.Substituting equation (9) into equations (10) and (11), the additional stress σzpi on the bottom of the pile microsegment unit is(12)σzpi=AiΔhi+BiΔri−σzpi−1,where Ai and Bi are calculating parameters, Ai=2Ep/ΔHi1+2μp2/1−μp−2μp2, Bi=4Epμp/1−μp−2μp2a.Take thei-th pile microsegment unit for force analysis, as shown in Figure 4, and from the vertical static balance of the pile, we can get(13)πa+Δri−12σzpi−1+2πa+ΔriΔHi−Δhiτi=πa+Δri2σzpi.Figure 4
Stress analysis of thei-th microelement of pile.In the formula,τi is the average frictional force on the pile side of the i-th pile microsection unit. From equation (6), we can see that Δri is a function of Δhi, and we can make Δri = f1(Δhi); substituting Δri for equation (12), we can see that σzpi is a function of Δhi, so that σzpi = f2(Δhi); equation (13) can obtain another functional relationship between σzpi and Δhi, and σzpi = f3(Δhi).Simultaneously, equations (12) and (13) can be used to obtain a univariate quadratic equation about Δhi; that is,(14)F1iΔhi2+F2iΔhi+F3i=0,where F1i, F2i, and F3i are the calculation parameters of the i-th pile microsegment unit, F1i=aτi/ΔHi+a2σzpi/4ΔHi2, F2i=aτi+a2σzpi/ΔHi−a2Ai−a3Bi/2ΔHi, F3i=2a2σzpi−2aτiΔHi.Δhi can be obtained from equation (14):(15)Δhi=−F2i+F2i2−4F1iF3i2F1i,τi=ca+k0pszitanφa,where ca is the cohesion between piles and soil; φa is the friction angle at the pile-soil interface; pszi is the vertical additional stress of soil around the pile at the depth zi in the middle of the microsegment element of the i-th pile; and k0 is the static earth pressure coefficient of soil around the pile. Because the soil around the pile has a certain shear strength, it is impossible to transfer excessive shear stress, so the pile lateral friction resistance is related to the soil’s shear strength and is τ finite value. It can be considered that the pile lateral friction resistance does not exceed the soil’s shear strength τ, and the pile lateral friction resistance of the whole reinforced section is(16)τz=ca+k0psztanφa≤τ.


## 3.3. Compression Deformation Analysis of Unreinforced Section

Similar to the above method, the unreinforced segment is divided intoN pile-soil microsegment units, and then the thickness of the j-th pile-soil microsegment unit ΔHj is(17)ΔHj=l1N.From the above analysis, it can be seen that the impact of vertical deformation and radial deformation should be taken into account in calculating the compression deformation of nonstiffened segment as well as that of stiffened segment, but the relative slip of pile and soil should be ignored in nonstiffened segment and analyzed according to the pile-soil element.Therefore, the compression deformation of nonstiffened segment can be analyzed by using the compression deformation method of stiffened segment. Thej-th pile and soil element was taken for force analysis, as shown in Figure 5.Figure 5
Stress analysis of thej-th pile-soil microelement in the unreinforced section.Considering the symmetry of pile-soil microsegment element and the stress coordination at the boundary of pile-soil element, the vertical stress at the boundary of pile-soil microsegment elementτj = 0, and then the vertical static equilibrium of pile-soil microsegment element can be obtained as follows:(18)πa+Δrj−12σzpj−1+πb2−a+Δrj−12σzsj−1=πa+Δrj2σzpj+πb2−a+Δrj2σzsj,where Δr(j−i) and Δrj are, respectively, the radial deformations of the (j − 1)th and j-th micropile soil elements; σzp(j−i) and σzpj are the (j − 1)th and j-th micropile soil element vertical stress at the bottom of the element pile: σzs(j−1) and σzsj are the vertical stress at the bottom of the (j − 1)th and j-th microelement soil element, and σzs(jj − i) = σzp(jj−i)/n(j−i), σzsj = σzpj/nj, where n(j−i) and nj are the pile-soil stress ratios at the bottom of the (j − 1)th and j-th pile-soil units, respectively.According to the reinforced section of compression deformation calculation, it is a cubic equation of one variable Δhj as follows:(19)λ3j⋅Δhj3+λ2j⋅Δhj2+λ1jΔhj=λj,where λj, λ1j, λ2j, and λ3j are parameters, λj=nja+Δrj−12+b2σzpj−1+njb2−a+Δrj−12σzsj−1, λ1j=b2+nj−1a2F1j+F2ja/2ΔZj−nj−1a2σzpj−1/ΔZj, λ2j=nj−1a2/ΔZjF1j+λ2ja/2ΔZj−σzpj−1/4ΔZj, λ3j=nj−1a2/4ΔZj2F1j+F2ja/2ΔZj.An analytical solution can be obtained from formula (19); that is,(20)λj=nja+Δrj−12+b2σzpj−1+njb2−a+Δrj−12σzsj−1,λ1j=b2+nj−1a2F1j+F2ja2ΔZj−nj−1a2σzpj−1ΔZj,λ2j=nj−1a2ΔZjF1j+λ2ja2ΔZj−σzpj−14ΔZj,λ3j=nj−1a24ΔZj2F1j+F2ja2ΔZj,Δhj=−qj2+qj22+pj333+−qj2−qj22+pj333−λ2j3λ3j,where qj and pj are parameters, qj=λ2j2λ2j2−9λ1jλ3j−27λjλ3j2/27λ3j3, pj=3λ1jλ3j−λ2j2/3λ3j2.


## 3.4. Calculation of Compressive Deformation in Reinforced Section

From the above analysis, it is known that the compression deformation amountS11 of the reinforced section in the reinforced area is S11=∑i=1MΔhi, and the compression deformation amount S12 of the unreinforced section in the reinforced area is S12=∑j=1NΔhj. The calculation formula of the compression deformation amount of the entire reinforced area is S1:(21)S1=S11+S12=∑i=1MΔhi+∑j=1NΔhj.


## 3.5. Determination of Related Calculating Parameters

The abovementioned analysis method for the settlement of the reinforced stone column reinforcement zone has been established. From equations (15) and (20), it can be known that the compression deformation S1 in the reinforced section is related to the following parameters:(22)S1=Fa,b,ΔHi,ΔHj,Δri,Δrj,σzpi,σzsi,σzpj,σzsj,nj,where a, b, ΔHi, and ΔHj are geometric parameters for composite foundation, which can be determined by the engineering condition and the calculation precision; Δri, σzpi, σzsi, Δrj, σzpj, σzsj, and nj are stress-strain parameters, which need to be calculated and determined. The determination process of parameters is as follows.Step 1.
Determining Δri and Δrj. In the reinforced section, Δri is the radial deformation of the top of the (i + 1)-th pile microsegment unit (that is the bottom of the i-th pile microsegment unit). Since the compression deformation analysis of the pile is recursive from top to bottom, the i-th element is calculated from the (i − 1)-th element, and Δri can be obtained by analogy from the radial initial deformation Δr0. Since the first microsegment element of the pile body is close to the bottom of the foundation or cushion, which has a greater constraint on the pile, Δr0 = 0 is assumed in order to simplify the calculation. Δr(j−1) can also be obtained by using the same method in the unreinforced section, where the first microelement of the pile body is the radial deformation of the interface between the reinforced section and the unreinforced section. The radial deformation of the top surface of the first pile-soil element is equal to the one of the bottom surfaces of the adjacent pile-soil element in the unreinforced section, that is, Δrj|j=0 = Δri|i=M.Step 2.
Determiningσzp and σzs. In the reinforced section, σzp(i−1) is the vertical additional stress of the top surface of the i-th pile-soil unit, and σzs(i-1) is the vertical additional stress of the surrounding soil. Likewise, Δr(i−1), σzp(i−1) and σzs(i−1) can be determined by the initial condition, that is, σzp0 and σzs0 of the first pile-soil unit, which are calculated as follows:(23)σzs0=ps=p1+mn−1,(24)σzp0=pp=nps,where pp and ps are the vertical additional stresses on the top of the pile and surrounding soil; p is the vertical pressure on the top of the composite foundation; m is the area replacement rate; and n is the stress ratio of the pile and soil, which can be calculated by the following formula (14) when there is no measured value:(25)n=Ep2kpEslnb/a+kskp,where Ep is the compression modulus of the pile, Es is the compression modulus of the soil, kp is the coefficient of lateral pressure of pile, and ks is the coefficient of lateral pressure of the soil around the pile.
In the unreinforced section, considering the continuity of stress, the vertical additional stress of the pile and soil at the top of the unreinforced section is equal to the one at the bottom of the reinforced section, that is,(26)σzpj|j=0=σzpi|i=M,σzsj|j=0=σzsi|i=M.Step 3.
Determining the pile-soil stress rationj of the element. Considering that the stone column is divided into many units and the thickness is small, the pile-soil stress ratio varies little in a pile-soil unit. To simplify the calculation, it can be assumed that the pile-soil stress ratio of the unit remains unchanged, that is,(27)nj=σzpjσzsj≈σzpj−1σzsj−1≈σzp0σzs0.


## 4. Engineering Case Analysis

In order to verify the rationality and feasibility of the above calculation method for the composite foundation of the reinforced stone column composite foundation, an engineering example is cited for analysis.


### 4.1. Case 1

When the length of the reinforced section of the reinforced stone column is zero, the settlement calculation method of the composite foundation of the reinforced stone column can be transformed into the settlement calculation method of the ordinary stone column composite foundation. In order to verify the versatility and rationality of the above calculation method for the settlement of the reinforced stone column composite foundation, an example of common stone column composite foundation engineering is first introduced for analysis.Taking a water conservancy project in [15] as an example, a sluice is planned to be built on silt and silty sand foundation, the surface layer is 2.0∼4.5 m which is silt, and the bearing capacity of the foundation is 100 kPa; below the silt is livery-grey silt sand with a foundation bearing capacity of 80 kPa. The groundwater is 2 m below the ground. The foundation of the lock chamber is reinforced with vibrating stone piles. The literature [15] used the finite element method to analyze the foundation strengthened by crushed stone piles. The simulated pile diameter was 1 m, the pile length was 11 m, and the total calculated depth of the foundation was 12 m. For other specific parameters, see the literature. When the upper load is 150 kPa, the settlement of the composite foundation is 4.76 cm. The above data are used to calculate the settlement of the project using the method in this paper. The length of the reinforced section of the composite foundation is set to zero, the unreinforced section is equally divided into 50 units, and the calculated S12 = 3.97 cm; the lower layer is calculated using the layered sum method to calculate S2 = 0.87 cm; the total settlement of the project is S = 4.84 cm, which is close to the calculation results of the finite element method, which shows that the method in this paper is reasonable and can be degraded to one of the ordinary composite foundations reinforced by stone columns.


### 4.2. Case 2

This project is a ramp of a highway interchange in Sichuan, which uses geogrid encasement, stone column, and sand cushion to reinforce the roadbed [16]. The parameters of the composite foundation are as follows: the thickness of cushion is h = 10 cm, the compression modulus of cushion is Ec = 20 MPa, the diameter of stone column is d = 0.6 m, the length of pile is L = 6m, the arrangement of piles is like a plum blossom, pile spacing is Sd = 2m, internal friction angle of stone is φp = 38o, and Poisson’s ratio is μp = 0.35, the length of geogrid encasement is l0 = 2m, and the compression modulus of reinforced section is Ep = 100 MPa and Ep = 20 MPa in the unreinforced section. The distribution of pile and soil layers is shown in Figure 6, and the calculating parameters are shown in Table 1. A uniformly distributed load p = 120 kPa is applied to the treated subgrade surface. The measured pile-soil stress ratio is n = 3.12, and the subgrade settlement is S = 11 cm.Figure 6
Distribution of pile and surrounding soil.Table 1
Calculating parameter.
SectionSoil typeφ (°)Es (MPa)ρs (g/cm3)μsc (kPa)Reinforced sectionFilled soil3161.700.414.9Unreinforced sectionWeak soil5.82.741.750.4511Underlying stratumPebble soil26501.600.48The settlement calculation of the subgrade project is carried out by using the above data in Table1 and the method in this paper. The specific process is as follows:(a)
Calculating the replacement rate of the composite foundation aream = 8.2% and the pile-soil stress ratio n = 3.12. From formulas (23) and (24), the load on the pile-soil top surface is σzp0 = 350.84 kPa and σzs0 = 112.45 kPa.(b)
CalculatingS11: the reinforced section is divided into 10 element bodies, and the element body is analyzed and calculated one by one from top to bottom. Finally, the compression deformation of the reinforced section is S11 = 2.547 cm.(c)
Calculating S12: the unreinforced section is divided into 20 element bodies, and the element body is analyzed and calculated one by one from top to bottom. The final compression deformation of the unreinforced section is S12 = 8.251 cm.(d)
The compression deformation of the underlying stratum: using the layer-wise summation method to obtain the settlementS2 = 1.072 cm.(e)
According to formula (3), it can be known that the total deformation of composite foundation S = 11.87 cm.In order to facilitate comparative analysis, other methods are used to calculate the settlement in Case 2, and the results are shown in Table2. It can be seen from Table 2 that the calculated results of this method and other methods are close to the measured values. The calculated results of this method are larger than those of other methods. Considering that theoretical derivation is generally the final settlement of foundation, while the measured value is the settlement at a certain time, and the value is generally less than the final settlement, the method in this paper is more in line with the actual situation and avoids the defect that the analysis results of other methods are too dangerous. Therefore, the method used in this paper to calculate the settlement of composite foundation is in good agreement with the actual situation. However, it is undeniable that there still exists certain calculation error of the method in this paper, the main reason is the research object is limited to by high tensile bidirectional geogrid on top of the stone column within the following 2∼3 times the depth of the pile diameter fan sleeve formed by confining hoop reinforcement stone column composite foundation, and the formula is derived using the continuous homogeneous medium assumption and the basic principle of mechanics of elasticity. We assume unit cell of constant stress ratio of pile-soil and add composite foundation reinforced by geogrid-encased stone column settlement and the actual engineering by the combination of many factors, such as stone column itself has large discreteness, and when the upper load is bigger, grille sleeve reinforced section will appear larger nonlinear deformation. The stress of pile and soil will increase with load-increasing.Table 2
Calculation results of different methods.
MethodCalculated value (cm)Measured values (cm)Relative error (%)Method from the literature [10]10.811.0−1.82Finite element method [16]10.911.0−0.91Proposed method11.8711.0+7.90


## 4.1. Case 1

When the length of the reinforced section of the reinforced stone column is zero, the settlement calculation method of the composite foundation of the reinforced stone column can be transformed into the settlement calculation method of the ordinary stone column composite foundation. In order to verify the versatility and rationality of the above calculation method for the settlement of the reinforced stone column composite foundation, an example of common stone column composite foundation engineering is first introduced for analysis.Taking a water conservancy project in [15] as an example, a sluice is planned to be built on silt and silty sand foundation, the surface layer is 2.0∼4.5 m which is silt, and the bearing capacity of the foundation is 100 kPa; below the silt is livery-grey silt sand with a foundation bearing capacity of 80 kPa. The groundwater is 2 m below the ground. The foundation of the lock chamber is reinforced with vibrating stone piles. The literature [15] used the finite element method to analyze the foundation strengthened by crushed stone piles. The simulated pile diameter was 1 m, the pile length was 11 m, and the total calculated depth of the foundation was 12 m. For other specific parameters, see the literature. When the upper load is 150 kPa, the settlement of the composite foundation is 4.76 cm. The above data are used to calculate the settlement of the project using the method in this paper. The length of the reinforced section of the composite foundation is set to zero, the unreinforced section is equally divided into 50 units, and the calculated S12 = 3.97 cm; the lower layer is calculated using the layered sum method to calculate S2 = 0.87 cm; the total settlement of the project is S = 4.84 cm, which is close to the calculation results of the finite element method, which shows that the method in this paper is reasonable and can be degraded to one of the ordinary composite foundations reinforced by stone columns.


## 4.2. Case 2

This project is a ramp of a highway interchange in Sichuan, which uses geogrid encasement, stone column, and sand cushion to reinforce the roadbed [16]. The parameters of the composite foundation are as follows: the thickness of cushion is h = 10 cm, the compression modulus of cushion is Ec = 20 MPa, the diameter of stone column is d = 0.6 m, the length of pile is L = 6m, the arrangement of piles is like a plum blossom, pile spacing is Sd = 2m, internal friction angle of stone is φp = 38o, and Poisson’s ratio is μp = 0.35, the length of geogrid encasement is l0 = 2m, and the compression modulus of reinforced section is Ep = 100 MPa and Ep = 20 MPa in the unreinforced section. The distribution of pile and soil layers is shown in Figure 6, and the calculating parameters are shown in Table 1. A uniformly distributed load p = 120 kPa is applied to the treated subgrade surface. The measured pile-soil stress ratio is n = 3.12, and the subgrade settlement is S = 11 cm.Figure 6
Distribution of pile and surrounding soil.Table 1
Calculating parameter.
SectionSoil typeφ (°)Es (MPa)ρs (g/cm3)μsc (kPa)Reinforced sectionFilled soil3161.700.414.9Unreinforced sectionWeak soil5.82.741.750.4511Underlying stratumPebble soil26501.600.48The settlement calculation of the subgrade project is carried out by using the above data in Table1 and the method in this paper. The specific process is as follows:(a)
Calculating the replacement rate of the composite foundation aream = 8.2% and the pile-soil stress ratio n = 3.12. From formulas (23) and (24), the load on the pile-soil top surface is σzp0 = 350.84 kPa and σzs0 = 112.45 kPa.(b)
CalculatingS11: the reinforced section is divided into 10 element bodies, and the element body is analyzed and calculated one by one from top to bottom. Finally, the compression deformation of the reinforced section is S11 = 2.547 cm.(c)
Calculating S12: the unreinforced section is divided into 20 element bodies, and the element body is analyzed and calculated one by one from top to bottom. The final compression deformation of the unreinforced section is S12 = 8.251 cm.(d)
The compression deformation of the underlying stratum: using the layer-wise summation method to obtain the settlementS2 = 1.072 cm.(e)
According to formula (3), it can be known that the total deformation of composite foundation S = 11.87 cm.In order to facilitate comparative analysis, other methods are used to calculate the settlement in Case 2, and the results are shown in Table2. It can be seen from Table 2 that the calculated results of this method and other methods are close to the measured values. The calculated results of this method are larger than those of other methods. Considering that theoretical derivation is generally the final settlement of foundation, while the measured value is the settlement at a certain time, and the value is generally less than the final settlement, the method in this paper is more in line with the actual situation and avoids the defect that the analysis results of other methods are too dangerous. Therefore, the method used in this paper to calculate the settlement of composite foundation is in good agreement with the actual situation. However, it is undeniable that there still exists certain calculation error of the method in this paper, the main reason is the research object is limited to by high tensile bidirectional geogrid on top of the stone column within the following 2∼3 times the depth of the pile diameter fan sleeve formed by confining hoop reinforcement stone column composite foundation, and the formula is derived using the continuous homogeneous medium assumption and the basic principle of mechanics of elasticity. We assume unit cell of constant stress ratio of pile-soil and add composite foundation reinforced by geogrid-encased stone column settlement and the actual engineering by the combination of many factors, such as stone column itself has large discreteness, and when the upper load is bigger, grille sleeve reinforced section will appear larger nonlinear deformation. The stress of pile and soil will increase with load-increasing.Table 2
Calculation results of different methods.
MethodCalculated value (cm)Measured values (cm)Relative error (%)Method from the literature [10]10.811.0−1.82Finite element method [16]10.911.0−0.91Proposed method11.8711.0+7.90


## 5. Conclusions

Based on the analysis of bearing and deformation mechanism of composite foundation reinforced by geogrid-encased stone columns, the settlement calculation method of this type of composite foundation is discussed. Theoretical analysis shows that the geogrid encasement effectively limits the radial deformation of the upper half of the stone column, making the half of the stone column body flexible or semirigid. The load transfer capacity of the stone column is stronger, and the deformation is more. Therefore, considering the relative slip of the pile and soil in the reinforced section and the coordinated characteristics of vertical and radial deformation of the pile and soil in the unreinforced section, the composite foundation reinforced by geogrid-encased stone column is divided into the reinforced section, unreinforced section, and underlying stratum.The settlement calculation model was established in the three sections of the horizontal layer, which accorded with the bearing and deformation characteristics of composite foundation reinforced by the geogrid-encased stone column. The analysis of engineering examples shows that the proposed method for calculating the settlement of the composite foundation of the geogrid-encased stone column composite foundation has good engineering accuracy and overcomes the shortcomings of the analysis results of the existing analysis methods, which are biased to danger, and it has good rationality and feasibility.In addition, the settlement calculation method of geogrid-encased stone column composite foundation proposed in this paper is mainly aimed at the reinforced gravel formed by the sleeve hoop within the depth range of 2∼3 times of the pile diameter below the top of the stone column using a two-way high-strength geogrid reinforcement, and the theoretical derivation uses the assumption of continuous homogeneous medium and the basic principles of elastic mechanics and the calculation assumption that the pile-soil stress ratio of the unit body is constant. In practice, the settlement of composite foundation reinforced by geogrid-encased stone column is affected by many factors such as the stiffness of stone column [17], the failure model of stone column [18], and the consolidation state of soil [19]. In addition, the method proposed in this paper still cannot consider the plastic state of pile and soil and needs to be further improved.



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*Source: 5576713-2021-04-23.xml*