# Analysis of Settlement of Group Pile Foundation in Linear Viscoelastic Soil

Authors: Long Li; Yousheng Deng
Journal: Advances in Civil Engineering
 (2023)

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## Abstract

The settlement of a group pile foundation is the result of elastic-plastic settlement of the pile and rheological settlement of the soil. Unfortunately, most research only considers elasto-plasticity settlement of piles or rheological settlement of soils. Classical theories like the Poynting-Thompson model only provide some prediction for the elasto-plasticity behavior of pile groups, which cannot satisfactorily reflect the total settlement characteristics of pile group foundation. In the current paper, we presented a method to study settlement behaviors of pile group foundation in linear viscoelastic soil based on the pile-pile interaction factors, the calculated equation of settlement of single piles and pile groups under vertical loads that are derived, the influence law of pile end resistance, pile spacing, slenderness ratio, elastic modulus ratio, and Poisson ratio on pile end load ratio, and the interaction factors that have been presented. Finally, some typical predictive examples have been presented for a 2 × 2, 3 × 3 group pile showing settlement from a field test. It is found that the pile-pile interaction factors increase with the increase of the pile-soil elastic modulus ratio and length-to-diameter ratio but decrease with the increase of the ratio of distance to diameter, and the ratio of distance to diameter has little effect on the interaction factor when pile spacing is more than 10 d (d is the diameter of the pile). The error between the field test results and computed data is 14.9%, the settlement calculated by the present method shows good agreement with the reported results, and the correlativity coefficients R2 are greater than 0.9223, which signifies that the present method is suitable for the theoretical calculation and prediction of the composite foundation settlement of linear viscoelastic soil when piles are used to strengthen the foundation. The method would provide a unique view to quantitative prediction of the settlement characteristic of group pile foundation.


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## Body

## 1. Introduction

Settlement characteristic is a common parameter in group pile foundation design, which plays a crucial role in engineering applications [1]. Several analytical [2], numerical [3], and experimental studies [4] have been made on the group piles that can predict immediate settlements with good accuracy. Mishra and Patra [5] investigated the creep settlement of group piles for line group piles, square group piles, and rectangular group piles undergoing creep settlements over a period of time and the fluctuations of interaction factors on parameters such as pile length to diameter ratio, pile spacing, Poisson’s ratio, and modulus ratio. The results of the analysis showed that the load rearrangement due to creep settlements causes about a 5% to 35% increase in base resistance over time. Interaction factors for group piles (2 × 1, 3 × 1, 2 × 2, and 3 × 2) undergoing creep settlement are about 15% to 55% higher than the interaction factors, considering only the immediate settlements for group pile spacing less than or equal to 5 d. Li et al. [6] proposed a semianalytical solution to analyze the long-term settlement of a single pile embedded in fractional derivative viscoelastic soils under time-dependent loading. Three well-documented cases were used to verify the correctness and reliability of the semianalytical solution, and the results indicate that the dimensionless settlement for fractional derivative viscoelastic models is larger in the early stage and smaller in the later stage than those for conventional viscoelastic models. Kumar et al. [7] investigated the application of relevance vector machines (RVM), generalized regression neural network (GRNN), genetic programming (GP) and adaptive-network-based fuzzy inference (ANFIS) in reliability analysis of settlement of group pile in the normally consolidated clay. The research concludes that the performance of ANFIS is found to be perfect, followed by RVM and GP, giving comparably good performance.The settlement behavior of a group pile foundation is primarily caused by the material properties of the piles [8–11], the load form of the foundation [12], and soil characteristics [13]. In general situations, it is known that the rheological properties of soils and the elastic-plastic settlement of piles should be considered in practice. Over the last decades, due to the complexity of rheological settlement of soil and the assumption that it only accounts for a small part of the total settlement of group piles, the analysis of group pile settlement in linear viscoelastic soil has been limited, and the research on group pile settlement considering the elastic-plastic settlement of piles is even less. It is however essential to investigate the settlement of linear viscoelastic soil for structures such as high-rise buildings and high-speed railway, bridges, and other important structures because excessive settlements may create severe safety issues.Based on the interaction factors, the finite element model of the interaction factors of group piles was established by Zhe et al. [14], and the method of calculating the plastic settlement of group piles by data regression is advanced. Several experimental results reveal that the total settlement of a group pile foundation under constant stress is the result of elasto-plasticity settlement of the pile and rheological settlement of the soil. However, the influence of two kinds of settlements on group pile foundations has not yet been elaborated. In this paper, we try to establish a settlement calculation method for group pile foundations by considering the interaction factors between piles. The new method would provide a new insight into settlement analysis and optimization of the bearing capacity of group pile foundations under constant stress.


## 2. Calculation for Rheological Settlement of Soil Around a Pile

The existing rules for settlement characteristic calculation, such as the Booker and Poulos laws [15], cannot guide the calculation and prediction of settlement characteristics well in soil around piles after decades of research. It is well known that the key process during pile foundation design is determining settlement characteristics. Among them, the generally accepted settlement characteristic theory was proposed by Mattes and Poulos [16].The settlements of soil around pile can be derived from the settlement characteristic theory as proposed by the Mattes and Poulos [16] and can be generalized as follows:(1)ρ=Pπd2c0E∞ISro+∑i=1nciISriE∞1−kEpλi/E∞,where [Is] is the displacement influence matrix; E∞ is the “long-term” elasto-plasticity modulus; {ro} and {ri} are the coefficient matrices; P is the load; c0 are ci is the logarithmic settlement rates; Ep is the elastic modulus; λi is the eigenvalue parameter; d is the pile diameter; k is the pile-soil elastic modulus ratio; and n is the number of piles.


## 3. Calculation of Interaction Factors

Figure1 shows a 2 × 2 group pile with pile spacing S. The interaction factors among piles are equal to the summation of the interaction effects between the elements of the same pile and the interaction effects between the elements of the adjacent piles [17]. The effect factors of pile-pile interaction factors mainly include pile length, pile diameter, elastic modulus and Poisson’s ratio of pile, elastic modulus and Poisson’s ratio of soil around pile, pile number and pile spacing. In order to simplify dimensional analysis, only two piles are considered. For other group pile configurations, the dimensional analysis of two piles can be used as an analogy.Figure 1
A 2 × 2 pile configuration depicting interaction between the piles.According to the three-parameter viscoelastic model and Guo’s theory [18], the pile-pile interaction factors can be written as a function of the following main physical quantities:(2)αij=fP,l1,d1,Ep1,νp1,l2,d2,Ep2,νp2,Es,νs,S,where αij is the interaction factor of piles; P is the constant load; l1 and l2 are the lengths of two piles; d1 and d2 are the diameters of two piles; Ep1 and Ep2 are the elastic moduli of two piles; Es is the elastic modulus of soil around the pile; νs is Poisson’s ratio of soil around the pile; and S is the pile spacing.Randolph and Wroth [19] pointed out that the Poisson’s ratio of the pile has little influence on the capacity and displacement of the pile and thus on the interaction factors of the pile-pile. Therefore, in order to simplify the analysis, the influence of Poisson’s ratio of pile is not considered. At the same time, considering that the pile diameter and elastic modulus are basically the same in actual pile foundation engineering, the same pile diameter and length are adopted. Therefore, equation (2) can be simplified as follows:(3)αij=fP,l,d,Ep,Es,νs,S.According to “П theorem” [20] (when a physical phenomenon can be described by the functional relationship of n physical quantities, and n physical quantities including m basic dimensions, then a dimensionless group with n = m can be obtained by dimensional analysis, and the characteristics of this phenomenon can be expressed in the form of n-m dimensionless groups), under the action of unit force, six dimensionless groups can be deduced (using d to eliminate the dimensions of l and S, and using Es to eliminate the dimensions of Ep)(4)Esαijd=fld,EpEs,Sdvs,where Esαij/d indicates the behavior of the pile-pile interaction factors; l/d is the slenderness ratio of pile; Ep/Es is the modulus ratio between the pile and the foundation soil; S/d is the distance-diameter ratio between two piles; and νs is Poisson’s ratio of foundation soil.Equation (4) illustrates that the interaction factors are related to the pile-soil elastic modulus ratio, slenderness ratio, distance-diameter ratio, and Poisson’s ratio of the foundation soil. The effects of pile-soil elastic modulus ratio, slenderness ratio, and pile spacing-to-diameter ratio have been statistically analyzed in numerous research studies [21].


### 3.1. Effect of Pile-Soil Elastic Modulus Ratio on Interaction Factors

It can be observed in Figure2 that when the pile spacing-to-diameter ratio S/d is constant, the pile-pile interaction factors increase with the increase of the pile-soil elastic modulus ratio k. When k is greater than 500, with the increase of pile-soil elastic modulus ratio, the pile-pile interaction factor has a slight increase.Figure 2
Influence of thek on pile-pile interaction factor (S/d = 3, S/d = 6).


### 3.2. Effect of Slenderness Ratio on Interaction Factors

Considering the effect of slenderness ratio on interaction factors, in 3 × 3 group piles withS/d = 3 and S/d = 6, l/d is 10, 15, 25, 35 and 50 respectively. It can be observed in Figure 3 that when the pile spacing-to-diameter ratio S/d is constant, the pile-pile interaction factors have steady increases with the increase of the length-to-diameter ratio l/d.Figure 3
Influence ofl/d on the pile-pile interaction factor (S/d = 3, S/d = 6).


### 3.3. Effect of Pile Spacing-to-Diameter Ratio on Interaction Factors

Considering the effect ofS/d on interaction factors, in 3 × 3 group piles with S/d = 3 and S/d = 6, l/d is 2, 3, 4, 5, 6, 10, and 20, respectively. It can be observed in Figure 4 that when the ratio of elastic modulus of pile to soil is constant, the interaction factors of pile and pile decrease with the increase of the ratio of distance to diameter S/d. The ratio of distance to diameter has little effect on the interaction factor when pile spacing is greater than 10 d, consistent with the reported literature (Guo [22] and Liang et al. [23]).Figure 4
Influence ofS/d on the pile-pile interaction factor (l/d = 10, l/d = 25).


### 3.4. Data Regression

The above discussion results show that the length-to-diameter ratiol/d has little influence on the pile-pile interaction factor. Therefore, the function relation of pile-pile interaction factor αij can be expressed as follows:(5)αij=fSd,k.According to date regression analysis, the parametersb1, b2, and b3 in the equation can be calculated, respectively, assuming that k is a constant value, as shown in Table 1, and the simplified equation of the pile-pile interaction factor is as follows:(6)αij=kb1+b2S/d3+b3lnS/d.Table 1
Parameters value.
kb1b2b31002.9280.0182.0815001.6030.0101.46210001.5500.0101.24730001.3150.0091.222


## 3.1. Effect of Pile-Soil Elastic Modulus Ratio on Interaction Factors

It can be observed in Figure2 that when the pile spacing-to-diameter ratio S/d is constant, the pile-pile interaction factors increase with the increase of the pile-soil elastic modulus ratio k. When k is greater than 500, with the increase of pile-soil elastic modulus ratio, the pile-pile interaction factor has a slight increase.Figure 2
Influence of thek on pile-pile interaction factor (S/d = 3, S/d = 6).


## 3.2. Effect of Slenderness Ratio on Interaction Factors

Considering the effect of slenderness ratio on interaction factors, in 3 × 3 group piles withS/d = 3 and S/d = 6, l/d is 10, 15, 25, 35 and 50 respectively. It can be observed in Figure 3 that when the pile spacing-to-diameter ratio S/d is constant, the pile-pile interaction factors have steady increases with the increase of the length-to-diameter ratio l/d.Figure 3
Influence ofl/d on the pile-pile interaction factor (S/d = 3, S/d = 6).


## 3.3. Effect of Pile Spacing-to-Diameter Ratio on Interaction Factors

Considering the effect ofS/d on interaction factors, in 3 × 3 group piles with S/d = 3 and S/d = 6, l/d is 2, 3, 4, 5, 6, 10, and 20, respectively. It can be observed in Figure 4 that when the ratio of elastic modulus of pile to soil is constant, the interaction factors of pile and pile decrease with the increase of the ratio of distance to diameter S/d. The ratio of distance to diameter has little effect on the interaction factor when pile spacing is greater than 10 d, consistent with the reported literature (Guo [22] and Liang et al. [23]).Figure 4
Influence ofS/d on the pile-pile interaction factor (l/d = 10, l/d = 25).


## 3.4. Data Regression

The above discussion results show that the length-to-diameter ratiol/d has little influence on the pile-pile interaction factor. Therefore, the function relation of pile-pile interaction factor αij can be expressed as follows:(5)αij=fSd,k.According to date regression analysis, the parametersb1, b2, and b3 in the equation can be calculated, respectively, assuming that k is a constant value, as shown in Table 1, and the simplified equation of the pile-pile interaction factor is as follows:(6)αij=kb1+b2S/d3+b3lnS/d.Table 1
Parameters value.
kb1b2b31002.9280.0182.0815001.6030.0101.46210001.5500.0101.24730001.3150.0091.222


## 4. Calculation of Total Settlement of Group Pile Foundation

We neglected the influence of plastic deformation on the bearing capacity of group piles in the process of analyzing pile-pile interaction factors in the previous section. It should be pointed out that the total settlement of group pile foundation includes both elastic settlement and plastic settlement. Therefore, the deformation calculation of a group pile foundation can be divided into three parts and is expressed as follows:(7)Spg=nRsSe+nSp+ρ,where Spg is the total settlement of group pile foundations; Se is the elastic deformation part of single pile displacement; Rs is the settlement ratio of group piles; Sp is the plastic deformation part of single pile displacement; {ρ} is the rheological settlement of soil around pile; and n is the number of piles.


### 4.1. Calculation of Elastic DeformationSe of Single Pile

In actual engineering, the load-displacement curve of a single pile includes two deformations: elastic deformation and irrecoverable plastic deformation [24] (Figure 5).Figure 5
Elastic and plastic deformation of a single pile.According to the pile configuration, the elastic deformation of a single pile can be determined by a load-displacement test. If there is no test data, the elastic deformation of a single pile can be determined through the settlement calculation method of a single pile proposed by Randolph and Wroth [19]. The specific calculation method can be expressed as follows:(8)Ke=Pw=Es1d41+ν4/1−νη/ζ+4πρ¯/ξl/dtanhμl/μl1+8/1−νη1/ζ1/πλl/dtanhμl/μl,where P is the top load of the pile; w is the top displacement of the pile; Es1 is the elastic modulus of the soil at the pile bottom; d is the diameter of the pile; l is the length of the pile; ν is Poisson’s ratio of soil; η = Es0/Es1; and(9)λ=21+νEpEs1;ζ=Es1Eb;ρ¯=E¯sEb,μl2=2ξλ2ld2,ξ=ln0.25+2.5ρ¯1−ν−0.25ζ2ld.The physical meaning of the parameters is shown in Figure6.Figure 6
Definition of pile-soil geometric conditions.Then, the elastic deformation of a single pile can be obtained(10)Se=PmaxKe.


### 4.2. Calculation of Settlement RatioRs of Group Piles

The settlement ratio of group piles is greatly influenced by the number of pilesn, and increases with the increase of n. Poulos and Davis [25] analyzed the relationship between Rs and n of group piles in homogeneous soil, and found that when the number of piles is large, the relationship between Rs and the square root of the number of piles n is linear. From the point of view of application, for group piles with a large number of piles (n > 25), the Rs can be approximately calculated according to the following equation:(11)Rs=R25n−4+5−nR16,where 16 and 25 are the numbers of piles.Based on equations (1)–(11), the total settlement of group piles could be derived as follows:(12)Spg=R25n−4+5−nR164nPmax1+νEs1d1+ψ2η1/ζ1/πλl/dϕψη/ζ+4πρ¯/ξl/dϕ+nSp+Pπd2c0E∞ISro+∑i=1nciISriE∞1−kEpλi/E∞,ψ=41−ν,ϕ=tanhμlμl.


## 4.1. Calculation of Elastic DeformationSe of Single Pile

In actual engineering, the load-displacement curve of a single pile includes two deformations: elastic deformation and irrecoverable plastic deformation [24] (Figure 5).Figure 5
Elastic and plastic deformation of a single pile.According to the pile configuration, the elastic deformation of a single pile can be determined by a load-displacement test. If there is no test data, the elastic deformation of a single pile can be determined through the settlement calculation method of a single pile proposed by Randolph and Wroth [19]. The specific calculation method can be expressed as follows:(8)Ke=Pw=Es1d41+ν4/1−νη/ζ+4πρ¯/ξl/dtanhμl/μl1+8/1−νη1/ζ1/πλl/dtanhμl/μl,where P is the top load of the pile; w is the top displacement of the pile; Es1 is the elastic modulus of the soil at the pile bottom; d is the diameter of the pile; l is the length of the pile; ν is Poisson’s ratio of soil; η = Es0/Es1; and(9)λ=21+νEpEs1;ζ=Es1Eb;ρ¯=E¯sEb,μl2=2ξλ2ld2,ξ=ln0.25+2.5ρ¯1−ν−0.25ζ2ld.The physical meaning of the parameters is shown in Figure6.Figure 6
Definition of pile-soil geometric conditions.Then, the elastic deformation of a single pile can be obtained(10)Se=PmaxKe.


## 4.2. Calculation of Settlement RatioRs of Group Piles

The settlement ratio of group piles is greatly influenced by the number of pilesn, and increases with the increase of n. Poulos and Davis [25] analyzed the relationship between Rs and n of group piles in homogeneous soil, and found that when the number of piles is large, the relationship between Rs and the square root of the number of piles n is linear. From the point of view of application, for group piles with a large number of piles (n > 25), the Rs can be approximately calculated according to the following equation:(11)Rs=R25n−4+5−nR16,where 16 and 25 are the numbers of piles.Based on equations (1)–(11), the total settlement of group piles could be derived as follows:(12)Spg=R25n−4+5−nR164nPmax1+νEs1d1+ψ2η1/ζ1/πλl/dϕψη/ζ+4πρ¯/ξl/dϕ+nSp+Pπd2c0E∞ISro+∑i=1nciISriE∞1−kEpλi/E∞,ψ=41−ν,ϕ=tanhμlμl.


## 5. Parametric Studies

### 5.1. Verify the Rationality of the Proposed Method

#### 5.1.1. Example I

According to the research of O’Neill et al. [26] reported, 11 piles were driven into linear viscoelastic soil. The diameter of the pile is 137 mm, and the buried length of the pile is 13.1 m. Among them, 4 piles are arranged in a 3 × 3 square and connected with a concrete cap, and the cap is not in contact with the soil below. The pile spacing is S = 3d, and the other two piles are 3.7 m on both sides of the center group pile.Specific parameters: the elastic modulus of soilEs = 1.95 × 105 kPa, Poisson’s ratio of soil ν = 0.4, and the elastic modulus of concrete pile Ep = 2.1 × 108 kPa. Average load of single pile in group pile is P=653kN. The load-displacement curve of single pile test is shown in Figure 7. Parameter S/d = 3, Ep = Es = 1077. Due to the lack of unloading curve in pile test, the method proposed in this paper is used to define the elastic stiffness of pile: Ke = 608 kN/mm, thus Se = P/Ke = 1.07 mm, and P=653kN. According to the above principle, the settlement of group pile can be calculated as S = 7.26 mm. The measured settlement of group pile is 6.63 mm, and the results demonstrate that the error between the field test results and computed data is 14.9%, which signifies that the present method is suitable for the theoretical calculation and predict the composite foundation settlement of linear viscoelastic soil when piles are used to strengthen the foundation.Figure 7
Comparison of the capacity of a single model pile.


#### 5.1.2. Example II

In order to further study the settlement characteristics of group piles in linear viscoelastic soil, based on the parameters of the soil layer and piles in a large-scale field group pile test [27] and according to the total settlement calculation method of our paper, the load-settlement curves and group pile settlement-interaction factor curves under different types of group piles, different pile lengths, and different pile spacing (Figure 8) are analyzed. The parameters of pile length, pile spacing, pile cap size, and slenderness ratio are shown in Tables 2, and Table 3 summarizes the soil properties of each soil layer crossed by the test piles.Figure 8
Layout plan of test piles and group piles.Table 2
Summary of field tests.
No. of pilesPile length (m)Pile spacingPile cap sizes (length × width × height; all in metres)Slenderness ratio (L/B)D-120—0.4 × 0.4 × 0.450D-224—0.4 × 0.4 × 0.4605–2202.5B1.8 × 0.8 × 0.8505–1243.0B2.0 × 0.8 × 0.860D-120—0.4 × 0.4 × 0.4504202.5B1.8 × 0.8 × 0.8502243.0B2.0 × 2.0 × 0.8603202.5B2.8 × 2.8 × 1.2501243.0B3.2 × 3.2 × 1.260Table 3
Soil properties.
SoilDirect shear test, curve-fit strength envelopeOne-dimensional consolidation testThickness (m)Unit weight,γ (kN/m3)Void ratio,e0Plasticity index,IpLiquidity index,ILCohesion (kPa)Internal friction angle (φ)α1 −2 (MPa−1)Es (MPa)Fill3817.40.306.072.619.20.7616.00.42Clay5018.50.237.652.019.60.7117.20.28Silt1122.50.257.565.618.50.855.80.60Silt intermixed with silty sand1323.70.208.887.318.80.795.00.64Soft clay158.30.673.4011.617.41.2421.70.96Based on the concept of elasticity theory, considering the rheological settlement of soils, we can get the implied interaction factors of group piles as follows:(13)αij=ΛλL,Ωlnrm/spijln2rm/B≥0,where(14)Ω=QbwbEpApλ=4Gbrs1−ν1λEpAp,λ=kEpAp,rm=0.25+2.51−vGGL−0.25GLGbL.Based on the interaction factor obtained by equation (13), the load-settlement relationship can be obtained as shown in Figure 9, and the relationship curve between the interaction factor and the settlement of different pile types is shown in Figure 10.Figure 9
Individual pile load versus group settlement relationship based on the interaction factor. (a) Pile lengthL = 20 m. (b) Pile length L = 24 m.
(a)(b)Figure 10
Interaction factor versus settlement for each group pile. (a) Two-group pile withL1 = 24 m and L2 = 20 m, (b) four-group pile with L1 = 24 m and L2 = 20 m, (c) nine-group pile with L2 = 20 m, and (d) nine-group pile with L1 = 24 m.
(a)(b)(c)(d)Figure9 presents the solutions from the present method and Guoliang Dai’s, where the present solutions consider the rheological settlement of soils. The total settlement calculated by the present method shows good agreement with the reported results, and the correlation coefficients R2 are greater than 0.9223.The relationship between settlement and interaction factor of different pile types is shown in Figure10αij indicates the ratio of the settlement of pile i under load to the settlement of adjacent pile j. For group pile types with different pile lengths (Figure 10(a)), the interaction factor increases and then decreases with the increase in group pile settlement, and the longer the pile length, the smaller the influence on the interaction factor, and the interaction coefficient for the two-group piles L1 is practically zero. This may be because of the spatial variability of the soil or other variability in the pile installation or pile cap. For the group pile types as shown in Figure 10(b), it is not difficult to see that pile spacing has a larger effect on interaction than pile length. For the 3 × 3 group pile types, it can be seen from Figure 10(c) that the settlement of the corner pile is the largest and that of the middle pile is the smallest, and the interaction factors are distributed proportionally to pile center-to-center spacing. Similar conclusions can also be drawn from Figure 10(d). Comparing Figures 10(a) and 10(b) with Figures 10(c) and 10(d), it can be seen that when the pile spacing is the same, the interaction factor of the piles in the four group piles is greater than that of the nine group piles.


### 5.2. Parametric Studies

The calculation method in this paper involves many design parameters, such as pile spacing, slenderness ratio, pile-soil elastic modulus ratio, Poisson’s ratio, etc. It is of great significance for optimal design to explore the influence law of each design parameter on the working performance of a group pile foundation. Therefore, the example of group pile foundation in Booker and Poulos [15] is selected for parameter analysis, and the influence laws of pile end resistance, pile spacing, slenderness ratio, elastic modulus ratio and Poisson ratio on pile end load ratio, and interaction factors are studied by the calculation method proposed in this paper.


#### 5.2.1. Pile Spacing

Figure11 shows the relationship of pile spacing and interaction factor when the ratio of pile spacing to interaction factor is S/d = 1, 2, 3, …, 19, the slenderness ratio l/d = 10, 50, 100, and the calculation elastic modulus ratio of pile-soil Ep/Es = 1000, Poisson’s ratio of soil is 0.5. It can be seen from Figure 11 that the interaction factor decreases with the increase of pile spacing, and the interaction factor decreases with the increase of pile length, but the influence of pile length on the interaction factor can be neglected with the increase of pile spacing.Figure 11
Effect of piles pacing for settlement on interaction factors of 2 × 2 group piles.


#### 5.2.2. Slenderness Ratio

Figure12 shows the relationship between interaction factor and slenderness ratio at different slenderness ratios l/d = 10, 20, …, 100, pile spacing, and diameter ratio S/d = 2, 3, 4, and 5. It can be seen from Figure 12 that the interaction factor decreases with the increase of slenderness ratio. When slenderness ratio is a constant value, the interaction factor decreases with the increase in pile spacing, which is the same as the conclusion in 5.2.1.Figure 12
Influence of slenderness ratiosl/d on interaction factor αs.


#### 5.2.3. Pile-Soil Elastic Modulus Ratio

Figure13 shows the relationship between interaction factor and pile-soil elastic modulus ratio at different pile spacing and diameter ratios. It can be seen from Figure 13 that the interaction factor decreases with the increase in pile spacing. When the pile spacing is constant, the interaction factor increases with the increase in the pile-soil elastic modulus ratio, which is consistent with the conclusion of Poulos and Davis [25].Figure 13
Influence of the pile-soil elastic modulus ratioEp/Es on interaction factor αs.


#### 5.2.4. Poisson’s Ratio

Figure14 shows the influence of different Poisson’s ratios on the interaction factor. When S/d = 15, Poisson’s ratio changes from 0 to 0.5, and the interaction factor decreases by 25%. The interaction factor decreases with the increase in pile spacing. At the same time, the larger the group pile spacing, the more obvious it is that the Poisson’s ratio affects the interaction factor, and the same conclusion can be found in [28].Figure 14
Influence of Poisson’s ratio on interaction factorαs.


## 5.1. Verify the Rationality of the Proposed Method

### 5.1.1. Example I

According to the research of O’Neill et al. [26] reported, 11 piles were driven into linear viscoelastic soil. The diameter of the pile is 137 mm, and the buried length of the pile is 13.1 m. Among them, 4 piles are arranged in a 3 × 3 square and connected with a concrete cap, and the cap is not in contact with the soil below. The pile spacing is S = 3d, and the other two piles are 3.7 m on both sides of the center group pile.Specific parameters: the elastic modulus of soilEs = 1.95 × 105 kPa, Poisson’s ratio of soil ν = 0.4, and the elastic modulus of concrete pile Ep = 2.1 × 108 kPa. Average load of single pile in group pile is P=653kN. The load-displacement curve of single pile test is shown in Figure 7. Parameter S/d = 3, Ep = Es = 1077. Due to the lack of unloading curve in pile test, the method proposed in this paper is used to define the elastic stiffness of pile: Ke = 608 kN/mm, thus Se = P/Ke = 1.07 mm, and P=653kN. According to the above principle, the settlement of group pile can be calculated as S = 7.26 mm. The measured settlement of group pile is 6.63 mm, and the results demonstrate that the error between the field test results and computed data is 14.9%, which signifies that the present method is suitable for the theoretical calculation and predict the composite foundation settlement of linear viscoelastic soil when piles are used to strengthen the foundation.Figure 7
Comparison of the capacity of a single model pile.


### 5.1.2. Example II

In order to further study the settlement characteristics of group piles in linear viscoelastic soil, based on the parameters of the soil layer and piles in a large-scale field group pile test [27] and according to the total settlement calculation method of our paper, the load-settlement curves and group pile settlement-interaction factor curves under different types of group piles, different pile lengths, and different pile spacing (Figure 8) are analyzed. The parameters of pile length, pile spacing, pile cap size, and slenderness ratio are shown in Tables 2, and Table 3 summarizes the soil properties of each soil layer crossed by the test piles.Figure 8
Layout plan of test piles and group piles.Table 2
Summary of field tests.
No. of pilesPile length (m)Pile spacingPile cap sizes (length × width × height; all in metres)Slenderness ratio (L/B)D-120—0.4 × 0.4 × 0.450D-224—0.4 × 0.4 × 0.4605–2202.5B1.8 × 0.8 × 0.8505–1243.0B2.0 × 0.8 × 0.860D-120—0.4 × 0.4 × 0.4504202.5B1.8 × 0.8 × 0.8502243.0B2.0 × 2.0 × 0.8603202.5B2.8 × 2.8 × 1.2501243.0B3.2 × 3.2 × 1.260Table 3
Soil properties.
SoilDirect shear test, curve-fit strength envelopeOne-dimensional consolidation testThickness (m)Unit weight,γ (kN/m3)Void ratio,e0Plasticity index,IpLiquidity index,ILCohesion (kPa)Internal friction angle (φ)α1 −2 (MPa−1)Es (MPa)Fill3817.40.306.072.619.20.7616.00.42Clay5018.50.237.652.019.60.7117.20.28Silt1122.50.257.565.618.50.855.80.60Silt intermixed with silty sand1323.70.208.887.318.80.795.00.64Soft clay158.30.673.4011.617.41.2421.70.96Based on the concept of elasticity theory, considering the rheological settlement of soils, we can get the implied interaction factors of group piles as follows:(13)αij=ΛλL,Ωlnrm/spijln2rm/B≥0,where(14)Ω=QbwbEpApλ=4Gbrs1−ν1λEpAp,λ=kEpAp,rm=0.25+2.51−vGGL−0.25GLGbL.Based on the interaction factor obtained by equation (13), the load-settlement relationship can be obtained as shown in Figure 9, and the relationship curve between the interaction factor and the settlement of different pile types is shown in Figure 10.Figure 9
Individual pile load versus group settlement relationship based on the interaction factor. (a) Pile lengthL = 20 m. (b) Pile length L = 24 m.
(a)(b)Figure 10
Interaction factor versus settlement for each group pile. (a) Two-group pile withL1 = 24 m and L2 = 20 m, (b) four-group pile with L1 = 24 m and L2 = 20 m, (c) nine-group pile with L2 = 20 m, and (d) nine-group pile with L1 = 24 m.
(a)(b)(c)(d)Figure9 presents the solutions from the present method and Guoliang Dai’s, where the present solutions consider the rheological settlement of soils. The total settlement calculated by the present method shows good agreement with the reported results, and the correlation coefficients R2 are greater than 0.9223.The relationship between settlement and interaction factor of different pile types is shown in Figure10αij indicates the ratio of the settlement of pile i under load to the settlement of adjacent pile j. For group pile types with different pile lengths (Figure 10(a)), the interaction factor increases and then decreases with the increase in group pile settlement, and the longer the pile length, the smaller the influence on the interaction factor, and the interaction coefficient for the two-group piles L1 is practically zero. This may be because of the spatial variability of the soil or other variability in the pile installation or pile cap. For the group pile types as shown in Figure 10(b), it is not difficult to see that pile spacing has a larger effect on interaction than pile length. For the 3 × 3 group pile types, it can be seen from Figure 10(c) that the settlement of the corner pile is the largest and that of the middle pile is the smallest, and the interaction factors are distributed proportionally to pile center-to-center spacing. Similar conclusions can also be drawn from Figure 10(d). Comparing Figures 10(a) and 10(b) with Figures 10(c) and 10(d), it can be seen that when the pile spacing is the same, the interaction factor of the piles in the four group piles is greater than that of the nine group piles.


## 5.1.1. Example I

According to the research of O’Neill et al. [26] reported, 11 piles were driven into linear viscoelastic soil. The diameter of the pile is 137 mm, and the buried length of the pile is 13.1 m. Among them, 4 piles are arranged in a 3 × 3 square and connected with a concrete cap, and the cap is not in contact with the soil below. The pile spacing is S = 3d, and the other two piles are 3.7 m on both sides of the center group pile.Specific parameters: the elastic modulus of soilEs = 1.95 × 105 kPa, Poisson’s ratio of soil ν = 0.4, and the elastic modulus of concrete pile Ep = 2.1 × 108 kPa. Average load of single pile in group pile is P=653kN. The load-displacement curve of single pile test is shown in Figure 7. Parameter S/d = 3, Ep = Es = 1077. Due to the lack of unloading curve in pile test, the method proposed in this paper is used to define the elastic stiffness of pile: Ke = 608 kN/mm, thus Se = P/Ke = 1.07 mm, and P=653kN. According to the above principle, the settlement of group pile can be calculated as S = 7.26 mm. The measured settlement of group pile is 6.63 mm, and the results demonstrate that the error between the field test results and computed data is 14.9%, which signifies that the present method is suitable for the theoretical calculation and predict the composite foundation settlement of linear viscoelastic soil when piles are used to strengthen the foundation.Figure 7
Comparison of the capacity of a single model pile.


## 5.1.2. Example II

In order to further study the settlement characteristics of group piles in linear viscoelastic soil, based on the parameters of the soil layer and piles in a large-scale field group pile test [27] and according to the total settlement calculation method of our paper, the load-settlement curves and group pile settlement-interaction factor curves under different types of group piles, different pile lengths, and different pile spacing (Figure 8) are analyzed. The parameters of pile length, pile spacing, pile cap size, and slenderness ratio are shown in Tables 2, and Table 3 summarizes the soil properties of each soil layer crossed by the test piles.Figure 8
Layout plan of test piles and group piles.Table 2
Summary of field tests.
No. of pilesPile length (m)Pile spacingPile cap sizes (length × width × height; all in metres)Slenderness ratio (L/B)D-120—0.4 × 0.4 × 0.450D-224—0.4 × 0.4 × 0.4605–2202.5B1.8 × 0.8 × 0.8505–1243.0B2.0 × 0.8 × 0.860D-120—0.4 × 0.4 × 0.4504202.5B1.8 × 0.8 × 0.8502243.0B2.0 × 2.0 × 0.8603202.5B2.8 × 2.8 × 1.2501243.0B3.2 × 3.2 × 1.260Table 3
Soil properties.
SoilDirect shear test, curve-fit strength envelopeOne-dimensional consolidation testThickness (m)Unit weight,γ (kN/m3)Void ratio,e0Plasticity index,IpLiquidity index,ILCohesion (kPa)Internal friction angle (φ)α1 −2 (MPa−1)Es (MPa)Fill3817.40.306.072.619.20.7616.00.42Clay5018.50.237.652.019.60.7117.20.28Silt1122.50.257.565.618.50.855.80.60Silt intermixed with silty sand1323.70.208.887.318.80.795.00.64Soft clay158.30.673.4011.617.41.2421.70.96Based on the concept of elasticity theory, considering the rheological settlement of soils, we can get the implied interaction factors of group piles as follows:(13)αij=ΛλL,Ωlnrm/spijln2rm/B≥0,where(14)Ω=QbwbEpApλ=4Gbrs1−ν1λEpAp,λ=kEpAp,rm=0.25+2.51−vGGL−0.25GLGbL.Based on the interaction factor obtained by equation (13), the load-settlement relationship can be obtained as shown in Figure 9, and the relationship curve between the interaction factor and the settlement of different pile types is shown in Figure 10.Figure 9
Individual pile load versus group settlement relationship based on the interaction factor. (a) Pile lengthL = 20 m. (b) Pile length L = 24 m.
(a)(b)Figure 10
Interaction factor versus settlement for each group pile. (a) Two-group pile withL1 = 24 m and L2 = 20 m, (b) four-group pile with L1 = 24 m and L2 = 20 m, (c) nine-group pile with L2 = 20 m, and (d) nine-group pile with L1 = 24 m.
(a)(b)(c)(d)Figure9 presents the solutions from the present method and Guoliang Dai’s, where the present solutions consider the rheological settlement of soils. The total settlement calculated by the present method shows good agreement with the reported results, and the correlation coefficients R2 are greater than 0.9223.The relationship between settlement and interaction factor of different pile types is shown in Figure10αij indicates the ratio of the settlement of pile i under load to the settlement of adjacent pile j. For group pile types with different pile lengths (Figure 10(a)), the interaction factor increases and then decreases with the increase in group pile settlement, and the longer the pile length, the smaller the influence on the interaction factor, and the interaction coefficient for the two-group piles L1 is practically zero. This may be because of the spatial variability of the soil or other variability in the pile installation or pile cap. For the group pile types as shown in Figure 10(b), it is not difficult to see that pile spacing has a larger effect on interaction than pile length. For the 3 × 3 group pile types, it can be seen from Figure 10(c) that the settlement of the corner pile is the largest and that of the middle pile is the smallest, and the interaction factors are distributed proportionally to pile center-to-center spacing. Similar conclusions can also be drawn from Figure 10(d). Comparing Figures 10(a) and 10(b) with Figures 10(c) and 10(d), it can be seen that when the pile spacing is the same, the interaction factor of the piles in the four group piles is greater than that of the nine group piles.


## 5.2. Parametric Studies

The calculation method in this paper involves many design parameters, such as pile spacing, slenderness ratio, pile-soil elastic modulus ratio, Poisson’s ratio, etc. It is of great significance for optimal design to explore the influence law of each design parameter on the working performance of a group pile foundation. Therefore, the example of group pile foundation in Booker and Poulos [15] is selected for parameter analysis, and the influence laws of pile end resistance, pile spacing, slenderness ratio, elastic modulus ratio and Poisson ratio on pile end load ratio, and interaction factors are studied by the calculation method proposed in this paper.


### 5.2.1. Pile Spacing

Figure11 shows the relationship of pile spacing and interaction factor when the ratio of pile spacing to interaction factor is S/d = 1, 2, 3, …, 19, the slenderness ratio l/d = 10, 50, 100, and the calculation elastic modulus ratio of pile-soil Ep/Es = 1000, Poisson’s ratio of soil is 0.5. It can be seen from Figure 11 that the interaction factor decreases with the increase of pile spacing, and the interaction factor decreases with the increase of pile length, but the influence of pile length on the interaction factor can be neglected with the increase of pile spacing.Figure 11
Effect of piles pacing for settlement on interaction factors of 2 × 2 group piles.


### 5.2.2. Slenderness Ratio

Figure12 shows the relationship between interaction factor and slenderness ratio at different slenderness ratios l/d = 10, 20, …, 100, pile spacing, and diameter ratio S/d = 2, 3, 4, and 5. It can be seen from Figure 12 that the interaction factor decreases with the increase of slenderness ratio. When slenderness ratio is a constant value, the interaction factor decreases with the increase in pile spacing, which is the same as the conclusion in 5.2.1.Figure 12
Influence of slenderness ratiosl/d on interaction factor αs.


### 5.2.3. Pile-Soil Elastic Modulus Ratio

Figure13 shows the relationship between interaction factor and pile-soil elastic modulus ratio at different pile spacing and diameter ratios. It can be seen from Figure 13 that the interaction factor decreases with the increase in pile spacing. When the pile spacing is constant, the interaction factor increases with the increase in the pile-soil elastic modulus ratio, which is consistent with the conclusion of Poulos and Davis [25].Figure 13
Influence of the pile-soil elastic modulus ratioEp/Es on interaction factor αs.


### 5.2.4. Poisson’s Ratio

Figure14 shows the influence of different Poisson’s ratios on the interaction factor. When S/d = 15, Poisson’s ratio changes from 0 to 0.5, and the interaction factor decreases by 25%. The interaction factor decreases with the increase in pile spacing. At the same time, the larger the group pile spacing, the more obvious it is that the Poisson’s ratio affects the interaction factor, and the same conclusion can be found in [28].Figure 14
Influence of Poisson’s ratio on interaction factorαs.


## 5.2.1. Pile Spacing

Figure11 shows the relationship of pile spacing and interaction factor when the ratio of pile spacing to interaction factor is S/d = 1, 2, 3, …, 19, the slenderness ratio l/d = 10, 50, 100, and the calculation elastic modulus ratio of pile-soil Ep/Es = 1000, Poisson’s ratio of soil is 0.5. It can be seen from Figure 11 that the interaction factor decreases with the increase of pile spacing, and the interaction factor decreases with the increase of pile length, but the influence of pile length on the interaction factor can be neglected with the increase of pile spacing.Figure 11
Effect of piles pacing for settlement on interaction factors of 2 × 2 group piles.


## 5.2.2. Slenderness Ratio

Figure12 shows the relationship between interaction factor and slenderness ratio at different slenderness ratios l/d = 10, 20, …, 100, pile spacing, and diameter ratio S/d = 2, 3, 4, and 5. It can be seen from Figure 12 that the interaction factor decreases with the increase of slenderness ratio. When slenderness ratio is a constant value, the interaction factor decreases with the increase in pile spacing, which is the same as the conclusion in 5.2.1.Figure 12
Influence of slenderness ratiosl/d on interaction factor αs.


## 5.2.3. Pile-Soil Elastic Modulus Ratio

Figure13 shows the relationship between interaction factor and pile-soil elastic modulus ratio at different pile spacing and diameter ratios. It can be seen from Figure 13 that the interaction factor decreases with the increase in pile spacing. When the pile spacing is constant, the interaction factor increases with the increase in the pile-soil elastic modulus ratio, which is consistent with the conclusion of Poulos and Davis [25].Figure 13
Influence of the pile-soil elastic modulus ratioEp/Es on interaction factor αs.


## 5.2.4. Poisson’s Ratio

Figure14 shows the influence of different Poisson’s ratios on the interaction factor. When S/d = 15, Poisson’s ratio changes from 0 to 0.5, and the interaction factor decreases by 25%. The interaction factor decreases with the increase in pile spacing. At the same time, the larger the group pile spacing, the more obvious it is that the Poisson’s ratio affects the interaction factor, and the same conclusion can be found in [28].Figure 14
Influence of Poisson’s ratio on interaction factorαs.


## 6. Conclusions

In the current paper, we present a method to study the settlement behaviors of group pile foundation in linear viscoelastic soil. In this method, the variation of linear viscoelastic soil settlement with pile spacing, length-diameter ratio, elastic modulus ratio, and Poisson’s ratio was investigated, and the results were validated through comparison with the field test results. Based on the results, we obtained the following key conclusions:(1)
The pile-pile interaction factors increase with the increase of the pile-soil elastic modulus ratio and length-to-diameter ratio but decrease with the increase of the ratio of distance to diameter, and the ratio of distance to diameter has little effect on the interaction factor when pile spacing is more than 10 d (d is the diameter of the pile).(2)
The settlement variation of a group pile foundation in linear viscoelastic soil calculated by the present method is consistent with field test results. Furthermore, according to the present method, the settlement of a group pile can be calculated at 7.26 mm, while the measured settlement of a group pile is 6.63 mm, and the results demonstrate that the error between the field test results and computed data is 14.9%, which signifies that the present method is suitable for the theoretical calculation and can predict the composite foundation settlement of linear viscoelastic soil when piles are used to strengthen the foundation.(3)
The interaction factor of a group pile can represent the settlement characteristics of a group pile foundation, and with the increase of pile number, the settlement of a group pile is linearly positively correlated with the interaction factor.(4)
The interaction factors-based method can estimate the total settlement of a group pile foundation under constant stress.It is worth noting that this study only pays attention to the total settlement of group pile foundations under static load, excluding the settlement caused by dynamic load.



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*Source: 3207304-2023-02-23.xml*