Calculation Model of Compaction Coefficient of Soil among SP−PSC Pile Group on Collapsible Loess Foundation

Abstract

Collapsible loess is a kind of soil with special properties, and is widely distributed in China. When it is not wetted by water, its strength is generally high and its compressibility is low. However, when collapsible loess is wetted by water under a certain pressure, the soil structure will be rapidly destroyed, resulting in large additional subsidence. Therefore, when engineering constructions on collapsible loess sites are carried out, appropriate foundation treatment measures must be taken to eliminate foundation collapsibility. Because of their advantages, static pressure plain soil compaction (SP−PSC) piles are widely used for collapsible loess foundation treatments in China. However, at present, there is still a lack of accurate understanding of the distribution of compaction coefficient of soil among SP−PSC pile groups on collapsible loess foundations. The present study systematically investigated the distribution of the soil compaction coefficient among SP−PSC pile groups based on SP−PSC pile group tests and finite element analyses. The effect of different factors on soil compaction coefficient was analyzed and explored, including the pile diameter and length of SP−PSC piles, the soil moisture content, the pile spacing within the SP−PSC pile group, and the depth to ground. Finally, the simplified calculation models of the compaction coefficient of the soil at the center of pile group and at the midpoints of adjacent piles were analytically formulated. These models established a theoretical basis for the design and construction of SP−PSC pile groups on collapsible loess foundations.

1. Introduction

Collapsible loess refers to soil subject to significant additional deformation due to the structural damage of soil after immersion under self-weight stress, or under the combined action of self-weight stress and additional stress. Collapsible loess is a form of special soil and is widely distributed in the Loess Plateau areas of Gansu, Ningxia, Shaanxi, and Shanxi in China. Therefore, when carrying out engineering constructions on collapsible loess foundations, it is necessary to consider the possible harm to the projects caused by additional settlement due to foundation collapsibility, the selection of appropriate foundation treatment methods, and the avoidance or elimination of the harm caused by foundation collapsibility.

Common foundation treatment measures to eliminate loess collapsibility include the replacement cushion method, heavy hammer compaction method, dynamic consolidation method, and compaction pile method. In view of its advantages, the compaction pile method is a popular foundation treatment method in collapsible loess regions. With the continuous advancement of pile driving technology, the jacked-in pile has become a popular pile type due to its fast construction speed, weak vibration, lower noise, low pollution, and many other advantages. At present, the penetration of jacked-in piles on clay foundations has been investigated by some scholars through model or field tests [1,2,3,4,5,6]. Furthermore, the interaction between piles and soil, and the stress and displacement distributions of soil around or among piles, have been analytically discussed. Based on model tests, some scholars also studied the penetration process of jacked-in piles on sandy soil foundations [7,8], and the pile–soil interaction and the foundation stress change were examined. There are also some studies on the penetration process of different compaction piles using various numerical methods, such as the finite element method and discrete element method [9,10,11,12]. In the theoretical analysis of foundation pile penetration, the most effective method is still considered to be the Cavity Expansion Method (CEM) [13,14], which provides theoretical support for the analysis of the change in mechanical properties of soil around piles. However, regarding penetration compaction piles in collapsible loess sites, the experimental, numerical, and theoretical studies are currently inadequate. In addition, there are still no reasonable calculation models of the compaction coefficient of soil within pile groups.

To improve the material properties of collapsible soils, some scholars have carried out a series of studies on the influence of different admixtures (such as nanomaterials or industrial wastes) on the properties of collapsible loess [15,16]. The results show that collapsible soils are very sensitive to the addition of different admixtures. The amount and type of admixtures added to the collapsible soils could have both positive and negative impacts on the desired properties. The relationship between the collapse potential of soils and the type and proportion of admixtures under different conditions has also been established. Therefore, in order to achieve the optimal performance improvement in collapsible soils, the most appropriate admixture type and percentage should be used. Unfortunately, the research results of the admixture improvement methods cannot be applied to the compaction pile improvement method.

In a part of jacked-in pile composite foundations, the reamed pile is not pulled out after penetration, and is used as a part of the composite foundation. This situation leads to a significant increase in the cost of foundation treatment. Therefore, a new effective and economical method for loess foundation treatment, namely, the static pressure plain soil compaction (SP−PSC) pile, has been proposed and widely applied in practical projects in recent years in China [17]. In this construction method, with the help of its dead weight and counterweight as the reaction force, the pile driver slowly drives the reamed steel pipe pile into the soil layer. When the formed hole reaches the design depth, the reamed steel pipe is slowly pulled out. Finally, the formed holes are filled with dry loess and compacted layer by layer with a heavy tamping hammer. However, there are few relevant research reports on this method at present. There is also a lack of in-depth understanding of the distribution of soil compaction coefficient among SP−PSC pile groups in the literature.

In this paper, during the construction of a SP−PSC pile group on a collapsible loess foundation, the stress of soil within the pile group was systematically investigated by SP−PSC pile group tests and finite element analyses. After the construction of the SP−PSC pile group, the compaction coefficient of the soil field among the pile group was measured, and the effect of different factors on it was analyzed. Finally, simplified calculation models of the compaction coefficient of soil at the center of the pile group and at the midpoint of adjacent piles were analytically formulated based on the theoretical analysis. These models can provide a basis for the accurate calculation of the compaction coefficient of soil at different positions within a SP−PSC pile group on a collapsible loess foundation.

2. Tests

2.1. Soil Tests

The pile group test site was located in Xianyang, Shaanxi, China, and the test loess samples were taken from the test site. The loess samples were divided into three groups according to the depths of the soil layer, which corresponded to 1, 3, and 5 m below the surface.

2.1.1. Density Tests

Figure 1 shows the density tests of the soil mass before compaction. The original soil samples (Figure 1a) were made into the standard specimens using a cutting ring (Figure 1b), and their wet density ${\rho }_{\mathrm{w}}$ was measured by weighing on a scale. Then, after drying in a dry oven for 24 h (Figure 1c), the dry loess specimens were obtained [18]. Finally, through measurement and calculation, the dry density ${\rho }_{\mathrm{d}}$ and the moisture content $\omega$ of the loess specimens before compaction were obtained.

Table 1. Soil material parameters before compaction.

Sample of Soil	Wet Density ρw (g/cm3)	Dry Density ρd (g/cm3)	Rate of Water ω Content (%)
S1	1.48	1.24	19.35
S2	1.56	1.25	24.80
S3	1.40	1.27	10.55

Note: S₁, S₂, S₃ soil samples were taken from 1, 3, and 5 m below the surface, respectively.

lists various material parameters of the soil mass before compaction.

2.1.2. Direct Shear Tests

The direct shear test is widely used in practical engineering due to its use of simple equipment, and convenient soil sample preparation and test operation. The direct shear tests of the soil samples before compaction are shown in Figure 2. The original soil samples (Figure 2a) were processed into standard specimens by cutting ring [18], which is a cylinder with a diameter of 61.8 mm and a height of 20 mm. Then, using the direct shear gauge (Figure 2b), the standard soil specimens were quickly sheared. The shear speed of the quick shear tests was 0.8 mm/min, which caused the sample to be damaged and sheared within 3 to 5 min. Based on the direct shear test results, the cohesion coefficient $c$ and the angle of internal friction $\phi$ of the loess were obtained.

Table 2. Cohesion coefficient and angle of internal friction of loess.

Sample of Soil	$\mathbf{Angle}\mathbf{of}\mathbf{Internal}\mathbf{Friction}\mathit{\phi }$ (°)	Cohesion Coefficient c (KPa)
S1	17.25	21.53
S2	7.74	12.24
S3	28.89	59.12

Note: S₁, S₂, S₃ soil samples were taken from 1, 3, and 5 m below the surface, respectively. The reason for the maximum φ and c of soil sample S₃ is that it has the minimum moisture content (see Table 1).

lists the cohesion coefficient and the angle of internal friction of the loess. In the direct shear tests, to maintain the stability of the samples and obtain $c$ and $\phi$ of the loess, vertical stresses of 100, 200, 300, and 400 kPa were applied to the samples. The moisture content of soil samples used for the direct shear tests should be consistent with that of the soil sampling sites as much as possible. Therefore, the soil samples were all unsaturated soils.

2.2. SP−PSC Pile Group Tests

In order to eliminate the collapsibility of the loess foundation, the SP−PSC piles is a very economic and effective method. In this study, the SP−PSC pile group (Figure 3) was constructed in strict accordance with the requirements [17]. At the initial stage of pile pressing, when the reamed pile head was at a depth of between 0.5 and 1 m underground, the reamed pile was corrected to be vertical and stable. When the formed hole reached the design depth, the reamed pile was slowly pulled out. After each of the three formed pile holes were completed, they were backfilled with dry loess and compacted layer by layer with a heavy tamping hammer. During the tamping of the plain loess layer by layer, each filling volume should be controlled within a height of between 500 and 800 mm in the hole. The tamping times of each layer must meet the requirements.

The SP−PSC pile group tests (Figure 3) were divided into two groups with identical design parameters, thus helping to avoid data loss caused by accidental factors and reducing test errors. In order to eliminate the mutual influence of the two groups of tests, the two groups of test sites were more than 20 m apart. In the SP−PSC pile group tests, the diameter of the reamed pile, namely, the diameter D of the SP−PSC pile, was 0.56 m, and the pile depth H was 7 m. The plane of the pile groups was an equilateral triangle with a side length of 2.24 m, and the depth of pile groups was 7 m.

Figure 3 shows the SP−PSC pile group tests, in which three piles form an equilateral triangle and the pile spacing is equal to four times the pile diameter (4D). The construction process of the SP−PSC pile group was Pile 1, Pile 2, and Pile 3, in turn. In the process of the tests, in order to measure the change in soil stress at different positions within the SP−PSC pile group, a total of 5 soil pressure cells were embedded in the soil within the SP−PSC pile group in each group of tests. At the midpoint A of Pile 1 and Pile 2, 3 soil pressure cells were respectively arranged 1, 3, and 5 m below the ground, as shown in Figure 3a. At the point B between Pile 2 and Pile 3, which was positioned 1 times the pile diameter from Pile 2, a soil pressure cell was embedded 3 m below the ground. At the point C between Pile 1 and the centroid of the pile group, which was one-third of the distance ($\frac{4\sqrt{3}}{9}$D) from Pile 1, a soil pressure cell was embedded 3 m below the ground. Before being embedded, the soil pressure cells were accurately fixed on flat and regular cork strips with glass glue (Figure 3d) to ensure their precise embedded depth. In addition, by adjusting the angle of the cork strips, it can be ensured that the measuring direction of the soil pressure cells was horizontal and pointed to the corresponding pile core. The measuring direction of soil pressure cells at A, B, and C pointed to Pile 1, Pile 2, and Pile 1, respectively.

In order to minimize the influence of embedded holes of the soil pressure cells on the soil mass, the diameter of embedded holes was controlled at about 80 mm, which was about twice the diameter of soil pressure cells. To ensure the hole density and avoid damage to the soil pressure cells, after the soil pressure cells were placed and positioned, the embedded holes were backfilled with dry fine sand.

Table 3. Soil stress results of pile group tests (Unit: kPa).

	A1			A3			A5
Process	Test1	Test2	FEM	Test1	Test2	FEM	Test1	Test2	FEM
P1	33.1	37.5	40.8	143.2	107.3	127.6	53.9	41.4	60.2
P2	48.3	40.1	58.3	156.3	131.7	178.4	56.2	69.3	80.2
	B			C
Process	Test1	Test2	FEM	Test1	Test2	FEM
P1	/			296.7	315.2	297.3
P2	263.5	245.4	287.5	/

Note: For more direct comparison between the finite element and the test results (soil stresses), the finite element results are listed in advance in Table 3, and the finite element analysis is described in Section 3.

shows the soil stress results of the SP−PSC pile group tests. In Table 3, A₁, A₃, and A₅ refer to the soil stresses measured at the depths of 1, 3, and 5 m, respectively, at location A. P₁ and P₂ correspond to the penetration of Pile 1 and Pile 2, respectively. It can be seen from Table 3 that the soil stresses measured at the depth of 3 m are the largest at location A. Irrespective of whether the depth increases or decreases, the soil stresses gradually decrease. Moreover, the soil stresses measured at A₁, A₃, and A₅ when Pile 2 was penetrated are obviously higher than the corresponding values when Pile 1 was penetrated. This is because the construction of Pile 1 makes the soil field denser. As the points B and C are close to their corresponding Pile 2 and Pile 1, the soil stresses measured at B and C are obviously large, as shown in Table 3.

2.3. Compaction Coefficient Tests

When the pile construction was completed in the SP−PSC pile group tests, in order to obtain the true compaction coefficient of soil at different locations among the SP−PSC pile group, we conducted borehole sampling with the drilling machine, and the compaction coefficient tests were carried out (Figure 4). In the borehole sampling method, before sampling, the soil layers above the sampling depths were removed by an auger with as little disturbance as possible. Then, the compacted soil samples at the predetermined depths were taken out by the thin-walled steel barrel of the drilling machine (Figure 4c). As shown in Figure 4d, the sampling points were distributed along two lines. Points a, b, and c were located on the connecting line between Pile 1 and Pile 3, and Point c was located at the midpoint between Pile 1 and Pile 3; the distances between Points a, b, and c were $\frac{2D}{3}$ and $\frac{2D}{3}$, respectively. Point f was located at the centroid of the pile group. Points d and e were located on the connecting line between Pile 2 and Point f, and the distances between Points d, e, and f were $\frac{4\sqrt{3}}{9}$D. The above six sampling points completely avoided the embedded positions of the soil pressure cells. At each sampling point, three groups of soil samples were collected from 1 , 3, and 5 m underground.

The compaction coefficient of soil samples can be derived from Equation (1).

$${\eta }_{\mathrm{c}}=\frac{{\rho }_{\mathrm{dc}}}{{\rho }_{\mathrm{dmax}}}$$

(1)

where ${\rho }_{\mathrm{dc}}$ is the dry density of the compacted soil samples, and is tested as Section 2.1.1; ${\rho }_{\mathrm{dmax}}$ is the maximum dry density of the soil samples, and is measured by indoor compaction tests. The compaction coefficient of the undisturbed soil was 0.74 in this study.

As shown in

Table 4. Compaction coefficients of compacted soil samples.

	a				b				c
Depth (m)	Test1	Test2	FEM	Error	Test1	Test2	FEM	Error	Test1	Test2	FEM	Error
1	0.9532	0.9235	0.9612	0.83%/3.92%	0.8347	0.8213	0.8375	0.33%/1.93%	0.7921	0.7793	0.7891	−3.80%/1.24%
3	0.987	0.9738	0.9796	−0.75%/0.59%	0.8631	0.8429	0.8693	0.71%/3.04%	0.8395	0.8354	0.8425	0.36%/0.84%
5	0.9316	0.8971	0.9482	1.75%/5.39%	0.8173	0.8256	0.8319	1.75%/0.76%	0.7512	0.7601	0.7664	1.98%/0.82%
	d				e				f
Depth (m)	Test1	Test2	FEM	Error	Test1	Test2	FEM	Error	Test1	Test2	FEM	Error
1	0.9443	0.9274	0.9392	−0.54%/1.26%	0.8413	0.8125	0.8177	−2.89%/0.64%	0.7691	0.7831	0.7724	0.43%/−1.38%
3	0.9713	0.9645	0.9617	−0.99%/−0.29%	0.8445	0.8329	0.8496	0.60%/1.96%	0.7815	0.7891	0.7896	1.02%/0.06%
5	0.9417	0.9359	0.9197	−2.39%/−1.76%	0.8436	0.8231	0.8079	−4.42%/−1.88%	0.7451	0.7483	0.7468	0.23%/−0.20%

Note: For more direct comparison between the finite element and the test results (compaction coefficients), the finite element results are listed in advance in Table 4, and the finite element analysis is described in Section 3.

, the compaction coefficient of the soil within the SP−PSC pile group was obtained based on the compaction test results and Equation (1). It can be found from Table 4 that with the increase in the distance from the pile core, the compaction coefficient of soil decreases gradually. Since the distance between the centroid (point f) of the pile group and the pile cores is the largest, the soil compaction coefficient at point f is the smallest. On the connecting line of two piles, the soil compaction coefficient at the midpoints is the minimum. At different depths of the same sampling point, it can be found that the soil compaction coefficient at the depth of 3 m is the largest. It can be seen from Table 4 that the variation law of the compaction coefficient of the soil was similar to that of the soil stress in both horizontal and vertical directions.

Compression tests have validated that loess collapsibility is closely related to its compaction coefficient [19]. When the loess compaction coefficient is larger than 0.9, its collapsibility coefficient is less than 0.015, and it can be assumed that the loess collapsibility is basically eliminated. Therefore, the compaction coefficient of the soil within the SP−PSC pile group is systematically discussed in the following section.

3. Finite Element Method (FEM)

3.1. Finite Element Model and Verification

Because of the limitations of test conditions, cost, and time, the mechanical property information of the soil mass obtained in the SP−PSC pile group tests was not enough. Therefore, in order to obtain more detailed results of soil mechanical properties under different conditions, a series of numerical studies on the SP−PSC pile group were carried out using ABAQUS software. The finite element analysis models are shown in Figure 5. Because the stiffness of the reamed steel pipes pile was far greater than that of the soil, the reamed steel pipe piles were assumed to be a rigid body in the numerical models. The Mohr–Coulomb model was adopted for the soil mass, and the material parameters were obtained through the soil tests (see Section 2.1). The surface-to-surface contact model was selected for the contact between the reamed steel pipe piles and the soil mass. In the contact normal direction, the hard contact mode was adopted. In the contact tangential direction, the penalty function method was selected. The friction coefficient was taken as 0.3, which could be calculated by the tangent value of the internal friction angle $\phi$ of soils (taken as the average value of different soil samples in Table 2, $\phi =17.96 °$) [20]. The meshes of the numerical models were divided using a hexahedron by a three-dimensional eight-node reduced integral cell (C3D8R). In the process of numerical analysis, the penetration of the reamed steel pipe piles was controlled by displacement loading. To avoid the boundary effects, the boundary of the semi-infinite soil mass was set at 20D. The distance between the pile hole boundary and the soil mass boundary was greater than 6D. Since the length of the piles was 7 m, the depth of the semi-infinite soil mass was set as 20 m. Thus, the boundary extent and size were enough. As described in the literature [21,22], the horizontal displacement constraint was applied only to the vertical boundaries of the model, while both horizontal and vertical constraints were applied simultaneously to the bottom boundary. Additionally, no constraints were applied to the model’s top boundary.

The compaction efficiency of the soil within the SP−PSC pile group in the process of penetration of the reamed steel pipe pile was much higher than that in the process of tamping the plain loess layer by layer, which was proved by the results of the SP−PSC pile group tests. For the above reasons, the results of finite element analysis were obtained from the state of the reamed steel pipe pile penetrating to the design depth. During the subsequent tamping of the plain loess, it was considered that only the collapsibility of the backfilling loess in the hole was eliminated, and the influence on the soil within the SP−PSC pile group was ignored.

The measurements of the stress and compaction coefficient of the soil within the SP−PSC pile group based on the finite element analysis are shown in Table 3 and Table 4. It can be observed that the numerical results are in good agreement with the test results.

3.2. Influencing Factors

In this section, the effect of different factors on the compaction coefficient of the soil mass among the SP−PSC pile group is systematically investigated using the finite element method, including the pile diameter and length of the SP−PSC pile, the soil moisture content, the pile spacing within the SP−PSC pile group, and the depth to ground.

In order to avoid the mutual coupling of various factors, the influence of the pile diameter, the pile length, and the soil moisture content was studied through single SP−PSC pile numerical analyses. At the same time, two groups of single SP−PSC pile tests were completed on the SP−PSC pile group test site to provide verification of the numerical analyses. The initial conditions of single SP−PSC pile tests and numerical analyses were the same as those of the SP−PSC pile group tests, and are not repeated here.

3.2.1. Pile Diameter

In this section, the pile length was 7 m and the moisture content was 18%. The relationship between the soil compaction coefficient and the pile diameter D is shown in Figure 6, in which the focused point was selected at the position 3 m from the ground and 0.56 m from the pile core. Figure 6 illustrates that the compaction coefficient gradually increased with increases in the pile diameter, and that the rate of increase first increased and then decreased. The variation range from 0.2 to 0.6 m was the sensitive area of the change in the pile diameter to the increase in the soil compaction coefficient.

3.2.2. Pile Length

During the finite element analysis of the pile length changes, the pile diameter and soil moisture content were fixed at 0.56 m and 18%, respectively. Figure 7 shows the relationship between the soil compaction coefficient and the pile length, in which the focused point was 3/7 of the pile length from the ground and 0.56 m from the pile core. It can be observed from Figure 7 that the soil compaction coefficient gradually increased with increases in the pile length, but the rate of increase gradually decreased.

3.2.3. Soil Moisture Content

In this section, the pile diameter and the pile length were fixed at 0.56 and 7 m, respectively. Figure 8 depicts the variation in the soil compaction coefficient with soil moisture content, and the focused point was 3 m from the ground and 0.56 m from the pile core. As shown in Figure 8, it can be noted that the compaction coefficient gradually decreased as the soil moisture content increased. When the soil moisture content exceeded 20%, the compaction coefficient decreased sharply with increases in soil moisture content.

3.2.4. Pile Spacing

The pile spacing was 4D of the fixed value in the SP−PSC pile group tests. In this section, the influence of pile spacing on the soil compaction coefficient among the pile group is systematically studied using the finite element method. In the numerical analyses, the pile spacing gradually increased from 1D to 8D, and the pile diameter, pile length, and soil moisture content were fixed at 0.56 m, 7 m, and 18%, respectively. As two special points in the pile group, the soil compaction coefficients at the centroid and the edge midpoint are given in the following.

Figure 9 and Figure 10 illustrate the variation in the soil compaction coefficients at the centroid and the edge midpoint, and the pile spacing, in which the focused points were taken at the place 3 m from the ground, where the abscissa is the multiple of the pile diameter. It can be found from Figure 9 and Figure 10 that the soil compaction coefficients gradually decreased with increases in the pile spacing, but the rate of decrease gradually decreased. When the pile spacing is greater than 3, the distance between the centroid (or edge midpoint) of the SP−PSC pile group and the compaction piles is relatively large. Therefore, based on the Saint-Venant principle, the impact of the pile spacing on the compaction coefficient is insignificant when the pile spacing exceeds 3. Comparing the soil compaction coefficients at the centroid and the edge midpoint, it is obvious that the compaction coefficient at the centroid is smaller than that at the edge midpoint, which is because the centroid is the farthest from each pile core.

4. Analysis and Discussion

Based on the results of the SP−PSC pile group tests and the finite element analyses, the mechanical models of the soil compaction coefficients at the centroid and the edge midpoint of the pile group are analyzed and subsequently discussed.

4.1. Pile Diameter

Using theoretical analysis and nonlinear regression analysis, the mechanical model of the change in the compaction coefficient with changes in the pile diameter was proposed as follows based on the numerical results of the change in the pile diameter (Figure 6):

$${\eta }_{\mathrm{c}}=1-\frac{0.28}{1+e^{\frac{D-0.3}{0.11}}} \left(D\ge 0\right)$$

(2)

As shown in Figure 6, the results of Equation (2) are consistent with the test and numerical results. When the pile diameter D equals zero, that is, there is actually no pile penetration, the soil compaction coefficient should theoretically be the compaction coefficient of undisturbed soil, that is, 0.74. By Equation (2), the model result of the soil compaction coefficient corresponding to zero pile diameter is also 0.74. On the other hand, when the pile diameter D tends to infinity, the soil compaction coefficient should theoretically be 1. The model value of the soil compaction coefficient corresponding to an infinite pile diameter derived from Equation (2) is also 1.

4.2. Pile Length

Based on the relationship between the soil compaction coefficient and the pile length (Figure 7), the mechanical model of soil compaction coefficient changing with pile length was proposed as follows:

$${\eta }_{\mathrm{c}}=0.98-\frac{0.33}{1+e^{\frac{H-2.27}{2.19}}} \left(H\ge 0\right)$$

(3)

It can be seen from Figure 7 that the results of Equation (3) are in good agreement with the test and numerical results. When the pile length equals zero, the soil compaction coefficient theoretically equals 0.74, that is, the compaction coefficient of undisturbed soil. When the pile length tends to infinity, the soil compaction coefficient theoretically equals a fixed value close to 1 based on Saint-Venant’s principle. On the other hand, based on Equation (3), the soil compaction coefficients corresponding to zero and infinite pile lengths are equal to 0.74 and 0.98, respectively.

4.3. Soil Moisture Content

The mechanical model of the soil compaction coefficient changing with the soil moisture content was proposed as Equation (4) based on the test and numerical results (as shown in Figure 8).

$${\eta }_{\mathrm{c}}=0.99-\frac{6.92}{{10}^5}{\mathrm{e}}^{0.33\omega }\left(25\ge \omega \ge 10\right)$$

(4)

As shown in Figure 8, it can be observed that the results of Equation (4) are consistent with the test and numerical results.

4.4. Pile Spacing

Figure 9 and Figure 10 show the relationship between the soil compaction coefficient and the pile spacing. However, due to possible coupling between the pile diameter and the pile spacing, in order to investigate the sensitivity of the influence of the pile spacing on the compaction coefficient to the pile diameter, three groups of supplementary numerical analyses (D = 0.5, 0.6, and 0.7 m) were carried out. It can be found from Figure 9 and Figure 10 that the variation in the pile diameter D has no effect on the variation rule of the soil compaction coefficient with the pile spacing.

The parameter $n$ was defined as the ratio of the pile spacing to the pile diameter, and the soil compaction coefficients at the centroid and the edge midpoint were defined as ${\eta }_{\mathrm{cc}}$ and ${\eta }_{\mathrm{cm}}$, respectively. Based on the SP−PSC pile group tests and numerical analyses, the mechanical model of the change in the soil compaction coefficient at the centroid and the edge midpoint with the change in the pile spacing was obtained as Equations (5) and (6).

$${\eta }_{\mathrm{cc}}=0.74+\frac{4.41}{1+e^{\frac{n+1.33}{0.86}}} \left(n\ge 1\right)$$

(5)

$${\eta }_{\mathrm{cm}}=0.74+\frac{69}{1+e^{\frac{n+13.65}{2.63}}} \left(n\ge 1\right)$$

(6)

Based on theoretical analysis, when the pile spacing n tends to 1, the soil compaction coefficients at the centroid and the edge midpoint should equal to 1. Furthermore, when the pile spacing n tends to infinity, based on Saint-Venant’s principle, the soil compaction coefficients at the centroid and the edge midpoint should equal 0.74, which is the compaction coefficient of undisturbed soil. According to the presented mechanical models of the soil compaction coefficient, it can be found that the soil compaction coefficients at the centroid and the edge midpoint are 1 and 0.74, corresponding to $n=0$ and $n~\infty$, respectively.

4.5. Depth to Ground

It can be observed from Table 4 that the soil compaction coefficients at the centroid and the edge midpoint are the largest in the middle of the pile length, and gradually decrease towards both sides. In this section, based on the results of numerical analysis, the variations in soil compaction coefficients at the centroid and the edge midpoint with the change in the depth to ground are given as shown in Figure 11 and Figure 12.

The parameter $m$ was taken as the ratio of the depth from the focused point to ground to the pile length. The mechanical models of the change in the soil compaction coefficient at the centroid and the edge midpoint with the change in the parameter $m$ were derived as Equations (7) and (8) from the SP−PSC pile group test and numerical results (as shown in Figure 11 and Figure 12).

$${\eta }_{\mathrm{cc}}=-0.14{\left(m-0.46\right)}^2+0.79 \left(1\ge m\ge 0\right)$$

(7)

$${\eta }_{\mathrm{cm}}=-0.36{\left(m-0.46\right)}^2+0.84 \left(1\ge m\ge 0\right)$$

(8)

According to Equations (7) and (8), it can be noted that the soil compaction coefficients at the centroid and the edge midpoint reach the maximum values in the depth direction when $m=0.46$.

4.6. Soil Compaction Coefficient Model

In view of the mutual independence of parameters $D$, $H$, $\omega$, $n$, and $m$, the five mechanical models of soil compaction coefficient derived above were coupled, and the general mechanical models of the soil compaction coefficient at the centroid and the edge midpoint were derived as Equations (9) and (10).

$$\begin{gathered} {\eta }_{\mathrm{cc}}=1.45\left(1-\frac{0.28}{1+e^{\frac{D-0.3}{0.11}}}\right)\left(0.98-\frac{0.33}{1+e^{\frac{H-2.27}{2.19}}}\right)\left(0.99-\frac{6.92}{{10}^5}{e}^{0.33\omega }\right) \\ \left(0.74+\frac{4.41}{1+e^{\frac{n+1.33}{0.86}}}\right)\left[-0.14{\left(m-0.46\right)}^2+0.79\right] \\ \left(D\ge 0, H\ge 0, 25\ge \omega \ge 10, n\ge 0, 1\ge m\ge 0\right) \end{gathered}$$

(9)

$$\begin{gathered} {\eta }_{\mathrm{cm}}=1.34\left(1-\frac{0.28}{1+e^{\frac{D-0.3}{0.11}}}\right)\left(0.98-\frac{0.33}{1+e^{\frac{H-2.27}{2.19}}}\right)\left(0.99-\frac{6.92}{{10}^5}{e}^{0.33\omega }\right) \\ \left(0.74+\frac{69}{1+e^{\frac{n+13.65}{2.63}}}\right)\left[-0.36{\left(m-0.46\right)}^2+0.84\right] \\ \left(D\ge 0, H\ge 0, 25\ge \omega \ge 10, n\ge 0, 1\ge m\ge 0\right) \end{gathered}$$

(10)

The comparisons between the presented mechanical model results and the test and numerical simulation results are shown in

Table 5. Comparison between model results and test and numerical results at the centroid.

H (m)	D (m)	ω (%)	n	m	Test/FEM	Model Results	Error (Test/FEM)
7	0.56	18	2	3/7	0.820(FEM)	0.846	3.12%(FEM)
6	0.56	18	4	3/7	0.762(FEM)	0.750	−1.59%(FEM)
8	0.56	18	4	3/7	0.778(FEM)	0.773	−0.67%(FEM)
7	0.5	18	4	3/7	0.760(FEM)	0.751	−1.12%(FEM)
7	0.6	18	4	3/7	0.758(FEM)	0.769	1.42%(FEM)
7	0.56	16	4	3/7	0.759(FEM)	0.773	1.89%(FEM)
7	0.56	20	4	3/7	0.741(FEM)	0.744	0.38%(FEM)
7	0.56	18	3	3/7	0.791(FEM)	0.783	−0.98%(FEM)
7	0.56	18	4	3/7	0.786/0.789	0.763	−2.26%/−3.33%
7	0.56	18	4	1/7	0.769/0.772	0.750	−2.49%/−2.92%
7	0.56	18	4	5/7	0.775/0.773	0.755	−2.62%/−2.32%

and

Table 6. Comparison between model results and test and numerical results at the edge midpoint.

H (m)	D (m)	ω (%)	n	m	Test/FEM	Model Results	Error (Test/FEM)
7	0.56	18	2	3/7	0.899(FEM)	0.920	2.35%(FEM)
6	0.56	18	4	3/7	0.822(FEM)	0.810	−1.45%(FEM)
8	0.56	18	4	3/7	0.826(FEM)	0.835	1.06%(FEM)
7	0.5	18	4	3/7	0.831(FEM)	0.825	−0.76%(FEM)
7	0.6	18	4	3/7	0.809(FEM)	0.812	0.36%(FEM)
7	0.56	16	4	3/7	0.838(FEM)	0.841	0.41%(FEM)
7	0.56	20	4	3/7	0.795(FEM)	0.804	1.08%(FEM)
7	0.56	18	3	3/7	0.856(FEM)	0.863	0.82%(FEM)
7	0.56	18	4	3/7	0.835/0.842	0.825	−1.29%/−2.12%
7	0.56	18	4	1/7	0.792/0.789	0.789	−0.34%/0.04%
7	0.56	18	4	5/7	0.811/0.806	0.802	−0.53%/−1.12%

. It can be found from Table 5 and Table 6 that the proposed model results, in which the errors are basically within 5%, are in good agreement with the test and numerical results.

5. Conclusions

This paper systematically studied the distribution of stress and the compaction coefficient of the soil within SP−PSC pile groups during construction based on tests and finite element analyses. The influence of different factors, such as the pile diameter, the pile length, the soil moisture content, the pile spacing, and the depth to ground, on the soil compaction coefficient was studied and discussed. Then, the simplified mechanical models of the soil compaction coefficient at the centroid and the edge midpoint of the pile group were established based on theoretical analysis and multiple nonlinear regression analysis. The following conclusions can be obtained:

(1)

The compaction coefficient of soils gradually increased with increasing the pile diameter, and the rate of increase first increased and then decreased. The variation range from 0.2 to 0.6 m was the sensitive area of the change in the pile diameter to the increase in soil compaction coefficient.

(2)

The compaction coefficient of soils gradually increased with increases in the pile length, but the rate of increase gradually decreased.

(3)

The soil compaction coefficient gradually decreased as the soil moisture content increased. When the soil moisture content exceeded 20%, the compaction coefficient decreased sharply with increases in soil moisture content.

(4)

The soil compaction coefficient gradually decreased with increases in the pile spacing, but the rate of decrease gradually decreased. It is obvious that the compaction coefficient at the centroid is smaller than that at the edge midpoint.

(5)

The soil compaction coefficients at the centroid and the edge midpoint were the largest near the middle of the pile length ($\mathrm{m}=0.46$), and gradually decreased towards both sides.

Based on the proposed mechanical models, the soil compaction coefficients at the centroid and the edge midpoint of the pile group under various conditions can be obtained. This can provide theoretical and technical guidance for the design and construction of SP−PSC pile groups on collapsible loess foundations.