# Estimating the Uplift Bearing Capacity of Belled Piers Adjacent to Sloping Ground by Numerical Modeling Based on Field Tests

Authors: Yangchun Han; Jiulong Cheng; Weifeng Zheng; Shijun Ding
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

In order to evaluate the uplift bearing capacity of belled piers beside slopes, a series of numerical simulations are carried out based on field tests data. First, a number of uplift loading tests of full-scale belled piers are carried out on the project site of transmission line in Anhui Province, China. Second, a slope-foundation model for numerical modeling is proposed and calibrated based on field tests data. The behavior of belled piers adjacent to slopes subject to uplift load is studied by numerical modeling. The impact of three parameters, including distance (a) from the belled pier to the crest of the slope, slope angle (β), and embedment depth (h) of the belled pier, has been investigated on the uplift capacity of the belled pier. Based on the simulation results, an attenuation coefficient (ω) is put forward for evaluating the reduction of uplift bearing capacity of the belled pier. The results show that the coefficient ω is negatively correlated with distance a and depth h, and the influence of distance a is greater than that of depth h according to the results of variance analysis, but the difference is not significant by F test. Moreover, the empirical equation between attenuation coefficient ω and three key factors a, β, and h had been presented by a series of fitting.


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## Body

## 1. Introduction

The rapid development of national economy puts forward higher demand for the construction of transmission lines. When the lines pass through mountainous and hilly areas, the foundations of line towers must be built at or near a sloping ground [1]. In either case, the existence of slopes may cause the attenuation of load-bearing capacity of foundations. The usual engineering solution is to increase the depth of the foundation while keeping the foundation as far away from the sloping ground as possible. However, due to the constraints of site topographic conditions and engineering costs, scientific analysis should be done to ensure the implementation of these measures. Therefore, it is of great significance to investigate the effect of slopes on the bearing capacity of adjacent foundations.Many theoretical, numerical, and experimental results have been proposed to determine the load-bearing capacity of the footings adjacent to slopes or resting on slope surfaces [2–9]. Based on the results of slope model footing tests, Castelli and Lentini [2, 10] studied the influence of the slope distance on the bearing capacity of the foundation and determined the modified bearing capacity factors. However, most of the literatures mentioned above focus on the compressive performance of foundations, and only a few focuses on the uplift or lateral bearing capacity of the foundations embedded in sloping ground. Jiang et al. [11] studied the behavior of piles in sloping ground under undrained lateral loading conditions by finite element analysis. In addition, the representative field tests were completed by Lu et al. [12] and Qian et al. [13]. Based on two full-scale uplift load tests conducted on belled piers in the arid loess sloping ground, Lu et al. discussed the uplift resistance of foundation on slope from the characteristics of load-displacement curve, failure mode of soil, and ultimate uplift load. Similarly, based on the comparative in situ load test results of two types of foundations under combined uplift and lateral loads on a slope, Qian et al. analyzed the uplift and lateral bearing capacity of the foundations. The foundations studied by these two scholars are constructed on the sloping ground, without considering the situation when the foundations are embedded near the slopes, which is inevitable in the process of tower foundation construction. Generally, increasing the distance between the foundation and the slope or embedding depth can increase the uplift resistance to match the unabated bearing capacity of the foundations that are far from slopes. And there are many qualitative analyses in this regard despite few quantitative ones.The engineering background of this paper is a 220 kV transmission line which will pass through the mountainous and hilly areas in southern Anhui Province, China. It is necessary to evaluate the influence of slopes on the bearing capacity of foundations. However, the field tests of bearing capacity were carried out only on the foundations embedded in horizontal ground far from slopes instead of on the belled piers embedded near sloping ground for various reasons. Therefore, according to field test data, the numerical simulations were performed to evaluate the uplift bearing capacity of belled piers near slopes in this area. At the beginning of this research, five belled piers were tested under axial uplift load on a flat ground far from slopes. Based on the data of field tests, a 3D slope-foundation model was developed, calibrated, and validated. The uplift performance of belled piers adjacent to sloping ground was simulated using the model. The simulation results were analyzed, and the attenuation coefficientω was taken as the index to quantitatively evaluate the uplift resistance of belled pier from three aspects: the distance a between belled pier and slope, the embedment depth h of the belled pier, and the slope angle β. These results will provide sufficient theoretical and data support for the construction of transmission line foundation in mountainous and hilly areas.


## 2. Experimental Study

In this section, the field tests of five full-scale belled piers under uplift loading are described. First of all, in order to determine the types and geomechanical properties of rock samples in the test area, a series of in situ tests and laboratory tests were carried out. Subsequently, the pouring and maintenance process of reinforced concrete foundation in site and the method and process of the loading test are illustrated. Finally, four commonly used failure criteria are introduced. One of them is selected through comparison to determine the ultimate uplift bearing capacity of foundation.


### 2.1. Test Site and Test Description

The field tests were carried out at a site along the 220 kV Huizhou–Wannan transmission line in Anhui Province, China. According to the preliminary investigation results of the project, the total length of the line is 35 km, with 91% in the mountains and 9% in the hills, respectively. Hence, the tower foundations of this line are mostly embedded in or near sloping ground. It is of great practical value to study the influence of slopes on bearing characteristics of adjacent foundations.To determine the rock properties of the test site, a series of laboratory and in situ tests were performed according to relevant standards [14–18], including rock identification, microstructure, moisture content, specific gravity, point loading test, and shear strength measurements. The test results are listed in Table 1. Rock samples from the test site are eventually identified as strongly weathered lithic quartz complex sandstone [19]. In addition, the in situ direct shear test and triaxial compression test in laboratory were carried out to determine the mechanical strength of rock samples.Table 1
Geotechnical properties of rock samples in the field test area.
Category
G
s (kN/m3)
w (%)
c (kPa)
φ (°)
Uniaxial saturated compressive strength of rocksRc (MPa)
Sandstone
17.4∼18.8
22.01∼32.01
22.45∼48.02
20.56∼33.82
4.97In situ direct shear tests were performed on the rock specimens in the pit shown in Figure1(a). The whole testing process was carried out according to the criteria proposed by ASTM D4554-12 [14], and the field diagram of the test instrument is shown in Figure 1(b). During the test, the loading device placed horizontally can provide shear force parallel to the preset shear plane of the specimen. As shown in Figure 1(a), four specimens were designed, each with a length, width, and height of 0.6 m, 0.6 m, and 0.4 m, respectively. The test results show that the peak shear stress is inversely proportional to the normal stress. The fitting parameters of cohesion and friction angle were calculated by the linear fitting method. The final results are listed in Table 1.Figure 1
In situ direct shear testing: (a) pit exploration illustration; (b) shearing test in progress.
(a)
(b)Meanwhile, laboratory triaxial compression tests were carried out on the rock samples. The undisturbed rock samples were obtained by drilling holes in the test site. Three sets of samples were taken at the depth of 1.0 m, 2.0 m, and 4.0 m which are tagged as A, B, and C, respectively. The original samples were processed into cylinders of suitable size to facilitate triaxial compression tests. Finally, the shear strength index of the test samples was obtained based on the Mohr–Coulomb theory. The test results are listed in Table2.Table 2
The results of triaxial compression tests on the rock samples in natural state.
Confining pressure (kPa)
Sample ID
Ultimate compressive strength (MPa)
Mean value (MPa)
Cohesion (MPa)
Internal friction angle (°)
100
A1
1.864
1.623
B1
1.754
C1
1.250
200
A2
2.235
2.023
0.407
29
B2
1.961
C2
1.874
300
A3
2.442
2.292
B3
2.381
C3
2.052


### 2.2. Foundation Installation Process

During the operation of transmission line tower foundation, due to wind and other factors, the foundation often bears the combined loads, including up-pull, down-compress, and lateral-push [20]. The uplift load has been proved to be the key design load of tower foundation [21]. Hence, as shown in Figure 2, the uplift loading test was conducted on five belled piers (named BP1∼BP5) in the pilot area. The shape and symbols of the belled piers are shown in Table 3.Figure 2
(a) Layout chart and (b) photo indicating the arrangement of belled piers within the implementation site.
(a)
(b)Table 3
Geometric dimensions of in situ test foundations.
No.
d (m)
b (m)
h (m)
e (m)
BP1
0.8
1.1
1
0.2
BP2
0.8
1.25
1.5
0.2
BP3
0.8
1.55
2.5
0.2
BP4
0.8
1.7
3
0.2
BP5
0.8
1.7
4
0.2
d is the top diameter; b is the bottom diameter; h is the embedment depth; e is the outcrop height.Similar to other field tests of cast-in-place concrete foundations [12, 13, 22, 23], the construction process of test foundations can be divided into three steps:(1)
The contour of the foundation is excavated from top to bottom slowly according to the design size. This step requires attention to the influence of groundwater. At the same time, in order to achieve the construction accuracy, dimensions are checked every 0.5 m in the process of downward excavation.(2)
The reinforcing frames of belled piers in the excavated contours are erected. The frame of the reinforcing cage is composed of vertical D28 (diameter = 28 mm) steel bar and transverse D14 (diameter = 14 mm) steel bar. The strength grade of the single steel bar is HRB335 (nominal yield stressfy = 335 MPa). Six D64 (diameter = 64 mm) high-strength alloy steel anchor bolts are symmetrically arranged on the top of the foundation along a circle of 0.5 m diameter. The specified yield strength and ultimate strength of a single bolt are 930 MPa and 1,080 MPa, respectively. The length of the anchor bolt in concrete increases with the depth of foundation, and the length of the anchor bolt remained outside is 0.7 m when loading ends.(3)
Pouring ready-mixed sulfate-resistant concrete into the reinforcing frames, the loading test can be carried out after 28 days of maintenance.


### 2.3. Field Load Test

Figure3 shows the schematic diagram of the axial uplift loading test. As illustrated in Figure 3(b), the clear distance between the reaction concrete blocks is 10 m, which is sufficient to avoid influencing the test results. For the shaft-head displacement measurement, two electronic displacement sensors are placed symmetrically on the top of the pier along the same diameter, as shown in Figure 3(a). All the electronic displacement sensors are attached to the reference beams, as shown in Figure 3(b). The reference beams are independent from test foundation and loading device to ensure the stability of electronic displacement sensors.Figure 3
Schematic diagram of the axial uplift loading test: (a) schematic diagram of the loading system; (b) uplift loading test in progress.
(a)
(b)The loading process was performed with static monotonic loading method without cycling, which is the typical procedure recommended by Chinese GB50007 [24]. The specific process is as follows: first, the initial load is applied at 20% of the predicted ultimate uplift load, and the incremental load per stage is half of the initial load; second, each load increment is maintained after loading until two continuous displacements are less than 0.1 mm per hour; finally, the loading process is repeated until the displacement surge occurs, at which point the foundation is considered to be completely unstable.


### 2.4. Test Results and Failure Criteria

Once the field tests are completed, the load-displacement curves of the five belled piers are counted as shown in Figure4. The characteristics of these curves are similar to those of the previous tests, including belled piers under axial uplift loading in Gobi gravel or loess sloping ground [12, 13, 22, 23], micropiled raft under uplift, and compression loading in cohesive soil [25]. Similarly, the approximation of these load-displacement curves can be divided into three stages, as shown in Figure 5: (1) an initial linear stage, (2) a curvilinear transition stage, and (3) a final linear stage. With different failure criteria, the value of ultimate bearing capacity is determined at different stages.Figure 4
The uplift load-displacement curves for BP1 to BP5.Figure 5
Sections of the load-displacement curve for theL1–L2 method.Currently, vast amount of literature has proposed various definition and criterion of ultimate bearing capacity. Fellenius [26] defined it as a point on the load-displacement curve for which rapid displacement takes place due to a little increase in the loading. In this subsection, four common and suitable approaches have been introduced to determine the uplift capacity of test piers, including mathematical modeling method [27, 28], slope tangent method [29], tangent intersection method [30], and L1–L2 method [31–33]. Table 4 describes these methods and lists the corresponding values. The values determined by the mathematical modeling method and tangent intersection method can be considered as upper and lower limits of ultimate bearing capacity, respectively. The results of the slope tangent method are close, despite more conservative, to that of the L1–L2 method, which may lead to waste of bearing capacity of foundation. Therefore, the L1–L2 method is chosen as the unified failure criterion in this paper.Table 4
Failure criteria for determining the uplift bearing capacity of test foundations.
Method
Failure load definition
Reference
Ultimate uplift loadT (kN)
BP1
BP2
BP3
BP4
BP5
Mathematical modeling method
For the mathematical fitting curve of the target curve, the tangent line is taken at final linear region and the tangent point is the value point
van der Veen [28]
230
450
1150
1700
2400
Chin [27]
Slope tangent method
Load corresponding to the tangent point of the initial linear stage
O’Rourke and Kulhawy [29]
120
300
550
1000
1400
Tangent intersection method
Load corresponding to the tangent intersection of initial and final linear stages
Housel [30]
200
430
1110
1480
2140
Tomlinson [34]
L
1–L2 method
Load corresponding to the tangent point of the final linear stage
Hirany and Kulhawy [31–33]
230
450
1250
1600
2400


## 2.1. Test Site and Test Description

The field tests were carried out at a site along the 220 kV Huizhou–Wannan transmission line in Anhui Province, China. According to the preliminary investigation results of the project, the total length of the line is 35 km, with 91% in the mountains and 9% in the hills, respectively. Hence, the tower foundations of this line are mostly embedded in or near sloping ground. It is of great practical value to study the influence of slopes on bearing characteristics of adjacent foundations.To determine the rock properties of the test site, a series of laboratory and in situ tests were performed according to relevant standards [14–18], including rock identification, microstructure, moisture content, specific gravity, point loading test, and shear strength measurements. The test results are listed in Table 1. Rock samples from the test site are eventually identified as strongly weathered lithic quartz complex sandstone [19]. In addition, the in situ direct shear test and triaxial compression test in laboratory were carried out to determine the mechanical strength of rock samples.Table 1
Geotechnical properties of rock samples in the field test area.
Category
G
s (kN/m3)
w (%)
c (kPa)
φ (°)
Uniaxial saturated compressive strength of rocksRc (MPa)
Sandstone
17.4∼18.8
22.01∼32.01
22.45∼48.02
20.56∼33.82
4.97In situ direct shear tests were performed on the rock specimens in the pit shown in Figure1(a). The whole testing process was carried out according to the criteria proposed by ASTM D4554-12 [14], and the field diagram of the test instrument is shown in Figure 1(b). During the test, the loading device placed horizontally can provide shear force parallel to the preset shear plane of the specimen. As shown in Figure 1(a), four specimens were designed, each with a length, width, and height of 0.6 m, 0.6 m, and 0.4 m, respectively. The test results show that the peak shear stress is inversely proportional to the normal stress. The fitting parameters of cohesion and friction angle were calculated by the linear fitting method. The final results are listed in Table 1.Figure 1
In situ direct shear testing: (a) pit exploration illustration; (b) shearing test in progress.
(a)
(b)Meanwhile, laboratory triaxial compression tests were carried out on the rock samples. The undisturbed rock samples were obtained by drilling holes in the test site. Three sets of samples were taken at the depth of 1.0 m, 2.0 m, and 4.0 m which are tagged as A, B, and C, respectively. The original samples were processed into cylinders of suitable size to facilitate triaxial compression tests. Finally, the shear strength index of the test samples was obtained based on the Mohr–Coulomb theory. The test results are listed in Table2.Table 2
The results of triaxial compression tests on the rock samples in natural state.
Confining pressure (kPa)
Sample ID
Ultimate compressive strength (MPa)
Mean value (MPa)
Cohesion (MPa)
Internal friction angle (°)
100
A1
1.864
1.623
B1
1.754
C1
1.250
200
A2
2.235
2.023
0.407
29
B2
1.961
C2
1.874
300
A3
2.442
2.292
B3
2.381
C3
2.052


## 2.2. Foundation Installation Process

During the operation of transmission line tower foundation, due to wind and other factors, the foundation often bears the combined loads, including up-pull, down-compress, and lateral-push [20]. The uplift load has been proved to be the key design load of tower foundation [21]. Hence, as shown in Figure 2, the uplift loading test was conducted on five belled piers (named BP1∼BP5) in the pilot area. The shape and symbols of the belled piers are shown in Table 3.Figure 2
(a) Layout chart and (b) photo indicating the arrangement of belled piers within the implementation site.
(a)
(b)Table 3
Geometric dimensions of in situ test foundations.
No.
d (m)
b (m)
h (m)
e (m)
BP1
0.8
1.1
1
0.2
BP2
0.8
1.25
1.5
0.2
BP3
0.8
1.55
2.5
0.2
BP4
0.8
1.7
3
0.2
BP5
0.8
1.7
4
0.2
d is the top diameter; b is the bottom diameter; h is the embedment depth; e is the outcrop height.Similar to other field tests of cast-in-place concrete foundations [12, 13, 22, 23], the construction process of test foundations can be divided into three steps:(1)
The contour of the foundation is excavated from top to bottom slowly according to the design size. This step requires attention to the influence of groundwater. At the same time, in order to achieve the construction accuracy, dimensions are checked every 0.5 m in the process of downward excavation.(2)
The reinforcing frames of belled piers in the excavated contours are erected. The frame of the reinforcing cage is composed of vertical D28 (diameter = 28 mm) steel bar and transverse D14 (diameter = 14 mm) steel bar. The strength grade of the single steel bar is HRB335 (nominal yield stressfy = 335 MPa). Six D64 (diameter = 64 mm) high-strength alloy steel anchor bolts are symmetrically arranged on the top of the foundation along a circle of 0.5 m diameter. The specified yield strength and ultimate strength of a single bolt are 930 MPa and 1,080 MPa, respectively. The length of the anchor bolt in concrete increases with the depth of foundation, and the length of the anchor bolt remained outside is 0.7 m when loading ends.(3)
Pouring ready-mixed sulfate-resistant concrete into the reinforcing frames, the loading test can be carried out after 28 days of maintenance.


## 2.3. Field Load Test

Figure3 shows the schematic diagram of the axial uplift loading test. As illustrated in Figure 3(b), the clear distance between the reaction concrete blocks is 10 m, which is sufficient to avoid influencing the test results. For the shaft-head displacement measurement, two electronic displacement sensors are placed symmetrically on the top of the pier along the same diameter, as shown in Figure 3(a). All the electronic displacement sensors are attached to the reference beams, as shown in Figure 3(b). The reference beams are independent from test foundation and loading device to ensure the stability of electronic displacement sensors.Figure 3
Schematic diagram of the axial uplift loading test: (a) schematic diagram of the loading system; (b) uplift loading test in progress.
(a)
(b)The loading process was performed with static monotonic loading method without cycling, which is the typical procedure recommended by Chinese GB50007 [24]. The specific process is as follows: first, the initial load is applied at 20% of the predicted ultimate uplift load, and the incremental load per stage is half of the initial load; second, each load increment is maintained after loading until two continuous displacements are less than 0.1 mm per hour; finally, the loading process is repeated until the displacement surge occurs, at which point the foundation is considered to be completely unstable.


## 2.4. Test Results and Failure Criteria

Once the field tests are completed, the load-displacement curves of the five belled piers are counted as shown in Figure4. The characteristics of these curves are similar to those of the previous tests, including belled piers under axial uplift loading in Gobi gravel or loess sloping ground [12, 13, 22, 23], micropiled raft under uplift, and compression loading in cohesive soil [25]. Similarly, the approximation of these load-displacement curves can be divided into three stages, as shown in Figure 5: (1) an initial linear stage, (2) a curvilinear transition stage, and (3) a final linear stage. With different failure criteria, the value of ultimate bearing capacity is determined at different stages.Figure 4
The uplift load-displacement curves for BP1 to BP5.Figure 5
Sections of the load-displacement curve for theL1–L2 method.Currently, vast amount of literature has proposed various definition and criterion of ultimate bearing capacity. Fellenius [26] defined it as a point on the load-displacement curve for which rapid displacement takes place due to a little increase in the loading. In this subsection, four common and suitable approaches have been introduced to determine the uplift capacity of test piers, including mathematical modeling method [27, 28], slope tangent method [29], tangent intersection method [30], and L1–L2 method [31–33]. Table 4 describes these methods and lists the corresponding values. The values determined by the mathematical modeling method and tangent intersection method can be considered as upper and lower limits of ultimate bearing capacity, respectively. The results of the slope tangent method are close, despite more conservative, to that of the L1–L2 method, which may lead to waste of bearing capacity of foundation. Therefore, the L1–L2 method is chosen as the unified failure criterion in this paper.Table 4
Failure criteria for determining the uplift bearing capacity of test foundations.
Method
Failure load definition
Reference
Ultimate uplift loadT (kN)
BP1
BP2
BP3
BP4
BP5
Mathematical modeling method
For the mathematical fitting curve of the target curve, the tangent line is taken at final linear region and the tangent point is the value point
van der Veen [28]
230
450
1150
1700
2400
Chin [27]
Slope tangent method
Load corresponding to the tangent point of the initial linear stage
O’Rourke and Kulhawy [29]
120
300
550
1000
1400
Tangent intersection method
Load corresponding to the tangent intersection of initial and final linear stages
Housel [30]
200
430
1110
1480
2140
Tomlinson [34]
L
1–L2 method
Load corresponding to the tangent point of the final linear stage
Hirany and Kulhawy [31–33]
230
450
1250
1600
2400


## 3. Numerical Modeling

### 3.1. Slope-Foundation Model

Given the goal of this paper, a numerical simulation scheme is proposed and illustrated in Figure6. The 3D sketch of the slope-foundation model is shown in Figure 7, where a is the distance from the edge of the belled pier to the crest of the slope, d is the top diameter of the belled pier, h is the embedment depth of the belled pier, and β is the slope angle. Considering the symmetry of the current model, the half-model is chosen to improve computational efficiency. Secondly, the model of a = 0.6 m is selected initially, whereas the increment of a is 0.5 m each time. The criterion of stopping simulation is that the uplift bearing capacity of model n reaches the unabated performance of the belled pier far from a slope. It should be pointed out particularly that in order to obtain the unabated bearing capacity of each foundation, distance a is set as a large value in the slope-foundation model to simulate the performance of the belled pier far away from the slope. After the preliminary debugging and verification, it is determined that a = 10 m can meet this requirement.Figure 6
Flow chart of the numerical analysis scheme.Figure 7
Geometrical dimensions of the slope-foundation model.


### 3.2. Geometric Modeling

All the meshes in the simulation are established using the built in grid editor of FLAC 3D software. To set boundary conditions, the recommendations by Choi et al. [35] and Zekavati et al. [25] are considered. In this simulation, as illustrated in Figure 7, the distance from the edge of the belled pier to the surface JAEF is set to 10 m after balancing the efficiency and correctness of calculation. Considering the vertical direction of loading, the distance from the bottom of belled pier to the surface EDGF is set to 2 m. Except for the top surface JIAB and IHCB, the displacements and velocities of the nodes in each surface are fixed. In addition, several debugging calculations are carried out to eliminate the influence of meshing and boundary effect on simulation accuracy. Figure 8 depicts a 3D mesh for the slope-foundation model of a = 2.1 m, h = 3 m, and β = 45°.Figure 8
A 3D mesh for the half-model ofa = 2.1 m, h = 3 m, and β = 45°.


### 3.3. Material Modeling

Material modeling involves setting up parameters of two materials, concrete and rock mass, and the interface between them. Accordingly, concrete foundation is set as linear elastic material, whereas strongly weathered rock is modeled as elastic-plastic continuous material according to the failure criterion of Mohr–Coulomb. The physical and mechanical parameters of rock and reinforced concrete are determined by in situ and laboratory tests (as listed in Tables1 and 2). It is necessary to point out that the foundations are buried in the shallow surface with less confining pressure. Therefore, it is more reasonable to choose the results of field direct shear tests listed in Table 1 as the simulation parameters. The selected values are shown in Table 5.Table 5
Basic physical and mechanical parameters of rock and reinforced concrete.
Material
γ (kN/m3)
E (GPa)
Poisson’s ratio
c (kPa)
φ (°)
Rock slope
18
0.34
0.3
48
25
Reinforced concrete foundation
25
30
0.2
—
—The intersection of the foundation zones and the rock zones is set as “hard” contact to limit the interlacing in between and prevent tension forming along the intersection. The specific way is to set the normal stiffness (kn) and tangential stiffness (ks) to larger values and set them to 10 times the smaller elastic modulus of both sides according to the manual of FLAC 3D software, i.e., 3.4 GPa/m. Meanwhile, the cohesion and friction angle of the interface are set to 40 kPa and 20°, respectively.


### 3.4. Calibration for Slope-Foundation Model

In this section, according to the data of field tests of five belled piers, the limited aspects of the slope-foundation model are calibrated, including material parameters, model size, mesh generation, and interface parameters. The specific process is to establish five slope-foundation models witha = 10 m according to the size of five-field test foundations and simulate loading. The calibration criteria refer to the scheme proposed in [25], which is used to evaluate the fitting accuracy of numerical simulations and field tests of micropile. There are two criteria: (1) the coincidence of load-displacement curve and (2) the relative error of ultimate uplift load. The verification results between the field and numerical results are illustrated in Figure 9: the maximum relative error of ultimate bearing capacity is 12.5%, the minimum is 4.3%, and the average is 6.0%. The result means that the numerical simulation results of the slope-foundation model are basically consistent with the field test ones.Figure 9
Calibration results between the numerical modeling and the field loading tests in Anhui site (RE means relative error).


### 3.5. Simulation of Belled Piers Adjacent to Slopes

In this part, eighteen sets of slope-foundation models, which consists of three slope anglesβ (15°, 30°, and 45°) and six embedment depths h (3 m, 4 m, 5 m, 6 m, 8 m, and 10 m), are established and loaded. A total of 126 belled piers are simulated of loading process in corresponding models. The uplift performance of these foundations is analyzed based on the load-displacement curve and ultimate uplift capacity in the next section.


### 3.6. Simulation Results

Figure10 illustrates the simulation results of three groups of models, consisting of one belled pier of h = 3 m and three slope angles of β = 15°, 30°, and 45°. The ultimate uplift load corresponding to each curve is obtained by the L1–L2 method and listed in the tables of Figure 10. For comparison, the model with a = 10 m is established to simulate uplift loading, whereas the ultimate uplift load is taken as the standard value, denoted by Tu. In order to quantitatively analyze the influence of slope on uplift performance of the belled pier, the concept of the attenuation coefficient is defined. The formula is as follows:(1)ω=Tu−TnTu,where ω is the attenuation coefficient, Tu is the ultimate uplift load of the belled pier far away from the slopes, and Tn is the ultimate uplift load of belled pier adjacent to the slopes.Figure 10
Simulation results of slope-foundation models withh of 3 m and β of (a) 15°, (b) 30°, and (c) 45°.
(a)
(b)
(c)The attenuation coefficients for foundations of 3 m embedment depth with three slope angles are shown in Figure10. The remaining fifteen sets of models are processed and analyzed in the same way. The 3D scatter plot of the variation of the attenuation coefficient ω wish respect to distance a and depth h is shown in Figure 11.Figure 11
The 3D sketches of the attenuation coefficientω with respect to distance a and embedment depth h with β of (a) 15°, (b) 30°, and (c) 45°.
(a)
(b)
(c)These data are analyzed qualitatively and quantitatively. First, the following observations can be made: (1) The attenuation coefficientω is negatively correlated with the distance a and the embedment depth h, as shown in Figure 11(a), Curve I and Curve II. (2) The attenuation coefficient ω is positively correlated with the slope angle β. For example, as shown in Figures 11(a)–11(c), with the increase of angle of 15°, 30°, and 45°, the corresponding fluctuation ranges of attenuation coefficient are 0∼0.02, 0∼0.05, and 0∼0.12, respectively. It means that the fluctuation of uplift bearing capacity of the belled pier increases along with the slope angle.The above conclusions are consistent with common sense that (1) the closer the foundation is to the sloping ground, the shallower the embedment depth is and (2) the steeper the slope is, the more obvious the attenuation of the uplift resistance of belled pier is. These qualitative conclusions are straight forward. So it is necessary to carry out further quantitative analysis.


### 3.7. Analysis of Variance and F Test

In this section, based on the data in Figure11, the influence of distance a and depth h on attenuation coefficient ω is compared by analysis of variance. By comparing the standard deviation of a and h, it can be observed which of the two factors has significant influence on the attenuation coefficient ω (the standard deviation of the main factor is greater than that of the minor factor). The calculated results are listed in Table 6. As shown in the table, the standard deviation of distance a is greater than that of embedment depth h, indicating that the former has greater influence on attenuation coefficient than the latter.Table 6
The computational results of variance analysis andF test.
β (°)
Source of variance
Standard deviation
F test
Degree of freedom
F calculated value
F critical value
15
a
0.011
7
1.40
h
0.006
5
30
a
0.123
7
1.53
4.88
h
0.060
5
45
a
0.574
7
1.33
h
0.323
5In order to test whether the difference of the influence is obvious, the joint hypothesis test (F test) is adopted. According to the process of F test, by comparing the F calculated value with the F critical value obtained by querying F table, we can determine whether the effect is significant. And as shown in Table 6, all the calculated values of F are less than the critical value of F, which means that the influence of distance a and depth h on uplift bearing capacity of belled pier is not significantly different.


### 3.8. Gaussian Function Fitting and Empirical Equation

After analyzing the data in Figure11, it is considered that the data conform to the characteristics of two-dimensional Gauss distribution [36]. The surface formed by nonzero points in Figure 11 can be approximated as a part of the surface of a two-dimensional Gauss function where the fitting equation is as follows:(2)ωβ=fa,h=k⋅e−a2/σa2⋅e−h2/σh2,where k is the parameter describing the range of ωβ and σa is the variance of distance a. In practical sense, σa describes the rate at which attenuation coefficient ωβ varies with distance a. σh is the variance of distance h. Similarly, σh describes the rate at which attenuation coefficient ωβ varies with depth h.The fitted parameters are illustrated in Figure12. The R-squared values of the results for three slope angles are 0.92, 0.97, and 0.98, respectively, which means that the fitting accuracy is high, and the choice of the Gauss function is appropriate. Furthermore, it is not difficult to find that the three fitting parameters k, σa, and σh are positively correlated with the slope angle β. Hence, the average values of these three fitting parameters were selected. The linear formulas between the fitting parameters and the slope angle β are fitted, as shown in Figure 12. The R-squared values are 0.98, 0.99, and 0.93, respectively. The empirical equation for calculating attenuation coefficient ω can be obtained by substituting the fitting parameters k, σa, and σh into equation (2), as shown in the following equation:(3)ω=0.0046β−0.0464⋅e−a/0.0694β+7.40672⋅e−h/0.0055β+1.9272,where 15≤β≤45, 0.5≤a≤4.1, and 3≤h≤10.Figure 12
Fitting results of (a)k, (b) σa, and (c) σh.
(a)
(b)
(c)According to the actual topographic conditions in the construction process, equation (3) as a reference can be flexibly applied to the design and site selection of belled piers along the transmission line in this area.


## 3.1. Slope-Foundation Model

Given the goal of this paper, a numerical simulation scheme is proposed and illustrated in Figure6. The 3D sketch of the slope-foundation model is shown in Figure 7, where a is the distance from the edge of the belled pier to the crest of the slope, d is the top diameter of the belled pier, h is the embedment depth of the belled pier, and β is the slope angle. Considering the symmetry of the current model, the half-model is chosen to improve computational efficiency. Secondly, the model of a = 0.6 m is selected initially, whereas the increment of a is 0.5 m each time. The criterion of stopping simulation is that the uplift bearing capacity of model n reaches the unabated performance of the belled pier far from a slope. It should be pointed out particularly that in order to obtain the unabated bearing capacity of each foundation, distance a is set as a large value in the slope-foundation model to simulate the performance of the belled pier far away from the slope. After the preliminary debugging and verification, it is determined that a = 10 m can meet this requirement.Figure 6
Flow chart of the numerical analysis scheme.Figure 7
Geometrical dimensions of the slope-foundation model.


## 3.2. Geometric Modeling

All the meshes in the simulation are established using the built in grid editor of FLAC 3D software. To set boundary conditions, the recommendations by Choi et al. [35] and Zekavati et al. [25] are considered. In this simulation, as illustrated in Figure 7, the distance from the edge of the belled pier to the surface JAEF is set to 10 m after balancing the efficiency and correctness of calculation. Considering the vertical direction of loading, the distance from the bottom of belled pier to the surface EDGF is set to 2 m. Except for the top surface JIAB and IHCB, the displacements and velocities of the nodes in each surface are fixed. In addition, several debugging calculations are carried out to eliminate the influence of meshing and boundary effect on simulation accuracy. Figure 8 depicts a 3D mesh for the slope-foundation model of a = 2.1 m, h = 3 m, and β = 45°.Figure 8
A 3D mesh for the half-model ofa = 2.1 m, h = 3 m, and β = 45°.


## 3.3. Material Modeling

Material modeling involves setting up parameters of two materials, concrete and rock mass, and the interface between them. Accordingly, concrete foundation is set as linear elastic material, whereas strongly weathered rock is modeled as elastic-plastic continuous material according to the failure criterion of Mohr–Coulomb. The physical and mechanical parameters of rock and reinforced concrete are determined by in situ and laboratory tests (as listed in Tables1 and 2). It is necessary to point out that the foundations are buried in the shallow surface with less confining pressure. Therefore, it is more reasonable to choose the results of field direct shear tests listed in Table 1 as the simulation parameters. The selected values are shown in Table 5.Table 5
Basic physical and mechanical parameters of rock and reinforced concrete.
Material
γ (kN/m3)
E (GPa)
Poisson’s ratio
c (kPa)
φ (°)
Rock slope
18
0.34
0.3
48
25
Reinforced concrete foundation
25
30
0.2
—
—The intersection of the foundation zones and the rock zones is set as “hard” contact to limit the interlacing in between and prevent tension forming along the intersection. The specific way is to set the normal stiffness (kn) and tangential stiffness (ks) to larger values and set them to 10 times the smaller elastic modulus of both sides according to the manual of FLAC 3D software, i.e., 3.4 GPa/m. Meanwhile, the cohesion and friction angle of the interface are set to 40 kPa and 20°, respectively.


## 3.4. Calibration for Slope-Foundation Model

In this section, according to the data of field tests of five belled piers, the limited aspects of the slope-foundation model are calibrated, including material parameters, model size, mesh generation, and interface parameters. The specific process is to establish five slope-foundation models witha = 10 m according to the size of five-field test foundations and simulate loading. The calibration criteria refer to the scheme proposed in [25], which is used to evaluate the fitting accuracy of numerical simulations and field tests of micropile. There are two criteria: (1) the coincidence of load-displacement curve and (2) the relative error of ultimate uplift load. The verification results between the field and numerical results are illustrated in Figure 9: the maximum relative error of ultimate bearing capacity is 12.5%, the minimum is 4.3%, and the average is 6.0%. The result means that the numerical simulation results of the slope-foundation model are basically consistent with the field test ones.Figure 9
Calibration results between the numerical modeling and the field loading tests in Anhui site (RE means relative error).


## 3.5. Simulation of Belled Piers Adjacent to Slopes

In this part, eighteen sets of slope-foundation models, which consists of three slope anglesβ (15°, 30°, and 45°) and six embedment depths h (3 m, 4 m, 5 m, 6 m, 8 m, and 10 m), are established and loaded. A total of 126 belled piers are simulated of loading process in corresponding models. The uplift performance of these foundations is analyzed based on the load-displacement curve and ultimate uplift capacity in the next section.


## 3.6. Simulation Results

Figure10 illustrates the simulation results of three groups of models, consisting of one belled pier of h = 3 m and three slope angles of β = 15°, 30°, and 45°. The ultimate uplift load corresponding to each curve is obtained by the L1–L2 method and listed in the tables of Figure 10. For comparison, the model with a = 10 m is established to simulate uplift loading, whereas the ultimate uplift load is taken as the standard value, denoted by Tu. In order to quantitatively analyze the influence of slope on uplift performance of the belled pier, the concept of the attenuation coefficient is defined. The formula is as follows:(1)ω=Tu−TnTu,where ω is the attenuation coefficient, Tu is the ultimate uplift load of the belled pier far away from the slopes, and Tn is the ultimate uplift load of belled pier adjacent to the slopes.Figure 10
Simulation results of slope-foundation models withh of 3 m and β of (a) 15°, (b) 30°, and (c) 45°.
(a)
(b)
(c)The attenuation coefficients for foundations of 3 m embedment depth with three slope angles are shown in Figure10. The remaining fifteen sets of models are processed and analyzed in the same way. The 3D scatter plot of the variation of the attenuation coefficient ω wish respect to distance a and depth h is shown in Figure 11.Figure 11
The 3D sketches of the attenuation coefficientω with respect to distance a and embedment depth h with β of (a) 15°, (b) 30°, and (c) 45°.
(a)
(b)
(c)These data are analyzed qualitatively and quantitatively. First, the following observations can be made: (1) The attenuation coefficientω is negatively correlated with the distance a and the embedment depth h, as shown in Figure 11(a), Curve I and Curve II. (2) The attenuation coefficient ω is positively correlated with the slope angle β. For example, as shown in Figures 11(a)–11(c), with the increase of angle of 15°, 30°, and 45°, the corresponding fluctuation ranges of attenuation coefficient are 0∼0.02, 0∼0.05, and 0∼0.12, respectively. It means that the fluctuation of uplift bearing capacity of the belled pier increases along with the slope angle.The above conclusions are consistent with common sense that (1) the closer the foundation is to the sloping ground, the shallower the embedment depth is and (2) the steeper the slope is, the more obvious the attenuation of the uplift resistance of belled pier is. These qualitative conclusions are straight forward. So it is necessary to carry out further quantitative analysis.


## 3.7. Analysis of Variance and F Test

In this section, based on the data in Figure11, the influence of distance a and depth h on attenuation coefficient ω is compared by analysis of variance. By comparing the standard deviation of a and h, it can be observed which of the two factors has significant influence on the attenuation coefficient ω (the standard deviation of the main factor is greater than that of the minor factor). The calculated results are listed in Table 6. As shown in the table, the standard deviation of distance a is greater than that of embedment depth h, indicating that the former has greater influence on attenuation coefficient than the latter.Table 6
The computational results of variance analysis andF test.
β (°)
Source of variance
Standard deviation
F test
Degree of freedom
F calculated value
F critical value
15
a
0.011
7
1.40
h
0.006
5
30
a
0.123
7
1.53
4.88
h
0.060
5
45
a
0.574
7
1.33
h
0.323
5In order to test whether the difference of the influence is obvious, the joint hypothesis test (F test) is adopted. According to the process of F test, by comparing the F calculated value with the F critical value obtained by querying F table, we can determine whether the effect is significant. And as shown in Table 6, all the calculated values of F are less than the critical value of F, which means that the influence of distance a and depth h on uplift bearing capacity of belled pier is not significantly different.


## 3.8. Gaussian Function Fitting and Empirical Equation

After analyzing the data in Figure11, it is considered that the data conform to the characteristics of two-dimensional Gauss distribution [36]. The surface formed by nonzero points in Figure 11 can be approximated as a part of the surface of a two-dimensional Gauss function where the fitting equation is as follows:(2)ωβ=fa,h=k⋅e−a2/σa2⋅e−h2/σh2,where k is the parameter describing the range of ωβ and σa is the variance of distance a. In practical sense, σa describes the rate at which attenuation coefficient ωβ varies with distance a. σh is the variance of distance h. Similarly, σh describes the rate at which attenuation coefficient ωβ varies with depth h.The fitted parameters are illustrated in Figure12. The R-squared values of the results for three slope angles are 0.92, 0.97, and 0.98, respectively, which means that the fitting accuracy is high, and the choice of the Gauss function is appropriate. Furthermore, it is not difficult to find that the three fitting parameters k, σa, and σh are positively correlated with the slope angle β. Hence, the average values of these three fitting parameters were selected. The linear formulas between the fitting parameters and the slope angle β are fitted, as shown in Figure 12. The R-squared values are 0.98, 0.99, and 0.93, respectively. The empirical equation for calculating attenuation coefficient ω can be obtained by substituting the fitting parameters k, σa, and σh into equation (2), as shown in the following equation:(3)ω=0.0046β−0.0464⋅e−a/0.0694β+7.40672⋅e−h/0.0055β+1.9272,where 15≤β≤45, 0.5≤a≤4.1, and 3≤h≤10.Figure 12
Fitting results of (a)k, (b) σa, and (c) σh.
(a)
(b)
(c)According to the actual topographic conditions in the construction process, equation (3) as a reference can be flexibly applied to the design and site selection of belled piers along the transmission line in this area.


## 4. Conclusions

In this research, field tests and numerical simulations are carried out on the uplift performance of belled piers constructed in two topographical grounds. The following conclusions can be drawn:(1)
For belled piers adjacent to sloping ground, the uplift load-displacement curves can be divided into three distinct stages: initial linear stage, curvilinear transition stage, and final linear stage. These characteristics are basically consistent with that of the curves obtained from field tests of belled piers far from slopes, which indicate that the existence of slopes has no obvious effect on the shape of uplift load-displacement curves.(2)
For belled piers adjacent to the sloping ground, the attenuation coefficientω is positively correlated with slope angle β and negatively correlated with distance a and depth h. The negative correlation between attenuation coefficient ω and distance a is stronger than that between attenuation coefficient ω and depth h. However, according to the results of F test, the difference is not significant. In the process of practical engineering, slope angle β is usually determined by the terrain. The additional distance a or the embedment depth h can increase the uplift resistance of belled pier to match the level-ground capacity. The effect of increasing the distance should be more obvious according to above the research results, which should be considered appropriately in practical designs.(3)
For belled piers adjacent to sloping ground, the empirical equation of attenuation coefficientω determined by slope angle β, distance a, and embedment depth h is proposed based on a series of fitting results. Using this formula, the attenuation coefficient can be estimated in combination with practical engineering conditions.(4)
To summarize, in view of the uplift bearing capacity of belled piers near the sloping ground, three representative slope angles are selected for modeling study. In future work, more factors, including more slope angles, more types of foundations, and more loading directions, will be considered to improve the existing research results.



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*Source: 1647197-2020-01-25.xml*