# Hole Wall Stability Analysis of Long-Distance Pipeline Horizontal Directional Crossing

Authors: Zhengwei Guo; Zezheng Chen; Yinjiang Deng; Tao Huang; Taotao chen; Jie Huang; Zhaohong Wang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Trenchless technology is a new technique for laying, repairing, and replacing underground pipelines in recent years, but various field accidents are often caused by the failure of hole wall in the process of trenchless construction, which leads to trenchless engineering accidents. Stress and deformation around hole are calculated by theoretical calculation. The numerical calculation method is also used to research the stability of hole wall with different diameters, buried depth, and soil parameters, and the curves of stress and deformation are fitted. The results show that stress and deformation of hole wall increase with the increase of the diameter. Under the condition of shallow burial depth, stress and deformation increase with the increase of burial depth of hole. The stress and deformation are different with different Poisson’s ratio and elastic modulus. This research has a new understanding for the instability of hole wall caused by variations of external parameters, which has important guiding significance for the selection of construction parameters and trenchless path.


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## Body

## 1. Introduction

Trenchless technology has been developed into a new construction technology in recent years for pipe laying, replacement, or repairing, which has a lot of advantages such as small surface interference, short construction period, accurate control, small occupation of land, and low comprehensive cost. It has been widely applied to oil and gas pipelines and city pipeline network construction [1, 2]. However, there are many risks in trenchless construction, among which the hole wall instability is a major one. The stability of hole wall is the key to the success of horizontal directional drilling, and the instability of hole wall in construction site would lead to hole wall shrinkage, collapse, and oversize, which cause surface collapse and underground cavity (Figure 1). In addition, the hole wall instability will cause the disruption of building and a series of engineering accidents [3].Figure 1
Accidents caused by the instability of hole wall.Hole wall easily collapse in sand, gravel, and broken formation. In some rock layers, the hole wall has no instability during construction with good adhesion and high strength even if drilling dry hole [4]. The instability of hole wall will seriously affect directional drilling construction, which cause the hole inside accident to be forced scrap hole and pipe resulting in waste of human and material resources [5]. For example, drilling hole length about 410 m was stuck, and 110 drill pipes were buried underground when across the Yellow River in Yongning County of Ningxia Hui Autonomous Region causing the direct financial loss more than 1 million Yuan [6]. The greater pipeline pull-back force lead to pipe failure when crossing the downstream of Haihe River. It was discovered that the instability of strata caused the high viscosity mud [7]. Pull-back force of drilling pipe reached the limit value in the 47 due to hole collapse causing the drilling pipe to get stuck [8].At present, the research of soil stability is mostly for oil drilling wellbore stability, vertical hole wall stability, and slope stability. A theoretical model of hole failure has been established by Bradley [9], which can predict the hole failure in vertical and horizontal directions. Numerical simulation strategies were proposed by Wang and Sterling [10]. The numerical analyses are performed to research the stability of hole wall during horizontal directional drilling in loose sand strata. A workflow is presented for estimation of geomechanical parameters and stress states for drilling anisotropic formations [11]. Aadnoy [12] developed a simulator to study the fracture and collapse behavior of hole, and it takes into account anisotropic elastic properties and directional shear and tensile strengths. Laboratory test data of rock compressive strength were analyzed by Zhang [13], and a new correlation was developed to allow for predicting uniaxial compressive strength in weak rocks by sonic velocities. Based on microstructure analysis and laboratory tri-axial test, physical and mechanical properties of conglomerate were obtained by Hou et al. [14], and the discrete element method was applied to simulate the failure process of the conglomerate formation and investigate the influence of gravel size distribution, drilling mud density, and fluid infiltration on the wellbore instability. Based on the poroelasticity theory and rock strength theory, Liu et al. [15] proposed a wellbore instability analysis model for horizontal wells to reveal the combined effects of hole trajectory, occurrence of stratification, and strength of weak planes on wellbore instability. All of these studies are related to geotechnical mechanicals, but few studied the influence of soil parameters to the stability of hole wall.In process of trenchless construction, stress and deformation around hole are different with different geological conditions and construction parameters. In this paper, numerical calculation and theoretical method are used to analyze the stress and deformation of hole wall. It is of great significance to predict the wall instability and select the trenchless path in process of trenchless.


## 2. Theoretical Analysis of Hole Wall Stability

The soil around hole has lost its original support because of underground excavation, and it has destroyed of original stress equilibrium state. There will be stress and deformation around hole wall under vertical and lateral pressure. The relative equilibrium between adjacent soil is changed due to movement of hole wall to central, which leads to readjustment of stress, strain and energy in a certain range and finally attain a new equilibrium.In the process of mechanical analysis, it is assumed that the surrounding rock is a homogeneous and isotropic elastic material, and the surrounding rock does not collapse. The deformation around hole section is simplified as a plane problem, and the pressure of vertical and lateral direction isP0 and λP0 separately (λ-lateral pressure coefficient), and the thickness of soil above hole is Z as shown in Figure 2 [16].Figure 2
Distribution of surrounding rock pressure around circular hole.


### 2.1. Elastic State of Soil around Hole

The surrounding rock was assumed to be elastic, which can use the superposition principle to decompose the equilibrium state into two kinds of states [17]. Suppose the surrounding rock pressures P0 and λP0 are the superpositions of P and P′, which satisfy the following formula:(1)P+P′=P0,P−P′=λP0.It can be solved:(2)P=121+λP0,P′=121+λP0,where λ=μ/1−μ is the lateral pressure coefficient and μ is Poisson’s ratio of soil.Through formula (1), the force around circular hole can be divided into two parts to solve, and then the loading state is solved, respectively. The final solution in hole periphery is the superposition of two loading states of I and II, and the loading state of surrounding rock is shown in Figure 3.Figure 3
Decomposition of the mechanical model of circular hole.Through the superposition of the two kinds of mechanical solutions, the total elastic stress solution around circular hole can be obtained:(3)σr=121+λP01−R02r2−121+λP01−4R02r2+3R04r4cos2θ,σθ=121+λP01+R02r2+121+λP01+3R04r4cos2θ,τrθ=121+λP01+2R02r2−3R04r4sin2θ.In the formula,δr is the radial stress of soil, and δθ is the tangential stress. τrθ is the shear stress. R is the distance from any point to the center of hole, and R0 is the hole radius. θ is the angle of between the diameter and the horizontal direction.The limit deformation of soil is a method to measure the stability of the hole wall, and the large deformation is very easy to cause shrinkage and collapse of hole wall. Through theoretical analysis, the strain equation of soil around hole can be expressed as follows:(4)εr=δr−μδθE,εθ=δθ−μδrE,where εr is the radial strain and εθ is the tangential strain of soil. According to geometrical relation of strain with displacement of εr=du/dr and εθ=u/r, the relative elastic displacement around hole wall can be obtained:(5)u=R1+μP02E1+λ−3−4μ1−λcos2θ,v=R1+μP02E3−4μ1−λsin2θ,where u and v are relative displacements of soil in horizontal and vertical directions.


### 2.2. Plastic State of Soil around Hole

According to the different stress states of soil, it is divided into elastic zone, plastic zone, and original rock stress zone, as shown in Figure4.Figure 4
Stress distribution in plastic state of circular hole.If the soil reaches plastic stress state, the stress of plastic zone around hole can be obtained as follows:(6)δrp=CcotφrR02sinφ/1−sinφ−1,δθp=Ccotφ2sinφ1−sinφrR02sinφ/1−sinφ−1.In formula (6), δrp is the radial stress; δθp is the tangential stress; C is the cohesion of soil; and φ is the internal friction angle of soil.The radius of plastic zone around hole isRp:(7)Rp=R0P0+Ccotφ1−sinφCcotφ1−sinφ/2sinφ.The stress of the elastic zone is as follows:(8)δre=P0−P0sinφ+CcosφP0+Ccotφ1−sinφCcotφ1−sinφ/sinφR0r2,δθe=P0+P0sinφ+CcosφP0+Ccotφ1−sinφCcotφ1−sinφ/sinφR0r2.The hole periphery is in the plastic stress state, and the elastoplastic displacement around hole can be obtained:(9)u=1+μ2ErRp2+1+λRpfθsinφ1+λP0+2Ccotφ1+1−λsinφRp1−sinφfθ−P01−λcos2θ.In the formula,fθ=2Rp1−sinφP0/1−λP0+2Ccotφsinφcos2θ; u is the elastoplastic displacement around hole.


## 2.1. Elastic State of Soil around Hole

The surrounding rock was assumed to be elastic, which can use the superposition principle to decompose the equilibrium state into two kinds of states [17]. Suppose the surrounding rock pressures P0 and λP0 are the superpositions of P and P′, which satisfy the following formula:(1)P+P′=P0,P−P′=λP0.It can be solved:(2)P=121+λP0,P′=121+λP0,where λ=μ/1−μ is the lateral pressure coefficient and μ is Poisson’s ratio of soil.Through formula (1), the force around circular hole can be divided into two parts to solve, and then the loading state is solved, respectively. The final solution in hole periphery is the superposition of two loading states of I and II, and the loading state of surrounding rock is shown in Figure 3.Figure 3
Decomposition of the mechanical model of circular hole.Through the superposition of the two kinds of mechanical solutions, the total elastic stress solution around circular hole can be obtained:(3)σr=121+λP01−R02r2−121+λP01−4R02r2+3R04r4cos2θ,σθ=121+λP01+R02r2+121+λP01+3R04r4cos2θ,τrθ=121+λP01+2R02r2−3R04r4sin2θ.In the formula,δr is the radial stress of soil, and δθ is the tangential stress. τrθ is the shear stress. R is the distance from any point to the center of hole, and R0 is the hole radius. θ is the angle of between the diameter and the horizontal direction.The limit deformation of soil is a method to measure the stability of the hole wall, and the large deformation is very easy to cause shrinkage and collapse of hole wall. Through theoretical analysis, the strain equation of soil around hole can be expressed as follows:(4)εr=δr−μδθE,εθ=δθ−μδrE,where εr is the radial strain and εθ is the tangential strain of soil. According to geometrical relation of strain with displacement of εr=du/dr and εθ=u/r, the relative elastic displacement around hole wall can be obtained:(5)u=R1+μP02E1+λ−3−4μ1−λcos2θ,v=R1+μP02E3−4μ1−λsin2θ,where u and v are relative displacements of soil in horizontal and vertical directions.


## 2.2. Plastic State of Soil around Hole

According to the different stress states of soil, it is divided into elastic zone, plastic zone, and original rock stress zone, as shown in Figure4.Figure 4
Stress distribution in plastic state of circular hole.If the soil reaches plastic stress state, the stress of plastic zone around hole can be obtained as follows:(6)δrp=CcotφrR02sinφ/1−sinφ−1,δθp=Ccotφ2sinφ1−sinφrR02sinφ/1−sinφ−1.In formula (6), δrp is the radial stress; δθp is the tangential stress; C is the cohesion of soil; and φ is the internal friction angle of soil.The radius of plastic zone around hole isRp:(7)Rp=R0P0+Ccotφ1−sinφCcotφ1−sinφ/2sinφ.The stress of the elastic zone is as follows:(8)δre=P0−P0sinφ+CcosφP0+Ccotφ1−sinφCcotφ1−sinφ/sinφR0r2,δθe=P0+P0sinφ+CcosφP0+Ccotφ1−sinφCcotφ1−sinφ/sinφR0r2.The hole periphery is in the plastic stress state, and the elastoplastic displacement around hole can be obtained:(9)u=1+μ2ErRp2+1+λRpfθsinφ1+λP0+2Ccotφ1+1−λsinφRp1−sinφfθ−P01−λcos2θ.In the formula,fθ=2Rp1−sinφP0/1−λP0+2Ccotφsinφcos2θ; u is the elastoplastic displacement around hole.


## 3. Soil Stress by Numerical Calculation

Based on the theory of geotechnical mechanics and numerical calculation, the paper chooses the suitable soil constitutive relation to determine the parameters of soil, and the numerical calculation method is used to establish the three-dimensional numerical model to consider the influence of rock and soil parameters, diameter, and buried depth to the stability of the hole. Maximum stress and deformation of holes with different diameters, buried depths, and soil parameters are analyzed statistically.Before numerical analysis, the following assumptions are made about the soil. Only stress and deformation around hole are analyzed with change of geotechnical parameters, diameter, and burial depth. The hole is infinitely long in direction of hole axis, and the deformation is considered in hole section. The surrounding rock is homogeneous, isotropic, and continuous.In order to analyze the influence of various parameters on the hole wall, the finite element model is established as shown in Figure5.Figure 5
Finite element model.In process of numerical calculation, the geotechnical material parameters are shown in Table1.Table 1
Parameters of geotechnical materials (unit: kg.m).
ParameterDensity (kg/m3)Young’s modulus (Pa)Poisson’s ratioCohesion (N)Internal friction angleDilatancy angleValue21002.6E70.4210000240290In the numerical analysis,D-P material model is used. The plastic state is assumed to be an ideal elastoplastic, and the yield strength increases with increasing hydrostatic pressure. In addition, the material considers the volume expansion due to yield, but does not take into account the effect of temperature variation to soil [18].


## 4. Effect of Diameter on Hole Wall Stability

After the excavation, the mechanical equilibrium of hole wall is destroyed, and the original balance condition is lost. The stress concentration of soil is caused around hole wall. Under the action of overburden pressure, the soil around hole moves to the hole central, and it may cause collapse of hole wall due to poor stability formation.


### 4.1. Deformation of Surrounding Rock

At present, the traditional limit equilibrium analysis method is widely used because it is relatively simple to calculate geotechnical engineering. Moreover, deformation analysis is also an important parameter to measure the stability of hole wall, and ultimate deformation can predict the stability of hole wall [19]. During process of trenchless, different diameters of hole have different deformation. The numerical method is used to calculate the deformation around hole with different hole diameters. The deformation nephogram of soil around hole is shown in Figure 6.Figure 6
Displacement nephogram of soil around the hole.When buried depth is 25 m, the hole diameter varies from 0.8 to 1.2 m, and stress and deformation around hole wall are inspected as diameter increases by 0.1 m. Through analysis and comparison, it is found that maximum deformation occurs on the top and side of hole. Moreover, the larger the hole diameter is, the greater the deformation of soil becomes. During analysis of stability, engineering calculation is only concerned with maximum stress and deformation around hole.When the hole diameter is 0.8 m, its maximum displacement is 93.91 mm. When the diameter of hole is 1.2 m, the maximum displacement of soil reaches 269.986 mm. In order to find out the influence of hole diameter to deformation, the deformation of soil at three positions is statistically analyzed as shown in Figure7 (separately the top, the side, and the bottom of hole).Figure 7
Variation curve of soil deformation with hole diameter.In process of trenchless, the diameter is changing from 0.8 m to 1.2 m. The deformation of soil is approximately parabola change. It can be concluded from the graph that displacement of the top around hole is greater than that of the lateral, and the lateral is greater than that of the bottom. Under the condition of certain buried depth, the deformation of hole is small when the diameter of hole is less than 1 m. The displacement of hole is relatively great when the diameter is greater than 1 m.


### 4.2. Soil Stress around the Hole

The maximum soil stress is located around the hole during the process of enlarging the hole diameter. The farther away from the hole, the less affected by the hole. After the excavation of soil, the stress nephogram of surrounding hole is shown in Figure8.Figure 8
Stress nephogram with hole diameter of 0.8 and 1.2.The hole periphery von Mises stress is symmetrical. The soil is closer to the hole wall, the greater of the soil stress. In order to analyze the variation regularity of soil stress under different hole diameters, the maximum von Mises stress and the maximum stress intensity of hole are statistically analyzed, and the maximum von Mises stress and maximum stress intensity curve of hole change with diameters are shown in Figure9.Figure 9
Variation curve of maximum von Mises stress and stress intensity with different diameters.By means of an analysis of stress-diameter curve, it is found that the influence of von Mises stress and stress intensity around the hole is small under the small diameter, and the von Mises stress and stress intensity of hole linearly change when the hole diameter exceeds 1 m. Therefore, the smaller the diameter of the hole is, the smaller the stress around hole becomes.


## 4.1. Deformation of Surrounding Rock

At present, the traditional limit equilibrium analysis method is widely used because it is relatively simple to calculate geotechnical engineering. Moreover, deformation analysis is also an important parameter to measure the stability of hole wall, and ultimate deformation can predict the stability of hole wall [19]. During process of trenchless, different diameters of hole have different deformation. The numerical method is used to calculate the deformation around hole with different hole diameters. The deformation nephogram of soil around hole is shown in Figure 6.Figure 6
Displacement nephogram of soil around the hole.When buried depth is 25 m, the hole diameter varies from 0.8 to 1.2 m, and stress and deformation around hole wall are inspected as diameter increases by 0.1 m. Through analysis and comparison, it is found that maximum deformation occurs on the top and side of hole. Moreover, the larger the hole diameter is, the greater the deformation of soil becomes. During analysis of stability, engineering calculation is only concerned with maximum stress and deformation around hole.When the hole diameter is 0.8 m, its maximum displacement is 93.91 mm. When the diameter of hole is 1.2 m, the maximum displacement of soil reaches 269.986 mm. In order to find out the influence of hole diameter to deformation, the deformation of soil at three positions is statistically analyzed as shown in Figure7 (separately the top, the side, and the bottom of hole).Figure 7
Variation curve of soil deformation with hole diameter.In process of trenchless, the diameter is changing from 0.8 m to 1.2 m. The deformation of soil is approximately parabola change. It can be concluded from the graph that displacement of the top around hole is greater than that of the lateral, and the lateral is greater than that of the bottom. Under the condition of certain buried depth, the deformation of hole is small when the diameter of hole is less than 1 m. The displacement of hole is relatively great when the diameter is greater than 1 m.


## 4.2. Soil Stress around the Hole

The maximum soil stress is located around the hole during the process of enlarging the hole diameter. The farther away from the hole, the less affected by the hole. After the excavation of soil, the stress nephogram of surrounding hole is shown in Figure8.Figure 8
Stress nephogram with hole diameter of 0.8 and 1.2.The hole periphery von Mises stress is symmetrical. The soil is closer to the hole wall, the greater of the soil stress. In order to analyze the variation regularity of soil stress under different hole diameters, the maximum von Mises stress and the maximum stress intensity of hole are statistically analyzed, and the maximum von Mises stress and maximum stress intensity curve of hole change with diameters are shown in Figure9.Figure 9
Variation curve of maximum von Mises stress and stress intensity with different diameters.By means of an analysis of stress-diameter curve, it is found that the influence of von Mises stress and stress intensity around the hole is small under the small diameter, and the von Mises stress and stress intensity of hole linearly change when the hole diameter exceeds 1 m. Therefore, the smaller the diameter of the hole is, the smaller the stress around hole becomes.


## 5. The Influence of Buried Depth to Hole Wall Stability

Buried depth is also a main factor to determining hole wall stability. Under different buried depth, the stress and deformation of hole are also different. However, elastic modulus of soil will be different with different burial depth, so it seems to be difficult to research the influence of buried depth to hole stability independently. Based on the mechanical analysis of geotechnical surroundings of oil drilling, Feng et al. carried out field measurements of elastic modulusE of soil, and the elastic modulus curve of hole was obtained with different burial depth. The results show that elastic modulus is approximate to logarithmic and linear variation. The variation functions are E=15.8lnH−42.95 and E=0.0766H+17.775, and the correlation coefficients are 0.67 and 0.63 [20]. However, the depth of hole is analyzed within dozens of meters in trenchless construction, and the elastic modulus of soil is very small in this condition, so it is assumed that the elastic modulus is a constant during numerical analysis. In the same hole diameter and rock-soil parameters, the buried depth of hole is adjusted to analyze the variation of stress and deformation. The diameter of trenchless hole is 1.2 m.


### 5.1. Displacement of Soil around Hole with Different Buried Depth

Under the condition of different buried depth and the same hole diameter, the deformation of surrounding soil is analyzed, as shown in Figure10. Through analysis of displacement nephogram, the maximum displacement around hole is found on the horizontal and the top position. According to statistical analysis of the displacements of the top, the bottom, and the side around hole, the changing curves of maximum displacement with thickness are shown in Figure 11.Figure 10
Soil displacement nephogram with different buried depth.Figure 11
The curve of maximum displacement around hole with different buried depth.According to Figure11, the soil thickness above the hole ranges from 10 m to 50 m, the displacement of the top is greater than that of the side, and the side is greater than that of the bottom. When the soil thickness is less than 20 m above the hole, the displacement of the top and the side is quite different, and the displacement of top is smaller than that of the side. When the soil thickness is less than 20 m, the maximum displacement around the hole is rather changeable with the soil thickness, and the displacement around the hole tends to be a constant when the soil thickness is greater than 20 m.


### 5.2. Soil Stress around the Hole

Based on analysis of soil stress around the hole with different buried depth, it is found that the equivalent stress in the horizontal position is greater than the other position around the hole. The farther from the hole, the smaller the stress around the hole. The stress nephogram around the hole is shown in Figure12. Through analysis of maximum stress, stress intensity, and maximum principal stress of hole changing with soil thickness, the variation curve is shown in Figure 13.Figure 12
Soil stress nephogram with different buried depth of hole.Figure 13
Variation curve of maximum stress with different buried depth of hole.As a result of analysis, it is indicated that the equivalent stress of hole wall changes a little when the soil thickness is less than 15 m, and the maximum stress intensity and equivalent von Mises stress of hole change almost linearly when the thickness of soil is between 10 m and 35 m. However, the stress intensity and the maximum von Mises stress around the hole tend to be constant when the soil thickness is greater than 35 m.


## 5.1. Displacement of Soil around Hole with Different Buried Depth

Under the condition of different buried depth and the same hole diameter, the deformation of surrounding soil is analyzed, as shown in Figure10. Through analysis of displacement nephogram, the maximum displacement around hole is found on the horizontal and the top position. According to statistical analysis of the displacements of the top, the bottom, and the side around hole, the changing curves of maximum displacement with thickness are shown in Figure 11.Figure 10
Soil displacement nephogram with different buried depth.Figure 11
The curve of maximum displacement around hole with different buried depth.According to Figure11, the soil thickness above the hole ranges from 10 m to 50 m, the displacement of the top is greater than that of the side, and the side is greater than that of the bottom. When the soil thickness is less than 20 m above the hole, the displacement of the top and the side is quite different, and the displacement of top is smaller than that of the side. When the soil thickness is less than 20 m, the maximum displacement around the hole is rather changeable with the soil thickness, and the displacement around the hole tends to be a constant when the soil thickness is greater than 20 m.


## 5.2. Soil Stress around the Hole

Based on analysis of soil stress around the hole with different buried depth, it is found that the equivalent stress in the horizontal position is greater than the other position around the hole. The farther from the hole, the smaller the stress around the hole. The stress nephogram around the hole is shown in Figure12. Through analysis of maximum stress, stress intensity, and maximum principal stress of hole changing with soil thickness, the variation curve is shown in Figure 13.Figure 12
Soil stress nephogram with different buried depth of hole.Figure 13
Variation curve of maximum stress with different buried depth of hole.As a result of analysis, it is indicated that the equivalent stress of hole wall changes a little when the soil thickness is less than 15 m, and the maximum stress intensity and equivalent von Mises stress of hole change almost linearly when the thickness of soil is between 10 m and 35 m. However, the stress intensity and the maximum von Mises stress around the hole tend to be constant when the soil thickness is greater than 35 m.


## 6. Effect of Soil Parameters on Hole Wall Stability

The most influential factor of hole wall stability is the property of soil, and the hole wall with high strength and strong bond between soil has great stability, so there will be no collapse during the process of trenchless.


### 6.1. Poisson’s Ratio

#### 6.1.1. Soil Deformation Changing with Poisson’s Ratio

The test of soil specimen indicates that the range of Poisson’s ratioμ varies from 0 and 0.5, and the horizontal lateral pressure coefficient λ is determined by Poisson ratio μ: λ=μ/1−μ, and the lateral pressure coefficient λ will become larger with increase of Poisson’s ratio. Generally, the softer the soil is, the greater the lateral pressure coefficient is and the more serious the horizontal direction deformation gets. The displacement nephogram around the hole with different Poisson’s ratio is shown in Figure 14.Figure 14
Displacement nephogram of soil with different Poisson’s ratio.Poisson’s ratio of soil varies from 0.17 to 0.49, and the maximum displacement variation curve of the top, the lateral, and the bottom around hole is shown in Figure15.Figure 15
Variation curve of maximum displacement with different Poisson’s ratio.From Figure15, it is shown that the deformation of the side and the bottom is not affected by the Poisson ratio when Poisson’s ratio is less than 0.33. In addition, the displacement of the side is approximately linearly reduced with Poisson’s ratio increasing. The displacement of the side is greater than the top, and the top is greater than that of the bottom. When Poisson’s ratio is less than 0.33, the displacement of hole is approximately 0. However, the displacement of the top and the bottom suddenly becomes larger when Poisson’s ratio is equal to 0.33, and the displacement of the top is greater than that of the side.


#### 6.1.2. Change of Soil Stress with Poisson’s Ratio

The soil stress is different with different Poisson’s ratio, and the equivalent von Mises stress nephogram around hole is shown in Figure16. Maximum von Mises stress and stress intensity of hole are statistically analyzed with the change of Poisson’s ratio, and the changing curve of von Mises stress and stress intensity is shown in Figure 17.Figure 16
Soil stress nephogram with different Poisson’s ratio.Figure 17
Variation curve of maximum stress and stress intensity with Poisson’s ratio.The influence of hole on surrounding soil is smaller with the increase of Poisson’s ratio from Figure16. It is found that maximum equivalent von Mises stress and stress intensity of soil have little change when the Poisson ratio is less than 0.33 from Figure 17. The maximum stress and stress intensity of surrounding soil have a sudden change when Poisson’s ratio is greater than 0.33, and maximum stress and stress intensity of surrounding soil increase with the increase of Poisson’s ratio. When Poisson’s ratio varies from 0.33 to 0.43, maximum stress and stress intensity around hole have a slight increase, and it increases linearly when the Poisson ratio is greater than 0.43.


### 6.2. Elastic Modulus

Different elastic modulus of soil has different stress and deformation around hole. Stress and deformation of hole are analyzed with different elastic modulus.


#### 6.2.1. Displacement around Hole with Different Elastic Modulus

The elastic modulus ranges from 20 MPa to 40 MPa, and stress and deformation of soil are statistically analyzed every 2 MPa variation. Only maximum stress and deformation around hole are considered during analysis of stress and deformation. If hole wall does not reach limited stress and deformation, it is indicated that the surrounding soil is stable. Stress and deformation around hole are calculated by the numerical method. The displacement nephogram is shown in Figure18, and the displacement variation curve of soil with different elastic modulus is shown in Figure 19.Figure 18
Soil displacement nephogram with different elastic modulus.Figure 19
Maximum displacement variation curve with different elastic modulus.It is found from Figure19 that the displacement of the top is greater than that of the side around hole, and the lateral is greater than that of the bottom with variation of elastic modulus. With increase of elastic modulus, the displacement of the top and the lateral around hole decreases, and the displacement of the bottom increases slightly.


#### 6.2.2. Stress Variation with Elastic Modulus around Hole

The stress around hole wall also changes with variation of elastic modulus. The von Mises stress nephogram around hole is shown in Figure20. The variation curves of equivalent stress and stress intensity of soil are shown in Figure 21.Figure 20
Soil stress nephogram under different elastic modulus.Figure 21
von Mises stress and stress intensity variation curve with elastic modulus.It is found from Figure21 that the elastic modulus of soil ranges from 20 MPa to 40 MPa, and maximum von Mises stress and stress intensity around hole increase linearly with elastic modulus.


## 6.1. Poisson’s Ratio

### 6.1.1. Soil Deformation Changing with Poisson’s Ratio

The test of soil specimen indicates that the range of Poisson’s ratioμ varies from 0 and 0.5, and the horizontal lateral pressure coefficient λ is determined by Poisson ratio μ: λ=μ/1−μ, and the lateral pressure coefficient λ will become larger with increase of Poisson’s ratio. Generally, the softer the soil is, the greater the lateral pressure coefficient is and the more serious the horizontal direction deformation gets. The displacement nephogram around the hole with different Poisson’s ratio is shown in Figure 14.Figure 14
Displacement nephogram of soil with different Poisson’s ratio.Poisson’s ratio of soil varies from 0.17 to 0.49, and the maximum displacement variation curve of the top, the lateral, and the bottom around hole is shown in Figure15.Figure 15
Variation curve of maximum displacement with different Poisson’s ratio.From Figure15, it is shown that the deformation of the side and the bottom is not affected by the Poisson ratio when Poisson’s ratio is less than 0.33. In addition, the displacement of the side is approximately linearly reduced with Poisson’s ratio increasing. The displacement of the side is greater than the top, and the top is greater than that of the bottom. When Poisson’s ratio is less than 0.33, the displacement of hole is approximately 0. However, the displacement of the top and the bottom suddenly becomes larger when Poisson’s ratio is equal to 0.33, and the displacement of the top is greater than that of the side.


### 6.1.2. Change of Soil Stress with Poisson’s Ratio

The soil stress is different with different Poisson’s ratio, and the equivalent von Mises stress nephogram around hole is shown in Figure16. Maximum von Mises stress and stress intensity of hole are statistically analyzed with the change of Poisson’s ratio, and the changing curve of von Mises stress and stress intensity is shown in Figure 17.Figure 16
Soil stress nephogram with different Poisson’s ratio.Figure 17
Variation curve of maximum stress and stress intensity with Poisson’s ratio.The influence of hole on surrounding soil is smaller with the increase of Poisson’s ratio from Figure16. It is found that maximum equivalent von Mises stress and stress intensity of soil have little change when the Poisson ratio is less than 0.33 from Figure 17. The maximum stress and stress intensity of surrounding soil have a sudden change when Poisson’s ratio is greater than 0.33, and maximum stress and stress intensity of surrounding soil increase with the increase of Poisson’s ratio. When Poisson’s ratio varies from 0.33 to 0.43, maximum stress and stress intensity around hole have a slight increase, and it increases linearly when the Poisson ratio is greater than 0.43.


## 6.1.1. Soil Deformation Changing with Poisson’s Ratio

The test of soil specimen indicates that the range of Poisson’s ratioμ varies from 0 and 0.5, and the horizontal lateral pressure coefficient λ is determined by Poisson ratio μ: λ=μ/1−μ, and the lateral pressure coefficient λ will become larger with increase of Poisson’s ratio. Generally, the softer the soil is, the greater the lateral pressure coefficient is and the more serious the horizontal direction deformation gets. The displacement nephogram around the hole with different Poisson’s ratio is shown in Figure 14.Figure 14
Displacement nephogram of soil with different Poisson’s ratio.Poisson’s ratio of soil varies from 0.17 to 0.49, and the maximum displacement variation curve of the top, the lateral, and the bottom around hole is shown in Figure15.Figure 15
Variation curve of maximum displacement with different Poisson’s ratio.From Figure15, it is shown that the deformation of the side and the bottom is not affected by the Poisson ratio when Poisson’s ratio is less than 0.33. In addition, the displacement of the side is approximately linearly reduced with Poisson’s ratio increasing. The displacement of the side is greater than the top, and the top is greater than that of the bottom. When Poisson’s ratio is less than 0.33, the displacement of hole is approximately 0. However, the displacement of the top and the bottom suddenly becomes larger when Poisson’s ratio is equal to 0.33, and the displacement of the top is greater than that of the side.


## 6.1.2. Change of Soil Stress with Poisson’s Ratio

The soil stress is different with different Poisson’s ratio, and the equivalent von Mises stress nephogram around hole is shown in Figure16. Maximum von Mises stress and stress intensity of hole are statistically analyzed with the change of Poisson’s ratio, and the changing curve of von Mises stress and stress intensity is shown in Figure 17.Figure 16
Soil stress nephogram with different Poisson’s ratio.Figure 17
Variation curve of maximum stress and stress intensity with Poisson’s ratio.The influence of hole on surrounding soil is smaller with the increase of Poisson’s ratio from Figure16. It is found that maximum equivalent von Mises stress and stress intensity of soil have little change when the Poisson ratio is less than 0.33 from Figure 17. The maximum stress and stress intensity of surrounding soil have a sudden change when Poisson’s ratio is greater than 0.33, and maximum stress and stress intensity of surrounding soil increase with the increase of Poisson’s ratio. When Poisson’s ratio varies from 0.33 to 0.43, maximum stress and stress intensity around hole have a slight increase, and it increases linearly when the Poisson ratio is greater than 0.43.


## 6.2. Elastic Modulus

Different elastic modulus of soil has different stress and deformation around hole. Stress and deformation of hole are analyzed with different elastic modulus.


### 6.2.1. Displacement around Hole with Different Elastic Modulus

The elastic modulus ranges from 20 MPa to 40 MPa, and stress and deformation of soil are statistically analyzed every 2 MPa variation. Only maximum stress and deformation around hole are considered during analysis of stress and deformation. If hole wall does not reach limited stress and deformation, it is indicated that the surrounding soil is stable. Stress and deformation around hole are calculated by the numerical method. The displacement nephogram is shown in Figure18, and the displacement variation curve of soil with different elastic modulus is shown in Figure 19.Figure 18
Soil displacement nephogram with different elastic modulus.Figure 19
Maximum displacement variation curve with different elastic modulus.It is found from Figure19 that the displacement of the top is greater than that of the side around hole, and the lateral is greater than that of the bottom with variation of elastic modulus. With increase of elastic modulus, the displacement of the top and the lateral around hole decreases, and the displacement of the bottom increases slightly.


### 6.2.2. Stress Variation with Elastic Modulus around Hole

The stress around hole wall also changes with variation of elastic modulus. The von Mises stress nephogram around hole is shown in Figure20. The variation curves of equivalent stress and stress intensity of soil are shown in Figure 21.Figure 20
Soil stress nephogram under different elastic modulus.Figure 21
von Mises stress and stress intensity variation curve with elastic modulus.It is found from Figure21 that the elastic modulus of soil ranges from 20 MPa to 40 MPa, and maximum von Mises stress and stress intensity around hole increase linearly with elastic modulus.


## 6.2.1. Displacement around Hole with Different Elastic Modulus

The elastic modulus ranges from 20 MPa to 40 MPa, and stress and deformation of soil are statistically analyzed every 2 MPa variation. Only maximum stress and deformation around hole are considered during analysis of stress and deformation. If hole wall does not reach limited stress and deformation, it is indicated that the surrounding soil is stable. Stress and deformation around hole are calculated by the numerical method. The displacement nephogram is shown in Figure18, and the displacement variation curve of soil with different elastic modulus is shown in Figure 19.Figure 18
Soil displacement nephogram with different elastic modulus.Figure 19
Maximum displacement variation curve with different elastic modulus.It is found from Figure19 that the displacement of the top is greater than that of the side around hole, and the lateral is greater than that of the bottom with variation of elastic modulus. With increase of elastic modulus, the displacement of the top and the lateral around hole decreases, and the displacement of the bottom increases slightly.


## 6.2.2. Stress Variation with Elastic Modulus around Hole

The stress around hole wall also changes with variation of elastic modulus. The von Mises stress nephogram around hole is shown in Figure20. The variation curves of equivalent stress and stress intensity of soil are shown in Figure 21.Figure 20
Soil stress nephogram under different elastic modulus.Figure 21
von Mises stress and stress intensity variation curve with elastic modulus.It is found from Figure21 that the elastic modulus of soil ranges from 20 MPa to 40 MPa, and maximum von Mises stress and stress intensity around hole increase linearly with elastic modulus.


## 7. Conclusions

The stress and deformation of circular hole are calculated by the numerical calculation method with different diameters, buried depth of hole, and soil parameters. It is shown that von Mises stress, shear stress, and deformation around hole become bigger with the increase of buried depth, and maximum shear stress of hole tends to be a constant when the buried depth reaches 35 m. The greater the diameter of hole is, the greater the stresses will be.Under the same diameter and burial depth, the deformation of hole causes high concentrated stress when the cohesion is less than 30 kN. The deformation around hole tends to be a constant when the cohesion is greater than the 30 kN. The stress of hole is the smallest when the cohesion of soil is 20 kN.Poisson’s ratio mainly affects the horizontal side pressure coefficient of hole. The displacement of the top and the side reduces slightly when Poisson’s ratio is less than 0.33, and the displacement of the side is greater than that of the top. The displacement of the bottom will suddenly increase when Poisson’s ratio is equal to 0.33, and the displacement of the top will be greater than that of the side. The stress of soil increases when Poisson’s ratio is greater than 0.33. In the process of the change of elastic modulus, the variation of stress and deformation is not significant.



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*Source: 1114417-2022-09-19.xml*