# Seismic Stability Analysis of Saturated and Unsaturated Soil Slopes Using Permanent Displacement

Authors: Shuai Huang; Yanju Peng
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

The permanent displacement has been widely used for slope seismic stability in practical engineering; however, the effect of the dynamic pore water pressure on the saturated and unsaturated soil slopes could not be neglected. In this paper, we propose a calculation method of dynamic pore water pressure by the hollow cylinder apparatus (GCTS) which is the most advanced and complicated device in lab testing on soil dynamics. Then, based on the proposed calculation method of dynamic pore water pressure combined with the limit equilibrium and finite element methods, we introduce a simple calculation method of permanent displacement, which avoids solving complex nonlinear equations and greatly simplifies the computational effort. Shaking table test results demonstrate the effectiveness and efficiency of the simple calculation method of permanent displacement, which could rapidly assess the soil slope seismic stability considering the effect of dynamic pore water pressure.


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## Body

## 1. Introduction

The landslide induced by earthquake is a common geological disaster. According to preliminary statistics, more than 15,000 landslides were triggered by the Wenchuan earthquake [1]. The slopes which were designed in accordance with the seismic codes were still destroyed a lot, and a series of new features appeared in slope failure. How to prevent the failure of slope under earthquake has become one of the key techniques in slope stability analysis [2, 3]. Safety factor is often used for the slope stability evaluation at present. In fact, the slope is considered to be damaged when the safety factor is less than 1. And at this time, the slope may be in a new equilibrium, which is common in practical engineering. Thus, the safety factor will not be used for further evaluation of the slope stability when it is less than 1. Fortunately, the permanent displacement of the slope is accumulated when the safety factor is less than 1. And permanent displacement could evaluate the slope stability when the safety factor is less than 1. In addition, with the further research on slope dynamic stability, the dynamic stability evaluation method using the single seismic coefficient has been found to be insufficient. The core issue of seismic resistance of geotechnical structures has gradually transformed from the strength standard into the deformation standard. The seismic design method based on deformation is currently one of the important design theories. Thus, permanent displacement is more suitable for slope seismic stability, and how to use permanent displacement for accurate assessment of the slope stability needs further research.Different soil slopes affected by different rainfall amounts are in different groundwater levels during the earthquake or aftershocks, and the slopes suffered from a combination of factors, including earthquake and groundwater. Seismic stability evaluation methods of slopes are the core of the slope seismic stability analysis. Therefore, it is especially important to use a reasonable safety evaluation method for slope seismic stability analysis. At present, the slope stability analysis methods mainly include the quasi-static method, Newmark sliding block method, and time-history method. The quasi-static method [4] transforms the dynamic problem into a static problem by using the dynamic coefficient, which avoids the solution of complex nonlinear equations and greatly simplifies the computational effort. The horizontal and vertical inertial forces transformed from dynamic loads act on the center of the slope gravity, and the safety factor can be calculated fast by the quasi-static method. The quasi-static method has been widely used in engineering projects and written into the regulations in many countries. However, seismic wave characteristics, such as vibration frequency and duration of earthquake, were not considered in the quasi-static method [5]. The time-history method [6] is used to analyze the dynamic property of the slope during the seismic process considering the soil dynamic property and seismic characters. And the dynamic safety factors of the slope could be calculated under earthquake. However, the instability criterion is mostly solved by statistical methods, such as the average safety coefficient method and reliability dynamic safety factor method. How to evaluate the slope stability with dynamic safety factors at different times of earthquake action needs further research. The Newmark sliding block method [7] was first put forward by Newmark, and the soil was assumed as the ideal one which is a rigid, perfectly plastic body. The yield acceleration of the landslide mass is a constant. The landslide mass tends to start sliding when the seismic accelerations exceed the yield acceleration. Then, the permanent displacement can be obtained by a double integration of the portions of the acceleration time-history exceeding the yield acceleration [8]. Rabie [9] researched the slope stability based on the quasi-static method and time-history method and found that the quasi-static method is more conservative for slope stability analysis; Lu et al. [10] researched the seismic stability of a three-dimensional slope using the permanent displacement and found that permanent displacement was more reasonable for slope stability analysis. Therefore, in this study, the seismic stability of the soil slope is evaluated by permanent displacement. Great attention will be paid to the issue of seismic stability of the saturated and unsaturated soil slopes using permanent displacement considering the influence of the groundwater level.In recent years, a large number of landslide disaster cases especially the slope failure phenomena in the Wenchuan earthquake show that the current evaluation method of slope stability could not meet the safety performance evaluation of the slope. The design method based on deformation is one of the most important theories of seismic design [11, 12]. The study for slope stability using permanent displacement is the hot issue in slope seismic response analysis, and the traditional method for evaluating the soil slope stability has many limits. Marzorati et al. [13] and Carro et al. [14] completed the seismic and landslide zoning maps of Umbria and Marche regions with the Newmark permanent displacement prediction model. Bray and Travasarou [15] proposed a fully coupled permanent displacement calculation method considering the dynamic effect of the slope. Shenglin et al. [16] proposed an algorithm for seismic permanent displacement of the slope considering structural surface degradation. Han et al. [17] analyzed the permanent displacement of the slope and found that the variation curve of the permanent displacement has a threshold value, and the permanent displacement increased and then became stable when the acceleration values were greater than the threshold value. Liu and Kong [18] calculated the permanent displacement of the slope using discontinuous deformation analysis; however, certain simplification has been made on the method. Although some research has been done on the slope stability using the permanent displacement, the effect of the dynamic pore water pressure under earthquake on the permanent displacement was neglected. The effect of the dynamic pore water pressure on the saturated or unsaturated soil slope stability is significant [19]. Thus, a soil slope located in Inner Mongolia Autonomous Region in China was selected to investigate the seismic stability analysis using permanent displacement.


## 2. Calculation Model of the Slope

### 2.1. Establishing the Slope Model

An actual slope which is along the Zhunshuo railway in Inner Mongolia Autonomous Region in China is selected as the research object, as shown in Figure1. The soil types of the slope are mainly sandy. The groundwater level changes with the season, and the mean water level is above 2.0 m. The slope length is 17 m, the slope height is 12 m, and the slope angle is 35°.Figure 1
Engineering field figure.The slope model in reference [20] is referred and established with the finite element software MIDAS GTS NX 2018, and the height of the finite element model is twice the slope height, as shown in Figure 2. In order to reduce the error of the calculation results, the maximum finite element mesh size is smaller than 1/10～1/8 of the input seismic wavelength. The Mohr–Coulomb constitutive model is adopted in this study. The free field boundaries are used for the model boundaries.Figure 2
Analysis model of the slope (reproduced from the study of Huang et al. [20], under the Creative Commons Attribution License/public domain).
(a)
(b)The calculation parameters of the slope model are obtained by the indoor test and referenced in theGeotechnical Engineering Handbook [21], as shown in Table 1.Table 1
Calculation parameters of the model.
Soil type
Gravity (kN·m−3)
Poisson’s ratio
Elastic modulus (MPa)
Friction angle (°)
Cohesion (kPa)
Permeability coefficient (cm·s−1)
Saturated water content (%)
Sandy
17.5
0.3
50.4
35.23
11.42
5e-5
30


### 2.2. Sliding Failure Evaluation Index of the Slope

By comparison of the slope codes of the earthquake-prone countries (China, Japan, European countries, and California in the United States), evaluation methods of the slope seismic stability in different specifications were determined at home and abroad, as shown in Table2.Table 2
Slope codes of the earthquake-prone countries.
Nation
Codes
Evaluation methods
Evaluation indictors
China
“Technical code for building slope engineering” (GB50330-2013)
Quasi-static method [22, 23]
Safety factor
“Design code for engineered slopes in water resources and hydropower projects” (SL386-2007)
Quasi-static method
Safety factor
“Specifications for design of highway Subgrades” (JTG D30-2004)
Quasi-static method
Safety factor
“Code for seismic design of railway engineering” (GB50111-2006)
Quasi-static method
Safety factor
“Code for design on subgrade of railway” (TB10001-2005)
Quasi-static method
Safety factor
“Code for design of high speed railway” (TB10621-2009)
Quasi-static method
Safety factor
Japan
“Design standards for railway structures”
Quasi-static method and Newmark method
Safety factor and permanent displacement
European countries
Eurocode 7: geotechnical design
Quasi-static method
Safety factor
California in the United States
California Geological Survey’s guidelines (2008)
Quasi-static method
Safety factorBy comparing slope codes in different countries, we used the safety factor and permanent displacement to evaluate the slope stability, which are closer to the actuals.


### 2.3. Numerical Procedure

Referring to JapaneseSpecifications for Highway Bridges, we used three real recorded seismic waves, including T1-II-1 which is a far-field seismic wave, T2-II-1 which is a near-field seismic wave, and El Centro seismic wave, to analyze the slope stability, as shown in Table 3. The peak acceleration values of the three seismic waves are scaled to 0.21g, while the frequencies remain unchanged, as shown in Figure 3. The earthquake motions are used as the horizontal excitations and input from the fixed bottom boundary.Table 3
Parameters of the earthquake motions.
Seismic wave
Recorded location
Earthquake name
Magnitude
Epicentral distance (km)
T1-II-1
Foundation of Itajima bridge
Hyūganada earthquake (1968)
7.5
100
El Centro
El Centro
Emperor Valley earthquake (1940)
7.7
12
T2-II-1
JR Takatori station
Kobe (1995)
7.2
16Figure 3
Time-history of earthquakes (reproduced from the study of Huang et al. [20], under the Creative Commons Attribution License/public domain).
(a)
(b)
(c)Different factors are changed, respectively, on the basis of the original slope. Effects of various factors including slope rates, slope heights, groundwater levels, peak accelerations, and earthquake types on the safety factor and the permanent displacement of the slope are studied, as shown in Table4.Table 4
Numerical procedure.
Slope rate
Slope height (m)
Underground water levelHw (m)
Peak acceleration (m·s−2)
Seismic waves
1 : 1.5
12
5
2, 3, 4, 5, 6, 7, 8, 9, 10
El Centro
1 : 1.4
12
5
4, 5, 6
El Centro
1 : 1.5
12
5
4, 5, 6
1 : 1.6
12
5
4, 5, 6
1 : 1.5
12
6
4, 5, 6
El Centro
7
4, 5, 6
8
4, 5, 6
1 : 1.5
12
5
4, 5, 6
El Centro
24
10
4, 5, 6
El Centro
36
15
4, 5,6
El Centro
1 : 1.5
12
5
4, 5, 6
El Centro, T1-II-1, T2-II-1


## 2.1. Establishing the Slope Model

An actual slope which is along the Zhunshuo railway in Inner Mongolia Autonomous Region in China is selected as the research object, as shown in Figure1. The soil types of the slope are mainly sandy. The groundwater level changes with the season, and the mean water level is above 2.0 m. The slope length is 17 m, the slope height is 12 m, and the slope angle is 35°.Figure 1
Engineering field figure.The slope model in reference [20] is referred and established with the finite element software MIDAS GTS NX 2018, and the height of the finite element model is twice the slope height, as shown in Figure 2. In order to reduce the error of the calculation results, the maximum finite element mesh size is smaller than 1/10～1/8 of the input seismic wavelength. The Mohr–Coulomb constitutive model is adopted in this study. The free field boundaries are used for the model boundaries.Figure 2
Analysis model of the slope (reproduced from the study of Huang et al. [20], under the Creative Commons Attribution License/public domain).
(a)
(b)The calculation parameters of the slope model are obtained by the indoor test and referenced in theGeotechnical Engineering Handbook [21], as shown in Table 1.Table 1
Calculation parameters of the model.
Soil type
Gravity (kN·m−3)
Poisson’s ratio
Elastic modulus (MPa)
Friction angle (°)
Cohesion (kPa)
Permeability coefficient (cm·s−1)
Saturated water content (%)
Sandy
17.5
0.3
50.4
35.23
11.42
5e-5
30


## 2.2. Sliding Failure Evaluation Index of the Slope

By comparison of the slope codes of the earthquake-prone countries (China, Japan, European countries, and California in the United States), evaluation methods of the slope seismic stability in different specifications were determined at home and abroad, as shown in Table2.Table 2
Slope codes of the earthquake-prone countries.
Nation
Codes
Evaluation methods
Evaluation indictors
China
“Technical code for building slope engineering” (GB50330-2013)
Quasi-static method [22, 23]
Safety factor
“Design code for engineered slopes in water resources and hydropower projects” (SL386-2007)
Quasi-static method
Safety factor
“Specifications for design of highway Subgrades” (JTG D30-2004)
Quasi-static method
Safety factor
“Code for seismic design of railway engineering” (GB50111-2006)
Quasi-static method
Safety factor
“Code for design on subgrade of railway” (TB10001-2005)
Quasi-static method
Safety factor
“Code for design of high speed railway” (TB10621-2009)
Quasi-static method
Safety factor
Japan
“Design standards for railway structures”
Quasi-static method and Newmark method
Safety factor and permanent displacement
European countries
Eurocode 7: geotechnical design
Quasi-static method
Safety factor
California in the United States
California Geological Survey’s guidelines (2008)
Quasi-static method
Safety factorBy comparing slope codes in different countries, we used the safety factor and permanent displacement to evaluate the slope stability, which are closer to the actuals.


## 2.3. Numerical Procedure

Referring to JapaneseSpecifications for Highway Bridges, we used three real recorded seismic waves, including T1-II-1 which is a far-field seismic wave, T2-II-1 which is a near-field seismic wave, and El Centro seismic wave, to analyze the slope stability, as shown in Table 3. The peak acceleration values of the three seismic waves are scaled to 0.21g, while the frequencies remain unchanged, as shown in Figure 3. The earthquake motions are used as the horizontal excitations and input from the fixed bottom boundary.Table 3
Parameters of the earthquake motions.
Seismic wave
Recorded location
Earthquake name
Magnitude
Epicentral distance (km)
T1-II-1
Foundation of Itajima bridge
Hyūganada earthquake (1968)
7.5
100
El Centro
El Centro
Emperor Valley earthquake (1940)
7.7
12
T2-II-1
JR Takatori station
Kobe (1995)
7.2
16Figure 3
Time-history of earthquakes (reproduced from the study of Huang et al. [20], under the Creative Commons Attribution License/public domain).
(a)
(b)
(c)Different factors are changed, respectively, on the basis of the original slope. Effects of various factors including slope rates, slope heights, groundwater levels, peak accelerations, and earthquake types on the safety factor and the permanent displacement of the slope are studied, as shown in Table4.Table 4
Numerical procedure.
Slope rate
Slope height (m)
Underground water levelHw (m)
Peak acceleration (m·s−2)
Seismic waves
1 : 1.5
12
5
2, 3, 4, 5, 6, 7, 8, 9, 10
El Centro
1 : 1.4
12
5
4, 5, 6
El Centro
1 : 1.5
12
5
4, 5, 6
1 : 1.6
12
5
4, 5, 6
1 : 1.5
12
6
4, 5, 6
El Centro
7
4, 5, 6
8
4, 5, 6
1 : 1.5
12
5
4, 5, 6
El Centro
24
10
4, 5, 6
El Centro
36
15
4, 5,6
El Centro
1 : 1.5
12
5
4, 5, 6
El Centro, T1-II-1, T2-II-1


## 3. Fitting Relationship of Permanent Displacement and Safety Factor

### 3.1. Determination of Permanent Displacement

The permanent displacement is calculated by the finite element method at a certain groundwater level considering the effect of the large deformation nonlinearity. And we calculated the permanent displacement of the slope under El Centro (2 m/s2) at the groundwater level 5 m, as shown in Figure 4.Figure 4
Permanent displacement calculated by the finite element method.As shown in Figure4, we could calculate the permanent displacement affected by different factors using the finite element method considering the effect of the large deformation nonlinearity. And the maximum permanent displacement of the slope under El Centro is 5.2 cm, when the groundwater level is 5 m.


### 3.2. Determination of the Safety Factor Calculation Method considering Dynamic Pore Water Pressure

Generation of the dynamic pore water pressure is mainly caused by the changes in the deviatoric stress and the average effective stress. Thus, the maximum dynamic pore water pressure could be calculated by the deviatoric stress and the average effective stress of the soil, as shown in Table5. The dynamic hollow cylinder torsional shear test is carried out by the American dynamic hollow cylinder apparatus (GCTS), as shown in Figure 5.Table 5
Test results under earthquakes and sine waves.
Soil compaction
Working condition
Cycle times
Vertical stress (kPa)
Horizontal stress (kPa)
Average effective stress (kPa)
Initial deviatoric stress (kPa)
Seismic waves
Maximum deviatoric stress (kPa)
Stress ratio
Dynamic pore water pressure (kPa)
300
150
200
150
T1-II-1
56.02
0.09
24
2
255
170
198
85
El Centro
51.56
0.26
25
2
300
150
200
150
62.36
0.10
29
2
255
170
198
85
T2-II-1
40.01
0.20
16
2
300
150
200
150
57.96
0.10
26
2
255
170
198
85
Sine wave 1 Hz
48.27
0.24
23
2
300
150
200
150
70.73
0.12
35
2
255
170
198
85
Sine wave 2 Hz
59.21
0.30
30
2
300
150
200
150
57.5
0.095
25
2
255
170
198
85
Sine wave 3 Hz
40.51
0.20
17
2
300
150
200
150
58.60
0.098
26
2
255
170
198
85
Sine wave 4 Hz
50.90
0.26
24
2
300
150
200
150
66.42
0.098
32
2
255
170
198
85
Sine wave 5 Hz
52.70
0.27
26
2
300
150
200
150
57.47
0.096
25
2
255
170
198
85
Sine wave 6 Hz
48.87
0.25
23
2
300
150
200
150
52.47
0.13
28
2Figure 5
Relationship curve of the maximum dynamic pore water pressure and the maximum deviatoric stress.The relations of the maximum dynamic pore water pressure and the maximum deviatoric stress and the average effective stress under different earthquakes and sine waves are analyzed, as shown in Figures5 and 6.Figure 6
Relationship curve of the maximum dynamic pore water pressure and the average effective stress.As shown in Figures5 and 6, the maximum dynamic pore water pressure increases linearly with the increasing maximum deviatoric stress, while the maximum dynamic pore water pressure is decreasing with the increasing average effective stress.Through the nonlinear regression analysis, the simple calculation method of the maximum dynamic pore water pressure could be obtained, as shown in the following equation:(1)Δu=9−0.07σpj′+0.7σy′−σx′d,where Δu is the maximum dynamic pore water pressure, σpj′ is the average effective stress, and σy′−σx′d is the maximum deviatoric stress.The simple calculation method of the maximum dynamic pore pressure is plugged into the limit equilibrium equation, and the safety factor which could consider the effect of the dynamic pore pressure is obtained, as shown in Figure7 and equation (2).Figure 7
Stress state of a free body in the slope.By analyzing the stress state of the isolator in the slope, the limit equilibrium method is shown as follows:(2)Fs=∑c′LWcosθ−u0L−ΔuL−WKhsinθtanφ∑Wsinθ+WKhcosθ,where Fs is the safety factor; W is the weight of the free body; θ is the slope angle; φ is the internal friction angle; c′ is the cohesion; L is the length of the free body parallel to the slope; u0 is the hydrostatic pressure; Δu is the maximum dynamic pore pressure; Kh is the horizontal seismic coefficient, Kh=a/g; N is the normal stress of the sliding plane including the normal stress produced by Wcosθ, WKhsinθ, u0, and Δu; and T is the tangential stress of the sliding plane including the tangential stress produced by c′L, Wsinθ, and WKhcosθ.By using balance equation (2) of the free body, σy′ and σx′ could be obtained, as shown in the following equations:(3)σy′=Wcosθ−u0L+Wsinθ+W′sinθ1−cos2θsin2θ,(4)σx′=Wcosθ−u0L+Wsinθ+W′sinθ1+cos2θsin2θ,where W′ is the thrust of the upper soil to the lower soil.By using equations (3) and (4), equations (5) and (6) could be obtained:(5)σpj′=σy′+2σx′3=2Wcosθ−u0L−Wsinθ+W′sinθ+3Wsinθ+W′sinθcos2θsin2θ,(6)σy′−σx′d=2WKhcosθsin2θ.Equations (5) and (6) are plugged into equation (1), and the maximum dynamic pore pressure Δu could be obtained. Then, Δu is plugged into equation (2), and the limit equilibrium method considering the effect of the maximum dynamic pore water pressure is obtained.


### 3.3. Simple Calculation Method of the Permanent Displacement

Fitting relationship of the permanent displacement and safety factor is shown in Figure8.Figure 8
(a) Relationship of the safety factor and the peak acceleration. (b) Fitting relationship of the permanent displacement and safety factor.
(a)
(b)As shown in Figure8, the safety factor and the permanent displacement change exponentially with the different factors. And the permanent displacements decrease with the increasing safety factor. By fitting analysis of the permanent displacement and the safety factor, the relationship between the permanent displacement and the safety factor could be obtained.Thus, fitting relationship of the permanent displacement and the safety factor could be obtained, as shown in the following equation:(7)δmax=1000e−6.6Fs−2,where δmax is the permanent displacement of the slope and Fs is the safety factor of the slope.Equation (7) considers the influence of peak acceleration, slope height, slope rate, groundwater levels, and spectrum characteristics of the seismic wave, so the permanent displacement could be calculated by this method, which could provide the reference for the seismic reinforcement of engineering designers.


## 3.1. Determination of Permanent Displacement

The permanent displacement is calculated by the finite element method at a certain groundwater level considering the effect of the large deformation nonlinearity. And we calculated the permanent displacement of the slope under El Centro (2 m/s2) at the groundwater level 5 m, as shown in Figure 4.Figure 4
Permanent displacement calculated by the finite element method.As shown in Figure4, we could calculate the permanent displacement affected by different factors using the finite element method considering the effect of the large deformation nonlinearity. And the maximum permanent displacement of the slope under El Centro is 5.2 cm, when the groundwater level is 5 m.


## 3.2. Determination of the Safety Factor Calculation Method considering Dynamic Pore Water Pressure

Generation of the dynamic pore water pressure is mainly caused by the changes in the deviatoric stress and the average effective stress. Thus, the maximum dynamic pore water pressure could be calculated by the deviatoric stress and the average effective stress of the soil, as shown in Table5. The dynamic hollow cylinder torsional shear test is carried out by the American dynamic hollow cylinder apparatus (GCTS), as shown in Figure 5.Table 5
Test results under earthquakes and sine waves.
Soil compaction
Working condition
Cycle times
Vertical stress (kPa)
Horizontal stress (kPa)
Average effective stress (kPa)
Initial deviatoric stress (kPa)
Seismic waves
Maximum deviatoric stress (kPa)
Stress ratio
Dynamic pore water pressure (kPa)
300
150
200
150
T1-II-1
56.02
0.09
24
2
255
170
198
85
El Centro
51.56
0.26
25
2
300
150
200
150
62.36
0.10
29
2
255
170
198
85
T2-II-1
40.01
0.20
16
2
300
150
200
150
57.96
0.10
26
2
255
170
198
85
Sine wave 1 Hz
48.27
0.24
23
2
300
150
200
150
70.73
0.12
35
2
255
170
198
85
Sine wave 2 Hz
59.21
0.30
30
2
300
150
200
150
57.5
0.095
25
2
255
170
198
85
Sine wave 3 Hz
40.51
0.20
17
2
300
150
200
150
58.60
0.098
26
2
255
170
198
85
Sine wave 4 Hz
50.90
0.26
24
2
300
150
200
150
66.42
0.098
32
2
255
170
198
85
Sine wave 5 Hz
52.70
0.27
26
2
300
150
200
150
57.47
0.096
25
2
255
170
198
85
Sine wave 6 Hz
48.87
0.25
23
2
300
150
200
150
52.47
0.13
28
2Figure 5
Relationship curve of the maximum dynamic pore water pressure and the maximum deviatoric stress.The relations of the maximum dynamic pore water pressure and the maximum deviatoric stress and the average effective stress under different earthquakes and sine waves are analyzed, as shown in Figures5 and 6.Figure 6
Relationship curve of the maximum dynamic pore water pressure and the average effective stress.As shown in Figures5 and 6, the maximum dynamic pore water pressure increases linearly with the increasing maximum deviatoric stress, while the maximum dynamic pore water pressure is decreasing with the increasing average effective stress.Through the nonlinear regression analysis, the simple calculation method of the maximum dynamic pore water pressure could be obtained, as shown in the following equation:(1)Δu=9−0.07σpj′+0.7σy′−σx′d,where Δu is the maximum dynamic pore water pressure, σpj′ is the average effective stress, and σy′−σx′d is the maximum deviatoric stress.The simple calculation method of the maximum dynamic pore pressure is plugged into the limit equilibrium equation, and the safety factor which could consider the effect of the dynamic pore pressure is obtained, as shown in Figure7 and equation (2).Figure 7
Stress state of a free body in the slope.By analyzing the stress state of the isolator in the slope, the limit equilibrium method is shown as follows:(2)Fs=∑c′LWcosθ−u0L−ΔuL−WKhsinθtanφ∑Wsinθ+WKhcosθ,where Fs is the safety factor; W is the weight of the free body; θ is the slope angle; φ is the internal friction angle; c′ is the cohesion; L is the length of the free body parallel to the slope; u0 is the hydrostatic pressure; Δu is the maximum dynamic pore pressure; Kh is the horizontal seismic coefficient, Kh=a/g; N is the normal stress of the sliding plane including the normal stress produced by Wcosθ, WKhsinθ, u0, and Δu; and T is the tangential stress of the sliding plane including the tangential stress produced by c′L, Wsinθ, and WKhcosθ.By using balance equation (2) of the free body, σy′ and σx′ could be obtained, as shown in the following equations:(3)σy′=Wcosθ−u0L+Wsinθ+W′sinθ1−cos2θsin2θ,(4)σx′=Wcosθ−u0L+Wsinθ+W′sinθ1+cos2θsin2θ,where W′ is the thrust of the upper soil to the lower soil.By using equations (3) and (4), equations (5) and (6) could be obtained:(5)σpj′=σy′+2σx′3=2Wcosθ−u0L−Wsinθ+W′sinθ+3Wsinθ+W′sinθcos2θsin2θ,(6)σy′−σx′d=2WKhcosθsin2θ.Equations (5) and (6) are plugged into equation (1), and the maximum dynamic pore pressure Δu could be obtained. Then, Δu is plugged into equation (2), and the limit equilibrium method considering the effect of the maximum dynamic pore water pressure is obtained.


## 3.3. Simple Calculation Method of the Permanent Displacement

Fitting relationship of the permanent displacement and safety factor is shown in Figure8.Figure 8
(a) Relationship of the safety factor and the peak acceleration. (b) Fitting relationship of the permanent displacement and safety factor.
(a)
(b)As shown in Figure8, the safety factor and the permanent displacement change exponentially with the different factors. And the permanent displacements decrease with the increasing safety factor. By fitting analysis of the permanent displacement and the safety factor, the relationship between the permanent displacement and the safety factor could be obtained.Thus, fitting relationship of the permanent displacement and the safety factor could be obtained, as shown in the following equation:(7)δmax=1000e−6.6Fs−2,where δmax is the permanent displacement of the slope and Fs is the safety factor of the slope.Equation (7) considers the influence of peak acceleration, slope height, slope rate, groundwater levels, and spectrum characteristics of the seismic wave, so the permanent displacement could be calculated by this method, which could provide the reference for the seismic reinforcement of engineering designers.


## 4. Application Analysis of the Permanent Displacement of a Slope

In order to determine the accuracy of the proposed simple calculation method, a shaking table test of a small soil slope is carried out, as shown in Figure9. The detailed test process can be obtained by referring to reference [20]. The groundwater levels of the slope model are 0 m, 0.6 m, 0.7 m, and 0.8 m. The length, width, and height of the model are 1.96 m, 0.96 m, and 1.4 m. The slope ratio is 1 : 1.5. We use sponge to reduce the seismic reflectance, and the sponge thickness is 20 mm.Figure 9
The small slope model (reproduced from the study of Huang et al. [20], under the Creative Commons Attribution License/public domain).
(a)
(b)In this study, a one-way shaking table (ES-15/KE-2000) is used for testing. There are four technical indicators in this equipment. The maximum test load and acceleration are 5000 kg and 20 m/s2, respectively. The rated speed is 0.5 m/s. And the equipment is shown in Figure 10.Figure 10
Vibration equipment (reproduced from the study of Huang et al. [20], under the Creative Commons Attribution License/public domain).
(a)
(b)The permanent displacements are obtained from the shaking table test under T1-II-1 when the peak acceleration value is 0.4 g, as shown in Figure11.Figure 11
Permanent displacements during the test. (a)Hw = 0 m. (b) Hw = 0.6 m. (c) Hw = 0.7 m. (d) Hw = 0.8 m.
(a)
(b)
(c)
(d)As shown in Figure11, the slope starts to slide at the groundwater level 0.6 m, and the safety factor is less than 1 at this time. Permanent displacement reaches the maximum value until the groundwater level is 0.8 m. The slope just slides but it does not collapse, and the slope is in a new equilibrium, which is usual in practical engineering. The safety factor will not be used for further evaluation of the slope stability when it is less than 1; however, the permanent displacement of the slope is accumulated. Therefore, the permanent displacement could be used for further evaluation of the slope stability. It is found that, with the increasing groundwater levels, the permanent displacement increases. The permanent displacement values will be 3.8 cm, 5.2 cm, and 8.3 cm, respectively, when the groundwater levels are 0.6 m, 0.7 m, and 0.8 m. It shows that the effect of the groundwater level on the permanent displacement should be paid more attention, and certain definitive steps should be taken to maintain the stability of the slope.The permanent displacement obtained from the shaking table test is compared with that calculated by the simple calculation method, as shown in Figure12.Figure 12
Permanent displacements of different methods.As shown in Figure12, the permanent displacements increase with the increasing groundwater levels. The permanent displacement obtained from the shaking table test at the groundwater level 0.8 m is about 4.0 times the permanent displacement at the groundwater level 0 m, which shows that the increase of the groundwater level has a significant effect on the permanent displacement of the slope. Also, we found that the permanent displacements obtained by the simple calculation method are greater than the permanent displacements obtained from the shaking table test. The maximum deviation value obtained from the two methods, respectively, is within 18% when the ground water level is 0.8 m. Therefore, the simple calculation method is feasible, and it could be used to calculate the permanent displacement of the soil slope in different groundwater levels.


## 5. Conclusion

The aim of this research is to lay a foundation for the stability evaluation of the saturated and unsaturated soil slopes using permanent displacement. A series of dynamic hollow cylinder torsional shear tests were conducted under different confining and deviatoric stresses, and a calculation method of dynamic pore water pressure associated with deviatoric and average effective stresses is proposed. The calculation method avoids the solution of complex nonlinear equations and greatly simplifies the computational effort.Based on the proposed calculation method of dynamic pore water pressure combined with the limit equilibrium and finite element methods, we introduced a simple calculation method of permanent displacement, which could provide a reference to the slope seismic reinforcement for engineering designers and be used as a rapid assessment method to the slope seismic stability. Unlike traditional calculation methods of permanent displacement, the proposed calculation method considered the effect of the dynamic pore water pressure.The superior performance of the simple calculation method of the permanent displacement was demonstrated based on the shaking table test. The shaking table test results indicated that the calculation method could rapidly assess the seismic stability of the soil slope considering the effect of dynamic pore water pressure. The permanent displacement values obtained from the simple calculation method were greater than the permanent displacement values obtained from the shaking table test; however, the maximum deviation was within 18%, which verified the simple calculation method is feasible.



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*Source: 1786392-2018-12-19.xml*