Stability analysis of unsaturated soil slopes based on ... - ISSMGE
Stability analysis of unsaturated soil slopes based on deformation characteristics Analyse de stabilité des pentes de sols non saturés basée sur les caractéristiques de déformation Yongmin Kim, Harianto Rahardjo School of Civil and Environmental engineering, Nanyang Technological University, Singapore, [email protected] & [email protected]
ABSTRACT: Matric suction contributes to the shear strength and factor of safety of unsaturated soil slopes. The reduction in shear strength due to the loss of matric suction associated with rainfall infiltration may cause slopes to become unstable. In this study, the results of shearing-infiltration tests were presented to investigate the mechanism associated with rainfall-induced slope failures. Numerical analyses were carried out to study the deformation characteristics of unsaturated soil slopes under infiltration condition. Both hydraulic and mechanical responses of the soil slope are modeled using the commercial software SEEP/W and SIGMA/W. Slope stability was investigated based on the deformation characteristics of the soil slope and the results were compared with those from the limit equilibrium method using SLOPE/W. RÉSUMÉ: La succion dans les pentes de sols non saturés contribue à l’amélioration de la résistance en cisaillement, ainsi que le coefficient de sécurité. La diminution de la résistance en cisaillement due à la réduction de succion associée aux infiltrations d’eaux de pluie, peut provoquer une instabilité de pente. Dans cette étude, les résultats d’essais de cisaillement sous infiltration sont présentés afin d’étudier le mécanisme des ruptures de pente lors de précipitations. De plus, des analyses numériques ont été menées pour déterminer les caractéristiques de déformation des pentes de sols non saturés dans des conditions d’infiltration. La réponse mécanique et hydraulique des pentes est modélisée en utilisant les logiciels commerciaux SEEP/W et SIGMA/W. La stabilité est analysée en se basant sur les caractéristiques de déformation de la pente et les résultats ont été comparés à ceux obtenus par la méthode des équilibres limites en utilisant SLOPE/W. KEYWORDS: Matric suction, Shearing-infiltration, Unsaturated soil, Slope stability, Slope deformation, Limit equilibrium 1 INTRODUCTION Residual soils are commonly found in tropical regions. A deep groundwater table and a significant thickness of unsaturated zone above groundwater table are general characteristics of steep residual soil slopes. Many of residual soil slopes remain stable due to the presence of matric suction within the unsaturated zone that contributes additional shear strength to the soil and factor of safety to the slope. During rainfall, as water infiltrates into the slope, the pore-water pressure in the slope increases and the additional shear strength due to matric suction will decrease or even disappear, causing the slope to be more susceptible to failure. In order to investigate the slope stability during rainfall, the understanding of the relationship between shear strength characteristics and water infiltration is thus essential. Many laboratory tests have been carried out to study failure mechanism of unsaturated soil slopes due to the rainfall infiltration (Melinda et al., 2004; Meilani et al., 2005; Rahardjo et al., 2009). Melinda et al. (2004) conducted direct shear tests under shearing-infiltration conditions to investigate the shear strength characteristics of a residual soil during infiltration. Meilani et al. (2005) and Rahardjo et al. (2009) have conducted triaxial shearing-infiltration tests to study the shear strength characteristics of saturated and unsaturated soils under infiltration condition. However, past research works focused only on experimental works. Although there were few studies on the numerical analysis of shearing-infiltration conditions and its application to the slope stability (Anderson and Sitar, 1995; Ng and Shi, 1998), the deformation characteristics of a soil were rarely incorporated into the assessment of slope stability. In this study, the results of shearing-infiltration tests were presented to investigate the mechanism of rainfall-induced slope failures and numerical analyses were carried out to study the deformation characteristics of an unsaturated soil slope under infiltration condition. Both hydraulic and mechanical responses of the soil slope are modeled using the commercial software SEEP/W and SIGMA/W. Slope stability was
investigated based on deformation of the soil slope and the results were compared with those from the limit equilibrium method using SLOPE/W. 2
SHEARING-INFILTRATION TEST
Shearing-infiltration tests on unsaturated soil specimens were conducted using a modified triaxial apparatus. Shearinginfiltration test is a shear strength test that consists of five stages such as: saturation, consolidation, matric suction equalization, shearing, and infiltration stages. A specific procedure of shearing-infiltration tests was reported by Rahardjo et. al. (2009). Figure 1 shows the stress path followed by the soil specimen in a consolidated drained (CD) triaxial test under a constant matric suction and in an infiltration test under a constant deviator stress. Point O’ represents the initial condition of the soil specimen when the soil specimen is placed in the triaxial cell. The soil specimen has a certain value of matric suction, but under a zero-vertical load. The stress path moves from point O’ to point O during the saturation stage. In the consolidation stage, the stress path moves along path OA. In the matric suction equalization stage, the stress path moves along path AB. The drying path AB represents the condition of a soil slope after a dry period where the soil above the groundwater table experiences a drying process that cause the pore-water pressure to become more negative. Loading of the soil specimen starts at point B until failure following path BB’. Path BC is the constant suction shearing of the soil specimen to a stress level which is 85-90% of peak deviator stress as obtained from the consolidated drained triaxial test. Prior to the shearinginfiltration test, independent consolidated drained triaxial test under a constant matric suction should be carried out to identify peak deviator stress on unsaturated specimen. Path CD represents the infiltration process whereby the matric suction is being reduced by increasing the water content. The soil specimen fails at point D under the infiltration condition.
- 1183 -
Proceedings of the 19th International Conference on Soil Mechanics and Geotechnical Engineering, Seoul 2017
Deviator stress, (σ1-σ3)/2
3 METHODLOGY B’
C D
O’
Constant matric suction plane B
O
A
Net normal stress, (σ1+σ3)/2 − ua
Figure 1. Stress paths of soil specimen during triaxial shearinginfiltration test
Figure 2a shows a typical variation in deviator stress in response to changes in axial strain throughout the shearinginfiltration test of a compacted kaolin (Rahardjo et al., 2009). Once failure occurred, the deviator stress could no longer be maintained by the force actuator. The deviator stress dropped gradually due to the infiltration process. Figure 2b shows the relationship between axial strain and elapsed time during the infiltration stage. The axial strain increased sharply after failure occurred. In order to find the axial strain at failure, two tangent lines can be drawn on the axial strain versus elapsed time curve. The intersection point indicates that the axial strain at failure is 7.1%. A similar procedure can also be applied to determine strain at failure in a soil slope under rainfall infiltration by observing deformation characteristics of the soil slope during the infiltration process.
In order to investigate the slope stability of unsaturated soils, three different soil types (ks = 6.03x10-6 m/s, 1.03x10-6 m/s, and 6.03x10-7 m/s) and slope angles (θ = 45°, 60°, and 75°) were selected to represent various soil slopes in Singapore. The soil slopes were subjected to a rainfall of 22 mm/h for a duration of 72 hours. Seepage analysis was performed to compute the pore-water pressure changes due to the rainfall. The computed pore-water pressures were then used as initial condition to conduct stressstrain analyses in order to assess the slope deformation and to conduct slope stability analyses in order to assess the factor of safety of the slope. The induced horizontal and vertical displacements at three different locations, such as slope crest, slope midpoint, and slope toe that corresponded to the location of a critical slip surface inferred from slope stability analyses were investigated. Based on the slope deformations, a dimensionless displacement (Esδ/γH2) given by Zienkiewicz et al (2005) was calculated to define a non-convergence of numerical solution, where Es is the Young’s modulus of the soil, δ is the displacement of the slope along the critical slip surface, γ is the unit weight, and H is the slope height. In order to find the time (or pore-water pressure) at failure, two tangent lines were drawn on the relationship between the dimensionless displacement versus elapsed time (or pore-water pressure). The intersection points of these tangent lines indicate the failure condition of the slope. The procedure for evaluating the unsaturated soil slope stability followed in this study is summarized succinctly using Figure 3. Stability analysis of unsaturated soil slopes
Seepage analysis (SEEP/W model) Pore-water pressure distribution
Slope stability analysis (SLOPE/W model) Limit equilibrium method Factor of safety & Critical slip surface
Stress-strain analysis (SIGMA/W model) Finite element method Deformation along the critical slip surface
Evaluation of slope stability based on deformation and limit equilibrium criteria
Figure 3. Stability analysis procedure based on stress-strain and limit equilibrium analysis
4 NUMERICAL MODELING (a) Variation in deviator stress in response to changes in axial strain
(b) Determination of failure point Figure 2. Laboratory tests results of shearing-infiltration under 200 kPa net normal stress and 100 kPa matric suction
First, seepage analyses were conducted using a numerical model of soil slope using SEEP/W (Geo-Slope, 2012) software. A 20m high model slope with an inclination of 60° was used in this study as a representative slope in Singapore, as shown in Figure 4. The groundwater table was assumed at a typical position in residual soil slopes in Singapore. The infiltration process of the rainwater into the soil was simulated using a saturated-unsaturated finite element analysis. The boundary conditions used for the transient seepage analysis are also shown in Figure 4. The boundary flux, q, equal to rainfall intensity, Ir = 22 mm/h was applied to the surface of the slope. The nodal flux, Q, equal to zero was applied along the sides of the slope above the groundwater table and along the bottom of the slope to simulate no flow zone. A boundary condition equal to total head, hw, was applied along the sides of the slope below the groundwater table. Initial condition for the slope model was taken as hydrostatic pore-water pressure condition with a limiting negative pore-water pressure of 50kPa. Second, stress-strain analyses were conducted using SIGMA/W (Geo-Slope, 2012) software for the load/deformation modelling of the slope. The finite element model was a two-dimensional plane strain model. The finite
- 1184 -
Technical Committee 106 / Comité technique 106
element mesh of the slope model in SEEP/W was imported to SIGMA/W. The pore-water pressure was computed by seepage analyses and used as initial pore-water pressure for load/deformation analyses. The fixed boundary condition of the slope model was assumed that horizontal displacement was fixed on both side boundaries and displacements in both directions were fixed on the bottom boundary of the model. The pore-water pressure distribution was selected for each time increment, and the corresponding deformation was calculated. For stress-strain analyses, the soil was modelled as an elastic-plastic material. The elasticity form of the volumetric strain constitutive relationship for an unsaturated soil under plane strain given by Fredlund and Rahardjo (1993) is written as follows: 2(1 )(12) 1 dv d(ave ua ) 2 d(ua uw) (1)
E
H
where dεv is the volumetric strain increment, d(σave-ua) and d(ua-uw) are the net normal stress and matric suction increment, respectively. In SIGMA/W, the E and H are related by Eq. 2.
12
No flow BC, Q=0m3/m3
11
E
Flux BC, q=22 mm/h, 72hr
9
60o
8
No flow BC, Q=0m3/m3
7
Elevation
(2) 12 This equation is only fundamentally correct in saturated condition as SIGNA/W has not included more complex unsaturated relationship condition between E and H. Third, slope stability analyses were conducted using SLOPE/W (Geo-Slope, 2012). The finite element mesh and pore-water pressure distribution of the slope model in SEEP/W were imported to SLOPE/W. The typical saturated and unsaturated shear strengths for the residual soils were used in the slope stability analyses using Morgenstern-Price method. The pore-water pressure distribution was selected for each time increment and the corresponding factor of safety was calculated. Permeability functions for three primary soil properties used in the seepage analyses are shown in Figure 5. The saturated coefficients of permeability of the three soils used in the analyses were equal to 6.03 10-6 m/s, 1.03 10-6 m/s, and 6.03 10-7 m/s. The SWCC parameters given by Fredlund and Xing (1994) were a = 28.4 kPa, n = 0.88, m = 0.26 for the three soils. Elastic and shear strength properties of the soil used in this study were selected based on laboratory and field tests of soils at Orchard boulevard in Singapore. The elastic modulus, E = 6,000 kPa and the Poisson’s ratio, μ = 0.33 were used in the stress-strain analyses. An effective cohesion, c’ = 5 kPa, effective angle of internal friction, ' = 28°, angle indicating the rate of change in shear strength relative to a change in matric suction, b = 28°, and unit weight of soil, γt = 22 kN/m3, were used in the stress-strain and slope stability analyses. All properties of the soils were kept constant for all cases to ensure that changes in stability of the slope were only attributed to pore-water pressure changes in the soil.
10
Groundwater table
6 5
Total head BC, H=10.5m
4 3 2
Total head BC, H=3.5m
1
No flow BC, Q=0m3/m3 0 -1 -1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Figure 4. Slope geometry and boundary conditions 1x10-5 1x10-6 1x10-7 1x10-8 1x10-9 1x10-10 1x10-11
6.03x10-6 m/s 1.03x10-6 m/s 6.03x10-7 m/s
1x10-12 0.01
0.1
1
10
100
1000
Matric suction, kPa
Figure 5. Permeability functions used in numerical modelling 2.2
5 NUMERICAL RESULTS AND DISSCUSSION
45o, 6.03x10-7 m/s 60o, 6.03x10-7 m/s 75o, 6.03x10-7 m/s
2
Factor of safety
Figure 6 provides the results obtained from the SLOPE/W analyses of various soil slopes with different soil permeabilities. Figure 6 shows that the rate of decrease in FS versus time was faster for the steep slope, followed by the average angle and gentle slope. This figure also shows that the rate of decrease in FS was faster for the more permeable soils compared to the less permeable soils. In other words, the steep slope with highly permeable soil resulted in the earliest time to reach the minimum FS. Figure 7 shows the slope deformations at slope crest, slope midpoint, and slope toe along the critical slip surface, as shown in Figure 8. The critical slip surface was determined from SLOPE/W analyses. Figure 7 shows horizontal displacements for slope angle of 60° and saturated permeability of 6.03x10-7
25
Distance
Coefficient of permeability, m/sec
H
m/s with time during rainfall. The results show that the deformation of the soil slope increased with time during rainfall, especially the rate of increase in deformation increased rapidly after 60 h which corresponded to FS of less than 1.0. Figure 9a shows the relationship between the dimensionless displacement and pore-water pressure and Figure 9b shows the relationship between the dimensionless displacement and elapsed time at different locations during the rainfall. In order to find the failure condition, two tangent lines were drawn on each curve in Figure 9. The dimensionless displacement at three locations increased sharply when the pore-water pressure reached 0 kPa, which can be considered as a saturated condition, as shown in Figure 9a. Similarly, the dimensionless displacement at three locations increased rapidly at an elapsed time of 58 h, as shown in Figure 9b. The intersection points indicate that the pore-water pressure at failure is 0 kPa and the time at failure is approximately 58 h.
6.03x10-7 m/s, 60o 1.03x10-6 m/s, 60o 6.03x10-6 m/s, 60o
1.8
1.6
1.4
1.2
1 0
12
24
36
48
60
Elapsed time, h
Figure 6. Factor of safety for various soil types and slope angles
- 1185 -
72
Proceedings of the 19th International Conference on Soil Mechanics and Geotechnical Engineering, Seoul 2017
SLOPE/W analyses was observed in a gentle slope (45°). This could be attributed to the fact that less rainwater infiltration took place in the soil slope with a steep slope and most rainfall became runoff on the steep slope. 6 CONCLUSIONS Based on the numerical analyses of unsaturated soil slopes subjected to rainfall infiltration, it was found that the deformation of soil slopes increased gradually with time during rainfall and increased rapidly when the slope failed. The slope with a highly permeable soil and a steep slope resulted in the earliest time to failure. In addition, slope failure can be estimated by plotting the results that represent the relationship between dimensionless displacement versus pore-water pressure and/or elapsed time. In addition, stress-strain finite element analysis is able to capture the progressive failure phenomena that result in an earlier time to failure than the limit equilibrium analysis. Therefore, slope stability during rainfall can be investigated using stress-strain analyses as well as limit equilibrium analyses.
Figure 7. Deformation of soil slope with time during rainfall
0
Slope crest
Slope middle 0.999
Slope angle, deg
0
12
Slope toe
45
11
75
Slope angle
9
20
SLOPE/W model SIGMA/W model
20
8
80
7
40
6
0 0
5 4
Failure time, h
Elevation
60
100
10
20
3
60
2
40
80
1
60
40
100
0 -1 -1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
20
25
Distance
Coefficient of permeability SLOPE/W model SIGMA/W model
Figure 8. Location of the observed deformations on the slope 0
Dimensionless diaplacement
8
1.03x10 1 -7 m/s
(a)
(b)
Slope crest Slope middle Slope toe
Slope crest Slope middle Slope toe
7
4
2
0 -20
-10
0
Pore-water pressure, kPa
10 0
6.03x10 3 -6 m/s
Figure 10. Factor of safety for various soil types and slope angles
6
-30
1.03x10 2 -6 m/s
Coefficient of permeability, m/s
20
40
60
80
Elapsed time, h
Figure 9. Dimensionless displacement of soil slope
Figure 10 shows the comparison results between SLOPE/W and SIGMA/W analyses. The results show that time to failure from SIGMA/W analyses tends to be earlier than SLOPE/W analyses. This could be due to the fact that the limit equilibrium solution provides only one average FS for the critical slip surface without considering the stress-strain distribution in the slope due to rainfall infiltration, whereas the stress-strain analyses performed in this study considers the deformation characteristics from the stress-strain distribution in the slope at several points in the slope. Therefore, the stress-strain analyses can detect potential failure of the slope earlier. In addition, a larger difference in failure time between SIGMA/W and
REFERENCES
Anderson, S., and Sitar, N. 1995. Analysis of rainfall-induced debris flows. Journal of Geotechnical Engineering, 121(7): 544–552. Fredlund D.G., Morgenstern, N.R., Widger, R.A. 1978. The shear strength of unsaturated soils. Canadian Geotechnical Journal, 15(3): 313–321. Fredlund, D.G., and Xing, A. 1994. Equations for the soil-water characteristic curve. Canadian Geotechnical Journal, 31(4):521– 532. Fredlund, G.G., and Rahardjo, H. 1993. Soil mechanics for unsaturated soils. Joh Wiley & Sons, Inc., New York. GeoSlope. 2012. User’s Manual, Geo-Slope International Ltd, Calgary, Alberta, Canada. Meilani, I., Rahardjo, H., and Leong, E.C. 2005. Pore-water pressure and water volume change of an unsaturated soil under infiltration conditions. Canadian Geotechnical Journal, 42(6): 1509–1531. Melinda, F., Rahardjo, H., Han, K.K., and Leong, E.C. 2004. Shear strength of compacted soil under infiltration condition. Journal of Geotechnical and Geoenvironmental Engineering, 130(8): 807–817. Ng, C.W.W., and Shi, Q. 1998. A numerical investigation of the stability of unsaturated soil slopes subjected to transient seepage. Computers and Geotechnics, 22(1): 1–28. Rahardjo, H., Meilani, I., and Leong, E. C. 2009. Shear strength characteristics of a compacted soil under infiltration conditions. Geomechanics and Engineering, 1(1): 35–52. Zienkiewicz, OC, Taylor, RL, Nithiarasu, P. 2005. The finite element method. 6th ed. Butterworth-Heinemann.
- 1186 -
ABSTRACT: Matric suction contributes to the shear strength and factor of safety of unsaturated soil slopes. The reduction in shear strength due to the loss of matric suction associated with rainfall infiltration may cause slopes to become unstable. In this study, the results of shearing-infiltration tests were presented to investigate the mechanism associated with rainfall-induced slope failures. Numerical analyses were carried out to study the deformation characteristics of unsaturated soil slopes under infiltration condition. Both hydraulic and mechanical responses of the soil slope are modeled using the commercial software SEEP/W and SIGMA/W. Slope stability was investigated based on the deformation characteristics of the soil slope and the results were compared with those from the limit equilibrium method using SLOPE/W. RÉSUMÉ: La succion dans les pentes de sols non saturés contribue à l’amélioration de la résistance en cisaillement, ainsi que le coefficient de sécurité. La diminution de la résistance en cisaillement due à la réduction de succion associée aux infiltrations d’eaux de pluie, peut provoquer une instabilité de pente. Dans cette étude, les résultats d’essais de cisaillement sous infiltration sont présentés afin d’étudier le mécanisme des ruptures de pente lors de précipitations. De plus, des analyses numériques ont été menées pour déterminer les caractéristiques de déformation des pentes de sols non saturés dans des conditions d’infiltration. La réponse mécanique et hydraulique des pentes est modélisée en utilisant les logiciels commerciaux SEEP/W et SIGMA/W. La stabilité est analysée en se basant sur les caractéristiques de déformation de la pente et les résultats ont été comparés à ceux obtenus par la méthode des équilibres limites en utilisant SLOPE/W. KEYWORDS: Matric suction, Shearing-infiltration, Unsaturated soil, Slope stability, Slope deformation, Limit equilibrium 1 INTRODUCTION Residual soils are commonly found in tropical regions. A deep groundwater table and a significant thickness of unsaturated zone above groundwater table are general characteristics of steep residual soil slopes. Many of residual soil slopes remain stable due to the presence of matric suction within the unsaturated zone that contributes additional shear strength to the soil and factor of safety to the slope. During rainfall, as water infiltrates into the slope, the pore-water pressure in the slope increases and the additional shear strength due to matric suction will decrease or even disappear, causing the slope to be more susceptible to failure. In order to investigate the slope stability during rainfall, the understanding of the relationship between shear strength characteristics and water infiltration is thus essential. Many laboratory tests have been carried out to study failure mechanism of unsaturated soil slopes due to the rainfall infiltration (Melinda et al., 2004; Meilani et al., 2005; Rahardjo et al., 2009). Melinda et al. (2004) conducted direct shear tests under shearing-infiltration conditions to investigate the shear strength characteristics of a residual soil during infiltration. Meilani et al. (2005) and Rahardjo et al. (2009) have conducted triaxial shearing-infiltration tests to study the shear strength characteristics of saturated and unsaturated soils under infiltration condition. However, past research works focused only on experimental works. Although there were few studies on the numerical analysis of shearing-infiltration conditions and its application to the slope stability (Anderson and Sitar, 1995; Ng and Shi, 1998), the deformation characteristics of a soil were rarely incorporated into the assessment of slope stability. In this study, the results of shearing-infiltration tests were presented to investigate the mechanism of rainfall-induced slope failures and numerical analyses were carried out to study the deformation characteristics of an unsaturated soil slope under infiltration condition. Both hydraulic and mechanical responses of the soil slope are modeled using the commercial software SEEP/W and SIGMA/W. Slope stability was
investigated based on deformation of the soil slope and the results were compared with those from the limit equilibrium method using SLOPE/W. 2
SHEARING-INFILTRATION TEST
Shearing-infiltration tests on unsaturated soil specimens were conducted using a modified triaxial apparatus. Shearinginfiltration test is a shear strength test that consists of five stages such as: saturation, consolidation, matric suction equalization, shearing, and infiltration stages. A specific procedure of shearing-infiltration tests was reported by Rahardjo et. al. (2009). Figure 1 shows the stress path followed by the soil specimen in a consolidated drained (CD) triaxial test under a constant matric suction and in an infiltration test under a constant deviator stress. Point O’ represents the initial condition of the soil specimen when the soil specimen is placed in the triaxial cell. The soil specimen has a certain value of matric suction, but under a zero-vertical load. The stress path moves from point O’ to point O during the saturation stage. In the consolidation stage, the stress path moves along path OA. In the matric suction equalization stage, the stress path moves along path AB. The drying path AB represents the condition of a soil slope after a dry period where the soil above the groundwater table experiences a drying process that cause the pore-water pressure to become more negative. Loading of the soil specimen starts at point B until failure following path BB’. Path BC is the constant suction shearing of the soil specimen to a stress level which is 85-90% of peak deviator stress as obtained from the consolidated drained triaxial test. Prior to the shearinginfiltration test, independent consolidated drained triaxial test under a constant matric suction should be carried out to identify peak deviator stress on unsaturated specimen. Path CD represents the infiltration process whereby the matric suction is being reduced by increasing the water content. The soil specimen fails at point D under the infiltration condition.
- 1183 -
Proceedings of the 19th International Conference on Soil Mechanics and Geotechnical Engineering, Seoul 2017
Deviator stress, (σ1-σ3)/2
3 METHODLOGY B’
C D
O’
Constant matric suction plane B
O
A
Net normal stress, (σ1+σ3)/2 − ua
Figure 1. Stress paths of soil specimen during triaxial shearinginfiltration test
Figure 2a shows a typical variation in deviator stress in response to changes in axial strain throughout the shearinginfiltration test of a compacted kaolin (Rahardjo et al., 2009). Once failure occurred, the deviator stress could no longer be maintained by the force actuator. The deviator stress dropped gradually due to the infiltration process. Figure 2b shows the relationship between axial strain and elapsed time during the infiltration stage. The axial strain increased sharply after failure occurred. In order to find the axial strain at failure, two tangent lines can be drawn on the axial strain versus elapsed time curve. The intersection point indicates that the axial strain at failure is 7.1%. A similar procedure can also be applied to determine strain at failure in a soil slope under rainfall infiltration by observing deformation characteristics of the soil slope during the infiltration process.
In order to investigate the slope stability of unsaturated soils, three different soil types (ks = 6.03x10-6 m/s, 1.03x10-6 m/s, and 6.03x10-7 m/s) and slope angles (θ = 45°, 60°, and 75°) were selected to represent various soil slopes in Singapore. The soil slopes were subjected to a rainfall of 22 mm/h for a duration of 72 hours. Seepage analysis was performed to compute the pore-water pressure changes due to the rainfall. The computed pore-water pressures were then used as initial condition to conduct stressstrain analyses in order to assess the slope deformation and to conduct slope stability analyses in order to assess the factor of safety of the slope. The induced horizontal and vertical displacements at three different locations, such as slope crest, slope midpoint, and slope toe that corresponded to the location of a critical slip surface inferred from slope stability analyses were investigated. Based on the slope deformations, a dimensionless displacement (Esδ/γH2) given by Zienkiewicz et al (2005) was calculated to define a non-convergence of numerical solution, where Es is the Young’s modulus of the soil, δ is the displacement of the slope along the critical slip surface, γ is the unit weight, and H is the slope height. In order to find the time (or pore-water pressure) at failure, two tangent lines were drawn on the relationship between the dimensionless displacement versus elapsed time (or pore-water pressure). The intersection points of these tangent lines indicate the failure condition of the slope. The procedure for evaluating the unsaturated soil slope stability followed in this study is summarized succinctly using Figure 3. Stability analysis of unsaturated soil slopes
Seepage analysis (SEEP/W model) Pore-water pressure distribution
Slope stability analysis (SLOPE/W model) Limit equilibrium method Factor of safety & Critical slip surface
Stress-strain analysis (SIGMA/W model) Finite element method Deformation along the critical slip surface
Evaluation of slope stability based on deformation and limit equilibrium criteria
Figure 3. Stability analysis procedure based on stress-strain and limit equilibrium analysis
4 NUMERICAL MODELING (a) Variation in deviator stress in response to changes in axial strain
(b) Determination of failure point Figure 2. Laboratory tests results of shearing-infiltration under 200 kPa net normal stress and 100 kPa matric suction
First, seepage analyses were conducted using a numerical model of soil slope using SEEP/W (Geo-Slope, 2012) software. A 20m high model slope with an inclination of 60° was used in this study as a representative slope in Singapore, as shown in Figure 4. The groundwater table was assumed at a typical position in residual soil slopes in Singapore. The infiltration process of the rainwater into the soil was simulated using a saturated-unsaturated finite element analysis. The boundary conditions used for the transient seepage analysis are also shown in Figure 4. The boundary flux, q, equal to rainfall intensity, Ir = 22 mm/h was applied to the surface of the slope. The nodal flux, Q, equal to zero was applied along the sides of the slope above the groundwater table and along the bottom of the slope to simulate no flow zone. A boundary condition equal to total head, hw, was applied along the sides of the slope below the groundwater table. Initial condition for the slope model was taken as hydrostatic pore-water pressure condition with a limiting negative pore-water pressure of 50kPa. Second, stress-strain analyses were conducted using SIGMA/W (Geo-Slope, 2012) software for the load/deformation modelling of the slope. The finite element model was a two-dimensional plane strain model. The finite
- 1184 -
Technical Committee 106 / Comité technique 106
element mesh of the slope model in SEEP/W was imported to SIGMA/W. The pore-water pressure was computed by seepage analyses and used as initial pore-water pressure for load/deformation analyses. The fixed boundary condition of the slope model was assumed that horizontal displacement was fixed on both side boundaries and displacements in both directions were fixed on the bottom boundary of the model. The pore-water pressure distribution was selected for each time increment, and the corresponding deformation was calculated. For stress-strain analyses, the soil was modelled as an elastic-plastic material. The elasticity form of the volumetric strain constitutive relationship for an unsaturated soil under plane strain given by Fredlund and Rahardjo (1993) is written as follows: 2(1 )(12) 1 dv d(ave ua ) 2 d(ua uw) (1)
E
H
where dεv is the volumetric strain increment, d(σave-ua) and d(ua-uw) are the net normal stress and matric suction increment, respectively. In SIGMA/W, the E and H are related by Eq. 2.
12
No flow BC, Q=0m3/m3
11
E
Flux BC, q=22 mm/h, 72hr
9
60o
8
No flow BC, Q=0m3/m3
7
Elevation
(2) 12 This equation is only fundamentally correct in saturated condition as SIGNA/W has not included more complex unsaturated relationship condition between E and H. Third, slope stability analyses were conducted using SLOPE/W (Geo-Slope, 2012). The finite element mesh and pore-water pressure distribution of the slope model in SEEP/W were imported to SLOPE/W. The typical saturated and unsaturated shear strengths for the residual soils were used in the slope stability analyses using Morgenstern-Price method. The pore-water pressure distribution was selected for each time increment and the corresponding factor of safety was calculated. Permeability functions for three primary soil properties used in the seepage analyses are shown in Figure 5. The saturated coefficients of permeability of the three soils used in the analyses were equal to 6.03 10-6 m/s, 1.03 10-6 m/s, and 6.03 10-7 m/s. The SWCC parameters given by Fredlund and Xing (1994) were a = 28.4 kPa, n = 0.88, m = 0.26 for the three soils. Elastic and shear strength properties of the soil used in this study were selected based on laboratory and field tests of soils at Orchard boulevard in Singapore. The elastic modulus, E = 6,000 kPa and the Poisson’s ratio, μ = 0.33 were used in the stress-strain analyses. An effective cohesion, c’ = 5 kPa, effective angle of internal friction, ' = 28°, angle indicating the rate of change in shear strength relative to a change in matric suction, b = 28°, and unit weight of soil, γt = 22 kN/m3, were used in the stress-strain and slope stability analyses. All properties of the soils were kept constant for all cases to ensure that changes in stability of the slope were only attributed to pore-water pressure changes in the soil.
10
Groundwater table
6 5
Total head BC, H=10.5m
4 3 2
Total head BC, H=3.5m
1
No flow BC, Q=0m3/m3 0 -1 -1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Figure 4. Slope geometry and boundary conditions 1x10-5 1x10-6 1x10-7 1x10-8 1x10-9 1x10-10 1x10-11
6.03x10-6 m/s 1.03x10-6 m/s 6.03x10-7 m/s
1x10-12 0.01
0.1
1
10
100
1000
Matric suction, kPa
Figure 5. Permeability functions used in numerical modelling 2.2
5 NUMERICAL RESULTS AND DISSCUSSION
45o, 6.03x10-7 m/s 60o, 6.03x10-7 m/s 75o, 6.03x10-7 m/s
2
Factor of safety
Figure 6 provides the results obtained from the SLOPE/W analyses of various soil slopes with different soil permeabilities. Figure 6 shows that the rate of decrease in FS versus time was faster for the steep slope, followed by the average angle and gentle slope. This figure also shows that the rate of decrease in FS was faster for the more permeable soils compared to the less permeable soils. In other words, the steep slope with highly permeable soil resulted in the earliest time to reach the minimum FS. Figure 7 shows the slope deformations at slope crest, slope midpoint, and slope toe along the critical slip surface, as shown in Figure 8. The critical slip surface was determined from SLOPE/W analyses. Figure 7 shows horizontal displacements for slope angle of 60° and saturated permeability of 6.03x10-7
25
Distance
Coefficient of permeability, m/sec
H
m/s with time during rainfall. The results show that the deformation of the soil slope increased with time during rainfall, especially the rate of increase in deformation increased rapidly after 60 h which corresponded to FS of less than 1.0. Figure 9a shows the relationship between the dimensionless displacement and pore-water pressure and Figure 9b shows the relationship between the dimensionless displacement and elapsed time at different locations during the rainfall. In order to find the failure condition, two tangent lines were drawn on each curve in Figure 9. The dimensionless displacement at three locations increased sharply when the pore-water pressure reached 0 kPa, which can be considered as a saturated condition, as shown in Figure 9a. Similarly, the dimensionless displacement at three locations increased rapidly at an elapsed time of 58 h, as shown in Figure 9b. The intersection points indicate that the pore-water pressure at failure is 0 kPa and the time at failure is approximately 58 h.
6.03x10-7 m/s, 60o 1.03x10-6 m/s, 60o 6.03x10-6 m/s, 60o
1.8
1.6
1.4
1.2
1 0
12
24
36
48
60
Elapsed time, h
Figure 6. Factor of safety for various soil types and slope angles
- 1185 -
72
Proceedings of the 19th International Conference on Soil Mechanics and Geotechnical Engineering, Seoul 2017
SLOPE/W analyses was observed in a gentle slope (45°). This could be attributed to the fact that less rainwater infiltration took place in the soil slope with a steep slope and most rainfall became runoff on the steep slope. 6 CONCLUSIONS Based on the numerical analyses of unsaturated soil slopes subjected to rainfall infiltration, it was found that the deformation of soil slopes increased gradually with time during rainfall and increased rapidly when the slope failed. The slope with a highly permeable soil and a steep slope resulted in the earliest time to failure. In addition, slope failure can be estimated by plotting the results that represent the relationship between dimensionless displacement versus pore-water pressure and/or elapsed time. In addition, stress-strain finite element analysis is able to capture the progressive failure phenomena that result in an earlier time to failure than the limit equilibrium analysis. Therefore, slope stability during rainfall can be investigated using stress-strain analyses as well as limit equilibrium analyses.
Figure 7. Deformation of soil slope with time during rainfall
0
Slope crest
Slope middle 0.999
Slope angle, deg
0
12
Slope toe
45
11
75
Slope angle
9
20
SLOPE/W model SIGMA/W model
20
8
80
7
40
6
0 0
5 4
Failure time, h
Elevation
60
100
10
20
3
60
2
40
80
1
60
40
100
0 -1 -1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
20
25
Distance
Coefficient of permeability SLOPE/W model SIGMA/W model
Figure 8. Location of the observed deformations on the slope 0
Dimensionless diaplacement
8
1.03x10 1 -7 m/s
(a)
(b)
Slope crest Slope middle Slope toe
Slope crest Slope middle Slope toe
7
4
2
0 -20
-10
0
Pore-water pressure, kPa
10 0
6.03x10 3 -6 m/s
Figure 10. Factor of safety for various soil types and slope angles
6
-30
1.03x10 2 -6 m/s
Coefficient of permeability, m/s
20
40
60
80
Elapsed time, h
Figure 9. Dimensionless displacement of soil slope
Figure 10 shows the comparison results between SLOPE/W and SIGMA/W analyses. The results show that time to failure from SIGMA/W analyses tends to be earlier than SLOPE/W analyses. This could be due to the fact that the limit equilibrium solution provides only one average FS for the critical slip surface without considering the stress-strain distribution in the slope due to rainfall infiltration, whereas the stress-strain analyses performed in this study considers the deformation characteristics from the stress-strain distribution in the slope at several points in the slope. Therefore, the stress-strain analyses can detect potential failure of the slope earlier. In addition, a larger difference in failure time between SIGMA/W and
REFERENCES
Anderson, S., and Sitar, N. 1995. Analysis of rainfall-induced debris flows. Journal of Geotechnical Engineering, 121(7): 544–552. Fredlund D.G., Morgenstern, N.R., Widger, R.A. 1978. The shear strength of unsaturated soils. Canadian Geotechnical Journal, 15(3): 313–321. Fredlund, D.G., and Xing, A. 1994. Equations for the soil-water characteristic curve. Canadian Geotechnical Journal, 31(4):521– 532. Fredlund, G.G., and Rahardjo, H. 1993. Soil mechanics for unsaturated soils. Joh Wiley & Sons, Inc., New York. GeoSlope. 2012. User’s Manual, Geo-Slope International Ltd, Calgary, Alberta, Canada. Meilani, I., Rahardjo, H., and Leong, E.C. 2005. Pore-water pressure and water volume change of an unsaturated soil under infiltration conditions. Canadian Geotechnical Journal, 42(6): 1509–1531. Melinda, F., Rahardjo, H., Han, K.K., and Leong, E.C. 2004. Shear strength of compacted soil under infiltration condition. Journal of Geotechnical and Geoenvironmental Engineering, 130(8): 807–817. Ng, C.W.W., and Shi, Q. 1998. A numerical investigation of the stability of unsaturated soil slopes subjected to transient seepage. Computers and Geotechnics, 22(1): 1–28. Rahardjo, H., Meilani, I., and Leong, E. C. 2009. Shear strength characteristics of a compacted soil under infiltration conditions. Geomechanics and Engineering, 1(1): 35–52. Zienkiewicz, OC, Taylor, RL, Nithiarasu, P. 2005. The finite element method. 6th ed. Butterworth-Heinemann.
- 1186 -
