Metamodel-Based Slope Reliability Analysis—Case of Spatially Variable Soils Considering a Rotated Anisotropy
Abstract
:1. Introduction
2. Random Field Generation
2.1. Spatial Correlation
2.2. K–L Expansion
2.3. Cross-Correlated Log-Normal Random Fields
3. Methodology
3.1. Deterministic Methods
3.1.1. Analytical Model: DKA
3.1.2. Numerical Model: FELA
3.2. Probabilistic Methods
3.2.1. Sparse Polynomial Chaos Expansion
3.2.2. Global Sensitivity Analysis
3.2.3. Combination of SPCE and GSA
4. Proposed Procedure and Studied Slope
4.1. Procedure of the Current Study
4.2. Reference Case and Conducted Results
5. Validation and Efficiency Investigation of the Proposed Procedure
5.1. Comparison with a Previous Study
5.2. DKA Accuracy Considering Spatially Varying Soils and Rotated Anisotropy
5.3. Comparison of SPCE/GSA and Direct MCS
6. Effects of Rotated Anisotropy Considering Different Influential Factors
6.1. Effect of the Rotation Angle
6.2. Effect of the Autocorrelation Length
6.3. Effect of the Cross-Correlation
6.4. Effect of the Coefficient of Variation
7. Conclusions
- (1)
- The proposed procedure DSG–MG provides a good insight for the probabilistic stability analyses of slopes by the fact that the analytical method DKA can capture accurately the spatially varied parameters within the random field generation and can give rational results efficiently compared to the FELA ones (5 s and 60 s, respectively, for one deterministic calculation with the two methods); the metamodel constructed using SPCE/GSA can reduce the problem dimension and also the number of deterministic simulations by comparing with the direct MCS; several interesting results (the failure probability, probability density function, statistical moments of the model response, and sensitivity index of each variable) can also be obtained effectively.
- (2)
- The rotation of the anisotropic soil fabric pattern has a significant effect on slope stability. The failure probability is increased drastically when the rotation angle approaches the slope inclination. Using the traditional horizontal random field will then overestimate greatly the slope reliability, particularly for the cases with larger values of autocorrelation length, cross-correlation, and coefficient of variation. Conversely, the slope is safer when the rotated stratification is perpendicular to the slope inclination.
- (3)
- The failure probability is increased with the increase of autocorrelation lengths, coefficient of variation, and cross-correlation coefficient, and the effects of these parameters are more significant when the soil stratification rotation angle is close to the slope inclination, which should be determined with caution.
- (4)
- The rotation of soil stratification and autocorrelation length have almost no influence on the sensitivity index of the cohesion and friction angle, and the influence of the friction angle on the model response variance is higher than the cohesion angle. Conversely, the cross-correlation coefficient and coefficient of variation influence significantly the sensitivity indices, and the Sobol index of cohesion is increased with the cross-correlation coefficient r and coefficient of variation of cohesion COVc increase. The friction angle case presents an opposite trend.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
FS | safety factor |
GSA | Global Sensitivity Analysis |
H | slope height (Figure 2) |
lh, lv | autocorrelation distances in the horizontal and vertical directions |
m | horizontal distance of CD (Figure 2) |
Pf | failure probability |
r0 | distance from point O to point C (Figure 2) |
r | cross-correlation coefficient |
SPCE | Sparse Polynomial Chaos Expansion |
vi | velocity vector at point Pi (Figure 2) |
ω | angular velocity (Figure 2) |
φi | friction angle at point Pi (Figure 2) |
δθ | angle between OPi and OPi + 1 (Figure 2) |
θ0 | angle between the x axis direction and line OC (Figure 2) |
θi | angle between the x axis direction and line OPi (Figure 2) |
θH | angle between the x axis direction and line OPn (Figure 2) |
β | rotation angle of anisotropy stratification |
μ | mean value |
μln | mean value of the log-normal random field |
σ | standard deviation |
σln | standard deviation of the log-normal random field |
ρ | autocorrelation function |
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Parameters | Notation | Statistics of Parameters | ||||
---|---|---|---|---|---|---|
Distribution | Mean | COV | Cross-Correlation Coefficient | Autocorrelation Length (m) | ||
Cohesion | c (kPa) | Lognormal | 10 | 0.3 | R = −0.7–0.5 | lh: 10–40 lv: 1–3 |
Friction angle | φ (°) | Lognormal | 30 | 0.2 | lh: 10–40 lv: 1–3 | |
Unit weight | γ (kN/m3) | - | 20 | - | - | - |
Deterministic Results | Probabilistic Results | ||
---|---|---|---|
FS | Pf | Number of Evaluations | |
Cho [21] | 1.204 | 0.0138 | 50000 |
DSG–MG | 1.201 | 0.0136 | 4200 |
FSG–MG | 1.202 | 0.0130 | 4200 |
Pf | Mean | Std | Number of Evaluations | |
---|---|---|---|---|
MCS | 0.056 | 1.192 | 0.130 | 10,000 |
DSG–MG | 0.053 | 1.190 | 0.124 | 4000 |
β (°) | Pf | Mean | Std | SC | Sφ |
---|---|---|---|---|---|
0 | 0.028 | 1.201 | 0.106 | 0.034 | 0.966 |
45 | 0.053 | 1.190 | 0.124 | 0.020 | 0.980 |
90 | 0.013 | 1.210 | 0.095 | 0.028 | 0.972 |
135 | 0.011 | 1.208 | 0.089 | 0.022 | 0.978 |
lv(m) | Pf | Mean | Std | Sc | Sφ |
---|---|---|---|---|---|
2 | 0.026 | 1.204 | 0.104 | 0.017 | 0.983 |
3 | 0.053 | 1.190 | 0.124 | 0.020 | 0.980 |
40 | 0.073 | 1.204 | 0.146 | 0.030 | 0.970 |
100 | 0.077 | 1.208 | 0.149 | 0.031 | 0.969 |
Random variables | 0.103 | 1.204 | 0.171 | 0.037 | 0.963 |
r | Pf | Mean | Std | SC | Sφ |
---|---|---|---|---|---|
−0.7 | 0.023 | 1.208 | 0.094 | 0.014 | 0.986 |
−0.5 | 0.053 | 1.190 | 0.124 | 0.020 | 0.980 |
−0.25 | 0.085 | 1.204 | 0.142 | 0.137 | 0.863 |
0 | 0.111 | 1.201 | 0.157 | 0.255 | 0.745 |
0.25 | 0.143 | 1.204 | 0.178 | 0.449 | 0.551 |
0.5 | 0.162 | 1.199 | 0.193 | 0.629 | 0.371 |
COV | Pf | Mean | Std | SC | Sφ |
---|---|---|---|---|---|
COVc = 0.3 | |||||
COVφ = 0.1 | 0.006 | 1.204 | 0.079 | 0.417 | 0.583 |
0.15 | 0.017 | 1.206 | 0.098 | 0.136 | 0.864 |
0.2 | 0.053 | 1.190 | 0.124 | 0.020 | 0.980 |
COVφ = 0.2 | |||||
COVc = 0.1 | 0.039 | 1.216 | 0.120 | 0.011 | 0.989 |
0.3 | 0.053 | 1.190 | 0.124 | 0.020 | 0.980 |
0.5 | 0.087 | 1.196 | 0.134 | 0.225 | 0.775 |
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Zhang, T.; Guo, X.; Baroth, J.; Dias, D. Metamodel-Based Slope Reliability Analysis—Case of Spatially Variable Soils Considering a Rotated Anisotropy. Geosciences 2021, 11, 465. https://doi.org/10.3390/geosciences11110465
Zhang T, Guo X, Baroth J, Dias D. Metamodel-Based Slope Reliability Analysis—Case of Spatially Variable Soils Considering a Rotated Anisotropy. Geosciences. 2021; 11(11):465. https://doi.org/10.3390/geosciences11110465
Chicago/Turabian StyleZhang, Tingting, Xiangfeng Guo, Julien Baroth, and Daniel Dias. 2021. "Metamodel-Based Slope Reliability Analysis—Case of Spatially Variable Soils Considering a Rotated Anisotropy" Geosciences 11, no. 11: 465. https://doi.org/10.3390/geosciences11110465
APA StyleZhang, T., Guo, X., Baroth, J., & Dias, D. (2021). Metamodel-Based Slope Reliability Analysis—Case of Spatially Variable Soils Considering a Rotated Anisotropy. Geosciences, 11(11), 465. https://doi.org/10.3390/geosciences11110465