Traditional slope stability analysis mostly adopts the limit equilibrium method, which predetermines the slope failure surface and assumes that failure occurs simultaneously at all points of the failure surface. The method is based on the balance of forces and torques. The slope stability is represented by the factor of safety. The lowest factor of safety obtained after repeated analysis indicates the most failure-prone slope surface. However, the factor of safety for only one slope failure surface is obtained when applying this method. The distribution and changes of factor of safety in the interior of the slope are not identified. In addition, the analysis of factor of safety is influenced by the uncertainty in soil mechanical parameters, whereas uncertainty is not quantified in the traditional deterministic analysis. Therefore, a probabilistic approach, which uses the probability distribution function to explain the randomness of parameters, is proposed for quantifying the uncertainty. Nonetheless, when the observation data are not sufficient for determining the probability distribution function, the fuzzy theory can be an alternative method for the analysis. The fuzzy theory is based on fuzzy sets. It expresses the ambiguity of incomplete sets of information using a membership function. Moreover, a correct judgment can be made without verbose iterations. Hence, the aim of this study is to examine the uncertainty in soil mechanical parameters. The membership functions between soil mechanical parameters, i.e., cohesion and angle of internal friction, were constructed based on the fuzzy theory. The fuzzy point estimation was used in combination with the hydrologic and mechanical coupling model on HYDRUS 2D and the Slope Cube Module. The local factor of safety at different depths of the slope was determined using the local factor of safety theory. The probability of failure at different depths was calculated through reliability analysis, which could serve as an early warning for subsequent slope failures.
Slope stability is affected by intrinsic and triggering factors. The intrinsic factors include soil, groundwater, vegetation, slope gradient, and lithology. The triggering factors include volcanic eruptions, earthquakes, and rainfall. A common trigger for natural slopes is rainfall [
Studies related to rainfall-induced slope failure can be divided into three types according to their theoretical basis: statistical-model-based [
Therefore, probabilistic analysis is used to quantify the uncertainty [
Traditional slope stability analysis adopts the limit equilibrium analysis, which discretizes the potential sliding soil mass into smaller vertical slices without considering soil deformation. It assumes that failure occurs simultaneously at all points of the failure surface. This method is based on the balance of forces and torques. The slope stability is represented by the factor of safety. Various analytical methods have been developed based on different assumptions on the balance of forces [
The above analytical methods based on the balance of forces or on the stress field usually seek a single general slope stability index. Hence, it is almost impossible to identify the changes in pore water pressure and effective stress owing to rainfall infiltration, or the actual slope failure surface and its geometry. Therefore, Lu et al. [
Hence, the aim of this study is to examine the uncertainty in soil mechanical parameters. The membership functions for the soil mechanical parameters, i.e., cohesion and angle of internal friction, were constructed based on the fuzzy theory. The fuzzy point estimation was used in combination with the hydromechanical coupling model on HYDRUS 2D and the Slope Cube Module. The local factor of safety at different depths of the slope was determined. The probability of failure at different depths was calculated through reliability analysis, which could serve as an early warning for subsequent slope failures.
In this study, the analytic solution of transient seepage in an unsaturated layer developed by Šimůnek et al. [
The soil water content and HCF of an unsaturated zone vary with the hydraulic head and are highly nonlinear. In this study, the relationship between soil water content and matric suction was predicted using the closed-form analytic solution proposed by van Genuchten [
We adopted the principle of effective stress proposed by Lu and Likos [
As each of the stress components in soil can be expressed as a function of matric suction
The local factor of safety is based on the Mohr–Coulomb failure criterion, and is defined by the ratio between the potential Coulomb stress and the current Coulomb stress as follows:
Conceptual illustration of the local factor of safety [
The following expression of LFS can be derived from equations (
Substituting equation (
Using modeling and finite element analysis, we can analyze the effect of changes in water content or suction stress on the stability of soil units at different locations or depths of the slope.
The fuzzy theory is also called the fuzzy set theory. The fuzzy number is a special case in a fuzzy set. If no assumption is specified (when limited data are available), the fuzzy number is assumed to be triangular and comprises maximum (
The
Fuzzy point estimation combines the vertex method and the point estimate method. The vertex method was proposed by Dong and Shah [
The probability of failure was calculated from the reliability index [
In this study, a two-dimensional numerical model was developed using HYDRUS 2D. We performed a transient seepage analysis based on the seepage theory proposed by Richards (1931). The Slope Cube Module was used to examine the stress change experienced by the soil. Slope stability analysis was performed using the local factor of safety theory. The probability of failure at different depths of the slope was calculated through reliability analysis. The slope is 18 m high, with a slope angle of 40°. Figure
Illustration of the slope conceptual model.
Illustration of the slope model grid.
We have considered loam and silt as examples in this study. The soil hydraulic properties and mechanical parameters are listed in Tables
Hydraulic properties of soil (Carsel and Parrish, 1988).
Soil type | |||||
---|---|---|---|---|---|
Loam | 0.43 | 0.078 | 0.36 | 1.56 | |
Silt | 0.46 | 0.034 | 0.16 | 1.37 |
Mechanical properties of soil (MnDOT Pavement Design Manual, 2007).
Soil type | |||||
---|---|---|---|---|---|
Loam | 2.65 | 10 | 35 | 15,000 | 0.30 |
Silt | 2.70 | 15 | 30 | 10,000 | 0.35 |
Hydraulic properties of loam soil and silt soil: (a) SWRC; (b) HCF; (c) SSCC.
The variables in this study include cohesion and the angle of internal friction. The values from Table
The degree of variation of parameters is described by the coefficient of variation (
Fuzzy number of (a) cohesion and (b) friction angle of loam soil.
Fuzzy number of (a) cohesion and (b) friction angle of silt soil.
Loam soil | Silt soil | |||||||
---|---|---|---|---|---|---|---|---|
0.1 | 6.40 | 13.60 | 16.10 | 53.90 | 9.60 | 20.40 | 13.80 | 46.20 |
0.2 | 6.80 | 13.20 | 18.20 | 51.80 | 10.20 | 19.80 | 15.60 | 44.40 |
0.3 | 7.20 | 12.80 | 20.30 | 49.70 | 10.80 | 19.20 | 17.40 | 42.60 |
0.4 | 7.60 | 12.40 | 22.40 | 47.60 | 11.40 | 18.60 | 19.20 | 40.80 |
0.5 | 8.00 | 12.00 | 24.50 | 45.50 | 12.00 | 18.00 | 21.00 | 39.00 |
0.6 | 8.40 | 11.60 | 26.60 | 43.40 | 12.60 | 17.40 | 22.80 | 37.20 |
0.7 | 8.80 | 11.20 | 28.70 | 41.30 | 13.20 | 16.80 | 24.60 | 35.40 |
0.8 | 9.20 | 10.80 | 30.80 | 39.20 | 13.80 | 16.20 | 26.40 | 33.60 |
0.9 | 9.60 | 10.40 | 32.90 | 37.10 | 14.40 | 15.60 | 28.20 | 31.80 |
In this study, the correlation between cohesion and angle of internal friction (
Average LFS, reliability index, and membership at observation points in the loam and silt soil slopes.
As the coefficient of permeability for loam was greater than that for silt in this study, the rainfall was likely to infiltrate into the interior of the slope, increasing the suction stress while decreasing the effective stress on the interior of the slope. Consequently, after 48 h of sustained rainfall, the factor of safety of loam was lower than that of silt at the observation points. The reliability index analysis reveals that, as the degree of membership increases, the reliability index increases. The results obtained from the observation points on the slope indicate that the probability of failure of a loam slope is 7.79% higher than that of a silt slope.
We investigated the change in suction stress owing to the change in soil water content in the slope at different times, as well as the change in the probability of failure after a sustained infiltration of rainfall into the interior of the soil. Observations were obtained at the 12th, 24th, and 48th hours. The variations in water content, suction stress, and the probability of failure of loam and silt at the top, middle, and toe of the slope are presented as follows:
In the loam soil slope, under the effect of sustained rainfall infiltration, the rainfall intensity exceeded the coefficient of permeability. Consequently, at the 12th hour, the surface layer of loam approached saturation with a water content of 0.43. As shown in Figure
Results of water content, suction stress, and failure probability in the loam slope at 0, 12, 24, and 48 hours under rainfall conditions (a) at the top of the slope, (b) middle of the slope, and (c) toe of the slope.
In the silt soil slope, under the effect of sustained rainfall infiltration, the rainfall intensity exceeded the coefficient of permeability. Therefore, the surface layer of silt approached saturation, with a water content of 0.46 at the 12th hour. Figure
Results of water content, suction stress, and failure probability in the silt slope at 0, 12, 24, and 48 hours under rainfall conditions (a) at the top of the slope, (b) middle of the slope, and (c) toe of the slope.
We observed that a greater change in probability of failure is associated with the infiltration depth and variation in suction stress. The variation of suction stress on the surface layer of silt was greater than that of loam. Nevertheless, the coefficient of permeability was lower for silt, limiting the rainfall infiltration depth. Consequently, under the same rainfall condition, the depth of the moisture band in silt was shallower than that of loam, as shown in Figures
Results of water content and suction stress in different soil slopes at 48 hours under rainfall conditions: (a) water content distribution in the loam slope; (b) suction stress distribution in the loam slope; (c) water content distribution in the silt slope; (d) suction stress distribution in the silt slope.
Results of slope failure probability in different soil slopes at 0, 24, and 48 hours under rainfall conditions: (a) 0 hours in the loam slope; (b) 24 hours in the loam slope; (c) 48 hours in the loam slope; (d) 0 hours in the silt slope; (e) 24 hours in the silt slope; (f) 48 hours in the silt slope.
We assessed the effect of the coefficient of correlation of the parameters on the probability of failure. The top of the slope was not investigated because of its low probability of failure. Only the middle and toe of the slope were discussed. Previous studies have reported a correlation between cohesion and the angle of internal friction and that the correlation is mostly negative [
Effect of parameter correlation on slope failure probability at the following observation points (at 48 hours): (a) observation point at the middle in the loam slope; (b) observation point at the toe in the loam slope; (c) observation point at the middle in the silt slope; (d) observation point at the toe in the silt slope.
In this study, we have examined the uncertainty in parameters. Fuzzy transform was performed on the cohesion and the angle of internal friction. Fuzzy point estimation was used in combination with the hydromechanical coupling model on HYDRUS 2D and the Slope Cube Module to examine the slope stability. The result shows that the fuzzy theory can effectively evaluate the fluctuation interval, mean, and standard deviation of the factor of safety and the reliability index. The probability of failure in the interior of the slope was computed through reliability analysis. At our observation points on the loam slope, the fuzzy reliability of loam was determined to be 1.3357, and the probability of failure was 0.0908. For silt, the fuzzy reliability was observed to be 2.2299, and the probability of failure was 0.0129. The results of the slope failure mechanism investigation is that, after rainfall infiltrates into the soil, the change in water content causes an increase in suction stress (a decrease in its absolute value). The resulting decrease in soil effective stress leads to slope instability. It has been determined in this study that the change in the probability of failure is spatially related to the depth of the moisture band caused by the soil hydraulic conductivity and to the suction stress change controlled by the water content. After 48 hr of rainfall, the infiltration depth into the loam slope was deeper than that into the silt slope. The area of the loam slope in which the probability of failure exceeded 50% was approximately twice as large as that of the silt slope. It suggests that, as the rainfall infiltrates deeper, the area of instability in the slope increases. This study was also aimed at determining the effect of correlation between the parameters on the probability of failure. It was shown that a stronger negative correlation between the mechanical parameters yields a lower calculated probability of failure when performing slope stability analysis. When the correlation was considered, the computed probability of failure at observed points decreased by <1%. It suggests that the correlation between parameters may be ignored when a conservative estimate of slope stability is required.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The authors are grateful for the support of the Research Project of the Ministry of Science and Technology, Taiwan (MOST 106-2625-M-006-014).