Chapter 2. Literature Review
2.7. Slope–Stability Analysis
2.7.2. Probabilistic Methods
numerical approaches, such as Monte Carlo simulation have been reported in the literature (Griffiths et al., 2007; Fenton and Griffiths, 2008)
Griffiths et al. (2009) presented the analysis of the probability of failure of slopes using both traditional and more advanced probabilistic analysis tools. The advanced method, called the random finite-element method (RFEM), uses elastoplasticity in a finite-element model combined with random field theory in a Monte-Carlo framework. RFEM enables the failure mechanism to seek out the weakest path through the heterogeneous soil mass that can lead to higher probabilities of failure than would be predicted by ignoring spatial variation.
Figure 2.19 shows typical failure mechanisms corresponding to different spatial correlation lengths of ΘCu = 0.2, and a relatively high spatial correlation length of ΘCu = 2. The dark regions depict higher cu values meaning higher shear strength while the light depict “weak”
soils or lower cu. Figure 2.20(a) shows the random finite element simulation results giving probability of failure (pf) of a 1:1 drained slope with FoS=1.47 (corresponding to the mean value) and cross correlation (ρ) of 0.5. Figure 2.20(b) shows the critical coefficient of variation υcrit against the spatial correlation length in dimensionless form Θ for different FoS values (corresponding to the mean value) for a 2:1 drained slope ρ=0.5. The traditional method, called the first-order reliability method, computes a reliability index that is the shortest distance (in units of directional equivalent standard deviations) from the equivalent mean-value point to the limit state surface and estimates the probability of failure from the reliability index. The study showed that simplified probabilistic analyses in which spatial variability of soil properties is not properly accounted for, could lead to unconservative estimates of the probability of failure if the coefficient of variation of the shear strength parameters exceeds a critical value (υcrit).
Figure 2.19 Influence of the spatial correlation length in RFEM analysis (Griffiths et al., 2009)
Figure 2.20 (a) RFEM results giving pf of a 1:1 drained slope with FoS=1.47 (based on the means) ρ=0.5 (b) υcrit vs Θ for different FoS values (based on the means) for a 2:1 drained slope ρ=0.5 (Griffiths et al., 2009)
Suchomel and Masin (2010) compared three probabilistic methods for calculation of slope stability using a well-documented case study of the slope failure in the Lodalen, Norway. A finite element method considering spatial random fields of uncorrelated parameters c (cohesion) and φ (friction angle) is compared with (FOSM) methods. Figure 2.21 shows a typical random field distribution for the case θ = 10 m. Figure 2.22(a) shows the evaluation of the probability of failure for θ = 10 m based on results of 250 Monte-Carlo realizations. The researchers showed that the density function could be fitted to Gaussian distribution. Figure 2.22(b) plots the probability of failure as a function of correlation lengths θ. The study showed that the FOSM method leads to different values of probability of failure as compared to the RFEM method. Several limitations of the FOSM method for calculating probability of a slope failure are identified. The researchers concluded that this is caused by the fact that the equivalent statistical distributions of soil properties, which effectively control the stability of a slope, are significantly different as compared to the original statistical distributions, which are used as an input into the basic FOSM simulations.
Figure 2.21 Typical realization of random fields of uncorrelated variables φ and c for θ = 10m (Suchomel and Masin, 2010)
Figure 2.22 (a) Evaluation of the probability of failure for θ = 10 m based on results of 250 Monte-Carlo realizations; (b) Probability of failure as a function of correlation lengths θ (Suchomel and Masin, 2010)
Srivastava et al. (2010) modelled the spatial variability of soil permeability parameters using random field theory. The parameter is considered spatially correlated log- normally distributed random variable and its influence on the steady state seepage flow and on the slope stability analysis are studied. The study investigated the influence of coefficient of spatial variation of permeability parameter for different soil slopes as well as effect of change in the mean value of permeability properties and its variations on the stability of the given slope. Figure 2.23(a) shows the permeability parameter modelled as a random field and the values distributed across the slope. Figure 2.23(b) shows the results of the stability analysis of the given soil slope (1 V: 1.5 H, height = 5.0 m) for spatially varying permeability parameters [correlation distance = 0.5 m]. The figure shows the influence of coefficients of variation (CoV %) and change in the mean value of permeability property on the stability of the given slope. The mean factor of safety increases with a decrease in the mean value of permeability parameter for a particular value of CoV.
Figure 2.23 (a) Numerical modelling of spatially variable permeability (m/s) parameter; (b) Influence of variation in mean value of spatially variable permeability on the stability of the given slope (1 V: 1.5 H) under steady state seepage conditions (Srivastava et al., 2010)
Griffiths et al. (2011) described a methodology in which parameters such as the soil strength, slope geometry and pore pressures, are generated using random field theory. Figure 2.24(a) shows a typical infinite slope as a column split into 100 equal slices parallel to slope surface and a 1-d random field of the required property is assigned to each slice. Figure 2.25 shows the computed probability density plots of FoS values for two different cases of coefficients of variation given by υcu = 0.1 and υcu = 0.5. Also included in the figures are the analytical normal and lognormal fits to the FoS distributions based on the computed mean and standard deviation values. Figure 2.24(b) shows the results of the random field analyses compared with FORM for a range of correlation lengths. The study showed that analytical methods (e.g. FOSM and FORM) applied to the infinite slope problem gave very similar results to each other, but inevitably underestimated the probability of failure compared with the random field analyses, because the failure plane in those cases is always assumed constant at a particular depth. Probabilistic slope stability methods that predefine the potential failure surface using deterministic methods are liable to overestimate the factor of safety or underestimate the probability of failure. The researchers concluded that the random field results converged on the first order values as the spatial correlation length was increased, because as the spatial correlation length increases, the greater homogeneity of the soil column means that the probability of the critical mechanism occurring above the base is reduced.
Figure 2.24 (a) Typical random fields of cu with large and small spatial correlation (b) Comparison of FORM and Random Field results showing the influence of the spatial correlation length Θ (log normally distributed tan φ') (Griffiths et al., 2011)
Figure 2.25 Histogram of FoS frequency distribution for normal and lognormal fits based on the computed mean and standard deviation (Griffiths et al., 2011)
Cho (2014) conducted a series of seepage and stability analyses of an infinite slope based on one-dimensional random fields to study the effects of uncertainty due to the spatial heterogeneity of hydraulic conductivity on the failure of unsaturated slopes due to rainfall infiltration. Figure 2.27(a) shows the infinite slope model with shallow impermeable layer.
Figure 2.26 shows the soil–water characteristic curve and the hydraulic conductivity function of a typical weathered granite soil, sampled in Seochang, used for in the study. Probabilistic stability analyses were conducted for a weathered residual soil slope with shallow impermeable bedrock to study the failure mechanism of rainfall-related landslides. Figure 2.28(a) shows the pore pressure and the, (b) variance of the factor of safety with propagation of the wetting front. Figure 2.27(b) shows the influence of the autocorrelation length of lnks
probability of failure for slope stability after 24 hours. The study showed that a probabilistic framework could be used to consider efficiently various failure patterns caused by spatial variability of hydraulic conductivity in rainfall infiltration assessment for a shallow infinite slope. It was observed from the study, if the duration of rainfall is sufficient, majority of the failure surfaces forms at the interface of the impermeable bedrock, irrespective of the random distribution of soil properties.
Figure 2.26 Hydraulic properties for analysis: (a) Soil–water characteristic curve; (b) Hydraulic conductivity function (Cho, 2014)
Figure 2.27 (a) Stability analysis of an infinite slope with shallow impermeable layer; (b) Influence of the lv and COVks on the estimated probability of failure for slope stability after 24 h (Cho, 2014)
Figure 2.28 (a) Pore pressure head profiles (b) Variance of the factor of safety with propagation of the wetting front (Cho, 2014)
Ering and Babu (2016) presented a methodology to back-analyze the Malin landslide occurrence to identify the mechanisms responsible for landslide initiation. Bayesian analysis is employed for the back analysis. The input parameters are considered as random variables in the analysis and are described by their probability distributions. The input parameters were updated based on the simulated slope behavior. Figure 2.29(a) shows the rainfall data recorded from different rain gauge stations located around Malin from 22 July to 30 July 2014. Figure 2.29(b) shows the soil water characteristic curve used in the analysis. Figure 2.30(a) shows the changes in FoS due to rainfall infiltration since the initial state of the slope where time t = 0 corresponds to 1st June and t = 60 corresponds to 30th July. Figure 2.30(b) shows the state of slope after the rainfall event. The researchers concluded that antecedent rainfall contributed in decreasing the factor of safety of the slope, but the low intensity and long duration rainfall infiltration was not sufficient to cause slope instability, it was high intensity and short duration rainfall infiltration at the end that triggered the slope failure. It is important to consider the uncertainties of soil parameters, pore pressures, field observations and that of method of analysis.
Figure 2.29 (a) Rainfall data; (b) Capillary pressure versus effective saturation (Ering and Sivakumar Babu, 2016)
Figure 2.30 (a) Factor of safety with time (b) Slope after triggering rainfall infiltration (Ering and Sivakumar Babu, 2016)