Reliability-Based Design of Spatially Variable
Undrained Slopes
Ghina FAOURa and Shadi NAJJARa
aDep. of Civil & Environmental Engin., American University of Beirut, Lebanon
Abstract. Spatial variability and model uncertainty are considered the major sources of geotechnical uncertainty. The primary objective of this paper is to provide slope stability investigators with a robust reliability framework that takes into consideration the combined uncertainty of spatial variability and model uncertainty. To achieve this objective, a quantification of the model uncertainty of common slope stability models is conducted by assembling and analyzing a database of historical failures of slopes. The database is also used to investigate the possibility of a lower-bound factor of safety and its impact on the reliability of slopes. It is concluded that both spatial variability and model uncertainty in the undrained shear strength of clays have a direct effect on the mean of the factor of safety and its coefficient of variation. Moreover, it is found that the lower-bound factor of safety can cause a significant increase in the calculated reliability for an undrained slope
Keywords. Reliability, Model Uncertainty, Spatial Variability, Lower-bound, Factor of Safety
1. Introduction
Slope stability analysis is a branch of geotechnical engineering that is highly amenable to uncertainties. Accordingly, numerous studies have been undertaken in recent years to introduce probabilistic approaches for slope stability analyses. Probabilistic analyses have the capability of incorporating the different sources of uncertainty that affect the design of slopes and could aid in rationalizing the process of decision making in slope stability problems.
In the last few decades, several research studies have targeted analyzing the effect of spatial variability in the soil properties on the stability of slopes in a reliability-based framework. For example, Li and Lumb (1987), Christian et al. (1994), Malkawi et al. (2000), El- Ramly et al. (2002), Low (2003), Babu and Mukesh (2004), and Cho (2007) used limit equilibrium methods and random field theory to achieve this objective.
To overcome some of the limitations of limit equilibrium methods, Griffiths and Fenton (2004), Griffiths et al. (2007), and Griffiths et al. (2010) pursued a more rigorous method of probabilistic geotechnical analysis which is based on the random finite element method (RFEM). The approach captures the effect of soil spatial variability and fully accounts for spatial correlation and averaging. It is also a powerful
slope stability analysis tool that does not require priori assumptions related to the shape or location of the failure mechanism.
In a recent study, Jha and Ching (2013) performed a robust and rigorous probabilistic slope stability analysis using the RFEM to study the effect of slope geometry, mean and coefficient of variation of the soil parameters, and the scale of fluctuation on the probability of failure of undrained slopes. The authors conducted the study by collecting a database for 34 real undrained engineered slope cases. An advanced model of spatial variability that takes into account vertical and horizontal spatial variability was adopted.
Published work on the reliability of slopes targets the issue of spatial variability while neglecting the contribution of model uncertainty to the probability of failure. In the absence of rigorous research efforts for quantifying the uncertainty in common limit equilibrium and numerical slope stability models, there is a lack of published statistics on the uncertainty of available models. Malkawi et al. (2000) attempted to estimate the model uncertainty of several limit equilibrium methods by comparing their performance to that of Spenser’s method. There is a need for characterizing the model uncertainty of available slope stability methods with published case histories of failed slopes.
© 2015 The authors and IOS Press. This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License. doi:10.3233/978-1-61499-580-7-722
The primary objective of this paper is to provide slope stability investigators with a robust reliability-based design procedure that incorporates the effect of spatial variability and model uncertainty in the probability distribution of the factor of safety of undrained slopes. Model uncertainty is quantified using a database of historical observations of failed slopes. The second objective is to investigate the existence of a physical lower-bound factor of safety of a slope. The lower-bound factor of safety represents the minimum, possible factor of safety for the undrained slope which if incorporated in the modeling of the probability distribution of the factor of safety, could provide a more realistic quantification of reliability and a more rational basis for design. The final objective is to propose a reliability-based design framework that would allow for recommending design factors of safety that would result in acceptable probabilities of failure for undrained slopes.
2. Database Collection
In this study, a database that is comprised of 52 case histories was compiled from documented failure cases of undrained slopes and embankments from the year 1956 to 2002. The details of the database are presented in Faour (2014). Slope heights range from 2 to 22 m, and slope angles range from 9 to 69 degrees. The subsoil natural materials are mostly clays and silty clays with unit weights ranging from 11 to 20 kN/m3. Embankment fills, when sandy or silty, were modeled as cohesionless soils in the LEM. The slope geometries preceding failure were used in the LEM due to the fact that all these cases are failure cases.
3. Quantification of Model Uncertainty (λ) and Investigation of the Presence of Lower-Bound Factor of Safety
The effectiveness of common limit equilibrium models in predicting the factor of safety has not been thoroughly quantified since databases of historical published records of slope failures are needed for evaluating biases and uncertainties in these models. In this study, the compiled
database is used to accomplish the following objectives: (1) quantify the model uncertainty of these slope stability models by evaluating the statistics {mean and coefficient of variation (COV)} and the probability distribution of the ratio of measured to predicted factor of safety for each method, and (2) investigate the presence of a lower-bound factor of safety that can be calculated using information on the slope geometry and site-specific soil properties.
3.1. Mean and COV of λ
The 52 cases studied were analyzed and the factors of safety evaluated using four different limit equilibrium methods (Simplified Bishop, Ordinary method of slices, Janbu, and Spencer) using SLIDE software. An automatic tension crack search procedure using circular slip surfaces was applied to eliminate tensile stresses in the upper portions of the analyzed slopes.
To quantify the model uncertainty, the ratio of the measured to predicted factor of safety λ was calculated for all the cases in the database and for the different slope stability methods considered. Since the cases are actual historical failed slopes, the measured factor of safety could be realistically assumed to be approximately equal to 1. The calculated values of λ are presented in Figure 1 for the 52 cases analyzed in this study. Results on Figure 1 indicate that the ratio of the measured to predicted factor of safety varies among the different cases with minimum and maximum values of about 0.5 and 1.8.
The mean value of λ is a direct measure of the model “bias” and is found to vary from 0.96 (for Spencer’s method) to 1.04 (for Janbu’s method) (Table 1), indicating that the limit equilibrium methods are relatively unbiased. The coefficient of variation (COV) of λ is an indication of model uncertainty. The COVs of λ range from 0.26 (ordinary method of slices) to 0.29 (Janbu). These COVs are significant and in line with model uncertainties encountered in other geotechnical engineering fields.
Table 1. Statistical Parameters of the model uncertainty All Cases λBishop λOMS λJanbu λSpencer
Mean 0.94 0.97 1.01 0.94
Standard Deviation 0.26 0.25 0.29 0.26 Coeff. of Variation 0.28 0.26 0.29 0.28
Figure 1. Values of the ratio of measured to predicted factor of safety for the 52 case histories
The above results indicate that uncertainty in slope stability models for undrained slopes is considerable and needs to be incorporated in any reliability-based design analysis that aims at characterizing the risk of failure of undrained slopes. The model uncertainty is not sensitive to the slope stability method utilized (see Figure 1) and could be assumed to be modeled realistically with a COV of about 0.27.
3.2. Probability Distribution of λ
The cumulative distribution function (CDF) of λ was determined for the four models and tested against theoretical normal and lognormal CDFs that could be used to model the data (Faour 2014). Results indicated that the lognormal distribution could provide a realistic representation of the actual data more than the normal distribution particularly at the left hand tail of the distribution.
3.3. Evidence of a Lower-bound Factor of Safety Results from 52 case histories of slope failures show significant scatter in the ratio of measured to predicted factor of safety. This uncertainty can be reduced by introducing a physical lower-bound factor of safety that represents the minimum, possible factor of safety for the undrained slope and which can be calculated by assuming that the undrained shear strength of soil reduces to the fully remolded undrained shear strength.
Figure 2. Evidence of a lower-bound factor of safety of 52 slope failure cases
To validate the hypothesis of a lower-bound factor of safety, a predicted lower-bound factor of safety was calculated using SLIDE by replacing the undisturbed undrained shear strength with the remolded shear strength. The remolded undrained shear strength represents the lowest possible strength for an undrained clay. For 46 out of 52 cases, information about the sensitivity of the clay was used to calculate the remolded shear strength. Sensitivity is defined as the ratio of the undisturbed strength to the remolded strength measured at the same water content. For the other cases, correlations between the liquidity index and sensitivity were used.
A predicted lower-bound factor of safety was calculated for the 52 cases using the 4 Limit Equilibrium methods and plotted on Figure 3. The data support the hypothesis of a lower-bound factor of safety because none of the data points fell above the measured factor of safety (assumed to be equal to 1.0 for a failed slope). The predicted lower-bound factors of safety were found to range from minimum values that are almost equal to zero (for highly sensitive quick clays) to maximum values of about 0.9, with a mean value ranging from 0.37 to 0.39, depending on the Limit Equilibrium Method. If the cases with high sensitivities are removed from the analysis, the range of the mean of the lower-bound factor of safety increases from 0.37 to 0.39 to a higher range of 0.45 to 0.47. This observation is important since the lower-bound factor of safety is expected to have a more considerable effect on the design of a slope as the magnitude of the lower bound increases. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0 5 10 15 20 25 30 35 40 45 50 55 λ(F S (m eas ured) / F S(predi cte d) ) Case Number Bishop OMS Janbu Spencer µ(Bishop)= 0.966 µ(OMS) = 0.99 µ(Janbu) = 1.036 µ(Spencer)= 0.964 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0 5 10 15 20 25 30 35 40 45 50 55 FS(lo w er B ou nd ) Case Number Bishop OMS Janbu Spencer µLB(BISHOP)= 0.380 µLB(OMS) = 0.372 µLB(Janbu) = 0.371 µLB(Spencer)= 0.393 FSmeasured
4. Impact of Spatial Variability on the Factor of Safety of Undrained Slopes
Spatial variability of soils contributes to the total uncertainty in geotechnical engineering systems. To investigate the effect of spatial variability in the undrained shear strength on the uncertainty in the FS of the slope, the work done by Jha and Ching (2013) will be used to estimate the mean and the COV of FS as:
µFS = (1-0.115(V/0.3)1.5- 0.06*V0.85*Z1)*FSd (1)
VFS = (0.2606* bδz/ Lf*bδx/ Lf*bv + 0.0466*Z2)*V (2)
where V is the coefficient of variation of the undrained shear strength, δz and δx represent the vertical and the horizontal scales of fluctuations of the strength, and Lf represents the length of the failure surface. Z1 and Z2 are standard normal random variables with a mean of zero and a standard deviation of one. Finally, FSd is the deterministic FS that is determined from one of the slope stability models analysed in this paper using mean values of undrained shear strength.
5. Combination of Model Uncertainty and Spatial Variability
The statistical parameters of the FS are estimated by combining the uncertainties in the factor of safety due to model uncertainty and spatial variability. To accomplish this objective, FS is assumed to be equal to the product of two random variables as indicated in Eq.3.
FS = λmodel. FSspatial (3) The first random variable (FSspatial) models the effect of spatial variability in the undrained shear strength on FS. The mean and the coefficient of variation of FSspatial are estimated using equations 1 and 2 as recommended by Jha and Ching (2013). The second random variable (λmodel ) represents the model uncertainty as reflected in the ratio of the measured to predicted FS of the slope. The mean and coefficient of variation of (λmodel ) are evaluated from the analysis of the database with real case histories of failed slopes as illustrated before. The distributions of the two random variables are assumed lognormal. Thus, exact solutions that
allow for combining the uncertainties in both parameters to calculate the parameters λ and ζ (parameters of a lognormal distribution) of the total FS are available and result in a total FS that is also lognormally distributed.
To incorporate the lower-bound FS into the reliability assessment, a simple approach is adopted through the use of a truncated lognormal probability distribution (Najjar and Gilbert 2009 and Gilbert et al. 2005). A Lognormal distribution that is truncated at a lower-bound factor of safety (FSLB) can be used to accomplish this purpose. A truncated lognormal distribution is convenient because the parameters describing the distribution are the same as those of the non-truncated distribution with the addition of one extra parameter, the lower-bound FS (FSLB).
6. Reliability of Undrained Slopes
The sensitivity of the probability of failure (Pf) to variations in the deterministic FS, coefficient of variation (V) of the undrained shear strength, vertical correlation distance, and lower-bound FS is investigated in this section. A number of spatially variable clay slopes that cover the typical range of design slope conditions are considered. For each analyzed slope, a reliability analysis is conducted to quantify Pf of the slope with and without the inclusion of the lower-bound FS. For the two cases, Pf is defined as the probability that the factor of safety is less than one. In the presence of FSLB, a truncated lognormal distribution is used to model FS. The probability of failure including the lower bound (LB) is calculated as:
⎟⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − Φ − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − Φ − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − Φ = ζ λ ζ λ ζ λ ) ( 1 ) ( ) 1 ( LB LN LB LN LN pf (4)
To study the effect of δz/Lf and the coefficient of variation of the undrained shear strength (V) on the probability of failure (Pf) of slopes, a number of spatially variable slopes with different slope geometries and different soil properties were considered. Since δx doesn’t have a significant effect on the probability of failure (Pf), a typical ratio of δx/δz of 20 (Phoon
and Kulhawy 1999) is adopted. The probability of failure of each case is calculated using the statistical parameters of Spencer’s method.
The variation of the Pf with the ratio of δz/Lf and V for deterministic FS of 1.5 and 2 is plotted on Figure 3. The deterministic FS is estimated based on the mean of the undrained shear strength. As a result, this design FS could be different from actual design FS that is conventionally used in slopes, whereby conservative estimates (rather than the mean) are generally adopted. For such studies, the design FS is expected to be lower than the FS adopted in this paper. Results on Figure 3 show that Pf depends on V, FS, and the scale of fluctuation.
The primary conclusion is that the spatial variability has a significant effect on Pf of slopes. The probability of failure increases significantly with the increase in V of the undrained shear strength. Consider the case where the factor of safety is equal to 1.5 and δz/Lf=0.15. For this case, Pf is found to increase from 15% for the case where the undrained shear strength is the least variable (V=0.1) to 48% for the case where the strength is highly variable (V=0.5). This increase in Pf diminishes with the increase in FS. Along the same lines, it is worth noting that the calculated Pf for the case with FS = 1.5 (which is a common design case) are relatively high (range from 15% to 50%) compared to typical Pf that are considered acceptable in engineering practice.
The second conclusion is that there is a significant decrease in Pf with the increase of FS. Consider the case where the ratio δz/Lf=0.15 and V=0.5. For this case, Pf decreases from 48% for FS=1.5 to 15% for FS=2. For the case with the lowest V of 0.1 and δz/Lf=0.15, Pf decreases from 14% for FS=1.5 to 1.7% for FS=2.
The third conclusion from Figure 3 is that there is a threshold value for the ratio of δz/Lf (δz/Lf=0.1) above which the Pf slightly increases until it reaches another threshold (δz/Lf=0.2) beyond which Pf remains constant as δz/Lf increases. This increase in the Pf between δz/Lf=0.1 and δz/Lf=0.2 is related to the variance reduction in the undrained shear strength due to averaging along the failure surface. As δz/Lf decreases (either due to small scale of fluctuation or long failure surface), there is more averaging in the undrained shear strength leading to variance reduction which ultimately translates
into a reduction in Pf. The effect of averaging is minor since model uncertainty masks the uncertainty due to spatial variability.
Figure 3. Variation of Pf with V and δz/Lf for FS =1.5 and 2
To illustrate the effect of the lower-bound factor of safety on the Pf of slopes, a number of homogenous slopes with spatially variable undrained shear strength are considered. Sensitivities of 1.5, 1.75, 2, 2.25, 2.5, and 3 are considered in the analysis for the calculation of the lower-bound factor of safety.
The variation of Pf with sensitivity of clays and the coefficient of variation of the undrained shear strength for slopes with ratio of δz/Lf=0.1 are shown in Figure 4 for factors of safety of 1.5 and 2. The primary conclusion is that a lower-bound FS can have a significant effect on the calculated Pf. For example, consider a typical case where the FS is 1.5. If the sensitivity of the soil is 1.75 (ratio of lower-bound to predicted FS of about 0.57), the probability of failure decreases to half of its magnitude compared to the case where there is no lower bound.
The second conclusion from Figure 4 is that there is a threshold value for the sensitivity
0.01 0.10 1.000.00 0.05 0.10 0.15 0.20 0.25 0.30 Pr ob ab ilit y of F ai lu re δz/Lf V=0.1 V=0.2 V=0.3 V=0.4 V=0.5 FS=1.5 0.01 0.10 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Pr ob ab ili ty of F ai lu re V=0.1 V=0.2 V=0.3 V=0.4 V=0.5 FS=2
(about 3), below which the lower-bound factor of safety affects Pf. Above this threshold, the lower-bound factor of safety has essentially no effect on the probability of failure.
The effect of the lower-bound FS on Pf is influenced by the magnitude of the deterministic FS and by the value of the coefficient of variation V; as FS increases, the lower-bound becomes more effective in reducing the Pf. Also, as the V increases, Pf becomes sensitive to the lower-bound FS. This is related to the fact that as V increases, the mean FS decreases. Thus, the ratio of the lower-bound FS to the mean FS becomes larger making Pf more sensitive to the lower-bound factor of safety.
7. Factor of Safety versus Pf
Relationships between FS and pf were established for cases with different lower-bound FS and different V. Plots showing these relationships for V of 0.1, 0.3, and 0.5 are shown in Figure 5 for the case with δz/Lf = 0.1. Results indicate that as the FS increases, Pf decreases as expected. For relatively small V (V=0.1), the relationship between FS and pf is unaffected by the lower-bound for cases with sensitivities ranging from 2 to 3. For sensitivities smaller than 2, the lower-bound FS starts to play a role in decreasing Pf for a given FS. The importance of the lower-bound becomes more significant for cases involving higher V (V=0.3 and 0.5), where considerable effects of lower-bound on the Pf are noticed from sensitivities as high as 2.5.
For the case where the lower-bound FS is not included in the analysis, results on Figure 5 indicate that the required FS increases significantly as the target reliability level of the slope increases. For example, for the case of intermediate spatial variability (V=0.3), the required FS decreases from a high value of 2.8 for the case with a target Pf of 0.001 (slopes that could result in loss of lives if failure occurs) to a low value of 1.76 for a target Pf of 0.1 (temporary slopes). These required FS values decrease when the lower-bound FS is incorporated in the analysis. For example, when a lower-bound FS for a sensitivity of 2 is incorporated, the required FS decreases from 2.8 to 2.0 (for the case with a target Pf of 0.001) and from 1.76 to 1.66 (for the case with a target Pf of
0.1). These results indicate that the lower-bound FS could play a significant role in reducing conservatism in design, particularly for slopes designed for a higher reliability level.
Figure 4. Variation of Pf with V and Sensitivityfor δz /Lf = 0.1, FS =1.5& FS=2
8. Conclusions
The following conclusions can be drawn from the study conducted in this paper:
1. The model uncertainty for LEM methods as reflected in the ratio of measured to predicted factor of safety, λ, has a mean of about 1.0 a coefficient of variation that is in the order of 0.27 to 0.29, and follows a lognormal distribution. 2. There is strong evidence of the existence of a lower-bound FS that could be estimated by the remolded undrained shear strength. For the failure cases with non-sensitive soils, the lower-bound FS has a mean value of about 0.46. 3. As the coefficient of variation in the undrained shear strength increases, the mean of FS decreases compared to its deterministic design
0.0 0.1 0.2 0.3 0.4 0.5 1.50 1.75 2.00 2.25 2.50 2.75 3.00 Pro ba bil ity o f Failure V=0.1 V=0.2 V=0.3 V=0.4 V=0.5 FS=1.5 0.00 0.05 0.10 0.15 0.20 2.00 2.25 2.50 2.75 3.00 Pr oba bility of Fa ilur e
Sensitivity of Homogenous Slopes V=0.1 V=0.2 V=0.3 V=0.4 V=0.5 FS=2.0
value and the coefficient of variation of FS increases. Spatial averaging along the failure surface was found to reduce the uncertainty in FS and is dictated by the ratio of the scale of fluctuation to the length of the failure surface.
Figure 5. Relationships between FS and Pf for different sensitivities (S) and different coefficients of variation (V)
4. When uncertainties in the spatial variability and model uncertainty are combined, Pf was found to decrease as (1) the design FS increases, (2) the coefficient of variation in the undrained shear strength decreases, (3) the ratio of δz/Lf
decreases below a threshold value of 0.2, and (4) the lower-bound factor of safety increases. 5. The effect of the lower-bound FS on the reliability of the slope was found to be significant and depends on the magnitude of the lower-bound FS relative to the design FS, the coefficient of variation of the undrained shear strength, and on the magnitude of the design FS. Pf could be reduced by more than half for cases where a lower bound FS is included in the analysis. This reduction in Pf translates into a reduction in the required FS for a target reliability level.
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