This paper aims to develop a rapid and practical procedure that can locate the slip surface for a slope with the minimum reliability index for limit equilibrium analysis at the minimum expense of time. The comparative study on the reliability indices from different sample numbers using the Monte Carlo simulation method has demonstrated that the results from a large enough sample number are related to those from a small sample number with high correlation indices. This observation has been tested for many homogeneous and heterogeneous slopes under various conditions in parametric studies. Based on this observation, the reliability index for a potential slip surface can be calculated with a small sample number, and the search for the minimum reliability index and the slip surface can be determined by a heuristic optimization algorithm. Based on the comparisons between the critical deterministic and probabilistic slip surfaces for many different cases, the use of the proposed fast method in locating the critical probabilistic slip surface is found to perform well, which is suitable for normal routine analysis and design works.

It is widely accepted that slopes with safety factors greater than unity are not necessarily safe because of the underlying geotechnical variability and uncertainty, as well as the simplifications assumed when using predictive methods. Hong Kong is well known for slope failures, with an average of approximately 300 such failures per year. Billions of dollars are spent on slope analysis and stabilization each year in Hong Kong. It has been noted by the Hong Kong Government that approximately 5 % of the stabilized slopes in Hong Kong have eventually failed, and that many slopes with safety factors greater than 1.0 still ultimately fail (Hong Kong SAR Government, 2000). The assessment of slope stability and the reliability of the assessment have become an important topic in Hong Kong, China, Taiwan and many developed cities elsewhere where collapse of slopes may have disastrous effects on human lives and properties.

Although the use of a deterministic approach for calculating the minimum safety factor is useful for design and stabilization purposes, the reliability of the results is also an important issue for many practical problems. A probabilistic or reliability approach that can deal with the uncertainty and variability in the problem will be complementary to the classical safety factor evaluation. One of the reasons that the reliability has not commonly been determined in the past is the long computation time required in the analysis.

The conventional deterministic approach is based on minimizing the safety factor (FS for “factor of safety”) over a range of potential slip surfaces, and the critical solution is called the critical deterministic slip surface (cdss) (Arai and Tagyo, 1985; Baker, 1980; Greco, 1996; Goh, 1999; Cheng, 2003; Bolton et al., 2003; Zolfaghari et al., 2005; Li et al., 2010, 2011; Cheng and Li, 2007; Cheng et al., 2008a, b). Based on the cdss, the failure probability and reliability index can be evaluated approximately, which is a relatively simple operation favoured by many engineers (Liu et al., 2015). There have been many attempts in recent years to use a probabilistic approach for analysing the safety of slopes. One common approach to determine the reliability of a slope is to assume it to be equal to the reliability index of the critical deterministic slip surface. Attempts to use this approach include Chowdhury et al. (1987), Honjo and Kuroda (1991), Christian et al., (1994) and many others. Another approach is to search for the slip surface with the minimum reliability index; this surface is known as the critical probabilistic slip surface (cpss) approach (e.g. Li and Lumb, 1987; Hassan and Wolf, 1999; Bhattacharya et al., 2003; Xue and Gavin, 2007). Several researchers have applied finite-element methods and random field theory to the probabilistic analysis of slopes. These methods considered the spatial variability that is inherent even in “homogeneous” slopes (Griffiths and Feton, 2004; Griffiths et al., 2009, 2011; Xu and Low, 2006). As mentioned by Cheng et al. (2007b), the use of finite-element methods is time-consuming in analysis with practical limitations in certain special cases. Finite-element analysis of slope stability is therefore still not favored by engineers for routine design work.

There are a number of approaches for probabilistic slope stability analysis that have differing assumptions, limitations and capabilities for handling problems with various levels of mathematical complexity. The approaches generally fall into one of two categories: (1) approximate methods – such as the first-order and second-order reliability method (FOSM, SORM) method, the improved point estimate method and the surrogate model methods – and (2) the Monte Carlo simulation method (MCSM). The former approach (approximate method) includes the works by Hasofer and Lind (1974), Li and Lumb (1987), Low et al. (1998), Low and Tang (2007), Oka and Wu (1990), Chowdhury and Xu (1995), Duncan (2000), El-Ramly et al. (2002), Hong and Roh (2008), Xue and Gavin (2007) and others. The surrogate method that includes the response surface method and kriging model (Yi et al., 2015; Zhang et al., 2013) can also provide a good estimation of the system reliability at reduced computation. The latter approach (MCSM) includes the works by Au and Beck (2001, 2003), Au et al. (2007, 2010), Ching et al. (2009) and others. The use of the MCSM can produce good results, although it can be computationally intensive, especially if the probability of failure is small. The FOSM and SOFM methods usually require the partial derivatives of the safety factor to be determinate, which may be not available for some slip surfaces. The widely used mean-value first-order second-moment method (MFOSM; Hassan and Wolff, 1999; Xue and Gavin, 2007) uses a finite-difference technique to form the gradient of the function. However, as discussed by Cheng et al. (2008c), because failure to converge during safety factor determination is common for slope stability analysis and is equivalent to the presence of discontinuities in the safety factor function, both finite-difference techniques and explicit partial derivatives in the first-order second-moment method encountered problems during use. Besides the above methods, there are also many other approximate methods to determine the system reliability of a slope (Zhang et al., 2011).

The classical assessment approach using a probabilistic slope analysis is usually computationally intensive, and there is a growing need for a more rapid assessment of the critical probabilistic slip surface. This requirement is particularly important for many highway projects in which there are hundreds of sections to be considered. It is generally recognized that the search for the critical probabilistic slip surface is similar in principle to that for the minimum FS surface in the deterministic approach. Hassan and Wolff (1999) have proposed a method to search for the critical slip surface associated with the minimum reliability index obtained by the MFOSM. To reduce the amount of computation, Cho (2009) has adopted the Monte Carlo simulation method with approximated limit state functions based on the ANN (Artificial Neural Network) model, with results comparable to those based on FORM or SORM, while Kang et al. (2015) have adopted the Gaussian process regression with Latin hypercube sampling method. The method was developed based on their observation that the critical probabilistic slip surface generally coincides with that obtained by setting one dominant parameter (random variable) to a low value. When the cohesion of soil, the friction angle and the location of water table are important variables in the problem, this empirical approach is cumbersome and tedious to manipulate. This paper aims to provide a fast and simple approach to finding the critical probabilistic slip surface based on MCSM results. The proposed method only requires two calculations of the safety factors within each iterative search step. Although the authors cannot establish the theoretical basis for the proposed approach, the authors have experimented with thousands of cases and find that this approach can be effective and highly efficient such that risk analysis can be simple and practical for engineers.

The traditional definition of the limit state function or performance
function as described in Eq. (1) is adopted this study:

As mentioned above, the reliability index can be calculated either by the
approximate methods or the MCSM. Griffiths and Fenton (2004) and Griffiths et
al. (2009) have implemented the MCSM with a random field model for
spatial distribution of shear strengths. The MCSM is adopted in the present
study, due to its simplicity of use. The slope may fail along any potential
slip surface; therefore, it is important to consider the slope stability
problem in terms of a system of multiple potential slip surfaces. The
procedure for using the MCSM to calculate the system reliability index (or,
more directly, the probability of system failure) is straightforward. Let

A counter denoted by

Generate

For each sample

Repeat step 3 for

A simple estimate of the system failure probability of the slope can be
defined as the ratio of

Calculating the reliability index for a given slip surface by the MCSM may
follow the following three steps:

Generate a trial slip surface (Cheng, 2003; Cheng and Li, 2007; Cheng et
al., 2007b; Cheng et al., 2008a, b) that can be either circular or
non-circular. Generate

For each sample

Repeat step 2 for

Thus, Ns safety factors

The second consideration is the determination of the value of Ns. It is
widely accepted that the output of the MCSM is sensitive to the number of
samples Ns. When Ns is large, the random samples generated for each input
variable are also large, and the match between the CDF (cumulative density
function) created by sampling and the original input CDF is better. Hence,
the level of noise in the simulation diminishes and the output becomes more
stable at the price of increasing computational time. The optimum number of
iterations depends on the sizes of the uncertainties in the input parameters
(case-dependent problem) and the correlations between the input variables and
the output parameter being estimated. A practical way to optimize the
simulation process is to repeat the simulation using the same seed value with
an increasing number of iterations. A plot of the number of iterations

The third consideration is the equivalent computational effort for the
following two approaches. Assume Nm total trial slip surfaces for the
deterministic critical search (Nm safety factors or Nm equivalent trial
slip surfaces). In one approach, Nm

It is noted that the evaluation of the system reliability index can be
notably time-consuming because Nm

Typical relation between failure intensity and number of simulations in typical Monte Carlo simulation modelling.

The critical deterministic slip surface for a slope is located by
systematically generating a series of trial surfaces and analysing each slip
surface with a set of soil parameters (Cheng, 2003; Cheng and Li, 2007; Cheng
et al., 2007b, 2008a, b). In most of these
algorithms, the location of the critical deterministic surface associated
with the minimum safety factor,

The actual procedures to search for the critical probabilistic slip surface
using the harmony search method (other methods are also possible) are the
following:

Generate a potential slip surface using the procedures given by Cheng (2003), Cheng and Li (2007), and Cheng et al. (2007b).

Calculate the reliability index for the potential slip surface by Eqs. (4) or (5).

Repeat steps 1 and 2 until several potential slip surfaces (

Initiate the parameters in the harmony search algorithm such as

Sort the

Generate a new potential slip surface using

Repeat step 5 and 6 until the maximum iteration number

Output the first-order potential slip surface in the harmony memory as the optimum slip surface together with its reliability index as the minimum reliability index of the slope

The Monte Carlo sampling technique includes the following steps (Ang and Tang
1984):

For each random variable, generate Ns random numbers

Next, generate random numbers

The procedures will then continue from

For variables

Take the unit weight

Sampling details for example 1.

Where

The authors have carried out many internal studies and have observed some interesting features which form the basis for the proposed rapid procedure. Before the discussion of the proposed rapid procedures, the observations will be illustrated by two examples. Based on the observations from these examples and many other examples not shown in the paper, it can be observed that a full MCSM may not be necessary for normal cases.

Sampling values of two independent variables with normal distribution.

The first problem example uses the work by Bhattacharya et al. (2003). The
cross section of the slope is shown in Fig. 3, and the statistical
geotechnical parameters are given in Table 2. In this example, four random
variables are considered: the unit weight of soil (

Cross section of the homogeneous slope in example 1.

Mean values and standard deviations for soil property parameters.

In Table 1,

Relations between pseudo reliability indices and true reliability indices.

Mean values and standard deviations for soil property parameters (soil number from top to bottom).

Malkawi et al. (2000) noted that random seeds do not affect MCSM results and that sample sizes over 700 are sufficient for the MCSM to converge to the reliability index. A sample size of 700 may be adequate for some cases (case-dependent), but this size is questionable for general conditions. It is more rational to expect that the value of the sample size (Ns in this paper) should depend on the reliability index of the trial slip surface or the system reliability index for the whole slope (Chen, 2003). Parametric studies are conducted for the problem in Fig. 3 to study the variation of results from the MCSM with various values of Ns, where the safety factor for each sampling trial is obtained by the simplified Bishop method. A series of values of Ns are assumed for this trial slip surface, and the results are given in Fig. 4; they are inconsistent with the general trend for normal MCSM. It is noticed from Fig.4 that there are fluctuation in the results with the change in Ns. When the value of Ns increases to 20 000, the reliability index tends to converge to a stable value of 2.02. Using a sample size of 700 slightly overestimates the reliability index in this case.

Numerical convergence of reliability index with different values of Ns.

The extensive computational effort required to apply the MCSM to the determination of a critical probabilistic slip surface is a primary reason that this approach has not been adopted by geotechnical engineers for routine analysis and design; this effort is also a reason why reliability assessment is not commonly performed in engineering practice. Most of the routine designs in Hong Kong require fast analysis not exceeding 1 to 2 h because there are too many sections to be considered. To overcome this limitation, decreasing the value of Ns would be an apparently simple solution. However, as shown in Fig. 4, the reliability index can be far from the stable value (2.02) if the value Ns is too small.

For the problem shown in Fig. 3, 100 trial circular slip surfaces are
randomly generated in the analysis, and the

100 centres of randomly generated trial slip surfaces.

Relations between pseudo reliability indices and true reliability
indices of 100 trial circular slip surfaces (normal distribution

Relations between pseudo reliability indices and true reliability
indices of 100 trial circular slip surfaces (log-normal distribution

The observations as discussed above are subsequently tested for the case of heterogeneous slopes. Consider a second example that consists of a stratified clay slope bounded by a hard stratum below and parallel to the ground surface (shown in Fig. 8). The statistical geotechnical properties of the soils are given in Table 4. One hundred non-circular slip surfaces are randomly generated, with 14 slip surfaces being kinematically unacceptable; therefore, 86 total trial slip surfaces are adopted in this example.

Summary of reliability indices for the problem in Fig. 10.

Note: cdss – critical deterministic slip surface; cpss – critical probabilistic slip surface.

Cross section of the heterogeneous slope in example 2.

The load factor method is used to calculate the safety factors for the 86 non-circular slip surfaces, and the relations between the true reliability indices and the pseudo reliability indices are given in Figs. 9 and 10 for the normal and log-normal distributions, respectively. Though the correlation coefficient for the normal distribution is lower than that for the homogeneous slope, the value is still 0.948. The observations about the correlation coefficients are therefore similar to those for the homogeneous slopes. The authors have also tested many other cases; in general, high correlation coefficients are obtained for many heterogeneous slopes, even though there is no theoretical background (at present) to model or describe this phenomenon.

Relationship between pseudo reliability indices and true reliability
indices of 86 non-circular trial slip surfaces (normal distribution

Relationship between pseudo reliability indices and true reliability
indices of 86 non-circular trial slip surfaces (log-normal distribution

Based on the above observations concerning the MCSM results for many
homogeneous and heterogeneous slopes with different geometries, the authors
propose a rapid analysis approach as follows that should be sufficient for
rapid engineering use. The pseudo reliability indices are used in the
search for the critical probabilistic slip surface; i.e. the optimization
problem can be summarized as

Only two safety factors (or more but limited, as chosen by the users) are required within each iteration step, and the smaller reliability index is then computed.

Instead of factor of safety, put the reliability as the objective function in the minimization harmony search.

It should be noted that, at the end of the search, the true reliability index for the critical slip surface should be recalculated using the larger value of Ns.

The proposed approach is then applied to the two above-mentioned examples, and the results are compared with those from the literature. Consider the first example, where both circular and non-circular slip surfaces are considered using the simplified Bishop method and the load factor method to determine the safety factors. The results by Bhattacharya et al. (2003) with the critical deterministic slip surface and the critical probabilistic slip surface are given in Fig. 11. The results from the proposed approach and the results by Bhattacharya et al. (2003) are given in Table 5. It can be noted from Table 5 that all of the reliability indices for the critical deterministic slip surface are greater than those for the critical probabilistic slip surface. In addition, the reliability indices for the two references slip surfaces by Bhattacharya et al. (2003) are recalculated using the MCSM, and the results are all greater than those determined by the present study. It is clear that the results as given by Bhattacharya et al. (2003) are not the minimum reliability index of the critical probabilistic surface.

Summary of reliability indices for the problem in Fig. 11 (soil number from top to bottom).

Summary of critical slip surfaces for example 1.

Mean values and standard deviations for soil property parameters (soil number from top to bottom).

The results for the second example are summarized in Table 6, as the unit
weight is not given by Bhattacharya et al. (2003). In the present study, two
combinations of unit weights for the two soil layers are assumed. In the
first combination, a unit weight of 18.0 kN m

Summary of critical slip surfaces for example 2.

The third example is a three-layer slope with a cross section, as given in Fig. 13, while the geotechnical statistical parameters are given in Table 7.

Cross section of the heterogeneous slope in example 3.

The critical deterministic slip surface is given in Fig. 13, while the corresponding safety factor is 1.392 by the simplified Bishop method. The reliability indices for the critical deterministic slip surface are 3.281 and 3.802 for the normal distribution and log-normal distribution assumptions, respectively. The critical probabilistic slip surface is located only within the first layer, and the minimum reliability indices are 1.918 and 2.264, corresponding to the normal and log-normal distribution assumptions, respectively. The considerable difference in the location of the critical deterministic slip surface and the critical probabilistic slip surface, as well as the reliability indices, is clearly noted in this third example. Using the critical deterministic slip surface as the critical probabilistic slip surface may be acceptable in certain cases, but it may also leads to a large error in other cases, and great care should be taken concerning this problem. A summary of the reliability indices are given in Table 8.

Summary of reliability indices for example 3 in Fig. 13.

The fourth example is considered by Zolfaghari et al. (2005). The cross section of the slope is given in Fig. 14, and the statistical parameters are given in Table 9.

Mean values and standard deviations for soil property parameters (soil number from top to bottom).

Cross section of Zolfaghari slope in example 4.

A problem with three soils and vertical pressure for non-circular slip surface analysis.

Summary of reliability indices for the problem in Fig. 14.

It can be seen from Fig. 14 that the left ends of the critical deterministic slip surface and the critical probabilistic slip surface are practically identical, but considerable differences can be found at the middle and the right exit ends of the slip surfaces. The results from the rapid method, as proposed in this paper, are actually better than those given by Zolfaghari et al. (2005), which is further support for the application of the fast method for routine analysis and design.

A further example in which vertical surcharge is applied is given for the problem in Fig. 15, while the soil parameters are given in Table 11. The analyses are carried out for the cases of circular and non-circular slip surfaces. This case is special in that the soil cohesion is notably low for soil layer 2, which creates a special slip surface and increases the difficulty of the optimization search. From the results as shown in Table 12, the reliability indices for cpss are always lower than those from cdss, which is similar to the above cases, and the differences are more pronounced for non-circular slip surfaces.

Mean values and standard deviations for soil property parameters (soil number from top to bottom).

Summary of reliability indices for the problem in Fig. 15.

For Hong Kong and other such places that are well known for frequent slope failures, where the slopes are composed of three to four layers of soils with varying soil parameters, the classical approach in evaluating the critical deterministic slip surface and determining the reliability index based on this slip surface is commonly practiced. A full analysis for the true reliability index using the full Monte Carlo simulation method is seldom applied, due to the excessive time requirement for the analysis. While this approach may be acceptable in some cases, the authors, as well as other researchers, have commented that there are many cases where the critical deterministic slip surface may not provide the critical reliability index. To attempt to solve this problem, the authors have constructed thousands of test problems with arbitrary geometry and soil parameters for a reliability study of slope based on this study.

By nature, slope stability analysis is a non-linear problem for the soil
parameters. The reliability index based on cdss is hence not necessarily the
true minimum reliability index. Based on the results from the MCSM for both
homogeneous and heterogeneous slopes (thousands of internal
studies but not shown in the present paper), an interesting phenomenon is
observed, and a rapid approach in reliability analysis is proposed. The main
advantage of the proposed fast approach is that two safety factor
calculations (or more if needed) are required within each iteration step
during the search for the critical probabilistic slip surface in the present
paper. Though the reliability index for the critical probabilistic slip
surface does not fully represent the reliability of the slope as a system,
the critical probabilistic slip surface and the reliability index are still
useful to many geotechnical engineers for the assessment. The proposed method
is applicable to any specific stability analysis method, and the Bishop and
load factor methods are adopted simply because of their simplicity and
popularity in Asia. Based on the present results for several examples, as
well as other results from internal studies, it is found that there is a high
correlation between the pseudo reliability indices and the true reliability
indices for different conditions. Although the pseudo reliability index
for a given slip surface is greatly different from the true reliability
index, the correlation coefficient between the pseudo and true series
of values is greater than 0.9 (usually greater than 0.95) for all of the
cases that have been tested by the authors, as well as many other cases not
shown in this paper. This result is the basis for the rapid search approach
proposed in this study. For those problems with a correlation less than 0.95
but greater than 0.9, they are usually problems with highly contrasting soil
parameters that may not be found for real cases. There are only a few test
cases with a correlation less than 0.9 in the experience of the authors, which
supports the use of the fast method as a practical tool for engineers in
routine analysis and design work. If the engineers intend to obtain better
results, the improvement in the result can be achieved by using more safety
factor calculations within each iteration step (

The authors have performed several thousands of tests in homogeneous and non-homogeneous slopes, and the performance of the fast method is actually good in nearly all cases. It is noticed that in most cases the fast method will give similar or smaller reliability indices as compared with cdss with only few exceptions. In actual application, the fast method is applied while the reliability index for cdss is also suggested to be evaluated as a counter-check for routine analysis and design. Determination of the reliability indices from the cdss and fast method approaches are very fast in operation (usually within 20 min) as compared with the full Monte Carlo simulation (may require 1 day of computation). The results from cdss or the fast method can be useful to the engineers in their works, particularly when there are a significant amount of construction works being undertaken in Asia.

The present fast approach can be incorporated into many research and commercial codes easily with minor effort, and a good approximation of the reliability index for a given problem can be determined within minutes, which is suitable for normal engineering use. At present, reliability analysis is not commonly considered for routine slope design work because of the long computation time, and it is suggested to adopt the present rapid approach that can provide an acceptable solution within an acceptable time period suitable for routine engineering analysis and design work. In fact, the fast method has already been used with satisfaction by some engineers for normal engineering works in Hong Kong.

Classically, cdss is used by the engineers for simplicity, while the full MCSM analysis is seldom performed, due to the lengthy computation required. In this paper, cdss is demonstrated to be a poor assessment of the reliability index of slope for certain cases from five examples (many more in the internal studies). Even though the proposed fast method for cpss, as suggested in the present paper, is based on the observations of many test problems without any theoretical background, the authors have carried out thousands of trial tests to confirm the applicability, and the results have supported this method for limit equilibrium analysis. For the full MCSM results, the analysis must be calculated with extensive computational effort that may require 1 or more days of computations, while the fast method requires less than half an hour for the analysis. For highly important cases or complicated problems, the full MCSM is still recommended. Conversely, the rapid approach, as proposed in the present study, is targeted toward the majority of slopes requiring routine analysis and design, and the test results, as given in the present study, support the adoption of the proposed rapid method for normal routine engineering work with a significant savings in computational time.

The present work is supported by the Research Grants Council of the Hong Kong SAR Government through the project PolyU 5128/13E, as well as the National Natural Science Foundation of China (grant no. 51008167), the S&T Plan Project (grant no. J10LE07) from the Shandong Provincial Education Department and the Research Fund for the Doctoral Program of Higher Education of China (grant no. 20103721120001). Edited by: P. Tarolli Reviewed by: F. Kang and two anonymous referees