Critical reliability slip surface in soil slope stability analysis using Monte Carlo simulation method

Seyyed Hossein Seyyed Alangi (Department of Civil Engineering, Kerman Graduate University of Advanced Technology, Kerman, Iran)
Saeed Nozhati (Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, Wisconsin, USA)
Seyyed Mohsen Vazirizade (Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, Arizona, USA)

International Journal of Structural Integrity

ISSN: 1757-9864

Publication date: 9 April 2018



Due to many different types of aleatory and epistemic uncertainty in soil properties, safety factor, which is assessed by deterministic analysis, is not reliable. The purpose of this paper is to determine the difference between critical slip surface in deterministic analysis and critical reliability slip surface in probabilistic analysis.


Deterministic analysis is formulated by the limit equilibrium methods, including Fellenius method, Bishop method, and Janbu’s simplified method. Then, the factor of safety is calculated for different slip surfaces. The stability of the soil is defined as the critical slip surface with the lowest factor of safety in each method. For probabilistic analysis, the value of reliability index, factor of safety, and probability of failure regarding given potential slip surface are considered as the stability index and obtained by the Monte Carlo simulation method.


To compare deterministic and probabilistic analysis as well as the influence of each of the aforementioned methods and stability index, a soil slope with three uncertainty parameters is analyzed and the results indicate that the critical slip surface is significantly different from critical reliability slip; however, the results from the above-mentioned methods are very close.

Research limitations/implications

There are many other methods that could be studied; however, the most usual ones were employed. Furthermore, this study just consider the most important factors as the uncertainty parameters; nevertheless, it can be extended to more geotechnical parameters.


Although there are many studies in this field, the authors conduct a succinct but very noteworthy research to show the difference between the results of mentioned methods as well as deterministic and probabilistic approaches and their influence on slip surface.



Seyyed Alangi, S., Nozhati, S. and Vazirizade, S. (2018), "Critical reliability slip surface in soil slope stability analysis using Monte Carlo simulation method", International Journal of Structural Integrity, Vol. 9 No. 2, pp. 233-240.

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1. Introduction

Due largely to the simplicity and extensive engineering experience of the equilibrium method, it is often the conventional method for slope stability analysis (Bishop, 1955; Price and Morgenstern, 1965; Sarma, 1979). Slope stability problems encounter many uncertainties in practice; however, deterministic methods are unable to handle these uncertainties. In fact, these methods employ the property variables such as pore water pressure, soil density, soil strength, etc. as single values. In deterministic methods, the factor of safety (FOS) of slopes is considered as an index of stability. In other words, this method presumes certain and definite values despite its rarity (Griffiths and Fenton, 2007). Deterministic methods show signs of weakness: they are unable to handle the uncertainties of the analyses; therefore, reliability analyses come into play in order to address the uncertainty problem in values (Chowdhury et al., 2009). In this regard, reliability analyses have been proven to be an alternative approach to the deterministic method (Phoon, 2008).

The Monte Carlo method (MCM) is used to determine the reliability index for a given slip surface of soil slope, although this method is rarely adopted due to its cumbersome calculation (Liang and Xue-song, 2012). However, thanks to modification of determining critical slip surface and the computer program SLOPE/W, MCM is employed to locate the critical reliability slip surface. This study combined both limit equilibrium methods and probabilistic methods for the stability analysis of slopes.

Liang and Xue-song (2012) assumed internal friction angle and cohesion of soil as variables and soil unit weight of soil as a constant parameter and considered critical reliability slip surface. Their method was based on mathematical procedures provided by Cheng (2003) in order to find the critical reliability slip surface.

According to the influence of the unit weight of soil on resistance and the stability of slope, in this research, not only internal friction angle and cohesion of soil but also the unit weight of soil are considered as variables. MCM is implemented to consider critical slip surface which is more accurate than other methods, e.g., first-order second-moment, first-order reliability method and second-order reliability method.

2. Deterministic analysis

Limit equilibrium methods are satisfactory methods for slope stability analysis. Most limit equilibrium methods are based on Coulomb’s failure criterion along an assumed failure surface (Shien, 2005). In this study three methods including Bishop, Ordinary and Janbu were used. The output of these three methods is FOS. As shown in Figure 1, forces are considered acting on a typical slice to formulate the algorithm in order to solve FOS based on the aforementioned methods. In Figure 1, b and H are the slope width and the piece height, respectively; W, N and τ are the slice weight, normal force, and shear strength, respectively; and E and X are horizontal and vertical internal force of pieces, respectively.

2.1 Ordinary method of slices or Fellenius method

Inter-slice forces are ignored in the Fellenius method in as much as this method assumes the inter-slice forces are parallel to the base of each slice and FOF is given as follows:

(1) FOF = c ' L + tan ϕ W cos α W sin α
where H, b, R, θ and W have been defined in Figure 1. L, which is the length of the slip surface, equals and α is the angle the normal vector acting on the slice makes with the vertical axis. In this formula, c’ and ϕ are cohesion and internal friction of soil, respectively. This method ignores forces on the base of each slice; therefore, it presents the results in a lower accuracy.

2.2 Bishop method

In the Bishop method, only the normal forces are employed to define the inter-slice forces, while shear forces are ignored (Bishop, 1955). The FOF is given as follows:

(2) FOF = ( c b + W Ψ ) W sin α


Ψ = cos α + sin α tan ϕ F

The equation has to be solved by successive approximation because of the presence of FOF in both sides of the equation.

2.3 Janbu’s simplified method

Despite the fact that the Janbu’s simplified method only satisfies overall horizontal force equilibrium rather than overall moment equilibrium, Janbu’s simplified method is similar to Bishop’s simplified method (Krahn, 2012). In this study, the average of the three methods was used to evaluate the results

3. Reliability or probabilistic analysis

3.1 Probability density function (PDF)

PDF shows frequencies of a parameter in different values. In other words, it expresses the estimation uncertainty. Distribution of many natural data sets follows a bell-shaped curve which is called normal PDF. Many geotechnical engineering material properties abide by normal PDF, too (Krahn, 2012).

3.2 Probability of failure (POF) and reliability index (β)

The ratio of the expected value of the performance function exceeds the limit state to standard deviations of the performance function known as reliability index (β). For uncorrelated normally distributed capacity, C, and demand, D, reliability index, β, are calculated using the following expression (Zhang et al., 2011):

(3) β = C D σ C 2 + σ D 2

US Army Corps of Engineers (USACE) (1997) stated that for good performance of a geotechnical system, β⩾3. Furthermore, the POF, pf, can be estimated by the reliability index. The following equation delineates the relation between pf and β where ϕ is the cumulative distribution function of the standard normal variety:

(4) p f = 1 ϕ ( β ) = ϕ ( β )

3.3 MCM

MCM is employed to generate results in a manner that is in some ways analogous to an although it is suitable for high-speed computers. Among different collections numerical methods, MCM is selected to generate the probability distribution for FOF numerically. In this approach, a large number of random variables are simulated by their known distribution such as normal distribution. N is the number of simulation and without reservation, by tending N to infinity, the estimation of the failure probability becomes more exact. It should be particularly noted that this process is time consuming, and there is an optimum number for N that not only satisfies required accuracy but also consumes decent processing time.

The relation between the sample size (the number of trials) and the reliability index in MCM is shown in Figures 2-4. It seems 1,000 is the appropriate number of trial and there is no significant difference after that. Furthermore, each figure includes different relations between the sample size and the reliability index in MCM which is based on the values in Table I (Husein Malkawi et al., 2000). In other words, the standard deviation and mean values of the soil properties are tabulated in Table I, and Figures 2-4 show that after sufficient number of trials reliability index is constant.

4. Finding critical slip surface

To find critical reliability slip surface, ten sliding surfaces having the least FOF in the deterministic analyses are extracted. Table II shows FOF from the three deterministic analyses (Ordinary, Bishop, and Janbu) and their mean for the following illustrative example. These slips are listed in the ascending order of the mean FOF.

Next, the probabilistic analysis is performed on these slip surfaces and the POF, FOF, and reliability index are obtained for each method and their means are shown in Tables III-V. The average FOF achieved from the three analyses (Ordinary, Bishop, and Janbu) is listed in the fifth column.

5. Illustrative example

The geotechnical parameters are based on mean and SD1 in Table I, and Figure 5 displays the cross-section of the slope.

In this table, γ is the unit weight of soil layer, c is the cohesion of soil layer, and Φ is the internal friction angle of soil layer. μc, σc, μΦ, σΦ, μγ and σγ are the mean value and standard deviation value of c, Φ and γ parameter types as mentioned before.

In order to compare the slip surface with the minimum FOF, minimum reliability index, and maximum POF in the probability analysis and minimum FOF in the deterministic analysis are performed.

Figure 6 shows the comparison of critical reliability slip surface and critical slip surface in the deterministic method. It is clearly noticed that the whole of critical reliability slip surface is entirely distinct from critical slip surface in the deterministic analysis and, moreover, there is a fine distinction between probabilistic analyses.

The FOF, reliability index, and POF of the most critical reliability slip surface are shown in Table VI. Although the geometric position of the slip surface in the probabilistic analyses is too close, the difference between the values is notable.

6. Conclusion

This paper has given an account of the difference between the results of the deterministic method and probabilistic method for a critical slip surface. The results showed a noteworthy difference in critical slip surface and critical reliability slip surface, which means the critical slip surface by using the deterministic method and probabilistic method is completely different. Furthermore, most studies are based on reliability index to determine the critical reliability slip surface, and this study showed that the critical reliability surface by using POF, reliability index, and FOF in a probabilistic analysis is subtly different from one other. Although there are no sharp distinctions in probabilistic analyses, further experimental investigations are needed to estimate which of them are more accurate.


Geometry of slope and forces acting on a typical slice

Figure 1

Geometry of slope and forces acting on a typical slice

Relationship between sample size and reliability index for the ordinary method

Figure 2

Relationship between sample size and reliability index for the ordinary method

Relationship between sample size and reliability index for the bishop method

Figure 3

Relationship between sample size and reliability index for the bishop method

Relationship between sample size and reliability index for the Janbu method

Figure 4

Relationship between sample size and reliability index for the Janbu method

Geometry of slope

Figure 5

Geometry of slope

Critical slip surfaces

Figure 6

Critical slip surfaces

Geotechnical parameters

Soil parameter Mean SD1 SD2 SD3 SD4
C Cohesion (kN/m2) 10 1 2 3 4
ϕ Friction angle (°) 10 0.5 1 1.5 2
γ Unit weight (kN/m3) 17.64 0.1764 0.3528 0.5292 0.7056

Factor of safety in deterministic analysis

Methods and deterministic analysis
Factor of safety
No. of slip surface Ordinary Bishop Janbu Mean
1 1.335 1.481 1.353 1.390
2 1.360 1.481 1.349 1.397
3 1.362 1.481 1.350 1.398
4 1.365 1.480 1.353 1.399
5 1.364 1.482 1.352 1.399
6 1.367 1.480 1.354 1.400
7 1.368 1.480 1.356 1.401
8 1.370 1.478 1.362 1.403
9 1.371 1.482 1.358 1.404
1 1.372 1.479 1.364 1.405

Factor of safety in probabilistic analysis

Method and probabilistic analysis
Factor of safety
No. of slip surface Ordinary Bishop Janbu Mean
 1 1.364 1.481 1.350 1.398
 2 1.360 1.481 1.346 1.396
 3 1.361 1.480 1.346 1.396
 4 1.366 1.481 1.351 1.399
 5 1.363 1.481 1.348 1.397
 6 1.366 1.480 1.351 1.399
 7 1.367 1.479 1.352 1.399
 8 1.370 1.478 1.359 1.402
 9 1.370 1.481 1.355 1.402
10 1.372 1.479 1.362 1.404

Reliability index in probabilistic analysis

Method and probabilistic analysis
Reliability index (β)
No. of slip surface Ordinary Bishop Janbu Mean
 1 4.114 5.392 3.835 4.447
 2 4.138 5.474 3.857 4.490
 3 4.135 5.462 3.848 4.482
 4 4.131 5.406 3.842 4.460
 5 4.127 5.429 3.836 4.464
 6 4.120 5.369 3.830 4.440
 7 4.113 5.341 3.820 4.425
 8 4.140 5.341 3.869 4.450
 9 4.118 5.332 3.822 4.424
10 4.139 5.319 3.866 4.441

Probability of failure in probabilistic analysis

Method and probabilistic analysis
Probability of failure (%)
No. of slip surface Ordinary Bishop Janbu Mean
 1 0.045 0.069 0.080 0.065
 2 0.070 0.088 0.105 0.088
 3 0.045 0.072 0.085 0.067
 4 0.070 0.089 0.120 0.093
 5 0.070 0.089 0.100 0.086
 6 0.055 0.063 0.070 0.063
 7 0.095 0.110 0.130 0.112
 8 0.060 0.062 0.065 0.062
 9 0.075 0.081 0.090 0.082
10 0.075 0.090 0.115 0.093

The most critical slip surfaces are obtained

Method Deterministic analysis Probabilistic or reliability analysis
Parameter FOS FOS β POF (%)
No. of critical slip surface 1 2 9 8
Value 1.390 1.396 4.424 0.062


Bishop, A.W. (1955), “The use of the slip circle in the stability analysis of slopes”, Géotechnique, Vol. 5 No. 1, pp. 7-17.

Cheng, Y.M. (2003), “Location of critical failure surface and some further studies on slope stability analysis”, Computers and Geotechnics, Vol. 30 No. 3, pp. 255-267.

Chowdhury, R., Flentje, P. and Bhattacharya, G. (2009), Geotechnical Slope Analysis, CRC Press.

Griffiths, D.V. and Fenton, G.A. (2007), “The random finite element method (RFEM) in slope stability analysis”, Probabilistic Methods in Geotechnical Engineering, Springer, Vienna, pp. 317-346.

Husein Malkawi, A.I., Hassan, W.F. and Abdulla, F.A. (2000), “Uncertainty and reliability analysis applied to slope stability”, Structural Safety, Vol. 22 No. 2, pp. 161-187.

Krahn, J. (2012), Stability Modeling with Slope/W, GEO-SLOPE International, Ltd, Calgary.

Liang, L. and Xue-song, C. (2012), “The location of critical reliability slip surface in soil slope stability analysis”, Procedia Earth and Planetary Science, Vol. 5, pp. 146-149, available at:

Phoon, K.K. (Ed.) (2008), “Reliability-based design in geotechnical engineering: computations and applications”, CRC Press.

Price, V.E. and Morgenstern, N.R. (1965), “The analysis of the stability of general slip surfaces”, Géotechnique, Vol. 15 No. 1, pp. 79-93.

Sarma, S.K. (1979), “Stability analysis of embankments and slopes”, Journal of the Geotechnical Engineering Division, Vol. 105 No. GT12, pp. 1511-1524.

Shien, N. (2005), Reliability Analysis on the Stability of Slope, UTM, Skudai.

US Army Corps of Engineers (USACE) (1997), “Introduction to probability and reliability methods for use in geotechnical engineering department of the army”, US Army Corps of Engineers, Engineering Technical Letter No. ETL 1110-2-547, Washington, DC.

Zhang, J., Zhang, L.M. and Tang, W.H. (2011), “New methods for system reliability analysis of soil slopes”, Canadian Geotechnical Journal, Vol. 48 No. 7, pp. 1138-1148.

Corresponding author

Seyyed Mohsen Vazirizade can be contacted at: