A novel probabilistic simulation approach for

Engineering with Computers https://doi.org/10.1007/s00366-018-0623-5

ORIGINAL ARTICLE

A novel probabilistic simulation approach for forecasting the safety factor of slopes: a case study S. Farid F. Mojtahedi1 · Sanaz Tabatabaee2 · Mahyar Ghoroqi3 · Mehran Soltani Tehrani4 · Behrouz Gordan5 · Milad Ghoroqi6 Received: 2 April 2018 / Accepted: 29 May 2018 © Springer-Verlag London Ltd., part of Springer Nature 2018

Abstract Stabilization of slopes is considered as the aim of the several geotechnical applications such as embankment, tunnel, highway, building and railway and dam. Therefore, evaluation and precise prediction of the factor of safety (FoS) of slopes can be useful in designing these important structures. This research is carried out to evaluate the ability of Monte Carlo (MC) technique for the forecasting the FoS of many homogenous slopes in the static condition. Moreover, the sensitivity of the FoS on the effective parameters was identified. To do this, the most important factors on FoS, such as angle of internal friction (�) , slope angle (𝛼) and cohesion (C) were investigated and used as the inputs to forecast the FoS. Then, a regression analysis was performed, and the results were used for the FoS prediction using MC. The obtained results of MC simulation were very close with the actual FoS values. The mean of the simulated FoS by MC was achieved as 1.32, while, according to actual FoSs, it was 1.27. These results showed that MC is an acceptable technique to estimate the FoS of slopes with high level of accuracy. Moreover, based on the results of correlation and regression sensitivity analyses, it was concluded that angle of internal friction, was the most influential one on the results of FoS in both types of sensitivity analyses. Keywords  Factor of safety · Monte Carlo simulation · Regression analysis · Sensitivity analysis Abbreviations ∅ Angle of internal friction AI Artificial intelligent C Cohesion CPD Continuous probability distributions FoS Factor of safety FD Finite difference FM Finite element LA Limit analysis LEM Limit equilibrium method

MC Monte Carlo MR Multiple regression PGA Peak ground acceleration RMSE Root mean squared error 𝛼 Slope angle H Slope height 𝛾 Unit weight VAF Variance account for

* Mahyar Ghoroqi [email protected]

1



Civil Engineering Department, Sharif University of Technology, Tehran, Iran

S. Farid F. Mojtahedi [email protected]

2



Faculty of Civil Engineering, Universiti Teknologi Malaysia (UTM), Skudai, Malaysia

Sanaz Tabatabaee [email protected]

3



Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran

Mehran Soltani Tehrani [email protected]

4



Department of Civil Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

Behrouz Gordan [email protected]

5



Department of Geotechnics and Transportation, Faculty of Civil Engineering, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia

6



Department of Civil Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran

Milad Ghoroqi [email protected]

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1 Introduction The evaluation and simulation of slope stability can be used in feasibility and design stage of civil engineering projects as well as their operations [1, 2]. The main purposes of slope stability analysis are designing a slope that is stable and economical, and having minimum chance of failure [3]. Reviewing the previous investigations on slope stability indicated that the evaluation of the factor of safety (FoS) is one of the most common approach to analyze the slope stability [4–6]. Generally, FoS˃1 and FoS˂1 signify the slope to be stable and unstable, in order [7, 8]. Several techniques like finite element (FM), limit analysis (LA), finite difference (FD), and limit equilibrium method (LEM) can be employed to evaluate the slope stability [9–12]. Among them, LEM is the most common method to simulate slope stability and it has been widely applied in previous researches, e.g., Davis et al. [13] and Helmstetter et al. [14]. According to Yang et al. [15], Shangguan et al. [16], and Saumi and Kothari [17] slope height ( H  ), cohesion (C) , unit weight (𝛾) , pore pressure ratio, slope angle (𝛼) , peak ground acceleration (PGA) and angle of internal friction (�) are the most influential factors on stabilization of slopes [18, 19]. Apart from the above-mentioned empirical methods to evaluate slope stability, the use of artificial intelligent (AI) techniques to simulate FoS has been highlighted in many studies [20–24]. For instance, Erzin and Cetin [25] employed artificial neural network (ANN) and multiple regression (MR) to simulation of the critical FoS of homogeneous slopes. In their research, 𝛾  , H  , 𝛼 , C and ∅ were utilized as the input parameters and obtained an amount of 0.99 for coefficient of determination (R2) between the calculated and predicted FoSs, which indicates that their ANN model is a strong application for the simulation of FOS. In another study, considering 𝛾 , H  , 𝛼 , C , ∅ and pore water pressure coefficient as model inputs, Samui and Kothari [17] established a new model based on Least square support vector machine (LSSVM) together with an ANN model (for comparison purpose) to predict FoS. The results showed that the simulation by LSSVM has better performance in comparison with the results of ANN model. Furthermore, a hybrid intelligent (HI) technique by particle swarm optimization (PSO) and ANN was proposed to predict FoS in the research carried out by Gordan et al. [26]. In their study, several performance indices, such as R2, were used to compare the ANN and PSO-ANN models. The results showed that PSO–ANN model can be introduced as a reliable tool to simulate the FoS, and its results were more precise than ANN. R2 of PSO–ANN model was obtained as 0.98, while the R2 of ANN was obtained as 0.91. In addition, the use of AI methods for the predicting and clustering aims have been highlighted in the different fields [27–35] and the

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results indicate the good performance for these methods. Recently, Monte Carlo (MC) simulation has been commonly developed in geotechnical engineering. For instance, MC was employed to simulate blast-induced flyrock, at the Sungun copper mine, Iran, by Ghasemi et al. [36]. They used spacing, burden, stemming, blast-hole diameter, blast-hole length, mean charge per blast-hole, and powder factor as the independent parameters. The results indicated that the MC can be used successfully for the simulation of flyrock induced by mine blasting. Moreover, Mahdiyar et al. [37] conducted a study to investigate the relation between FoS and six involved variables for slope stability under seismic condition through MC simulation. Based on the results, MC was effective to simulate FoS under seismic condition. In this study, using MR predictive model, an equation is developed and then, the created equation will be utilized for the simulating and risk analysis of FoS values by MC technique. In the following, after introducing the source of database, implementation of MR technique in predicting FoS will be described. Then, MC background and its development are discussed, and finally, results of sensitivity analyses are presented.

2 Database source To receive a reasonable result for evaluation and prediction of slope stability, an appreciate database is required. Considering a series of analysis, a database (224 datasets) based on the most influential factors on slopes has been prepared, and used in the present research. For the mentioned analyses, a LEMbased system (GeoStudio software) was utilized. According to previous investigations for example Samui and Kothari [17], and Choobbasti et al. [38], the most effective factors on a slope under static condition identified as slope angle (𝛼) , slope height (H), angle of internal friction (�) and cohesion (C) . Hence, they were used in the modeling process of this study for the FoS prediction. In the modeling procedure, slope height amount of 15, 20, 25 and 30 m, slope angle amount of 20°, 25°, 30° and 35°, cohesion amount of 20, 30, 40 and 50 kPa and angle of internal friction amount of 30°, 35° and 40° were used. All slopes were considered as homogenous with 𝛾 of 18 kg/m3. Furthermore, crest width was considered to be 8 m, and bedrock with rigid behavior was considered for the location of the slopes. All slopes were placed on the bedrock with rigid behavior. Mohr–Coulomb failure criterion was considered as failure criteria of the slopes in GeoStudio software environment. To calculate FoS values, technique of grid and radius with 30 slices as slip surfaces was selected and implemented. A view of embankment model used in this study can be seen in Fig. 1. At the end of this section, 224 datasets number with four inputs and one output (FoS) was established for covering the aims of this study.

Engineering with Computers Fig. 1  A schematic view of embankment used in this research

3 A statistical predictive model Multiple regression (MR) model attempts to fit an equation between two or more independent factors and a dependent factor. The use of the MR in various engineering disciplines has been expanding in the recent years [39, 40]. For example, Jahed Armaghani et al. [40] employed a MR for forecasting the blast-induced airblast. Based on their results, the value of R-square (R2) between the measured and predicted airblast (s) was 0.82 which proves that MR is an acceptable method to estimate the airblast. In the presented study, MR is applied to develop an equation to forecast the FoS. To achieve this purpose, H, 𝛼, C, and ∅ were considered as independent factors, while, FoS was the dependent factor. In total, a database including 224 datasets were prepared and used to develop an equation. To develop the mentioned equation, the SPSS v16 software was used and the Eq. 1 was proposed as follows:

( ) ( ) SF = 1.146 − 71 × 10−6 × H 2 − 0.0011 × 𝛼 2 ( ) ( ) + 195 × 10−7 × C2 + 976 × 10−6 × �2

(1)

The behavior of the developed MR can be evaluated by calculating the following performance indices: R 2, root mean squared error (RMSE), variance account for (VAF) and median absolute error (MEDAE).

� R2 =

n � ∑

xi − xmean

i=1

�2

�

−

�

n � ∑

xi − xp

i=1

�

n � �2 ∑ xi − xmean

�

�2

(5) where xp and xi are the predicted and measured FoS, respectively. Note that, the R2, RMSE, VAF and MEDAE equal to 1, 0, 100(%) and 0, respectively, show the best approximation. Figure 2 shows the measured values of FoS versus the predicted values of FOS by Eq. 1. Eventually, the values R2, RMSE, VAF and MEDAE were computed as 0.91, 0.11, 91.2, and 0.08, respectively. These values reveal that the developed MR can forecast the FOS with high accuracy level.

MEDAE = median(xi − xp ),

4 Probabilistic simulation for FOS prediction 4.1 Background MC simulation considers the influences of risk and uncertainty in various forecasting models on a variety of issues, for example, project management as well as problems regarding decision-making, and financial issues. This simulation model applies a repeated random sampling method supported by statistical analysis to the achievement of a probabilistic approximation in case of an arithmetic

� (2)

i=1

√ √ n [ √1 ∑ ( )2 ] RMSE = √ × xi − xp n i=1 [

( )] var xi − xp VAF = 1 − × 100 ( ) var xi

(3)

(4)

Fig. 2  Measured vs. predicted values of FoS using MR model

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equation or developed model [41]. To develop a predictive model (which creates results about the future time), it is important to consider a series of the problem-related assumptions and to make necessary estimation about the expected value(s). Two important goals are typically held by the MC simulation; firstly, quantitatively deciding the uncertainty and variability during the simulation of exposure of risk. Secondly, exploring the key agents of uncertainties and variability [37]. Note that, in conventional forecasting models, only fixed values are estimated. However, in MC simulation, a range of estimated values are employed as inputs, and a range of values are also presented as the system’s output. Considering the range of estimates, the MC simulation selects a random value for each input; then it calculates an output based on the randomly selected values and records the outputs. This process is iterated for many times (as defined in the settings) using a variety of the chosen values. In most of the studies, the number of iterations for a MC simulation is considered between 1000 and 10,000. Implementing the simulation process leads to the achievement of numerous results as output ranges [42, 43]. In terms of the dependency of variables in a MC model, two variables are taken into account as independent variables if the value that has been selected for a variable has no effect on the value of the other variable. Since the random variables are independent, uncertainty can be represented and analyzed more easily. MC simulations has contributed to different studies conducted for geotechnical engineering and slope stability, e.g., Mahdiyar et al. [37], Zhu et al. [44] and Li et al. [45]. In a study conducted by Calamak and Yanmaz [46], MC simulation was employed to predict the slope stability of earth-fill dams. Hydraulic conductivity, C and ∅ of the soil were set as the inputs and uncertainties used in the MC model. In another study, Danka [47] applied MC simulation to calculate the probability of failure of soil dikes in a flood accident. Moreover, EL-Ramly et al. [48] discussed the importance of using probabilistic techniques for estimating FoS in slope stability studies. The reason of using probabilistic techniques is the impact of uncertainty regarding the reliability of slope design. In another study, reliability of slope stability was examined using MC simulation in the study carried out by Husein Malkawi et al. [49]. In another study Ma and Wang [50] performed a probabilistic analysis using MC simulation for analyzing the slope stability. Moreover, Using the Kuz–Ram fragmentation model, Morin and Ficarazzo [51] used the MC simulator for simulating the full fragmentation size distribution. As showed by the obtained results, this simulator could be appropriately applied to the simulation of rock fragmentation. Different software and add-in features such as @RISK, SPSS, and Analytic Solver Platform, and

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Geo-slope can be used for MC simulation, and different settings can be applied based on the requirements of the simulation. Following section explains the way that MC simulated the FoS in this research.

4.2 Development of MC simulation Figure 3 illustrates the flowchart of developing MC model for FoS prediction. As the first step, to simulate the range of FoS, there is a need to use FoS equation, which depends on some variables. Equation 1 shows the equation and the dependent variables. A stochastic model was developed based on the available data, and the distribution function of the model inputs (H, 𝛼 , C, and ∅ ) were considered as continuous probability distributions (CPD). It is notable that Best-fit function in Risk Solver platform has been used for determining the distribution function of the inputs. Uniform

Fig. 3  Flowchart for MC simulation for FOS prediction

Engineering with Computers Table 1  Ranges of variables used in MC simulation Input

Unit

Slope height ( H ) Slope angle ( 𝛼) Cohesion ( C) Friction angle ( ∅)

meter degree KPa degree

Minimum

15 20 20 25

Maximum

30 35 50 40

Distribution function Uniform Uniform Uniform Uniform

Table 2  Correlation coefficients of Spearman’s rho for model inputs Input

H

𝛼

H 𝛼 C ∅

1 − 1.7E−17 0 0

1 0 0.408232

C

1 0

∅

1

distribution has constant probability and considered as the probability distribution functions of all input variables used in the MC simulation. Table 1 shows the inputs and their distribution functions. This research employed Risk Solver Platform as the tool of MC simulation. Risk Solver Platform was installed as a MS-Excel™ add-in features which enables the MC functions and random sampling. Two different sampling methods can be used in MC simulation, namely Latin Hypercube sampling and simple random sampling. In Latin hypercube sampling, a stratified sampling scheme is assumed to be able to appropriately represent upper and lower ends of distributions that are used in the analysis. Some studies have confirmed that the Latin hypercube sampling outperforms the simple one since, for generation of the same accuracy, the Latin hypercube requires fewer number of simulations [52]. To make a condition in which all combinations could be chosen randomly, 10,000 iterations were executed by means of the Latin hypercube. It is noticeable that these combinations were randomly chosen from the defined distributions. There are some relationships amongst the input variables and may affect the results of FoS. Therefore, for the improvement of MC modeling in a way to be used for simulating FoS range, the relationships among the variables should be considered. As it is shown in Table 2, there are some relationships among H, 𝛼 , ∅ and C. During input sampling, the correlations that are given in Table 2 were considered in simulation. Considering these correlations leads in producing meaningful combinations rather than completely random sampling. The program is composed of major rank-order correlations (Spearman’s rho) using a correlation matrix among the model inputs. To

estimate the amount of FOS stochastically, following stages were executed: 1. The static FoS data were obtained for different conditions using GeoStudio software. 2. The best fitted distribution functions were applied to the inputs through Risk Solver Platform. 3. Correlations among the variables were applied to be considered during random sampling. 4. 10,000 was considered as the number of iterations in MC simulation 5. The range of FoS were obtained as the results of MC simulation. Figure 4 illustrates the range of FoS which is the results of MC simulation. Moreover, the details of statistics of FoS range is shown. As it can be seen, Weibull was considered as the best distribution function by Risk Solver Platform for the results of FoS. As it can be seen, there is a considerable variation in the range of FoS. The ranges of FoS were calculated from 0.37 to 2.26, as the minimum and maximum amounts, respectively. Additionally, the average amount of FoS is 1.32. Furthermore, Fig. 5 illustrates the obtained results from MC simulation, together with measured results gained by GeoStudio software, and predicted results obtained from MR model. As it can be seen, considering the simulation results, with around 80% confidence, the amount of FoS is more than 1. Moreover, Fig. 6 shows the input values used in MC simulation and their corresponding amounts of the FoS. According to this figure, ∅ and 𝛼 have the direct and indirect relationships with the FoS, respectively.

5 Sensitivity analysis All variables that shown in Table 2 affect the value of FoS. Two different types of sensitivity analyses, correlation and regression sensitivity analyses, were conducted to understand the level of effectiveness of each variable on the result of FoS. The former can identify the variables with highest influence on FoS. On the other hand, the latter was investigated to find out the changes in FoS results in case of changing the value of any variables. In correlation sensitivity analyses, Risk Solver Platform was used to find the rank-order correlations that were obtained through the simulated results. Rank-order values for correlation ranges from − 1 and + 1. The closer the value to − 1, the stronger the negative correlation between the variable and FoS. Moreover, the higher the absolute amount of correlation coefficient of the variables, the more effective on the amount of FoS. As shown in Table 3, FoS is sensitive to, in descendent order, ∅ , 𝛼 , C, and H. Consequently, ∅ with

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Fig. 4  Distribution model of FoS results

Fig. 5  Results of the measured, predicted, and simulated FoS

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Fig. 6  Scatter plots of the variables’ values and their corresponding FoS Table 3  Sensitivity ranking of input variables according to correlation analysis

Variable

Correlation coefficient

∅ 𝛼 C H

+ 0.71 − 0.69 + 0.05 − 0.03

correlation coefficient of + 0.71 has the deepest impact on the amount of FoS. In addition, 𝛼 is the second significant variable in FoS prediction; however, it has negative relation with FoS. The differences between the amounts of correlation coefficients among the variables prove the different levels of effectiveness of the variables on the amount of FoS. On the other hand, in regression sensitivity analyses, which mentioned as conventional method of sensitivity analysis [53], FoS calculation should be repeated, while the value of a selected variable is changing, and the rest of variables were kept constant. The range of changing the value of

variables is considered between − 50 and + 50%, and same processes were carried out for each variable. Figure 7 shows the regression sensitivity of FoS of inputs variations. As it can be seen, variation of ∅ has the highest impact on FoS in comparison with other inputs. 50% changes in the value of ∅ results in around 55% changes in the value of FoS. The 𝛼 can be considered as the second influential variable with a significant difference in the level of effectiveness on the value of FoS. It should be noted that ∅ and 𝛼 has indirect and direct relation with the FoS, respectively. Moreover, changes in the value of H and C do not affect the result of the FoS significantly. It is notable that the correlation sensitivity analysis is presented as one of the results of MC simulation. It is obtained through MC simulation using Risk Solver platform. In other words, Risk Solver platform calculated these values based on the sensitivity of each input on the results. However, in regression sensitivity analysis, we are looking to identify how sensitive is the result based on the changes in the inputs. So, each of these types present different results.

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Fig. 7  Regression sensitivity analysis for the variables affecting FoS

6 Discussion and conclusions

ships with the FoS. Finally, from the obtained results of both correlation and regression sensitivity analyses, it was concluded that ∅ has the highest impact on the value of FoS compared with other independent factors used in this study, and changed on the value of ∅ results in significant change in the value of FoS. 3. Note that, owing to the nature of this specific problem, the equation proposed in the present research cannot be used directly for other conditions. If simulation and prediction are required in slopes with different features, it is recommended to reconsider the process offered in this study.

FoS is the most important descriptor of the slope stability analysis. The aims of this study were to develop a MC model for simulating the FoS, and identify the sensitivity of FoS on each variable. For this aim, GeoStudio software environment was used, and 224 datasets were prepared. In the prepared datasets, H, 𝛼, C, and ∅ are the independent parameters, while, FoS is the dependent parameter. In the MC modeling, the first task is to develop an accurate equation based on the prepared datasets. In the present study, MR technique was used for creating an accurate equation. Two types of sensitivity analyses were conducted to understand the level of effectiveness of each variable on the result of FoS. The conclusions that can be extracted from this study are as follows:

References

1. Based on obtained results, the developed MR equation in this paper, can simulate the FoS with high degree of accuracy and confidence. Thus, the developed equation is proper to be used in MC simulation. Additionally, the mean of the simulated FOS using MC is 1.32, while according to actual measured FoS, it is 1.27. Hence, the proposed MC presented in this study can be introduced as an acceptable method to simulate the FoS. 2. According to the results of sensitivity analyses, ∅ and C have a direct relationship with the value of FoS. On the other words, increase in mentioned factors leads to increase in FoS, while H and 𝛼 have indirect relation-

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