Microsoft Word - ETASR_V12_N2_pp8236-8240 Engineering, Technology & Applied Science Research Vol. 12, No. 2, 2022, 8236-8240 8236 www.etasr.com Kar & Roy: Probabilistic Based Reliability Slope Stability Analysis Using FOSM, FORM, and MCS Probabilistic Based Reliability Slope Stability Analysis Using FOSM, FORM, and MCS Saurav Shekhar Kar Department of Civil Engineering National Institute of Technology, Patna Patna, India sauravk.phd17.ce@nitp.ac.in Lal Bahadur Roy Department of Civil Engineering National Institute of Technology, Patna Patna, India lbroy@nitp.ac.in Received: 13 December 2021 | Revised: 11 January 2022| Accepted: 18 January 2022 Abstract-Soil uncertainties play an important part in the analysis and design of geotechnical structures. The effect of uncertainties on the geotechnical structures and their influence on the probability of failure or reliability of the structure is of great interest for geotechnical researchers. Probabilistic-based slope stability analysis incorporates the uncertainties present in the soil, as expressed in terms of mean, variance, and autocorrelation. In this paper, reliability analysis of a finite cohesive soil slope based on the probabilistic approach is presented using the First Order Second Moment (FOSM) method, First Order Reliability Method (FORM), and Monte Carlo Simulation (MCS) method. Stability analysis has been performed using the ordinary method of slices to calculate the Factor Of Safety (FOS) of the slope under undrained conditions. The reliability analysis has been implemented in the MS-excel spreadsheet environment and was mainly focused on the two models, namely the deterministic model for calculating the FOS of the slope and the uncertainty model for generating the random variables of uncertain soil parameters. The reliability index (β) of the soil slope and its corresponding probability of failure (Pf) was calculated using the above methods. The obtained result shows that the MCS method has significantly shown better performance than FOSM and FORM because of its robustness and simple approach to calculate Pf and β of the slope. Keywords-reliability index; FOSM; FORM; MCS; probability of failure I. INTRODUCTION Soil characteristics are influenced by various factors (e.g. characteristics of their origin rock, erosion and weathering actions, and sedimentation condition) and therefore, its properties vary spatially at different depths, something that is also referred to as inherent spatial variation of the soil. These inherent variations of the soil cannot be reduced as they are independent in nature, therefore, classified as aleatory uncertainty [1]. Apart from the inherent variability of the soil, several other uncertainties such as measurement uncertainty, statistical uncertainty, and transformation model uncertainty affect the design and analysis work of geotechnical structures. The measurement uncertainties [2] arise due to system errors, sample mishandling, and testing errors which can be reduced by improving the knowledge on testing techniques and equipment, therefore, are classified as epistemic uncertainties [1]. The statistical uncertainty [2] is a part of the measurement uncertainty, which may arise due to the unavailability of the adequate number of sample data. The insufficiency of the model to represent the system's actual conditions, results into transformational model uncertainties [3]. These uncertainties can be reduced if proper correlations between the relevant parameters can be established. It has been shown that the epistemic uncertainties do not affect the geotechnical structures. The response of geotechnical structures is significantly affected by the inherent spatial variability of the soil [4-6]. These uncertainties affect the soil properties and ground stratification and subsequently influence the design of geotechnical structures. Slope stability determination is based on FOS. In the deterministic method of analysis, FOS is expressed as the ratio of the resisting moment to the overturning moment. When FOS > 1, the slope is considered as safe [7]. In the probabilistic approach of slope analysis, the uncertainty present in the soil is expressed in terms of its mean and standard deviation and is modeled using the autocorrelation function [8]. Probabilistic slope stability analysis is used to address the various uncertainties present in the soil [9, 10] and calculate the �� and � of the slope. Many probabilistic based reliability methods have been developed to estimate the values of � and �� for geotechnical structures specially for the slope problem, such as FOSM [1, 11-13], FORM [1, 13-15], and MCS [13, 16-18]. In this study, probabilistic based reliability analysis of a finite cohesive soil slope is done using FOSM, FORM, and MCS. The uncertainties present in the undrained shear strength of the soil at vertical depth have been considered. The spatial variations in undrained shear strength of the soil ���� are modeled by the one-dimensional random field theory and the autocorrelation function. The saturated unit weight �γ ��, ��, coordinates of the centre of slip surface �� ,� � and the radii of the slip surface �� � are used to prepare the sample data. The slope stability model has been prepared and analyzed in a MS- Excel spreadsheet which is divided into two parts, the deterministic and the uncertainty model. The obtained � and �� were compared. Corresponding author: Saurav Shekhar Kar Engineering, Technology & Applied Science Research Vol. 12, No. 2, 2022, 8236-8240 8237 www.etasr.com Kar & Roy: Probabilistic Based Reliability Slope Stability Analysis Using FOSM, FORM, and MCS II. METHODOLOGY A. Deterministic Model Deterministic modeling is the process of estimating the ��� for a defined set of parameters using limit equilibrium methods. No concept of probability is used in the deterministic analysis. The ��� is calculated using the ordinary method of slices [19]: ��� � ∑ � ∆�� ∑�� � �� � � ∅� ∑ � �� � (1) where ′ is the effective cohesion, ∆" is the length of the arc, ∅′ is the effective frictional angle, # is the weight of the slice, $ is the inclination of the slice base. For cohesive soil under undrained condition, (1) can be further modified as: ��� � ∑ %& ∆� ∑ � �� � (2) where �� is the undrained shear strength of the soil. Visual Basic Application (VBA) codes have been written for determining the ��� values with respect to different center coordinates ��,��and radii of the slip surface ��� and the lowest value as ��� which corresponds to the critical slip surface having coordinates �� ,� � and radius �� � were identified. B. Ucertainty Model The uncertainty model is used to generate the uncertain parameters which are considered as random variables in reliability analysis. Based on the distribution type, correlation details, and statistics of the random variables, the random samples of the uncertain parameters are obtained in the spreadsheet. In this study, the �� of the soil is taken as an uncertain parameter with respect to the depth '� . Let �̅ �)���'*�,���'+�,…,���'��-. represent the vector of �� at different depths'� � '*,'+,…,'� . When �� is log-normally distributed, it can be represented as [9,20-21]: �̅ � /�0 �113 4 5" 676� (3) where 1 and 5 are the mean and standard deviation (SD) of 89)���'�- , 13 is the 9 -column unit vector, 76 is the 9 - dimensional standard normal vector, "3 is a 9 : 9 dimensional lower triangular matrix generated by the Cholesky decomposition of ;3 � "3"3.. The correlation between 89)���'��- and 89<���'=�> at respective depth '� and '= is given as: ;3 � ;�= � /?@ A|CDECF| G H (4) where ;3 is the correlation matrix and I is the correlation length. I is defined as the length up to which the soil parameters are fully correlated. III. RELIABILITY ANALYSIS A. FOSM The FOSM is a very simple method for calculating reliability based on the Taylor’s first-order series expansion. The � is calculated using FOSM [1,8,13,22-23] as: � � JKLM@*NKLM (5) where 1OP% and 5OP% are the mean and SD of FOS. B. FORM In FORM, � is calculated in terms of length as the shortest distance measured from the origin of the failure surface and the design point. � is expressed in matrix formulation [22] as: � � QR9S T∈O VWxD@JDND Y . )R6-@* WxD@JDND Y , R � 1,2,…,9 (6) where � represents the failure domain, �� represents a set of random variables, 1� represents the mean of the random variable, 5� represents the SD of the random variable and )R6- is the correlation matrix of uncertain parameters. C. MCS MCS is a mathematical procedure of continuously evaluating an empirical operator having a random variable of known probability distribution. For obtaining preferred accuracy level ��, the number of samples to be generated by MCS should be at least equal to 10/�� [18]. For example, for obtaining �� of 0.001 accuracy, the total number of samples to be generated by MCS should be at least equal to 10,000. Figure 1 shows the flowchart of the slope stability analysis using MCS. The �� of the slope is calculated as the ratio of number of samples having ��� \ 1 to the total number of generated samples. The � corresponding to the �� is calculated as: � � ]@*�1 ^ ��� (7) where ] is the standard normal cumulative distribution function. Fig. 1. Systematic representation of MCS for slope stability analysis. IV. PROBLEM STATEMENT A finite cohesive slope [21], has been taken in this study to assess its reliability having uncertainty in undrained shear Engineering, Technology & Applied Science Research Vol. 12, No. 2, 2022, 8236-8240 8238 www.etasr.com Kar & Roy: Probabilistic Based Reliability Slope Stability Analysis Using FOSM, FORM, and MCS strength in vertical depth. The cross-section and soil properties of the slope are shown in Figure 2. The slope stability analysis is carried out using the ordinary method of slices in undrained situation. Hard stratum is assumed to be present at 15.0m below the soil. The total 15.0m soil layer is divided into 30 layers, each having thickness equal to 0.5m. The undrained shear strength with depth is log-normally distributed having an exponential correlation. The correlation length is taken as 2.0m. Fig. 2. Cross-section for the slope stability problem. V. RESULTS AND DISCUSSION The critical slip circle is obtained by choosing different combination of center coordinates ��,�� and radius of the slip surface ��� to obtain minimum ���. Table I shows the range of ��,�� and � taken in this study. TABLE I. RANGE OF CENTER COORDINATES AND RADIUS OF THE SLIP CIRCLE Parameter Minimum Maximum Range Coordinate x (m) 1.0 4.0 3.0 Coordinate y (m) 7.0 10.0 3.0 Radius r (m) 11.0 16.0 5.0 Fig. 3. Deterministic model developed in MS-Excel. Based on this data, an area having the lowest ��� has been identified, and a grid of small intervals of 0.2m has been taken to further identify the critical slip surface. Figure 3 shows the deterministic model worksheet of the example problem. Table II shows the ���, the center coordinates �� ,� �, and the radius ��� of the slip surface obtained with the deterministic model. TABLE II. CRITICAL SLIP CIRCLE AND FOS Method _`a b (m) cd (m) ed (m) Ordinary method (MS-Excel) 1.248 16.0 2.6 8.8 After obtaining the ��� and critical slip surface, the uncertainty model is developed. Figure 4 shows the uncertainty model worksheet. Fig. 4. Uncertainty model developed in MS-Excel. Fig. 5. ��� histogram showing the mean and the SD of the samples used in FOSM. The �� values obtained in the uncertainty model are copied to the �� values in the deterministic model through linking of their input/output cell. By doing this task, the values of ��� generated in the deterministic model will become random and using the F9 key in MS-Excel will produce random values of ���. By doing so, we could easily perform MCS, FORM, and FOSM by continuously pressing F9 key, but instead, a VBA Engineering, Technology & Applied Science Research Vol. 12, No. 2, 2022, 8236-8240 8239 www.etasr.com Kar & Roy: Probabilistic Based Reliability Slope Stability Analysis Using FOSM, FORM, and MCS macro code has been run in the MS-Excel to calculate the random values of ���. For this study, a total of 1850 FOS samples were generated and reliability analysis has been performed using FOSM, FORM, and MCS. Figure 5 shows the mean and SD obtained from the samples of FOS in FOSM. The value of � was calculated as 2.56 and the value of �� as 1 ^ ](2.56) = 0.52%. The � is estimated using (6). Each term in (6) is computed using a code written in Matlab. The correlation matrix )R6-, with dimensions of 31×31, was obtained using (4) and is shown in Figure 6. The reliability is calculated for all the 1850 FOS samples and the minimum value is reported as the reliability index. The � obtained using FORM is 2.62 which corresponds to a �� value of 0.44%. Table III summarizes the result of MCS obtained for I = 2m. All the 1850 samples were taken for MCS. For a specified target FOS (say 1.2), out of the 1850 samples, 374 samples got values less than 1.2. In other words, it can be stated that these 374 samples failed. Therefore, the �� is calculated as 374/1850 = 2.76%. This corresponds to a value of � equal to 1.92. Fig. 6. Correlation matrix )R6-. TABLE III. SUMMARY OF MCS RESULTS f Simulation technique Samples Target FOS Number of Samples < target FOS gh (%) i 2m MCS 1850 1.1 51 2.76 1.92 1850 1.2 374 20.22 0.83 1850 1.3 977 52.81 - 1850 1.4 1538 83.14 - 1850 1.5 1764 95.35 - 1850 1.6 1839 99.40 - 1850 1.7 1848 99.89 - Fig. 7. ��� histogram from MCS. The histogram of the ��� obtained from the 1850 MCS samples is illustrated in Figure 7. It can be seen that out of the 1850 samples, 13 have FOS values less than 1. The �� is calculated as 13/1850 = 0.7%. This corresponds to a value of � equal to 2.46. Table IV summarizes the results of the obtained � and �� using the 3 different reliability methods. The value of �� varies from 0.44% to 0.70% having maximum relative difference among different methods of about 37%. There is a decrease of 26% in �� in FOSM as compared to MCS. Similarly, a difference of 37% in �� in FORM as compared to MCS is observed. The � of the slope varies from 2.46 to 2.62. TABLE IV. TESULTS OBTAINED FROM DIFFERENT RELIABILITY METHODS Method Samples β Pf (%) Relative difference in gh (%) FOSM 1850 2.56 0.52 -26.0 FORM 1850 2.62 0.44 -37.0 MCS 1850 2.46 0.70 # # Base value for calculating relative difference VI. RESULT COMPARISON Authors in [22] assessed the reliability of a cohesive soil slope having spatial inherent variation in the undrained shear strength. The height of the slope is considered 10m having a slope angle of 26.6°. The hard stratum is present 20m below the top of the soil. They analyzed the slope and calculated the FOS using the ordinary method of slices. Further, they calculated the reliability index of the slope using FOSM, FORM, and MCS. They concluded that various uncertainties can be taken into account rationally in probabilistic slope stability analysis through MCS. MCS method provides a robust and conceptually simple way to estimate the reliability index or slope failure probability. Authors in [13] studied the reliability-based probabilistic method of analysis of a finite soil slope by considering the uncertainties in cohesion and angle of the internal friction of the soil. The height of the slope is considered to be 5m having a slope inclination of 1V:2H. The unit weight of the soil is taken as a constant and the hard stratum is present 15m below the top of the soil. They analyzed the slope using the ordinary method of slices and calculated the reliability index and its corresponding probability of failure of the slope using FOSM, FORM, and MCS. They found that the MCS method performed significantly better than FOSM and FORM. VII. CONCLUSION AND FUTURE WORK The current study mainly focuses on the reliability analysis of a finite cohesive soil slope in MS-Excel spreadsheet environment based on the probabilistic approach. The effect of uncertainties arising due to the spatial variability of the soil was examined. A comparative study on the results of slope analysis using the FOSM, FORM, and MCS reliability methods was presented. The obtained results show that MCS exhibited improved performance in comparison with the other methods due to its robustness and simple approach. When using the MCS method, it becomes very easy to generate any number of samples of the FOS and calculate their failure probability and its corresponding reliability index. Engineering, Technology & Applied Science Research Vol. 12, No. 2, 2022, 8236-8240 8240 www.etasr.com Kar & Roy: Probabilistic Based Reliability Slope Stability Analysis Using FOSM, FORM, and MCS The study also presents the involvement of a large number of random variables modeled with the random field theory. The research also shows that the MCS method can assist in the understanding of the nature of complex problems and assessing the associated risk. This study will help in guiding the geotechnical practitioners dealing with slope stability analysis when various uncertainties are encountered in the soil and will help them in their decision-making process. In the future, the study will be further extended for a cohesive-frictional soil slope and layered soil slope using other sophisticated limit equilibrium methods, i.e. the Bishop’s simplified method and the Morgenstern-Price method. REFERENCES [1] G. B. Baecher and J. T. Christian, Reliability and Statistics in Geotechnical Engineering, 1st ed. Hoboken, NJ, USA: Wiley, 2007. [2] F. H. Kulhawy and C. H. Trautmann, "Estimation of In-Situ Test Uncertainty," presented at the Uncertainty in the Geologic Environment: from Theory to Practice, 1996, pp. 269–286. [3] K. K. Phoon and F. H. Kulhawy, "Evaluation of geotechnical property variability," Canadian Geotechnical Journal, vol. 36, no. 4, pp. 625– 639, 1999, https://doi.org/10.1139/t99-039. [4] K.-K. Phoon and F. H. Kulhawy, "Characterization of geotechnical variability," Canadian Geotechnical Journal, vol. 36, no. 4, pp. 612– 624, 1999, https://doi.org/10.1139/T99-038. [5] G. A. Fenton and D. V. Griffiths, "Bearing-capacity prediction of spatially random c ϕ soils," Canadian Geotechnical Journal, Jan. 2011, https://doi.org/10.1139/t02-086. [6] G. A. Fenton and D. V. Griffiths, "Probabilistic Foundation Settlement on Spatially Random Soil," Journal of Geotechnical and Geoenvironmental Engineering, vol. 128, no. 5, pp. 381–390, May 2002, https://doi.org/10.1061/(ASCE)1090-0241(2002)128:5(381). [7] D. A. Mangnejo, S. J. Oad, S. A. Kalhoro, S. Ahmed, F. H. Laghari, and Z. A. Siyal, "Numerical Analysis of Soil Slope Stabilization by Soil Nailing Technique," Engineering, Technology & Applied Science Research, vol. 9, no. 4, pp. 4469–4473, Aug. 2019, https://doi.org/ 10.48084/etasr.2859. [8] H. Benzeguir, S. M. Elachachi, D. Nedjar, and M. Bensafi, "Reliability of Buried Pipes in Heterogeneous Soil Subjected to Seismic Loads," Engineering, Technology & Applied Science Research, vol. 11, no. 1, pp. 6708–6713, Feb. 2021, https://doi.org/10.48084/etasr.4000. [9] A. H.-S. Ang and W. H. Tang, Probability Concepts in Engineering Planning and Design, Vol. 2: Decision, Risk, and Reliability, 1st ed. New York, NY, USA: John Wiley & Sons Inc, 1984. [10] K.-K. Phoon, Ed., Reliability-Based Design in Geotechnical Engineering: Computations and Applications. London, UK: CRC Press, 2008. [11] W. H. Tang, M. S. Yucemen, and A. H.-S. Ang, "Probability-based short term design of soil slopes," Canadian Geotechnical Journal, vol. 13, no. 3, Aug. 1976, https://doi.org/10.1139/t76-024. [12] J. T. Christian, C. C. Ladd, and G. B. Baecher, "Reliability Applied to Slope Stability Analysis," Journal of Geotechnical Engineering, vol. 120, no. 12, pp. 2180–2207, Dec. 1994, https://doi.org/ 10.1061/(ASCE)0733-9410(1994)120:12(2180). [13] S. S. Kar and L. B. Roy, "Reliability Analysis of a Finite Slope Considering the Effects of Soil Uncertainty," International Journal of Performability Engineering, vol. 17, no. 5, May 2021, Art. no. 473, https://doi.org/10.23940/IJPE.21.05.P7.473483. [14] B. K. Low and W. H. Tang, "Efficient Spreadsheet Algorithm for First- Order Reliability Method," Journal of Engineering Mechanics, vol. 133, no. 12, pp. 1378–1387, Dec. 2007, https://doi.org/10.1061/(ASCE)0733- 9399(2007)133:12(1378). [15] B. K. Low and W. H. Tang, "Probabilistic Slope Analysis Using Janbu’s Generalized Procedure of Slices," Computers and Geotechnics, vol. 21, no. 2, pp. 121–142, Jan. 1997, https://doi.org/10.1016/s0266-352x(97)0 0019-0. [16] R. Y. Rubinstein and D. P. Kroese, Simulation and the Monte Carlo Method, 3rd ed. Hoboken, NJ, USA: Wiley-Interscience, 2016. [17] J. M. Hammersley and D. C. Handscomb, Monte Carlo methods. London, UK: Methuen & Co., 1965. [18] C. P. Robert and G. Casella, Monte Carlo Statistical Methods. New York, NY, USA: Springer, 2010. [19] W. Fellenius, "Calculation of stability of earth dam," in Transactions. 2nd Congress Large Dams, Washington DC, USA, 1936, vol. 4, pp. 445–462. [20] Y. Wang, Z. Cao, and S. K. Au, "Practical reliability analysis of slope stability by advanced Monte Carlo simulations in a spreadsheet," Canadian Geotechnical Journal, vol. 48, no. 1, pp. 162–172, Jan. 2011, https://doi.org/10.1139/T10-044. [21] S. K. Au, Z. J. Cao, and Y. Wang, "Implementing advanced Monte Carlo simulation under spreadsheet environment," Structural Safety, vol. 32, no. 5, pp. 281–292, Sep. 2010, https://doi.org/10.1016/j.strusafe. 2010.03.004. [22] Z. Cao, Y. Wang, and D. Li, Probabilistic Approaches for Geotechnical Site Characterization and Slope Stability Analysis, 1st ed. New York, NY, USA: Springer, 2016. [23] N. M. Okasha, "Reliability-Based Design Optimization of Trusses with Linked-Discrete Design Variables using the Improved Firefly Algorithm," Engineering, Technology & Applied Science Research, vol. 6, no. 2, pp. 964–971, Apr. 2016, https://doi.org/10.48084/etasr.675. AUTHORS PROFILE Saurav Shekhar Kar is a Ph.D. scholar of the Department of Civil Engineering, National Institute of Technology Patna, Patna, India. His research interests include probabilistic slope stability analysis and reliability analysis. Lal Bahadur Roy is a professor at the Department of Civil Engineering, National Institute of Technology Patna, Patna, India. His research interests include stability analysis of soil and rocks.