Influence of slope angle and height on the failure mode and safety factor of a homogeneous slope in 3D using the Finite Element Method

Influence of slope angle and height on the failure mode and safety factor of a homogeneous slope in 3D using the Finite Element Method | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Influence of slope angle and height on the failure mode and safety factor of a homogeneous slope in 3D using the Finite Element Method Dhananjai Verma, Megotsohe Chasie, Akshay Mishra This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5036259/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The bearing of slope geometry on the safety factor of a predefined homogeneous slope is analysed in 3D following the finite element-based strength reduction technique for establishing the relationships and the resultant failure modes. The type of failure and shape of the sliding mass is found guided by the slope geometry. The results indicate that FOS decreases nonlinearly with increasing height but decreases linearly with increasing slope angle. This insight can guide practicing professionals from making informed designs on geometry alteration of slopes for effective stabilization. Earth and environmental sciences/Environmental social sciences Earth and environmental sciences/Natural hazards Slope stability 3D FOS Slope angle Slope height Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 1: Introduction Unstable slopes can cause significant social and financial consequences especially in areas where human settlement and infrastructures are concentrated. Modified slopes like road cuts, dams, landfills, embankments are particularly vulnerable to slope stability issues (Pourkhosravani and Kalantari, 2011 ). As such, slope stability is a common concern for geoscientists and engineers alike in populated areas and in large civil engineering projects. Here, thorough geological and geotechnical investigations backed by reliable slope stability analysis can only serve as the cardinal task for project safety. Soil type, slope geometry, stratification, groundwater and infiltration all contribute to different types of slope failures (Chatterjee and Murali, 2019; Shiferaw, 2021 ). While most of the contributory factors for slope instability are intrinsic to the slope, the slope geometry (height and angle) can be considered extrinsic which can be manipulated to a certain extent for safety of the slopes. By altering the geometry of the slope, the probability to failure can be reduced. Types of slope failure on a soil slope can be translational, rotational, compound or flow (Rotaru et.al, 2007 ; Petley et.al, 2002 ). In fine-grained homogeneous soils, rotational failure may occur in three different ways: Face failure or Slope failure, Toe failure and Base failure (Fig. 1 ). The geometric shape of a slope surface significantly influences slope stability (Sabzevari et al., 2009 ; Askari et al., 2012; Sharma, 2013; Sun et al., 2017; Wang et al., 2019; Yang et al., 2020; Zan et al., 2022). Failure to fully evaluate the impact of irregular morphology on slope stability can result in significant human and material losses, particularly in slope reinforcement design and construction. Shiferaw ( 2021 ) analysed three homogeneous slopes of different soil characteristics (sandy-clay, clay and sandy soils) in 2D and concluded that decreasing slope angle increases the factor of safety nearly linearly while a decrease in height increases the factor of safety at a parabolic rate. Kumar et.al ( 2023 ) carried out a similar study in 3D following the limit equilibrium method (LEM) and found that decrease in slope height raises the safety factor nonlinearly while a decrease in slope angle increases the safety factor almost linearly. Their study with respect to different slope material also indicated that toe slip is the predominant slope collapse for clay and sandy-clay soils, while slope failure is the dominant failure mode for sandy soils. Recent advancements in computing technology have propelled the three-dimensional (3D) finite element numerical approaches as a popular alternative to traditional limit equilibrium methods (LEM) for slope stability analysis. The advantages of 3D methodologies over 2D approaches in determining slope stability are well established (Griffith and Marquez, 2007; Zabuski, 2005 ). This study describes the results of a 3D Finite Element investigation to the influence of slope morphology in terms of slope height and angle on its stability conditions and different modes of failure for a pre-defined set of material properties. 2: Slope stability analysis in 3D using Finite Element Method (FEM) Slope stability refers to the resistance of inclined slopes (man-made or natural) to failure by sliding. Any exposed inclined slope, in general, may be prone to deformation and mass movement under the influence of gravity. Slope failures are determined by the slope angle, slope height, soil type, soil stratification, groundwater, and seepage. Calculating the factor of safety (FOS) is the standard approach in slope stability analysis (Halder et.al, 2020 ). FOS is computed as the ratio of the soil's shear strength (τf) to the mobilized shear (τm) at the failure surface, as shown in Eq. 1. FS = τf/τm…………………………………………………………………………………. (1) There are several approaches for analysing slope stability and computing the factor of safety. FEM is a computational technique used to obtain approximate solutions of boundary value problems in science (Hutton, 2004 ). The most common approach to 3D slope stability analysis is still the LEM, which is usually a direct extension of the various 2D slices method to 3D columns method; for example, the ordinary, modified Bishop, Morgenstern and Price, Spencer, and Janbu methods (Hungr, 1987 ; Hungr et.al, 1989 ; Lam and Fredlund, 1993 ; Yiang and Yamagami, 2004 ; Razdolsky, 2009 ), and numerous improvements to these methods (Hovland, 1977 ; Chen and Chameau, 1983 ; Duncan, 1996 ; Cheng and Yip, 2007 ; Zhang, 2013). The 3D LEM is simple and quick. Unfortunately, the 3D LEM relies on a number of assumptions and cannot be easily adjusted to account for true third-dimension boundary constraints. Moreover, another limitation of the 3D LEM is the difficulty of locating the critical general 3D failure surface in both shape and location (Cheng et.al, 2005 ; Wei et.al, 2009 ). A primary advantage of FEM in slope stability analysis is that no prior assumptions regarding the critical failure surface are required. In this study, FEM-based stability modelling in 3D was carried out by adopting the strength reduction technique wherein the shear strength parameters (c, Φ) of the slope are gradually reduced by a safety factor called the strength reduction factor (SRF). The analysis is repeated for different values of SRF; until the model becomes unstable. The slope's FS is then calculated as the proportion of the soil's real shear strength characteristics to its reduced (critical) shear strength parameters. The strength reduction approach defines the FS as in Eq. (2) (Griffiths & Lane, 1999 ). 𝐹𝑆= 𝑡𝑎𝑛 ф 𝑖𝑛𝑝𝑢𝑡 /𝑡𝑎𝑛ф 𝑟𝑒𝑑𝑢𝑐𝑒𝑑 = 𝑐 𝑖𝑛𝑝𝑢𝑡 / 𝑐 𝑟𝑒𝑑𝑢𝑐𝑒𝑑 ……………………………………………… (2) 3: Methodology 3.1: Aim The objective of this paper is to understand the influence of slope height and slope angle in the computation of safety factor of a slope in 3D. By altering the slope height and slope angle, the changes in the FS values and variation in the predicted failure modes are to be understood by establishing their inter-relationships. 3.2: Approach Past experimental on the relationship between slope and height has been carried out at only for low height slope mainly for embankment and cut slope stability cases (Kumar et al 2023 ). The research of slope stability along the highways and roads in the Himalayan areas is of great interest since slope instability has resulted in transportation issues, human deaths and injuries, property loss, and environmental degradation. In this a natural slope from 10m to 50m height and angle up to 63m have been taken where homogeneous exposure debris material are has been taken as case and numerical modelling has been performed with different combination of slope angle and height. In this study examines the safety factor in 3D and resultant failure surfaces at various heights and base inclination angles for a certain soil type. Three analytical cases with soil properties as presented in Table 1 were adopted and simulated. For simplification, a homogeneous medium is assumed in the study. Height variation to compute the safety factor at a specific slope angle. Progressively change the slope angle while keeping the slope height constant. Simultaneous increase in slope height and angle. Table 1 Geotechnical Properties of Soil Sl no. Properties Values 1 Unite Weight (Dry) 12.85 kN/m3 2 Unite Weight (Saturated) 16.48 kN/m3 3 Cohesion 22.5 kN/m2 4 Angle of Internal friction 37.4° 5 Poisson Ration 0.33 6 Modulus of elasticity 10000 kN/m2 3.3: Geometry and modelling Majority of the parameters in the study of slope through FEM are slope geometry specially height and angle, soil mechanical characteristics, and groundwater conditions to estimate the factor of safety assuming all other parameters to be constant (Shepheard et.al., 2018). This underlines the significance of slope geometry in determining the safety factor of a slope. A 3D slope geometry (d imensions, 100m x 120m x 50m) created in CAD environment was used as input file in MIDAS’ GTS NX software, a finite element-based program, to compute the factor of safety for several slope models in respect of different slope height and angle (Fig. 2 a & b). From the developed geometry, a plain strain model with a hybrid Mesher type with medium coarse mesh element of 5 m size as illustrated in Fig. 2 c is generated and used in the study. 4: Result 5.1. Deformation Failure type/mode on slopes are determined by a number of factors such as soil type and slope geometry. By keeping the soil type and properties constant, the mode of failure and safety factors were examined for various heights and inclination angles. At 50 m fixed slope height, base failure has been observed for slope angle up to 45 degrees (Fig. 3 a). For slope angle greater than 45 degrees up to 57 degrees, toe failure (Fig. 3 b) has been observed. Face/slope failure mode has been observed for slope angles of 63 degrees or more (Fig. 3 c). However, with decrease in the slope height base failure occurs at greater angle. For instance, at 20 m fixed slope height (Fig. 4 ), base failure occurred up to an angle of 51 degrees (Fig. 5 ) while toe failure occurred at 57 degrees (Fig. 7 ). At higher angles of 63 degrees and more, face/slope failure is observed (Fig. 8 ). No failure plane (NFP) has been consistently observed at 10 degrees slope angle for any slope height from 5 m to 50 m. The changing failure scenarios (mode of failure) with changes in the slope angle and slope height is summarized in Table 2 . Table 2 Mode of failure based on height and slope angle. Slope angle (degree) 10 20 30 45 51 57 63 69 75 Slope height (m) 05 NFP Base Base Base Base Toe Slope Slope Slope 10 NFP Base Base Base Toe Toe Slope Slope Slope 20 NFP Base Base Base Base Toe Slope Slope Slope 30 NFP Base Base Base Toe Toe Slope Slope Slope 40 NFP Base Base Base Toe Toe Slope Slope Slope 50 NFP Base Base Base Toe Toe Slope Slope Slope 5.2. Slope height and FOS In the first analytical case, the slope height is gradually increased from 05 m to 50 m at intervals of 5–10 m to see its variation at a defined slope angle. This variation was analysed at 10, 20, 30, 45, 51, 57, 63, 69 and 75 degree angles. Table 3 shows that the safety factor decreases with increasing slope height for specific soil types at a constant slope angle. Table 3 Slope height and FOS at a specific slope angle Slope angle (degree) 10 20 30 45 51 57 63 69 75 Slope height (m) 05 13.04 9.44 7.84 6.1 5.52 4.58 4.21 3.13 2.51 10 7.22 5.3 4.7 4.26 4.01 3.68 3.51 2.33 2.31 20 4.04 3.56 3.12 2.6 2.43 2.29 2.16 1.94 1.76 30 3.46 3.12 2.53 1.97 1.79 1.73 1.61 0.99 1.15 40 3.18 2.89 2.21 1.68 1.47 1.13 1.33 1.32 1.16 50 3.32 2.82 2.03 1.47 1.35 1.09 1.22 1.12 1.05 5.3. Slope angle and FOS In the second case, the slope angle was progressively increased while maintaining the slope's height and soil characteristics constant. This investigation was conducted at 10, 20, 30, 45, 51, 57, 63, 69, and 75 degrees with heights ranging from 05 to 50 meters. The factor of safety decreases as the slope angle increases as illustrated in Table 3 . 5.4 Slope height and angle In the third analytical case, the slope height and angle were simultaneously increased while keeping the soil parameters constant. Modelling was carried out from a height of 40 m to 68 m (failure point) with a 2 m interval. Similarly, the slope angle was increased from 35 degrees to 63 degrees (failure point) with a 2-degree interval. Table 4 shows the computed safety factor for various slope heights and angles and the associated failure modes. Table 4 Slope height, slope angle, FOS and failure mode Slope Height (m) Slope Angle (degree) FOS Failure mode 40 35 1.876 Base 42 37 1.80 Base 44 39 1.72 Base 46 41 1.56 Base 48 43 1.57 Base 50 45 1.47 Base 52 47 1.36 Base 54 49 1.31 Toe 56 51 1.29 Toe 58 53 1.19 Toe 60 55 1.19 Toe 62 57 1.14 Toe 64 59 1.15 Toe 66 61 1.04 Slope 68 63 0.89 Slope 5: Discussion 6.1: Influence of slope height on FOS and Failure modes As seen in Fig. 5 , decreasing slope height beyond 20m always shows sharp rise in factor of safety. This is more pronounced for the 10, 20, 30-degree slopes. At heights ranging from 5 to 20 meters of the graph is extremely steep and abrupt in relation to the safety factor. For slope heights ranging from 30 to 50 meters, the slope of the line remains relatively shallow giving rise to a half-parabola. This indicates that the change in FOS change up to a height of 20 m, a very small rate from 20 m to 30 m, and fairly constant from 30 m onwards for any particular slope angle. Also, FOS is influenced at higher rate up to the angle of 51 degree and smaller rate from 63 degree onwards in the height between 5 m to 20 m. So, if height increases from 30 m onwards, there will be very little influence on FOS. In terms of failure types, it has been noted that at any slope height, base failure occurs up to 45 degrees, mixed base and toe failure up to 57 degrees and all face/slope failure occurs above this angle. This implies that steeper slopes are more vulnerable to slope collapse. 6.2: Influence of slope angle on FOS and Failure modes Figure 6 indicates that the relationship between slope angle and factor of safety is almost linear for any slope height i.e., 5, 10, 20, 30, 40, and 50 metres. FOS has a lesser rate of influence when the slope height increases beyond 20 m. It also indicates that at a given height of 10, 20, 30, 40 and 50 m there is fairly constant FOS beyond the 69 degree slope. In terms of failure types, gentle slopes are indicating base failure whereas steeper slopes are more vulnerable to slope collapse. 6.3: Influence of slope height-slope angle on FOS and Failure modes Table 4 shows how increasing slope height and angle concurrently affects the factor of safety and vice-versa. In general, the desired parameters of a slope to be considered stable is illustrated which is vital for slope geometry alteration activities like scaling. The results also show that the slope's height and angle determine how and when specific soil types collapse. Base failure occurs most frequently when the slope angle is less than 49 degrees and the height less than 54 m (Fig. 8 ). Toe failure is the most common cause of slope collapse on mild slopes (Fig. 9 ). Slope failure occurs most commonly on steep slopes greater than 59 degrees (Fig. 10 ). This analysis therefore gives an insight to optimizing slope height and angle for maximum safety factor. 6: Conclusion The study presents the variation in the safety factor of a 3D homogeneous slope and the associated failure modes under changing slope angle and height. Base failure occurs most frequently when the slope angle is less than 45 degrees and the height is low. Toe failure is the most common cause of slope collapse on moderately steep slopes (45–55 degrees). Slope failure occurs most commonly on steep slopes greater than 55 degrees. Slope safety normally improves linearly as the slope angle reduces, although the safety factor climbs at different rates when the slope height drops. At heights between 5 and 20 meters, the slope of the curve drastically changes in response to the safety factor. As the height grows to 30 meters, the factor of safety drops more quickly but the FOS remains reasonably consistent throughout slope heights ranging from 30 to 50 meters. The results demonstrates that although the FOS decreases nonlinearly with increasing height (half-parabolic), it decreases linearly with increasing slope angle. This understanding of the influence of slope height and angle on the safety factor of a slope is vital for practicing geoscientists and geotechnical engineers for reducing the probability of a slope towards failure or for adopting effective geometric alteration techniques for slope stabilization. Declarations Author Contribution Dhananjai Verma : Prepared the data set and wrote the manuscriptMegotsohe Chasie: Given technical impute, modified figure and tables and accordingly re-aaragne Akshay Kumar Mishra: critically review and suggested technical points Acknowledgement The authors wish to thank DG, GSI, NMH-IV, and DDG MIVA for providing software and facilities. Also, the authors would like to thank all of their GHRM and GSI colleagues for their kind assistance with this research. Data Availability The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.The datasets generated and/or analysed during the current study are not publicly available due [since it huge file, yet to be published , once published will be pulically avaialabe] but are available from the corresponding author on reasonable request. References Askari, F. 3D stability analysis of convex slopes in plan view using lower bound linear finite element Int. J. Civ. Eng. 10 (2), 112–123 (2012). Chaowei, S., Junrui, C., Zengguang, X. & Qin, Y. 3D Stability Charts for Convex and Concave Slopes in Plan View with Homogeneous Soil Based on the Strength-Reduction Method. Int. J. Geomech. 17 (5). 10.1061/(ASCE)GM.1943-5622.0000809 (2017). Chatterjee, D. & Murali Krishna, A. Effect of slope angle on the stability of a slope under rainfall infiltration. Indian Geotech. J. 49 , 708–717 (2019). Chen, R. & Chameau, J. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5036259","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":361077956,"identity":"a7c22918-5ae1-4d91-97bc-b803f182e476","order_by":0,"name":"Dhananjai Verma","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEklEQVRIie2QsUoDQRCG5xjYNGPSbshBXmHhQAmGu1dRFlJtwDKgYEDYNKe2EQtfwUeILMQmJO3B2aWxPA0EQRA3RkhhNmcpuN+yy1/Mx84MgMfzF8lWD9mzpg2A60QO4YfS+a2ywZT3Vc0vHxYnYRwCl+PX7GwW3w0An180hAcOpf40lY0hSQLekbdqnMt7A6x1o4Fa/e2KyJRoEKGdZhKhYrkUCCyiCZAYOZXonejcKrMFqo+pbF6UK/v2F0NQSRG7ehTbDeCcem6lbpVDokdiFR1h90oeCRPoYNjjTqVqG8v30tOkhjhHtYyT5rUxRSHaiUv5IkgB2Hc+7kOgOQDfUb/ibRMTe7Eoqfd4PJ7/xSfp4U6k5497yAAAAABJRU5ErkJggg==","orcid":"","institution":"Geological Survey of India","correspondingAuthor":true,"prefix":"","firstName":"Dhananjai","middleName":"","lastName":"Verma","suffix":""},{"id":361077957,"identity":"6eb74d12-0d7d-43d6-a03b-049158f09556","order_by":1,"name":"Megotsohe Chasie","email":"","orcid":"","institution":"Geological Survey of India","correspondingAuthor":false,"prefix":"","firstName":"Megotsohe","middleName":"","lastName":"Chasie","suffix":""},{"id":361077958,"identity":"de3b241b-7612-40f7-a69c-e67692ea9c3c","order_by":2,"name":"Akshay Mishra","email":"","orcid":"","institution":"Geological Survey of India","correspondingAuthor":false,"prefix":"","firstName":"Akshay","middleName":"","lastName":"Mishra","suffix":""}],"badges":[],"createdAt":"2024-09-05 07:53:23","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5036259/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5036259/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":69563648,"identity":"ae6a5517-27b8-499f-9be3-83883e4b5d0a","added_by":"auto","created_at":"2024-11-21 16:57:01","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":51097,"visible":true,"origin":"","legend":"\u003cp\u003eTypes of rotational failure in a slope.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/975e3e562d39d58ed21c338f.png"},{"id":69563646,"identity":"0053f44a-b9bc-4ee5-ac5f-8099f78641ed","added_by":"auto","created_at":"2024-11-21 16:57:01","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":509841,"visible":true,"origin":"","legend":"\u003cp\u003eThe slope geometry used in modelling, a) Slope geometry at constant 10° angle with changing height at 10 m interval, b) Slope geometry with constant height and changing slope angle at 10° interval, c) Medium coarse meshing of the model.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/9ce9b852eb98eac0b0cbf9f7.png"},{"id":69563645,"identity":"2cc81ac3-3b4e-40bb-b06c-c96705509880","added_by":"auto","created_at":"2024-11-21 16:57:01","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":205899,"visible":true,"origin":"","legend":"\u003cp\u003eCritical failure surface at 50 m slope height with varying slope angles; a) base failure at 20° slope angle, b) toe failure at 51° slope angle, c) face/slope failure at 75° slope angle.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/1e6c09cc23a74f88b128fa48.png"},{"id":69563640,"identity":"b711e209-c9bb-41c1-86b5-52461f7a1b9e","added_by":"auto","created_at":"2024-11-21 16:57:01","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":112415,"visible":true,"origin":"","legend":"\u003cp\u003eCritical failure surface at 20 m slope height with varying slope angles; a) base failure at 20° slope angle, b) toe failure at 57° slope angle, c) face/slope failure at 75° slope angle.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/95280268084a48a0e90c24c1.png"},{"id":69563885,"identity":"6e6f7cee-b1a4-4969-83d7-6a4e6f7963d7","added_by":"auto","created_at":"2024-11-21 17:05:01","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":138231,"visible":true,"origin":"","legend":"\u003cp\u003eRelationship between slope height and FOS at different degrees of slope inclination.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/f7ab0e87421f43c26f65e01a.png"},{"id":69564660,"identity":"85cd65a8-f559-4eac-a2b4-daa4793d5334","added_by":"auto","created_at":"2024-11-21 17:13:01","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":114457,"visible":true,"origin":"","legend":"\u003cp\u003eRelationship between slope angle and FOS at different slope height.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/b7921a88c6ea5c2f8e38fa64.png"},{"id":69563883,"identity":"e8f4d704-47ee-461f-8c50-633ed598fa5b","added_by":"auto","created_at":"2024-11-21 17:05:01","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":92057,"visible":true,"origin":"","legend":"\u003cp\u003eComparative analysis of the relationship between slope angle, slope angle and FOS.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/214ecaa9f752ac1618639490.png"},{"id":69563649,"identity":"a1b3e841-c2ec-43f0-9f29-6b821eb360ab","added_by":"auto","created_at":"2024-11-21 16:57:02","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":183402,"visible":true,"origin":"","legend":"\u003cp\u003eBase failure at 47° angle and 52 m height.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/235d5023afea61329048e4f6.png"},{"id":69563642,"identity":"5c5465ac-d03b-4ee5-aafa-a074387e4f42","added_by":"auto","created_at":"2024-11-21 16:57:01","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":162944,"visible":true,"origin":"","legend":"\u003cp\u003eToe failure at 49° angle and 54 m height.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/5942566044908ddba2f4c3df.png"},{"id":69563644,"identity":"b1acdcda-5a07-4857-9be2-b021e769ed76","added_by":"auto","created_at":"2024-11-21 16:57:01","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":154473,"visible":true,"origin":"","legend":"\u003cp\u003eSlope failure at 61° angle and 66 m height.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/0f2db11479b66366df9403ab.png"},{"id":95894619,"identity":"5db60ef1-075e-46cc-926a-ab20add2e6a1","added_by":"auto","created_at":"2025-11-14 07:09:06","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2385763,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5036259/v1/01eea43e-d359-4d38-8517-4a28305f0bbd.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Influence of slope angle and height on the failure mode and safety factor of a homogeneous slope in 3D using the Finite Element Method","fulltext":[{"header":"1: Introduction","content":"\u003cp\u003eUnstable slopes can cause significant social and financial consequences especially in areas where human settlement and infrastructures are concentrated. Modified slopes like road cuts, dams, landfills, embankments are particularly vulnerable to slope stability issues (Pourkhosravani and Kalantari, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). As such, slope stability is a common concern for geoscientists and engineers alike in populated areas and in large civil engineering projects. Here, thorough geological and geotechnical investigations backed by reliable slope stability analysis can only serve as the cardinal task for project safety. Soil type, slope geometry, stratification, groundwater and infiltration all contribute to different types of slope failures (Chatterjee and Murali, 2019; Shiferaw, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). While most of the contributory factors for slope instability are intrinsic to the slope, the slope geometry (height and angle) can be considered extrinsic which can be manipulated to a certain extent for safety of the slopes. By altering the geometry of the slope, the probability to failure can be reduced.\u003c/p\u003e \u003cp\u003eTypes of slope failure on a soil slope can be translational, rotational, compound or flow (Rotaru et.al, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Petley et.al, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2002\u003c/span\u003e). In fine-grained homogeneous soils, rotational failure may occur in three different ways: Face failure or Slope failure, Toe failure and Base failure (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe geometric shape of a slope surface significantly influences slope stability (Sabzevari et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Askari et al., 2012; Sharma, 2013; Sun et al., 2017; Wang et al., 2019; Yang et al., 2020; Zan et al., 2022). Failure to fully evaluate the impact of irregular morphology on slope stability can result in significant human and material losses, particularly in slope reinforcement design and construction. Shiferaw (\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) analysed three homogeneous slopes of different soil characteristics (sandy-clay, clay and sandy soils) in 2D and concluded that decreasing slope angle increases the factor of safety nearly linearly while a decrease in height increases the factor of safety at a parabolic rate. Kumar et.al (\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) carried out a similar study in 3D following the limit equilibrium method (LEM) and found that decrease in slope height raises the safety factor nonlinearly while a decrease in slope angle increases the safety factor almost linearly. Their study with respect to different slope material also indicated that toe slip is the predominant slope collapse for clay and sandy-clay soils, while slope failure is the dominant failure mode for sandy soils.\u003c/p\u003e \u003cp\u003eRecent advancements in computing technology have propelled the three-dimensional (3D) finite element numerical approaches as a popular alternative to traditional limit equilibrium methods (LEM) for slope stability analysis. The advantages of 3D methodologies over 2D approaches in determining slope stability are well established (Griffith and Marquez, 2007; Zabuski, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2005\u003c/span\u003e). This study describes the results of a 3D Finite Element investigation to the influence of slope morphology in terms of slope height and angle on its stability conditions and different modes of failure for a pre-defined set of material properties.\u003c/p\u003e"},{"header":"2: Slope stability analysis in 3D using Finite Element Method (FEM)","content":"\u003cp\u003eSlope stability refers to the resistance of inclined slopes (man-made or natural) to failure by sliding. Any exposed inclined slope, in general, may be prone to deformation and mass movement under the influence of gravity. Slope failures are determined by the slope angle, slope height, soil type, soil stratification, groundwater, and seepage. Calculating the factor of safety (FOS) is the standard approach in slope stability analysis (Halder et.al, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). FOS is computed as the ratio of the soil's shear strength (τf) to the mobilized shear (τm) at the failure surface, as shown in Eq.\u0026nbsp;1.\u003c/p\u003e \u003cp\u003eFS\u0026thinsp;=\u0026thinsp;τf/τm\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;. (1)\u003c/p\u003e \u003cp\u003eThere are several approaches for analysing slope stability and computing the factor of safety. FEM is a computational technique used to obtain approximate solutions of boundary value problems in science (Hutton, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). The most common approach to 3D slope stability analysis is still the LEM, which is usually a direct extension of the various 2D slices method to 3D columns method; for example, the ordinary, modified Bishop, Morgenstern and Price, Spencer, and Janbu methods (Hungr, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1987\u003c/span\u003e; Hungr et.al, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e1989\u003c/span\u003e; Lam and Fredlund, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e1993\u003c/span\u003e; Yiang and Yamagami, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Razdolsky, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2009\u003c/span\u003e), and numerous improvements to these methods (Hovland, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1977\u003c/span\u003e; Chen and Chameau, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1983\u003c/span\u003e; Duncan, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e1996\u003c/span\u003e; Cheng and Yip, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Zhang, 2013). The 3D LEM is simple and quick. Unfortunately, the 3D LEM relies on a number of assumptions and cannot be easily adjusted to account for true third-dimension boundary constraints. Moreover, another limitation of the 3D LEM is the difficulty of locating the critical general 3D failure surface in both shape and location (Cheng et.al, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Wei et.al, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). A primary advantage of FEM in slope stability analysis is that no prior assumptions regarding the critical failure surface are required. In this study, FEM-based stability modelling in 3D was carried out by adopting the strength reduction technique wherein the shear strength parameters (c, Φ) of the slope are gradually reduced by a safety factor called the strength reduction factor (SRF). The analysis is repeated for different values of SRF; until the model becomes unstable. The slope's FS is then calculated as the proportion of the soil's real shear strength characteristics to its reduced (critical) shear strength parameters. The strength reduction approach defines the FS as in Eq.\u0026nbsp;(2) (Griffiths \u0026amp; Lane, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1999\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e\u0026#119865;\u0026#119878;= \u0026#119905;\u0026#119886;\u0026#119899; ф\u003csub\u003e\u0026#119894;\u0026#119899;\u0026#119901;\u0026#119906;\u0026#119905;\u003c/sub\u003e/\u0026#119905;\u0026#119886;\u0026#119899;ф\u003csub\u003e\u0026#119903;\u0026#119890;\u0026#119889;\u0026#119906;\u0026#119888;\u0026#119890;\u0026#119889;\u003c/sub\u003e= \u0026#119888;\u003csub\u003e\u0026#119894;\u0026#119899;\u0026#119901;\u0026#119906;\u0026#119905; /\u003c/sub\u003e \u0026#119888;\u003csub\u003e\u0026#119903;\u0026#119890;\u0026#119889;\u0026#119906;\u0026#119888;\u0026#119890;\u0026#119889;\u003c/sub\u003e \u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip;\u0026hellip; (2)\u003c/p\u003e"},{"header":"3: Methodology","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1: Aim\u003c/h2\u003e \u003cp\u003eThe objective of this paper is to understand the influence of slope height and slope angle in the computation of safety factor of a slope in 3D. By altering the slope height and slope angle, the changes in the FS values and variation in the predicted failure modes are to be understood by establishing their inter-relationships.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2: Approach\u003c/h2\u003e \u003cp\u003ePast experimental on the relationship between slope and height has been carried out at only for low height slope mainly for embankment and cut slope stability cases (Kumar et al \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). The research of slope stability along the highways and roads in the Himalayan areas is of great interest since slope instability has resulted in transportation issues, human deaths and injuries, property loss, and environmental degradation. In this a natural slope from 10m to 50m height and angle up to 63m have been taken where homogeneous exposure debris material are has been taken as case and numerical modelling has been performed with different combination of slope angle and height. In this study examines the safety factor in 3D and resultant failure surfaces at various heights and base inclination angles for a certain soil type. Three analytical cases with soil properties as presented in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e were adopted and simulated. For simplification, a homogeneous medium is assumed in the study.\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eHeight variation to compute the safety factor at a specific slope angle.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eProgressively change the slope angle while keeping the slope height constant.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eSimultaneous increase in slope height and angle.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eGeotechnical Properties of Soil\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSl no.\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003eProperties\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003eValues\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eUnite Weight (Dry)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e12.85 kN/m3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eUnite Weight (Saturated)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e16.48 kN/m3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCohesion\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e22.5 kN/m2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAngle of Internal friction\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e37.4\u0026deg;\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePoisson Ration\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.33\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eModulus of elasticity\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e10000 kN/m2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3: Geometry and modelling\u003c/h2\u003e \u003cp\u003eMajority of the parameters in the study of slope through FEM are slope geometry specially height and angle, soil mechanical characteristics, and groundwater conditions to estimate the factor of safety assuming all other parameters to be constant (Shepheard et.al., 2018). This underlines the significance of slope geometry in determining the safety factor of a slope. A 3D slope geometry (d imensions, 100m x 120m x 50m) created in CAD environment was used as input file in MIDAS\u0026rsquo; GTS NX software, a finite element-based program, to compute the factor of safety for several slope models in respect of different slope height and angle (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ea \u0026amp; b). From the developed geometry, a plain strain model with a hybrid Mesher type with medium coarse mesh element of 5 m size as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ec is generated and used in the study.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4: Result","content":"\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e5.1. Deformation\u003c/h2\u003e \u003cp\u003eFailure type/mode on slopes are determined by a number of factors such as soil type and slope geometry. By keeping the soil type and properties constant, the mode of failure and safety factors were examined for various heights and inclination angles. At 50 m fixed slope height, base failure has been observed for slope angle up to 45 degrees (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea). For slope angle greater than 45 degrees up to 57 degrees, toe failure (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb) has been observed. Face/slope failure mode has been observed for slope angles of 63 degrees or more (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ec). However, with decrease in the slope height base failure occurs at greater angle. For instance, at 20 m fixed slope height (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e), base failure occurred up to an angle of 51 degrees (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e) while toe failure occurred at 57 degrees (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e). At higher angles of 63 degrees and more, face/slope failure is observed (Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e). No failure plane (NFP) has been consistently observed at 10 degrees slope angle for any slope height from 5 m to 50 m. The changing failure scenarios (mode of failure) with changes in the slope angle and slope height is summarized in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMode of failure based on height and slope angle.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"10\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSlope angle (degree)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e45\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e51\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e57\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e63\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e69\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c10\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e75\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSlope height (m)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNFP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eToe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNFP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eToe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eToe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNFP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eToe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNFP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eToe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eToe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNFP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eToe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eToe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNFP\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eToe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eToe\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e5.2. Slope height and FOS\u003c/h2\u003e \u003cp\u003eIn the first analytical case, the slope height is gradually increased from 05 m to 50 m at intervals of 5\u0026ndash;10 m to see its variation at a defined slope angle. This variation was analysed at 10, 20, 30, 45, 51, 57, 63, 69 and 75 degree angles. Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows that the safety factor decreases with increasing slope height for specific soil types at a constant slope angle.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSlope height and FOS at a specific slope angle\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"10\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSlope angle (degree)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e45\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e51\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e57\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e63\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c9\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e69\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c10\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e75\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSlope height (m)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e13.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e9.44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.84\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e6.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e5.52\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e4.58\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e4.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e3.13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e2.51\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7.22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e4.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e4.01\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e3.68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e3.51\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e2.33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e2.31\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e2.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e2.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e2.16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1.94\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1.76\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.46\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.79\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.73\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e0.99\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1.15\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.18\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.89\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.21\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1.32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1.16\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.32\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.82\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e1.09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1.22\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c9\"\u003e \u003cp\u003e1.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c10\"\u003e \u003cp\u003e1.05\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e5.3. Slope angle and FOS\u003c/h2\u003e \u003cp\u003eIn the second case, the slope angle was progressively increased while maintaining the slope's height and soil characteristics constant. This investigation was conducted at 10, 20, 30, 45, 51, 57, 63, 69, and 75 degrees with heights ranging from 05 to 50 meters. The factor of safety decreases as the slope angle increases as illustrated in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e5.4 Slope height and angle\u003c/h2\u003e \u003cp\u003eIn the third analytical case, the slope height and angle were simultaneously increased while keeping the soil parameters constant. Modelling was carried out from a height of 40 m to 68 m (failure point) with a 2 m interval. Similarly, the slope angle was increased from 35 degrees to 63 degrees (failure point) with a 2-degree interval. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows the computed safety factor for various slope heights and angles and the associated failure modes.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eSlope height, slope angle, FOS and failure mode\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSlope Height (m)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSlope Angle (degree)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFOS\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eFailure mode\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e35\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.876\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e42\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e37\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e39\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e46\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e43\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e52\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e47\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBase\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e54\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e49\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.31\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003eToe\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e51\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003eToe\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e58\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.19\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003eToe\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e60\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.19\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003eToe\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003eToe\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e64\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e59\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cb\u003eToe\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e66\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e68\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.89\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSlope\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"5: Discussion","content":"\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e6.1: Influence of slope height on FOS and Failure modes\u003c/h2\u003e \u003cp\u003eAs seen in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, decreasing slope height beyond 20m always shows sharp rise in factor of safety. This is more pronounced for the 10, 20, 30-degree slopes. At heights ranging from 5 to 20 meters of the graph is extremely steep and abrupt in relation to the safety factor. For slope heights ranging from 30 to 50 meters, the slope of the line remains relatively shallow giving rise to a half-parabola. This indicates that the change in FOS change up to a height of 20 m, a very small rate from 20 m to 30 m, and fairly constant from 30 m onwards for any particular slope angle. Also, FOS is influenced at higher rate up to the angle of 51 degree and smaller rate from 63 degree onwards in the height between 5 m to 20 m. So, if height increases from 30 m onwards, there will be very little influence on FOS. In terms of failure types, it has been noted that at any slope height, base failure occurs up to 45 degrees, mixed base and toe failure up to 57 degrees and all face/slope failure occurs above this angle. This implies that steeper slopes are more vulnerable to slope collapse.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e6.2: Influence of slope angle on FOS and Failure modes\u003c/h2\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e indicates that the relationship between slope angle and factor of safety is almost linear for any slope height i.e., 5, 10, 20, 30, 40, and 50 metres. FOS has a lesser rate of influence when the slope height increases beyond 20 m. It also indicates that at a given height of 10, 20, 30, 40 and 50 m there is fairly constant FOS beyond the 69 degree slope. In terms of failure types, gentle slopes are indicating base failure whereas steeper slopes are more vulnerable to slope collapse.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e6.3: Influence of slope height-slope angle on FOS and Failure modes\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows how increasing slope height and angle concurrently affects the factor of safety and vice-versa. In general, the desired parameters of a slope to be considered stable is illustrated which is vital for slope geometry alteration activities like scaling. The results also show that the slope's height and angle determine how and when specific soil types collapse. Base failure occurs most frequently when the slope angle is less than 49 degrees and the height less than 54 m (Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e). Toe failure is the most common cause of slope collapse on mild slopes (Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e). Slope failure occurs most commonly on steep slopes greater than 59 degrees (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e). This analysis therefore gives an insight to optimizing slope height and angle for maximum safety factor.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"6: Conclusion","content":"\u003cp\u003eThe study presents the variation in the safety factor of a 3D homogeneous slope and the associated failure modes under changing slope angle and height. Base failure occurs most frequently when the slope angle is less than 45 degrees and the height is low. Toe failure is the most common cause of slope collapse on moderately steep slopes (45\u0026ndash;55 degrees). Slope failure occurs most commonly on steep slopes greater than 55 degrees. Slope safety normally improves linearly as the slope angle reduces, although the safety factor climbs at different rates when the slope height drops. At heights between 5 and 20 meters, the slope of the curve drastically changes in response to the safety factor. As the height grows to 30 meters, the factor of safety drops more quickly but the FOS remains reasonably consistent throughout slope heights ranging from 30 to 50 meters. The results demonstrates that although the FOS decreases nonlinearly with increasing height (half-parabolic), it decreases linearly with increasing slope angle. This understanding of the influence of slope height and angle on the safety factor of a slope is vital for practicing geoscientists and geotechnical engineers for reducing the probability of a slope towards failure or for adopting effective geometric alteration techniques for slope stabilization.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eDhananjai Verma : Prepared the data set and wrote the manuscriptMegotsohe Chasie: Given technical impute, modified figure and tables and accordingly re-aaragne Akshay Kumar Mishra: critically review and suggested technical points\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eThe authors wish to thank DG, GSI, NMH-IV, and DDG MIVA for providing software and facilities. Also, the authors would like to thank all of their GHRM and GSI colleagues for their kind assistance with this research.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe datasets used and/or analysed during the current study available from the corresponding author on reasonable request.The datasets generated and/or analysed during the current study are not publicly available due [since it huge file, yet to be published , once published will be pulically avaialabe] but are available from the corresponding author on reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAskari, F. 3D stability analysis of convex slopes in plan view using lower bound linear finite element Int. \u003cem\u003eJ. Civ. 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(2005).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Slope stability, 3D, FOS, Slope angle, Slope height","lastPublishedDoi":"10.21203/rs.3.rs-5036259/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5036259/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe bearing of slope geometry on the safety factor of a predefined homogeneous slope is analysed in 3D following the finite element-based strength reduction technique for establishing the relationships and the resultant failure modes. The type of failure and shape of the sliding mass is found guided by the slope geometry. The results indicate that FOS decreases nonlinearly with increasing height but decreases linearly with increasing slope angle. This insight can guide practicing professionals from making informed designs on geometry alteration of slopes for effective stabilization.\u003c/p\u003e","manuscriptTitle":"Influence of slope angle and height on the failure mode and safety factor of a homogeneous slope in 3D using the Finite Element Method","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-11-21 16:56:56","doi":"10.21203/rs.3.rs-5036259/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"444e4a6d-5198-49a2-b752-eeb4ba4d62f5","owner":[],"postedDate":"November 21st, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":38414515,"name":"Earth and environmental sciences/Environmental social sciences"},{"id":38414516,"name":"Earth and environmental sciences/Natural hazards"}],"tags":[],"updatedAt":"2025-11-14T07:08:42+00:00","versionOfRecord":[],"versionCreatedAt":"2024-11-21 16:56:56","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5036259","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5036259","identity":"rs-5036259","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}