Slope Stability Evaluations Using Limit Equilibrium and Finite Element Methods
Mohamad Ayob1, Anuar Kasa2, Mohd Sofiyan Sulaiman3*, Nirwani Devi Miniandi4 and Abdul Hafidz Yusoff5
1,2Structural and Civil Engineering Department, Faculty of Engineering and Built Environment, National University of Malaysia, 43600 Bangi, Selangor, Malaysia
3,4Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia
5Faculty of Bioengineering and Technology, Universiti Malaysia Kelantan, Jeli Kampus, Locked Bag No. 100, 17600 Jeli, Kelantan, Malaysia
Abstract
Landslide occurrences in Malaysia cause huge economic losses each year and have resulted in over 600 recorded casualties from 1961 to 2011. It is therefore overdue for practitioners and researchers alike in Malaysia to re-evaluate slope stability in high-risk areas prior to any mitigation work. In this study, the case study is focused on one such slope stability evaluation that was conducted at a landslide-prone location, i.e., Maktab Rendah Sains MARA (MRSM), Bentong, Pahang. The evaluation uses limit equilibrium and finite element methods; more specifically, SLOPE/W software for the limit equilibrium method and PLAXIS software for the finite element method. The primary distinction between these two analytical approaches is that finite element methods are based on the stress–strain relationship of the soil whereas limit equilibrium methods are based on static equilibrium that divide sliding mass into smaller slices. Here limit equilibrium methods collectively represent a conventional approach in which the fundamental principles of static equilibrium and interslice forces are used in the past.
Conversely, finite element approaches serve as a more realistic indicator for the factor of safety in the absence of stress distribution data. The simulations showed that both methods produce results that are not significantly different, but the use of the finite element method proves to be best-suited for complicated geometries, as those experienced in Malaysia.
Keywords: factor of safety, finite element method, limit equilibrium method, Plaxis, slope stability, slope/W
1. Introduction
Limit equilibrium (LE) methods are widely used by researchers and engineers in conducting slope stability analyses [23]; however, the use of finite element (FE) methods has gained acceptance within the geotechnical community to better assess slope stability.
Two key methods fall under the FE approach, these being the enhanced limit strength method [43] and the Strength Reduction Method [44]. Regardless of the efficacy of these methods, the LE approach has remained the method of choice for researchers since 1930.
This approach makes full use of a variety of methods depending on the type of problem to be solved (e.g., circular versus non-circular) and the required accuracy of the results [24].
Past researchers have devoted their work toward developing methods that make use of LE analysis [6], [13], [21], [26], [31], and LE methods are widely used because of their simplicity and accuracy, as opposed to FE methods, which require computer models and specific operators to handle the complex variables. In [24] conducted a detailed study comparing the applicability of LE and FE methods. In their work, they used a homogeneous slope with no foundation layer as the basis for their analysis and findings.
Similarly, published examples from [9], [15]-[16] were re-examined using both types of
methods. They found out that problems with complex geometries or those requiring an analysis of seepage, consolidation, and other coupled hydrological and mechanical behaviors could be consistently better-handled using FE methods rather than LE methods.
More recently, in [23] found out that the critical slip surfaces for both FE and LE methods are in good agreement; however, the factor of safety (FOS) obtained from the LE method is slightly lower than those obtained from two FE methods. These findings lead to an early conclusion that both LE and FE approaches are not too different from one another, though the use of the LE approach is very much accepted for practical purposes. In Malaysia, instability issues in natural slopes are common challenges that both researchers and professionals face. A direct consequence to slope instability is a landslide, which can have a tragic and extremely costly impact. According to the National Slope Master Plan (NSMP) 2009–2023, Selangor and the Federal Territories have experienced the most landslides since the 1970s, followed by Pahang, Penang, and Sabah [40]. Through November 2011, approximately 600 deaths have been recorded since 1961. The highest fatality count for a single landslide was recorded on December 26, 1996, and killed 300 people in Keningau, Sabah. Economic losses from landslides totaled almost RM3 billion (S$1.2 billion) from 1961 to 2007. Overall, reports for decades have described tragedies that have not only taken innocent lives but also affected social demographics, political influence, economic calamities, and environmental blows [40]. Table 1 summarizes the loss of life and estimated economic losses from 1961 to 2008. As noted above, the highest recorded death toll was in 1996 when 300 souls were lost; the estimated financial loss stood at RM450 million for this single event, which signifies the degree of catastrophe possible. Over the past few years, the Ampang district within the territory of Kuala Lumpur has been recognized as a hotspot for slope failures. Inadequate slope analysis design has led to slope failure in many instances. Slope analysis techniques must be evaluated using a correct and effective method before implementation or mitigation work can then begin.
Table 1. Major Landslides in Malaysia
Location Year Consequences
Estimated Loss (RM Millions) Ringlet, Cameron
Highlands
1961 16 deaths 35
Highland Tower, Selangor
1993 48 deaths 185
Genting Sempah Jalan ke Genting Highlands, Selangor
1995 20 deaths 48
Lebuh Raya Utara Selatan di Gua Tempurung, Perak
1996 1 deaths, expressway closed for 15 days
17
Aliran Puing Keningau, Sabah
1996 302 deaths 456
Simunjan, Sarawak 2002 16 deaths 32
Km 44 Jalan Spg.
Pulai Cameron Highlands
2000 Road opened in 2004 (4-year delay)
355
Bukit Lanjan, Lebuhraya NKVE
2003 Expressway closed for 6 months
860 Kg. Pasir, Ulu
Kelang
2006 4 deaths 21
Bukit 2008 5 deaths 200
Antarabangsa, Selangor
Given the scale of the problem, in this paper, FOS for selected terrain in Malaysia, specifically in Maktab Rendah Sains MARA (MRSM) Bentong, Malaysia was given special attention. Both LE and FE methods were applied to determine the FOS results and compare these results to observe similarities and differences between these two approaches. The primary aim of this paper is to analyze slope stability using both LE and FE approaches for hilly terrain at MRSM Bentong, Pahang, in Malaysia. The FOSs obtained from both approaches are compared to identify the suitability of one approach over another in terms of slope stability. It is expected that the FE approach will yield reliable results that agree with those obtained via the LE method. This study has been carried out in sliding area at Maktab Rendah Sains MARA (MARA) Bentong, a boarding school in Malaysia. The first visit was carried out on June 19, 2014 due to a reported landslide issue on December 2009.
2. Methodology
For this study, soil samples were collected for determination of the shear strength parameters as part of the slope stability evaluations prior to modeling in the Slope/W and PLAXIS software packages. In addition, the determination of soil classification based on the grain size distribution and Atterberg limits for identifying the mechanical properties of soil are explained. The subsections that follow detail study methodology.
2.1. Study Area
MRSM Bentong, Pahang, is a secondary public school located along the Karak Highway. In addition to a main highway connecting the east and west coasts of Peninsular Malaysia, there is a major river called the Benus River flowing nearby the school compound. The school area was originally a rubber plantation but was redeveloped as an academic area because of a government policy. The school area is a hilly terrain, but slopes were cut to flatten the terrain and provide ample space for the school compound. Unfortunately, as shown in Figs. 1 and 2, in December 2009, a failure occurred on the southeast cut slopes that affected the edge of the sports facility and destroyed the sewage treatment plant. Luckily, neither injury nor fatality was reported. Soil samples were collected from the crest of the slope at depths ranging from 1.5 to 1.95 m for further laboratory analysis and software model execution.
Figure 1. Topography of the Study Area Showing the Location of Slope Failure
Figure 2. A Photograph of the December 2009 Slope Failure in MRSM Bentong
2.2. Laboratory Analysis
Laboratory analysis was performed to determine the particle size distributions, moisture content (MC), Atterberg limit, and shear strength parameters. The MC was measured to determine the water content of the soil sample in accordance with the BS1377-1:1990 standard. The Atterberg limits, plastic limit (PL), liquid limit (LL), and plasticity index (PI) were also obtained in accordance with the MC of the soil. Note that the PL, LL, and PI are very much correlated to one another. Apart from these tests, the shrinkage limit must be obtained as well to determine the behavior of the soil volume with regard to MC. These laboratory tests were conducted by following the standard laboratory procedures presented in BS1377-2:1990. Then, the particle size distribution test was conducted to determine the grain size distribution percentage. For coarser particles, dry sieving was performed to produce a finer percentage. If more than 10% of the soil samples were within the standard clay or silt range, then hydrometer test is commenced.
The investigations here were conducted again by adhering to the standard laboratory procedures stated in BS1377-2:1990. Finally, it is important to note that the shear strength test is crucial in determining for slopes or cuts, the bearing capacity, and pressure exerted on the retaining structure. Direct shear tests is conducted to determine the angle of friction and cohesion values of the soil. This test is performed by following the procedures stated in BS1377-7:1990.
2.3. Analysis Methodology
The data obtained from the site visit and laboratory investigations were analyzed primarily via two software packages, i.e., SLOPE/W and PLAXIS. The SLOPE/W software is used to implement the LE method, whereas the PLAXIS software is used to implement the FE method. Each of these methods is described in more detail below.
SLOPE/W can effectively analyze both simple and complex geometrical shape for a variety of slip surface shapes, pore-water pressure conditions, soil properties, analysis methods, and loading conditions [18]. The software is able to perform a variety of LE methods, including the aforementioned Ordinary Method of Slices [13], Bishop’s Simplified Method [6], Janbu’s Simplified Method [45], Morgenstern and Price’s Method [26], and Spencer’s Method [33]; see Table 2. Further, the Mohr–Coulomb soil strength properties obtained from the laboratory investigation results can be loaded into the software to perform each of these analysis methods. The slices involved in the critical slip surface are then analyzed and displayed together with the free-body diagram and force polygon. A half-sine function is used to compute the interslice forces based on the assumption that there are 30 slices, no phreatic level correction, no tension cracks, and no
optimization of the critical shear surface. Further, the slip surface option is defined to be the entry and exit method, and the direction of movement is from left to right. The PLAXIS software, developed by PLAXIS BV in the Netherlands, uses the incremental tangent stiffness approach to FE method analysis in which the load is divided into a number of small increments that are determined simultaneously. During each load increment, stiffness properties appropriate for the current stress level are employed in the numerical analysis [7]. Next, the Shear Strength Reduction technique [11], [16]-[17]
enables the FE method to calculate the FOS for the given slopes. The above method provides numerous benefits by simultaneously incorporating the ability to forecast stresses and deformations for the given slope model. This method also enables users to envision the progression of failure mechanisms. The model used in the PLAXIS software is the plain-strain model, with a 15-node element containing 12 stress points to provide accurate calculations of stresses and failure loads. In addition, no tension cracks and a 1%
tolerance are defined in the model. Finally, the mesh global coarseness is specified to be a medium for element distribution.
3. Results and Discussion
3.1. Laboratory Results
The laboratory tests was performed to obtain MC, the Atterberg limits, particle size distributions, and shear strength parameters of the soil sample from MRSM Bentong. As noted above, the soil samples were obtained from MRSM Bentong, Pahang, during a site visit on June 19, 2014. More specifically, soil samples were taken from the location of the slope failure in the vicinity of Borehole 2 at depths of 1.5 to 1.95 m. From simple visual observations, the obtained soil was light brown in color and was perceived to be composed primarily of sand. The parameters of a specific sample (sample #5) from our laboratory results are shown in Table 2.
Table 2. Sample of The Laboratory Test Results, Including Soil Characteristics
Sample No.
MC (%)
γd
(kN/m3)
γ (kN/m3)
PL (%)
LL (%)
PI (%)
Φ (°) c (kPa)
Gs
#5 13.61 19.11 21.71 23 33 10 30.84 1.1685 2.65
From Table 2, the angle of friction ()) and cohesion (c) values were obtained via the graph of the peak shear stress versus the normal stress for 4.5, 9.0, and 18.0 kg loading on the direct shear equipment. The respective shear stress is manually observed as shown in Fig. 3, which shows peaks of 28, 16, and 7 kN/m2, respectively. These three values are then plotted to obtain, shown in Fig. 4.
Figure 3. Shear Stress Versus Horizontal Displacement
Figure 4. Peak Shear Stress Versus Normal Stress
The soil can be classified as A-2-4 in the American Association of State Highway and Transportation Officials (AASHTO) Classification System, indicating that the typical significant constituent materials are silty or clayey gravel and sand. Further, the general subgrade rating is from excellent to good. Therefore, using the Unified Soil Classification System, the soil can be categorized as SC (i.e., sands with fines) with less than 15%
gravel; thus, the soil is clayey sand. Previous slope assessment report of MRSM Bentong is acquired during the 2014 site visit. From this report, the critical information pertaining to the soil sample collected from Borehole 2 are able to obtained. More specifically, these measures included depth, gradient, the modulus of elasticity, and water level. From the laboratory investigations, as described in the previous subsection, unit weight γ, dry unit weight γd, angle of friction Φ, cohesion c, and the MC of the upper soil layer are determined. Combining the slope assessment report and laboratory results, the pertinent parameters for the middle and lower soil layers from the previous report, as summarized in Table 3, were applied to the stability evaluation methods. Note that the above input parameters belong to the Mohr–Coulomb soil modeling technique obtained from laboratory tests conducted to determine the shear strength parameters. Finally, Young’s modulus was obtained from the previous site assessment report, and Poisson’s ratio is assumed to be 0.33.
Table 3. Input Parameters for the Stability Analysis
Soil Layer
Depth (m)
γd, (kN/m3)
γ, (kN/m3)
Φ (°)
c (kPa)
Gradient of Slope
(°)
MC (%)
E,
MPa ν Water Level
(m) Upper 7 19.11 21.71 30.84 1.1685 45.00 13.61 15.00 0.33 Nil Mid 7 15.16 18.50 33.00 18.000 45.00 22.00 30.00 0.33 Nil Lower 15 16.52 19.00 34.00 20.000 - 15.00 50.00 0.33 Nil
To model a slope, a simplified model is relied in which the existing ground level and angle of inclination are assumed to be flat even planes with no jagged surface features.
Further, the slope is modeled using three soil layers according to the input parameters listed in Table 3, with the boundaries defined as straight planes using our simplified approach.
3.2. LE Method Results
Five LE methods are applied using the aforementioned SLOPE/W software. Soil properties were defined using a Mohr–Coulomb soil model at each layer. In the resultant slope model, which is shown in Fig. 5, the dark green region represents the slip surface, whereas the red region symbolizes the safety map overlaying on the profile to indicate a zone in which slip surfaces with very similar FOS measures could develop, i.e., where similar problems could occur.
Figure 5. MRSM Bentong Slope Model Obtained From the SLOPE/W Software Next, the free-body diagram and force polygon are compared for each of the LE methods of slice 10, which produced the largest force in the critical slip surface. Table 4 shows these comparisons, revealing that the weight of slice 10 for all approaches was the same, i.e., 57.056 kN, since the volume of the soil is the same for all methods. Further, it is observed that there was up to a 17.33% difference between the largest values obtained from the Bishop’s Simplified Method (BSM) and the smallest values obtained from the M-PM of the base normal force for each of the LE methods; note that these differences originate from the formula for base normal force N. It is also observed that the largest difference between base shear forces occurs with M-PM and Janbu’s Simplified Method (JSM), which have base shear forces of −30.334 and −38.166 kN, respectively, producing a 20.52% difference due to the dissimilar FOS measures for all methods. More specifically, the base shear force is calculated as Eq. (1).
Sm FOS* . (1)
All LE methods, except for OM, produce left- and right-side normal forces; of particular note, BSM and JSM produced the same left- and right-side normal force values, whereas M-PM and Spencer’s Method (SM) produced different values because of differences in the base normal and shear force calculations. Only M-PM and SM correctly computed the left- and right-side shear forces, with similar values obtained because M- PM relates the shear force to the normal force via equation T = λf(x)E whereas SM assumes interslice normal and shear forces to be parallel, with the difference between the right-side shear forces reaching a maximum of 5.23%.
Table 4. A comparison of forces on slice 10 using different LE methods LE
Method
OM BSM JSM M-PM SM
Free- body diagram
Slice 10 - Ordinary Method
57.056
35.25 39.733
Slice 10 - Bishop Method
57.056
35.681 45.306 150.64
154.71
Slice 10 - Janbu Method
57.056
38.166 42.908 150.64
154.71
Slice 10 - Morgenstern-Price Method
57.056
30.334 34.906 124.07
95.55
128.09
106.53
Slice 10 - Spencer Method
57.056
33.458 40.705 121.13
96.255
127.05
100.96
Weight, kN (Blue)
57.056 57.056 57.056 57.056 57.056
Base Normal
Force, kN (Red)
39.733 45.306 42.908 34.906 40.705
Base Shear Force, kN (Green)
−35.250 −35.681 −38.166 −30.334 −33.458
Left- Side Normal
Force, kN (Black)
- 150.64 150.64 124.07 121.13
Right- Side Normal
Force, kN (Black)
- 154.71 154.71 128.09 127.05
Left- Side Shear Force, kN (Black)
- - - 95.550 96.255
Right- Side Shear Force, kN
- - - 106.53 100.96
(Black)
OM = Ordinary method; BSM = Bishop’s method; JSM = Janbu’s simplified method; M-PM = Morgenstern and price method; SM = Spencer’s method Table 5 summarizes the results obtained for the five LE methods. From the table, it is proven that in using five distinct approaches for the LE methods, the conventional method for calculating the FOS is essentially JSM because of its simplicity, which has a FOS value of 1.097, whereas the highest FOS is produced by BSM, with a FOS value of 1.217.
Morgenstern and Price’s and SMs of analysis are more comprehensive as they yield both the moment and force equilibrium through their computation of the FOS. Both methods produced a FOS value of 1.209. Therefore, it is opted to use Morgenstern and Price’s Method and Spencer;s Method to compare the FOS generated using the FE approach.
Table 5. Summary of Results Obtained for Each of the LE Methods
LE
Method FOS
Total Volume
(m3)
Total Weight
(kN)
Total Resisting
Moment (kNm)
Total Driving Moment,
(kNm)
Total Resisting
Force (kN)
Total Driving
Force (kN)
OM 1.130 53.79 1087.7 19174 16974 - -
BSM 1.217 53.79 1087.7 20664 16974 - -
JSM 1.097 53.79 1087.7 - - 626.89 571.30
M-PM 1.209 53.79 1087.7 20528 16974 660.93 545.64
SM 1.209 53.79 1087.7 20521 16974 661.01 545.13
3.3. FE Method Results
For the FE method, a mesh model is generated using the PLAXIS software for the upper, middle, and lower soil layers. The analysis is then conducted using the c–Φ reduction method. Note that the slope is modeled in the PLAXIS software to better illustrate the mesh used for our FE method of analysis. There were a total of 154 elements within the mesh, with pore-water distributions based on determined phreatic levels. In the modeling, standard fixities is applied to the boundaries of the slope model and added closed boundary conditions to the left and right sides of the model.
Figure 6 shows the active pore-water pressures acting on the slope model, with the extreme active pore pressure reaching 290.00 kN/m2 at the highest level of the slope. Note that groundwater levels were not detected via piezometer readings for all layers; therefore, the groundwater level was assumed to be located at the base of the slope model.
Figure 6. Pore-Water Pressure of the Slope Model Determined Via the PLAXIS Software
Given the steep 45° angle of inclination of the slope, the soil initiates movement based on gravity, pore-water pressure, and other factors, all of which combine and cause the upper soil layer of the slope to partially collapse at the top of the slope. From the slope model, we observed an extreme total predicted displacement of 2.18 m. Fig. 7 and 8 show the deformed mesh and total displacement of the slope after the model is applied via the PLAXIS software. From these figures, it is observed that the most critical area for displacement is at the peak of the inclination of the slope. This displacement does not affect the middle and lower layers of the soil, as is evident from the given model.
Therefore, it is conclude that upper layer of the soil is not having any displacement as well.
Figure 7. Deformed Mesh of the Slope After the Analysis Using the PLAXIS Software is Complete
Figure 8. Total Displacement of the Slope After the Analysis Using the PLAXIS Software is Complete
Finally, Figure 9 shows the FOS values produced by the PLAXIS software for our FE method with a value of 1.060. In the figure, we progressively decreased the strength parameters of the soil model until the calculation phase produces a completely developed failure. Note that the green shaded region identifies the critical slip surface of the slope.
Figure 9. FOS Produced by the PLAXIS Software Using the FE Method 3.4. Comparing the Slope Stability Analysis Methods
The existing LE methods take into account a variety of factors, including interslice forces and Mohr–Coulomb principle to determine the shear strength along the failure surface. Conversely, the FE method analyzes the slope model via the stress distribution of the soil. The fundamental principles of both the LE and FE methods are illustrated in Table 6. In the rightmost column of the table, we present the percentage differences for the slope model across all methods as evaluated against M-PM are presented.
Table 6. A Comparison of the Resultant FOS Values for the Five LE Methods and One FE Method
Method Analysis FOS
% Difference vs Morgenstern and Price’s
Method
Limit equilibrium method by slope/W
Ordinary method of
slices (OM) 1.130 −6.53%
(discrepancy ratio of 1.09) Bishop’s simplified
method (BSM) 1.217 0.66%
(discrepancy ratio of 0.99) Janbu’s simplified
method (JSM) 1.097 −9.26%
(discrepancy ratio of 1.10) Morgenstern and
price’s method (MPM) 1.209 -
Spencer’s method (SM) 1.209 -
Finite element method by PLAXIS
Strength reduction method
(c–Φ)
1.060 −12.32%
(discrepancy ratio of 1.14) From Table 6 and Figure 10, it is observed that BSM was the only method that overestimated the FOS, with a value of 1.217, i.e., from Table 6, 0.66% higher as compared to Morgenstern and Price’s Method. Conversely, the method that provided the lowest FOS was the Strength Reduction Method (i.e., the selected FE method), which
yielded results 12.32% lower than those produced by Morgenstern and Price’s Method.
The Ordinary Method of Slices and JSM had differences of 6.53% and 9.26%, respectively, versus that of Morgenstern and Price’s Method, and SM yielded the same results as Morgenstern and Price’s Method.
Figure 10. A Comparison of the Analysis Results Versus Those Produced by Morgenstern and Price’s Method
The Ordinary Method of Slices is the simplest LE method of the five, where the FOS is computed via the moment equilibrium by dividing the total resisting moment against the total driving moment; further, only the weight of the slice with the shear forces acting on the base of the slice is considered here, thereby ignoring the horizontal and vertical interslice forces. Even so, the FOS computed from the Ordinary Method of Slices is the second lowest compared to all other LE methods; thus, it is conclude that less accurate prediction was obtained since the method ignores a number of factors that affect the moment equilibrium. Given these omissions, the Ordinary Method of Slices is seldom used, producing consistently lower FOS results and poor precision. Unlike the Ordinary Method of Slices, BSM reflects the FOS by computing the moment equilibrium by dividing the total resisting moment by the total activating moment. Here, the weight of the slice with the shear forces acting on the base of the slice impacts the results, with the interslice forces assumed to be horizontal and the vertical interslice forces ignored. The calculated FOS from BSM was the highest among all other methods because of the overestimation in the calculation of the effective base normal acting on the shear surface of the slice, thereby producing a higher total resisting moment. As a result, this method is typically used more frequently than the Ordinary Method of Slices, but caution must be exercised, as the produced FOS is the highest of all five methods to avoid overprediction.
Next, JSM determines the FOS from force equilibrium conditions by dividing the total resisting force by the total driving force. Similar to BSM, the interslice forces are assumed to be horizontal, and the vertical interslice forces are ignored; thus, the model considers the weight of the slice and the shear forces acting on the base of the slice. Here, the calculated FOS was the lowest among all other LE methods, which is attributable to the underestimation of the total resisting force and the overestimation of the total driving force. Given these results, JSM is typically used more frequently than the Ordinary Method of Slices for composite soils, though usage is rather limited as the produced FOS was the lowest as compared to all other methods. The value of the FOS in Morgenstern and Price’s Method is achieved from the force and moment equilibrium requirements by dividing the total resisting force by the total activating force and then dividing the total resisting moment by the total activating force, taking the smaller value of the two as the resultant FOS. Further, both horizontal and vertical interslice forces are considered on top of the weight of the slice and the shear forces acting on the base of the slice. Morgenstern
and Price’s Method is the most comprehensive of all methods, taking into account all static equilibrium conditions and all interslice forces, relating the shear force to the normal force. Not surprisingly, this method had the second-highest FOS as compared to the other methods, and the FOS is justified here as it satisfies all force and moment conditions for the slices in the critical slip surface. Therefore, in general, Morgenstern and Price’s Method is well-received and extensively used in slope stability evaluations at hilly terrain environment. Moving to SM, we found it to be similar to Morgenstern and Price’s Method, with its FOS calculation using force and moment equilibriums, dividing the total resisting force by the total driving force and then dividing the total resisting moment by the total driving force, taking the smaller value of the two as the resultant FOS. Here, both horizontal and vertical interslice forces are considered together, with the weight of the slice and the shear forces acting on the base of the slice. The only difference between SM and Morgenstern and Price’s Method is that SM assumes that all interslice forces are parallel with an unknown inclination, which is computed using an iterative approach. Not surprisingly, SM produced the same FOS value as that of Morgenstern and Price’s Method given the negligible differences in total resisting moment. Correspondingly, the FOS is defensible since it entirely fulfills force and moment settings for the slices in the critical slip surface. Given these results, SM is indeed the second-most widespread LE method used after Morgenstern and Price’s Method. Finally, for the FE method, the focus of this paper, the Strength Reduction Method was used, finding it able to effectively accommodate changes in the stress–strain distribution, which is in stark contrast to the LE methods described above. The excessive strains in the mesh generation are localized within the elements of the soil model; thus, we calculate the FOS using a c–Φ reduction system. Further, using the FE method, it is able to efficiently compute the total displacement of the soil model from the input data. The Strength Reduction Method procedure uses an iterative approach in which the strength parameters are gradually decreased until the slope becomes unstable, the FOS obtained by dividing the base strength by the lowest strength at which the slope is stable. From the analysis, the FOS computed by this method was the lowest among all LE methods. It can be concluded that this was due to the shear stress concentrations successfully captured in the FE method but not in the LE analyses in which the normal forces at the base are principally a consequent of the weight of the slices. Hence, the stress distribution in the slope model of the FE method is more representative of real-life conditions. The required FOS from local authority or public works department i.e: Jabatan Kerja Raya Malaysia (JKR) is 1.2. As such, the FOS values computed via Morgenstern and Price’s Method, SM, and BSM are all independently sufficient to fulfill the JKR requirement. Since the other LE methods and the proposed FE method do not satisfy this FOS requirement, it is recommended that the slope is reinforced with soil nailing, geosynthetics, a retaining wall, or a horizontal drain. The addition of such slope reinforcements will greatly increase the FOS computed in stability analyses.
4. Conclusion
In this paper, a study on both LE and FE methods for slope stability is performed.
From the finding, it is conclude that the studied slope stability evaluation methods should be obtained collectively as part of a larger slope stability analysis to determine the resultant FOS. LE methods have the advantage of widespread use and proven methodologies of static equilibrium from numerous scientific research studies; in particular, the most extensively used method, i.e., Morgenstern and Price’s Method, covers both force and moment requirements of the static equilibrium. In general, LE methods are well-acknowledged as a result of easy-to-use interfaces, straightforward modeling, and common fundamental principles that are thoroughly established throughout the field. LE methods use the philosophies of static equilibrium in relation to slices in the critical slip surface, and LE methods have been broadly applied since they produce
satisfactory FOS results that can be corroborated from its basic ideologies. Nonetheless, it is found out that LE methods are lacking in that they do not fully consider the stress–
strain relationship of the soil, which is also essential in slope stability evaluation. The proposed FE method can therefore be used as an additional slope stability analysis tool to support the LE methods. In particular, it is showed that the Strength Reduction Method is well-suited for complicated geometry and further representative normal stress distributions, successfully generating consequential FOS values. The FE method considers the basic physics of the stress–strain relationship of a given soil model, generating a mesh with elements and stresses on the nodes to produce an accurate stress distribution result. Therefore, a more realistic stress situation can be computed from the stress–strain behavior of the soil and provide even more accurate FOS results, as shown by the modeling experiments. Finally, it is noted that the FE method can be used to handle the absence of stress concentration problems present in LE methods.
Acknowledgments
The authors convey their gratitude to Kumpulan Ikram Sdn Bhd and Infrastructure University Kuala Lumpur for providing the MS ISO/IEC 17025:2005 certified laboratory to conduct some of the geotechnical tests.
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