## Stability analysis for two-layered slopes by using the strength reduction method

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- Published: 04 September 2021
- Volume 12 , article number 24 , ( 2021 )

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- Sourav Sarkar 1 &
- Manash Chakraborty ORCID: orcid.org/0000-0003-3526-5116 1

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The aim of this article is to present the slope stability charts for two layered soil slopes by using the strength reduction method (SRM). The primary focus is to provide a quantitative estimation of the improvement of slope stability when a stronger layer is placed over the weaker layer. The SRM carried in this work comprises a series of finite element lower bound (LB) and upper bound (UB) limit analysis in conjunction with nonlinear optimization. Unlike the limit equilibrium method (LEM), there is no need to consider any prior assumptions regarding the failure surface in SRM. The study is carried out for different combinations of (i) slope angles ( β ), (ii) strength properties of the top and the bottom layer ( c , ϕ ) and (iii) different thickness of the top layer. The failure patterns are shown for a few cases.

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## Introduction

Since the last eight decades several researchers [ 3 , 8 , 12 , 19 , 33 , 39 , 44 , 46 ] had drawn their attention for analyzing the stability of soil slopes. Nevertheless, the problem still remains to be one of the most interesting and challenging problems in geotechnical engineering. Most of the previous studies related to slope stability analysis were mainly carried out by using the limit equilibrium method (LEM). However, in LEM, prior to the analysis, shape of the slip surface, distribution of the normal stress along the slip surface and nature of the interslice forces are require to pre-assume. Moreover, strain and displacement compatibility are not being considered in LEM and hence, the method suffers serious limitations as discussed by Duncan [ 11 ] and Krahn [ 23 ]. Several other researchers [ 1 , 2 , 14 , 21 , 25 , 32 , 48 , 49 ] used various other analytical and numerical methods for analyzing the homogenous slopes and thereby removing the limitations associated with LEM to some extent. Zaki [ 49 ] and Griffiths and Lane [ 15 ] used strength reduction method (SRM) in the framework of conventional displacement based finite element method for analyzing the homogenous slope. Yu et al. [ 48 ] and Kim et al. [ 21 ] used finite element limit analysis to analyze the homogenous slopes with and without considering the effect of pore water pressure. Michalowski [ 32 ] proposed stability charts for uniform slopes subjected to pore- water pressure and horizontal seismic force by using kinematic approach of limit analysis. Baker et al. [ 1 ] produced stability charts for homogeneous slopes by applying the variational method and the strength reduction technique considering pseudostatic analysis.

The stability of layered slopes was also studied rigorously especially in the past few years [ 4 , 5 , 7 , 17 , 18 , 24 , 27 , 28 , 29 , 37 , 42 ]. By assuming the log-spiral failure mechanism, Chen et al. [ 7 ] evaluated the upper bound stability of non-homogeneous slopes corresponding to varying cohesion with depth. Kumar and Samui [ 24 ] evaluated the stability of layered soil slopes subjected to pore-water pressure and horizontal seismic force by using the rigid block upper bound limit analysis. Hammouri et al. [ 17 ] carried out the stability analysis of the layered slopes by considering the effects of rapid drawdown and tension cracks with the aid of PLAXIS 8.0 and SAS-MCT 4.0 software. By using the finite element limit analysis, [ 27 , 28 ], Shiau et al. [ 42 ], and Qian et al. [ 37 ] analyzed the undrained stability of the non-homogenous cohesive soil slopes. Chang-yu et al. [ 18 ] considered rotational mechanism with logarithm helicoids surface and assessed the 3D stability of non-homogeneous slopes. Lim et al. [ 29 ] proposed stability charts for frictional fill material placed on purely cohesive soil by using finite element LB limit analysis method. By using SRM, Chatterjee and Krishna [ 5 ] analyzed two and three-layered soil slopes considering a fixed slope angle (26.57°) and different combinations of three different chosen soils. The literature review clearly indicates that the rigorous analysis for the two-layered cohesive-frictional soil slope is quite limited. Although a few stability studies [ 24 , 38 , 40 , 41 ] were previously carried out by considering weaker layer overlying on strong layer, however, as per the authors’ findings except the work of Sazzad et al. [ 40 ] hardly any study seems to be available for the case where strong layer is considered to be placed over weak stratum. The work of Sazzad et al. [ 40 ] also pertains to a specific combination of layered system (Top Soil: c 1 = 10 kPa, ϕ 1 = 18° and Bottom Soil: c 2 = 6 kPa, ϕ 2 = 10°). Hence, there is a requirement to carry out an extensive and rigorous investigation to estimate the improvement in stability by placing a stronger layer over a weaker layer. This is the prime motivation to carry out the present work. In this present article, strength reduction method is employed to analyze the two-layered soil slopes and to determine the factor of safety. The factor of safety is obtained for different combinations of (i) slope geometry (i.e. slope angle, β ), (ii) strength properties of the top ( c 1 , ϕ 1 ) and bottom layer ( c 2 , ϕ 2 ) (iii) and the thickness of the top layer ( t ). The effect of placing different stronger layers over the weaker bottom layers is thoroughly investigated.

## Strength reduction method (SRM)

The work of Zienkiewicz et al. [ 52 ] appears to be the first where SRM was used to solve the slope stability problem. Following his work, many further studies [ 5 , 9 , 10 , 13 , 15 , 16 , 20 , 26 , 30 , 31 , 34 , 35 , 40 , 45 , 47 , 50 , 51 ] were performed for slope stability problems by using SRM. This method is mainly based on the frame work of finite element method (FEM) and hence, all the advantages of FEM are retained in this method. The significant advantages of this method are the following: (i) the method is suitable to apply for complex geometries, complicated boundary and loading conditions, and (ii) there is no need to consider any assumptions regarding interslice shear forces and the critical failure surface. The critical failure surface is obtained automatically from the shear strength reduction. The additional information regarding stresses, strains, and pore pressures can also be obtained from this method. SRM is also applied on the basis of finite difference method [ 10 ]. Generally, SRM is applied to determine the factor of safety by successively reducing or increasing the shear strength of the material until the slope reaches the limiting equilibrium state.

The SRM is commonly used with the linear Mohr–Coulomb criterion where the failure strength is characterized by cohesion ( c ) and the internal friction angle ( ϕ ). The Mohr–Coulomb model is expressed in Eq. ( 1 ).

where τ is the maximum amount of shear stress the soil can resist for a certain applied normal stress ( σ n ). The analysis is carried out by reducing the strength parameters ( c , ϕ ) progressively until the slope becomes unstable. In conventional SRM, both parameters are reduced by the same factor, or in other words, the reduction path of the cohesion and the friction are identical. The reduced cohesion and the friction angle are computed from Eq. ( 2 ).

where (i) c r and ϕ r are the reduced strength parameters and (ii) F s is the strength reduction factor. These reduced parameters are then reinserted into the model till the failure occurs. The main objective of SRM approach is to compute the strength reduction factor and the reduced material parameters that lead to collapse state.

In the present analysis, Optum G2 [ 36 ] is used for estimating the factor of safety of the slope through strength reduction method. OptumG2 is a finite element limit analysis (FELA) based software developed by OptumCE. For obtaining the limiting solutions, OptumG2 uses second order cone programming to solve the plane-strain stability problems. This scheme works by infeasibility detection in a very controllable way [ 22 , 43 ]. Following steps are adopted for the formulation:

Step 1: Assuming F min and F max ; where, F min and F max are the minimum and maximum value of factor of safety. Generally, F min is chosen to be zero and F max is taken to be a large number within the range of machine precision.

Step 2: Initializing the value of F s and computing reduced strength parameters with the help of Eq. ( 2 ).

Step 3: Checking feasibility through the interior point method by using the reduced strength parameters.

Step 4: If the problem is feasible, assign F min = F S and evaluate a new factor of safety by using the harmonic mean as depicted in Eq. ( 3 ).

Otherwise, if the problem is infeasible, assign F max = F S and evaluate a new factor of safety by following the arithmetic mean as expressed in Eq. ( 4 ).

Step 5: Continue the iterative process (Step 1–Step 4) until the following convergence condition as mentioned in Eq. ( 5 ) is fulfilled:

here, the tolerance limit, T L , is kept as 0·01.

It is worth mentioning that based on the solution process, F min and F max provides the limiting extremities of the bound theorem. F min and F max represent rigorous lower and upper bound on the factor of safety corresponding to the statically admissible stress field domain and kinematically admissible velocity field domain, respectively. The numerical values presented herein are the average of both the limiting values.

## Problem statement and methodology

Figure 1 shows a two-layered soil slope having an angle, β . The strength parameters of the top and the bottom layer are represented by c 1 , ϕ 1 and c 2 , ϕ 2 , respectively. Slope height ( H ) in the present analysis is taken as 20 m for representing the high cut slopes. With the aid of strength reduction method, it is intended to analyze and subsequently compute the factor of safety for three different two-layered slopes (namely, 25°, 35° and 45°) consisting of different soil materials.

Schematic diagram and the boundary conditions of a two-layered soil slope

For performing the analysis, the size of the domain is considered adequately high so that the failure surface remains contained well within the domain. Based on trials, the height ( D ) and length ( L ) of the domain are kept as 2 H and 9 H , respectively. The boundary conditions are mentioned in Fig. 1 . Vertical and horizontal displacements are restrained along the base of the considered domain. Along the left and right boundaries, horizontal displacement is not allowed to occur. The soil mass is discretized by using three nodded linear triangular elements. The soil plasticity is governed by the Mohr–Coulomb failure criterion and associated flow rule. Adaptive mesh refinement based on plastic shear dissipation has been used. Three iterations of adaptive meshing with 10,000 elements have been considered for all analyses. A nonlinear optimizer named sonic is used in Optum G2 software for optimization.

## Results and discussions

In the present work, solutions are presented in terms of factor of safety for different combinations of the (a) slope angles ( β ), (b) soil strength properties of the top and the bottom layers ( c 1 , ϕ 1 and c 2 , ϕ 2 ) and (c) top layer thickness ( t ). Stability charts are presented in tabular form. Tables 1 , 2 , 3 show the factor of safety values computed for three different two-layered slopes ( β = 25°, 35° and 45°). In this article, nine different stronger soil layers [(35,0), (35,5), (35,8), (40,0), (40,5), (40,8), (45,0), (45,5), (45,8)] were considered to be placed over twelve different weaker bottom layer [(20,5), (20,10), (20,15), (20,20), (25,5), (25,10), (25,15), (25,20), (30,5), (30,10), (30,15), (30,20)]; the first and second part within the parenthesis indicate the frictional (in degrees) and cohesive strength (in kPa), respectively. The thickness of the stronger layer was varied between 20%–80% of the domain height. Both the limiting values (lower and upper) were obtained. A total number of 3240 computations were performed. Following observations are made from the numerical results:

As expected, the factor of safety ( F s ) decreases with an increase in slope angle ( β ). However, this decrement depends on the strength of the soil layers. For an example, as β varies from 25° to 45°, F s reduces markedly. This reduction differs by 14% (from 53 to 39%) as the cohesive strength of the top layer, ( ϕ 1 = 35°, and, t/D = 0.4) which is rested upon a certain bottom layer ( c 2 = 20 kPa, ϕ 2 = 25°), increases up to 8 kPa from 0 kPa.

Placing a stronger layer over a weaker stratum undoubtedly improves the stability of the slope and this improvement is more significant for steep slopes. For the previous example, (i) if the cohesive strength of the top layer rises from 0 to 8 kPa (keeping ϕ 1 equals to be 35°), the improvement in F s for 25° and 45° slope occurs by 19% and 56%, respectively; and, (ii) if the frictional strength increases from 35° to 45°, F s improves by 26% and 42% for 25° and 45° slope, respectively.

When the thickness of the top layer is within a certain limit, the strength of the bottom layer also influences the stability of the slope. There is almost a linear relationship between the improvement of F s with the increase in cohesive strength of the bottom layer. However, the relation between the improvement of F s with the increase in frictional strength of the bottom layer is highly nonlinear.

The tabulated data clearly reveals that the improvement in F s is quite significant as the top layer thickness changes from 0.2 D to 0.4 D . On the contrary, when t/D ratio varies from 0.6 to 0.8, the improvement in stability is almost negligible. The effect of the thickness of the top layer is further studied graphically.

Figure 2 illustrates the variation of F s with t/D for a 25° slope. Figures 2 (a) and (b) represent the cases where weaker bottom layer ( ϕ 2 = 25°) is strengthened by placing two different cohesionless layer of friction angle 40° and 45°. Figures 2 (c) and (d) displays the cases where the cohesion of the top layer is considered as 8 kPa. It is quite evident that higher the strength of the top layer higher would be the safety factor, F s . These figures depict that there is a certain t/D beyond which there is hardly any improvement in stability of the slopes. This particular top layer thickness is termed as optimum thickness and is referred here as dimensionless parameter, ‘ t opt /D ’. The value of t opt /D increases with the increase in the strength of the top layer. The figure shows that the dependence of t opt /D on the cohesive strength of the bottom layer is further influenced by the frictional strength of the top layer; when ϕ 1 = 40°, t opt /D decreases with increase in c 2 however, when ϕ 1 = 45° there is no impact of c 2 on the computed value of t opt /D .

The variation of F s with t/D for a two-layered slope ( β = 25°) corresponding to varying c 2 and ( a ) c 1 = 0 kPa, ϕ 1 = 40°, ϕ 2 = 25°, b c 1 = 0 kPa, ϕ 1 = 45°, ϕ 2 = 25°, c c 1 = 8 kPa, ϕ 1 = 40°, ϕ 2 = 25° and ( d ) c 1 = 8 kPa, ϕ 1 = 45°, ∅ 2 = 25°

Figure 3 shows the variation of F s with t/D for β = 25° and 45°, corresponding to two different ϕ 2 , namely, 20° and 30°. The properties of the top layer are kept to be constant and the cohesive strength of the bottom layer is varied within the range of 5–20 kPa. The figures clearly reveal that for the same soil properties, optimum thickness of the top layer is significantly smaller for the steeper slopes. As the frictional strength of the bottom layer increases, the magnitude of t opt /D further reduces. The numerical solutions give an impression that the impact of the strength of the top layer on the stability is much higher than the strength of the bottom layer.

The variation of F s with t/D for two different layered slopes ( β = 25° and 45°) corresponding to varying c 2 and ( a ) c 1 = 8 kPa, ϕ 1 = 35°, ϕ 2 = 20° and b c 1 = 8 kPa, ϕ 2 = 35°, ϕ 2 = 30°

Figure 4 shows the mesh pattern at the collapse state for three different slope angles, namely, 25°, 35° and 45°. The soil profiles for these three cases are kept to be the same. It is to be noted that adaptive mesh refinement technique continuously updates the sizes of all the elements in an optimal fashion by computing the variations of stresses and velocities. Finer elements were automatically placed in the shear failure zone. Hence, these meshes indirectly depict the failure patterns. The figure shows that the size of the failure zone decreases with the increase in slope angle. Moreover, as the slope angle increases the failure is likely to become toe failure. This observation is in accordance with the studies of Lim et al. [ 29 ] and Sazzad, et al. [ 40 ] who had earlier reported that if the top layer is considered to be stronger than the bottom layer and the slope angle is considered to be less than equal to 45° the incipient state of collapse in the soil slope will be triggered by developing base failure. As the steepness of the slope increases, the extent of the failure zone seems to be restricted closer to the slope surface.

Adaptive mesh patterns at the collapse state for three different slopes: a β = 25°, b β = 35° and ( c ) β = 45°

Figure 5 illustrates failure state corresponding to three different thickness of the top layer. The soil properties of the top layer as well as the bottom layer are the same for all the three cases. The figure demonstrates that as the thickness of the top layer increases the type of failure surface turns from toe to base. However, beyond a certain thickness, the slope collapses by developing the toe failure surface and the shear zone seems to be confined within the top layer. This observation substantiates the existence of t opt /D .

Adaptive mesh patterns at the collapse state by varying the top layer thickness: a t = 0.2 D , b t = 0.5 D and c t = 0.8 D

Figure 6 depicts the mesh pattern at the collapse state corresponding to three different frictional angle of the top layer. All other geometrical and material strength parameters are kept to be the same. The figure illustrates that as the frictional strength of the top layer increases the failure zone grows in size. However, the extent at which the finer elements are laid at the collapse state goes thinner with the increase in ϕ 1 . It gives an impression that the thickness of the shearing zone (i.e. shear band) becomes smaller with the placement of stronger layer over a weaker stratum.

Adaptive mesh patterns at the collapse state by varying the top layer frictional strength: a ϕ 1 = 35°, b ϕ 1 = 40° and c ϕ 1 = 45°

## Comparison of results

Comparisons of both the limiting solutions, for the homogenous and layered slopes, are presented in Tables 4 and 5 , respectively. In most of the cases the difference seems to be in the second decimal place. Closeness of the lower and upper bound solutions further depicts the accuracy in the computed solutions. Limit theorems suggest that the true solution will lie somewhere between these bounding values. It should be recalled that the safety factors charts presented in Tables 1 , 2 , 3 , are the average value of the two extremities.

Table 6 shows the comparison of the present solutions computed with the numerical results provided by Dawson et al. [ 10 ] for a homogenous slope of 10 m height having unit weight of soil, γ = 20 kN/m 3 and cohesion, c = 12.38 kPa. Dawson et al. [ 10 ] had employed strength reduction method by using the explicit finite difference code, FLAC . As the frictional strength of the soil increases the present method provides higher stability value. For the same soil, the difference between these two solutions reduces as the steepness of the slope increases. The reason may be attributed not only to the methodology but also to the choice of elements. Dawson et al. [ 10 ] had discretized the chosen domain with four nodded rectangular elements, whereas, in the present work three nodded linear triangular elements are used. The same trend is also observed while the solutions of Dawson et al. [ 10 ] are compared with the upper bound solutions (assuming log spiral mechanism) obtained by Chen [ 6 ]. It is to be noted that the present finite element limit solutions are quite smaller than those rigid block upper bound solutions provided by Chen [ 6 ]. It shows the improvement of the solutions when finite element limit analysis is employed for the analysis.

Table 7 shows the comparison of the present solutions with the results provided by Kumar and Samui [ 24 ] by using the rigid block upper bound method considering log-spiral failure mechanism. The comparison is carried out for 45° slope corresponding to different soil layer properties and varying top layer thickness. Similar to the previous observation, it is well noted that the present solutions become quite smaller than the reported solutions of Kumar and Samui [ 24 ] as the strength of the soil layer increases.

Table 8 illustrates the comparison of current solutions with the solutions provided by Chatterjee and Krishna [ 5 ] for non-homogeneous slopes. Chatterjee and Krishna [ 5 ] used (i) SLIDE v6 and Morgenstern and Price [ 33 ] method for performing the LEM analysis and (ii) PHASE v9 for obtaining the FE solutions. The present solutions are quite agreeable with the reported FE solutions. Table 9 depicts the comparison of present solutions with limit equilibrium solutions presented by Sazzad et al. [ 41 ] for layered soil slopes. Sazzad et al. [ 41 ] used Bishop Method [ 3 ] for LEM analysis. The present solutions appear to be smaller than the LEM solutions. This can be attributed to the fact that LEM solutions generally overestimate the factor of safety due to the usage of statical and kinematical assumptions. This is also observed in earlier studies as well [ 48 ].

## Conclusions

In the present article, the stability of two-layered soil slopes is analyzed by using strength reduction method. A series of upper and lower bound limit analyses are carried out in Optum G2 software by placing different stronger layers of varied thickness over the weaker stratum. Stability charts are prepared in the form of the factor of safety for different soil properties, slope geometries and top layer thickness. The amount of improvement in the stability by placing a layer of stronger soil over the weaker stratum is numerically investigated. The optimum thickness of the top layer is reported for a wide range of slopes. The extent and the type of the failure zones are presented for several cases. The obtained solutions compared quite well with the available solutions. The proposed design charts may be useful to the practicing engineers.

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## Acknowledgements

The corresponding author acknowledges the support of “Department of Science and Technology (DST), Government of India” under Grant Number DST/INSPIRE/04/2016/001692.

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Sarkar, S., Chakraborty, M. Stability analysis for two-layered slopes by using the strength reduction method. Geo-Engineering 12 , 24 (2021). https://doi.org/10.1186/s40703-021-00153-4

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Received : 06 June 2020

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DOI : https://doi.org/10.1186/s40703-021-00153-4

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## Factorial experiment of slope stability under slope-top loading and heavy rainfall

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1 Civil Engineering College, University of south China, 421001, China

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Because the slope-top loading and heavy rainfall are both important factors inducing slope landslides and another geologic hazard. it is necessary to analyze factorial experiment of the slope stability under the slope-top loading and heavy rainfall. Taking a lope engineering of a municipal roads as an example, the stability of the slope was calculated based on the strength reduction method with FEM and Midas-GTS (SRM), and then the double-factor factorial experiment with repeated tests was used to analyze the stability of the slope. The stability results show that, when rainfall conditions remain unchanged, the slope-top loading is applied, and the safety coefficient is reduced by 16%∼21%. When the slope-top loading remains unchanged and the rainfall is strengthened, the security coefficient decreases by 10%∼19%.The analysis shows that the maximum shear strain of the slope is significantly affected by the slope- top loading and rainfall on the top of the slope. The effect of the slope-top loading on the maximum horizontal displacement of the slope is not significant. The effect of heavy rainfall on the maximum displacement of slope is very significant. The combined effect of slope-top loading and heavy rainfall on the maximum horizontal displacement of the slope is significant. Generally speaking, heavy rainfall is the main factor affecting the slope safety. Under the interaction of two factors, the slope stability is poor. In the future, attention should be paid to some local projects such as slope drainage in landslide control and prevention.

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## Slope Stability

Slope stability analysis, slope stability: example analysis.

- Slope Stability Analysis Slope Stabilization Slope Stability

Contents [ hide show ]

## Slope Stabilization

Web class assignments.

In this chapter of the Geoengineer.org series on Slope Stability , a simple example of slope stability analysis will be illustrated. The analysis will be conducted in Slide2 v6.0, a commercial 2D slope stability software by Rocscience that utilizes the Limit Equilibrium Method (LEM) and can be used for the analyses of soil and rock slopes, dams, embankments and retaining walls. Since the analysis will be two-dimensional, a representative cross-section of an idealized slope is developed to derive meaningful insights.

More specifically, a 1:1 slope (45 degrees inclination) that can be either natural or man-made will be considered. The slope height and width will both be 10 meters and the ground conditions will be comprised by two soil layers, a silty clay and a clay . The shear strength of both soils will be assessed via the Mohr-Coulomb failure criterion, as:

Where t is the shear strength, σ n is the effective stress, σ t is the total stress, u is the pore-water pressure, c is the cohesion and φ is the friction angle. In actual real-world cases, the strength parameters of the soil can be determined in drained, triaxial or direct shear tests. The analysis will be conducted assuming long-term conditions; therefore, the effective stresses will be used. In case of short-term analysis, and if undrained conditions are presumed, the undrained shear strength of the ground c u is used ( c=c u and φ=0 °) in the LEM analysis.

The soil parameters that are assumed for the slope stability analysis are given in Table 1 .

Table 1 . The soil parameters used in the example analysis

The boundaries of the model are placed at a distance of 15 meters from the toe and the crown of the slope, respectively, so that the critical failure surface search will not be affected. The initial geometry of the ground, as inserted in Slide2, is illustrated in Figure 1 .

The objective of a slope stability analysis is to detect the critical failure surface and derive its Factor of Safety (FoS) . If the FoS is greater than the minimum acceptable FoS, as defined by the codes/standards/regulations and the significance of the project, the slope is considered stable and no further actions may take place. On the contrary, if the FoS is found to be inadequate, support measures need to be implemented. For the purposes of the present simplified example, an indicative FoS greater than 1.25 will be required for the slope to be considered stable.

Slide2 enables the utilization of multiple limit equilibrium methods (e.g., Ordinary , Bishop , Janbu , Spencer , Morgenstern-Price). Here, the Bishop and the Spencer methods will be used, for illustration purposes. The number of slices for each surface analyzed is set to 30 and the maximum number of iterations to 50 .

The groundwater table level may significantly vary based on the time of the year and other environmental conditions. In this example, 2 groundwater table cases will be assumed:

- A favorable condition, in which groundwater table is relatively low and does not affect the failure surfaces, and
- An unfavorable condition, in which most of the ground lies below the groundwater table ( Figures 2a and 2b ).

In the second case, saturated conditions lead to pore-water pressure build-up, a fact that reduces the shear strength of the ground and consequently the FoS of the slope. For simplification reasons, the first conditions will be referred to as “ Dry conditions ” and the second as “ Partially saturated conditions ”.

At this stage, if any additional external load impacts the stability of the slope, it should be added in the analysis. In our case, we do not consider any external loads. Finally, an auto-generated or a manual grid based on which Slide2 will derive the failure surfaces, is determined. The grid should be large enough to ensure that the critical surface will be included in it.

The problem is currently well-defined and the computational process may begin. The FoS of the slope in dry and partially saturated conditions is derived via the Bishop and Spencer methods and the findings on the critical search surface are given in Figures 3a to 3d . The factors of safety for each case are given in Table 2 .

Table 2 . Factors of safety for the example slope, given the water table conditions and the limit equilibrium methods considered.

Based on the results of the Limit Equilibrium analyses, we derive the following conclusions:

- In all cases (dry and partially saturated conditions), the critical surface is the same. It initiates about 3 meters behind the slope’s crown and ends up at its toe.
- The factors of safety computed by either the Bishop or the Spencer method are almost identical, given the groundwater conditions (dry or partially saturated).
- In all cases, the failure surface propagates through both soil layers. Its depth is significant and cannot be neglected (there are cases that shallow failure surfaces are not taken into consideration due their minor impact on the engineering project).
- Given that the limit value of the FoS in the example project is indicatively set to 1.25, the slope is considered marginally stable in dry conditions and unstable in partially saturated conditions, hence support measures should be implemented.

In practice, an engineer must always ensure the stability of a slope given the worst-case scenario. In our simplified example, the slope is unstable in partially saturated conditions; hence, support measures must be applied so that its FoS exceeds the indicative limit value (1.25).

Many techniques to stabilize a slope and increase its FoS have been developed and employed over the years. Some of the most notable of them include the installation of drainage features, the construction of retaining walls or buttresses, the utilization of geotextiles or anchors, etc.

An optimal design also accounts for the financial feasibility of an engineering project thus, the objective of applying support measures is to increase the FoS with the minimum required and/or available resources. Hence, the selection of the most appropriate support measures is a case-specific issue .

Here, a ground stabilization technique known as soil nailing will be used to increase the FoS of the example slope. This method involves drilling holes for steel bars to be inserted into the face of the slope. The holes are subsequently grouted. As passive supports elements, soil nails are frequently utilized to stabilize slopes in soils and weak rocks. The technical characteristics that need to be defined for each soil nail are the following:

- Length : Soil nails usually measure 0.8 to 1.2 of the slope’s height, however, practically, they should not exceed 15 meters due to the difficulty in forming and maintaining the stability of the drillhole (Bridges, 2017).
- Tensile Capacity : The maximum tensile load that the soil nail can withstand, a values that depends on the material’s properties (steel) and its cross-sectional area.
- Plate Capacity : The maximum load that the plate which connects the soil nail and the ground can bear.
- Shear and Compression Capacities (optional): The maximum shear and compression loads that the soil nail can withstand without failing.
- Pullout Strength : The pullout stripping force which can be produced by a soil nail.

Regarding the geometry of the grid of soil nails that will be established to support a slope, additional parameters need to be determined. These are:

- Out-of-plane spacing : It defines how many meters will separate the soil nails in the 3 rd dimension (perpendicular to the 2D cross-sectional view).
- Longitudinal spacing : The longitudinal distance between soil nails measured i) along the boundary, ii) horizontally or iii) vertically.
- Orientation of the soil nails : The angle of the soil nails usually measured from the horizontal direction.
- Spatial distribution of the soil nails : At which points on the slope the support measures begin and end.

The assumed values of the aforementioned characteristics used for the simplified example analysis presented herein are shown in Table 3 .

Table 3 . Soil nail characteristics used in the example analysis

Table 4 . New factors of safety for the slope, after the support measures are applied.

The results of the analyses yield the following findings:

- The new FoS considering both the Bishop and the Spencer method is above the limit value (1.25), hence, the support measures are considered acceptable.
- Both methods yield the same critical failure surface.
- The critical failure surface does not intersect with the soil nails and thus, it is longer and deeper than the one found by assuming an un-reinforced slope ( Figure 3 ).
- The installation of the soil nails increased the FoS of the slope by 59% and 61% considering the utilization of the Bishop and Spencer method, respectively.

To complete the assessment of the applied support measures, the new FoS in dry conditions is also derived. It can be reasonably assumed that since the measures are adequate in partially saturated conditions, the slope will also be stable in dry conditions. Nonetheless, it is useful to derive its FoS for comparison reasons. The results are given in Figure 6a and 6b and in Table 4 .

When the support measures are applied, the factors of safety in dry conditions increase by 59% and 60% considering the Bishop and Spencer method, respectively.

The support measures are acceptable in both groundwater conditions considered in this example analysis and could be implemented to stabilize the slope.

We emphasize that all the parameters given in this example (shear strength characteristics of the ground, tensile and shear strength of the soil nails, plate capacity etc.) refer to their design values after safety factors have been applied in accordance with the relevant design standards/codes/regulations .

Alsubal, S., Harahap, I.S.H. and Babangida, N.M. (2017). A Typical Design of Soil Nailing System for Stabilizing a Soil Slope: Case Study. Volume: 10, Issue: 4, Pages: 1-7. DOI: 10.17485/ijst/2017/v10i4/110891

Bridges, C. (2017). Design of Soil Nailed Walls According to AS4678-2002. 8th Australian Small Bridges Conference

Tutorial: Rocscience/Slide

Tutorial: Rocscience/Soil Nails

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## Investigating slope failure and landscape evolution with red beans and rice!

Thomas Hickson , University of St. Thomas, St. Paul, MN Author Profile

This activity was selected for the On the Cutting Edge Reviewed Teaching Collection

This activity has received positive reviews in a peer review process involving five review categories. The five categories included in the process are

For more information about the peer review process itself, please see https://serc.carleton.edu/teachearth/activity_review.html .

Expand for more detail and links to related resources

## Activity Classification and Connections to Related Resources Collapse

Grade level.

Audience : I use the full-blown lab activity in a 3 hour lab in a geomorphology course (a sophomore level course). I have also used the experiment as a demo in a large (100 person) introductory geology course by placing a video camera in front of the experiment and projecting it on a large screen.

Skills and concepts that students must have mastered : To my knowledge, none. It is a great way to introduce and explore in-depth many ideas surrounding slope stability and landscape evolution. It is also a great lab to introduce data collection and analysis, as well as measurement. I do not assume that students know anything about slope stability when I start the lab or the activity in class.

How the activity is situated in the course : This is a stand-alone lab exercise or an in-class demo. As a lab, it forms the central motivation for about 1.5 weeks of class material and in-class discussion.

National or State Education Standards addressed by this activity? :

## Description of the activity/assignment

This lab or in-class exercise is based on an actual experiment that appeared in the journal "Science" (Densmore et al., 1997) that used an analog model to simulate the role of bedrock landslides in long-term landscape evolution. Students essentially replicate this experiment by using an acrylic-walled slope failure box filled with either red beans or rice to simulate the landscape. A sliding door on one long side of the box simulates downcutting of the landscape by a river. Lowering the door leads to 'rock' avalanches; students weigh the material that fails and trace slope profiles on the sidewall of the box. As a result of this data collection process, they create a time series of slope failures and slope profiles that shows (a) that large landslides are interspersed with smaller landslides; (b) that oversteepened toes of slope may be an equilibrium landform; and (c) that the type of material that fails will in part govern the nature of the failures, slope profiles, and time series that they detect. Their task is to hypothesize about the behavior of the experiment before running it, then replicate the Densmore et al. (1997) experiment. They then analyze their data and submit a lab report that examines their results in light of the 'Science' article. The activity allows students to read the scientific literature, to collect and analyze real data, to investigate the nature of equilibrium, and to gain a gut feeling for the controls on slope failure.

## Determining whether students have met the goals

Students submit a lab write-up that is graded based on a grading rubric. They must also submit the results of the experiment as a series of Excel plots/charts.

## Download teaching materials and tips

- Activity Description/Assignment (Acrobat (PDF) 93kB May11 05)
- Instructors Notes (Acrobat (PDF) 61kB May11 05)
- Solution Set (Excel 19kB May11 05)

## Other Materials

- Examples of slope profiles (Acrobat (PDF) 2.5MB May11 05)
- Data collection sheet (Acrobat (PDF) 30kB May11 05)
- A short of the exercise for use as an in-class demo (Acrobat (PDF) 41kB May11 05)
- Grading rubric for lab write-up (Excel 9kB May11 05)

## Supporting references/URLs

Densmore, A.L., Anderson, R.S., McAdoo, B.G., and Ellis, M.A., 1997, Hillslope Evolution by Bedrock Landslides, Science, Vol. 275, p. 369-372.

See more Teaching Resources »

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## JSmol Viewer

Research and experiment on cruise control of a self-propelled electric sprayer chassis.

## 1. Introduction

2. materials and methods, 2.1. test platform construction, 2.2. analysis of the overall cruise control model architecture, 2.3. modeling of cruise control dynamics, 2.3.1. modeling of inverse longitudinal dynamical systems, 2.3.2. four-wheel torque distribution strategy, 2.3.3. modeling of motor systems, 2.3.4. chassis dynamics modeling, 2.4. design of constant speed cruise control method based on fuzzy pid, 2.4.1. general architecture of cruise control system for self-propelled sprayers, 2.4.2. upper and lower level controller design, 3. analysis of simulation and field test results, 3.1. analysis of simulation test results, 3.1.1. four-wheel torque distribution strategy validation, 3.1.2. cruise control function verification, 3.2. analysis of field vehicle test results, 3.2.1. transit transportation test, horizontal straight-line acceleration, straight-line uphill driving, 3.2.2. fieldwork trials, 4. discussion, 5. conclusions, author contributions, institutional review board statement, data availability statement, conflicts of interest.

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Parameter Name | Parameter Unit | Description |
---|---|---|

Chassis drive form | / | Four-wheel independent motor drive |

Steering form | / | Four-wheel independent motor steering |

Vehicle full load mass | kg | 450 |

Sprayer wheelbase | mm | 1200 |

Ground clearance | mm | 600 |

Maximum working speed | km/h | 20.0 |

Minimum working speed | km/h | 3.0 |

e | e | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

NB | PB/NB/PS | PB/NB/NS | PM/NM/NB | PM/NM/NB | PS/NS/NB | ZO/ZO/NM | ZO/ZO/PS |

NM | PB/NB/PS | PB/NB/NS | PM/NM/NB | PS/NS/NM | PS/NS/NM | ZO/ZO/NS | NS/ZO/ZO |

NS | PM/NB/ZO | PM/NM/NS | PM/NS/NM | PS/NS/NM | ZO/ZO/NM | NS/PS/NS | NS/PS/ZO |

ZO | PM/NM/ZO | PM/NM/NS | PS/NS/NS | ZO/ZO/NS | NS/PS/NS | NM/PM/NS | NM/PM/ZO |

PS | PS/NM/ZO | PS/NS/ZO | ZO/ZO/ZO | NS/PS/ZO | NS/PS/ZO | NM/PM/ZO | NM/PB/ZO |

PM | PS/ZO/PB | ZO/ZO/NS | NS/PS/NS | NM/PS/PS | NM/PM/PS | NM/PB/PS | NB/PB/PB |

PB | ZO/ZO/PB | ZO/ZO/PM | NM/PS/PM | NM/PM/PM | NM/PM/PS | NB/PB/PS | NB/PB/PB |

Test Scene | Test Grouping | Road Conditions | Driving Conditions |
---|---|---|---|

Transit transportation | Group 1 | Asphalt concrete roads | Accelerate in a straight line to a constant speed |

Group 2 | Concrete pavement | Driving straight uphill | |

Field operations | Group 3 | Compacted dirt road | Accelerate in a straight line to a constant speed |

Parameter Name | Maximum Values | Minimum Value | RMS Value |
---|---|---|---|

Speed (m/s) | 2.13 | 1.85 | 1.91 |

Acceleration (m/s ) | 0.08 | −0.01 | 0.06 |

Parameter Name | Maximum Values | Minimum Value | RMS Value |
---|---|---|---|

Speed when traveling horizontally (m/s) | 2.39 | 2.14 | 2.24 |

Speed when traveling uphill (m/s) | 2.38 | 2.11 | 2.28 |

Acceleration while traveling horizontally (m/s ) | 0.08 | −0.09 | 0.063 |

Acceleration when traveling uphill (m/s ) | 0.27 | −0.03 | 0.221 |

Pitch angle when traveling horizontally (°) | 1.55 | 0.12 | 1.47 |

Pitch angle when traveling uphill (°) | 9.35 | 7.64 | 8.42 |

Parameter Name | Maximum Values | Minimum Value | RMS Value |
---|---|---|---|

Speed (m/s) | 1.41 | 1.28 | 1.35 |

Acceleration (m/s ) | 0.64 | −0.03 | 0.013 |

Pitch angle (°) | 9.24 | 2.89 | 13.74 |

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## Share and Cite

Zhou, L.; Hu, C.; Chen, Y.; Guo, P.; Zhang, L.; Liu, J.; Chen, Y. Research and Experiment on Cruise Control of a Self-Propelled Electric Sprayer Chassis. Agriculture 2024 , 14 , 902. https://doi.org/10.3390/agriculture14060902

Zhou L, Hu C, Chen Y, Guo P, Zhang L, Liu J, Chen Y. Research and Experiment on Cruise Control of a Self-Propelled Electric Sprayer Chassis. Agriculture . 2024; 14(6):902. https://doi.org/10.3390/agriculture14060902

Zhou, Lingxi, Chenwei Hu, Yuxiang Chen, Peijie Guo, Liwei Zhang, Jinyi Liu, and Yu Chen. 2024. "Research and Experiment on Cruise Control of a Self-Propelled Electric Sprayer Chassis" Agriculture 14, no. 6: 902. https://doi.org/10.3390/agriculture14060902

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## Geological and geotechnical assessment of slope stability along proposed road of Ayah-Jladri section, Kebumen, Central Java

- Indrawan, I. Gde Budi
- Warmada, I. Wayan

Geological and geotechnical assessments play a significant role in geotechnical work and slope stability analysis. This study aimed to assess the stability conditions of the proposed road cut along the Ayah-Jladri section, Kebumen, Central Java, using parameters from the geological and geotechnical survey. The parameters were defined through geological mapping on a scale of 1:25.000 and measurement of the depth of the water table, the weathering grade, estimated UCS value, and GSI value. Besides, the input parameters during the computation were obtained from laboratory test results, such as value UCS and unit density, and using the strength behavior of a rock mass approach. The slope stability assessment was performed based on deterministic analysis by the limit equilibrium method (LEM) of Morgenstern & Price assuming a seismic load of the study area is 0.196. The results showed that the slope composed of sparitic limestones with slightly weathered (73.62 MPa) has a safety factor 7.314 using circular slip surface and 6.461 using non circular slip surface, which means the slope is very safe. While, the andesitic lava that is undergoing moderate weathering near the surface (13.59-138.16 MPa) has a safety factor 1.095 using circular slip surface and 1.556 using non circular slip surface, which means the slope failure will occur through the circular slip surface. Based on the above results, it is inferred that the weathering condition that affects the activity of swelling clay minerals has a great effect on the stability conditions of the study area. There are two recommendations to improve the stability of the unstable slope: 1) change the geometry of slope to get a gentler overall slope angle in the weathered andesite zone and 2) added two supports on each bench in the weathered andesite zone.

- INNOVATION IN CIVIL ENGINEERING SOLUTIONS

## COMMENTS

The aim of this article is to present the slope stability charts for two layered soil slopes by using the strength reduction method (SRM). The primary focus is to provide a quantitative estimation of the improvement of slope stability when a stronger layer is placed over the weaker layer. The SRM carried in this work comprises a series of finite element lower bound (LB) and upper bound (UB ...

Later, a replicative experiment was carried out simultaneously on a vegetated slope with the uniform root morphology, a bare soil slope, and an earthen slope reinforced by soil nails. Similar results occurred again that the vegetated slope with the uniform root morphology maintained the stability for a long time in the early stage until a ...

A factor of safety greater than 1.00 suggests that the slope is stable. Slope stability analysis is implemented in numerous applications of civil engineering projects such as dams, embankments, excavated slopes, and natural slopes. Slope stability involves both static and dynamic analyses. The stability techniques include limit equilibrium ...

Method of slices. Slope stability analysis is a static or dynamic, analytical or empirical method to evaluate the stability of slopes of soil- and rock-fill dams, embankments, excavated slopes, and natural slopes in soil and rock. It is performed to assess the safe design of a human-made or natural slopes (e.g. embankments, road cuts, open-pit mining, excavations, landfills etc.) and the ...

This study reviews the existing methods used for slope stability analysis. These methods are divided into five different groups which are; (a) Limit equilibrium method, (b) Numerical simulation ...

This study aims to evaluate the stability of the soil-nailed slope using an improved discretisation-based failure model in the framework of the upper-bound limit analysis theorem and the ...

The stability of the nailed slope is governed by various factors such as slope geometry, nail parameters, the soil-nail interaction, etc. [7]. Slope angle, backslope gradient, nail inclination, nail length, the spacing of the nails—these are some of the important parameters that directly affect the stability of slopes.

Taking a lope engineering of a municipal roads as an example, the stability of the slope was calculated based on the strength reduction method with FEM and Midas-GTS (SRM), and then the double-factor factorial experiment with repeated tests was used to analyze the stability of the slope. The stability results show that, when rainfall conditions ...

Secondly, after the fourth scouring experiment, the depth changed greatly, ranging from -32 to 1 cm. Due to the increase in the scouring flow in the fourth experiment, the slope surface was eroded seriously, and the water flow carried a large amount of sediment, so most of the depths were lower than those in the third experiment.

Compared with the mild slope angle, the lithological factors have more significant influence on the subaqueous slope stability (McAdoo et al., 2000). For instance, the medium or coarse sandy slopes are difficult to lose the overall stability under ocean waves. ... Fig. 6, the incident waves in this experiment have notable nonlinear behaviour ...

Abstract. Limit equilibrium analysis of slope stability is comprised of two coupled problems: kinematical (i.e., find the critical slip surface) and statical (i.e., assure the existence of global equilibrium at the defined limit state). Determination of the general‐shaped critical slip surface can be attained by available optimization ...

Slope Stability Analysis. In this chapter of the Geoengineer.org series on Slope Stability, a simple example of slope stability analysis will be illustrated.The analysis will be conducted in Slide2 v6.0, a commercial 2D slope stability software by Rocscience that utilizes the Limit Equilibrium Method (LEM) and can be used for the analyses of soil and rock slopes, dams, embankments and ...

In combination with the video recorded for the experiment, the slope collapsed when the earth pressure was less than before the experiment. 3.3. ... Liqun Ma, Zhiying Lv, Hongyu Qin, and Jianbo Guo. 2023. "Experimental and Numerical Calculation Study on the Slope Stability of the Yellow River Floodplain from Wantan Town to Liuyuankou" Toxics 11 ...

Experiment 1: Investigating slopes made from different materials. You will need: some containers with at least three different types of sediment (for example: pea gravel, washed/dried sand, builders sand, rock chippings), a plastic tray, something to measure slope angles. Slowly pour the sediment of your choice to make a hill in the plastic tray.

Some other finite element computations for slope stability problems were performed by Laouafa and Darve (2002), ... The sand for the experiment was selected because it showed a fairly small permeability, in order to approach natural permeabilities as much as possible. Its shear parameters were known and, hence, the results of this benchmark ...

The paper describes a field experiment in which a 9 m deep cut slope in Gault Clay was brought to failure by pore pressure recharge. The geology of the site is described and the procedures used to form the slope are explained. A novel feature was the use of low-friction panels at each end of the study section to form isolation trenches and thus encourage primarily two-dimensional displacements ...

This lab or in-class exercise is based on an actual experiment that appeared in the journal "Science" (Densmore et al., 1997) that used an analog model to simulate the role of bedrock landslides in long-term landscape evolution. Students essentially replicate this experiment by using an acrylic-walled slope failure box filled with either red ...

How does water impact slope stability? Find out in this simple experiment. For more information - https://ausearthed.blogspot.com/2020/06/slope-stability-ii....

Limit equilibrium-based Morgenstern and Price method used for the determination of factor of safety of slopes. The results showed that all the slopes are stable under normal conditions except the Ramsoo slope. However, under the transient conditions with a maximum daily rainfall scenario, the stability of the slopes is reduced.

The study proves the performance of the Vetiver grass in improving the unsaturated soil behavior and stability of highway slopes built on highly expansive clay. Due to cyclic wetting and drying, the hydro-mechanical behavior of unsaturated soil is impacted significantly. In order to assess the soil strength parameters, knowing the unsaturated ...

Slope deformation, a key factor affecting slope stability, has complexity and uncertainty. It is crucial for early warning of slope instability disasters to master the future development law of slope deformation. In this paper, a model for point prediction and probability analysis of slope deformation based on DeepAR deep learning algorithm is proposed.

The data needed to analyze slope stability include the internal shear angle, cohesion, and soil unit weight; analysis and modeling of slope stability using Geostudio SLOPE/W software. The results of the slope stability analysis at point 1 show that the average failure line is at a depth of 6.00 m with the smallest Safety Factor (SF) value of 0. ...

In order to address the issues of poor stability in vehicle speed and deteriorated spraying quality caused by changes in road slope and the decrease in overall mass due to liquid spraying, this study focuses on analyzing the structure and longitudinal dynamic characteristics of a 4WID high ground clearance self-propelled electric sprayer. By utilizing MATLAB/Simulink software, three subsystems ...

Geological and geotechnical assessments play a significant role in geotechnical work and slope stability analysis. This study aimed to assess the stability conditions of the proposed road cut along the Ayah-Jladri section, Kebumen, Central Java, using parameters from the geological and geotechnical survey. The parameters were defined through geological mapping on a scale of 1:25.000 and ...