# Sustainable Slope Stability Analysis: A Critical Study on Methods

^{1}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Information on Slope Stability Analysis

#### 1.2. Historical Context, Understanding Primary Methods

- a.
- The Swedish circle approach (Ordinary or Fellenius method, 1936) applies to homogeneous soils, stratified soils, fully or partially submerged soils, non-uniform soils, and cases where seepage and pore pressure exist within the soil slope. Fellenius is used to analyse the stability of a slope assuming a circular failure surface. Shear strength along the slip surface contributes by the frictional component, which depends on normal stress. Normal stress changes horizontally at any point on the slip surface. Therefore, the analysis divides the wedge into many vertical slices. The distance between the centroid of the potential moving wedge and the centre of rotation “O” is calculated by dividing the algebraic sum of the weight moment for each slice about centre “O” by the wedge weight. The safety factor is generated from the momentum equilibrium equation concerning the centre of the potential slip surface (O). Investigations are repeated on different slip surfaces to define the factor of safety.$$FoS=\frac{{\sum}_{i}\left[{c}_{i}\xb7{l}_{i}+\left({N}_{i}-{u}_{i}\xb7{l}_{i}\right)tan{\varphi}_{i}\right]}{{\sum}_{i}\left({W}_{i}\xb7sin{\alpha}_{i}\right)}$$
- b.
- Bishop’s simplified method (1955) considers the normal interslice forces but ignores the interslice shear forces. That satisfies the overall equilibrium of moments but not the general equilibrium of horizontal forces, assuming a circular slip surface. The value of the factor of safety is determined by successive iterations.

^{2}x/dt

^{2}= A(x) or equivalently of the form $\dot{v}$ = dv/dt = A(x), $\dot{x}$ = dx/dt = v [13].

- c.
- For a minimal-width slice, the assumption of Janbu’s method is that the vertical component of the interslice forces depends on the numerical approximation of the differential equation of the moment equilibrium [15]. The method considers normal interslice forces but ignores interslice shear forces. It satisfies the overall horizontal force equilibrium, not the general moment equilibrium.
- d.
- In the Spencer method (1967), the interslice forces are parallel, and the normal force (N) acts on the centre of the base of each slice [16]. Spencer has developed two equations for the factor of safety; the first one of moment equilibrium and the other one of horizontal force equilibrium. This method adopts a constant relationship between the interslice forces (shear and normal forces). By iterative procedures, the interslice shear alters to a normal ratio until the two safety factors are equal. Finding a shear-normal portion equalizing the two safety factors means that the balance between moment and force is met [17].
- e.
- Morgenstern and Price calculus is similar to Spencer’s method but allows various specified interslice force functions.

#### 1.3. Objectives of the Current Research

^{®}software [27]. This study compares the limit equilibrium results with those determined by the shear strength reduction method. In particular, the slip surface assumes a circular shape in the case of the limit equilibrium method and a logarithmic spiral in the upper-bound limit analysis. However, in this paper, some limitations of those shapes in comparison to the other shapes of slip surfaces such as damped sinusoid (damped sine wave) and parabola are presented. Some case studies have identified the position of the slipping centre, sampling it according to the initial hypothesis of each method.

## 2. Materials and Methods

#### 2.1. Limit Equilibrium Methods

- Determining the coordinates of the centre of the slip surface.

_{f}(x), has a different factor of safety that can be evaluated by the safety function, F[Y

_{f}(x)]. The factor of safety can be found by searching for the critical slope failure surface, Yc(x), associated with the lowest factor of safety.

#### 2.2. The Upper-Bound Limit Method

#### 2.3. Methodology for the Centre and Slipping Surface Identification in FEM Analysis

- Based on the slope stability problem results, identify a set of points on the slip surface. These points can be identified as the ones separating the slip block (large displacement) from the unmovable soil masses (Figure 4).
- Using the coordinates of the set points in 1, find the best fit using the least-squares method and the slip surface equations. Construct a system of two equations: the general equation of the second-order surface and a spiral logarithmic one using a non-linear least-squares procedure.
- Using the coefficients of surfaces identified in 2, find the nature of the slip surface and its parameters.
- For the slip surface identified in step 3, estimate the centre of the slip surface.

## 3. Results

#### 3.1. Case Studies for the Comparative Study

#### 3.2. Results

_{i}, y

_{i}), i = 1, …, n, where ”x” is an independent variable, and ”y” is a dependent variable, with the values found by observation. The fitting of the model distribution to a data point was measured by its residual, which is defined as the difference between the observed value of the dependent variable and the fitting value predicted by the model used on the result of each equation. The least-squares method approximated the solutions, finding the optimal parameter values by minimizing the sum of squared residuals.

Soil Type | S1 | S2 | |||||||
---|---|---|---|---|---|---|---|---|---|

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 1.994 | 1.600 | 4.296 | 4.711 | 2.602 | 1.646 | 4.357 | 4.741 |

Bishop simplified | 2.081 | 1.508 | 5.210 | 5.432 | 2.698 | 1.600 | 4.966 | 5.216 | |

Janbu corrected | 2.102 | 1.554 | 4.783 | 5.094 | 2.743 | 1.646 | 4.783 | 5.062 | |

Spencer | 2.079 | 1.508 | 5.210 | 5.432 | 2.696 | 1.600 | 4.905 | 5.170 | |

Morgenstern–Price | 2.077 | 1.508 | 5.210 | 5.432 | 2.695 | 1.600 | 4.905 | 5.170 | |

FEM (Plaxis)—circular | 2.063 | 1.374 | 5.626 | 5.919 | 2.689 | 1.558 | 5.598 | 6.048 | |

Soil Type | S3 | S4 | |||||||

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 3.314 | 1.692 | 4.357 | 4.726 | 5.203 | 1.866 | 4.513 | 6.110 |

Bishop simplified | 3.431 | 1.646 | 4.783 | 5.062 | 5.329 | 1.720 | 4.513 | 4.834 | |

Janbu corrected | 3.495 | 1.646 | 4.783 | 5.062 | 5.521 | 1.866 | 5.123 | 6.539 | |

Spencer | 3.429 | 1.646 | 4.783 | 5.062 | 5.330 | 1.720 | 4.513 | 4.834 | |

Morgenstern–Price | 3.427 | 1.646 | 4.783 | 5.062 | 5.328 | 1.720 | 4.513 | 4.834 | |

FEM (Plaxis)—circular | 3.423 | 1.564 | 5.366 | 5.908 | 5.282 | 1.986 | 5.417 | 6.793 |

Soil Type | S1 | S2 | |||||||
---|---|---|---|---|---|---|---|---|---|

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 1.170 | 2.270 | 13.973 | 14.159 | 1.464 | 2.710 | 13.364 | 13.638 |

Bishop simplified | 1.224 | 1.390 | 15.040 | 15.104 | 1.532 | 1.976 | 14.354 | 14.490 | |

Janbu corrected | 1.223 | 1.976 | 14.354 | 14.490 | 1.532 | 2.563 | 13.592 | 13.830 | |

Spencer | 1.222 | 1.390 | 15.040 | 15.104 | 1.528 | 1.976 | 14.354 | 14.490 | |

Morgenstern–Price | 1.221 | 1.390 | 15.040 | 15.104 | 1.528 | 1.976 | 14.354 | 14.490 | |

FEM (Plaxis) | 1.195 | 1.161 | 17.484 | 17.706 | 1.509 | 1.859 | 16.742 | 17.024 | |

Soil Type | S3 | S4 | |||||||

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 1.844 | 2.930 | 13.059 | 13.381 | 2.513 | 3.957 | 11.230 | 11.912 |

Bishop simplified | 1.930 | 2.050 | 14.278 | 14.422 | 2.601 | 3.737 | 11.687 | 12.270 | |

Janbu corrected | 1.931 | 2.563 | 13.592 | 13.830 | 2.655 | 3.664 | 11.840 | 12.390 | |

Spencer | 1.925 | 2.050 | 14.278 | 14.422 | 2.599 | 3.590 | 11.992 | 12.510 | |

Morgenstern–Price | 1.925 | 2.050 | 14.278 | 14.422 | 2.597 | 3.737 | 11.687 | 12.270 | |

FEM (Plaxis) | 1.844 | 2.930 | 13.059 | 13.381 | 2.604 | 3.915 | 14.845 | 15.915 |

Soil Type | S1 | S2 | |||||||
---|---|---|---|---|---|---|---|---|---|

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 1.752 | 0.565 | 4.864 | 4.897 | 2.298 | 0.703 | 4.620 | 4.674 |

Bishop simplified | 1.785 | 0.382 | 5.169 | 5.183 | 2.338 | 0.382 | 5.169 | 5.183 | |

Janbu corrected | 1.851 | 0.382 | 5.169 | 5.183 | 2.440 | 0.382 | 5.169 | 5.183 | |

Spencer | 1.784 | 0.382 | 5.169 | 5.183 | 2.338 | 0.382 | 5.169 | 5.183 | |

Morgenstern–Price | 1.784 | 0.382 | 5.169 | 5.183 | 2.337 | 0.382 | 5.169 | 5.183 | |

FEM (Plaxis) | 1.678 | 0.113 | 5.237 | 5.260 | 2.210 | 0.283 | 5.183 | 5.225 | |

Soil Type | S3 | S4 | |||||||

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 2.931 | 0.703 | 4.620 | 4.674 | 4.707 | 0.978 | 4.071 | 4.190 |

Bishop simplified | 2.980 | 0.565 | 4.864 | 4.897 | 4.738 | 0.978 | 4.071 | 4.190 | |

Janbu corrected | 3.116 | 0.382 | 5.169 | 5.183 | 5.106 | 0.565 | 4.864 | 4.897 | |

Spencer | 2.980 | 0.565 | 4.864 | 4.897 | 4.878 | 0.290 | 5.291 | 5.306 | |

Morgenstern–Price | 2.978 | 0.565 | 4.864 | 4.897 | 4.794 | 0.428 | 5.108 | 5.121 | |

FEM (Plaxis) | 2.821 | 0.334 | 5.159 | 5.195 | 4.489 | 0.782 | 5.210 | 5.325 |

Soil Type | S1 | S2 | |||||||
---|---|---|---|---|---|---|---|---|---|

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 0.962 | −0.754 | 12.235 | 12.255 | 1.222 | −0.020 | 11.778 | 11.771 |

Bishop simplified | 0.994 | −1.047 | 12.388 | 12.431 | 1.258 | −0.387 | 12.007 | 12.011 | |

Janbu corrected | 1.011 | −0.901 | 12.311 | 12.343 | 1.287 | −0.387 | 12.007 | 12.011 | |

Spencer | 0.992 | −1.047 | 12.388 | 12.431 | 1.256 | −0.387 | 12.007 | 12.011 | |

Morgenstern−Price | 0.992 | −0.974 | 12.388 | 12.413 | 1.255 | −0.460 | 12.083 | 12.081 | |

FEM (Plaxis) | 0.947 | −2.204 | 15.199 | 15.376 | 1.190 | −1.716 | 14.472 | 14.705 | |

Soil Type | S3 | S4 | |||||||

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 1.545 | −0.020 | 11.778 | 11.771 | 2.193 | 1.464 | 10.518 | 11.016 |

Bishop simplified | 1.589 | −0.387 | 12.007 | 12.011 | 2.227 | 0.950 | 10.975 | 11.016 | |

Janbu corrected | 1.630 | −0.387 | 12.007 | 12.011 | 2.353 | 0.143 | 14.785 | 14.786 | |

Spencer | 1.588 | −0.387 | 12.007 | 12.011 | 2.222 | 1.317 | 10.670 | 10.736 | |

Morgenstern−Price | 1.586 | −0.387 | 12.007 | 12.011 | 2.227 | 0.950 | 10.975 | 11.016 | |

FEM (Plaxis) | 1.501 | −1.315 | 13.789 | 13.944 | 2.130 | 0.032 | 14.438 | 14.537 |

Soil Type | S1 | S2 | |||||||
---|---|---|---|---|---|---|---|---|---|

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 1.405 | −1.350 | 4.681 | 4.866 | 1.869 | −1.029 | 4.437 | 4.554 |

Bishop simplified | 1.396 | −1.350 | 4.681 | 4.866 | 1.853 | −1.029 | 4.437 | 4.554 | |

Janbu corrected | 1.512 | −1.625 | 4.864 | 5.122 | 2.034 | −1.533 | 4.803 | 5.036 | |

Spencer | 1.639 | −2.166 | 7.790 | 7.873 | 2.203 | −0.799 | 7.729 | 7.768 | |

Morgenstern−Price | 1.631 | −1.212 | 7.790 | 7.881 | 2.186 | 0.576 | 2.365 | 2.672 | |

FEM (Plaxis) | 1.389 | −1.473 | 5.429 | 5.666 | 1.860 | −1.267 | 5.293 | 5.481 | |

Soil Type | S3 | S4 | |||||||

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 2.392 | −1.029 | 4.437 | 4.554 | 3.985 | −0.799 | 4.254 | 4.328 |

Bishop simplified | 2.371 | −1.029 | 4.437 | 4.554 | 3.941 | −0.249 | 3.767 | 3.772 | |

Janbu corrected | 2.610 | −1.441 | 4.742 | 4.951 | 4.421 | −1.808 | 7.851 | 8.052 | |

Spencer | 2.812 | 0.668 | 2.243 | 2.565 | 4.752 | −0.020 | 7.546 | 7.549 | |

Morgenstern−Price | 2.790 | 0.622 | 2.304 | 2.618 | 4.635 | 0.255 | 1.938 | 2.702 | |

FEM (Plaxis) | 2.379 | −1.268 | 5.424 | 5.612 | 3.970 | −0.535 | 4.955 | 5.106 |

Soil Type | S1 | S2 | |||||||
---|---|---|---|---|---|---|---|---|---|

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 0.745 | −4.059 | 10.172 | 10.951 | 0.961 | −3.619 | 10.324 | 10.934 |

Bishop simplified | 0.741 | −2.959 | 8.724 | 9.211 | 0.951 | −2.812 | 8.724 | 9.160 | |

Janbu corrected | 0.792 | −4.573 | 11.544 | 12.404 | 1.022 | −4.426 | 11.620 | 12.433 | |

Spencer | 0.762 | −4.206 | 11.848 | 12.544 | 1.007 | −3.106 | 12.382 | 12.759 | |

Morgenstern−Price | 0.766 | −3.986 | 11.925 | 12.565 | 1.009 | −3.106 | 12.458 | 12.806 | |

FEM (Plaxis) | 0.598 | −7.835 | 17.389 | 19.126 | 0.886 | −7.202 | 16.031 | 17.663 | |

Soil Type | S3 | S4 | |||||||

Method | FoS | XC | YC | R | FoS | XC | YC | R | |

LEM (Slide) | Fellenius | 1.219 | −4.426 | 11.620 | 12.433 | 1.803 | −1.749 | 10.461 | 10.605 |

Bishop simplified | 1.206 | −2.739 | 8.724 | 9.134 | 1.762 | −1.162 | 8.327 | 8.407 | |

Janbu corrected | 1.299 | −4.426 | 11.620 | 12.433 | 1.956 | −2.996 | 12.442 | 12.788 | |

Spencer | 1.290 | −2.812 | 12.534 | 12.833 | 2.038 | 0.085 | 13.509 | 13.504 | |

Morgenstern−Price | 1.285 | −2.886 | 12.534 | 12.838 | 2.015 | −0.135 | 13.509 | 13.487 | |

FEM (Plaxis) | 1.171 | −7.384 | 15.778 | 17.502 | 1.786 | −3.974 | 14.841 | 15.487 |

_{s}(x) and Y

_{f}(x) describe slope surface and failure surface, respectively. Y

_{f}(x) corresponds to a log spiral, described by ‘r’ in polar coordinates. Coordinates (Xc, Yc) represent the pole of the log spiral of the Cartesian system with the origin at the toe of the slope. That is the reason why (Xs, Ys) and (Xe, Ye) are the coordinates at which the failure surface intersects the slope surface, associated with angles θs and θe of polar coordinates [33]. The log-spiral failure surface predicts the critical failure surface of different 2D slope geometries studied using an optimization process.

## 4. Discussion and Conclusions

- The stability analysis compared the safety factor values obtained by the limit equilibrium method with those resulting from the upper-bound analysis by connecting the advantages provided by the upper-bound theorem with the finite element method through a strength reduction method with displacement-based finite element method (strength reduction finite element analysis).
- Numerical models run in PLAXIS 2D using these data led to accurate results regarding the factor of safety compared to those obtained in Slope2 Rocscience using five LE methods (Fellenius, Bishop simplified, Janbu corrected, Spencer, and Morgenstein–Price). The comparison has significant importance for verifying slope stability requirements.
- The results being in good agreement in the cases taken into account show the influence of the slope geometry on the safety factor. Varying the slope angle gradually while keeping the height of the slope constant, the factor of safety increases as the slope angle decreases for the same type of soil, as Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8 show. The decrease in slope increases the factor of safety almost linearly. There is a strong and opposite relation between slope angle and factor of safety for the four types of soils. Results also show that the FoS (stability, implicitly) is dependent of the slope height, even in the case of homogeneous slopes. The factor of safety increases as the slope height decreases, as the pair of Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8 show for the same type of soil and gradient. Additionally, at a lower height, i.e., 3 m, the failure mode in clays (S4) tends to be base slide for all the three gradients, while at 8 m height, the failure mode tends to be toe slide. All these results indicate a strong relationship between the slope height and the factor of safety. In this regard, the geometry of the slope (height and angle) may and should be optimized to maximize the slope factor of safety.
- The mesh shape has a limited influence on the slope stability in FEM. However, both mesh size and density significantly influence the shape of the slip surface. Hence, the study adopted a local mesh refinement in the critical area and sparse mesh in the other areas. Furthermore, the adaptive mesh refinement influences the error margins.
- Even though the comparison of the factor of safety resulting from the analysed methods shows a slight difference, this approach indicates a fundamental difference in their basic principles. While the LE method relies on the formulations of the limit equilibrium dependent on a static force or moment equilibrium, the other formulation depends on the stress–strain relationship. It finds a critical slip surface where the excessive strains are localised and computes the FoS by a c-phi reduction procedure for the Mohr–Coulomb model. This analysis calculates the safety factor for each element along with the CSS. That makes the FoS more reliable than in the LE methods.
- Compared to LEM, numerical analysis does not require any a priori definition of the failure mechanism and provides accurate upper bounds of the (FoS) but is limited by the associated flow rules.
- Even though both methods provide tight values for safety factor estimation, potential slip surface, and the presumed centre of the slip surface, they have their advantages and limitations. The values achieved indicate that the frequency of using the upper-bound limit method may be similar to using LEM in routine analysis and design, considering the limitations of each method in evaluating the results.
- Graphs of the regression curves for five distinct shapes of the critical slip surface, including the benchmark (the circular slip surface), show that the shape of the slip surface is not necessarily the same in the case of the same slope example. Depending on the slope geometry and material, a critical slip surface may develop layouts closer to a logarithmic spiral, damped sinusoid, parabola, or circle. Presented results could serve as a starting point for further research on the shape of the CSS. They should further progress to provide relevant interpretations summarized in the form of regression curves for some other shapes, defining the most probable shape of the CSS using various equations.
- The conclusion from the above discussion states that reliable methods are available for searching for critical slip surfaces and calculating the safety factor for a slope. However, no approach can involve all the uncertainties that emerged in the safety factor and CSS calculus, on the one hand, and the shape of the CSS still needs to be attentively studied, on the other hand. Thus, slope stability analysis requires an increasingly accurate strategy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**General slope stability problem for the Fellenius method: (

**a**) geometry of the problem, (

**b**) convention of vector orientations for equilibrium state analysis.

**Figure 6.**(

**a**–

**e**). The circular slip surface for the minimum factor of safety achieved through an optimization algorithm of the Slide2 Rocscience Program for a slope with the geometry 1:1.5, H = 3 m, S4, using five distinct limit methods.

**Figure 8.**Mesh construction: plane strain and 15-node triangular elements with a fine mesh around the slip surface for the slope 1:1, H = 3 m, S1.

**Figure 22.**The log-spiral failure surface (after [33]).

**Figure 23.**Graphs of regression curves for five distinct shapes of the potential slip surface and three distinct slope gradients, H = 3, S1.

Method | Circular | Non-Circ. | ƩM = 0 | ƩF = 0 |
---|---|---|---|---|

Ordinary (also known as Swedish, Fellenius) | ✓ | - | ✓ | - |

Bishop simplified | ✓ | - | ✓ | vertical only |

Jambu corrected | ✓ | ✓ | ✓ | ✓ |

Morgenstern–Price | ✓ | ✓ | ✓ | ✓ |

Spencer | ✓ | ✓ | ✓ | ✓ |

Type of Soil | $\mathsf{\gamma}$ $\left[kN/{m}^{3}\right]$ | ${c}^{\prime}$ $\left[\mathrm{kPA}\right]$ | ${\mathsf{\varphi}}^{\prime}$ $[\xb0]$ |
---|---|---|---|

$\mathrm{S}1\u2014\mathrm{clayey}\mathrm{silt}({I}_{P}=12,{I}_{C}=0.50,e=0.90$) | 18.0 | 10 | 18 |

$\mathrm{S}2\u2014\mathrm{sandy}-\mathrm{clayey}\mathrm{silt}({I}_{P}=14,{I}_{C}=0.70,e=0.80)$ | 19.0 | 15 | 20 |

$\mathrm{S}3\u2014\mathrm{sandy}-\mathrm{silty}\mathrm{clay}({I}_{P}=16,{I}_{C}=0.90,e=0.60)$ | 19.5 | 20 | 24 |

$\mathrm{S}4\u2014\mathrm{clay}({I}_{P}=28,{I}_{C}=0.80,e=0.55$) | 20.0 | 40 | 20 |

**Table 9.**Fitting parameters for the damped sinusoid and second-degree parabola for all the considered cases.

Slope | Hight | Soil Type | Damped Sinusoid $y=A\xb7{e}^{-\gamma t}\xb7\mathrm{cos}\left(\mathsf{\omega}t+\mathsf{\varphi}\right)-B$ | Second−Degree Parabola $y=A\xb7{x}^{2}+B\xb7x+C$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

A | ω | ϕ | γ | B | A | B | C | |||

1:1.5 | 3.0 m | S1 | 50.024 | 0.052 | 3.849 | −0.052 | −37.952 | 0.124 | −0.402 | −0.044 |

S2 | 57.596 | 0.054 | 3.633 | −0.036 | −50.601 | 0.121 | −0.426 | −0.162 | ||

S3 | 60.563 | 0.056 | 3.535 | −0.030 | −55.660 | 0.123 | −0.427 | −0.280 | ||

S4 | 74.381 | 0.054 | 2.939 | 0.005 | −71.736 | 0.104 | −0.422 | −1.108 | ||

8.0 m | S1 | 77.753 | 0.032 | 3.220 | −0.005 | −77.576 | 0.043 | −0.219 | 0.062 | |

S2 | 63.134 | 0.029 | 3.683 | −0.021 | −54.063 | 0.044 | −0.268 | 0.067 | ||

S3 | 62.264 | −0.038 | 3.769 | 0.020 | −49.316 | 0.036 | −0.275 | −1.140 | ||

S4 | 32.854 | −0.044 | 2.866 | −0.023 | −31.173 | 0.047 | −0.434 | −0.342 | ||

1:1 | 3.0 m | S1 | 3.572 | 0.000 | 4.765 | −0.583 | 0.260 | 0.161 | −0.219 | 0.062 |

S2 | 8.327 | 0.016 | 4.689 | −0.360 | −0.126 | 0.157 | −0.243 | 0.044 | ||

S3 | 1.699 | 0.000 | 4.798 | −0.613 | 0.229 | 0.159 | −0.262 | 0.063 | ||

S4 | 11.259 | 0.033 | 4.557 | −0.238 | −1.664 | 0.145 | −0.322 | 0.018 | ||

8.0 m | S1 | 65.402 | 0.037 | 3.673 | −0.021 | −56.557 | 0.063 | −0.057 | 0.213 | |

S2 | 65.545 | 0.037 | 3.665 | −0.021 | −56.799 | 0.063 | −0.066 | 0.052 | ||

S3 | 66.168 | 0.039 | 3.637 | −0.022 | −58.359 | 0.070 | −0.142 | 0.190 | ||

S4 | 4.203 | 0.000 | 4.852 | −0.207 | 0.825 | 0.059 | −0.201 | 0.170 | ||

2:1 | 3.0 m | S1 | 3.016 | −0.019 | 4.894 | −0.648 | 0.603 | 0.195 | 0.075 | 0.014 |

S2 | 2.378 | −0.023 | 4.877 | −0.772 | 0.433 | 0.201 | 0.016 | 0.038 | ||

S3 | 3.110 | −0.017 | 4.873 | −0.650 | 0.568 | 0.193 | 0.030 | 0.020 | ||

S4 | 0.993 | −0.021 | 4.826 | −1.134 | 0.212 | 0.193 | −0.120 | −0.039 | ||

8.0 m | S1 | 7.766 | −0.012 | 4.876 | −0.337 | 1.182 | 0.067 | 0.291 | 0.081 | |

S2 | 9.021 | −0.009 | 4.833 | −0.368 | 1.026 | 0.077 | 0.250 | 0.075 | ||

S3 | 8.493 | −0.010 | 4.840 | −0.381 | 0.080 | 0.267 | 0.073 | 0.080 | ||

S4 | 9.082 | −0.005 | 4.783 | −0.378 | 0.639 | 0.072 | 0.052 | 0.090 |

**Table 10.**Fitting errors of regression curves for five distinct shapes of the potential slip surface (the circular benchmark slip surface included) and three distinct slope gradients for the discussed cases.

Slope | Hight | Soil Type | Fitting Shape | ||||
---|---|---|---|---|---|---|---|

Shape 1 | Shape 2 | Shape 3 | Shape 4 | Shape 5 | |||

1:1.5 | 3.0 m | S1 | 0.438 | 0.057 | 0.104 | 0.069 | 0.162 |

S2 | 0.860 | 0.109 | 0.147 | 0.092 | 0.307 | ||

S3 | 1.273 | 0.132 | 0.149 | 0.108 | 0.296 | ||

S4 | 7.541 | 0.460 | 0.465 | 0.471 | 1.014 | ||

8.0 m | S1 | 0.573 | 0.480 | 0.527 | 0.082 | 0.263 | |

S2 | 0.792 | 0.272 | 0.682 | 0.136 | 0.445 | ||

S3 | 9.166 | 0.808 | 1.752 | 1.238 | 1.828 | ||

S4 | 4.831 | 0.745 | 1.140 | 0.544 | 1.638 | ||

1:1 | 3.0 m | S1 | 0.143 | 0.182 | 0.358 | 0.128 | 0.117 |

S2 | 0.106 | 0.161 | 0.331 | 0.093 | 0.088 | ||

S3 | 0.097 | 0.136 | 0.362 | 0.077 | 0.085 | ||

S4 | 0.087 | 0.172 | 0.340 | 0.077 | 0.068 | ||

8.0 m | S1 | 0.389 | 0.230 | 0.379 | 1.504 | 0.345 | |

S2 | 0.895 | 1.489 | 1.704 | 0.662 | 0.926 | ||

S3 | 0.696 | 1.276 | 1.354 | 0.363 | 0.538 | ||

S4 | 1.750 | 2.132 | 3.572 | 1.438 | 1.781 | ||

2:1 | 3.0 m | S1 | 0.139 | 0.054 | 0.028 | 0.135 | 0.057 |

S2 | 0.057 | 0.016 | 0.020 | 0.041 | 0.029 | ||

S3 | 0.208 | 0.095 | 0.041 | 0.130 | 0.101 | ||

S4 | 0.536 | 0.042 | 0.053 | 0.081 | 0.152 | ||

8.0 m | S1 | 0.673 | 0.069 | 0.187 | 0.594 | 2.232 | |

S2 | 1.036 | 0.061 | 0.207 | 0.234 | 0.626 | ||

S3 | 1.237 | 0.037 | 0.183 | 0.206 | 0.764 | ||

S4 | 1.537 | 0.114 | 0.341 | 0.887 | 0.779 |

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**MDPI and ACS Style**

Rotaru, A.; Bejan, F.; Almohamad, D.
Sustainable Slope Stability Analysis: A Critical Study on Methods. *Sustainability* **2022**, *14*, 8847.
https://doi.org/10.3390/su14148847

**AMA Style**

Rotaru A, Bejan F, Almohamad D.
Sustainable Slope Stability Analysis: A Critical Study on Methods. *Sustainability*. 2022; 14(14):8847.
https://doi.org/10.3390/su14148847

**Chicago/Turabian Style**

Rotaru, Ancuța, Florin Bejan, and Dalia Almohamad.
2022. "Sustainable Slope Stability Analysis: A Critical Study on Methods" *Sustainability* 14, no. 14: 8847.
https://doi.org/10.3390/su14148847