# Stability of Embankments Resting on Foundation Soils with a Weak Layer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Position of the Problem

_{i}from the foundation plane of the reinforced earth wall (Figure 1) and it is characterized by thickness t

_{0}. Considering the values reported in Table 1, the values of z

_{i}/H range from 0 to 2.

_{x}≠ 0; ε

_{z}≠ 0), and along the longitudinal axis (out of plane direction) the strain is assumed to be zero (ε

_{y}= 0). These conditions agree with the geometry of the original slope and the embankment (length significantly larger than the width, [31]). The reference scheme and the boundary conditions of the FE model are shown in Figure 1.

_{i}, Figure 1), and considering the variations of the constitutive parameters of materials (Table 1), about 800 FE simulations were carried out. One of the FEM meshes used, the one relating to the case of z

_{i}/H = 1.18 is shown in Figure 2.

## 3. Numerical Results

_{0}= 0.5 m and depth z

_{i}ranging from 0 to 17 m. In all analysed cases, the shear strength parameters of the involved soils (foundation soil and weak layer) varied according to the ranges reported in Table 1.

#### 3.1. Failure Mechanisms

_{1}= 10 kPa and φ′

_{1}= 30°, extended down to the bottom of the FE model. This figure clearly shows that, in the case of the homogeneous foundation soil, the failure mechanism is almost circular in type and involves a superficial plasticized soil volume below the foundation of the earth wall and the embankment. Furthermore, from Figure 3, it can be seen that in the backfill a volume very close to that of the Rankine active limit state is affected by significant deformations; that is, it is involved in the failure process. The foundation soils are affected by a failure mechanism, which involves only the soils directly below the base of the reinforced earth wall.

_{i}= 4 m (z

_{i}/H = 0.47, Figure 5a), z

_{i}= 6 m (z

_{i}/H = 0.71, Figure 5b) and z

_{i}= 9 m (z

_{i}/H = 1.06, Figure 5c) and z

_{i}= 10 m (z

_{i}/H = 1.18, Figure 5d) and its shear strength parameters are c′

_{2}= 0.2 kPa and φ′

_{2}= 10°. In these analyses, the foundation soil has the same mechanical properties considered before for the case of homogeneous soil mass (c′

_{1}= 10 kPa and φ′

_{1}= 30°).

_{i}/H = 0.47 (Figure 5a), the failure mechanism is clearly mixtilinear in shape and develops for a long distance in the weak layer. It is very different from the one relative to the case of homogeneous foundation soils (Figure 3).

_{i}/H = 0.47, 0.71, 1.06, the failure mechanism is mixtilinear in shape and develops for a significant length within the weak layer. For z

_{i}/H > 1.06 m (Figure 5d), the failure mechanism does not reach the weak layer and it develops in the soil foundations above. The geometry of the failure mechanism is very similar to that obtained in the absence of a weak layer (Figure 3). So, in this case, the extension of the mobilised soil volume is no longer affected by the presence of the weak layer.

_{i}/H = 0.71 and 1.06), the failure mechanism is still strongly affected by the weak layer. However, in the case of Figure 5c, the depth of the weak layer is enough to make the embankment less vulnerable compared to the first two cases. In fact, on the one hand, the foundation soil suffers the highest effect in terms of deepening of the failure mechanism, but on the other hand, the shear strains are less concentrated in the weak layer. Hence, a negligible effect on the safety factor could be expected. Finally, when the weak layer is located at the greatest depth (z

_{i}/H = 1.18, Figure 5d), the failure mechanism is no longer affected by the presence of the weak layer and it is once again very similar to that of the homogeneous soil foundation.

#### 3.2. Influence of the Weak Layer on the Safety Factor

_{1}, for different values of the shear strength angle φ′

_{1}(26°, 30° and 34°) of foundation soils. Obviously, the safety factor increases both with intercept cohesion and with the shear strength angle. The similar trend of the curves proves that, in this case, the failure mechanism is not dependent on the shear strength parameters.

_{i}/H, with variation in the different shear strength parameters for both the foundation soil and the weak layer (see Table 1).

_{i}/H lower than about 0.1 ÷ 0.5 (weak layer located very close to the foundation of the reinforced earth wall), stability is not possible for the parameters of the weak layer considered in the numerical analyses. Obviously, if the shear strength parameters of the weak layer were very similar to those of the foundation soils, this case would also present stability conditions. These cases are not represented in the results because they are not significant for the purpose of this study, and because, in the authors’ opinion, they are not significant from a practical point of view.

## 4. Discussion

_{1}= 10 kPa and φ′

_{1}= 30° are represented, while the shear strength parameters of the weak layer are c′

_{2}= 0.2 kPa and φ′

_{2}= 10°. Similar failure mechanisms were obtained for other values of the shear strength parameters of the weak layer and foundation soils. The first mechanism (Figure 11) considers the homogeneous foundation soil and the safety factor resulting from the stability analysis is higher than 1 (SF

_{0}>1). Overall stability conditions are ensured by the mechanical properties of the foundation soil and by the characteristics of the embankment (soil compaction, reinforcement geomembrane, depth of the foundation of the embankment). The failure mechanism is almost circular in shape.

_{i}equal to z*, which is defined as the critical depth. As defined here, z* is the minimum depth of the weak layer, below which the overall stability condition is not ensured. In this condition, the safety factor SF is equal to 1. The failure mechanism largely affects the weak layer and becomes mixtilinear in shape. In the third scenario (Figure 12b), the weak layer is located at an intermediate depth between z* and z

_{max}, and the calculated safety factor is between 1 and SF

_{0}. The depth z

_{max}can be defined by means of the fourth scenario depicted in Figure 12c, which represents the case in which the weak layer is precisely located at depth equal to z

_{max}. In this condition, the extension of the failure mechanism is the maximum, but it affects the weak layer to a limited extent. Hence, a reduced influence of the weak layer on the safety factor can be attained (SF ≈ SF

_{0}), even though the mixtilinear shape is maintained. The condition in which the layer is located at a depth higher than z

_{max}is represented by the fifth scenario (Figure 12d). The failure mechanism mirrors the one observed in the case of homogeneous foundation soils, and the safety factor is the same (SF = SF

_{0}).

_{max}depend on the shear strength of the involved soils (both foundation soil and weak layer soil). According to this, the ratio between the critical depth z*and the height of the embankment H (z*/H) is plotted as a function of the intercept cohesion of the foundation soil c’

_{1}and of the shear strength angle of the foundation soil (φ′

_{1}= 26° in Figure 13a, φ′

_{1}= 30° in Figure 13c and φ′

_{1}= 34° in Figure 13e). The different values of the shear strength angle of the weak layer (φ′

_{2}= 5°, 10° and 5°) are considered in the same figures. Similarly, the values of the ratio between the maximum depth z

_{max}and the height of the embankment H (z

_{max}/H) are plotted in Figure 13b (φ′

_{1}= 26°), Figure 13d (φ′

_{1}= 30°) and Figure 13f (φ′

_{1}= 34°).

_{max}rises with the increase in the intercept cohesion c′

_{1}. Conversely, the shear strength angle of the foundation soils φ′

_{1}has a limited influence on the geometrical variable z*/H, while it almost does not affect parameter z

_{max}/H. Hence, the domain in which the weak layer affects the overall stability of the embankment expands with the intercept cohesion of the foundation soil c′

_{1}.

_{2}raises both the geometrical variables z*/H and z

_{max}/H. In other words, the soil volume within the foundation soil in which the weak layer can affect both the deformation mechanism and the safety factor increases with the difference between the shear strength of the weak layer and the foundation soil. A maximum value of z

_{max}/H equal to about 1.3 was obtained in this study.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Scheme for formulation of the problem: the weak layer is horizontal and is located at depth z

_{i}. Depth z

_{i}is measured from the foundation plane of the reinforced earth wall. The boundary conditions and the dimensions of the finite-element (FE) model are also represented in the figure.

**Figure 3.**Distribution of the total displacements at failure calculated for the case of homogeneous foundation soils (c′

_{1}= 10 kPa and φ′

_{1}= 30°), by means of the phi-c reduction stage. The colorimetric scale is not plotted in the figure because, as is well known, the intensity of displacement at failure calculated by the model is an unrealistic result, due to the elastoplastic constitutive soil model adopted in the simulations. However, this representation is effective in pointing out the volume of materials involved in the failure (volume in which the displacements and the deformation are concentrated).

**Figure 4.**Total strain at failure calculated for the case of homogeneous foundation soils (c′

_{1}= 10 kPa and φ′

_{1}= 30°) by means of the phi-c reduction stage. The length and the direction of the arrows represent the intensity and the direction, respectively, of the principal strains. The concentration of the strains identifies a global failure mechanism.

**Figure 5.**Spatial distribution of the total displacements at failure calculated for the case of a horizontal weak layer with c′

_{2}= 0.2 kPa and φ′

_{2}= 10° located at depth equal to (

**a**) z

_{i}= 4 m (z

_{i}/H = 0.47), (

**b**) z

_{i}= 6 m (z

_{i}/H = 0.71), (

**c**) z

_{i}= 9 m (z

_{i}/H = 1.06) and (

**d**) z

_{i}= 10 m (z

_{i}/H = 1.18).

**Figure 6.**Total strain at failure for the case of the horizontal weak layer with c′

_{2}= 0.2 kPa and φ′

_{2}= 10° located at depth equal to (

**a**) z

_{i}= 4 m (z

_{i}/H = 0.47), (

**b**) z

_{i}= 6 m (z

_{i}/H = 0.71), (

**c**) z

_{i}= 9 m (z

_{i}/H = 1.06) and (

**d**) z

_{i}= 10 m (z

_{i}/H = 1.18).

**Figure 7.**Safety factor SF as a function of the intercept cohesion c′

_{1}, for different values of the shear strength angle φ′

_{1}(26°, 30° and 34°) in the case of homogeneous foundation soils.

**Figure 8.**Safety factor SF as a function of the geometrical variable z

_{i}/H for φ′

_{1}= 26° and c′

_{1}= 0.2 kPa (

**a**), c′

_{1}= 5 kPa (

**b**), c′

_{1}= 10 kPa (

**c**) and c′

_{1}= 30 kPa (

**d**), for different values of the shear strength angle φ′

_{2}(5°, 10° and 15°).

**Figure 9.**Safety factor SF as a function of the geometrical variable z

_{i}/H for φ′

_{1}= 30° and c′

_{1}= 0.2 kPa (

**a**), c′

_{1}= 5 kPa (

**b**), c′

_{1}= 10 kPa (

**c**) and c′

_{1}= 30 kPa (

**d**), for different values of the shear strength angle φ′

_{2}(5°, 10° and 15°).

**Figure 10.**Safety factor SF as a function of the geometrical variable z

_{i}/H for φ′

_{1}= 34° and c′

_{1}= 0.2 kPa (

**a**), c′

_{1}= 5 kPa (

**b**), c′

_{1}= 10 kPa (

**c**) and c′

_{1}= 30 kPa (

**d**), for different values of the shear strength angle φ′

_{2}(5°, 10° and 15°).

**Figure 11.**Significant scenario representing the failure mechanism and the stability condition (safety factor SF

_{0}> 1) of the embankment in the case of homogeneous foundation soil. The failure mechanism has been drawn on the basis of the numerical results, in particular of the concentration of the shear strain in the phi-c reduction phase.

**Figure 12.**Significant scenario representing the failure mechanism and the stability condition of the embankment in the case of the weak layer located at different depths. The weak layer affects both the failure mechanism and the safety factor: (

**a**) weak layer located at depth z

_{i}equal to the critical depth z* (SF = 1); (

**b**) weak layer located at depth z

_{i}between the critical depth z* and the maximum depth z

_{max}(1 < SF < SF

_{0}, where SF

_{0}is the safety factor in case of homogenous foundation soil); (

**c**) weak layer located at depth equal to the maximum depth z

_{max}(SF ≈ SF

_{0}); (

**d**) weak layer located at a depth higher than the maximum depth z

_{max}(SF = SF

_{0}).

**Figure 13.**Evolution of the ratio between the critical depth z*and the height of the embankment H and of the ratio between the maximum depth z

_{max}and the height of the embankment H as a function of the intercept cohesion of the foundation soil c

_{1}’ and of the shear strength angle of the foundation soil. (

**a**,

**b**) φ

_{1}′ = 26°, (

**c**,

**d**) φ

_{1}′ = 30°, (

**e**,

**f**) φ

_{1}′ = 34°.

L_{1}(m) | L_{2}(m) | L_{3}(m) | L_{4}(m) | L_{5}(m) | L_{6}(m) | H (m) | β (°) | a (m) | z_{i}(m) | t_{0}(m) | B (m) |
---|---|---|---|---|---|---|---|---|---|---|---|

28 | 37 | 45 | 42.5 | 11.5 | 32 | 8.5 | 11 | 3 | 0 ÷ 17 | 0.5 | 130 |

**Table 2.**Geotechnical parameters of materials used in the numerical analyses. γ

_{d}: dry unit weight; γ

_{sat:}saturated unit weight; c′: effective intercept cohesion; φ′: effective shear strength angle; Ψ′: dilation angle; E′: Young modulus; ν′: Poisson’s ratio. The elastic–perfectly plastic Mohr–Coulomb constitutive model was used for all modelled soils.

Material | γ_{d}(kN/m ^{3}) | γ_{sat}(kN/m ^{3}) | φ′ (°) | Ψ′ (°) | c′ (kPa) | E′ (MPa) | ν′ (-) |
---|---|---|---|---|---|---|---|

Embankment | 17 | 20 | 36 | 10 | 1 | 20 | 0.30 |

Reinforced earth | 17 | 20 | 36 | 10 | 100 | 20 | 0.30 |

Foundation soil | 16 | 18 | 26, 30, 34 | 0 | 0.2, 5, 10, 20, 30 | 10 | 0.35 |

Weak layer | 16 | 18 | 5, 10, 15 | 0 | 0.2 | 10 | 0.35 |

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**MDPI and ACS Style**

Ziccarelli, M.; Rosone, M.
Stability of Embankments Resting on Foundation Soils with a Weak Layer. *Geosciences* **2021**, *11*, 86.
https://doi.org/10.3390/geosciences11020086

**AMA Style**

Ziccarelli M, Rosone M.
Stability of Embankments Resting on Foundation Soils with a Weak Layer. *Geosciences*. 2021; 11(2):86.
https://doi.org/10.3390/geosciences11020086

**Chicago/Turabian Style**

Ziccarelli, Maurizio, and Marco Rosone.
2021. "Stability of Embankments Resting on Foundation Soils with a Weak Layer" *Geosciences* 11, no. 2: 86.
https://doi.org/10.3390/geosciences11020086