# An Approach to Assess the Stability of Unsaturated Multilayered Coastal-Embankment Slope during Rainfall Infiltration

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{a}-u

_{w}) and the suction angle (φ

^{b}) had significant effects on the safety factor of the embankment slope. Basically, linear distribution of (u

_{a}-u

_{w}) along the depth and linear relationship between φ

^{b}and (u

_{a}-u

_{w}) should be adopted in assessing the stability of the unsaturated multilayered coastal-embankment slope.

## 1. Introduction

_{a}-u

_{w}), which were critical to the stability of unsaturated-soil slopes, were dependent on the practical environmental conditions. For the sake of simplicity, the distribution of matric suction in general is considered to be uniformed or linearly decreased along the buried depth [26,27]. Recently, Zhang et al. [28] investigated the effect of the different distributions of matric suction on the overturning stability of the retaining wall with homogeneous, continuous and non-layered surrounding soils. It was found that the influence of uniformed suction was more remarkable than that of linear suction. Xu and Yang [21] found that there is a little difference between the effect of the distribution pattern of matric suction on stability of three-dimensional unsaturated, homogeneous, continuous and non-layered soil slope. However, in practice, the ground conditions in the unsaturated region are generally multilayered, which results in the uncertainty of influence of the matric suction distribution patterns on the stability of unsaturated-soil slope. Moreover, the existence of the multilayered behavior will significantly increase the difficulty in the determination of strength parameters when the rigorous limit equilibrium method was adopted to investigate the safety of the unsaturated-soil slope, which will be analyzed in detail in a later section.

## 2. A Simple Approach to Simulate Rainfall Infiltration

## 3. Limit Equilibrium Method of Unsaturated-Multilayered-Soil Slope

#### 3.1. Failure Criterion of Unsaturated-Soil

_{u}= the shear strength; c’ = the effective cohesion; σ

_{n}= the total normal stress; u

_{w}= the pore-water pressure; φ’ = the effective internal friction angle associated with (σ

_{n}-u

_{w}); (u

_{a}-u

_{w}) = the matric suction; φ

^{b}= the suction angle. (u

_{a}-u

_{w}) and φ

^{b}are the key parameters of unsaturated-soil failure criterion. Furthermore, the value of φ

^{b}is always dependent on (u

_{a}-u

_{w}) [44], the (u

_{a}-u

_{w}) has become a crucial issue unsaturated soil theory [25,26,27,28].

#### 3.2. Distributions of Matric Suction

_{a}-u

_{w}) and φ

^{b}are the key parameters of unsaturated-soil failure criterion. Furthermore, the value of φ

^{b}is always dependent on (u

_{a}-u

_{w}) [44], the accurate determination of (u

_{a}-u

_{w}), therefore, is of great important to the failure criterion of unsaturated soil, which has become a crucial issue in unsaturated-soil theory.

_{a}-u

_{w})

_{g}can be easily tested by tensiometer, the underground matric suctions are difficult to test in practical engineering. For the sake of simplicity, the matric suction is assumed to be distributed along the buried depth by following two patterns [21,25,26,27,28]:

#### 3.3. Determination of the Strength Parameters under Different Distributions of Matric Suction

_{a}-u

_{w})tanφ

^{b}.

_{i}, φ’

_{i}(i = 1,2,3). Because of the existence of (u

_{a}-u

_{w}) in the soil layers above the groundwater level, the correspondent strength parameters can be determined as follows:

_{i}, φ’

_{i}(j = I, II, III) = the strength parameters of the original soil layers; (u

_{a}-u

_{w})

_{g}= the matric suction at the ground surface; C

_{i}, φ’

_{i}(i = 1,2,…,m+n+2) = the strength parameters of the divided small layers; H = the distance between the groundwater level and the top of the slope, h

_{i}(i = 1,2,…,m+n+2) = the distance between the ith small soil layers and the top of the slope.

#### 3.4. Modified M-P Method

_{s}= the safety factor of the trial sliding surface; λ = the scaling factor; E

_{j}

_{-1}and E

_{j}= the inter-slice force for slice j-1 and j; W

_{j}= the weight of jth slice; Q

_{j}= the external loading; S

_{j}= the shear force; N

_{j}= the normal pressure; u

_{j}= the pore-water pressure at slice-base; φ’

_{j}and C

_{j}= the mobilized internal friction angle and total cohesion along; f

_{j}

_{-1}and f

_{j}= the inter-slice force function for slice j-1 and j which can be determined by relating them with the horizontal coordinate. Other detailed information about the model parameters can be found in Reference [34]. The safety factor of a trial sliding surface in the unsaturated multilayered coastal-embankment slope can be performed as shown in Figure 5.

## 4. A Global Algorithm to Search for the Critical Slip Surface of Unsaturated Coastal-Embankment Slope

_{L}, y

_{L}), R(x

_{R}, y

_{R})) and a radius (R

_{c}) shown in Figure 6, in which the potential sliding surface should be addressed in order to satisfy the boundary conditions represented by y = g(x) and y = r(x), the hydraulic conditions y = w(x) and the discontinuity between the layers demonstrated by y = l

_{j}(x). Then, the potential sliding surface represented by S can be expressed as functions of x

_{L}, y

_{L}, x

_{R}, y

_{R}and R

_{c}in the following form:

_{s}) with respect to the most dangerous sliding surface should be minimized among all of the trial failure surfaces. The issue in terms of locating the most dangerous slip surface can be mathematically represented by addressing the minimum of F

_{s}with regard to S as follows:

_{L}, y

_{L}), R(x

_{R}, y

_{R})) and R

_{c}. However, most of these are invalid due to their impossible occurrence in practical engineering. To improve the searching efficiency, the control parameters should be empirically constrained as:

_{Lmax}and x

_{Lmin}= the upper and lower limits of horizontal ordinate of point L; x

_{Rmax}and x

_{Rmin}= the upper and lower limits of horizontal ordinate of point R; R

_{cmax}and R

_{cmin}= the upper and lower limits of radius. It should be noted that definitions of these parameters should be dependent on the experience of the researcher.

**LR‖**= the length of

**LR**; y

_{c}= the vertical ordinate of the center of the circular slip surface.

_{c}shown in Figure 6) within the range of [x

_{Lmin}, x

_{Lmax}], [x

_{Rmin}, x

_{Rmax}] and [R

_{cmin}, R

_{cmax}], in which points L and R can be located stochastically in the following forms:

_{1}, r

_{2}and r

_{3}= stochastic numbers which can be obtained randomly from 0 to 1.

_{c}= the horizontal ordinate of O’. There are two groups of solution for Equation (22), and to make sure the trial slip surface to be reliable, the smaller value of x

_{c}and the lager value of y

_{c}are used.

## 5. Application

#### 5.1. Comparison with the Existing Model

^{b}. This is because of the amplification of φ

^{b}which significantly increases the total cohesion (shown in Equations (3)–(8), and enhances the shear strength (Equation (2)), which in turn improves the safety factor. Moreover, both the critical slip surfaces (marked by the solid line) and the associated safety factor always agree well with those (marked by other line type) predicted by Zhang et al. [50]. Therefore, the present method performs well in investigating the stability of the unsaturated-soil slope.

#### 5.2. Case Study

#### 5.3. Discussion on the Effect of Distribution of (u_{a}-u_{w}) on F_{smin}

_{smin}) under different groundwater level shown in Figure 10 was adopted to investigate the effects of the different distributions of matric suction (u

_{a}-u

_{w}) on the stability of unsaturated-soil slope. As shown in Figure 11, F

_{smin}is intensively related to the distribution of (u

_{a}-u

_{w}) along the buried depth. Overall, all of the three values of F

_{smin}corresponding to different distributions show gradual decrease with the rise of the groundwater level. However, the reduced rate of Distribution III without suction, which keeps constant (approaches to 0) under the condition that the groundwater level is below 6 m, is remarkable lower that of the other two distributions. Moreover, both of the amplitude and reduced rate of Distribution I are greater than those of Distribution II. When the groundwater level approaches to 15 m, F

_{smin}of the three distributions coincides with each other, in which the soil is fully saturated. The effects of distributions of (u

_{a}-u

_{w}) on F

_{smin}varies accordingly to the distributions. For Distribution I and II with consideration of (u

_{a}-u

_{w}), the reduction in F

_{smin}is due to the decrease of distributed depth of (u

_{a}-u

_{w}) (shown in Figure 1) and the increase of the pore-water pressure (Equations (13) and (16)) with the increase of groundwater level. However, the decrease of F

_{smin}for Distribution III regardless of (u

_{a}-u

_{w}) is mainly attributed to the increase of the pore-water pressure. Therefore, the reduced rate and amplitude of F

_{smin}for Distribution I and II are much greater than that for Distribution III.

_{a}-u

_{w}) is always greater than that for linear distribution (Distribution II), which may result in an unsafe design of the unsaturated-soil slope. With the comprehensive consideration of safety and economy, it is suggested that Distribution II can be used in addressing the stability of unsaturated coastal-embankment slope in practical engineering.

#### 5.4. Discussion on the Relationship between φ^{b} and (u_{a}-u_{w})

^{b}was conventionally supposed to be constant and independent on (u

_{a}-u

_{w}) in the previous literature [19,51,52,53,54]. However, the multistage direct shear test results [44] showed that the relationship between φ

^{b}and (u

_{a}-u

_{w}) could be depicted by Curve 3, as shown in Figure 12. When (u

_{a}-u

_{w}) is small, φ

^{b}is supposed to be the maximum value φ

^{b}

_{max}. With the increase of (u

_{a}-u

_{w}), φ

^{b}gradually decreases and when (u

_{a}-u

_{w}) increases to a certain value φ

^{b}reaches the minimum value φ

^{b}

_{min}. Some researchers [25,55,56] used the Curve 1 and Curve 2 to simulate the relationship between φ

^{b}and (u

_{a}-u

_{w}) as shown in Figure 12. Curve 1, Curve 2 and Curve 3 can be described by the following functions:

^{b}

_{max}and φ

^{b}

_{min}are the upper and lower limit respectively. Herein, φ

^{b}

_{max}is 30°and φ

^{b}

_{min}is 5°.

^{b}on safety factor in different (u

_{a}-u

_{w})

_{g}is shown in Figure 13. It can be seen that when φ

^{b}is supposed to be constant, the safety factor(F

^{1}

_{s}) associated with Curve 1 is almost linearly increased with the increasing of (u

_{a}-u

_{w})

_{g}. When the distribution of φ

^{b}is linear (Curve 2) and nonlinear (Curve 3) both the safety factor (F

^{2}

_{s}) associated with Curve 2 and (F

^{3}

_{s}) associated with Curve 3 show the same change trend with the increase of (u

_{a}-u

_{w})

_{g}. At first, F

^{2}

_{s}and F

^{3}

_{s}have a dramatic rise until reaching the maxima. This is because that (u

_{a}-u

_{w})

_{g}is smaller in this stage and the matric suction on the slip surface is smaller than 50 kPa. Meanwhile, the correspondent value of φ

^{b}always equals to φ

^{b}

_{max}and the total cohesion on the slip surface increases with the increasing of (u

_{a}-u

_{w})

_{g}. Therefore, F

^{2}

_{s}and F

^{3}

_{s}become larger and larger. When (u

_{a}-u

_{w})

_{g}increases to a certain value (about 450 kPa in the example) the matric suction on the slip surface will exceed to 50kPa and the correspondent φ

^{b}will decrease accordingly to Equation (24) and Equation (25). The value of (u

_{a}-u

_{w}) will increase but the value of φ

^{b}will decrease on the slip surface and the total cohesion slightly decreases. Therefore, F

^{2}

_{s}and F

^{3}

_{s}decrease slightly at this stage. When (u

_{a}-u

_{w})

_{g}is kept increasing to a certain value (about 750 kPa in the example), (u

_{a}-u

_{w}) in the slip surface always exceed to 500 kPa and the correspondent value of φ

^{b}no longer change. Then, the total cohesion keeps increasing with the increase of (u

_{a}-u

_{w})

_{g}and F

^{2}

_{s}and F

^{3}

_{s}will increase at this stage.

^{b}and Curve 2 and Curve 3 are more rational than Curve 1 shown in Figure 13. For the sake of simplicity, it is more reasonable to adopt the linear relationship between φ

^{b}and (u

_{a}-u

_{w}) in addressing the unsaturated-soil slope’s safety factor.

## 6. Conclusions

^{b}) on the safety factor was investigated. It was found that the fluctuation of the groundwater level has a significant influence on the location of the most dangerous sliding surface. The associated minimum safety factor and the sliding modes of unsaturated-soil slope gradually change from deep sliding to shallow sliding with the rise of groundwater level. Moreover, the traditional slope stability method regardless of the matric suction is conservative to the predicted results. It is more reasonable to adopt the linear distribution of matric suction in practical calculation of the safety factor (F

_{s}). In addition, F

_{s}is sensitive to the distribution of φ

^{b}and the linear relationship between φ

^{b}and (u

_{a}-u

_{w}) is more beneficial in addressing the stability of the unsaturated multilayered coastal slope. It should be noted that the present approach is simple and a lot of factors such as the effect of soil layering on the matric suction and non-circular failure surface are not covered, which will be addressed in detail in future work.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Cross section of unsaturated multilayered coastal-embankment slope under rainfall infiltration.

**Figure 3.**Distributions of the (u

_{a}-u

_{w}) along the depth in unsaturated-multilayered-soil slope (

**a**) Distribution I: uniform distribution of (u

_{a}-u

_{w}); (

**b**) Distribution II: linear distribution of (u

_{a}-u

_{w}).

**Figure 4.**Discrete model and inter-slice forces in the slip surface of an unsaturated-soil slope (

**a**) discrete model of a slip surface, (

**b**) inter-slice forces.

**Figure 10.**Critical slip surfaces associated with each distribution pattern of matric suction under rainfall infiltration.

**Figure 11.**Effect of distribution pattern of matric suction on safety factor under rainfall infiltration.

**Figure 13.**The influence of the distribution of φ

^{b}on the safety factor, (

**a**) Underground water level = 6 m; (

**b**) Underground water level = 12 m.

Analysis Case | γʹ kN/m^{3} | cʹ/kPa | φʹ/° | (u_{a}-u_{w})_{g}/kPa | φ^{b}/° |
---|---|---|---|---|---|

Case 1 | 18 | 10 | 34 | 200 | 0 |

Case 2 | 18 | 10 | 34 | 200 | 15 |

Case 3 | 18 | 10 | 34 | 200 | 34 |

Layer | γʹ kN/m^{3} | cʹ/kPa | φʹ/° | (u_{a}-u_{w})_{g}/kPa | φ^{b}/° |
---|---|---|---|---|---|

1 | 20 | 0.5 | 40.0 | 200 | 15 |

2 | 20 | 1 | 42.0 | ||

3 | 19.5 | 0.0 | 38.0 | ||

4 | 17 | 15.0 | 21.0 |

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**MDPI and ACS Style**

Zhu, J.-f.; Chen, C.-f.; Zhao, H.-y.
An Approach to Assess the Stability of Unsaturated Multilayered Coastal-Embankment Slope during Rainfall Infiltration. *J. Mar. Sci. Eng.* **2019**, *7*, 165.
https://doi.org/10.3390/jmse7060165

**AMA Style**

Zhu J-f, Chen C-f, Zhao H-y.
An Approach to Assess the Stability of Unsaturated Multilayered Coastal-Embankment Slope during Rainfall Infiltration. *Journal of Marine Science and Engineering*. 2019; 7(6):165.
https://doi.org/10.3390/jmse7060165

**Chicago/Turabian Style**

Zhu, Jian-feng, Chang-fu Chen, and Hong-yi Zhao.
2019. "An Approach to Assess the Stability of Unsaturated Multilayered Coastal-Embankment Slope during Rainfall Infiltration" *Journal of Marine Science and Engineering* 7, no. 6: 165.
https://doi.org/10.3390/jmse7060165