# Prediction Model of Residual Soil Shear Strength under Dry–Wet Cycles and Its Uncertainty

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Overview of Soil Samples

**Figure 1.**Grain size distribution curve of granite residual soil. The classification of soil refers to the “Standard for Engineering Classification of Soil” [44].

#### 2.2. Experimental Plan

#### 2.2.1. Indoor Wet and Dry Cycle Simulation Test

- According to the physical properties of the soil samples given in Table 1 and the water content and density of the soil samples, calculate the mass of the soil when the water content reaches 12%, 20%, 25%, 30%, and 35%.
- Use an electric fan to accelerate the air-drying speed of the soil samples. Measure the quality of the soil samples every hour to detect whether the water content of the soil samples has reached the predetermined value. When the water content reaches 12%, turn off the electric fan, stop air-drying dehydration immediately, wrap the soil with plastic wrap, and seal it for 24 h to ensure that the moisture inside the soil is evenly distributed.
- Put the soil sample with an initial water content of 12% into a stacked saturation device, and use a vacuum suction device combined with a vacuum suction saturation method to perform suction saturation on the soil sample. After the soil is saturated, use step 2 of the method, using an electric fan to air-dry and dehydrate the saturated soil samples. When the water content of the soil sample reaches 20%, stop air-drying dehydration and wrap it with plastic wrap for sealed curing for 24 h. At this point, a soil sample with a water content of 20% has undergone a complete dry–wet cycle.
- According to steps 2–3, continue to perform dry–wet cycle tests on soil samples with a water content of 20% 2, 4, 6, 8, and 10 times.
- According to steps 1–3, perform indoor simulated dry–wet cycle tests on soil samples with a water content of 25%, 30%, and 35%.

#### 2.2.2. Quick Shear Test

## 3. Results

#### 3.1. Relationship among Cohesion, Moisture Content, and Drying and Wetting Cycles of Granite Residual Soil

^{B},

_{1}, b

_{1}, a

_{2}, and b

_{2}are the fitting parameters, and their values are shown in Table 3.

#### 3.2. Relationship among Internal Friction Angle, Moisture Content, and Number of Drying and Wetting Cycles of Granite Residual Soil

^{D},

_{3}, b

_{3}, a

_{4}, and b

_{4}are the fitting parameters, and their values are shown in Table 3.

#### 3.3. Establishment of Prediction Model of Soil Shear Strength

## 4. Uncertainty Analysis of Shear Strength Prediction Model

#### 4.1. Fundamentals of Point Estimation

_{0}is:

_{0}= 0 and m = 1 in Formula (11), the average value $\overline{x}$ of x can be obtained; when x

_{0}= $\overline{x}$ and m = 2, the variance V[x] of x can be obtained. The square root of V[x] is the standard deviation σ[x] of x. For a bivariate function y = y(x

_{1}, x

_{2}) containing variables x

_{1}and x

_{2}, its m-th order matrix is:

_{± ±}is the value of the bivariate function y considering the uncertainty of variables x

_{1}and x

_{2}:

_{± ±}is:

_{1}and x

_{2}:

_{1}, x

_{2}) is the covariance between x

_{1}and x

_{2}. The standard deviation of function y is:

#### 4.2. Uncertainty in Predicting Shear Strength τ

_{i}is the i-th actual value, and $\overline{{x}_{i}}$ is the corresponding predicted value of x

_{i}. The root mean square errors δ[c] and δ[φ] of Formulas (8) and (9) are 1.330 and 0.044, respectively. Combined with Formulas (7) and (13), considering the uncertainty of c and φ, τ can be expressed as:

## 5. Conclusions

- (1)
- The cohesion c of granite residual soil decreases with an increase in the water content W, and the approximate relationship between the two satisfies the power function. The fitting parameters of the power function also satisfy a power function relationship with the number of dry–wet cycles N. A prediction formula for c considering the influence of N and W was obtained.
- (2)
- The internal friction angle φ of granite residual soil decreases with an increase in water content W, and the nonlinear relationship between the two can be described using a power function. The fitting parameters of this power function and the number of wet and dry cycles N also satisfy a power function relationship. A prediction formula for φ considering the influence of N and W was obtained.
- (3)
- A prediction formula for soil shear strength τ considering the influence of N and W was established.
- (4)
- The uncertainty of τ predicted by the formula jointly caused by the uncertainty of c and φ and the univariate uncertainty of τ predicted by only c or φ increases first and then decreases with an increase in N, and both increase with an increase in water content W.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Slope soil sample information diagram: (

**a**) Soil layer distribution of the slope; (

**b**) slope soil layer borehole column diagram.

**Figure 3.**Relationship between cohesion and water content under different dry–wet cycles: (

**a**) the relationship between cohesion and the number of dry–wet cycles; (

**b**) the relationship between cohesion and moisture content.

**Figure 5.**Relationship between internal friction angle and water content under different dry–wet cycles: (

**a**) the relationship between the internal friction angle and the number of dry–wet cycles; (

**b**) the relationship between internal friction angle and moisture content.

**Figure 8.**Frequency distribution of the ratio between predicted and test value of shear strength: (

**a**) the relationship between the predicted value and the experimental value; (

**b**) frequency distribution of the ratio of predicted and experimental values.

**Figure 9.**Flow chart for determining the uncertainty of the predicted τ caused by the uncertainties of c and φ.

**Figure 10.**Influence of the uncertainties of c and φ on the total uncertainty and single univariate uncertainty of the predicted τ of soil at various W and N: (

**a**) moisture content W = 20%; (

**b**) moisture content W = 25%; (

**c**) moisture content W = 30%; (

**d**) moisture content W = 35%.

Moisture Content W/% | Dry Density ρ _{d}/(g/m^{3}) | Proportion G _{s} | Void Ratio e | Saturation Level S _{r}/% |
---|---|---|---|---|

23.2 | 1.56 | 2.78 | 0.71 | 86.59 |

Number of Cycles | Cohesion c | Coefficient of Determination R^{2} | Internal Friction Angle φ | Coefficient of Determination R^{2} | ||
---|---|---|---|---|---|---|

A | B | C | D | |||

2 | 463.86 | −1.05 | 0.981 | 80.75 | −0.43 | 0.931 |

4 | 439.97 | −1.12 | 0.892 | 87.73 | −0.44 | 0.944 |

6 | 425.99 | −1.11 | 0.902 | 102.31 | −0.51 | 0.905 |

8 | 410.94 | −1.13 | 0.861 | 108.66 | −0.53 | 0.872 |

10 | 390.95 | −1.16 | 0.871 | 114.32 | −0.55 | 0.956 |

Prediction Equation | Coefficient | Coefficient of Determination R ^{2} | |
---|---|---|---|

a_{i} | b_{i} | ||

A = a_{1}N^{b}^{1} | 500.52 | −0.098 | 0.972 |

B = a_{2}N^{b}^{2} | −1.01 | 0.05 | 0.912 |

C = a_{3}N^{b}^{3} | 66.75 | 0.23 | 0.981 |

D = a_{4}N^{b}^{4} | −0.37 | 0.10 | 0.953 |

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## Share and Cite

**MDPI and ACS Style**

Ding, J.; Wang, S.; Huang, H.; Pan, F.; Wu, Y.; Gu, Y.; Zhang, Y.
Prediction Model of Residual Soil Shear Strength under Dry–Wet Cycles and Its Uncertainty. *Water* **2023**, *15*, 3931.
https://doi.org/10.3390/w15223931

**AMA Style**

Ding J, Wang S, Huang H, Pan F, Wu Y, Gu Y, Zhang Y.
Prediction Model of Residual Soil Shear Strength under Dry–Wet Cycles and Its Uncertainty. *Water*. 2023; 15(22):3931.
https://doi.org/10.3390/w15223931

**Chicago/Turabian Style**

Ding, Jiefa, Shijun Wang, Haoran Huang, Fengqian Pan, Yunxing Wu, Yanchang Gu, and Yan Zhang.
2023. "Prediction Model of Residual Soil Shear Strength under Dry–Wet Cycles and Its Uncertainty" *Water* 15, no. 22: 3931.
https://doi.org/10.3390/w15223931