# Modified Equation of Shear Strength with Respect to Saturation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Bishop Shear Strength Model

#### 2.2. Van Genuchten SWCC Model

#### 2.3. Khalili Model

#### 2.4. The New Model

## 3. Materials and Experimental System

#### 3.1. Soil–Water Characteristic Curve Test

^{3}, 1.6 t/m

^{3}, and 1.7 t/m

^{3}. Before loading, the soil specimens in the rings were saturated with water. After loading, the specimen’s soil–water characteristic test at a temperature of 25 °C (normal temperature) was measured using the pressure plate method (Figure 5) or the pressure plate test; the test started with a saturated sample. The degree of saturation was determined by weighing the wet soil sample.

#### 3.2. Shear Strength Test

## 4. Results and Discussion

#### 4.1. Comparison with Experimental Data

^{3}and the confining pressure of 150 kPa under the saturation of 30%, 40%, 50%, 60%, and 70% is shown in Figure 11. The curve with a saturation of 70% and confining pressure of 150 kPa under the dry density of 1.5 t/m

^{3}, 1.6 t/m

^{3}, and 1.7 t/m

^{3}is plotted in Figure 12. If the specimen is made up of saturated soil, the shear strength is constant under different cell pressures. Consequently, the failure envelope will be parallel to the horizontal axis because the angle of friction is zero. The other situation is that if the specimen is not saturated, the shear strength will increase as the confining pressure increases. When the result was plotted, it can be seen that the size of the circles increased and the failure envelope was an inclined line.

#### 4.2. Comparison with Literature Data from Clayed-Silty Sand and Ankara Clay

**Clayed-Silty sand**. The clayed-silty sand (sand 0%, silt 92%, clay 8%) has a relatively uniform grain-size distribution with a median grain size D50 of approximately 0.1 mm. Tomoyoshi [26] also adopted a pressure-plate system to test the soil–water characteristic curve, a method similar to this research. The difference was that he used the direct shear apparatus to obtain the shear strength. The relationship between matric suction, effective stress parameter, and saturation of various compaction pressures for the modeled data and measured data are presented in Figure 18 and Figure 19, respectively. Figure 20 and Figure 21 show the prediction of the proposed model for the clayed-silty sand for various compaction pressures. The computed results were compared with the reported data in that study. The correlation coefficients between the experimental data and fitting data were 0.95 and 0.98 at a compaction pressure of 100 kPa and 600 kPa, respectively.

**Clayey soil**. The clayey soil (clay fraction 67.9%) researched by Erdal [27] was obtained from METU (Middle East Technical University). The specific gravity was 2.73, the liquid limit was 48%, the plastic limit was 21%, and the plasticity index was 27%. Figure 22 and Figure 23 show the modeled and experimental data for the relationship between matric suction and saturation as well as the effective stress parameter, respectively, with good agreement. The relationship between the shear strength and saturation under various net normal stresses is shown in Figure 24. The calculated values were close to the experimental points in Erdal’s paper. All correlation coefficients were larger than 0.98.

## 5. Sensitivity Analysis of Four Parameters

^{3}as an example, the ranges of four parameters are shown as follows: ${\xi}_{2}$, [38912.99, 41929.02]; ${\xi}_{3}$, [−4.184, −4.049]; ${\xi}_{4}$, [19883.32, 20763.47]; m, [0.2397, 0.2429]. By comparing the range of shear strength under the variation of the one parameter, we found that this model was much more sensitive regarding the parameter m, which is also related to the material characteristic. When soil is close to a saturated status, the shear strength tends to a stable value. This is because the softening effect of soil against shear strength is much more obvious when soil tends to the saturated status.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 9.**The effective stress parameter $\chi $ of the calculated points and fitted curve at various dry densities.

**Figure 10.**The schematic diagram of the soil shear strength parameter calculation. (

**a**) Stress–strain curve. (

**b**) Mohr’s circle.

**Figure 11.**Stress–strain curve under various saturations (${\mathsf{\rho}}_{\mathrm{d}}$ = 1.6 t/m

^{3}, $\mathsf{\sigma}-{\mathsf{\mu}}_{\mathrm{a}}$ = 150 kPa).

**Figure 12.**Stress–strain curve under various dry densities (Saturation = 70%, $\mathsf{\sigma}-{\mathsf{\mu}}_{\mathrm{a}}$ = 150 kPa).

**Figure 14.**Experimental points of the relationship between the internal friction angle and saturation.

**Figure 16.**Experimental points and fitted curve of the shear strength at various net normal stresses.

**Figure 18.**Soil–water characteristic curve of the experimental points and fitted curve at different compaction pressures.

**Figure 19.**Effective stress parameter $\chi $ of the calculated points and fitted curve at different compaction pressures.

**Figure 22.**Soil–water characteristic curve based on experimental points [27] and the fitted curve.

**Figure 23.**Effective stress parameter $\chi $ based on the calculated data [27] and the fitted curve.

**Figure 24.**Comparison of the calculated shear strength of the proposed model and the experimental data [27].

Specific Gravity (G_{s}) | ω (%) | $\mathbf{e}$ | $\mathbf{K}\times {10}^{-10}\phantom{\rule{0ex}{0ex}}(\mathbf{m}{\mathbf{s}}^{-1})$ | Particle Composition (%) | ||
---|---|---|---|---|---|---|

0.05–2 (mm) | 0.002–0.05 (mm) | <0.002 (mm) | ||||

2.70 | 32.5 | 1.45 | 1.74 | 87.8 | 10.6 | 1.6 |

$\mathbf{Parameter}$ | ${\mathsf{\xi}}_{2}^{}$ | ${\mathsf{\xi}}_{3}^{}$ | ${\mathsf{\xi}}_{4}^{}={\mathsf{\xi}}_{1}^{}\cdot {\mathbf{P}}_{0}$ | $\mathbf{m}$ |
---|---|---|---|---|

${\mathsf{\rho}}_{\mathrm{d}}=1.5\mathrm{t}/{\mathrm{m}}^{3}$ | 40,009.57 | −4.21 | 20,295.99 | 0.24 |

${\mathsf{\rho}}_{\mathrm{d}}=1.6\mathrm{t}/{\mathrm{m}}^{3}$ | 40,017.51 | −4.51 | 11,675.79 | 0.21 |

${\mathsf{\rho}}_{\mathrm{d}}=1.7\mathrm{t}/{\mathrm{m}}^{3}$ | 64,735.01 | −5.01 | 9886.80 | 0.20 |

$\mathbf{Parameter}$ | ${\mathsf{\xi}}_{2}^{}$ | ${\mathsf{\xi}}_{3}^{}$ | ${\mathsf{\xi}}_{4}^{}={\mathsf{\xi}}_{1}^{}\cdot {\mathbf{P}}_{0}$ | $\mathbf{m}$ |
---|---|---|---|---|

Clayed-silty sand (e = 1.31) | 2.63 | −1.42 | 6.52 | 0.82 |

Clayed-silty sand (e = 1.08) | 89.07 | −1.79 | 551.05 | 0.58 |

Ankara clay | 52.59 | −3.42 | 160.82 | 0.38 |

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**MDPI and ACS Style**

Tian, W.; Peiffer, H.; Malengier, B.; Liu, G.; Cheng, L. Modified Equation of Shear Strength with Respect to Saturation. *Appl. Sci.* **2023**, *13*, 4305.
https://doi.org/10.3390/app13074305

**AMA Style**

Tian W, Peiffer H, Malengier B, Liu G, Cheng L. Modified Equation of Shear Strength with Respect to Saturation. *Applied Sciences*. 2023; 13(7):4305.
https://doi.org/10.3390/app13074305

**Chicago/Turabian Style**

Tian, Wenjing, Herman Peiffer, Benny Malengier, Gang Liu, and Liangliang Cheng. 2023. "Modified Equation of Shear Strength with Respect to Saturation" *Applied Sciences* 13, no. 7: 4305.
https://doi.org/10.3390/app13074305