# Effects of Principal Stress Rotation on the Fluid-Induced Soil Response in a Porous Seabed

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models

#### 2.1. Flow Model

#### 2.2. Seabed Model

## 3. Model Verification

#### 3.1. Comparison with Hollow Cylinder Apparatus (HCA) Element Tests

#### 3.2. Comparison with Laboratory Experiments for the Seabed Response to Waves and Currents

#### 3.3. Comparison with Centrifuge Tests and Previous Numerical Model for the Seabed Response To Waves

## 4. Results and Discussion

#### 4.1. Seabed Liquefaction

#### 4.2. Effect of Currents

#### 4.3. Effect of Principal Stress Rotation with Various Wave and Soil Parameters

## 5. Conclusions

- (1)
- Principal stress rotation (PSR) has a significant effect on the soil liquefaction depth. It accelerates the growth of pore pressures and reduces the vertical effective stress, so that the soil is easier to liquefy.
- (2)
- The existence of ocean currents has an important impact on the development of the liquefaction potential of a seabed foundation. When considering the interactions between waves and currents, the soil pore pressure and effective force change significantly and have a significant impact on soil liquefaction. The following current aggravates the soil reaction and promotes soil liquefaction. On the contrary, the opposing current reduces soil instability and plays a positive role in soil stability.
- (3)
- With the combined action of waves and current, the seabed with porous media shows pronounced lateral expansion and vertical settlement.
- (4)
- The liquefaction potential of the elastoplastic seabed foundation increases with time and decreases with depth. This indicates that liquefaction is more likely to occur in the upper layer of the seabed foundation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Comparison of the present model with the HCA tests [56] under continuous rotation of the major principal stress axis.

**Figure 3.**Validation of the present model with the experimental data [33] (wave only): (

**a**) water surface elevation ($\eta $) and (

**b**) the wave-induced pore pressure in seabed (${p}_{s}$). Input data: wave height (H) = 5 cm, wave period (T) = 1 s, water depth (d) = 50 cm, seabed thickness (h) = 50 cm, degree of saturation (${S}_{r}$) = 1, Shear modulus (G) = 10${}^{7}$ N/m${}^{2}$, Poisson’s ratio ($\mu $) = 0.3, soil permeability (${K}_{s}$) = 1.88 × 10${}^{-4}$ m/s, and soil porosity (${n}_{s}$) = 0.771.

**Figure 4.**Validation of the present model with experimental data resulting from combining the wave and the following current loading [33]: (

**a**) water surface elevation ($\eta $) and (

**b**) the wave-induced pore pressure in seabed (${p}_{s}$). Input data: H = 5 cm, T = 1 s, d = 50 cm, h = 50 cm, ${S}_{r}$ = 1, G = 10${}^{7}$ N/m${}^{2}$, $\mu $ = 0.3, ${K}_{s}$ = 1.88 $\times {10}^{-4}$ m/s, ${n}_{s}$ = 0.771, current velocity (U) = 0.05 m/s.

**Figure 5.**Comparison of the distribution of excessive pore pressure between the present model (solid line) and the centrifuge tests (dashed line) [40].

**Figure 6.**Excess pore pressure at point A subject to progressive wave loading. Notation: red lines = the present model, blue lines = Sassa’s model [35].

**Figure 7.**Liquefaction process in a sandy seabed according to (

**a**) the original PZIII (Pastor–Zienkiewicz Mark-III) model and (

**b**) the present model.

**Figure 8.**Cyclic response of seabed under various current conditions at different locations: (

**a**) pore pressures; (

**b**) vertical effective stresses.

**Figure 9.**Displacements under different current conditions: (

**a**) horizontal displacement, (

**b**) vertical displacement.

**Figure 10.**Effect of currents ${U}_{0}$ on the vertical distribution of liquefaction potential on z = −5 m at four typical time points.

**Figure 11.**Effect of wave height H on the vertical distribution of liquefaction potential on $z=$−5 m at four typical time points. (

**a**) The original PZIII model without PSR (principal stress rotation) and (

**b**) the present model with PSR.

**Figure 12.**Effect of wave period T considering PSR on the vertical distribution of liquefaction potential on $z=$−5 m at four typical time points. (

**a**) The original PZIII model without PSR and (

**b**) the present model with PSR.

**Figure 13.**Effect of soil permeability ${K}_{s}$ considering PSR on the vertical distribution of liquefaction potential on $z=$−5 m at four typical time points. (

**a**) The original PZIII model without PSR and (

**b**) the present model with PSR.

**Figure 14.**Effect of the degree of $Sr$ considering PSR on the vertical distribution of liquefaction potential on $z=$−5 m at four typical time points. (

**a**) The original PZIII model without PSR and (

**b**) the present model with PSR.

**Table 1.**Present constitutive model’s parameters for comparison with the HCA (Hollow Cylinder Apparatus) element test of sand [56].

${K}_{ev0}$ (kPa) | 24,727.3 | ${G}_{ev0}$ (kPa) | 34,000 |

${\beta}_{0}$ | 0.3 | ${\beta}_{1}$ | 5.5 |

${H}_{0}$ (kPa) | 600 | ${H}_{u0}$ (kPa) | 1000 |

${\gamma}_{U}$ | 6.0 | ${M}_{g0}$ | 0.7 |

$\phantom{\rule{4pt}{0ex}}{M}_{f0}$ | 0.42 | ${\alpha}_{0}$ | 0.005 |

a | 0.25 | b | 0.65 |

${p}_{0}^{\prime}$ (kPa) | 4 |

Wave and Seabed Characteristics | Parameters for PZIII Model ${}^{\#}$ | ||
---|---|---|---|

T (s) | 4.55 | ${H}_{0}$ (kPa) | 700 |

h (m) | 1.7 | ${H}_{u0}$ (kPa) | 1000 |

d (m) | 4.5 | ${K}_{ev0}$ (kPa) | 660.8 |

${L}_{0}$ (m) | 25 | ${G}_{ev0}$ (kPa) | 770.0 |

H (m) | 5.0 | ${\gamma}_{U}$ | 6.0 |

${S}_{r}$ (%) | 100 | ${\gamma}_{DM}$ | 4.0 |

${K}_{s}$ (m/s) | 0.00015 | ${M}_{g0}$ | 1.2124 |

${\beta}_{0}$ | 0.2 | ||

${\beta}_{1}$ | 2.5 | ||

${M}_{f0}$ | 0.75 | ||

${\alpha}_{0}$ | 0.01 | ||

${p}_{0}^{\prime}$ (kPa) | 4 | ||

a | 0.3 | ||

c | 0.5 |

Parameters | Original PZIII | The Present Model | Unit |
---|---|---|---|

${K}_{(}ev0)$ | 2000 | 2000 | kPa |

${G}_{(}ev0)$ | 2600 | 2600 | kPa |

${p}_{0}^{\prime}$ | 4.0 | 4.0 | kPa |

${M}_{g}$ | 1.32 | - | - |

${M}_{f}$ | 1.3 | - | - |

${\alpha}_{g}$ | 0.45 | - | - |

${\alpha}_{f}$ | 0.45 | - | - |

${\beta}_{0}$ | 4.2 | 4.2 | - |

${\beta}_{1}$ | 0.2 | 0.2 | - |

${H}_{0}$ | 750 | 750 | kPa |

${H}_{(}u0)$ | 40,000 | 40,000 | kPa |

${\gamma}_{U}$ | 4 | 4 | - |

${S}_{r}$ | 0.98 | 0.98 | - |

n | 0.397 | 0.397 | - |

${M}_{(}g0)$ | - | 1.32 | - |

${M}_{(}f0)$ | - | 1.3 | - |

${\alpha}_{0}$ | - | 0.45 | - |

a | - | 0.1 | - |

c | - | 0.1 | - |

${e}_{0}$ | - | 0.4286 | - |

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**MDPI and ACS Style**

Li, Z.; Jeng, D.-S.; Zhu, J.-F.; Zhao, H.
Effects of Principal Stress Rotation on the Fluid-Induced Soil Response in a Porous Seabed. *J. Mar. Sci. Eng.* **2019**, *7*, 123.
https://doi.org/10.3390/jmse7050123

**AMA Style**

Li Z, Jeng D-S, Zhu J-F, Zhao H.
Effects of Principal Stress Rotation on the Fluid-Induced Soil Response in a Porous Seabed. *Journal of Marine Science and Engineering*. 2019; 7(5):123.
https://doi.org/10.3390/jmse7050123

**Chicago/Turabian Style**

Li, Zhengxu, Dong-Sheng Jeng, Jian-Feng Zhu, and Hongyi Zhao.
2019. "Effects of Principal Stress Rotation on the Fluid-Induced Soil Response in a Porous Seabed" *Journal of Marine Science and Engineering* 7, no. 5: 123.
https://doi.org/10.3390/jmse7050123