# Lateral Dynamic Response of Helical Pile in Viscoelastic Foundation Considering Shear Deformation

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## Abstract

**:**

## 1. Introduction

## 2. Equivalent Model of Helical Pile

#### 2.1. Establishment of Equivalent Model

_{p}denote the elastic modulus and cross-section inertia moment of the helical pile, respectively. ${E}_{\mathrm{p}}^{\prime}$ and ${I}_{\mathrm{p}}^{\prime}$ represent the elastic modulus and cross-section inertia moment of the equivalent cylindrical pile, respectively.

#### 2.2. Validation of Equivalent Model

## 3. Governing Equations and Their Solutions

#### 3.1. Dynamic Equation of Soil

#### 3.2. Dynamic Equation of Helical Pile

- (1)
- The pile’s surrounding soil is a homogeneous, isotropic, and viscoelastic medium. Therefore, the stiffness and damping coefficients of the soil layers are both constants along the vertical direction.
- (2)
- The pile body is a combination of circular and rectangular sections, and only the bending deformation of the pile is considered after simplification.
- (3)
- The pile–soil system is subjected to small deformations and strains during lateral vibration, and the longitudinal displacement of the pile’s surrounding soil is ignored.
- (4)
- No relative sliding occurs at the pile–soil interface.
- (5)
- The influence of pile cap is not considered.
- (6)
- The harmonic excitation acts horizontally on the pile top.

#### 3.3. Solutions of the Equations

## 4. Verification of the Present Solutions

#### 4.1. Element Division Accuracy of Pile–Soil System

#### 4.2. Comparison with Existing Analytical and FEM Solutions

## 5. Parametric Study

#### 5.1. Space Response Analysis of Helical Pile

#### 5.1.1. Influence of Helix Inclination Angle

#### 5.1.2. Influence of Dimensionless Frequency

#### 5.1.3. Influence of Pile–Soil Stiffness Ratio

#### 5.2. Time response Analysis of Helical Pile

#### 5.2.1. Analysis of Displacement Time Response

#### 5.2.2. Analysis of Bending Moment Time Response

#### 5.2.3. Analysis of Shear Force Time Response

## 6. Conclusions

- (1)
- The LDD of the pile top increases with the increase in the helix inclination angle, but the BM of the pile top and pile end, as well as the SF of the middle part of the helical pile, increase with the decrease in the helix’s inclination angle. The smaller the helix’s inclination angle, the greater the change degree.
- (2)
- The LDD of the pile top decreases with the increase in the dimensionless frequency, while the SF of the pile end increases with the increase in the dimensionless frequency.
- (3)
- The LDR of the pile body basically increases significantly with the increase in the pile–soil stiffness ratio.
- (4)
- The smaller the dimensionless frequency, the more time it takes for the LDR to reach its maximum values, and the smaller the dimensionless frequency, the greater the change degree. The reason for these phenomena is that the change in dimensionless frequency results in nonlinear change in the damping coefficient and period.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Influence of element division accuracy on LDR of helical pile. (

**a**) Lateral displacement envelope. (

**b**) BM envelope. (

**c**) SF envelope.

**Figure 6.**Comparison between the present solutions and the solutions obtained by Hu et al. [49]. (

**a**) Lateral displacement envelope. (

**b**) BM envelope. (

**c**) SF envelope.

**Figure 7.**Comparison between the present solutions and the solutions obtained by Wang et al. [50]. (

**a**) Lateral displacement envelope. (

**b**) BM envelope. (

**c**) SF envelope.

**Figure 8.**Influence of helix inclination angle on LDR of helical pile. (

**a**) Lateral displacement envelope. (

**b**) BM envelope. (

**c**) SF envelope.

**Figure 9.**Influence of dimensionless frequency on LDR of helical pile. (

**a**) Lateral displacement envelope. (

**b**) BM envelope. (

**c**) SF envelope.

**Figure 10.**Influence of pile–soil stiffness ratio on LDR of helical pile. (

**a**) Lateral displacement envelope. (

**b**) BM envelope. (

**c**) SF envelope.

**Figure 11.**Analysis of displacement time response of helical pile. (

**a**) Displacement time response at z = 0 m. (

**b**) Displacement time response at z = 2 m. (

**c**) Displacement time response at z = 4.5 m.

**Figure 12.**Analysis of BM time response of helical pile. (

**a**) BM time response at z = 0 m. (

**b**) BM time response at z = 2 m. (

**c**) BM time response at z = 4.5 m.

**Figure 13.**Analysis of SF time response of helical pile. (

**a**) SF time response at z = 0 m. (

**b**) SF time response at z = 2 m. (

**c**) SF time response at z = 4.5 m.

Parameters | Symbol | Value | Unit |
---|---|---|---|

Pile diameter | D | 0.3 | m |

Pile length | L | 4.5 | m |

Helical extension ratio | D_{0}/D | 1.5 | - |

Helical inclination angle | $\phi $ | 30 | ° |

Helical tooth width | b | 0.075 | m |

Dimensionless frequency | a_{0} | 0.5 | - |

Elastic modulus of pile | E_{p} | 2.0 × 10^{10} | Pa |

Elastic modulus of soil | E_{s} | 4.0 × 10^{6} | Pa |

Pile density | ${\rho}_{\mathrm{p}}$ | 2.5 × 10^{3} | kg/m^{3} |

Soil density | ${\rho}_{\mathrm{s}}$ | 2.0 × 10^{3} | kg/m^{3} |

Pile’s Poisson’s ratio | ${\nu}_{\mathrm{p}}$ | 0.17 | - |

Soil’s Poisson’s ratio | ${\nu}_{\mathrm{s}}$ | 0.4 | - |

Soil’s damping ratio | ${\beta}_{\mathrm{s}}$ | 0.05 | - |

External load amplitude | ${Q}_{0}$ | 100 | kN |

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**MDPI and ACS Style**

Yang, X.; Wang, C.; Cao, S.; Wang, F.; Wu, W.
Lateral Dynamic Response of Helical Pile in Viscoelastic Foundation Considering Shear Deformation. *Appl. Sci.* **2023**, *13*, 12220.
https://doi.org/10.3390/app132212220

**AMA Style**

Yang X, Wang C, Cao S, Wang F, Wu W.
Lateral Dynamic Response of Helical Pile in Viscoelastic Foundation Considering Shear Deformation. *Applied Sciences*. 2023; 13(22):12220.
https://doi.org/10.3390/app132212220

**Chicago/Turabian Style**

Yang, Xiaoyan, Chaozhe Wang, Sheng Cao, Fengxi Wang, and Wenbing Wu.
2023. "Lateral Dynamic Response of Helical Pile in Viscoelastic Foundation Considering Shear Deformation" *Applied Sciences* 13, no. 22: 12220.
https://doi.org/10.3390/app132212220