# Influence of the Three-Dimensional Effect of Pile-Soil System on the Vertical Dynamic Response of Large-Diameter Piles in Low-Strain Integrity Testing

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Assumptions

_{h}.

- (1)
- The pile surrounding the soil is homogenous, and its top surface is free. Meanwhile, the pile surrounding the soil is infinite in the radial direction, and the displacement and stress of the soil at infinity are zero.
- (2)
- During the vibration, only the vertical displacement of the hammer-pile-soil system is taken into account, and the radial displacement is ignored. Both the vertical and radial differentials of the vertical displacement are considered in this paper.
- (3)
- There is no big deformation in the hammer-pile-soil system; thus, the stress and displacement at the interface between the pile and the surrounding soil are equal. Obviously, pile I also remains in perfect contact with pile II. The hammer-pile-soil system is located on rigid rock.
- (4)
- The hammer-pile-soil system is static at the beginning.

## 3. Vibration Equations and Their Solutions

#### 3.1. Vertical Vibration Equation of Pile Surrounding Soil and Its Solution

- (1)
- The stress at the surface of the pile surrounding the soil top is zero:

- (2)
- The vertical displacement of the pile surrounding soil at the top of the rigid rock is zero:

- (3)
- The stress and displacement of the pile surrounding soil are zero at infinity:

- (4)
- The stress and displacement of the pile surrounding the soil are zero at the beginning:

#### 3.2. Vertical Vibration Equations of the Pile

- (1)
- The boundary condition of the pile top:

- (2)
- The pile bottom is fixed:

- (3)
- The interface between the pile and the surrounding soil is continuous:

- (4)
- The interface between Pile I and Pile II is continuous:

- (5)
- The pile is static at the beginning:

#### 3.3. The Solution for Pile I

#### 3.4. The Solution for Pile II

## 4. Comparison

#### 4.1. Comparison with Finite Element Model

_{h}= 0.01 m. Figure 3a–e show the velocity curves in the time domain of different test points along the radius of the large-diameter pile top of the present solution and the calculated results using Abaqus. It is observed that the present solution can match very well with the Abaqus solution at different test points. The phenomena indicate that the present solution can accurately reflect the 3D effect of the axisymmetric large-diameter pile-soil system. Meanwhile, it also proves the correctness of the derivation process of the present solution.

#### 4.2. Comparison with Other Existing Solutions

_{h}= 0.02 m. In the model of Zheng et al. [29], the 3D continuum model is adopted for the pile body, which can reflect the 3D effect of the pile body. However, the plane strain model is used to simulate the pile surrounding soil, which cannot reflect the 3D effect of the pile surrounding soil. Meanwhile, in the process of Zheng et al.’s solution [29], only the displacement continuity at the pile center between the interface of each pile segment is considered, which cannot reflect the displacement continuity at any point between the interface of each pile segment. However, in the present model, the 3D continuum model is used to simulate the pile body and the pile surrounding soil; therefore, the present model can reflect the 3D effect of the pile-soil system, which is more rigorous. Figure 4a–e display the velocity curves in the time domain of different test points along the radius of the large-diameter pile top of the present solution and the solution of Zheng et al. [29]. It is captured that the trend of the velocity curves in the time domain of the present solution can match well with those of Zheng et al. [29]. However, compared with the solution of Zheng et al. [29], the present solution at different test points exhibits more obvious high-frequency interference characteristics, the velocity curves in the time domain of different test points are more irregular and less smooth, and the incident wave signal and the first reflected wave signal are also significantly affected by the high-frequency interference.

## 5. Parametric Study

#### 5.1. The Influence of the 3D Effect of the Pile-Soil System

#### 5.1.1. Within the Excitation Region

_{h}= 0.01 m, 0.02 m, and 0.04 m. The location of the test point is set as r = 0, 0.25r

_{h}, 0.5r

_{h}, 0.75r

_{h}, and 1.0r

_{h}. Figure 5, Figure 6, Figure 7 and Figure 8 illustrate the vertical velocity response in the time domain of different test points within the excitation region for different pile radii. As shown in Figure 5, Figure 6, Figure 7 and Figure 8, the incident wave signal of different test points in the excitation region is composed of the positive wave signal and the evidently negative wave signal after the positive wave signal. In the FEM study of Chow et al. [26], the significantly negative wave signal after the positive wave signal was also observed at the edge of the excitation region, a result which was considered to be the influence of the 3D effect of the pile body on the vertical dynamic response of the pile top. This phenomenon proves again that the present solution in this paper is reasonable for evaluating the 3D effect of the pile-soil system.

_{h}= 0.04 m, A* sharply decreases when the test point is close to the edge of the excitation region. Meanwhile, comparing Figure 5d, Figure 6d, Figure 7d, and Figure 8d, it is noted that A* increases gradually with the decrease in the radius of the excitation hammer and the increase in the pile radius. The increase in A* indicates that the negative wave signal in the incident wave signal becomes more and more obvious. Thus, the above results prove that the larger radius of the pile and the smaller radius of the hammer lead to the more obvious negative wave signal in the incident wave signal.

#### 5.1.2. At the Edge of the Excitation Region

#### 5.1.3. Outside the Excitation Region

#### 5.2. The Mechanism of the Negative Wave Signal in the Incident Wave Signal during Low-Strain Integrity Testing

#### 5.2.1. Influence of the Ratio of Pile Radius to the Radius of Hammer

_{h}) to make the following analyses more convenient.

#### 5.2.2. From the Perspective of Wave Propagation

#### 5.2.3. The Vertical Velocity Response in Time Domain along the Depth of the Pile

_{h}and r = 0.2R.

_{h}. Figure 14b depicts the velocity curves in the time domain of different test points along the depth of the pile with r = 0.2R. From Figure 14a,b, the amplitude of the negative wave signal in the time-domain velocity curve of each test point gradually decreases with the increase of depth. However, the negative wave signal in the incident wave signal of each test point basically disappears when the depth reaches z = 0.24R, i.e., z = 0.12 m.

## 6. Conclusions

- The rationality of the present solution is verified by comparing it with the FEM results obtained using Abaqus software. Meanwhile, the reasonableness and advantages of the present solution in evaluating the 3D effect of the pile-soil system are verified by comparing it with the other existing solutions.
- It is reasonable to assume pile I (impacting area) as a 1D rod in the research of the low-strain integrity testing of large-diameter piles.
- In the time-domain velocity curves of different test points at the pile top along the radial direction, the negative wave signal in the incident wave signal only occurs at the test point inside and near the excitation region. Meanwhile, the negative wave signal at the edge of the excitation region is almost unaffected by the variation in the radius of the general engineering pile.
- During the low-strain integrity testing, the test point should be located near 0.6R–0.8R, which can reduce the high-frequency interference of the large-diameter pile top.
- In the time-domain velocity curves of the test points inside and near the excitation region, the negative wave signal in the incident wave signal is influenced by the pile-hammer-ratio and is caused by the larger pile-hammer-ratio. There is a critical value of the pile-hammer-ratio: when the pile-hammer-ratio is smaller than 3, that negative wave signal will totally vanish.
- From the perspective of wave propagation, in the time-domain velocity curves of the test points inside and near the excitation region, the negative wave signal in the incident wave signal is caused by the superposition of the direct R-wave on the direct S-wave. It is suggested to filter the R wave signal in the detection signal to eliminate the negative wave peak by adding a device to filter the R wave signal.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Wu, W.B.; Liu, H.; Yang, X.Y.; Jiang, G.S.; El Naggar, M.H.; Mei, G.X.; Liang, R.Z. New method to calculate apparent phase velocity of open-ended pipe pile. Can. Geotech. J.
**2020**, 57, 127–138. [Google Scholar] [CrossRef] - Yang, X.Y.; Wang, L.X.; Wu, W.B.; Liu, H.; Jiang, G.S.; Wang, K.H.; Mei, G.X. Vertical dynamic impedance of a viscoelastic pile in arbitrarily layered soil based on the fictitious soil pile model. Energies
**2022**, 15, 2087. [Google Scholar] [CrossRef] - Zhang, Y.P.; Jiang, G.S.; Wu, W.B.; El Naggar, M.H.; Liu, H.; Wen, M.J.; Wang, K.H. Analytical solution for distributed torsional low strain integrity test for pipe pile. Int. J. Numer. Anal. Meth. Geomech.
**2022**, 46, 47–67. [Google Scholar] [CrossRef] - Song, Y.; Zhang, M.J.; Øiseth, O.; Rønnquist, A. Wind deflection analysis of railway catenary under crosswind based on nonlinear finite element model and wind tunnel test. Mech. Mach. Theory
**2022**, 168, 104608. [Google Scholar] [CrossRef] - Yang, X.Y.; Jiang, G.S.; Liu, H.; Wu, W.B.; Mei, G.X.; Yang, Z.J. Horizontal vibration characteristics of a tapered pile in arbitrarily layered soil. Energies
**2022**, 15, 3193. [Google Scholar] [CrossRef] - Yang, Z.J.; Wu, W.B.; Liu, H.; Zhang, Y.P.; Liang, R.Z. Flexible support of a pile embedded in unsaturated soil under Rayleigh waves. Earthq. Eng. Struct. Dyn.
**2022**, 52, 226–247. [Google Scholar] [CrossRef] - Wu, J.T.; El Naggar, M.H.; Wang, K.H.; Wu, W.B. Lateral vibration characteristics of an extended pile shaft under low-strain integrity test. Soil Dyn. Earthq. Eng.
**2019**, 126, 105812. [Google Scholar] [CrossRef] - Meng, K.; Cui, C.Y.; Liang, Z.M.; Li, H.J.; Pei, H.F. A new approach for longitudinal vibration of a large-diameter floating pipe pile in visco-elastic soil considering the three-dimensional wave effects. Comput. Geotech.
**2020**, 128, 103840. [Google Scholar] [CrossRef] - Wu, W.B.; Wang, Z.Q.; Zhang, Y.P.; El Naggar, M.H.; Wu, T.; Wen, M.J. Semi-analytical solution for negative skin friction development on deep foundations in coastal reclamation areas. Int. J. Mech. Sci.
**2022**, 241, 107981. [Google Scholar] [CrossRef] - Liu, H.; Li, J.X.; Yang, X.Y.; Chen, L.B.; Wu, W.B.; Wen, M.J.; Jiang, M.J.; Guo, C.J. Lateral dynamic response of offshore pipe piles considering effect of superstructure. Energies
**2022**, 15, 6759. [Google Scholar] [CrossRef] - Zhang, J.; Yuan, G.K.; Zhu, S.Y.; Gu, Q.; Ke, S.T.; Lin, J.H. Seismic analysis of 10 MW offshore wind turbine with large-diameter monopile in consideration of seabed liquefaction. Energies
**2022**, 15, 2539. [Google Scholar] [CrossRef] - Chen, L.B.; Li, J.X.; Wu, W.B.; Liu, H.; Yao, Y.; Zhang, P. New method to calculate the kinematic response of offshore pipe piles under seismic S-waves. Soil Dyn. Earthq. Eng.
**2023**, 165, 107651. [Google Scholar] [CrossRef] - Zhang, Y.P.; El Naggar, M.H.; Wu, W.B.; Wang, Z.Q.; Yang, X.Y.; Jiang, G.S. Dynamic torsional impedance of large-diameter pipe pile for offshore engineering: 3D analytical solution. Appl. Math. Model.
**2022**, 111, 664–680. [Google Scholar] [CrossRef] - Cui, C.Y.; Meng, K.; Xu, C.S.; Wang, B.L.; Xin, Y. Vertical vibration of a floating pile considering the incomplete bonding effect of the pile-soil interface. Comput. Geotech.
**2022**, 150, 104894. [Google Scholar] [CrossRef] - Pasqual, R.P.S.; Kormann, A.C.M. Low strain integrity tests in piles –1-D and 3-D numerical modeling and comparisons with results obtained in the field. Multi-Sci. J.
**2020**, 2, 1–8. [Google Scholar] [CrossRef] [Green Version] - Ding, W.; Chai, H.Y.; Liu, S.H.; Chen, C.; Nie, T.; Hu, Z. Analysis of influence of source and pile-soil interaction in low strain pile integrity testing. J. Civil Environ. Eng.
**2020**, 42, 73–79. [Google Scholar] - Smith, E.A.L. Pile driving analysis by the wave equation. J. Soil Mech. Found. Div.
**1960**, 86, 35–64. [Google Scholar] [CrossRef] - Ni, S.H.; Lehmann, L.; Charng, J.J.; Lo, K.F. Low-strain integrity testing of drilled piles with high slenderness ratio. Comput. Geotech.
**2006**, 33, 283–293. [Google Scholar] [CrossRef] - Wang, K.H.; Wu, W.B.; Zhang, Z.Q.; Leo, C.J. Vertical dynamic response of an inhomogeneous viscoelastic pile. Comput. Geotech.
**2010**, 37, 536–544. [Google Scholar] [CrossRef] - Wang, X.F.; Chen, P.; Huang, Y.H.; Gan, Y. Analysis of 3D effect of dynamic test along pile tip. Chin. J. Rock Mech. Eng.
**2007**, 26, 3209–3214. [Google Scholar] - Steinbach, J.; Vey, E. Caisson evaluation by stress wave propagation method. J. Geotech. Geoenviron. Eng. Div.
**1975**, 101, 361–368. [Google Scholar] [CrossRef] - Wu, W.B.; Wang, K.H.; Zhang, Z.Q.; Leo, C.J. Soil-pile interaction in the pile vertical vibration considering true three-dimensional wave effect of soil. Int. J. Num. Anal. Meth. Geomech.
**2013**, 37, 2860–2876. [Google Scholar] [CrossRef] - Zhang, Y.P.; Wang, Z.Q.; El Naggar, M.H.; Wu, W.B.; Wang, L.X.; Jiang, G.S. Three-dimensional wave propagation in a solid pile during torsional low strain integrity test. Int. J. Num. Anal. Meth. Geomech.
**2022**, 46, 2398–2411. [Google Scholar] [CrossRef] - Fukuhara, T.; Kakurai, M.; Sugimoto, M. Analytical evaluation of defective piles. In Application of Stress-Wave Theory to Piles; Barends, Frans, B.J., Eds.; Routledge: London, UK, 1992; pp. 563–569. [Google Scholar]
- Chen, F.; Wang, R.J. Dimension effect on low strain integrity testing of piles. Chin. J. Geotech. Eng.
**1998**, 20, 92–96. [Google Scholar] - Chow, Y.K.; Phoon, K.K.; Chow, W.F.; Wong, K.Y. Low strain integrity testing of piles: Three-dimensional effects. J. Geotech. Geoenviron. Eng.
**2003**, 129, 1057–1062. [Google Scholar] [CrossRef] - Li, L.C.; Liu, X.; Liu, H.; Wu, W.B.; Lehane, B.M.; Jiang, G.S.; Xu, M.J. Experimental and numerical study on the static lateral performance of monopile and hybrid pile foundation. Ocean Eng.
**2022**, 255, 111461. [Google Scholar] [CrossRef] - Wu, W.B.; Yang, Z.J.; Liu, X.; Zhang, Y.P.; Liu, H.; El Naggar, M.H.; Xu, M.J.; Mei, G.X. Horizontal dynamic response of pile in unsaturated soil considering its construction disturbance effect. Ocean Eng.
**2022**, 245, 110483. [Google Scholar] [CrossRef] - Zheng, C.J.; Kouretzis, G.P.; Ding, X.M.; Liu, H.L.; Poulos, H.G. Three-dimensional effects in low-strain integrity testing of piles: Analytical solution. Can. Geotech. J.
**2016**, 53, 225–235. [Google Scholar] [CrossRef] [Green Version] - Zheng, C.J.; Liu, H.L.; Ding, X.M.; Kouretzis, G.P.; Sloan, S.W.; Poulos, H.G. Non-axisymmetric response of piles in low-strain integrity testing. Geotechnique
**2017**, 67, 181–186. [Google Scholar] [CrossRef] - Zheng, C.J.; Ding, X.M.; Kouretzis, G.P.; Liu, H.L.; Sun, Y. Three-dimensional propagation of waves in piles during low-strain integrity tests. Geotechnique
**2018**, 68, 358–363. [Google Scholar] [CrossRef] - Liu, X.; El Naggar, M.H.; Wang, K.H.; Wu, J.T. Three-dimensional axisymmetric analysis of pile vertical vibration. J. Sound Vib.
**2021**, 494, 115881. [Google Scholar] [CrossRef] - Zhang, Y.P.; Di, T.Y.; El Naggar, M.H.; Wu, W.B.; Liu, H.; Jiang, G.S. Modified Rayleigh-Love rod model for 3D dynamic analysis of large-diameter thin-walled pipe pile embedded in multilayered soils. Comput. Geotech.
**2022**, 149, 104853. [Google Scholar] [CrossRef] - Chai, H.Y.; Liu, M.G.; Li, Q.; Chen, X.Y. Propagation of stress waves in a plate-pile system: Experimental studies. Rock Soil Mechan.
**2002**, 23, 459–464. [Google Scholar] - Chai, H.Y.; Liu, M.G.; Bai, S.W.; Li, Q. Numerical analysis of wave propagation in platform-pile system. Chin. J. Geotech. Eng.
**2003**, 25, 624–628. [Google Scholar] - Chai, H.Y.; Phoon, K.K.; Zhang, D.J. Effects of the Source on Wave Propagation in Pile Integrity Testing. J. Geotech. Geoenviron. Eng.
**2010**, 136, 1200–1208. [Google Scholar] [CrossRef] - Chai, H.Y.; Li, Q.; Liu, M.G.; He, H.J. Analysis of wave propagation on pile top. In Proceedings of the Second National Geotechnical and Engineering Academic Conference, Wuhan, China, 28 October–November 2006. [Google Scholar]

**Figure 1.**Computational model of the hammer-pile-soil system: (

**a**) Three-dimensional model; (

**b**) Top view of the pile.

**Figure 3.**Comparison of the present solution and Abaqus solution at different testing points along the radius of the large-diameter pile top: (

**a**) r = r

_{h}; (

**b**) r = 0.2R; (

**c**) r = 0.5R; (

**d**) r = 0.7R; (

**e**) r = 1.0R.

**Figure 4.**Comparison of the present solution and the solution of Zheng et al. [29] at different testing points along the radius of the large-diameter pile top: (

**a**) r = 0.2R; (

**b**) r = 0.4R; (

**c**) r = 0.6R; (

**d**) r = 0.8R; (

**e**) r = 1.0R.

**Figure 5.**Velocity response in time domain of different test points along the radial direction within the excitation region for pile with R = 0.25 m: (

**a**) r

_{h}= 0.01 m; (

**b**) r

_{h}= 0.02 m; (

**c**) r

_{h}= 0.04 m; (

**d**) Variation of the negative-positive-ratio.

**Figure 6.**Velocity response in time domain of different test points along the radial direction within the excitation region for pile with R = 0.5 m: (

**a**) r

_{h}= 0.01 m; (

**b**) r

_{h}= 0.02 m; (

**c**) r

_{h}= 0.04 m; (

**d**) Variation of the negative-positive-ratio.

**Figure 7.**Velocity response in time domain of different test points along the radial direction within the excitation region for pile with R = 0.75 m: (

**a**) r

_{h}= 0.01 m; (

**b**) r

_{h}= 0.02 m; (

**c**) r

_{h}= 0.04 m; (

**d**) Variation of the negative-positive-ratio.

**Figure 8.**Velocity response in time domain of different test points along the radial direction within the excitation region for pile with R = 1.0 m: (

**a**) r

_{h}= 0.01 m; (

**b**) r

_{h}= 0.02 m; (

**c**) r

_{h}= 0.04 m; (

**d**) Variation of the negative-positive-ratio.

**Figure 9.**Vertical velocity time histories at the edge of the excitation region for different pile radii.

**Figure 10.**Vertical velocity time histories of the different test points at pile top along the radial direction outside the excitation region: (

**a**) R = 0.25 m; (

**b**) R = 0.5 m; (

**c**) R = 0.75 m; (

**d**) R = 1.0 m.

**Figure 11.**Vertical velocity time histories at the edge of the excitation region with different pile-hammer-ratios: (

**a**) R = 0.25 m; (

**b**) R = 0.5 m; (

**c**) R = 0.75 m; (

**d**) R = 1.0 m.

**Figure 12.**Vertical velocity time histories of different test points near the excitation region on the pile top: (

**a**) R = 0.25 m; (

**b**) R = 0.5 m; (

**c**) R = 0.75 m; (

**d**) R = 1.0 m.

**Figure 14.**Vertical velocity time domain curves at different test points along the depth of pile: (

**a**) r = r

_{h}; (

**b**) r = 0.2R.

Pile (I, II) | ${G}_{i}^{\mathrm{p}}$(GPa) | ${\rho}_{i}^{\mathrm{p}}$(kg/m^{3}) | ${\upsilon}_{\mathrm{p}}$ | H (m) |

40 | 2500 | 0.2 | 10 | |

Surrounding soil | E_{s} (Mpa) | ${\rho}_{\mathrm{s}}$(kg/m^{3}) | ${\upsilon}_{\mathrm{s}}$ | |

54 | 1800 | 0.35 |

**Table 2.**The parameters of the pile-soil system of Zheng et al. [29].

Pile (I, II) | ${G}_{i}^{\mathrm{p}}$(GPa) | ${\rho}_{i}^{\mathrm{p}}$(kg/m^{3}) | ${\upsilon}_{\mathrm{p}}$ | H (m) |

25 | 2500 | 0.15 | 10 | |

Surrounding soil | E_{s} (MPa) | ${\rho}_{\mathrm{s}}$(kg/m^{3}) | ${\upsilon}_{\mathrm{s}}$ | |

6 | 1800 | 0.35 |

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## Share and Cite

**MDPI and ACS Style**

Guan, W.; Zhang, M.; Wang, Z.; Jiang, G.; Liu, W.; Cao, S.; Leo, C.J.; An, E.; Gao, X.; Wu, W.
Influence of the Three-Dimensional Effect of Pile-Soil System on the Vertical Dynamic Response of Large-Diameter Piles in Low-Strain Integrity Testing. *Energies* **2022**, *15*, 9548.
https://doi.org/10.3390/en15249548

**AMA Style**

Guan W, Zhang M, Wang Z, Jiang G, Liu W, Cao S, Leo CJ, An E, Gao X, Wu W.
Influence of the Three-Dimensional Effect of Pile-Soil System on the Vertical Dynamic Response of Large-Diameter Piles in Low-Strain Integrity Testing. *Energies*. 2022; 15(24):9548.
https://doi.org/10.3390/en15249548

**Chicago/Turabian Style**

Guan, Wenjie, Meixia Zhang, Zekun Wang, Guosheng Jiang, Wenqi Liu, Sheng Cao, Chin Jian Leo, Elieen An, Xiaodong Gao, and Wenbing Wu.
2022. "Influence of the Three-Dimensional Effect of Pile-Soil System on the Vertical Dynamic Response of Large-Diameter Piles in Low-Strain Integrity Testing" *Energies* 15, no. 24: 9548.
https://doi.org/10.3390/en15249548