Elsevier

Ocean Engineering

Volume 219, 1 January 2021, 108009
Ocean Engineering

Vertical vibration of an end bearing pile interacting with the radially inhomogeneous saturated soil

https://doi.org/10.1016/j.oceaneng.2020.108009Get rights and content

Highlights

  • The construction disturbance of the saturated soil is simulated accurately.

  • The analytical solution for the vertical vibration of a pile embedded in radially inhomogeneous saturated soil is built.

  • The effect of the construction disturbance for different pile–soil parameters is investigated.

Abstract

This study aims at portraying the vertical vibration characteristics of an end bearing pile interacting with the saturated soil that is of distinctly inhomogeneous behavior in the radial direction due to the construction disturbance. The pile is considered a one-dimensional bar. The saturated soil is considered by the Biot's two-phase linear theory and its radial inhomogeneity is simulated by the gradual variation of soil properties radially in a linear way. In practical terms, it is partitioned into a number of vertical ring-shaped zones in the radial direction with the soil properties approximately remain the same within the same soil zone. The pile–soil dynamic interaction is then obtained by solving the governing equations of each soil zone and is substituted into the governing equation of the pile to derive the analytical solution for the dynamic impedance at the pile head. The present solution is validated by comparing with other solutions. Meanwhile, it is employed to investigate the effect of the construction disturbance on the vertical vibration of the pile for different pile–soil parameters.

Introduction

Pile foundations have been increasingly adopted in engineering as the soil base is insufficient to support the load from the superstructures in most cases (Liang et al., 2018a, 2018b; He et al., 2019; Li et al., 2019b). Except the static loads, pile foundations usually withstand vertical dynamic loadings (Liu et al., 2019). Investigating the behavior of pile foundations under vertical dynamic loadings can provide theoretical basis for aseismic design (Yang et al., 2020), dynamic foundation design and non-destructive testing of the piles (Meng et al., 2020). Given this, many researchers have been engaged in this field in the past few decades. Despite some achievements based on the numerical methods (Shi et al., 2014; Ai and Wang, 2017; Li et al., 2019a), most of the studies are focused on simulating the pile–soil dynamic interaction by introducing more accurate models and on this basis solving the dynamic characteristics of the piles by analytical or semi-analytical methods. Under this guidance, many types of models have been proposed, such as the Winkler model (Yesilce and Catal, 2008; Ding et al., 2014; Liu et al., 2018), plane strain model (Novak et al., 1978; Zheng et al., 2016, 2017; Wu et al., 2020; Luan et al., 2020) and three dimensional axisymmetric model (Nogami and Novak, 1976; Wu et al., 2014; Zheng et al., 2015a). These models with increasing accuracy greatly promoted the development of the pile vibration theory.

In coastal areas, the soil can be simplified as a two-phase medium as it is generally saturated by ground water (Shi and Zhao, 2020). As a result, the pile vibration theories for the single-phase soil case are not appropriate for the piles interacting with the two-phase soil as they cannot take the effect of the permeability into account. Under this background, some researchers developed the two-phase medium theory in which the two-phase soil is considered using the dynamic consolidation theory built by Biot (1956). Under the proposed idea, Zeng and Rajapakse (1999) examined the vertical vibration of a pile partially embedded in homogeneous saturated soil. Then, Lu et al. (2009), Zheng et al. (2015b), and Ai et al. (2016, 2018) established the corresponding solutions allowing for the stratification of the soil. Further, some researchers extended the studies to more complicated conditions. For example, Yang and Pan (2010) took the end bearing pile into account based on the hypothesis that the pile overlies a rigid bedrock. Liu et al. (2014) and Ding et al. (2017) extended the corresponding solution for the solid pile to that for the open-ended pipe pile by considering the dynamic action of the soil plug. Xiao et al. (2018) introduced the Rayleigh-Love rod theory to approximately simulate the 3D effect of the wave propagation of a large-diameter pile. Zhang et al. (2019c) considered the transversely isotropic behavior of the soil. Li et al. (2019d) extended the solution for the saturated soil to that for the frozen saturated soil by introducing the conception of the temperature.

In most cases, the soil would be compacted or relaxed during pile installation. This ubiquitous phenomenon is known as the construction disturbance. As the disturbance degree of the soil is inversely related to its distance from the pile, i.e., the soil closer to the pile is more strongly disturbed, the soil surrounding the pile is of distinctly inhomogeneous behavior in the radial direction (Hu et al., 2018; Ji et al., 2018; Zhang et al., 2019a, 2019b). This well-nigh inevitable question plays an important role in the static capacity (Zhou et al., 2017) and dynamic characteristic of the pile (Li and Gao, 2019a, 2019b). Given this, it has been a focus problem in the field of pile vibration theory over the years. Based on the review of the related literatures, it can be derived that they simulated this phenomenon in much the same way, i.e., varying the soil properties radially in a certain way such as a linear way, parabolic way, exponential way and so on. Based on these train of thoughts, a range of models have been put forward. From the weakened annular boundary zone model (Veletsos and Dotson, 1988) and non-reflective boundary zone model (Dotson and Veletsos, 1990; Han and Sabin, 1995), to the district boundary zone model (El Naggar, 2000; Wang et al., 2019) and the shear complex stiffness transfer model (Yang et al., 2009; Zhang and Pan, 2017; Li and Gao, 2019b), the accuracy of the models is continuously improved with a wider application range.

Similarly, the saturated soil would also be inevitably disturbed during the installation of the pile (Li et al., 2019c). However, the aforementioned studies were mainly focused on the analysis of the piles in the single phase soil, thus resulting in insufficient research on those in the saturated soil. In view of this, this paper simulates the radial inhomogeneity of the saturated soil using a more accurate model and on this basis addresses the vibration of an end bearing pile subjected to vertical dynamic loading by analytical method. Based on the proposed solutions, the effect of the construction disturbance for different parameters of the pile (length and longitudinal wave velocity) and the soil (porosity and Darcy permeability coefficient) are investigated.

Section snippets

Mathematical model

A mathematical model presented in Fig. 1 is established to examine the vertical vibration of an end bearing pile interacting with the radially inhomogeneous saturated soil. The vertical exciting force, P0eiωt, is acted on the head of the visco-elastic solid pile with a length of Hp and a radius of rp. In practice, it can represent many types of dynamic loads, such as the exciting force inspired by the handheld hammer in non-destructive testing of pile, the exciting force inspired by the dynamic

Vibration of the soil

By introducing the decoupling strategies to Eqs. (6), (7), uzj can be rewritten asuzj=uz1j+uz2jwhere uz1j and uz2j satisfy the following two partial differential equations, respectively.2uz1jz2β1j2uz1j=0(γ2j2uz2jz2γ1juz2j)+(2uz2jr2+1ruz2jr)=0where β1j2=Mjω2mj+iωbj; γ1j=ω2ρjGjω4ρfj2Gj(ω2mj+iωbj) and γ2j=λjGj+2.

The solution for Eq. (19) can be expressed asuz1j=Ejeβ1jz+Fjeβ1jzwhere Ej and Fj are undetermined constants.

Based on the theory of the separation variable, uz2j can be

Parametric study

First, the present solution is compared with the solution for a pile embedded in radially inhomogeneous single-phase medium proposed by Yang et al. (2009) and the plane strain solution for a pile embedded in radially inhomogeneous saturated soil developed by Wang et al. (2019) to verify its reliability. Then, a systematic sensitivity analysis, which aims at revealing the effect of the construction disturbance under different conditions, is conducted with the aid of the proposed solution. In the

Conclusions

An analytical solution is built and is employed to examine the effect of the construction disturbance of the saturated soil on the vertical vibration characteristics of an end bearing pile under different conditions. Comparisons with other solutions confirmed that the present solution is reliable and is more accurate than the plane strain solution. The following conclusions can be condensed:

  • (1)

    Both the dynamic stiffness and damping above the cut-off frequency decrease (increase) as the weakening

CRediT authorship contribution statement

Zhenya Li: Methodology, Writing - original draft, Investigation, Visualization. Yufeng Gao: Conceptualization, Supervision. Kuihua Wang: Writing - review & editing, Validation.

Declaration of competing interest

The authors declare that they have no known competing financialinterestsor personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 51808190, 41630638, 51779217), the Fundamental Research Funds for the Central Universities (Grant No. 2019B08014, B200204032), the China Postdoctoral Science Foundation funded project (Grant No. 2018M630501).

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