Vertical vibration of an end bearing pile interacting with the radially inhomogeneous saturated soil
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, , 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), can be rewritten aswhere and satisfy the following two partial differential equations, respectively.where ; and .
The solution for Eq. (19) can be expressed aswhere and are undetermined constants.
Based on the theory of the separation variable, 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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2022, Ocean EngineeringCitation Excerpt :Zhang et al. (2021) further presented a modified additional mass model to investigate the torsional impedance of thin-wall pipe piles in vertically and radially inhomogeneous soil. Li et al. (2019, 2021) investigated the vertical and torsional impedance of an end-bearing pile in radially inhomogeneous saturated soil. Wang et al. (2019) and Wang and Gao (2018) presented the vertical impedances of a straight pile and a tapered pile in radially inhomogeneous saturated soil with multilayers.