Longitudinal Dynamic Response of a Large-Diameter-Bored Pile Considering the Bottom Sediment and Radial Unloading of the Surrounding Soil

Abstract

The longitudinal dynamic response of a large-diameter-bored pile is investigated considering the bottom sediment and the radial unloading of the surrounding soil. First, the sediment between the pile tip and the bedrock is treated as a fictitious soil pile with a cross-sectional area similarto that of the pile tip. The large-diameter-bored pile (including the fictitious soil pile) is considered as a Rayleigh–Love rod and is divided into finite segments. Under theseconditions, the three-dimensional (3D) effect of the wave propagation along the pile is indirectly simulated by considering the transverse inertia of the pile to avoid complicated calculations. Meanwhile, the surrounding soil is divided into finite annular zones in the radial direction, with the soil properties varying radially as well to simulate the radial unloading of the surrounding soil during construction. The governing equation for each soil zone is built and solved, from zone to zone, to obtain the shear stress acting on the pile. Then, the governing equation for the fictitious soil pile (i.e., the sediment) is solved to derive the dynamic action at the pile tip. In a similar manner to that ofthe fictitious soil pile and together with the recursion method, the governing equation for the pile is solved to obtain the pile’s complex impedance and velocity response. The proposed solution is verified and then introduced to portray the coupling effect of the sediment, pile parameters and radial unloading of the surrounding soil on the longitudinal dynamic response of the large-diameter-bored pile.

1. Introduction

Large-diameter-bored piles, devoid of issues related to vibration, soil squeezing, and noise during their installation, have been commonly utilized in the establishment of foundations for energy infrastructures, such as power stations, transmission towers and wind turbines, both onshore and offshore. Some researchers have emphasized the distinctiveness of large-diameter-bored piles concerning static performance [1,2,3]. Accordingly, the dynamic behavior of this type of pile also deserves special attention, from both nondestructive detection and dynamic foundation design aspects, given the crucial role of vibration analysis in engineering [4,5,6].

Because of the inherent limitations of construction technology, a certain thickness of sediment is present at the bottom of alarge-diameter-bored pile, whose properties differ greatly from the bedrock, as well as the soil at the pile tip [7,8]. The accurate simulation of the dynamic pile–sediment interaction is crucial for understanding the vibration characteristics of the large-diameter-bored pile. Owing to their simplicity and convenience, the rigid model [9,10,11,12,13,14,15,16,17] and the spring–dashpot model [18,19,20,21,22,23] are commonly utilized for simulating the support conditions at the pile tip. However, it should be noted that these models have their limitations. The rigid model, in which the boundary condition of zero displacement is imposed onthe pile tip, only applies to pilesthat lie directly on rigid bedrock. The spring–dashpot model wasdeveloped for floating piles. In this model, a spring and a dashpot are connected in parallel to provide consideration of the elasticity and viscosity of the bottom soil. Unfortunately, this commonly used model ignores the stress equilibrium and displacement continuity conditions of the bottom soil [24]. In addition, it does notdeal with the layer properties and thickness of the bottom soil, which can exhibit significant impacts on pile dynamics. In fact, all of these factors merit careful consideration to accurately capture pile vibration characteristics when it comes to sediment.

To address this issue, Yang and Wang [25] established a fictitious soil pile model to consider the dynamic pile tip–bottom soil interaction. The core idea of this model is to consider the soil column between the pile tip and the underlying bedrock as a soil pile, the characteristics of which are determined based on the soil parameters. On this basis, the governing equation for the soil pile can be established similar to that for a concrete pile, enabling the dynamic action at the pile tip to be derived by solving the equation. With this approach, the thickness of the bottom soil can be directly reflected by the length of the soil pile, and the stratification of the bottom soil can be considered by segmenting the soil pile. Subsequently, a series of follow-up studies wereconducted. Wu et al. [26] analyzed pile vibrations under different lengths of fictitious soil piles. Yu et al. [27] discussed the influence of sediment thickness on pile complex impedance. Yang et al. [28] analyzed the stratification of the pile end soil. Some researchers [29,30] introduced the concept of cone angle and, on this basis, developed an improved model to account for stress diffusion in the bottom soil. The corresponding studies involving the two-phase medium [31], three-phase medium [32], pipe pile [33] and incomplete bonding effect of the pile–soil interface [34] have further expanded the application range of the fictitious soil pile model.

The vibration characteristics of large-diameter-bored piles are further complicated by two factors in addition to sediment. The first factor is radial unloading of the soil, which weakens the soil around the pile during or after the hole’sformation, making it radially inhomogeneous. This phenomenon, which is positively related to pile diameter, has an adverse effect on both the bearing capacity [35,36,37] and dynamic response of the pile [38,39,40,41]. Therefore, it deserves extra attention when dealing with the vibration of large-diameter-bored piles.

The second factor to consider is the 3D effect of the pile. Historically, pile vibration studies have primarily relied on one-dimensional (1D) theory [42], which is suitable for slender rods. However, for large-diameter-bored piles, the 1D theory is not strictly applicable due to the 3D effect of the pile [43]. A notable discrepancy has been observed between measured and calculated results derived usingthe 1D theory [44,45,46], and this phenomenon is ascribed to the wave dispersion effect when propagating in a large-diameter-bored pile [47]. Given the challenge of solving the rigorous 3D wave equation for the pile, an alternative approach was proposed. In this method, the pile is treated as a Rayleigh–Love rod, and its governing equation can be easily established by considering the pile’s transverse inertia on the basis of the 1D wave equation for the pile. This approach significantly reduces the computational requirements while maintaining precision. It has been adopted by a series of researchers, including Lü et al. [48], Gao et al. [49], Xiao et al. [50], Zhang et al. [51], and Liu et al. [52].

For large-diameter-bored piles, both the radial unloading of soil and the 3D effect of wave propagation should be considered simultaneously, when accounting for the action of bottom sediment. However, the correlative research has not yet been reported to the authors’ knowledge. Therefore, this paper presents a rigorous solution to address this gap. The bottom sediment is modeled as a fictitious soil pile and is simulated by the Rayleigh–Love rod model in conjunction with the large-diameter-bored pile. The radial unloading of the surrounding soil is considered by the gradual variation in thesoil properties in the radial direction. On the basis of the solution, the coupling effect of the sediment, pile parameters, and radial unloading of the soil on the longitudinal dynamic response of the large-diameter-bored pile is analyzed.

2. Model Establishment

The problem to be solved in this study can be depicted usinga schematic, as shown in Figure 1. The large-diameter-bored pile, with a length of H_p, is subjected to a longitudinal dynamic load of q(t) at its head. The bottom sediment, with a thickness of H_s, is regarded as a fictitious soil pile. According to this train of thoughts, the pile (including the fictitious soil pile) is divided into n segments. The fictitious soil pile (i.e., the sediment) is numbered as 1, while the pile segments are numbered as 2, 3, …, n. l_i is the length of the ith segment, and h_i is the distance between the head of the ith segment and the ground surface. Radial unloading is simulated by the gradual variation in thesoil properties along the radial direction. For computational convenience, each soil layer is partitionedinto m vertical annular zones. The soil within the same zone is approximately considered as homogeneous. The accuracy of such treatment is the positive correction with m. The dynamic interactions at the interfaces of the adjacent soil layers are simulated using Voigt models, regardless of the radial distance. The stiffness and damping coefficients at the upper surface of the jth zone within the ith layer are denoted by $k_{\left(i+1\right)j}$ and ${\delta }_{\left(i+1\right)j}$, respectively, and those at the lower surface are $k_{ij}$ and ${\delta }_{ij}$, respectively. Although this simplified treatment may not be strictly rigorous, it provides a straightforward and efficient approach to derive the solution for pile vibration.The feasibility of the approach has been demonstrated with a satisfactory level of precision [53].

The following assumptions are indispensable during the analysis:

• (1). The large-diameter-bored pile, as well as the bottom sediment, is an elastic Rayleigh–Love rod;

• (2). The top surface of the surrounding soil is free, while its bottom surface is in direct contact with the rigid bedrock;

• (3). The displacement of the surrounding soil in the vertical direction is considered while that in the radial direction is neglected;

• (4). The pile and soil, as well as the adjacent soil zones, are in continuous contact;

• (5). The displacement and velocity of the soil–pile system are zero when t = 0;

• (6). The study is conducted under the conditions of small deformations and strains, and the conclusions are valid only under axisymmetric condition.

3. Governing Equations

According to the elastodynamic theory [54], the dynamic equilibrium equation for the soil in an axisymmetric cylindrical coordinate system can be established. By disregarding the radial component of the soil displacement to simplify the equation, the dynamic governing equation for the soil with the hysteresis damping can be written as follows. It is also worth noting that the accuracy requirement can be satisfied according to Nogami and Novak [9].

(1)$\left[\left({\lambda }_{ij}+2G_{ij}\right)+\mathrm{i}\left({\lambda }_{ij}^{\prime }+2G_{ij}^{\prime }\right)\right]\frac{{\partial }^2}{\partial z^2}{u}_{ij}+\left(G_{ij}+\mathrm{i}{G}_{ij}^{\prime }\right)\left(\frac{1}{r}\frac{\partial }{\partial r}+\frac{{\partial }^2}{\partial r^2}\right)u_{ij}={\rho }_{ij}\frac{{\partial }^2}{\partial t^2}{u}_{ij}$

where $u_{ij}=u_{ij}\left(r,z,t\right)$ is the vertical displacement of the soil; ${\lambda }_{ij}$, $G_{ij}$, $E_{ij}$, ${\mu }_{ij}$, and ${\rho }_{ij}$ represent the Lame constant, shear modulus, elastic modulus, Poisson’s ratio, and density of the soil, respectively; ${\lambda }_{ij}^{\prime }$ and $G_{ij}^{\prime }$ are viscosity coefficients corresponding to ${\lambda }_{ij}$ and $G_{ij}$; $\mathrm{i}=\sqrt{-1}$ and denotes the imaginary unit.

In the 1D theory, the pile is treated as an Euler rod. As mentioned above, this approach is not suitable for the large-diameter-bored pile due to its 3D effect. The 3D wave equation for the pile is complex and impractical in engineering. To address this issue, Rayleigh and Love established the Rayleigh–Love rod theory by considering the transverse inertia effect of the pile in the premise of the assumptions adopted in the 1D theory. According to the Rayleigh–Love rod theory, the longitudinal vibration of the large-diameter-bored pile (including the fictitious soil pile, i.e., the sediment) can be represented by the following equation.

(2)$E_{\mathrm{p}i}{A}_{\mathrm{p}i}\frac{{\partial }^2{u}_{\mathrm{p}i}}{\partial z^2}-{\rho }_{\mathrm{p}i}{A}_{\mathrm{p}i}\left[\frac{{\partial }^2}{\partial t^2}-{\left({\mu }_{\mathrm{p}i}{r}_{\mathrm{p}i}\right)}^2\frac{{\partial }^4}{\partial z^2\partial t^2}\right]u_{\mathrm{p}i}-2\pi r_{\mathrm{p}i}{f}_i=0$

where $u_{\mathrm{p}i}=u_{\mathrm{p}i}\left(z,t\right)$ denotes the vertical displacement of the ith pile segment; $E_{\mathrm{p}i}$, $A_{\mathrm{p}i}$, ${\rho }_{\mathrm{p}i}$, and ${\mu }_{\mathrm{p}i}$ represent the elastic modulus, cross-sectional area, density, and Poisson’s ratio of the ith pile segment, respectively; and $f_i=f_i\left(r_{\mathrm{p}i},z,t\right)$ is the frictional resistance at the soil–pile interface.

4. Solutions for the Equations

4.1. Solution for the Soil

By introducing the concept of local coordinate system and the Laplace transform technique, Equation (1) can be solved to derive the following equation. The derivation procedure can be seen in Supplementary Materials.

(3)$U_{ij}=\left[A_{ij}{\mathrm{I}}_0\left(q_{ij}r\right)+B_{ij}{\mathrm{K}}_0\left(q_{ij}r\right)\right]\cdot \left[C_{ij}\sin \left(h_{ij}{z}^{\prime }\right)+D_{ij}\cos \left(h_{ij}{z}^{\prime }\right)\right]$

where $U_{ij}$ is the Laplace transform of $u_{ij}$; ${\mathrm{I}}_0\left(q_{ij}r\right)$ and ${\mathrm{K}}_0\left(q_{ij}r\right)$ are modified Bessel functions; $z^{\prime }=z-h_i$; $A_{ij}$, $B_{ij}$, $C_{ij}$, and $D_{ij}$ are undetermined constants; $q_{ij}$ and $h_{ij}$ satisfy the relation

(4)$q_{ij}^2=\frac{\left\{{\eta }_{ij}^2+\mathrm{i}\left[D_{\mathrm{v}ij}\left({\eta }_{ij}^2-2\right)+2D_{\mathrm{s}ij}\right]\right\}{h}_{ij}^2-{\left(\omega /v_{ij}\right)}^2}{1+\mathrm{i}{D}_{\mathrm{s}ij}}$

where ${\eta }_{ij}=c_{ij}/v_{ij}$; $c_{ij}$ and $v_{ij}$ are longitudinal wave velocity and shear wave velocity of the soil, respectively; $D_{\mathrm{s}ij}=G_{ij}^{\prime }/G_{ij}$ and $D_{\mathrm{v}ij}={\lambda }_{ij}^{\prime }/{\lambda }_{ij}$ denote the hysteretic-type damping coefficients.

The boundary conditions at the upper and lower surfaces of the soil layer can be respectively expressed as

(5)${\left.\frac{\partial u_{ij}\left(r,z^{\prime },t\right)}{\partial z^{\prime }}\right|}_{z^{\prime }=0}={\left.\left[\frac{k_{(i+1)j}{u}_{ij}\left(r,z^{\prime },t\right)}{E_{ij}}+\frac{{\delta }_{(i+1)j}}{E_{ij}}\frac{\partial u_{ij}\left(r,z^{\prime },t\right)}{\partial t}\right]\right|}_{z\prime =0}$

(6)${\left.\frac{\partial u_{ij}\left(r,z\prime ,t\right)}{\partial z}\right|}_{z\prime =l_i}={\left.-\left[\frac{k_{ij}{u}_{ij}\left(r,z\prime ,t\right)}{E_{ij}}+\frac{{\delta }_{ij}}{E_{ij}}\frac{\partial u_{ij}\left(r,z\prime ,t\right)}{\partial t}\right]\right|}_{z\prime =l_i}$

Specifically, the stiffness and damping coefficient at the lower surface of the first soil layer tend to infinity according to Assumption (2), while those at the upper surface of the nth soil layer can be considered to be zero, as it is a free surface.

Applying the Laplace transform technique to Equations (5) and (6) and together with Equation (3), it can be derived that

(7)$\tan \left(h_{ij}{l}_i\right)=\frac{\left(\overline{K_{ij}}+\overline{K_{ij}^{\prime }}\right)h_{ij}{l}_i}{{\left(h_{ij}{l}_i\right)}^2-\overline{K_{ij}}\overline{K_{ij}^{\prime }}}$

where $\overline{K_{ij}}=\frac{k_{ij}+\mathrm{i}\omega {\delta }_{ij}}{E_{ij}}{l}_i$ and $\overline{K_{ij}^{\prime }}=\frac{k_{(i+1)j}+\mathrm{i}\omega {\delta }_{(i+1)j}}{E_{ij}}{l}_i$ denote the dimensionless complex stiffness.

Equation (7) can be solved by means of the bisection method to derive a series of eigenvalues (denoted by $h_{ijk}$, where k = 1, 2, …). On this basis, $q_{ijk}$ (k = 1, 2, …) can be obtained according to Equation (4). Further, Equation (3) can be rewritten as

(8)$U_{ij}={\displaystyle \sum _{k=1}^{\infty }\left[A_{ijk}{\mathrm{I}}_0\left(q_{ijk}r\right)+B_{ijk}{\mathrm{K}}_0\left(q_{ijk}r\right)\right]\cos \left(h_{ijk}z\prime -{\phi }_{ijk}\right)}$

where ${\phi }_{ijk}=\arctan \left[\overline{K_{ij}^{\prime }}/\left(h_{ijk}{l}_i\right)\right]$. Specifically, the displacement of the outermost zone (i.e., the mth zone) decays to zero at infinity. So, $A_{ijk}=0$, when j = m.

The shear stress of the jth zone acting on the (j−1)th zone can be expressed as

(9)${\tau }_{ij}=G_{ij}^{\ast }{\displaystyle \sum _{k=1}^{\infty }\left[A_{ijk}{q}_{ijk}{\mathrm{I}}_1\left(q_{ijk}{r}_{ij}\right)-B_{ijk}{q}_{ijk}{\mathrm{K}}_1\left(q_{ijk}{r}_{ij}\right)\right]\cos \left(h_{ijk}z\prime -{\phi }_{ijk}\right)}$

where $G_{ij}^{\ast }=G_{ij}\left(1+\mathrm{i}{D}_{\mathrm{s}ij}\right)$; ${\mathrm{I}}_1\left(q_{ijk}{r}_{ij}\right)$ and ${\mathrm{K}}_1\left(q_{ijk}{r}_{ij}\right)$ represent modified Bessel functions.

4.2. Solution for the Pile

The boundary conditions at the top and bottom surfaces of the ith pile segment are, respectively, expressed as

(10)${\left.\left[E_{\mathrm{p}i}{A}_{\mathrm{p}i}\frac{\partial u_{\mathrm{p}i}\left(z\prime ,t\right)}{\partial z}+{\rho }_{\mathrm{p}i}{A}_{\mathrm{p}i}{\left({\mu }_{\mathrm{p}i}{r}_{\mathrm{p}i}\right)}^2\frac{{\partial }^3{u}_{\mathrm{p}i}\left(z\prime ,t\right)}{\partial z\partial t^2}\right]\right|}_{z\prime =0}={\left.-z_{\mathrm{p}i}{u}_{\mathrm{p}i}\left(z\prime ,t\right)\right|}_{z\prime =0}$

(11)${\left.\left[E_{\mathrm{p}i}{A}_{\mathrm{p}i}\frac{\partial u_{\mathrm{p}i}\left(z\prime ,t\right)}{\partial z}+{\rho }_{\mathrm{p}i}{A}_{\mathrm{p}i}{\left({\mu }_{\mathrm{p}i}{r}_{\mathrm{p}i}\right)}^2\frac{{\partial }^3{u}_{\mathrm{p}i}\left(z\prime ,t\right)}{\partial z\partial t^2}\right]\right|}_{z\prime =l_i}={\left.-z_{\mathrm{p}(i-1)}{u}_{\mathrm{p}i}\left(z\prime ,t\right)\right|}_{z\prime =l_i}$

where $z_{\mathrm{p}i}$ and $z_{\mathrm{p}(i-1)}$ represent the displacement impedance at the top and bottom surface of the ith pile segment, respectively. Specifically, $z_{\mathrm{p}0}$ tends to infinity as the sediment directly rests on the rigid bedrock.

According to Assumption (4) and Equation (9), the frictional resistance acting on the pile $f_i$ can be derived. Then, it can be obtained by applying the Laplace transform technique to Equation (2) that

(12)$\begin{array}{l}{\left(c_{\mathrm{p}1}\right)}^2\left[1-\frac{{\left({\mu }_{\mathrm{p}1}{r}_{\mathrm{p}1}\right)}^2{\omega }^2}{{\left(c_{\mathrm{p}1}\right)}^2}\right]\frac{{\partial }^2{U}_{\mathrm{p}1}}{\partial {\left(z\prime \right)}^2}+{\omega }^2{U}_{\mathrm{p}1}+\\ \frac{2\pi r_{\mathrm{p}1}{G}_{11}^{\ast }}{{\rho }_{\mathrm{p}1}{A}_{\mathrm{p}1}}{\displaystyle \sum _{k=1}^{\infty }\left\{\left[A_{11k}{q}_{11k}{\mathrm{I}}_1\left(q_{11k}{r}_{\mathrm{p}1}\right)-B_{11k}{q}_{11k}{\mathrm{K}}_1\left(q_{11k}{r}_{\mathrm{p}1}\right)\right]\cos \left(h_{11k}z\prime -{\phi }_{11k}\right)\right\}}=0\end{array}$

where $U_{\mathrm{p}1}$ is the Laplace transform of $u_{\mathrm{p}1}$; $c_{\mathrm{p}1}$ is the longitudinal wave velocity of the sediment.

$U_{\mathrm{p}1}$ can be obtained as follows by solving Equation (12). The derivation procedure can be seen in Supplementary Materials.

(13)$U_{\mathrm{p}1}=a_1\left[\cos \left(\overline{{\lambda }_1}\overline{z\prime }\right)+{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{1k}^{\prime }}\cos \left(\overline{h{}_{11k}}\overline{z\prime }-{\phi }_{11k}\right)\right]+b_1\left[\sin \left(\overline{{\lambda }_1}\overline{z\prime }\right)-{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{1k}^{″}}\cos \left(\overline{h{}_{11k}}\overline{z\prime }-{\phi }_{11k}\right)\right]$

where ${\zeta }_{1k}^{\prime }={\zeta }_{1k}\left[\frac{\sin \left[\left(\overline{{\lambda }_1}-\overline{h_{11k}}\right)+{\phi }_{11k}\right]-\sin \left({\phi }_{11k}\right)}{\overline{{\lambda }_1}-\overline{h_{11k}}}+\frac{\sin \left[\left(\overline{{\lambda }_1}+\overline{h_{11k}}\right)-{\phi }_{11k}\right]+\sin \left({\phi }_{11k}\right)}{\overline{{\lambda }_1}+\overline{h_{11k}}}\right]$;

${\zeta }_{1k}^{″}={\zeta }_{1k}\left[\frac{\cos \left[\left(\overline{{\lambda }_1}+\overline{h_{11k}}\right)-{\phi }_{11k}\right]-\cos \left({\phi }_{11k}\right)}{\overline{{\lambda }_1}+\overline{h_{11k}}}+\frac{\cos \left[\left(\overline{{\lambda }_1}-\overline{h_{11k}}\right)+{\phi }_{11k}\right]-\cos \left({\phi }_{11k}\right)}{\overline{{\lambda }_1}-\overline{h_{11k}}}\right]$;

${\zeta }_{1k}=-\frac{\overline{G_{11}^{\ast }}\overline{q_{11k}}}{\overline{r_{\mathrm{p}1}}\left({\theta }_1{}^2-{\overline{h_{11k}}}^2{\theta }_1{}^2/{\overline{{\lambda }_1}}^2\right){\varphi }_{1k}{\displaystyle {\int }_0^1{\cos }^2\left(\overline{h_{11k}}\overline{z\prime }-{\phi }_{11k}\right)\mathrm{d}}\overline{z\prime }}\left[M_{11k}{\mathrm{I}}_1\left(\overline{q_{11k}}\overline{r_{\mathrm{p}1}}\right)-{\mathrm{K}}_1\left(\overline{q_{11k}}\overline{r_{\mathrm{p}1}}\right)\right]$;

${\varphi }_{1k}=M_{11k}\left[{\mathrm{I}}_0\left(\overline{q_{11k}}\overline{r_{\mathrm{p}1}}\right)+\frac{2\overline{G_{11}^{\ast }}\overline{q_{11k}}}{\overline{r_{\mathrm{p}1}}\left({\theta }_1{}^2-{\overline{h_{11k}}}^2{\theta }_1{}^2/{\overline{{\lambda }_1}}^2\right)}{\mathrm{I}}_1\left(\overline{q_{11k}}\overline{r_{\mathrm{p}1}}\right)\right]{+\mathrm{K}}_0\left(\overline{q_{11k}}\overline{r_{\mathrm{p}1}}\right)-\frac{2\overline{G_{11}^{\ast }}\overline{q_{11k}}}{\overline{r_{\mathrm{p}1}}\left({\theta }_1{}^2-{\overline{h_{11k}}}^2{\theta }_1{}^2/{\overline{{\lambda }_1}}^2\right)}{\mathrm{K}}_1\left(\overline{q_{11k}}\overline{r_{\mathrm{p}1}}\right)$; $\overline{{\lambda }_1}=\sqrt{\frac{{\theta }_1{}^2}{1-{\mu }_{\mathrm{p}1}^2{\overline{r_{\mathrm{p}1}}}^2{\theta }_1{}^2}}$; ${\theta }_1=\omega t_1$, $\overline{h_{11k}}=h_{11k}{l}_1$, $\overline{q_{11k}}=q_{11k}{l}_1$, $\overline{G_{11}^{\ast }}=G_{11}^{\ast }/E_{\mathrm{p}1}$, $\overline{z\prime }=z\prime /l_1$, and $\overline{r_{\mathrm{p}1}}=r_{\mathrm{p}1}/l_1$ are all dimensionless coefficients; $M_{11k}=A_{11k}/B_{11k}$ and can be derived according to Assumption (4).

Applying the Laplace transform to Equation (10) and substituting Equation (13) into it, the dynamic impedance at the top of the fictitious soil pile (i.e., the bottom of the pile) can be obtained as

(14)$Z_{\mathrm{p}1}=-\frac{E_{\mathrm{p}1}{A}_{\mathrm{p}1}-{\rho }_{\mathrm{p}1}{A}_{\mathrm{p}1}{\left({\mu }_{\mathrm{p}1}{r}_{\mathrm{p}1}\right)}^2{\omega }^2}{l_1}\frac{\frac{a_1}{b_1}{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{1k}^{\prime }}\overline{h_{11k}}\sin {\phi }_{11k}+\overline{{\lambda }_1}-{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{1k}^{″}\overline{h_{11k}}\sin {\phi }_{11k}}}{\frac{a_1}{b_1}\left(1+{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{1k}^{\prime }}\cos {\phi }_{11k}\right)-{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{1k}^{″}\cos {\phi }_{11k}}}$

where $\frac{a_1}{b_1}=-\frac{\sin \overline{{\lambda }_1}-{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{1k}^{″}\cos \left(\overline{h_{11k}}-{\phi }_{11k}\right)}}{\cos \overline{{\lambda }_1}+{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{1k}^{\prime }\cos \left(\overline{h_{11k}}-{\phi }_{11k}\right)}}$ is derived according to Equation (11).

Using a similar approach and together with the recursive method, the displacement impedance at the head of the large-diameter-bored pile, $Z_{\mathrm{p}n}$, can be derived. It can be further written as

(15)$Z_{\mathrm{p}n}=K_{\mathrm{p}}+\mathrm{i}\cdot C_{\mathrm{p}}$

where $K_{\mathrm{p}}$ and $C_{\mathrm{p}}$ denote the dynamic stiffness and damping, respectively.

Further, the pile velocity admittance is derived as

(16)$H_v\left(\mathrm{i}\omega \right)=\frac{\mathrm{i}\omega }{Z_{\mathrm{p}n}\left(\mathrm{i}\omega \right)}=-\frac{1}{{\rho }_{\mathrm{p}n}{A}_{\mathrm{p}n}{c}_{\mathrm{p}n}}{H}_{v0}$

where $H_{v0}=\frac{\mathrm{i}{\theta }_n}{1-{\left({\mu }_{\mathrm{p}n}\overline{r_{\mathrm{p}n}}\right)}^2{\theta }_n{}^2}\frac{\frac{a_n}{b_n}\left(1+{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{nk}^{\prime }}\cos {\phi }_{n1k}\right)-{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{nk}^{″}\cos {\phi }_{n1k}}}{\frac{a_n}{b_n}{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{nk}^{\prime }}\overline{h_{n1k}}\sin {\phi }_{n1k}+\overline{{\lambda }_n}-{\displaystyle \sum _{k=1}^{\infty }{\zeta }_{nk}^{″}\overline{h_{n1k}}\sin {\phi }_{n1k}}}$ is the dimensionless form of $H_v\left(\mathrm{i}\omega \right)$.

The velocity response in the time domain, $v\left(t\right)$, is derived using the inverse Fourier transform and convolution integral theorem. It can be written as follows if q(t) is a half-sine exciting force.

(17)$v\left(t\right)=-\frac{Q_{\max }}{{\rho }_{\mathrm{p}n}{A}_{\mathrm{p}n}{c}_{\mathrm{p}n}}{v}_0\left(t\right)$

where $Q_{\max }$ is the amplitude of the half-sine exciting force; $v_0\left(t\right)$ is the dimensionless form of $v\left(t\right)$ and can be expressed as

(18)$v_0\left(t\right)=\frac{1}{2}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\frac{H_v^{\prime }T}{{\pi }^2-T^2{\omega }^2}}\left(1+e^{-\mathrm{i}\omega T}\right)e^{\mathrm{i}\omega t}\mathrm{d}\omega$

where T is the width of the half-sine exciting force.

5. Parametric Analysis

In this section, the solution established in Section 4 is compared with those of Li et al. [55] and Cai et al. [56] to demonstrate its reliability. Then, a parametric analysis is conducted to reveal the coupling effect of the sediment thickness, pile parameters, and radial unloading of the surrounding soil on the longitudinal dynamic response of the large-diameter-bored pile. The parameters of the pile and the sediment adopted here are given in Table 1. The density, Poisson’s ratio, and disturbance range are 1850 kg/m³, 0.4, and r_d= 0.5 r_p. The shear wave velocity of the soil within the outermost undisturbed zone is v_m = 180 m/s, while that within the innermost zone is v₁ = 144 m/s. Under this condition, the soil disturbance degree is x = v₁/v_m = 0.8. The shear wave velocity of the soil in the disturbed zone is assumed to change in a linear way for convenience of calculation; m is assigned as 40 allowing for the precision requirement and computational efficiency. A declaration ismade once the soil–pile parameters change during the analysis.

5.1. Comparison with Other Solutions

Let H_s = 0 mm, the proposed solution is reduced to that for a large-diameter-bored pile resting on the rigid bedrock and interacting with the radially inhomogeneous soil. The comparison between the reduced solution and the solution of Li et al. [55] is given in Figure 2. One can derive that the two solutions agree well with each other.

The solution established in Section 4 can also be reduced to that for an Euler rod satisfying 1D theory when the pile Poisson’s ratio is assigned as 0. The reduced solution is compared with that established by Cai et al. [56]. As presented in Figure 3, the reduced solution coincides with Cai’s solution.

In the study by Li et al. [55], the reaction of the bottom soil is considered using a very simple model, which fails to account for the thickness and layered properties of the soil at the pile tip. In Cai’s [56] solution, the pile is considered as an Euler rod satisfying 1D theory, making it inapplicable to bored pile with a large diameter. In contrast, this paper presents a more rigorous solution with a broader application range, surpassing the limitations of Li’s and Cai’s works. The problems addressed in these two studiesare special instances of research that is presented in this paper.

5.2. Parametric Analysis and Discussion

5.2.1. Coupling Effect of Sediment Thickness and Pile Parameters

The coupling effect of the sediment thickness and pile parameters (i.e., the pile Poisson’s ratio, radius, and longitudinal wave velocity)is investigated here. The corresponding results are portrayed in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9.

Figure 4 presents the coupling effect of the sediment thickness and the pile Poisson’s ratio on the pile complex impedance. Compared to the strictly end-bearing pile (i.e., H_s = 0 mm), the presence of sediment significantly reduces the support stiffness at the pile tip. As a result, the oscillation amplitude and resonant frequency of K_p and C_p become lower, especially at higher frequencies. As the sediment thickness increases, the oscillation amplitude and resonant frequency of the pile complex impedance decrease with the decreasing rate, which decreases with frequency. When the pile’s transverse inertia is taken into account, the oscillation amplitude and resonant frequency of K_p and C_p within the high-frequency range decrease, especially at higher frequencies. In addition, the decrease in the oscillation amplitude of the complex impedance due to the sediment becomes less pronounced to some degree when the pile’s transverse inertia is considered while that in the resonant frequency of the complex impedance remains unchanged.

The coupling effect of the sediment thickness and pile Poisson’s ratio on the velocity response is illustrated in Figure 5. When there exists a certain thickness of sediment, the signal reflected by the pile tip weakens because of the greater dissipation of the elastic wave in the sediment compared to the rigid bedrock. In addition, the time lag of the reflected signal increases notably, leading to the overestimation of the pile length calculated using the low-strain reflected wave method.

It is also noted that a synthetic reflected signal can be observed before the reverse reflected signal, which may interfere with the detection of the support condition at the pile tip. The aforementioned phenomena are found to become more pronounced with a larger sediment thickness. Furthermore, when considering the pile’s transverse inertia (μ_p = 0.2), it takes longer to receive the signal reflected by the pile tip, and the signal is less weakened by the sediment, indicating the decreased influence of the sediment.

Given the notable influence of the sediment presented in Figure 4 and Figure 5, the pile hole should be cleared as thoroughly as possible in engineering.

Figure 6 presents the coupling effect of the sediment thickness and pile radius on the pile complex impedance. The oscillation amplitudes of K_p and C_p are higher for the pile with a radius of 0.5 m compared to one with a radius of 0.4 m. In addition, the resonant frequency is found to be lower for the pile with a larger radius, which demonstrates that the pile’s transverse inertia plays a more apparent role in its dynamic response. It should also be noted that the oscillation amplitude of the complex impedance is more notably decreased by the increase in the sediment thickness when the pile radius is larger, indicating the greater influence of sediment. However, the decrease in the resonant frequency due to the increase in the sediment thickness is not affected by the pile radius.

The coupling effect of the sediment thickness and pile radius on the velocity response is presented in Figure 7. For a fixed sediment thickness, the peak value of the signal reflected by the pile tip is higher and its time delay is longer when the pile radius is larger. In addition, it is observed that the reflected signal for r_p = 0.5 m weakens at a faster rate with an increasing sediment thickness compared to that for r_p = 0.4 m. This implies that the sediment thickness would exert greater influence on the pile dynamic response as r_p increases. Additionally, the delay in the reflected signal due to an increase in sediment thickness remains largely unaffected by variations in pile radius.

Figure 8 portrays the coupling effect of the sediment thickness and the pile’s longitudinal wave velocity on the pile complex impedance. As v_p increases, both the oscillation amplitude and resonant frequency of the complex impedance increase. When the longitudinal wave velocity of the pile is higher, an increase in the sediment thickness would result in a more notably decreased oscillation amplitude and an unchanged resonant frequency of the pile complex impedance.

The coupling effect of the sediment thickness and the pile’s longitudinal wave velocity on the velocity response is providedin Figure 9. When v_p is higher, the signal reflected by the pile tip becomes more notable and reaches the pile head more quickly. Thisapplies to the synthetic reflected signal before the reversed reflected signal is caused by the sediment. This would cause more serious interference in the judgment of the reflected signal and the support condition at the pile tip. As the sediment thickness increases, the peak value of the signal reflected by the pile tip decreases while that of the synthetic reflected signal before the reversed reflected signal increases. These phenomena can be more easily observed in a pile with a larger v_p. It is worth noting that the delay in the peak time of the reflected signal with the sediment thickness is not affected by v_p. When comparing curves on the same horizontal coordinate scale, a secondary reflected signal from the pile tip is observed for those corresponding to a v_p of 3800 m/s. Moreover, because of the sediment interference at the bottom of the pile, the secondary reflected signal is not an ideal co-directional reflected signal like that of a fully end-bearing pile. In fact, when using the low-strain method to judge the quality of pile foundations, the first reflected signal at the bottom of the pile is more meaningful, while the reference value of the secondary reflected signal is not significant.

5.2.2. Coupling Effect of Sediment Thickness and the Radial Unloading of the Surrounding Soil

The coupling effect of the sediment thickness and radial unloading of the surrounding soil, characterized by the softening range and degree, is involved and the corresponding results are presented in Figure 10, Figure 11, Figure 12 and Figure 13.

Figure 10 shows the coupling effect of the sediment thickness and soil softening range on the pile complex impedance. For a given sediment thickness, the oscillation amplitudes of K_p and C_p corresponding to a larger soil softening range are higher. Additionally, the decrease in the resonant frequency of the complex impedance driven by the increase in the sediment thickness becomes more notable as the soil softening range increases. One more point, the decrease in the resonant frequency of K_p and C_p caused by the increase in the sediment thickness remains consistent regardless of the soil softening range. Generally speaking, the influence of the sediment thickness is found to be more notable as the soil softening range increases.

The coupling effect of the sediment thickness and soil softening range on the velocity response is portrayed in Figure 11. A higher soil softening range corresponds to a stronger signal reflected by the pile tip and a more notable weakness of the reflected signal caused by the increase in the sediment thickness. Figure 11 also shows that the delay in the reflected signal attributed to the sediment remains almost the same under different soil softening ranges.

Figure 12 and Figure 13 present the coupling effect of the sediment thickness and soil softening degree on the pile complex impedance and velocity response, respectively. As the soil softening degree increases, the oscillation amplitude of the complex impedance becomes higher, leading to a more pronounced reflection of the elastic wave by the pile tip. Concurrently, the aforementioned factors representing the pile dynamic response are more notably diminished with the increase in the sediment thickness. It is also noted that the decrease in the resonant frequency of the complex impedance and the delay of the signal reflected by the pile tip remain unaffected by the variation insoil softening degree.

6. Conclusions

The solution for the longitudinal dynamic response of a large-diameter-bored pile is established considering the reaction of the sediment and radial unloading of the surrounding soil. The parametric analysis drew the following conclusions:

• (1). When there exists a certain thickness of sediment at the pile tip, the oscillation amplitude and resonant frequency of the complex impedance decrease, the signal reflected by the pile tip weakens with an increase in time lag, and a synthetic reflected signal occurs before the reverse reflected signal. Under this condition, the pile length would be overestimated and the support condition at the bottom of the pile would be misjudged. These phenomena become more apparent for increases in the sediment thickness.

• (2). When the transverse inertia of the pile is taken into account, the oscillation amplitude and resonant frequency of the complex impedance within the high-frequency range decrease, especially at higher frequencies. Meanwhile, it takes more time to receive the signal reflected by the pile tip.

• (3). As the pile radius, pile longitudinal wave velocity, soil softening range, or degree increases, the oscillation amplitude and resonant frequency of the complex impedance increase, the signal reflected by the pile tip becomes more pronounced.

• (4). The decrease in the oscillation amplitude of the complex impedance and the weakness of the reflected signal due to the increased sediment thickness become less notable as the pile Poisson’s ratio increases. Conversely, these effects are exacerbated by the increasing pile radius, longitudinal wave velocity, soil softening range, and degree. It is also noted that the decrease in the resonant frequency of the complex impedance and the time delay of the reflected signal are not affected during this process.

• (5). The aforementioned conclusions can provide a theoretical basis for the judgment of low-strain test results of pile foundation in actual engineering, thus avoiding misjudgment and evaluating the length and integrity of piles more accurately.

Author Contributions

Conceptualization, Z.L. and C.Z.; methodology, Z.L.; software, Y.Z.; validation, C.Z. and P.Z.; formal analysis, Y.Z.; investigation, C.Z. and P.Z.; resources, Z.L.; data curation, C.Z.; writing—original draft preparation, Z.L. and C.Z.; writing—review and editing, Z.L.; visualization, C.Z.; supervision, Z.L.; project administration, Z.L.; funding acquisition, C.Z. and Z.L. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Table

Figure 1. Schematic model.

Figure 2. Comparison with Li et al. [55].

Figure 3. Comparison with Cai et al. [56].

Figure 4. Coupling effect of the sediment thickness and pile Poisson’s ratio on the pile complex impedance.

Figure 5. Coupling effect of the sediment thickness and pile Poisson’s ratio on the velocity response.

Figure 6. Coupling effect of the sediment thickness and pile radius on the pile complex impedance.

Figure 6. Coupling effect of the sediment thickness and pile radius on the pile complex impedance.

Figure 7. Coupling effect of the sediment thickness and pile radius on the velocity response.

Figure 8. Coupling effect of the sediment thickness and pile’s longitudinal wave velocity on the pile complex impedance.

Figure 9. Coupling effect of the sediment thickness and the pile’s longitudinal wave velocity on the velocity response.

Figure 10. Coupling effect of the sediment thickness and soil softening range on the pile complex impedance.

Figure 11. Coupling effect of the sediment thickness and soil softening range on the velocity response.

Figure 12. Coupling effect of the sediment thickness and soil softening degree on the pile complex impedance.

Figure 13. Coupling effect of the sediment thickness and soil softening degree on the velocity response.

Table 1

Parameters of the pile and sediment.

	Length (m)	Radius (m)	Density (kg/m3)	Longitudinal Wave Velocity (m/s)	Poisson’s Ratio
Pile	10	0.5	2500	3800	0.2
Sediment	0.01	0.5	1700	1200	0.35

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/app132413252/s1.

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