Abstract

In this paper, an analytical solution is developed to investigate soil consolidation around a pile under earthquake loading. The solution is validated using finite element method. The influence of various parameters on excess pore water pressure is analyzed. The results show that excess pore water pressure increases with depth and is positively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}}

/Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 , while negatively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} . The values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}} /Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} 

affect excess pore water pressure during and after the earthquake, while the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
only affects excess pore water pressure after the earthquake. The growth rate of excess pore water pressure during the earthquake is positively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}} /Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 , and negatively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d}

. The dissipation rate of excess pore water pressure during the earthquake is positively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}} /Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} , and negatively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} . Additionally, a formula for calculating the reconsolidation settlement of pile-soil foundation after an earthquake is proposed, and the effects of pile and soil parameters on pile-soil foundation reconsolidation settlement are analyzed. The results indicate that reconsolidation settlement can be divided into two stages: rapid settlement stage and slow settlement stage. The total settlement of the foundation is positively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}} /Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 , while negatively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} .

Keywords: Earthquake, pile-soil foundation, excess pore water pressure, reconsolidation settlement, analytical solution

1. Introduction

The use of pile foundation in high-rise and super high-rise buildings has become increasingly common with the continuous development of modern urbanization [1]. However, frequent earthquakes around the world have raised concerns about the instability of buildings caused by liquefaction and reconsolidation of sandy soil foundations [2,3]. To address this issue, it is necessary to analyze the anti-liquefaction capacity and post-earthquake reconsolidation behavior of pile foundation treatment in sandy soil foundations to provide guidance for the safe and effective application of pile foundation in sandy soils.

After the implementation of pile foundation treatment in sandy soils, the pile and the sand collaboratively bear the upper loading. However, during strong earthquake activity, the pore water pressure inside the saturated sand may increase sharply, leading to a rapid decrease in effective stress and the continuous disappearance of the sand’s bearing capacity [4]. In such cases, the pile foundation can offer better stability and support to reduce the risk of upper structure collapse [5]. Knappett and Madabhushi [6] conducted centrifuge test to analyze the development of excess pore water pressure caused by liquefaction in pile-sandy soil foundation and the post-liquefaction reconsolidation settlement of the foundation. Bao et al. [7] analyzed the earthquake performance and post-earthquake consolidation settlement of multi-layered pile foundation buildings on liquefiable ground using finite element software.

Similarly, other researcher such as Xu et al. [1], Rollins and Strand [8], Chen et al. [9], Hussein and El Naggar [10], Chiou et al. [11], Xu et al. [12], Chen et al. [13] have analyzed the dynamic response of pile-sandy soil foundations under earthquake loading using experimental methods (such as shake table tests) or numerical simulation methods. Xu et al. [1] provided a comprehensive introduction to the existing experimental situations. Although these experimental and numerical simulation results are of great significance in guiding the earthquake design of pile-sandy soil foundations, they also reflect that the current research methods for the earthquake behavior of pile-sandy soil foundation mainly rely on experiments and numerical simulation, lacking analytical model. Compared to experiments and numerical simulations, analytical models are simpler and more convenient for engineering application. Additionally, analytical model can verify the accuracy of experiments and numerical simulations to some extent.

This paper proposes a theoretical model for soil consolidation around a pile considering earthquake effects, using the pile-soil strain compatibility assumption. After analyzing and solving the problem using Fourier transforms, the accuracy of the analytical model is verified using numerical simulations. Based on this, the influence of key parameters in the analytical model on the development and distribution of excess pore water pressure is analyzed using graphical methods. Finally, based on the obtained analytical model for soil consolidation around a pile considering earthquake effects, a calculation formula for reconsolidation settlement of pile-soil foundation is proposed, and the effects of pile and soil parameters on pile-soil foundation reconsolidation settlement are analyzed.

2. Mathematical model

2.1 Model background

Figure 1(a) shows two common types of pile layout: rectangular distribution and triangular distribution. Both types can be utilized to extract the pile-soil element illustrated in Figure 1(b). Furthermore, the pile-soil element can be further depicted on a coordinate axis as shown in Figure 1(c). The radius of the pile is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_p} , the influence radius of the pile is denoted as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_e} , and the thickness of the pile-soil foundation is represented as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H} . Since the pile is an impermeable pile, only vertical drainage of the soil is considered, and the vertical permeability coefficient is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} . The compressive modulus of the pile and the soil are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_p} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_s}

, respectively.

Review 979873928567-image1.png
Figure 1. Diagram of pile-soil foundation. (a) Pile layout. (b) Pile-soil element. (c) Mathematical model of pile-soil element

2.2 Governing equation

Under earthquake loading, the equilibrium relationship can be expressed as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \pi \left(r_e^2-r_{p}^2\right)\overline{\sigma }_{s}+\pi r_{p}^2{\overline{\sigma }}_{p}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \pi r_{e}^2\sigma

(1)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{\sigma }}_{s} 

and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{\sigma }}_{p}
denote the radial average total stresses of the soil and pile, respectively, and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma 
represents the loading induced by earthquake.

Based on the assumption of equal strain, one can obtain:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\overline{\sigma }}_{s}-{\overline{u}}_{s}}{E_{s}}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\overline{\sigma }}_{p}}{E_{p}}={\epsilon }_{v}

(2)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{u}}_{s} 

denote the radial average excess pore water pressure of the soil , and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\epsilon }_{v}
is the vertical strain of pile-soil foundation.

Combining Eqs. (1) and (2), the vertical strain rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \partial {\epsilon }_{v}/\partial t 

is expressed as follows:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial {\epsilon }_{v}}{\partial t}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): -\frac{\left(1-\lambda \right)\frac{\partial {\overline{u}}_{s}}{\partial t}-\frac{\partial \sigma }{\partial t}}{E_{s}\left(1-\lambda +\lambda \epsilon \right)}

(3)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \lambda =\pi r_0^2/\left(\pi r_\mbox{e}^2\right) , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \chi =E_{p}/E_{s} .

The rate of change of total stress Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \partial \sigma /\partial t 

is regarded as the rate of excess pore pressure induced by earthquake [14], that is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial \sigma }{\partial t}=\frac{N_{{eq}}{\gamma }^{{'}}z}{N_{l}t_{d}}H\left(t_{d}- t\right)
(4)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_{{eq}} 

related to magnitude, as shown in Table 1 by Seed and Booker [15];  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\gamma }^{{'}}
is effective unit weight of soil;  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_{l}
is number of liquefaction cycles;  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t_{d}
is the duration of the earthquake, and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H\left(t\right)=\left\{ \begin{array}{cc} 0, & t<0\\ 1, & t\geq 0 \end{array}\right.

.

Table 1. The relationship between Magnitude and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_{{eq}} [15]
Magnitude Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_{{eq}}
6.5 8
7 12
8 30


The mass conservation equation for the water phase can be obtained using Darcy’s law by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial {\epsilon }_{v}}{\partial t}=-\frac{k_\mbox{v}}{{\gamma }_{w}}\frac{{\partial }^2{\overline{u}}_{s}}{\partial z^2}
(5)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\gamma }_{w}} 

unit weight of water, 9.8 kN/m3.

Combing Eqs. (3)~(5), one can obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial {\overline{u}}_{s}}{\partial t}=\frac{\alpha k_{v}E_{s}}{{\gamma }_{w}}\frac{{\partial }^2{\overline{u}}_{s}}{\partial z^2}+

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{1-\lambda }\frac{N_{{eq}}{\gamma }^{{'}}z}{N_{l}t_{d}}H\left(t_{d}- t\right)

(6)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha =\left(1-\lambda +\lambda \chi \right)/\left(1- \lambda \right)} 

.

2.3 Solving conditions

The initial excess pore water pressure is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {{\overline{u}}_{s}\Bigg| }_{t=0}=0
(7)


The top surface of soil is pervious, and the bottoms are impervious. The vertical boundary conditions of soil can be expressed as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {{\overline{u}}_{s}\Bigg| }_{z=0}={\frac{\partial {\overline{u}}_{s}}{\partial z}\Bigg| }_{z=H}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0

(8)

2.4 Normalization

For convenience, normalization is generally conducted to simplify problems to eliminate the influence of measured units. In this model, dimensionless variables or parameters are given as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Z=\frac{z}{H},T_{v}=\frac{k_{v}E_{s}t}{{\gamma }_{w}H^2},T_{d}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{k_{v}E_{s}t_{d}}{{\gamma }_{w}H^2},{\overline{U}}_{s}= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\overline{u}}_{s}}{{\gamma }^{{'}}H}.

(9)


Using the dimensionless parameters defined in Eq. (9), Eqs. (6)~(8) in dimensionless form are, respectively, listed below:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial {\overline{U}}_{s}}{\partial T_{v}}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha \frac{{\partial }^2{\overline{U}}_{s}}{\partial Z^2}+ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{1}{1-\lambda }\frac{N_{{eq}}Z}{N_{l}T_{{vd}}}H\left(T_{d}- T_{v}\right)

(10)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {{\overline{U}}_{s}\Bigg| }_{T_{v}=0}=0
(11)

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {{\overline{U}}_{s}\Bigg| }_{Z=0}={\frac{\partial {\overline{U}}_{s}}{\partial Z}\Bigg| }_{Z=1}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0

(12)

2.5 Solution of dimensionless excess pore water pressure

Applying the finite Fourier transform to Eqs. (10) and (11) with the boundary conditions of Eq. (12), one can obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial {\tilde{\overline{U}}}_{s}}{\partial T_{v}}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): -\alpha M_m^2{\tilde{\overline{U}}}_{s}+\frac{1}{1-\lambda }\frac{{\left(-1\right)}^{m-1}}{M_m^2}\frac{N_{{eq}}}{N_{l}T_{d}}H\left(T_{d}- T_{v}\right)

(13)

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {{\overline{U}}_{s}\Bigg| }_{T_{v}=0}=0
(14)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{\overline{U}}}_{s}={\int }_0^1{\overline{U}}_{s}\sin\left(M_mZ\right){d}Z , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): M_m=\frac{2m-1}{2}\pi .

Eq. (13) is a first-order differential equation that can be solved using the initial condition of Eq. (14). The solution is given by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{\overline{U}}}_{s}=\left\{ \begin{array}{lc} \displaystyle\frac{N_{{eq}}}{N_{l}T_{d}}\displaystyle\frac{{\left(-1\right)}^{m-1}}{\left(1-\lambda \right)\alpha M_m^4}\left(1-{e}^{-\alpha M_m^2T_{v}}\right), & T_{v}\leq T_{d}\\ \displaystyle\frac{N_{{eq}}}{N_{l}T_{d}}\displaystyle\frac{{\left(-1\right)}^{m-1}}{\left(1-\lambda \right)\alpha M_m^4}\left[{e}^{\alpha M_m^2\left(T_{d}-T_{v}\right)}-{e}^{-\alpha M_m^2T_{v}}\right], & T_{v}>T_{d}\end{array}\right.

,

(15)


The general solution of this model can be obtained by applying the inverse finite Fourier transform to Eq. (14), that is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{U}}_{s}=\left\{ \begin{array}{lc} \displaystyle 2\sum_{m=1}^{\infty }\displaystyle\frac{N_{{eq}}}{N_{l}T_{d}}\displaystyle\frac{{\left(-1\right)}^{m-1}}{\left(1-\lambda \right)\alpha M_m^4}\left(1-{e}^{-\alpha M_m^2T_{v}}\right)\sin\left(M_mZ\right), & T_{v}\leq T_{d}\\ \displaystyle2\sum_{m=1}^{\infty }\frac{N_{{eq}}}{N_{l}T_{d}}\frac{{\left(-1\right)}^{m-1}}{\left(1-\lambda \right)\alpha M_m^4}\left[{e}^{\alpha M_m^2\left(T_{d}-T_{v}\right)}-{e}^{-\alpha M_m^2T_{v}}\right]\sin\left(M_mZ\right), & T_{v}>T_{d} \end{array}\right.
(16)

4. Numerical verification

The finite element software COMSOL Multiphysics is used to solve the model numerically to verify the correctness of the analytical model. The parameters used in numerical simulation are given as follows: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r_{p}=0.25 

m,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r_{e}=1\mbox{ m}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H=3.75\mbox{ m} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): k_{v}={10}^{-4}\mbox{ m}/\mbox{s} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): E_{s}=3\mbox{ MPa} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): E_{p}=150\mbox{ MPa} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): t_{d}=30\mbox{ s} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_{{eq}}/N_1=2 . Figure 2 shows the distribution of excess pore water pressure around a pile simulated by COMSOL Multiphysics. The legend unit in Figure 2 is “Pa”. The data of excess pore water pressure calculated by COMSOL Multiphysics are extracted and compared with the analytical results, as shown in Figure 3. It is found that the numerical results coincide perfectly with the analytical results, which proves the correctness of the analytical solution.

Review 979873928567-image43-c.png
Figure 2. The distribution of excess pore water pressure around a pile by COMSOL Multiphysics (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t= 30} 
s)


Review 979873928567-image44.png
Figure 3. Comparison of numerical results and analytical results


5. Excess pore water pressure during earthquake

Figure 4 shows the distribution and development of excess pore water pressure in soil around a pile under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n } 

values (or influence radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_e}

). Firstly, it can be seen from Figure 4(a) that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n } 

value mainly has an obvious influence on the excess pore water pressure at the bottom of soil. Specifically, the excess pore water pressure increases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  n }
value. Figure 4(b) shows the development of excess pore water pressure at the bottom of soil. It can be found that the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  n }
value has little influence on the excess pore water pressure in the earthquake stage, but has obvious influence on the excess pore water pressure in the post-earthquake stage. Specifically, the dissipation rate of excess pore water pressure is accelerated with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  n }
value.
Review 979873928567-image45.png
Review 979873928567-image46.png
(a) (b)
Figure 4. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n } 
value on excess pore water pressure. (a) Distribution along depth. (b) Development over time


Figure 5 shows the distribution and development of excess pore water pressure in soil around a pile under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } 

values (or the thickness of the pile-soil foundation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  H }

). It can be seen from Figure 5(a) that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } 

value has a significant influence on the excess pore water pressure along the depth direction. At the same depth, the excess pore water pressure increases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \eta }
value. However, it is noted that the increase of excess pore water pressure at the same depth decreases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \eta }
value. Figure 5(b) shows the development of excess pore water pressure at the bottom of soil. It can be found that the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \eta }
value has a significant effect on the excess pore water pressure during and after the earthquake. Firstly, it can be found that the maximum excess pore water pressure generated by earthquake increases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \eta }
value. Then, the growth rate of excess pore water pressure during the earthquake and the dissipation rate of excess pore water pressure after earthquake increase with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \eta }
value.
Review 979873928567-image47.png
Review 979873928567-image48.png
(a) (b)
Figure 5. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } 
value on excess pore water pressure. (a) Distribution along depth. (b) Development over time


Figure 6 shows the distribution and development of excess pore water pressure in soil around a pile under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi } 

values (or the compression moduli of the pile Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_p}

). It can be seen from Figure 6(a) that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi } 

value has a significant influence on the excess pore water pressure along the depth direction. At the same depth, the excess pore water pressure decreases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \chi }
value. Moreover, the increase of excess pore water pressure at the same depth decreases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \chi }
value. Figure 6(b) shows the development of excess pore water pressure at the bottom of soil. It can be found that the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \chi }
value has a significant effect on the excess pore water pressure during and after the earthquake. Firstly, it can be found that the maximum excess pore water pressure generated by earthquake increases with the decrease of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \chi }
value. Then, the dissipation rate of excess pore water pressure after earthquake increase with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \chi }
value.
Review 979873928567-image49.png
Review 979873928567-image50.png
(a) (b)
Figure 6. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi } 
value on excess pore water pressure. (a) Distribution along depth. (b) Development over time


Figure 7 shows the distribution and development of excess pore water pressure in soil around a pile under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} . It can be seen from Figure 7(a): when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} 

is small, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v}
mainly has an obvious influence on the excess pore water pressure at the bottom of soil; when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v}
is high, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v}
value has a significant influence on the excess pore water pressure along the depth direction. The effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v}
on the excess pore water pressure shows that the excess pore water pressure decreases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v}

. Figure 7(b) shows the development of excess pore water pressure at the bottom of soil. It can be found that the maximum excess pore water pressure generated by earthquake increases with the decrease of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} 

value. Then, the growth rate of excess pore water pressure during the earthquake and the dissipation rate of excess pore water pressure after earthquake increase with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v}
value.
Review 979873928567-image51.png
Review 979873928567-image52.png
(a) (b)
Figure 7. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} 
on excess pore water pressure. (a) Distribution along depth. (b) Development over time


Figure 8 shows the distribution and development of excess pore water pressure in soil around a pile under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}/N_1} . It can be seen from Figure 8(a) that the excess pore water pressure increases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}/N_1} . Figure 8(b) shows the development of excess pore water pressure at the bottom of soil. It can be found that the maximum excess pore water pressure generated by earthquake increases with the decrease of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}/N_1} 

value. Then, the growth rate of excess pore water pressure during the earthquake and the dissipation rate of excess pore water pressure after earthquake increase with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}/N_1}
value.
Review 979873928567-image53.png
Review 979873928567-image54.png
(a) (b)
Figure 8. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}/N_1} 
on excess pore water pressure. (a) Distribution along depth. (b) Development over time


Figure 9 shows the distribution and development of excess pore water pressure in soil around a pile under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} 

stands for the duration of an earthquake. If the frequency of earthquakes remains constant, a shorter duration implies a higher frequency. It can be seen from Figure 9(a) that the excess pore water pressure decreases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d}

. Figure 9(b) shows the development of excess pore water pressure at the bottom of soil. It can be found that the maximum excess pore water pressure generated by earthquake increases with the decrease of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} 

value. Then, it is noted that the effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d}
on the excess pore water pressure during the earthquake is obviously greater than that after the earthquake.
Review 979873928567-image55.png
Review 979873928567-image56.png
(a) (b)
Figure 9. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} 
on excess pore water pressure. (a) Distribution along depth. (b) Development over time

6. Reconsolidation settlement of goundation after earthquake

Due to the assumption of equal strain, the reconsolidation settlement of pipe-soil foundation can be calculated by the reconsolidation settlement of soil around the pile. The reconsolidation settlement of soil around the pile can be expressed as follows [4]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{c} \displaystyle S\left(T_{v}\right)=\frac{1}{E_{s}}{\int }_{T_{{vd}}}^{T_{v}}{\int }_0^H\frac{\partial {\gamma }_{w}H{\overline{U}}_{s}}{\partial t}{d}z{d}t\\ \displaystyle =\frac{{\gamma }_{w}H}{E_{s}}\sum_{m=1}^{\infty }\frac{2N_{{eq}}}{N_{l}T_{{vd}}}\frac{{\left(-1\right)}^{m-1}\left[{e}^{\alpha M_m^2\left(T_{{vd}}-T_{v}\right)}-{e}^{-\alpha M_m^2T_{v}}+{e}^{-\alpha M_m^2T_{{vd}}}-1\right]}{\left(1-\lambda \right)\alpha M_m^5} \end{array}
(16)


Figure 10 shows the reconsolidation settlement of foundation after earthquake under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} 

values. It can be found that reconsolidation settlement is mainly divided into two stages: one is the rapid settlement stage, in which reconsolidation settlement develops rapidly; then there is the slow settlement stage in which reconsolidation settlement develops slowly. It can be seen from Figure 10 that the larger the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
value is, the slower the settlement rate in the early stage and the faster the settlement rate in the later stage. In general, the total settlement of foundation increases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
value.
Review 979873928567-image58.png
Figure 10. Effect of n on reconsolidation settlement of foundation after earthquake


Figure 11 shows the reconsolidation settlement of foundation after earthquake under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta} 

 values. It can be seen that the total settlement of the foundation increases significantly as the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta}
increases.
Review 979873928567-image59.png
Figure 11. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta} 
on reconsolidation settlement of foundation after earthquake


Figure 12 shows the reconsolidation settlement of foundation after earthquake under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi } 

values. It can be seen from Figure 12 that the larger the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \chi }
value is, the faster the settlement rate in the early stage and the slower the settlement rate in the later stage. In general, the total settlement of foundation decreases with the increase of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \chi }
value.
Review 979873928567-image60.png
Figure 12. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi } 
on reconsolidation settlement of foundation after earthquake


Figure 13 shows the reconsolidation settlement of foundation after earthquake under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} 

values. It can be seen that the total settlement of the foundation decreases significantly as the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v}
increases.
Review 979873928567-image61.png
Figure 13. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} 
on reconsolidation settlement of foundation after earthquake


Figure 14 shows the reconsolidation settlement of foundation after earthquake under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}} /Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 

values. It can be seen that the total settlement of the foundation increases significantly as the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}}

/Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 

increases.
Review 979873928567-image62.png
Figure 14. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}}

/Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 

on reconsolidation settlement of foundation after earthquake


Figure 15 shows the reconsolidation settlement of foundation after earthquake under different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} 

values. It can be seen that the total settlement of the foundation decreases significantly as the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d}
increases.
Review 979873928567-image63.png
Figure 15. Effect of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} 
on reconsolidation settlement of foundation after earthquake

7. Conclusions

In this paper, an analytical solution to soil consolidation around a pile considering earthquake effect is proposed, and the accuracy of the analytical model is verified by finite element software. Finally, the effects of soil parameters, pile parameters and earthquake parameters in the analytical model on the excess pore water pressure and reconsolidation settlement after earthquakes are analyzed and discussed. The main conclusions are as follows:

(1) Distribution of excess pore water pressure in soil around the pile: The excess pore water pressure increased with depth, and is positively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}}

/Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 , and negatively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} .

(2) Development of excess pore water pressure in soil around the pile: The values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}} /Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} 

affect the excess pore water pressure during and after the earthquake, and the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
only obvious affects the excess pore water pressure after the earthquake; The growth rate of excess pore water pressure during the earthquake is positively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}} /Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 , and negatively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d}

. The dissipation rate of excess pore water pressure during the earthquake is positively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}} /Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} , and negatively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} .

(3) Reconsolidation settlement of foundation after earthquake: Reconsolidation settlement is mainly divided into two stages: rapid settlement stage and slow settlement stage; The total settlement of foundation is positively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta } , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{eq}} /Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N_1 , and negatively correlated with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_v} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_d} .

Data Availability Statement

All data, models, and code generated or used during the study appear in the submitted article.

Acknowledgements

This work had been supported by the Guangdong Key Areas R&D Program Projects, China (Grant number: 2020B0101130005), Research Project of Guangdong Power Grid Co. (Grant number: 031000QQ00220012).

References


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[8] Rollins K.M., Strand S.R. Downdrag forces due to liquefaction surrounding a pile. Proc. 8th National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Paper No. 1646, San Francisco, CA, April 18-22 2006.

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Published on 09/06/23
Accepted on 22/05/23
Submitted on 03/05/23

Volume 39, Issue 2, 2023
DOI: 10.23967/j.rimni.2023.05.007
Licence: CC BY-NC-SA license

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