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An Analytical Solution for Wave Propagation in a Pipe Pile with Multiple Defects

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Abstract

An analytical solution is developed in this paper to conduct the low-strain integrity testing for a pipe pile with multiple defects. The derived solution allows simulating the pipe pile as a three-dimensional model by considering the wave propagation in the vertical, circumferential and radial directions. Analytical solutions of the pile are obtained by the Laplace transform and separation of variables. Accordingly, time-domain responses of the solution are deduced by the inverse Fourier transform numerically. The solution is validated against the published solutions for an intact pile and a pile with a single defect. Parametric studies are conducted to identify and characterize the velocity responses on the top of pipe piles with multiple defects. Numerical results suggest that the reflected waves generated by the deep defects are affected by the secondary reflections from the shallow defects. A new detecting method is proposed to decrease the influence of high-frequency interferences and to predict the defective depth, which suggests putting the receiver at the point of \(90^{\circ }\) along the circumferential direction.

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Abbreviations

\(r, \theta , z\)::

Radial, circumferential and vertical directions in the cylindrical coordinate system

k::

Number of the pile segments

h::

Wall thickness

\(R_{O}\)::

Outer radius of pile

\(R_{I}\)::

Inner radius of pile

E::

Young’s modulus of pile

Subscript i::

Number of pile segments

\(R_{0}\)::

Average radius of pile

u::

Soil displacement

t::

Time

\(f_{s}\)::

Soil resistance

k::

Elastic coefficient of soil

c::

Damping coefficient of soil

Subscripts O, I and p::

Corresponding coefficients of outer soil, inner soil and pile tip

T::

Time period

Q::

Peak value

\(\rho \) ::

Mass density of pile

\(\lambda , G\)::

Lame’s constants of pile

\(\nu \)::

Poisson’s ratio

\(\eta \)::

Damping coefficient of pile

H::

Pile length

\(\delta ()\)::

The Dirac delta function

\(H_{i}\)::

Depth of the i-th segment

\(U(z, r, {\theta }, s)\)::

Laplace transform of \(u(z, r, \theta , t)\)

\(Z(z), \varPhi (\theta ), R(r), \beta _{im}^2 , \zeta _{inm}, \alpha _{inm}^2 , \chi _{nm}, R_{n}(r),R_{l}(r), \bar{{h}}_i , \bar{{g}}_i:\) :

Temporary variables \(M, N, A, B, C_{inm}, D_{inm}, C_{i+1nm}, D_{i+1nm}, C_{knm}, D_{knm}\): Temporary constants

Jm(), Ym()::

The first and second kinds of Bessel function of order m

F(s)::

Laplace transformation of f(t) \(\left[ {T_{inm} } \right] , \left[ {\bar{{T}}_i } \right] :\) Coefficient transfer matrix

\(U(z,r, {\theta }, \hbox {i}\omega \))::

One-sided Fourier transform of \(U(z, r, \theta , s)\)

\(\omega \)::

Angular frequency

\(v_{i}(z, r, {\theta }, t)\)::

Velocity response of the i-th pile shaft in the time domain

\({[} {\bar{T}}_{k} {]}:\) :

Unit matrix

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Acknowledgements

This work was supported by the National Key Research and Development Program of China with Grant Number 2016YFE0200100, the National Natural Science Foundation of China with Grant Numbers 51622803 and 51708064, and the Fundamental Research Funds for the Central Universities with Grant Numbers 106112017CDJXY200002 and 106112016CDJXZ208821.

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Authors and Affiliations

  1. Key Laboratory of New Technology for Construction of Cities in Mountain Area, College of Civil Engineering, Chongqing University, Chongqing, 400045, China

    Xuanming Ding, Lubao Luan & Hang Zhou

  2. School of Civil Engineering, Fujian University of Technology, Fuzhou, 350118, China

    Changjie Zheng

  3. Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education, College of Civil Engineering and Architecture, Guangxi University, Nanning, 530004, China

    Guoxiong Mei

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  1. Xuanming Ding
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Correspondence to Xuanming Ding.

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Ding, X., Luan, L., Zheng, C. et al. An Analytical Solution for Wave Propagation in a Pipe Pile with Multiple Defects. Acta Mech. Solida Sin. 33, 251–267 (2020). https://doi.org/10.1007/s10338-019-00123-5

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