Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 495253, 13 pages

http://dx.doi.org/10.1155/2015/495253

## Dynamic Response of an Inhomogeneous Viscoelastic Pile in a Multilayered Soil to Transient Axial Loading

^{1}Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China^{2}College of Urban Construction, Zhejiang Shuren University, Hangzhou 310015, China^{3}Key Laboratory of Soft Soils and Geoenvironmental Engineering, Ministry of Education, Zhejiang University, Hangzhou 310027, China^{4}Department of Civil Engineering, Zhejiang Ocean University, Zhoushan 316004, China^{5}School of Civil Engineering and Architecture, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received 15 October 2014; Accepted 15 April 2015

Academic Editor: Igor Andrianov

Copyright © 2015 Zhiqing Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A quasi-analytical solution is developed in this paper to investigate the mechanism of one-dimensional longitudinal wave propagating in inhomogeneous viscoelastic pile embedded in layered soil and subjected to a transient axial loading. At first, the pile-soil system is subdivided into several layers along the depth direction in consideration of the variation of cross-sectional acoustic impedance of the pile or differences in soil properties. Then, the dynamic governing equation of arbitrary soil layer is established in cylindrical coordinates and arbitrary viscoelastic pile segment is modeled using a single Voigt model. By using the Laplace transform and boundary conditions of the pile-soil system, the vertical impedance at the top of arbitrary pile segment is defined in a closed form in the frequency domain. Then by utilizing the method of recursion typically used in the Transfer Function technique, the vertical impedance at the pile top can be derived in the frequency domain and the velocity response of an inhomogeneous viscoelastic pile subjected to a semi-sine wave exciting force is obtained in a semi-analytical form in the time domain. Selected numerical results are obtained to study the mechanism of longitudinal wave propagating in a pile with a single defect or double defects.

#### 1. Introduction

Pile vibration theory can provide valuable guidance for both the dynamic design of embedded foundations and dynamic nondestructive integrity testing of piles. For the dynamic design of pile foundations, the most concerned problem is the study of vibration characteristics of embedded piles in the low frequency range, and accordingly the dynamic reaction from the surrounding soil is an essential component in the relevant study. In light of this, many soil models, such as simplified continuum model [1–3], continuum model [4–6], and finite element model [7], have been developed to investigate the pile-soil dynamic interaction. For the dynamic nondestructive integrity testing of piles, low strain integrity testing technique has received wide application in assessing the construction quality of piles due to its relatively low-cost and simplicity. This testing technique is mainly based on the stress wave propagation theory through a bar. As a result, the theoretical study of a pile embedded in the soil and subjected to a dynamic vertical load under small deformation condition is increasingly becoming more important. For instance, Davis and Dunn [8] firstly proposed the mechanical admittance (or mobility function) method to determine the length and cross-sectional area of a pile by applying a steady-state harmonic excitation at a specified set of frequencies on the pile top. In subsequent studies, Davis and Robertson [9] extended the mechanical admittance method to determine the pile head stiffness. Accounting for the convenience of transient excitation, Higgs [10] modified the mechanical admittance method by applying a vertical impact load on the pile top instead of steady-state harmonic excitation. After that, Lin et al. [11] introduced the impact-echo method to assess the integrity of piles by using the amplitude Fourier spectrum of the displacement record at the pile head instead of the mobility function. Watson et al. [12] developed a wavelet transform signal processing method instead of the conventional Fourier based methods to locate the position of the pile tip. It is noted that, in the low strain integrity pile testing where the hammer is relatively small compared to the pile dimension, Rayleigh and shear waves will radiate from the impact loading and the effects of three-dimensional (3D) waves on the near field responses are obvious. Subsequently, several researchers conducted relevant studies and proposed several methods to diminish the effects of 3D waves on the dynamic response of the pile top. Liao and Roesset [13, 14] investigated the influence of 3D waves on the dynamic response at the top of intact and defective pile by comparing one-dimensional (1D) wave theory and 3D axisymmetric finite element simulation results. It is shown from their studies that 3D effects are mainly influenced by the frequency and are more strongly manifested at high frequencies. Chow et al. [15] found that the velocity response curves resemble that of a pile with a defect near the pile head when considering 3D effects and further proposed that the potential source of error can be removed by maintaining a distance between hammer and receiver that is greater than 50% of the pile radius. Chai et al. [16] found that when the ratio of the characteristic length of an impact pulse to the pile radius is large enough, the components of Rayleigh waves in the wave field at the pile top are diminished. In this study, Chai et al. still proposed that the receiver should be placed at positions between and ( = pile radius) from the pile axis to diminish the influence of the multireflections. Lu et al. [17] investigated the 3D characteristics of wave propagation in pipe-pile using elastodynamic finite integration technique and found that the interferences of Rayleigh waves are weakest at an angle of 90° from where hammer hits. Furthermore, for the drilled piles with high slenderness ratio, it is difficult to detect the pile length and deep flaw from the traditional low strain pile integrity testing technique due to insufficient impact energy, testing signal decay, and soil-pile interaction. To solve this problem, Ni et al. [18] adjusted the testing devices for acquiring a lower frequency signal and developed a new numerical signal process method to enhance the reflection signals from the pile tip. It is also shown from the experimental results that the testing signal identification abilities can be improved by the modified method.

Most of the previous studies on the low strain pile integrity testing did not consider the effect of pile material damping, abrupt variation of surrounding soil properties, and multidefects in pile on the dynamic response. It is worth noting that the material damping indeed exists in a pile, the properties of the surrounding soil may change greatly in certain embedment depth, and a pile may contain several defects. Wang et al. [19] investigated the vertical dynamic response of an inhomogeneous viscoelastic pile and analyzed the effect of pile material damping and soil properties on the mechanical admittance and velocity response of the pile top. However, in this study, the surrounding soil reaction on the pile is approximately simulated by a general Voigt model which cannot veritably and accurately reflect the pile-soil interaction. Therefore, the objective of this paper is to develop a practical solution to evaluate the theoretical capabilities of the nondestructive dynamic response method in detecting the existence and location of single or double defects in a viscoelastic pile embedded in a multilayered soil. Using the solution developed, a parametric study has been undertaken to investigate the mechanism of one-dimensional elastic longitudinal wave propagating in a defective pile. Finally, the theoretical model developed in the present paper is validated by comparison of the theoretical fitted curve and field measured curve of velocity response.

#### 2. Formulation of the Problem

##### 2.1. Geometry and Basic Assumption

The system examined is an inhomogeneous viscoelastic pile embedded in a multilayered soil and the geometric model is shown in Figure 1. To portray the variation of cross-sectional acoustic impedance (the product of density, cross-sectional area, and the one-dimensional elastic longitudinal wave velocity) of a pile or differences in soil properties, the pile-soil system is subdivided into a total of segments (layers) numbered by from pile tip to pile head. The thickness of the th soil layer is equal to the length of the th pile segment and is denoted by . In order to derive an analytical or quasi-analytical solution for this problem, the assumptions are made as follows: the surrounding soil is a linearly viscoelastic layer and the pile is vertical, elastic, and circular in cross-section. The pile and soil layer properties are assumed to be homogeneous within each segment or layer, respectively, but may change from segment to segment or layer to layer; the pile-soil system is subjected to small deformations and strains during the vibration; the pile has a perfect contact with the surrounding soil during the vibration; the free surface of the soil has no normal and shear stresses and the soil is infinite in the radial direction; the soil at the base of the pile is modeled using a spring with elastic constant and a dashpot with damping coefficient ; the contact traction acting at the th soil layer due to th and th soil layers is treated as the distributed Winkler subgrade model independent of the radial distance (see Figure 2).