J. Cent. South Univ. Technol. (2010) 17: 1370-1375
DOI: 10.1007/s11771-010-0644-8
Comparative analysis of seismic response characteristics of pile-supported structure
KONG De-sen(孔德森)1, 2, 3, LI Chun-jie(李纯洁)1, ZHANG Wei-wei(张伟伟)1, MENG Qing-hui(孟庆辉)1
1. School of Civil and Architectural Engineering, Shandong University of Science and Technology,
Qingdao 266510, China;
2. Shandong Provincial Key Laboratory of Civil Engineering Disaster Prevention and Mitigation,
Shandong University of Science and Technology, Qingdao 266510, China;
3. School of Civil Engineering, Tongji University, Shanghai 200092, China
? Central South University Press and Springer-Verlag Berlin Heidelberg 2010
Abstract: In order to analyze the seismic response characteristics of pile-supported structure, a computational model considering pile-soil-structure interaction effect was established by finite element method. Then, numerical implementation was made in time domain. At the same time, a simplified approximation for seismic response analysis of pile-soil-structure system was briefly presented. Furthermore, comparative study was performed for an engineering example. Through comparative analysis, it is shown that the results obtained by the simplified method well agree with those achieved by the finite element method. These results show that spectrum characteristics and intensity of input earthquakes are two important factors that can notablely influence the seismic response characteristics of superstructure. When the input ground motion acceleration amplitude gradually increases from 1 to 4 m/s2, the acceleration of pier top will increase, but it will not be simply proportional to the increase of input acceleration amplitude.
Key words: pile-supported structure; pile-soil interaction; seismic response; spectrum characteristics; finite element method
1 Introduction
The study on earthquake-resistant performance of pile-soil-structure interaction system is a relatively complicated and primarily important issue in civil engineering practice because many factors such as pile-soil interaction effect, infinite ground base, superstructure and other coupling effects are involved in the system. DOU [1], KUCUKARSLAN et al [2] and ZHAO et al [3] all pointed out that the dynamic interaction characteristics and the interaction mechanism of pile-soil-structure system were not recognized definitely. So far, there have not been systematic analytic method and sophisticated computational model in engineering design.
At present, substructure method has become a conventional method and been widely used. In this conventional earthquake-resistance design method for pile-supported structures, rigid ground assumption is widely adopted. Superstructure, pile foundation and ground base are all computed individually, and these three parts are associated with each other only by load transfer. Otherwise the coordinations of stiffness and deformation of these three parts are not involved in the pile-soil-structure interaction system. The dynamic interaction effects of pile-soil-structure system are not considered. Many theoretical and experimental investigations at home and abroad indicate that the earthquake-resistant design of superstructure based on rigid ground assumption is not always secure. Therefore, the dynamic interaction effect of pile-soil-structure system must be considered adequately in the design.
In order to analyze the seismic response characteristics of pile-supported structure, a computational model that can consider the pile-soil- structure interaction effect was established by the finite element method. Then, numerical implementation was made in time domain. At the same time, a simplified method for seismic response analysis of pile-soil- structure system was also presented. Furthermore, a comparative study was performed for an engineering example. Some results were obtained respectively by the finite element method and the simplified method.
2 Finite element method
The computational model of pile-soil-structure interaction system for computation by finite element method is illustrated in Fig.1.
Under lateral seismic loads, pile foundation bears a relatively great lateral shearing force and the lateral flexure is the main deformation behavior of pile foundation. In order to simulate the lateral flexure deformation of pile foundation accurately, the incompatibility element, H11, was adopted in the analysis. And the displacement interpolating function of H11 can be formulated as follows [4-6]:
(1)
(2)
(3)
where u, v and w are the displacement components of incompatibility element; uq is the compatibility displacement vector; uλ is the incompatibility displacement vector; Ni and Pi are respectively the formal functions of compatibility displacement and incompatibility displacement; ui, vi and wi are the compatibility displacement components of node i; αi, βi and γi are the incompatibility displacement components of node i; ξ, η and ζ are the coordinate variables of element; 
and i=1, 2, …, 8.
As a result of the introduction of incompatibility term, the quadratic term of the interpolating function of three-dimensional eight-node-element becomes integrated. Accordingly, the computational accuracy of the element can be enhanced. The computational accuracy of three-dimensional eight-node incompatibility element is equivalent to that of three-dimensional twenty-node compatibility element. However, the node number of the incompatibility element is only 2/5 as large of that of twenty-node compatibility element. Therefore, the computational efficiency is greatly increased [7-8].
Soils were simulated with ordinary eight-node isoparametric element. Furthermore, the single yield surface constitutive model [9-11] was adopted to analyze the elasto-plastic behavior of soils.
The pile-soil interface is one of the most important factors that influence the seismic response characteristics of pile-soil-structure interaction system. In this work, the dynamic interface element was used to simulate the behavior of pile-soil interface. The relationship between
dynamic shearing rigidity and dynamic shearing strain is nonlinear and can be formulated as follows:
(4)
where
and
are respectively the maximum dynamic shearing modulus and damage shear stress; 
and 
are respectively the normal stress and the friction angle of interface element; m and c are empirical coefficients; and Kj and γ are respectively the dynamic shearing rigidity and the dynamic shearing strain of interface element. These parameters can be determined on the basis of engineering experience. Under normal conditions, δ=23?, c=22, and m=2. The damping ratio of interface element takes the form as follows:
(5)
where 
is the maximum damping ratio with a value of 20%.
Fig.1 Computational model of pile-soil-structure system
Limited by many negative factors during the process of numerical computation, a certain finite area around pile foundation is often cut from semi-infinite subgrade when analyzing the interaction characteristics of pile-soil-structure system by finite element method. In order to reasonably simulate the radiation damping of the system during earthquake, an artificial boundary was set at the computational boundary [12-13]. In this work, a fixed boundary and a viscous boundary were respectively specified at the bottom and on the bilateral sides of the computational boundary as illustrated in Fig.1.
3 Simplified analysis method
Based on the principle of substructure method, the pile-soil-structure interaction system illustrated in Fig.1 can be partitioned into three parts, i.e. the bridge pier structure, the rigid bearing platform and the pile group foundation [14-15]. The governing equation of the bridge pier structure is expressed as follows:
(6)
or
(7)
where Ms, Cs and Ks are respectively the mass matrix, the damping matrix and the stiffness matrix of the bridge pier structure; xs is the displacement vector of bridge pier relative to that of bearing platform; xf is the displacement vector of bearing platform relative to that of free soil field; xg is the displacement vector of free soil field; and T is the quasi-static transition matrix and can be expressed as
(8)
where hi (i=1, 2, …, n) is the height of the degree of freedom i above the bottom surface of bearing platform.
The governing equation of the rigid bearing platform takes the form:
(9)
where
, 
(10)
where mf and If are respectively the mass and the mass moment of inertia matrices of rigid bearing platform; and Fc and Mc are the counterforce and the countermoment of pile group acting on the base surface of rigid bearing platform, respectively.
Substituting Eq.(7) into Eq.(9) gives the following formulation:
(11)
where x is supposed to be the displacement vector of bridge pier structure relative to that of free soil field and x=xs+Txf.
Solving Eqs.(6) and (11), the governing equation of bridge pier structure-rigid bearing platform system is obtained:
(12)
In frequency domain, the relationship between displacement and counterforce of pile group is formulated as follows:
(13)
where 
and 
are respectively the Fourier transformations of Ff and xf; and 
is the dynamic impedance of pile group, which takes the form
(14)
The dynamic response characteristics of pile group-soil-structure system can be obtained by solving Eqs.(12) and (13).
4 Illustrative example and comparative study
In order to verify the compatibility of the finite element method and the simplified method presented in this work, an engineering example shown in Fig.2 was chosen for the comparative analysis. The layout of a 2×3 pile group used in this example was illustrated by cross section C-C and the pile diameter was 1.2 m. In the analysis, three different ground acceleration records (the El-Centro earthquake, Tangshan earthquake and an artificial earthquake) were chosen as the input ground motions.
Firstly, the effect of spectrum characteristics of input earthquakes on the seismic response behavior of superstructure was investigated by two methods presented above. During the analysis, the peak values of three different ground acceleration records were all chosen as 2 m/s2. On the one hand, the finite element method was used to study the seismic response of pile-supported structures under the conditions that three different earthquakes given above were respectively inputted into the system. On the other hand, when the peak value of ground acceleration record of El-Centro earthquake was 2 m/s2, the simplified method was adopted to analyze the seismic response characteristics of pile-soil-structure system. Through computation and comparative analysis, the lateral acceleration amplification factors, the peak value of lateral displacements and the acceleration response spectra of typical nodes are obtained and shown in Figs.3-4.
It is indicated from Figs.3-4 that the results gained by the simplified method well agree with the computational results achieved by the finite element method. Because the predominant periods of these three earthquakes are all smaller than the natural vibration period of pile-soil-structure system, the differences of the lateral acceleration amplification factors and the peak lateral relative displacements of the pile-soil-structure system under the three different earthquakes are not obvious. However, the differences between the acceleration response spectra at the typical nodes under these three different input earthquakes are relatively great. Compared with the spectrum characteristics of Tangshan earthquake, the magnification of pier top acceleration spectrum increases slightly and the peak value shifts to the lower-frequency range, with low-frequency components being more abundant. But for El-Centro earthquake, the magnification of pier top acceleration spectrum and the peak value are much closer to the lower-frequency range compared with those for Tangshan earthquake. Furthermore, the low- frequency components of acceleration spectra of the top of the rigid bearing platform are much smaller.
Fig.2 Illustrative example used for analysis (Unit: m)
Fig.3 Distributions of lateral acceleration amplification factor (a) and peak lateral relative displacement (b) for different earthquakes
Fig.4 Acceleration response spectra at typical nodes of structure for different earthquakes: (a) For bridge pier top; (b) For bearing platform top
Next, the effect of earthquake intensity on the seismic response characteristics of pile-supported structure was studied respectively by the simplified method and the finite element method. In the analysis, the ground acceleration record of El-Centro earthquake was inputted into the pile-soil-structure system and the maximum acceleration amplitudes were respectively assumed to be 1, 2 and 4 m/s2. The seismic response characteristics of the pile-soil-structure system under these three different earthquake intensities were all computed by the finite element method. For comparison, the simplified method was used for the case that the ground acceleration amplitude was 4 m/s2. The lateral response spectra of acceleration and displacement of the bridge pier top achieved by the above two methods are shown in Figs.5-6, respectively. The bending moments and shear forces at typical sections of structure under different amplitudes of input ground motion are listed in Table 1, where SM denotes the results obtained by the simplified method.
It is shown from Figs.5-6 and Table 1 that the results obtained by the simplified method are in agreement with the computational results achieved by the finite element method. In addition, the effects of the amplitude of input ground motion on the acceleration, the displacement and the internal forces of pile-soil- structure system are remarkable. The acceleration, displacement and internal force of bridge pier top all tend to increase with the increase of the amplitude of input ground motion. But the increases of these parameters are not monotonic and proportional. Furthermore, the displacement time history exhibits a phenomenon of excursion when the amplitude of input ground motion is 4 m/s2, which is the result of accumulation of plastic deformation and generation of permanent deformation.
Fig.5 Lateral response spectra of acceleration of pier top under different amplitudes of input ground motion
Fig.6 Lateral response spectra of displacement of pier top under different amplitudes of input ground motion
Table 1 Bending moments and shear forces at typical sections under different amplitudes of input ground motion
5 Conclusions
(1) The seismic response characteristics of pile-soil- structure interaction system are comparatively analyzed by the finite element method and the simplified method.
(2) Considering the interaction effect, a computational model and computation procedures for pile-supported structures are established by the finite element method. Through comparative analysis, it is shown that the computational model and computation procedures are pretty good.
(3) The spectrum characteristics and the intensity of input earthquakes are two important factors that can notablely influence the seismic response behavior of superstructure. When the input ground motion acceleration amplitude gradually increases from 1 to 4 m/s2, the acceleration of pier top increases, but it is not simply proportional to the increase of input acceleration amplitude.
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(Edited by CHEN Wei-ping)
Foundation item: Project(Y2007F48) supported by the Natural Science Foundation of Shandong Province, China; Project(SDTS20080422) supported by the Specialized Development Foundation for Taishan Scholars of Shandong Province, China; Project(SDVS20090525) supported by the Specialized Foundation for Domestic Visiting Scholars of Shandong Province, China
Received date: 2010-03-16; Accepted date: 2010-06-30
Corresponding author: KONG De-sen, PhD, Associate Professor; Tel: +86-532-86057631; E-mail: dskong828@163.com
