J. Cent. South Univ. Technol. (2010) 17: 642-647
DOI: 10.1007/s11771-010-0534-0 
Distribution of acceleration and empirical formula for
calculating maximum acceleration of rockfill dams
ZHOU Hui(周晖)1, 2, LI Jun-jie(李俊杰)1, 2, KANG Fei(康飞)1, 2
1. Institute of Hydraulic and Hydroelectric Engineering, Dalian University of Technology, Dalian 116024, China;
2. Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China
? Central South University Press and Springer-Verlag Berlin Heidelberg 2010
Abstract: To find the distribution patterns of dynamic amplification coefficients for dams subjected to earthquake, 3D seismic responses of concrete-faced rockfill dams with different heights and different shapes of river valley were analyzed by using the equivalent-linear model. Statistical analysis was also made to the seismic coefficient, and an empirical formula for calculating the maximum acceleration was provided. The results indicate that under the condition of the same dam height and the same base acceleration excitations, with the increase of the river valley width, the position of the maximum acceleration on the axis of the top of the dam moves from the center to the riversides symmetrically. For the narrow valleys, the maximum acceleration occurs in the middle of the axis at the top of the dam; for wide valleys the maximum acceleration appears near the riversides. The result negates the application of 2D dynamical computation for wide valleys, and shows that for the seismic response of high concrete-faced rockfill dams, the seismic coefficient along the axis should be given, except for that along the dam height. Seismic stability analysis of rockfill dams using pseudo-static method can be modified according to the formula.
Key words: concrete-faced rockfill dam; 3D dynamical response analysis; equivalent-linear method; acceleration; seismic coefficient; statistical analysis
1 Introduction
Concrete-faced rockfill dams with a height level of 200 m have been built in high intensity regions in recent years, which brings some problems still unsolved. Serious seismic catastrophe was caused to Zipingpu concrete-faced rockfill dam in Wenchuan earthquake of China in 2008 [1]. Therefore, it is very important and necessary to further study the seismic responses of earth-rockfill dams [2-4].
Slope stability analysis was paid much attention [5-7] and many methods were proposed to solve this problem. The general methods are the limit equilibrium method [8-9] and finite element method [10-11]. The limit equilibrium method is usually used as pseudo-static method [12] for seismic stability analysis of earth- rockfill dams. The seismic coefficient is the key parameter for seismic stability analysis by pseudo-static method. The code (SL203—1997) [13] for earthquake resistant design of hydraulic structures is only applicable to earth-rockfill dams with the height less than 150 m. Meanwhile, the code has been established quite a long time and the data are mainly adopted from 2D calculation, so it cannot be applied to the seismic stability calculation of present rockfill dams. ZHANG et al [14-15] studied the seismic coefficient of earth- rockfill dams with the height more than 150 m. However, they only considered the effect of dam height and neglected the effect of valley shape and seismic characteristics are not studied by them. In this work, considering different seismic characteristics, dam heights and valley shapes, systematic analysis about the distribution of the maximum acceleration along the dam height and axis was conducted. The amplification effect orderliness and an empirical formula for calculating the seismic coefficient of concrete-faced rockfill dams were also obtained.
2 Theory for dynamic analysis of rockfill dams
According to the theory of structural dynamics, the dynamic equilibrium equation for concrete-faced rockfill dams can be written as
(1)
where [M], [C] and [K] are the mass, damping and stiffness matrixces, respectively; 
and 
are the column matrices of acceleration, velocity and displacement, respectively; and {R(t)} is the vector of seismic load.
In the time domain, Eq.(1) was solved by step-by- step integration procedure. In this work, Wilson-θ step-by-step integration procedure was adopted.
The dynamic stress-strain relationship was used to express the dynamic characteristic of rockfill materials. The equivalent-linear visco-elastic model corresponded with the soil dynamic characteristic of rockfill materials very well, thus it was widely applied to dynamic response analysis of concrete faced rockfill dams [16].
Shear modulus G of rockfill materials at a random point can be expressed as
(2)
where 
is the shearing strain, 
is the reference shearing strain, and 
is the maximum shear modulus of rockfill materials.
Damping ratio 
can be expressed as
(3)
where 
can be obtained by experimental and statistical analysis.
The relationship between the normalized dynamic equivalent shear modulus Geq and dynamic shearing strain of rockfill materials is shown in Fig.1.
Fig.1 Relationship between normalized dynamic equivalent shear modulus and dynamic shearing strain of rockfill materials
From Eqs.(2)-(3), it can be seen that the dynamic shear modulus and damping ratio change with the variation of shearing strain. Shearing strain in the equivalent-nonlinear visco-elastic model can be used as the product of the maximum shearing strain and a discount coefficient. For the earthquake, the discount coefficient of 0.65 is generally adopted. However, Eqs.(2)-(3) still differ from the real dynamic characteristic curves of rockfill materials. Thus, the experimental dynamic characteristic curves were adopted for dynamic calculation of concrete-faced rockfill dams [17].
The relationship between equivalent damping ratio λeq and dynamic shearing strain of rockfill materials is shown in Fig.2.
Fig.2 Relationship between equivalent damping ratio and dynamic shearing strain of rockfill materials
3 Calculation cases
3D seismic response analysis was carried out to study a concrete-faced rockfill dam. The effects of earthquake wave spectrum, seismic intensity a, dam height H, valley width B and bank slope ratio m on the dynamic response were studied. In order to get some regularity conclusions, 1 200 cases (shown in Table 1), were calculated.
Table 1 Parameters of concreted-faced rockfill dam for computational cases
The crest width of the dam was 10 m, the slope ratio of the upstream and downstream was 1:1.6, the density of the rockfill was 2 000 kg/m3 and the density of the concrete faceplate was 2 180 kg/m3.
Four seismic waves with different frequency spectra were adopted, including Taft wave (the first wave), El Centro wave (the second wave), Banshen wave (the third wave) and Shenhu wave (the fourth wave). The response acceleration curves of the four waves are shown in Fig.3. The primary periods of the four waves were 0.34, 0.26,0.20 and 0.30 s, respectively. The durative time of the four waves were 16, 20, 20 and 25 s, respectively, and the maximum accelerations considered were 0.98, 1.96 and 3.92 m/s2. The earthquake waves were inputted along the river flow direction.
Fig.3 Response spectra of input waves at damping ratio of 4%
4 Results analysiss
The variation of acceleration along dam axis at the top of dam is shown in Fig.4, when dam height is 300 m and the first earthquake wave is inputted. From Fig.4 it can be seen that for Taft wave with the intensity of grade 7 and high dams (the dam height is 300 m) with narrow valley (the width is 150 m), the horizontal acceleration along the axis symmetrically distributes on the center line of the dam and the maximum amplification effect appears at the center point. When the slope ratio is varied from 0.4 to 1.5, the seismic coefficient β varies from 7.7 to 4.0. When the valley width increases to 500 m, no peak value occurs and the maximum seismic coefficient β fluctuates between 3.3 and 3.8. The distribution of seismic coefficients of narrow valley and wide valley are not the same, mainly because the difference in modes of vibration. When the earthquake intensity is equal to grades 8 and 9, the distribution of horizontal acceleration along the axis behaves like the one with earthquake intensity of grade 7, but the seismic coefficient β becomes smaller accordingly.
Fig.4 Variation of acceleration along dam axis at top of dam when dam height is 300 m and the first earthquake wave is inputted: (a) B=150 m, a=grade 7; (b) B=150 m, a=grade 8; (c) B=150 m, a=grade 9; (d) B=500 m, a=grade 7; (e) B=500 m, a=grade 8; (f) B= 500 m, a=grade 9 5 Empirical formulas for amplification effect of maximum acceleration
The amplification effect of acceleration in a concrete-faced rockfill dam is correlated with the following parameters. The first one is the dam height H. Different dam heights correspond to different natural vibration periods Tb and different sensibility to earthquakes. The second one is the predominant period Td of the inputted earthquake wave, because if parameter Td becomes close to parameter Tb, the dynamic response of the dam will be very large. Even though parameter Tb varies during the earthquake process, the pertinence between Td and Tb is still very large. The third parameter is the broad band range of the acceleration spectrum 
, but is a fuzzy concept, and how to use it needs special research. The fourth parameter is earthquake intensity a. Seismic coefficient β will become smaller when the earthquake intensity becomes larger. Seismic coefficient βi can be written as [18]
(4)
Dynamic response analysis of five types of dams with different heights is carried out and four earthquake waves are adopted in the calculation. Statistical analysis is made for the calculated results and an empirical formula for calculating seismic coefficient is concluded as
(5)
where βi is the seismic coefficient of the dam crest; Ai is the coefficient to reflect different earthquake intensities, corresponding to earthquake intensities of grades 7, 8 and 9, the values of Ai are 35, 16 and 5; Ki is the earthquake coefficient, corresponding to earthquake intensities of grades 7, 8 and 9, and the values of Ki are 0.1, 0.2 and 0.4, respectively; η1, η2, and η3 are the participate coefficients of the first three natural vibration periods of the dam, η1=0.7, η2=0.2, and η3=0.1. β1, β2 and β3 are the corresponding seismic coefficient of the first three natural vibration periods in the acceleration spectrum when the damping ratio is 4%. The first three natural vibration periods Tb1, Tb2, Tb3 of the dam can be calculated according to empirical equations [18]:
(6)
Eq.(5) is suitable for low dams in broad valleys, but when the dam is higher than 200 m, the value calculated by Eq.(5) will be smaller than the actual one. So it should be modified according to the dam height as follows:
(7)
Eq.(7) is the empirical formula for calculating the maximum seismic coefficient, considering dam height H, the spectrum of the wave and the earthquake intensity. If the valley width and bank slope ratio are considered, the seismic coefficient can be calculated as follows:
(1) When H≤200 m,
B≤300 m
B＞300 m
(8)
(2) When H＞200 m,
(9)
where βmax is the maximum seismic coefficient.
The distribution of acceleration difference between values calculated by finite element method and those by the empirical formula is shown in Fig.5, where sequence numbers of valley parameters are shown in Table 2, and the sequence numbers of dam parameters are shown in Table 3. From the figure, it can be seen that, accelerations calculated by Eqs.(8)-(9) approach the values calculated by finite element method. So the proposed empirical formula can be applied to estimating the maximum seismic coefficient of concrete-faced rockfill dams.
The seismic coefficient along the dam height is shown in Fig.6. The regularity is concluded from the
Fig.5 Distribution of acceleration difference between results calculated by FEM and these by empirical formula: (a) Taft seismic wave; (b) El Centro seismic wave; (c) Banshen seismic wave; (d) Shenhu seismic wave
Table 2 Sequence numbers of valley parameters
Table 3 Sequence numbers of dam parameters
calculation results of 1 200 cases, which are shown in Table 1. Fig.6 cooperating with Eqs.(8)-(9) can be applied to seismic stability analysis of rockfill dams by pseudo-static method.
Fig.6 Seismic coefficients at different positions of dam: (a) At center of dam; (b) At slope of dam
6 Conclusions
(1) When the dam height is the same, the seismic coefficient will become smaller if the inputted acceleration amplitude becomes larger. But the calculated seismic coefficients are larger than those in the code. The seismic spectrum has large effect on the acceleration response. If the vibration period of the dam is similar to the predominant period of the wave, the seismic coefficient will be great. Seismic waves with wide spectrum band have a great effect on the acceleration magnification of the concrete-faced rockfill dams with different heights.
(2) For the dam with the same height and the same inputted accelerations, the location where the maximum acceleration occurs moves to the bank with the increase of the valley width, which implies that 3D effects are very important for the calculation. 3D seismic coefficient should be provided in a graph. The maximum acceleration of the dam crest becomes smaller with the increase of the bank slope ratio. The effect of bank slope ratio on the maximum acceleration becomes smaller with the increase of the valley width.
(3) An empirical formula for calculating the maximum acceleration of concrete-faced rockfill dams is proposed according to the statistical analysis of many calculation cases.
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Foundation item: Project(90815024) supported by the National Natural Science Foundation of China
Received date: 2009-08-24; Accepted date: 2009-12-13
Corresponding author: ZHOU Hui, PhD, Associate Professor; Tel: +86-15941139590; E-mail: zhoubridge18@163.com
(Edited by CHEN Wei-ping)
