Mathematical Problems in Engineering

Volume 2019, Article ID 6958643, 12 pages

https://doi.org/10.1155/2019/6958643

## Seismic Fragility Analysis of High Earth-Rockfill Dams considering the Number of Ground Motion Records

^{1}State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China^{2}Institute of Earthquake Engineering, Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China

Correspondence should be addressed to Congcong Jin; moc.361@3263oatuil

Received 18 October 2018; Revised 21 January 2019; Accepted 30 January 2019; Published 12 February 2019

Academic Editor: Daniela Addessi

Copyright © 2019 Congcong Jin and Shichun Chi. 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

This study analyzes the impact of the number of ground motions on the seismic fragility of a high earth-rockfill dam and the estimation of reasonable fragility parameters based on a sufficient number of earthquake records. In this paper, the vertical deformation is obtained using the three-dimensional finite element program DYNE3WAC combined with the Pastor–Zienkiewicz–Chan model and Biot dynamic consolidation theory. The relative seismic settlement rate is considered the damage index for the seismic fragility analysis of the dam. The fragility curves of the high earth-rockfill dam are determined by the multiple stripe analysis (MSA) method. A set of seismic waves is chosen based on the spectrum in the Chinese hydraulic structure seismic code. With an increasing number of earthquake records, the coefficients of variation (COV) of the mean and standard deviation (STD) of the relative seismic settlement rate decrease and tend to stabilize when the number of earthquake records reaches 34. The estimated fragility parameters and are constant when the number of earthquake records exceeds 34. The requisite number of earthquake records for an accurate fragility estimation is determined by analyzing the lower and upper confidence intervals for the estimated and . The 95% and 90% confidence interval can accurately estimate the fragility of a high earth-rockfill dam when the number of ground motion records reaches 36 and 32, respectively. The results of the fragility analysis demonstrate that the DYNE3WAC program and MSA method can provide an effective basis for determining fragility curves. Furthermore, the sensitivity analysis of earthquake records is essential for assessing the seismic fragility of high earth-rockfill dams.

#### 1. Introduction

High earth-rockfill dams are an important part of the China’s infrastructure and represent sources of massive amounts of sustainable, renewable and clean hydropower energy [1]. China is one of the countries most affected by natural disasters, especially earthquakes, due to its specific location and large territory. The 1999 Chi-Chi earthquake, the 2008 Wenchuan earthquake, and the 2010 Yushu earthquake have demonstrated that the dams are vulnerable to earthquakes. The great Wenchuan earthquake that occurred in China caused earthquake damage to more than 2500 earth-rockfill dams. High earth-rockfill dams with heights of 200-300 meters are located in areas prone to strong earthquakes [2, 3]. Therefore, seismic performance assessments must be performed to assess the safety of these dams.

Seismic fragility analysis is one of the most effective methods for evaluating seismic performance and can effectively estimate the risk to dams affected by earthquakes; thus, this technique has gradually becomes an important method in seismic safety evaluation [4]. Fragility analysis can predict the probability of damage when structures reach or exceed a certain limit state probability under different strengths of seismic action and can employ a fragility curve to describe the probability distributions of the limit states [5, 6]. The fragility curves can be used to describe the probability of selected structure responses exceeding a certain critical level with a ground motion intensity measure (IM). To construct a fragility curve, incremental dynamic analysis (IDA) [7] and multiple strip analysis (MSA) [8] have been extensively applied as seismic analysis tools. The IDA method is employed to estimate the limit state capacity and seismic performance by performing a series of nonlinear time history analyses for a suite of ground motion records. In contrast to IDA, the MSA approach is a nonlinear dynamic analysis method and involves a sufficient set of stripes that are used to analyze and evaluate structural seismic responses under different IM levels. Baker [9] found that MSA produces more efficient fragility estimates than IDA for a given number of structural analyses and proposed an assessment approach. During the past several decades, many studies have been conducted to research the seismic fragility of arch dams and concrete dams [10, 11]. However, few papers have analyzed the seismic fragility of high earth-rockfill dams. Pang and Xu* et al*. [12] recently evaluated the seismic fragility of concrete-faced rockfill dams (CFRDs) subjected to the acceleration response spectra of 11 earthquake waves. However, these fragility analyses did not consider a large number of earthquake records, which might affect the parameter estimation for the fragility function. Several studies have been performed to promote the accurate estimation of the probability of exceeding a certain limit state in terms of earthquake IMs. Cimellaro* et al.* [13] investigated the impact of the number of ground motions on elastic shear-type buildings using a multidegree of freedom (MDOF) system and concluded that the minimum number of earthquake records needed to accurately estimate the parameters of the fragility functions is 23. Baker and Cornell [14] investigated a method of selecting ground motions and concluded that selecting earthquake records according to spectral shape can reduce the variance and bias of structural responses.

Deformation and stability are the two major aspects of the earthquake destruction patterns of earth-rockfill dams and are typically regarded as seismic performance assessment indexes [15–17]. Generally, the deformation index, especially the relative seismic settlement ratio [18, 19], is the most common seismic assessment index for earth-rockfill dams. The relative seismic settlement ratio is an important physical parameter when evaluating the seismic performance of earth-rockfill dams [20–22]. Furthermore, the index can illustrate the safety of the earth-rockfill dam and can be easily found in seismic records. Therefore, it is reasonable and feasible to regard the relative seismic settlement rate as an evaluation index of the seismic performance of earth-rockfill dams.

This paper aims to investigate the effect of the number of earthquake records on the fragility of high earth-rockfill dams and to perform a sensitivity analysis of the fragility parameters. The seismic fragility assessment of the dam is studied based on the MSA method. First, 60 seismic waves are selected according to the site conditions and the spectrum in China’s hydraulic structure seismic code. The relative seismic settlement rate of the high earth-rockfill dam is selected as the damage index. The three-dimensional elastoplastic finite element program DYNE3WAC (DYNamic Earthquake Analysis Program 3D Window Version for Academic), which is the 3D version of SWANDYNE II [23], can effectively calculate geotechnical engineering under static and dynamic conditions. The DYNE3WAC program and MSA method can reasonably determine the fragility curves of high earth-rockfill dams. Considering the number of ground motion inputs, the appropriate number of seismic records can be determined according to the fragility equation parameters. The coefficient of variation (COV) of the mean and standard deviation (STD) of the relative seismic settlement rate and the estimated and are analyzed. The estimated fragility parameters and tend to be constant when the number of earthquake records increases. The confidence interval can accurately estimate the fragility of a high earth-rockfill dam when the number of earthquake records is sufficiently high.

#### 2. Seismic Fragility Method Based on MSA

Performance-based earthquake engineering (PBEE) methods, such as IDA and MSA, have been proposed by many scholars. The difference between IDA and MSA lies in the record selection. In the MSA method, ground motion records are selected to fit all predefined IM levels. Furthermore, the MSA method is a sufficient set of single stripes used to analyze and evaluate structural seismic responses at different IM levels [24]. Compared with IDA, MSA is less time-consuming and relatively accurate. The seismic fragility curves provide the conditional probabilities of the structural response reaching or exceeding certain limit states that correspond to the seismic capacity under different earthquake intensities. The probability that the structure response exceeds specified limit states can be described as a function of the IM of ground motion. For fragility curves, peak ground acceleration (PGA) is commonly exploited as a convenient IM. The variation in the IM values of ground motions is commonly assumed to be related to the damage state of the selected structure following a lognormal distribution. The fragility function is generally defined as a lognormal cumulative distribution function and is shown in where is the probability that a ground motion with results in a structure that attained a damaged state, is the standard normal cumulative distribution function, and and are the median and STD of , respectively.

The MSA approach does not suggest that estimates of IM amplitudes for all the selected ground motions result in the exceedance of limit states. To estimate the fragility parameters, the likelihood function can be defined as follows:where is the number of levels, is a product over all levels, and and are the number of times the limit state is exceeded and the number of ground motions with , respectively.

The parameters and can be estimated by maximizing this likelihood function. It is equivalent and numerically easier to maximize the logarithm of the likelihood, as shown in

##### 2.1. Seismic Records

An important step in seismic fragility analysis is the selection of a representative set of earthquake motions at different levels of ground motion intensity. For ground motion selection, the magnitude, distance, and soil condition should be considered. There are uncertainties in the magnitude and distance, so it is reasonable to select earthquake records within appropriate ranges. The general consensus is to select ground motions whose magnitude and distance are within a close range of the target magnitude and distance. The dynamic peak acceleration with a transcendental probability of 2% in 100 years is selected as the seismic response spectrum. Shome [25] notes that the use of 10~20 seismic waves can produce an accurate estimation of the structure's earthquake response. However, due to the complex structure of high earth-rockfill dams, the geological conditions are different from those of ordinary buildings.

Because randomness is an inherent characteristic of earthquake ground motions [26], seismic randomness analysis should be researched and applied. The seismic performance assessment of the earth-rockfill dam employs fragility analysis based on 10–20 ground motion records from the PEER database, which has tens of thousands of ground motions [27, 28]. To analyze the relationship between the seismic fragility of high earth-rockfill dams and the number of earthquake records, the number of ground motions is selected to be 3 times the upper limit of Shome’s recommended value. Therefore, the selection of the ground motion input should take full account of the randomness of earthquakes. The response spectra based on the site conditions of a high earth-rockfill dam are selected as the target response spectra. Furthermore, 60 actual seismic records that agree well with the target response spectra based on site conditions are selected in PEER. These records are obtained from 60 different earthquakes with magnitudes ranging from 5.0 to 7.0 and with epicentral distances ranging from 10 to 55 km. The ground records are scaled and matched to the 5% damped target spectrum using SeismoMatch software. The distributions of PGAs, magnitudes and epicentral distances of these seismic waves are shown in Figure 1(a), and the seismic response spectrum of acceleration is shown in Figure 1(b):