# Embankment Seismic Fragility Assessment under the Near-Fault Pulse-like Ground Motions by Applying the Response Surface Method

Authors: Fa Che; Chao Yin; Jilei Zhou; Zhinan Hu; Xingkui Zhao; Lu Sheng; Qian Zhang; Xiaoyu Cheng; Gangting Tang; Han Zhang; Dong Liu
Journal: Shock and Vibration
 (2021)

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## Abstract

Uncertainties of the ground motions and structural parameters are the main factors that limit the accuracy of embankment seismic fragility assessment. In response to the uncertainties of the ground motions, artificial synthesizing method of the near-fault pulse-like ground motions was proposed, and 15 ground motions with the rupture fault distances ranging from 1 to 15 km were synthesized by taking the Chi-Chi earthquake in Taiwan, China, as an example. The Xi’an-Baoji expressway K1125 + 470 embankment was taken as the research object, and a total of 12 structural parameters were selected as the design variables, namely, the elastic modulus, bulk modulus, shear modulus, density, cohesion force, and internal friction angle of the embankment fill and soil foundation, respectively. In response to the uncertainties of these parameters, 3 principal components with large impacts on the embankment seismic fragility were extracted based on the principal component analysis. Mapping relationships among the principal components and embankment seismic damages were analyzed using the uniform design response surface method, and the seismic fragility assessment was carried out and the fragility curves were plotted. The research results are consistent with the actual embankment seismic damage conditions of the Chi-Chi earthquake, indicating that the proposed method is scientific and reasonable. It also shows that it would obviously overestimate the seismic performance in the embankment seismic fragility assessment without considering the uncertainties of the ground motions and structural parameters.


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## Body

## 1. Introduction

Strong earthquakes have occurred frequently in the world; for example, the 2004 Sumatra earthquake in Indonesia, the 2008 Wenchuan earthquake in China, the 2010 earthquakes in Haiti and Chile, the 2011 east Japan earthquake, and the 2017 Jiuzhaigou Valley’s earthquake in China resulted in severe structural damages, causing major human casualties and economic losses [1–3]. Near-fault ground motions generally occur at sites located near active faults and are characterized by the following main features: (1) large amplitude and long-period velocity pulses; (2) high Peak Ground Velocity/Peak Ground Acceleration (PGV/PGA) and Peak Ground Displacement/Peak Ground Acceleration (PGD/PGA) ratios; and (3) concentration of energy in one or few pulses. The pulse-like ground motions are caused primarily by the forward directivity effect, which are observed at a site when the fault rupture propagates towards the site with a velocity close to shear wave velocity [4–6]. The near-fault ground motions with long-period pulses have much more energies input to long-span linear structures than the far-fault ground motions. Therefore, researches on the fragility assessment of structures such as embankments under the near-fault pulse-like ground motions are of great significance for performance-based seismic design [7–9].Uncertainties of the ground motions and structural parameters are the main factors that limit the accuracy of the seismic fragility assessment [10, 11]. The uncertainties of the ground motions are influenced by aleatory uncertainties (uncertainty due to the random nature of the processes under consideration such as ground motion record-to-record variability) and epistemic uncertainties (uncertainty due to incomplete knowledge and data such as model parameter uncertainty, omissions, and errors) [12–15]. There are two main methods to solve the problem of the uncertainties of the ground motions. The first selects the actual recorded ground motions provided by the Pacific Earthquake Engineering Research Center (PEER). For example, Yin et al. [16] selected 15 ground motions in medium-hard sites to carry out the embankment seismic fragility assessment and verified the impact of the retaining wall on improving the embankment seismic performance. Bao et al. [17] selected 10 near-fault ground motions and 10 far-fault ground motions to carry out the fragility assessment of the nuclear reactor shell structure under the main shock and aftershocks, respectively. Li et al. [7] investigated the dynamic response characteristics of the 1/30 scale model of the Wuhan Shimao Center subjected to the El Centro wave and Taft wave by analyzing the shaking table test data. He et al. [18] adopted 22 actual recorded ground motions according to the recommendations of the Applied Technology Council (ATC) in the FEMA P695 report in the seismic fragility analysis of multiage buried steel pipes. Since it is difficult to find the ground motions that fully comply with the lithology properties of where the structure is located and with the law of seismic wave propagation, the accuracy of the seismic fragility assessment based on actual recorded ground motions is relatively low [19, 20]. The second is synthesizing ground motions. For example, Sheng et al. [21] calculated the fragility indexes of the frequent earthquake, fortification earthquake, and rare earthquake of group structures combined with the synthesized ground motions and empirical earthquake damage indexes. Salami et al. [22] carried out the seismic fragility assessment of low-rise reinforced concrete structures under the main shock and aftershocks based on the synthesized ground motions. Abyani et al. [23] used a genetic algorithm to analyze the impact of the number of synthesized ground motions on the fragility assessment results of jacket offshore platforms. For near-fault pulse-like ground motions, the maximum values of the parameters in different directions vary significantly. For example, Yang and Zhou [24] studied the impacts of the pulse-like ground motions on the response spectrum in different directions and pointed out that the ground motions on the strongest pulse direction comprehensively considered the impacts of the pulse period, peak velocity, and duration, and the energies input to the structure were maximized. However, existing artificial near-fault ground motion models do not take into account the directivity of pulse-like ground motions in history analysis, and the seismic performance may be overestimated in the seismic fragility assessment [15, 25–27].Some scholars ignore the impacts of the uncertainties of the structural parameters on the seismic performance and build models using deterministic values directly, while some introduce the response surface method to characterize the mapping relationships among the structural parameters and seismic responses [28]. There are 3 steps of the seismic fragility assessment based on the response surface method: (1) experimental design, that is to determine the sample points and structural response values [29]; (2) select the response surface function that meets the structural functional requirements [30]; and (3) fit the response surface function and verify the model accuracy [31, 32]. For example, Saha et al. [33] used the Monte Carlo method to study the discreteness of the parameters of the tank structure and carried out the seismic fragility assessment based on the response surface method. Li and Li [29] proposed a uniform design response surface method to study the impacts of the multiple correlations of the design variables on the bridge seismic fragility. Tran et al. [34] carried out the seismic fragility assessment of nuclear power plants based on the response surface method and plotted the fragility curves by the maximum likelihood estimation method and linear regression method, respectively. Since the response surface functions often take the form of summation of several complete polynomials, when there are too many structural parameters, the applicability of the model greatly reduces and the solution of the response surface function tends to fall into a “curse of dimensionality” [30, 35–37].In view of this, the Xi’an-Baoji expressway K1125 + 470 embankment was taken as the research object, the stochastic pulse models with different rupture fault distances were established by considering the directional impacts of the pulse-like ground motions, and the linear correlation of the structural parameters was eliminated based on the principal component analysis. In addition, embankment seismic fragility assessment using 15 artificial pulse-like ground motions with the rupture fault distances ranging from 1 to 15 km was carried out by applying the response surface method, aiming to provide a theoretical basis for the performance-based design of embankment.


## 2. Artificial Synthesizing Method of the Near-Fault Pulse-Like Ground Motions

Since the orientation of the apparatus in a seismic station is arbitrary, the recorded ground motions at the time of the earthquake are not necessarily the maximum, especially for the near-fault pulse-like ground motions, the seismic intensities in different directions are obviously different [38]. In this paper, the linear combination of the continuous wavelet transformation results of two horizontal components (fault-normal velocity history and fault-parallel velocity history) of the pulse-like ground motions was used to build the motion model on the strongest pulse direction, and the following methods were applied for stochastic modeling and artificial synthesizing of the near-fault pulse-like ground motions [24, 26, 39–41]:(1)
Apply wavelet transformation to synthesize the fault-normal velocity history and fault-parallel velocity history of the pulse-like ground motions into the velocity history on the strongest pulse direction and divide the velocity history into low-frequency components and high-frequency components.(2)
Fit the low-frequency components using the Gabor wavelet analytic function and propose the correlations among the pulse parameters, i.e., seismic moment, rupture fault distance and site condition, and seismological parameters. The regression model between the PGV on the strongest pulse direction and rupture fault distance is calculated, and the low-frequency stochastic pulse function is established after obtaining the probability distribution of the low-frequency pulse parameters.(3)
Differentiate the residual velocity history containing the high-frequency components to obtain the residual acceleration history and employ the improved Kanai–Tajimi power spectrum model to fit the statistical power spectrum containing the high-frequency components to obtain the power spectrum parameters.(4)
Use the envelope function to fit the residual acceleration history after Hilbert transformation to reflect the nonstationarity of the high-frequency components and obtain the probability distribution of the corresponding envelope parameters via statistics.(5)
Generate the velocity history of the near-fault pulse-like ground motions by superimposing the long-period velocity pulse and velocity history with the high-frequency components and obtain the acceleration history through differentiation, as shown in Figure1.Figure 1
Stochastic modeling and artificial synthesizing method of the near-fault pulse-like ground motions.The pulse-like ground motions of the station CHY080 of the Chi-Chi earthquake on September 21, 1999, in Taiwan, China, were adopted to synthesize 15 ground motions with the rupture fault distances ranging from 1 to 15 km, respectively. The rupture fault distance of the CHY080 station was 10.9 km, and the PGA was 1.193 g [42–44]. The regression model between the PGV on the strongest pulse direction and rupture fault distance was calculated according to the recorded ground motions, as shown in Figure 2.Figure 2
Attenuation of PGV with the rupture fault distanceR for the Chi-Chi earthquake (the dashed lines correspond to mean values ± σ1).For the synthesized ground motions were generated from the same earthquake event, i.e., Chi-Chi earthquake, the magnitudeMw = 7.6 was not included in the regression model. The standard deviation σ1 reflects epistemic modeling uncertainty regarding the factors controlling variety of PGVs with the same rupture fault distance R and follows a log-normal distribution. A representative sample under the scenario of rupture fault distance R = 15 km was used to generate the synthesized ground motions according to Figure 1, as shown in Figure 3. It is indicated that the synthetic velocity history represents the pulse characteristic of the near-fault ground motions properly, and the corresponding acceleration history can be used as the stochastic excitation input for the embankment seismic fragility assessment. It is worth noting that the seismic fragility assessment in this paper is only based on the Chi-Chi earthquake, and the results are valid for earthquakes with the same characteristics as the Chi-Chi earthquake.Figure 3
Artificial synthesizing results of the pulse-like ground motion with the rupture fault distanceR = 15 km. (a). Acceleration history of high-frequency components. (b) Velocity history of high-frequency components. (c) Random velocity pulse. (d) Pulse-like velocity history. (e) Synthetic acceleration history. (f) Synthetic displacement history.
(a)(b)(c)(d)(e)(f)


## 3. Response Surface Method

### 3.1. Embankment Model

FLAC can simulate structures made of soil, rock, and other materials that flow plastically when reaching their yield limits [45]. FLAC adopts the finite difference scheme to solve the governing differential equation, which can accurately simulate the yield, plastic flow, softening, and large deformation of materials, especially has unique advantages in the fields of elastic-plastic analysis, large deformation analysis, and simulation of construction process [46]. A finite difference model of the embankment fill-soil foundation system of the research object was established by using FLAC. The width of the embankment fill was 24.5 m, the left side was 6.1 m high, the right side was 2.6 m high, and the slope ratio was 1:1.5. The slope of the soil foundation was 24°, the thickness was 30 m, and the width was 120 m. Among them, the vehicle loads have been converted to the thickness of the embankment fill according to the elastic layer theory, as shown in Figure 4.Figure 4
Embankment model.A total of 12 structural parameters were selected as the design variables, namely, the elastic modulus, bulk modulus, shear modulus, density, cohesion force, internal friction angle of the embankment fill, and soil foundation. Considering the uncertainties of the material sources and construction qualities, the values of the above parameters were highly random [29]. According to the geological survey data of the Xi’an-Baoji expressway, the laboratory test results of the embankment fill and soil foundation, and other existing researches [18, 47], the elastic modulus, bulk modulus, shear modulus, cohesion force of the embankment fill and soil foundation followed the log-normal distribution; the density, internal friction angle of the embankment fill, and soil foundation followed the normal distribution; the probability distribution parameters of these structural parameters are shown in Table 1, and the probability distribution curves of these structural parameters are shown in Figure 5.Table 1
Probability distribution parameters of the structural parameters.
Sequence numbersStructural parametersDistribution patternsMean valuesCoefficients of variationX1Elastic modulus of the embankment fillLog-normal distribution48.00 MPa0.20X2Elastic modulus of the soil foundationLog-normal distribution42.00 MPa0.20X3Bulk modulus of the embankment fillLog-normal distribution50.00 MPa0.20X4Bulk modulus of the soil foundationLog-normal distribution43.75 MPa0.20X5Shear modulus of the embankment fillLog-normal distribution17.91 MPa0.33X6Shear modulus of the soil foundationLog-normal distribution15.67 MPa0.33X7Density of the embankment fillNormal distribution1970 kg/m30.20X8Density of the soil foundationNormal distribution1630 kg/m30.20X9Cohesion force of the embankment fillLog-normal distribution34.00 kPa0.20X10Cohesion force of the soil foundationLog-normal distribution31.00 kPa0.20X11Internal friction angle of the embankment fillNormal distribution33.00°0.20X12Internal friction angle of the soil foundationNormal distribution28.00°0.20Figure 5
Probability distribution curves of the structural parameters.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


### 3.2. Selection of the Embankment Seismic Damage Parameter

According to Song et al. [48] and Li et al. [49], PGA is selected as the intensity parameter of the ground motions, and the embankment seismic damage levels are divided into 5 grades: basically intact, minor damage, moderate damage, severe damage, and destruction. Since the embankment is a statically indeterminate structure, damages caused by earthquakes would be reflected by the displacement. Therefore, εmax is selected as the embankment seismic damage parameter based on the displacement failure criterion, and the calculation method of εmax is shown in the following equation:(1)εmax=100dmaxD,where dmax is the lateral maximum permanent displacement on the top surface of the embankment and D is the embankment width; D = 24.5 m. According to the investigation results of the Wenchuan earthquake, embankments at the epicenter (Yingxiu town) suffered from the most severely damage, i.e., destruction, and εmax reached 2.059% [50–52]; therefore, considering εmax = 2.0% as the boundary of severe damage and destruction was reasonable. Besides, the boundaries of εmax among other embankment seismic damage grades were further established based on the equidistant classifying method [17, 53], as summarized in Table 2.Table 2
Boundaries ofεmax among the embankment seismic damage grades.
Embankment seismic damage gradesBasically intactMinor damageModerate damageSevere damageDestructionεmax0 ≤εmax＜0.30.30 ≤εmax ＜0.600.60 ≤εmax ＜1.201.20 ≤εmax ＜2.00εmax ≥2.00


### 3.3. Principal Component Analysis Results

Since the solution of the response surface function easily falls into a “curse of dimensionality” when there are too many structural parameters and the parameters may not be linearly independent, the principal component analysis was used to reduce the number of variables [36, 54]. According to the probability distribution parameters listed in Table 1, 10,000 sets of structural parameter combinations were randomly generated via MATLAB. Due to the different dimensions and vast differences in data amplitude of the structural parameters, data normalization was conducted before further calculations. The calculation method is shown in the following equation:(2)vXin=2Xin−uXiXimax−Ximin,where Xi is the ith structural parameter, i is a natural number from 1 to 12, Ximax and Ximin are the maximal and minimal values of Xi, and uXi is the average value of Xi, as shown in Table 1. Xin is the nth value of Xi, n is a natural number from 1 to 10000, v(Xin) is the normalized value of Xin, and v(Xin)∈[−1, 1]. The correlation coefficient ρij of v(Xj) and v(Xj) is calculated according to the following equation:(3)ρij=COVvXi,vXjσvXi⋅σvXj=EvXin−μvXi⋅vXjn−μvXjσvXi⋅σvXj,where μvXi, σvXi, μvXj, and σvXj are the mean values and standard deviations of v(Xi) and v(Xj), respectively, and other symbols have the same meaning. The calculation results are shown in Table 3.Table 3
Correlation coefficient matrixρij of v(Xi) and v(Xj).
ρijv(X1)v(X2)v(X3)v(X4)v(X5)v(X6)v(X7)v(X8)v(X9)v(X10)v(X11)v(X12)v(X1)1v(X2)0.2791v(X3)0.5010.0991v(X4)0.376−0.1970.7421v(X5)0.1060.0830.2030.2341v(X6)−0.099−0.1040.2970.1370.1381v(X7)0.0420.097−0.251−0.074−0.056−0.0761v(X8)0.0750.1040.1050.960.2710.1340.1571v(X9)−0.087−0.0380.218−0.0330.1030.2540.0460.8641v(X10)0.043−0.103−0.5480.2570.095−0.358−0.1850.3580.6541v(X11)−0.075−0.0970.087−0.159−0.2780.5450.104−0.567−0.149−0.4521v(X12)−0.3890.0800.3640.242−0.152−0.2210.2890.3250.3120.4890.5451The weight vector, eigenvalue, original information contribution rate, and cumulative original information contribution rate of each structural parameter corresponding to the principal component were further calculated based on SPSS22.0, and 3 principal componentsF1, F2, and F3 were extracted according to the eigenvalue not less than 1, as shown in Table 4.Table 4
Principal component analysis results of the structural parameters.
Calculation resultsNormalized valuesPrincipal componentsF1F2F3Weight vectorsv(X1)0.156−0.7010.307v(X2)0.5910.113−0.198v(X3)0.0620.798−0.095v(X4)−0.0430.8840.113v(X5)0.2380.1700.817v(X6)−0.2270.2490.917v(X7)−0.1140.182−0.243v(X8)0.963−0.048−0.132v(X9)0.8140.1970.039v(X10)0.798−0.2010.135v(X11)0.2080.198−0.816v(X12)−0.1090.2340.295Eigenvalues6.2743.1281.574Original information contribution rates54.385%29.667%9.578%Cumulative original information contribution rate93.630%The equations ofF1, F2, and F3 are shown in equations (4)–(6) according to Table 4.(4)F1=0.156vX1+0.591vX2+0.062vX3−0.043vX4+0.238vX5−0.227vX6−0.114vX7+0.963vX8+0.814vX9+0.798vX10+0.208vX11−0.109vX12,(5)F2=−0.701vX1+0.113vX2+0.798vX3+0.884vX4−0.170vX5+0.249vX6+0.182vX7−0.048vX8+0.197vX9−0.201vX10+0.198vX11+0.234vX12,(6)F3=0.307vX1−0.198vX2−0.095vX3+0.113vX4+0.817vX5+0.917vX6−0.243vX7−0.132vX8+0.039vX9+0.135vX10−0.816vX11+0.295vX12,The following conclusions can be obtained from equations (4) to (6):(1)
F1 mainly reflects the elastic modulus, density, and cohesion force of the soil foundation, and the cohesion force of the embankment fill; F2 mainly reflects the elastic modulus and bulk modulus of the embankment fill, and the bulk modulus of the soil foundation; F3 mainly reflects the shear modulus and internal friction angle of the embankment fill, and the shear modulus of the soil foundation.(2)
WhenX1 to X12 all take the mean values, F1, F2, and F3 can also take the mean value 0. When X4, X6,X7, and X12 take the minimal values and other structural parameters take the maximal values, F1 takes the maximal value, or otherwise, the minimal value. When X1, X8, and X10 take the minimal values and other structural parameters take the maximal values, F2 takes the maximal value, or otherwise, the minimal value. When X2, X3, X7, X8, and X11 take the minimal values and other structural parameters take the maximal values, F3 takes the maximal value, or otherwise, the minimal value.


### 3.4. Calculation Results of the Response Surface Method

Mapping relationships betweenεmax and F1, F2, F3, PGA, and standard deviation σ2 of εmax and F1, F2, F3, PGA were calculated based on the uniform design response surface method [17], as shown in the following equations:(7)εmax=α0+α1F1+α2F2+α3F3+α4PGA+α5F12+α6F22+α7F32+α8PGA2+α9F1F2+α10F1F3+α11F1PGA+α12F2F3+α13F2PGA+α14F3PGA+ε1,(8)σ2=β0+β1F1+β2F2+β3F3+β4PGA+β5F12+β6F22+β7F32+β8PGA2+β9F1F2+β10F1F3+β11F1PGA+β12F2F3+β13F2PGA+β14F3PGA+ε2,where α0–α14, β0–β14 are undetermined coefficients and ε1 and ε2 are regression coefficients of εmax and σ2, respectively. The main object of this paper was to reveal the dynamic response regulars of the embankment under the near-fault pulse-like ground motions with different rupture fault distances.In dynamic response analysis, more accurate results are often obtained when more than one earthquake are used, but according to the stochastic modeling and artificial synthesizing method of the near-fault pulse-like ground motions, it is necessary to calculate the attenuation relationship of PGV with the rupture fault distance, which requires a large number of seismic records [55]. In this paper, we calculated the attenuation relationship of Chi-Chi earthquake with Mw = 7.6, and the 15 synthesized ground motions generated from the Chi-Chi earthquake were employed to the dynamic response analysis and the results were valid for earthquakes with the same characteristics [56]. Dynamic response analysis based on more earthquakes would be conducted in future researches. Since the PGA of the Chi-Chi earthquake was 1.193 g, three values of 0.2 g, 0.7 g, and 1.2 g were assumed to cover the recorded PGA of the Chi-Chi earthquake. PGAs of the 15 synthesized ground motions were adjusted to 0.2 g, 0.7 g, and 1.2 g, respectively, and 45 ground motions were obtained. 26 combinations were acquired through the experimental design of 3 levels of F1, F2, F3 and PGA, the above 45 ground motions were used to conduct 390 dynamic response analysis for each combination. In addition, a 27th combination was added, and the 15 original ground motions were used for dynamic response analysis, and a total of 405 dynamic response analysis were conducted. The analysis results are shown in Table 5.Table 5
Dynamic response analysis results of the embankment.
Sequence numbersF1F2F3PGA/gMean values and standard deviationsσ2 of εmaxMean valuesStandard deviationsσ21−4.323−3.975−4.1071.21.80560.47522−4.32304.1070.20.28240.07543−4.3233.97500.70.86370.21534−4.323−3.97500.70.74180.19805−4.32304.1071.21.92110.49906−4.3233.975−4.1070.20.21570.05687−4.323−3.9754.1071.22.07700.53278−4.323000.20.30430.0770903.975−4.1070.70.92180.2364100−3.9754.1070.20.21950.0543110001.21.86890.46581203.97500.20.28850.0763130−3.9754.1070.70.85170.2129140000.70.70610.18341500−4.1071.22.07920.51721603.9754.1070.20.37120.0928170−3.97500.70.91070.2397180000.20.29670.0771194.3233.975−4.1070.70.78270.1957204.32304.1071.21.67410.4293214.323−3.97500.20.18580.0478224.3233.97500.70.75490.2455234.32304.1070.20.32170.0847244.323−3.975−4.1071.22.03820.5155254.3233.9754.1070.20.24890.0655264.323000.70.84380.207327000Original value1.92780.5130According to the dynamic response analysis results of the first 26 combinations, equations (7) and (8) were solved, and the regression coefficients are shown in Table 6.Table 6
Regression coefficients of equations (7) and (8).
Regression coefficientsα0α1α2α3α4α5α6α7Values0.250650.016120.041870.022900.35272−0.00330−0.00527−0.00253Regression coefficientsα8α9α10α11α012α13α14ε1Values0.91883−0.00332−0.00396−0.02488−0.00384−0.06253−0.033490.0342Regression coefficientsβ0β1β2β3β4β5β6β7Values0.063460.004240.01102−0.006540.090440.000830.00132−0.00067Regression coefficientsβ8β1β2β3β4β5β6ε2Values0.24502−0.00085−0.00102−0.00479−0.000970.01645−0.008370.0169


## 3.1. Embankment Model

FLAC can simulate structures made of soil, rock, and other materials that flow plastically when reaching their yield limits [45]. FLAC adopts the finite difference scheme to solve the governing differential equation, which can accurately simulate the yield, plastic flow, softening, and large deformation of materials, especially has unique advantages in the fields of elastic-plastic analysis, large deformation analysis, and simulation of construction process [46]. A finite difference model of the embankment fill-soil foundation system of the research object was established by using FLAC. The width of the embankment fill was 24.5 m, the left side was 6.1 m high, the right side was 2.6 m high, and the slope ratio was 1:1.5. The slope of the soil foundation was 24°, the thickness was 30 m, and the width was 120 m. Among them, the vehicle loads have been converted to the thickness of the embankment fill according to the elastic layer theory, as shown in Figure 4.Figure 4
Embankment model.A total of 12 structural parameters were selected as the design variables, namely, the elastic modulus, bulk modulus, shear modulus, density, cohesion force, internal friction angle of the embankment fill, and soil foundation. Considering the uncertainties of the material sources and construction qualities, the values of the above parameters were highly random [29]. According to the geological survey data of the Xi’an-Baoji expressway, the laboratory test results of the embankment fill and soil foundation, and other existing researches [18, 47], the elastic modulus, bulk modulus, shear modulus, cohesion force of the embankment fill and soil foundation followed the log-normal distribution; the density, internal friction angle of the embankment fill, and soil foundation followed the normal distribution; the probability distribution parameters of these structural parameters are shown in Table 1, and the probability distribution curves of these structural parameters are shown in Figure 5.Table 1
Probability distribution parameters of the structural parameters.
Sequence numbersStructural parametersDistribution patternsMean valuesCoefficients of variationX1Elastic modulus of the embankment fillLog-normal distribution48.00 MPa0.20X2Elastic modulus of the soil foundationLog-normal distribution42.00 MPa0.20X3Bulk modulus of the embankment fillLog-normal distribution50.00 MPa0.20X4Bulk modulus of the soil foundationLog-normal distribution43.75 MPa0.20X5Shear modulus of the embankment fillLog-normal distribution17.91 MPa0.33X6Shear modulus of the soil foundationLog-normal distribution15.67 MPa0.33X7Density of the embankment fillNormal distribution1970 kg/m30.20X8Density of the soil foundationNormal distribution1630 kg/m30.20X9Cohesion force of the embankment fillLog-normal distribution34.00 kPa0.20X10Cohesion force of the soil foundationLog-normal distribution31.00 kPa0.20X11Internal friction angle of the embankment fillNormal distribution33.00°0.20X12Internal friction angle of the soil foundationNormal distribution28.00°0.20Figure 5
Probability distribution curves of the structural parameters.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


## 3.2. Selection of the Embankment Seismic Damage Parameter

According to Song et al. [48] and Li et al. [49], PGA is selected as the intensity parameter of the ground motions, and the embankment seismic damage levels are divided into 5 grades: basically intact, minor damage, moderate damage, severe damage, and destruction. Since the embankment is a statically indeterminate structure, damages caused by earthquakes would be reflected by the displacement. Therefore, εmax is selected as the embankment seismic damage parameter based on the displacement failure criterion, and the calculation method of εmax is shown in the following equation:(1)εmax=100dmaxD,where dmax is the lateral maximum permanent displacement on the top surface of the embankment and D is the embankment width; D = 24.5 m. According to the investigation results of the Wenchuan earthquake, embankments at the epicenter (Yingxiu town) suffered from the most severely damage, i.e., destruction, and εmax reached 2.059% [50–52]; therefore, considering εmax = 2.0% as the boundary of severe damage and destruction was reasonable. Besides, the boundaries of εmax among other embankment seismic damage grades were further established based on the equidistant classifying method [17, 53], as summarized in Table 2.Table 2
Boundaries ofεmax among the embankment seismic damage grades.
Embankment seismic damage gradesBasically intactMinor damageModerate damageSevere damageDestructionεmax0 ≤εmax＜0.30.30 ≤εmax ＜0.600.60 ≤εmax ＜1.201.20 ≤εmax ＜2.00εmax ≥2.00


## 3.3. Principal Component Analysis Results

Since the solution of the response surface function easily falls into a “curse of dimensionality” when there are too many structural parameters and the parameters may not be linearly independent, the principal component analysis was used to reduce the number of variables [36, 54]. According to the probability distribution parameters listed in Table 1, 10,000 sets of structural parameter combinations were randomly generated via MATLAB. Due to the different dimensions and vast differences in data amplitude of the structural parameters, data normalization was conducted before further calculations. The calculation method is shown in the following equation:(2)vXin=2Xin−uXiXimax−Ximin,where Xi is the ith structural parameter, i is a natural number from 1 to 12, Ximax and Ximin are the maximal and minimal values of Xi, and uXi is the average value of Xi, as shown in Table 1. Xin is the nth value of Xi, n is a natural number from 1 to 10000, v(Xin) is the normalized value of Xin, and v(Xin)∈[−1, 1]. The correlation coefficient ρij of v(Xj) and v(Xj) is calculated according to the following equation:(3)ρij=COVvXi,vXjσvXi⋅σvXj=EvXin−μvXi⋅vXjn−μvXjσvXi⋅σvXj,where μvXi, σvXi, μvXj, and σvXj are the mean values and standard deviations of v(Xi) and v(Xj), respectively, and other symbols have the same meaning. The calculation results are shown in Table 3.Table 3
Correlation coefficient matrixρij of v(Xi) and v(Xj).
ρijv(X1)v(X2)v(X3)v(X4)v(X5)v(X6)v(X7)v(X8)v(X9)v(X10)v(X11)v(X12)v(X1)1v(X2)0.2791v(X3)0.5010.0991v(X4)0.376−0.1970.7421v(X5)0.1060.0830.2030.2341v(X6)−0.099−0.1040.2970.1370.1381v(X7)0.0420.097−0.251−0.074−0.056−0.0761v(X8)0.0750.1040.1050.960.2710.1340.1571v(X9)−0.087−0.0380.218−0.0330.1030.2540.0460.8641v(X10)0.043−0.103−0.5480.2570.095−0.358−0.1850.3580.6541v(X11)−0.075−0.0970.087−0.159−0.2780.5450.104−0.567−0.149−0.4521v(X12)−0.3890.0800.3640.242−0.152−0.2210.2890.3250.3120.4890.5451The weight vector, eigenvalue, original information contribution rate, and cumulative original information contribution rate of each structural parameter corresponding to the principal component were further calculated based on SPSS22.0, and 3 principal componentsF1, F2, and F3 were extracted according to the eigenvalue not less than 1, as shown in Table 4.Table 4
Principal component analysis results of the structural parameters.
Calculation resultsNormalized valuesPrincipal componentsF1F2F3Weight vectorsv(X1)0.156−0.7010.307v(X2)0.5910.113−0.198v(X3)0.0620.798−0.095v(X4)−0.0430.8840.113v(X5)0.2380.1700.817v(X6)−0.2270.2490.917v(X7)−0.1140.182−0.243v(X8)0.963−0.048−0.132v(X9)0.8140.1970.039v(X10)0.798−0.2010.135v(X11)0.2080.198−0.816v(X12)−0.1090.2340.295Eigenvalues6.2743.1281.574Original information contribution rates54.385%29.667%9.578%Cumulative original information contribution rate93.630%The equations ofF1, F2, and F3 are shown in equations (4)–(6) according to Table 4.(4)F1=0.156vX1+0.591vX2+0.062vX3−0.043vX4+0.238vX5−0.227vX6−0.114vX7+0.963vX8+0.814vX9+0.798vX10+0.208vX11−0.109vX12,(5)F2=−0.701vX1+0.113vX2+0.798vX3+0.884vX4−0.170vX5+0.249vX6+0.182vX7−0.048vX8+0.197vX9−0.201vX10+0.198vX11+0.234vX12,(6)F3=0.307vX1−0.198vX2−0.095vX3+0.113vX4+0.817vX5+0.917vX6−0.243vX7−0.132vX8+0.039vX9+0.135vX10−0.816vX11+0.295vX12,The following conclusions can be obtained from equations (4) to (6):(1)
F1 mainly reflects the elastic modulus, density, and cohesion force of the soil foundation, and the cohesion force of the embankment fill; F2 mainly reflects the elastic modulus and bulk modulus of the embankment fill, and the bulk modulus of the soil foundation; F3 mainly reflects the shear modulus and internal friction angle of the embankment fill, and the shear modulus of the soil foundation.(2)
WhenX1 to X12 all take the mean values, F1, F2, and F3 can also take the mean value 0. When X4, X6,X7, and X12 take the minimal values and other structural parameters take the maximal values, F1 takes the maximal value, or otherwise, the minimal value. When X1, X8, and X10 take the minimal values and other structural parameters take the maximal values, F2 takes the maximal value, or otherwise, the minimal value. When X2, X3, X7, X8, and X11 take the minimal values and other structural parameters take the maximal values, F3 takes the maximal value, or otherwise, the minimal value.


## 3.4. Calculation Results of the Response Surface Method

Mapping relationships betweenεmax and F1, F2, F3, PGA, and standard deviation σ2 of εmax and F1, F2, F3, PGA were calculated based on the uniform design response surface method [17], as shown in the following equations:(7)εmax=α0+α1F1+α2F2+α3F3+α4PGA+α5F12+α6F22+α7F32+α8PGA2+α9F1F2+α10F1F3+α11F1PGA+α12F2F3+α13F2PGA+α14F3PGA+ε1,(8)σ2=β0+β1F1+β2F2+β3F3+β4PGA+β5F12+β6F22+β7F32+β8PGA2+β9F1F2+β10F1F3+β11F1PGA+β12F2F3+β13F2PGA+β14F3PGA+ε2,where α0–α14, β0–β14 are undetermined coefficients and ε1 and ε2 are regression coefficients of εmax and σ2, respectively. The main object of this paper was to reveal the dynamic response regulars of the embankment under the near-fault pulse-like ground motions with different rupture fault distances.In dynamic response analysis, more accurate results are often obtained when more than one earthquake are used, but according to the stochastic modeling and artificial synthesizing method of the near-fault pulse-like ground motions, it is necessary to calculate the attenuation relationship of PGV with the rupture fault distance, which requires a large number of seismic records [55]. In this paper, we calculated the attenuation relationship of Chi-Chi earthquake with Mw = 7.6, and the 15 synthesized ground motions generated from the Chi-Chi earthquake were employed to the dynamic response analysis and the results were valid for earthquakes with the same characteristics [56]. Dynamic response analysis based on more earthquakes would be conducted in future researches. Since the PGA of the Chi-Chi earthquake was 1.193 g, three values of 0.2 g, 0.7 g, and 1.2 g were assumed to cover the recorded PGA of the Chi-Chi earthquake. PGAs of the 15 synthesized ground motions were adjusted to 0.2 g, 0.7 g, and 1.2 g, respectively, and 45 ground motions were obtained. 26 combinations were acquired through the experimental design of 3 levels of F1, F2, F3 and PGA, the above 45 ground motions were used to conduct 390 dynamic response analysis for each combination. In addition, a 27th combination was added, and the 15 original ground motions were used for dynamic response analysis, and a total of 405 dynamic response analysis were conducted. The analysis results are shown in Table 5.Table 5
Dynamic response analysis results of the embankment.
Sequence numbersF1F2F3PGA/gMean values and standard deviationsσ2 of εmaxMean valuesStandard deviationsσ21−4.323−3.975−4.1071.21.80560.47522−4.32304.1070.20.28240.07543−4.3233.97500.70.86370.21534−4.323−3.97500.70.74180.19805−4.32304.1071.21.92110.49906−4.3233.975−4.1070.20.21570.05687−4.323−3.9754.1071.22.07700.53278−4.323000.20.30430.0770903.975−4.1070.70.92180.2364100−3.9754.1070.20.21950.0543110001.21.86890.46581203.97500.20.28850.0763130−3.9754.1070.70.85170.2129140000.70.70610.18341500−4.1071.22.07920.51721603.9754.1070.20.37120.0928170−3.97500.70.91070.2397180000.20.29670.0771194.3233.975−4.1070.70.78270.1957204.32304.1071.21.67410.4293214.323−3.97500.20.18580.0478224.3233.97500.70.75490.2455234.32304.1070.20.32170.0847244.323−3.975−4.1071.22.03820.5155254.3233.9754.1070.20.24890.0655264.323000.70.84380.207327000Original value1.92780.5130According to the dynamic response analysis results of the first 26 combinations, equations (7) and (8) were solved, and the regression coefficients are shown in Table 6.Table 6
Regression coefficients of equations (7) and (8).
Regression coefficientsα0α1α2α3α4α5α6α7Values0.250650.016120.041870.022900.35272−0.00330−0.00527−0.00253Regression coefficientsα8α9α10α11α012α13α14ε1Values0.91883−0.00332−0.00396−0.02488−0.00384−0.06253−0.033490.0342Regression coefficientsβ0β1β2β3β4β5β6β7Values0.063460.004240.01102−0.006540.090440.000830.00132−0.00067Regression coefficientsβ8β1β2β3β4β5β6ε2Values0.24502−0.00085−0.00102−0.00479−0.000970.01645−0.008370.0169


## 4. Embankment Seismic Fragility Assessment

### 4.1. Embankment Seismic Damage Analysis

#### 4.1.1. Impact of PGA

Based on the dynamic response analysis results, the embankment seismic damage grades are basically intact and minor damage when PGA = 0.2 g, but due to the impacts of the structural parameters and rupture fault distances,εmax has a high dispersion, ranging from 0.1219 to 0.4532. With the increase of PGA, the embankment seismic damage is intensified, most of the damage grades are minor damage and moderate damage when PGA = 0.7 g, and εmax ranges from 0.5791 to 1.0816; the damage grades are all severe damage and destruction when PGA = 1.2 g, and εmax ranges from 1.5007 to 2.3301. Figure 6 shows the embankment seismic damages of R = 10 km for the 11th, 14th, 18th, and 27th combinations. It can be seen that when the structural parameters and rupture fault distances are constant, the damage grade and PGA have a positive correlation. Embankment seismic damages are mainly manifested by the lateral displacement on the top surface of the embankment and cracks (red solid lines in the figures). Among them, all the cracks originate from the deformation of the slope and extend to the interior of the embankment. The lateral displacements and length of the cracks are also positively correlated [57–59].Figure 6
Embankment seismic damages ofR = 10 km. (a) Seismic damages of the 11th combination (F1 = F2 = F3 = 0, PGA = 1.2 g, and εmax = 1.8433). (b) Seismic damages of the 14th combination (F1 = F2 = F3 = 0, PGA = 0.7 g, and εmax = 0.6874). (c) Seismic damages of the 18th combination (F1 = F2 = F3 = 0, PGA = 1.2 g, and εmax = 0.2687). (d) Seismic damages of the 27th combination (F1 = F2 = F3 = 0, PGA = 1.197 g, and εmax = 1.8075).
(a)(b)(c)(d)


#### 4.1.2. Impact of the Rupture Fault Distance

Figure7 shows the embankment seismic damages of the 11th combination with R = 1 km, 4 km, 7 km, 10 km, 13 km, and 15 km. Combined with the analysis results of other combinations, it is noticeable that the correlation between the rupture fault distance and εmax is not obvious when the structural parameters and PGA are constant. Figure 8 shows the embankment seismic damages of the 27th combination with R = 1 km, 4 km, 7 km, 10 km, 13 km, and 15 km. It can be seen that different rupture fault distances correspond to different seismic damages to the same structure in the same earthquake, i.e., the smaller the rupture fault distances, the more severe the damages; and the larger the rupture fault distances, the milder the damages.Figure 7
Embankment seismic damages of the 11th combination (F1 = F2 = F3 = 0 and PGA = 1.2 g). (a) Seismic damages of R = 1 km (εmax = 1.7913). (b) Seismic damages of R = 4 km (εmax = 1.8931). (c) Seismic damages of R = 7 km (εmax = 1.9736). (d) Seismic damages of R = 10 km (εmax = 1.8433). (e) Seismic damages of R = 13 km (εmax = 1.8408). (f) Seismic damages of R = 15 km (εmax = 1.8297).
(a)(b)(c)(d)(e)(f)Figure 8
Embankment seismic damages of the 27th combination (F1 = F2 = F3 = 0 and PGA = original value). (a) Seismic damages of R = 1 km (PGA = 1.233 g and εmax = 0.2687). (b) Seismic damages of R = 4 km (PGA = 1.221 g and εmax = 2.0836). (c) Seismic damages of R = 7 km (PGA = 1.209 g and εmax = 1.9533). (d) Seismic damages of R = 10 km (PGA = 1.197 g and εmax = 1.8075). (e) Seismic damages of R = 13 km (PGA = 1.182 g and εmax = 1.6656). (f) Seismic damages of R = 15 km (PGA = 1.173 g and εmax = 1.5447).
(a)(b)(c)(d)(e)(f)


#### 4.1.3. Energy Dissipation

Figure9 shows the embankment energy dissipations of the 27th combination with R = 1 km to 15 km, respectively. It can be seen that the energy dissipations are between 3.048 × 106 and 4.626 × 106 N·m; the smaller the rupture fault distances, the greater the energy dissipations; and the larger the rupture fault distances, the smaller the energy dissipations [59].Figure 9
Embankment energy dissipations of the 27th combination (F1 = F2 = F3 = 0 and PGA = original value).


### 4.2. Embankment Seismic Fragility Curves

By inputting the 10,000 sets of structural parameter combinations obtained in Section3.3 to equation (7), obtain the probabilistic seismic demand model of the embankment [60–62], as shown in the following equation:(9)lnεmax=ln1.5897+1.3457lnPGA.Substituting equation (7) into the classical calculation formula of the seismic fragility, the following equation is obtained:(10)Rf=Φln1.5897PGA1.3457/Sf0.5,where Rf is the exceeding probability of the embankment seismic damage grade f, Sf is the structural performance level shown in Table 2, that is, Sf = 2.00 when f is destruction, Sf = 1.20 when f is severe damage, Sf = 0.60 when f is moderate damage, and Sf = 0.30 when f is minor damage. The embankment seismic fragility curves plotted from equation (10) are shown in Figure 10.Figure 10
Embankment seismic fragility curves.Exceeding probabilities of each embankment seismic damage grade under different PGAs were calculated according to Figure10 [63, 64], as shown in Table 7.Table 7
Exceeding probabilities of each embankment seismic damage grade under different PGAs.
Embankment seismic damage gradesMinor damageSevere damageDestructionExceeding probabilities of each embankment seismic damage grade under different PGAs0.1 g0.0021040.0000010.0000000.0000000.2 g0.1594780.0085880.0000080.0000000.3 g0.5377090.0982430.0037040.0001080.4 g0.8075580.3024510.0284770.0017210.5 g0.9291510.5331560.0962720.0100430.6 g0.9750140.7169840.2082840.0333240.7 g0.9912270.8386160.3454970.0779260.8 g0.9968760.9111990.4847960.1446240.9 g0.9988610.9520610.6098320.2288101.0 g0.9995740.9743370.7130940.3230441.1 g0.9998360.9862820.7935970.4196901.2 g0.9999350.9926450.8538630.5125631.3 g0.9999730.9960330.8977040.5975161.4 g0.9999890.9978440.9289520.6723381.5 g0.9999950.9988170.9509070.7363281.6 g0.9999980.9993450.9661810.7898111.7 g0.9999990.9996330.9767360.8337101.8 g1.0000000.9997930.9840000.8692311.9 g1.0000000.9998820.9889870.8976512.0 g1.0000000.9999320.9924080.9201872.1 g1.0000000.9999600.9947560.9379322.2 g1.0000000.9999770.9963680.9518292.3 g1.0000000.9999860.9974770.9626652.4 g1.0000000.9999920.9982420.9710882.5 g1.0000000.9999950.9987710.9776202.6 g1.0000000.9999970.9991380.982676It can be seen from Table7 that the exceeding probability of the severe damage is 0.713094 and of the destruction is 0.323044 when PGA = 1.0 g, while according to Yin et al. [47], the two exceeding probabilities were 0.634172 and 0.265767, respectively. According to Yu et al. [65], Zhang et al. [66], and Luo et al. [67], the exceeding probability of each seismic damage grade of embankments at the epicenter of the Chi-Chi earthquake (PGA = 0.977 g) is shown in Table 8.Table 8
Exceeding probability of each seismic damage grade at the epicenter of the Chi-Chi earthquake [65–67].
Embankment seismic damage gradesMinor damageModerate damageSevere damageDestructionExceeding probabilities0.9998510.9638460.6724830.298452By comparing Tables7 and 8, it can be seen that the research results of this paper are consistent with the actual embankment seismic damage conditions of the Chi-Chi earthquake, which indicates that the method proposed in this paper is scientific and reasonable. It also shows that it would obviously overestimate the seismic performance in the embankment seismic fragility assessment without considering the uncertainties of the ground motions and structural parameters.


## 4.1. Embankment Seismic Damage Analysis

### 4.1.1. Impact of PGA

Based on the dynamic response analysis results, the embankment seismic damage grades are basically intact and minor damage when PGA = 0.2 g, but due to the impacts of the structural parameters and rupture fault distances,εmax has a high dispersion, ranging from 0.1219 to 0.4532. With the increase of PGA, the embankment seismic damage is intensified, most of the damage grades are minor damage and moderate damage when PGA = 0.7 g, and εmax ranges from 0.5791 to 1.0816; the damage grades are all severe damage and destruction when PGA = 1.2 g, and εmax ranges from 1.5007 to 2.3301. Figure 6 shows the embankment seismic damages of R = 10 km for the 11th, 14th, 18th, and 27th combinations. It can be seen that when the structural parameters and rupture fault distances are constant, the damage grade and PGA have a positive correlation. Embankment seismic damages are mainly manifested by the lateral displacement on the top surface of the embankment and cracks (red solid lines in the figures). Among them, all the cracks originate from the deformation of the slope and extend to the interior of the embankment. The lateral displacements and length of the cracks are also positively correlated [57–59].Figure 6
Embankment seismic damages ofR = 10 km. (a) Seismic damages of the 11th combination (F1 = F2 = F3 = 0, PGA = 1.2 g, and εmax = 1.8433). (b) Seismic damages of the 14th combination (F1 = F2 = F3 = 0, PGA = 0.7 g, and εmax = 0.6874). (c) Seismic damages of the 18th combination (F1 = F2 = F3 = 0, PGA = 1.2 g, and εmax = 0.2687). (d) Seismic damages of the 27th combination (F1 = F2 = F3 = 0, PGA = 1.197 g, and εmax = 1.8075).
(a)(b)(c)(d)


### 4.1.2. Impact of the Rupture Fault Distance

Figure7 shows the embankment seismic damages of the 11th combination with R = 1 km, 4 km, 7 km, 10 km, 13 km, and 15 km. Combined with the analysis results of other combinations, it is noticeable that the correlation between the rupture fault distance and εmax is not obvious when the structural parameters and PGA are constant. Figure 8 shows the embankment seismic damages of the 27th combination with R = 1 km, 4 km, 7 km, 10 km, 13 km, and 15 km. It can be seen that different rupture fault distances correspond to different seismic damages to the same structure in the same earthquake, i.e., the smaller the rupture fault distances, the more severe the damages; and the larger the rupture fault distances, the milder the damages.Figure 7
Embankment seismic damages of the 11th combination (F1 = F2 = F3 = 0 and PGA = 1.2 g). (a) Seismic damages of R = 1 km (εmax = 1.7913). (b) Seismic damages of R = 4 km (εmax = 1.8931). (c) Seismic damages of R = 7 km (εmax = 1.9736). (d) Seismic damages of R = 10 km (εmax = 1.8433). (e) Seismic damages of R = 13 km (εmax = 1.8408). (f) Seismic damages of R = 15 km (εmax = 1.8297).
(a)(b)(c)(d)(e)(f)Figure 8
Embankment seismic damages of the 27th combination (F1 = F2 = F3 = 0 and PGA = original value). (a) Seismic damages of R = 1 km (PGA = 1.233 g and εmax = 0.2687). (b) Seismic damages of R = 4 km (PGA = 1.221 g and εmax = 2.0836). (c) Seismic damages of R = 7 km (PGA = 1.209 g and εmax = 1.9533). (d) Seismic damages of R = 10 km (PGA = 1.197 g and εmax = 1.8075). (e) Seismic damages of R = 13 km (PGA = 1.182 g and εmax = 1.6656). (f) Seismic damages of R = 15 km (PGA = 1.173 g and εmax = 1.5447).
(a)(b)(c)(d)(e)(f)


### 4.1.3. Energy Dissipation

Figure9 shows the embankment energy dissipations of the 27th combination with R = 1 km to 15 km, respectively. It can be seen that the energy dissipations are between 3.048 × 106 and 4.626 × 106 N·m; the smaller the rupture fault distances, the greater the energy dissipations; and the larger the rupture fault distances, the smaller the energy dissipations [59].Figure 9
Embankment energy dissipations of the 27th combination (F1 = F2 = F3 = 0 and PGA = original value).


## 4.1.1. Impact of PGA

Based on the dynamic response analysis results, the embankment seismic damage grades are basically intact and minor damage when PGA = 0.2 g, but due to the impacts of the structural parameters and rupture fault distances,εmax has a high dispersion, ranging from 0.1219 to 0.4532. With the increase of PGA, the embankment seismic damage is intensified, most of the damage grades are minor damage and moderate damage when PGA = 0.7 g, and εmax ranges from 0.5791 to 1.0816; the damage grades are all severe damage and destruction when PGA = 1.2 g, and εmax ranges from 1.5007 to 2.3301. Figure 6 shows the embankment seismic damages of R = 10 km for the 11th, 14th, 18th, and 27th combinations. It can be seen that when the structural parameters and rupture fault distances are constant, the damage grade and PGA have a positive correlation. Embankment seismic damages are mainly manifested by the lateral displacement on the top surface of the embankment and cracks (red solid lines in the figures). Among them, all the cracks originate from the deformation of the slope and extend to the interior of the embankment. The lateral displacements and length of the cracks are also positively correlated [57–59].Figure 6
Embankment seismic damages ofR = 10 km. (a) Seismic damages of the 11th combination (F1 = F2 = F3 = 0, PGA = 1.2 g, and εmax = 1.8433). (b) Seismic damages of the 14th combination (F1 = F2 = F3 = 0, PGA = 0.7 g, and εmax = 0.6874). (c) Seismic damages of the 18th combination (F1 = F2 = F3 = 0, PGA = 1.2 g, and εmax = 0.2687). (d) Seismic damages of the 27th combination (F1 = F2 = F3 = 0, PGA = 1.197 g, and εmax = 1.8075).
(a)(b)(c)(d)


## 4.1.2. Impact of the Rupture Fault Distance

Figure7 shows the embankment seismic damages of the 11th combination with R = 1 km, 4 km, 7 km, 10 km, 13 km, and 15 km. Combined with the analysis results of other combinations, it is noticeable that the correlation between the rupture fault distance and εmax is not obvious when the structural parameters and PGA are constant. Figure 8 shows the embankment seismic damages of the 27th combination with R = 1 km, 4 km, 7 km, 10 km, 13 km, and 15 km. It can be seen that different rupture fault distances correspond to different seismic damages to the same structure in the same earthquake, i.e., the smaller the rupture fault distances, the more severe the damages; and the larger the rupture fault distances, the milder the damages.Figure 7
Embankment seismic damages of the 11th combination (F1 = F2 = F3 = 0 and PGA = 1.2 g). (a) Seismic damages of R = 1 km (εmax = 1.7913). (b) Seismic damages of R = 4 km (εmax = 1.8931). (c) Seismic damages of R = 7 km (εmax = 1.9736). (d) Seismic damages of R = 10 km (εmax = 1.8433). (e) Seismic damages of R = 13 km (εmax = 1.8408). (f) Seismic damages of R = 15 km (εmax = 1.8297).
(a)(b)(c)(d)(e)(f)Figure 8
Embankment seismic damages of the 27th combination (F1 = F2 = F3 = 0 and PGA = original value). (a) Seismic damages of R = 1 km (PGA = 1.233 g and εmax = 0.2687). (b) Seismic damages of R = 4 km (PGA = 1.221 g and εmax = 2.0836). (c) Seismic damages of R = 7 km (PGA = 1.209 g and εmax = 1.9533). (d) Seismic damages of R = 10 km (PGA = 1.197 g and εmax = 1.8075). (e) Seismic damages of R = 13 km (PGA = 1.182 g and εmax = 1.6656). (f) Seismic damages of R = 15 km (PGA = 1.173 g and εmax = 1.5447).
(a)(b)(c)(d)(e)(f)


## 4.1.3. Energy Dissipation

Figure9 shows the embankment energy dissipations of the 27th combination with R = 1 km to 15 km, respectively. It can be seen that the energy dissipations are between 3.048 × 106 and 4.626 × 106 N·m; the smaller the rupture fault distances, the greater the energy dissipations; and the larger the rupture fault distances, the smaller the energy dissipations [59].Figure 9
Embankment energy dissipations of the 27th combination (F1 = F2 = F3 = 0 and PGA = original value).


## 4.2. Embankment Seismic Fragility Curves

By inputting the 10,000 sets of structural parameter combinations obtained in Section3.3 to equation (7), obtain the probabilistic seismic demand model of the embankment [60–62], as shown in the following equation:(9)lnεmax=ln1.5897+1.3457lnPGA.Substituting equation (7) into the classical calculation formula of the seismic fragility, the following equation is obtained:(10)Rf=Φln1.5897PGA1.3457/Sf0.5,where Rf is the exceeding probability of the embankment seismic damage grade f, Sf is the structural performance level shown in Table 2, that is, Sf = 2.00 when f is destruction, Sf = 1.20 when f is severe damage, Sf = 0.60 when f is moderate damage, and Sf = 0.30 when f is minor damage. The embankment seismic fragility curves plotted from equation (10) are shown in Figure 10.Figure 10
Embankment seismic fragility curves.Exceeding probabilities of each embankment seismic damage grade under different PGAs were calculated according to Figure10 [63, 64], as shown in Table 7.Table 7
Exceeding probabilities of each embankment seismic damage grade under different PGAs.
Embankment seismic damage gradesMinor damageSevere damageDestructionExceeding probabilities of each embankment seismic damage grade under different PGAs0.1 g0.0021040.0000010.0000000.0000000.2 g0.1594780.0085880.0000080.0000000.3 g0.5377090.0982430.0037040.0001080.4 g0.8075580.3024510.0284770.0017210.5 g0.9291510.5331560.0962720.0100430.6 g0.9750140.7169840.2082840.0333240.7 g0.9912270.8386160.3454970.0779260.8 g0.9968760.9111990.4847960.1446240.9 g0.9988610.9520610.6098320.2288101.0 g0.9995740.9743370.7130940.3230441.1 g0.9998360.9862820.7935970.4196901.2 g0.9999350.9926450.8538630.5125631.3 g0.9999730.9960330.8977040.5975161.4 g0.9999890.9978440.9289520.6723381.5 g0.9999950.9988170.9509070.7363281.6 g0.9999980.9993450.9661810.7898111.7 g0.9999990.9996330.9767360.8337101.8 g1.0000000.9997930.9840000.8692311.9 g1.0000000.9998820.9889870.8976512.0 g1.0000000.9999320.9924080.9201872.1 g1.0000000.9999600.9947560.9379322.2 g1.0000000.9999770.9963680.9518292.3 g1.0000000.9999860.9974770.9626652.4 g1.0000000.9999920.9982420.9710882.5 g1.0000000.9999950.9987710.9776202.6 g1.0000000.9999970.9991380.982676It can be seen from Table7 that the exceeding probability of the severe damage is 0.713094 and of the destruction is 0.323044 when PGA = 1.0 g, while according to Yin et al. [47], the two exceeding probabilities were 0.634172 and 0.265767, respectively. According to Yu et al. [65], Zhang et al. [66], and Luo et al. [67], the exceeding probability of each seismic damage grade of embankments at the epicenter of the Chi-Chi earthquake (PGA = 0.977 g) is shown in Table 8.Table 8
Exceeding probability of each seismic damage grade at the epicenter of the Chi-Chi earthquake [65–67].
Embankment seismic damage gradesMinor damageModerate damageSevere damageDestructionExceeding probabilities0.9998510.9638460.6724830.298452By comparing Tables7 and 8, it can be seen that the research results of this paper are consistent with the actual embankment seismic damage conditions of the Chi-Chi earthquake, which indicates that the method proposed in this paper is scientific and reasonable. It also shows that it would obviously overestimate the seismic performance in the embankment seismic fragility assessment without considering the uncertainties of the ground motions and structural parameters.


## 5. Conclusion

In this paper, stochastic modeling and artificial synthesizing method of the near-fault pulse-like ground motions was proposed, and a total of 12 structural parameters were selected as the design variables by taking the Xi’an-Baoji expressway K1125 + 470 embankment as the research object. In order to eliminate the linear correlations of these parameters, 3 principal components with large impacts on the embankment seismic fragility were extracted based on the principal component analysis. Mapping relationships among the principal components and embankment seismic damages were analyzed using the uniform design response surface method, and the seismic fragility assessment was carried out and the fragility curves were plotted. The following conclusions can be drawn:(1)
Since the orientation of the apparatus in a seismic station is arbitrary, the recorded ground motions at the time of the earthquake are not necessarily the maximum, especially for the near-fault pulse-like ground motions, the seismic intensities in different directions are obviously different. The stochastic pulse models of different rupture fault distances were established by considering the directional impacts of the pulse-like ground motions, and 15 ground motions with the rupture fault distance ranging from 1 to 15 km were synthesized by taking the Chi-Chi earthquake in Taiwan, China, as an example.(2)
Embankment seismic damages are mainly manifested by the lateral displacement on the top surface of the embankment and cracks. All the cracks originate from the deformation of the slope and extend to the interior of the embankment, and the lateral displacements and length of the cracks are positively correlated. It can also be seen that different rupture fault distances correspond to different seismic damages and energy dissipations to the same structure in the same earthquake, i.e., the smaller the rupture fault distances, the more severe the damages and the greater the energy dissipations; and the larger the rupture fault distances, the milder the damages and the smaller the energy dissipations.(3)
The research results of the embankment seismic fragility assessment are consistent with the actual embankment seismic damage conditions of the Chi-Chi earthquake, indicating that the method proposed in this paper is scientific and reasonable. It also shows that it would obviously overestimate the seismic performance in the embankment seismic fragility assessment without considering the uncertainties of the ground motions and structural parameters.(4)
Supporting structures such as retaining wall and antislide pile have positive effects on improving the embankment seismic performance. Defining the characteristics of the damages of the supporting structure-embankment fill-soil foundation system under different ground motions is a prerequisite to uncover the dynamic coupling mechanism between the supporting structure and the embankment, but relevant researches have not been carried out yet.



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*Source: 8849043-2021-03-25.xml*