Earthquakes can have significant impacts on transportation networks because of the physical damage they can cause to bridges. Hence, it is essential to assess the seismic risk of a bridge transportation network accurately. However, this is a challenging task because it requires estimating the performance of a bridge transportation network at the system level. Moreover, it is necessary to deal with various possible earthquake scenarios and the associated damage states of component bridges considering the uncertainty of earthquake locations and magnitudes. To overcome these challenges, this study proposes a new method of systemlevel seismic risk assessment for bridge transportation networks employing probabilistic seismic hazard analysis (PSHA). The proposed method consists of three steps: (1) seismic fragility estimation of the bridges based on PSHA; (2) systemlevel performance estimation using a matrixbased framework; and (3) seismic risk assessment based on the total probability theorem. In the proposed method, PSHA enables the seismic fragility estimation of the component bridges considering the uncertainty of earthquake locations and magnitudes, and it is systemically used to carry out a posthazard bridge network flow capacity analysis by employing the matrixbased framework. The proposed method provides statistical moments of the network performance and component importance measures, which can be used by decision makers to reduce the seismic risk of a target area. To test the proposed method, it is applied to a numerical example of an actual transportation network in South Korea. In the seismic risk assessment of the example, PSHA is successfully integrated with the matrixbased framework to perform system reliability analysis in a computationally efficient manner.
Natural disasters can have significant impacts on infrastructure, such as transportation, electricity, gas, and water distribution networks, causing structural damage and economic losses in commercial and residential activities. In particular, earthquakes that occurred in recent years caused serious physical damage and disruption to transportation networks. Damage to transportation system is a major concern because it can impose extra burden on the other lifelines [
The objective of a seismic risk assessment is to obtain useful information for riskinformed decision making on seismic hazard mitigation and disaster management. For this reason, seismic risk assessment of critical infrastructure has been conducted extensively. Nuti et al. [
For accurate seismic risk assessment, it is necessary to deal with the uncertainties associated with earthquakes and infrastructure responses [
To overcome these challenges, a few nonsamplingbased approaches have been developed. Li and He [
Meanwhile, most of the previous studies conducted seismic risk assessments without probabilistic seismic hazard analysis (PSHA). PSHA can play a crucial role in considering earthquake uncertainty because it is useful for determining the uncertainty in earthquake locations and magnitudes in a target region [
For this reason, this paper proposes a new method of systemlevel seismic risk assessment for bridge transportation networks employing PSHA. The proposed method consists of three steps: (1) seismic fragility estimation of the bridges based on PSHA; (2) systemlevel performance estimation using the matrixbased framework of the MSR method; and (3) seismic risk assessment based on the total probability theorem. In the proposed method, PSHA enables the seismic fragility estimation of the component bridges considering the uncertainty of earthquake locations and magnitudes, and it is systemically used to carry out a posthazard bridge network flow capacity analysis employing the matrixbased framework of the MSR method. The proposed method provides statistical moments of the network performance and component importance measures, which are useful for decision makers to reduce the seismic risk of the target area.
In PSHA, a seismic hazard is defined as a physical phenomenon such as ground shaking or failure affected by an earthquake which can cause serious effects on human activities [
In the proposed method, the uncertainty of earthquake locations and magnitudes are identified using PSHA based on past earthquake records. The uncertainty of earthquake locations can be considered by using the epicenters of past earthquakes for the seismic risk assessment. Meanwhile, to account for the uncertainty of earthquake magnitudes, a series of modeling procedures is required. In this paper, one such modeling procedure is briefly introduced, whereas more details on PSHA can be found in Baker [
To account for the uncertainty of earthquake magnitudes, as the first step, the occurrence rate of earthquakes in a target region is assumed to follow the GutenbergRichter (GR) recurrence law [
When the minimum and maximum magnitudes are determined using equation (
Once the bounded PDF is obtained, to generate possible earthquake scenarios with varying magnitudes as an input of seismic risk assessment, the continuous distribution of earthquake magnitudes needs to be converted into a discrete set of magnitudes of interest. The probabilities of occurrence, according to this discrete set of magnitudes, represent only a partial distribution of magnitude at a site. Then, a normalizing process that divides all of the cumulated values by their sum is needed so that the sum of the probability distribution in the partial magnitudes becomes 1.0.
After modeling the probability distribution of earthquake magnitudes, the associated distances from the earthquake source to the bridges (i.e., sourcetosite distances) of the target transportation network and ground motion intensities need to be analyzed. Given the earthquake magnitude and location, the ground motion intensities at different bridges must be analyzed based on a seismic attenuation model. A representative model is the ground motion prediction equation (GMPE). In this equation, the ground motion intensity is expressed as a function of several parameters including earthquake magnitude, distance, and local site effects [
In this study, as in HAZUSMH [
In addition, the inter and intraevents parameters in equation (
As described so far, PSHA enables the consideration of earthquake uncertainty. For the given location and magnitude of an earthquake, the ground shaking intensity of individual bridges is expressed in terms of SA using the GMPE in equation (
As previously mentioned, samplingbased approaches are often used for the performance estimation of lifeline networks subjected to earthquakes. However, these approaches may prevent rapid risk assessment and make providing useful probabilistic measures difficult [
A bridge transportation network is considered, consisting of
Let
In the MSR method, to carry out a posthazard flow capacity analysis, a certain corresponding quantity can be estimated by using the matrixbased framework. Therefore, a new column vector
As results of the probability vector
A potential drawback of the MSR method is that the sizes of the vectors and matrices increase exponentially as the number of components (e.g., the number of bridges in a transportation network) increases [
Furthermore, an importance measure of components can also be evaluated using a matrixbased formulation. For regional authorities who need to make decisions for allocating budgets and other resources, it is important to identify the important locations of a transportation network [
The proposed method aims at estimating the performance of a bridge transportation network after a seismic event considering earthquake uncertainty. For this purpose, the proposed method introduces PSHA to deal with the uncertainty of earthquake magnitudes and locations. The performance of a bridge transportation network which is subjected to earthquakes also becomes uncertain, and the proposed method employs the matrixbased framework of the MSR method to evaluate this uncertainty at the system level.
The MSR method was originally developed to perform system reliability analyses of various structures [
As stated previously, by assuming that all the components are statistically independent, the basic MECE events can be easily computed by using the matrix calculation proposed in equation (
In the proposed method, the earthquake magnitude and location are introduced as CSRVs that influence the damage states of individual bridges. When the uncertainty of earthquake magnitudes and locations is characterized by PSHA and defined as CSRVs, using the total probability theorem, the probability of a system can be expressed as
Since the damage states of individual bridges become conditionally statistically independent given the earthquake magnitude and location, the conditional probability vector
Hence, using the total probability theorem, the statistical parameters can be obtained as
Similarly, the reduction factor RF can be calculated as
The proposed method consists of three steps: (1) seismic fragility estimation of the bridges based on PSHA, (2) systemlevel performance estimation using the matrixbased framework of the MSR method, and (3) seismic risk assessment based on the total probability theorem. In the proposed method, PSHA enables the seismic fragility estimation of the component bridges considering the uncertainty of earthquake locations and magnitudes, and it is systemically used to carry out a postearthquake bridge network flow capacity analysis employing the MSR method.
The MSR method provides an efficient framework for seismic risk assessment which performs separate calculations for seismic hazard and network flow capacity analyses. PSHA contains the investigation of earthquake generation using several proposed relations provided in equations (
Flow chart of the proposed seismic risk assessment employing PSHA and the MSR method.
To start the analysis employing the proposed method, it is first necessary to collect input data in the form of information related to the study area. The input data can be classified into three groups: exposure data, hazard data, and structural vulnerability data. The exposure data contains the topology information of the target transportation network such as the number of nodes and links in the target region. The past earthquake data are also necessary to identify the earthquake uncertainty using PSHA. The hazard data contain the mean occurrence rate of earthquakes with varying magnitudes and the seismic attenuation law (i.e., GMPE) with the spatial correlations described in equations (
The next step is to perform seismic risk assessment employing PSHA and the matrixbased framework of the MSR method. First, from the past earthquake data, earthquake sources are analyzed using the GR relation law. The purpose of PSHA is to identify uncertainties related to the earthquake itself and calculate the resulting intensity of ground motion. Therefore, final results are the PDF of earthquake magnitude and the probability of the different damage states of the component bridges. After performing the seismic hazard analysis, the proposed approach predicts the posthazard flow capacity of a transportation network for given magnitudes and locations of earthquake. In this process, the quantity vector
Lastly, the conditional probability vector construction is repeated for various earthquake scenarios with different earthquake magnitudes (
To test the proposed method, it is applied to an actual transportation network around Pohang city, South Korea. The study area is located in the southeast of the Korean Peninsula, and it experienced a 5.4magnitude earthquake in 2017 [
Figure
Network map of the Pohang bridge transportation network.
Locations of the bridges.
Bridge no.  Connecting nodes 

1  (14,16) 
2  (19,20) 
3  (25,26) 
4  (20,21) 
5  (29,30) 
6  (33,36) 
7  (15,23) 
8  (28,29) 
9  (4,7) 
10  (10,12) 
Structural information on the bridges.
Bridge no.  Total length (m)  Width (m)  Maximum span length (m)  Type  Year of construction  Design 

1  169.7  20.5  25.7  PSCI  1992  Conventional 
2  345  7.5  60  Steel box  2006  Seismic 
3  455  28  35  PSCI  2009  Seismic 
4  300  21  40  PSCI  2011  Seismic 
5  25  8  12.5  RC slab  1990  Conventional 
6  140  20  50  Steel box  2012  Seismic 
7  102  9.5  14.6  RC slab  1992  Conventional 
8  125.2  24  26  Steel plate  1975  Conventional 
9  480  12.1  60  Steel box  2004  Seismic 
10  115  8.3  45  Steel box  2004  Seismic 
Maximum flow capacity of links in the Pohang transportation network.
Maximum flow capacity (number of vehicles per hour)  Link numbers 

2200  (2,5), (4,5), (5,6), (6,11), (10,11), (11,14), (14,15), (15,23), (22,23), (23,25), (22,28), (28,29), (29,30), (30,32) 
4400  (1,2), (2,3), (2,4), (4,7), (7,10), (13,14), (14,16), (16,17), (17,18), (17,21), (18,19), (19,20), (19,24), (20,21), (21,22), (24,25), (25,26), (26,27), (27,35), (29,31), (31,32), (31,33), (32,34), (33,34), (33,36), (35,36), (35,37) 
6520  (8,9), (9,10), (10,12), (12,18), (27,37) 
To account for uncertainty in the seismic damage states of bridges, seismic fragility curves are introduced. Seismic fragility is defined as the conditional probability that the demand of a structure exceeds a specified threshold for a given earthquake intensity [
In this example, five damage states of no, slight, moderate, extensive, and complete damage are assumed (i.e.,
Damage states and associated flow capacities.
Damage states  Description  Flow capacities (%) 

No  —  100 
Slight  Any column experiencing minor cracking  75 
Moderate  Any column experiencing moderate cracking  50 
Extensive  Any column degrading without collapse  25 
Complete  Any column collapsing and connection  0 
To identify the earthquake uncertainty in the target region, past earthquake data (with magnitude,
Earthquake events with magnitude 3.0 and above in the study area.
No.  Date  Origin time  Latitude  Longitude  Depth (km) 


1  14 Apr. 1981  11 : 47  35.90  130.10  7  4.8 
2  27 Aug. 1981  21 : 35  35.80  129.80  7  3.5 
3  10 Dec. 1985  21 : 42  35.80  129.70  7  3.2 
4  17 Mar. 1986  11 : 52  35.90  129.50  7  3.2 
5  6 Oct. 1987  7 : 04  35.90  129.90  7  3.1 
6  6 Oct. 1987  23 : 36  36.20  130.10  7  3.5 
7  22 Oct. 1990  18 : 09  35.90  130.00  7  3.4 
8  24 Apr. 1999  1 : 35  36.00  129.30  7  3.2 
9  9 Jul. 2002  4 : 01  35.90  129.60  7  3.8 
10  28 Mar. 2011  13 : 50  35.97  129.95  7  3.2 
11  15 Apr. 2017  11 : 31  36.11  129.36  7  3.1 
12  15 Nov. 2017  14 : 29  36.11  129.37  7  5.4 
13  15 Nov. 2017  14 : 32  36.10  129.36  8  3.6 
14  15 Nov. 2017  15 : 09  36.09  129.34  8  3.5 
15  15 Nov. 2017  16 : 49  36.12  129.36  10  4.3 
16  16 Nov. 2017  9 : 02  36.12  129.37  8  3.6 
17  19 Nov. 2017  23 : 45  36.12  129.36  9  3.5 
18  20 Nov. 2017  6 : 05  36.14  129.36  12  3.6 
19  25 Dec. 2017  16 : 19  36.11  129.36  10  3.5 
20  11 Feb. 2018  5 : 03  36.08  129.33  9  4.6 
Locations of the earthquake epicenters and bridges around Pohang, South Korea.
Based on these earthquake records, the earthquake uncertainty is identified using the GR law given in equation (
Exceedance rate of earthquake magnitude and the regression function.
Then, using equation (
Discrete occurrence probabilities for a range of 4.5 and 7.5.
Once the occurrence probability of earthquake magnitude is constructed, the earthquake intensities at the ten bridge locations are calculated using equation (
First, for each set of earthquake magnitude and location, the proposed method provides the probability mass function (PMF) and cumulative distribution function (CDF) of the maximum flow capacity of the network. For example, for EQ8’s epicenter, Figure
PMF of the maximum flow capacity of the network for an earthquake magnitude of
CDFs of the maximum flow capacity of the network for
To verify the analysis results of the proposed method, an MCS with 3 × 10^{5} samples was conducted for
Mean (a) and standard deviation (b) of the maximum flow capacity of the network from the proposed method and the MCS with an increasing number of samples.
For earthquake magnitudes between 4.5 and 7.5, the statistical moments of the maximum flow capacity according to the twenty earthquake sources are also obtained from the proposed method. Figure
Mean of the maximum flow capacity for all earthquake scenarios with varying earthquake magnitudes for (a) Earthquakes 1 to 5, (b) Earthquakes 6 to 10, (c) Earthquakes 11 to 15, and (d) Earthquakes 16 to 20.
For a better comparison, Figure
Mean flow capacities for given earthquake locations and for uncertain location.
Similarly, the standard deviation and the c.o.v. of the flow capacities of the twenty earthquake scenarios and their averages (colored blue) for uncertain earthquake location can be calculated, which are presented in Figures
The standard deviation of the maximum flow capacity of the network for given earthquake locations and for uncertain location.
The c.o.v. of the maximum flow capacity of the network for given earthquake locations and for uncertain location.
Furthermore, the mean, standard deviation, and c.o.v. of the maximum flow capacity of the network for an uncertain earthquake (i.e., uncertain magnitudes and locations of earthquake) can be calculated using equation (
Statistical moments of the maximum flow capacity of the network for uncertain earthquake.
Statistical moments  Results 

Mean ( 
4076.077 
Standard deviation ( 
57.263 
Coefficient of variation ( 
0.015 
It is noteworthy that samplingbased approaches would be inefficient for this sort of parametric study because network flow capacity analysis should be conducted for all of the individual magnitude values and locations of earthquake. On the contrary, the proposed method enables to perform this parametric study efficiently.
The proposed method also enables the computation of the reduction factor RF using equation (
Reduction factors for EQ8, EQ20, and uncertain earthquake.
Furthermore, one important advantage of the proposed method is its efficiency, which makes it feasible to calculate the sensitivity of the statistical moments with respect to each parameter of the bridge fragility curves. Although samplingbased approaches such as the MCS may not be feasible for such a sensitivity analysis, the proposed method enables a sensitivity calculation, by applying the method with the finite difference method. A fragility curve is determined by a median and a logstandard deviation [
Mean sensitivity of the maximum flow capacity of the network with respect to the median of the fragility curve.
Bridge  Fragility curve  Mean sensitivity  

For uncertain earthquake  For EQ8  For EQ20  
3  Slight  5.69  5.56  5.65 
Moderate  6.14  15.82  5.00  
Extensive  6.53  24.37  5.64  
Complete  2.44  7.30  2.21  
4  Slight  5.60  5.53  5.62 
Moderate  3.66  7.45  3.52  
Extensive  2.90  9.12  2.74  
Complete  1.91  6.79  1.78 
In this study, a new method has been proposed for systemlevel seismic risk assessment of bridge transportation networks considering earthquake uncertainty. The proposed method consists of three steps: (1) component failure probability calculation of bridges based on PSHA; (2) systemlevel performance estimation of the transportation network using the matrixbased framework of the MSR method; and (3) seismic risk assessment based on the total probability theorem. In the proposed method, PSHA enables the seismic fragility estimation of the component bridges considering the uncertainty of earthquake locations and magnitudes, and it is systemically used to carry out the estimation of the postearthquake performance (i.e., maximum flow capacity) of the target bridge network by employing the matrixbased framework. The matrixbased framework enables efficient evaluations of the network performance with various magnitudes and locations of earthquake, without performing deterministic analyses of maximum flow capacity repeatedly.
The proposed method has been tested through its application to a numerical example of an actual transportation network in Korea, considering various earthquake scenarios with twenty locations and a range of earthquake magnitude from 4.5 to 7.5, and ten bridges with five damage states. As a result, the probabilistic distributions and statistical moments of the maximum flow capacity are obtained. It was observed that as the earthquake magnitude increases, the mean value of the maximum flow capacity of the network decreases but the uncertainty of the maximum flow capacity represented by the c.o.v. increases. In addition, the mean, standard deviation, and c.o.v. of the maximum flow capacity of the network for an uncertain earthquake are computed. The proposed method also enables the computation of the reduction factors for all bridges, and it was observed that Bridges 3, 5, 6, 9, and 10 are relatively important in the target transportation network. The proposed method can also provide the mean sensitivity of the maximum flow capacity of the network with respect to the median of each fragility curve. Based on the analysis results, it has been successfully demonstrated that the proposed method enables the systemlevel seismic risk assessment of bridge transportation networks considering the earthquake uncertainty.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was supported by a grant (19SCIPB14694602) from the Construction Technology Research Program funded by the Ministry of Land, Infrastructure and Transport of Korean government. This work was also supported by the 2019 Research Fund (1.190011.01) of UNIST (Ulsan National Institute of Science and Technology).