Correlated Damage Probabilities of Bridges In Seismic Risk Assessment of Transportation Networks

This paper aims to identify and use a logistic regression approach to model the spatial correlation of damage 4 probabilities in expanded transportation networks. This paper uses Bayesian theory and the multinomial logistic 5 model to analyze the different damage states and damage probabilities of bridges by considering the damage 6 correlation. The correlation of the damage probabilities is considered both in different bridges of a network and in 7 the different damage states of each bridge. The correlation model of the damage probabilities is considered in the 8 seismic assessment of a part of the Tehran transportation network with 26 bridges. Moreover, the extra daily traffic 9 time (EDTT) is selected as an operational parameter of the transportation network, and the shortest path algorithm is 10 considered to select the path between two nodes. The results show that including the correlation of the damage 11 probabilities decreases the travel time of the selected network. The average decreasing in the correlated case 12 compared to the uncorrelated case, using two selected EDTT models are 53% and 71%, respectively. 13


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Urban lifeline networks are critical systems which play an important role in the recovery activity after extreme 18 events such as earthquakes, fires, and terrorist attacks (Borzoo et al., 2020b). Evaluating the seismic risk of 19 spatially-distributed systems must include seismic hazard and damage assessment of lifeline networks (Erdik, 2017).

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The estimation of direct and indirect losses in lifeline networks is generally accompanied by uncertainties. This is 21 because of uncertainties in estimating the ground-motion intensity measures (IM) and structural damages of

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Several studies investigated the effects of spatial correlation of ground-motion IMs on the loss assessment of 24 spatially distributed systems. It is shown that ignoring the spatial correlation may lead to overestimate the evaluation 25 of seismic risk in most events and underestimate in rare ones (Abbasnejadfard et al., 2020;Adachi & Ellingwood, 2008;Dong & Frangopol, 2017;Dong et al., 2014;Garakaninezhad, 2018;Jayaram & Baker, 2009;Silva, 2016). On 27 the other hand, it is shown that the spatial correlation of the earthquake IMs yields to precise risk estimation in 28 extreme earthquake events (Borzoo et al., 2020a;Garakaninezhad & Bastami, 2017;Park et al., 2007).
Semivariogram is a statistical tool that is used extensively to model the spatial correlation of ground-motion IMs 30 (Garakaninezhad & Bastami, 2017;Garakaninezhad et al., 2017;Jayaram & Baker, 2010b;Miller & Baker, 2015;31 Park et al., 2007). The semivariogram is a function of separation distance (h), and different models can be addressed 32 as the valid function for a semivariogram from which the exponential model is commonly used for the spatial 33 correlation of the ground motion IMs. The exponential semivariogram model is as: where a and b are the sill and range of the semivariogram, respectively. The relation between the semivariogram 36 and the correlation coefficient is as: (2)

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In addition to the correlation of the ground motion IMs, the correlation of damages in the components of a lifeline 39 network is considerable. The importance of damage correlation is due to the similarity of bridges in age, type of 40 structure and design codes. Also, the proximity of the bridges to others and similarity in environment condition,

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Despite the different studies that take the damage correlations into account, the cross-correlation of damage 57 probability among different damage states is not considered in any of the existing damage correlations models.

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Damage states are generally incorporated in the risk assessment process via fragility functions. Fragility functions 59 represent the probability of the damage being exceeded from a predefined threshold which is known as damage 60 state. In other words, the reliability of a structure for an input variable is presented using the fragility function

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In this paper, the multinomial models are used to investigate the effect of considering the spatial correlation of the where IM is the intensity measure and 1, 2, 3, 4 Ds Ds Ds Ds are limit states of the damage states corresponding to 89 IMs. To calculate the damage probability in the different damage states by considering the correlation of damage 90 probabilities, the multinomial model is developed as below: where, () ki ysis the number of seismic ground-motion maps that caused damage state k and lower at site i s , 93 n is the number of seismic ground-motion maps and m is the number of considered sites. The form of the 94 selected link function is as: where ij  is the spatial correlation term between observations of two sites i and j , ik  states the effects of 97 variables at the site i , () kj ys is the number of seismic ground-motion maps with damage state k and lower 98 For the spatial correlation term, the 99 exponential form was adopted as:      12 are prior independents. Therefore, we considered two normal prior distributions for these 116 parameters. So the posterior distribution of the parameters is: Therefore, since there is no closed-form expression for the posterior distribution, a Gibbs sampler is 119 constructed by using the following full conditional distributions   Figure 1.

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In this article, the proposed methodology is applied to a transportation network and the EDTT is adopted to . . Moreover, the EDTT model presented by Shinozuka et al. (2003) is also utilized in the current study. The 142 mentioned model is adopted as: x t x t x 1 0.15   (Table 1).

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To explain the effect of correlation more clearly, Table 2

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In the intended network, using Equations 11 and 12, the amounts of the EDTTs for two models are calculated.

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Based on the damage states of the bridges, four passing traffic states are considered separately which is 194 explained in Table 3

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In this paper, the spatial correlation of the damage probabilities of bridges is analyzed using the multinomial 204 models. In addition to the spatial correlation of intensities, the cross-correlation of the damage states