## 2 Methodology  
The framework depicted in Figure 1 shows the proposed methodology for a parametric fragility assessment of MAB under local scouring. Initially, a case study is defined, and the preliminary data is gathered from maintenance records, previous assessment studies, and rehabilitation projects performed on the case study [25]. Subsequently, a 2D numerical model is built based on the collected information using limit state-based software. In parallel, a scour analysis is performed, in which several random variables related to the scouring process are reviewed, assessed, and introduced for the computation of the probabilistic distribution of the scour depth [26]. For the probabilistic analysis, UQlab, a general-purpose Uncertainty Quantification framework running on MATLAB, was employed [27].  
It should be noted that several scour depth prediction formulas are available in the literature (for an overview, refer to [28]). For the present study, the formulation from the Hydraulic Engineering Circular Number 18 (HEC-18) from the Federal Highway Administration is used to estimate the local scour depth at bridge piers [26]. The HEC-18 pier scour depth given in Equation (1) is computed based on the upstream flow depth \(y_{0}\), pier geometry \(D\), Froude number \(Fr\), flow velocity \(V\), flow depth \(y_{0}\), and four correction  
Figure 1: Framework for parametric fragility assessment of MAB under local scouring.  
factors. The correction factors are related to pier shape \(K_{1}\), angle of attack \(K_{2}\), bed condition \(K_{3}\) and sediment size \(K_{4}\)[26].  
\[\frac{y_{s}}{y_{0}}=2.0K_{1}K_{2}K_{3}K_{4}\bigg{(}\frac{D}{y_{0}}\bigg{)}^{0.6.5}Fr^{0.43} \tag{1}\]  
Subsequently, in the variable definition stage, after scour analysis is performed, the physical properties used for describing the mechanical behavior of materials are considered. Then, a surrogate model is constructed, which is later used to perform a sensitivity analysis that identifies the most influential variables on the ultimate capacity of the bridge, which enables the reduction of the computational costs of the full probabilistic analysis [9; 24; 29]. Finally, the reliability index and the failure probability are computed using the subset simulation technique [30] by using the performance function, \(G\), where \(R\) and \(S\) are the capacity and demand, respectively. Therefore, when the load actions overcome the load-carrying capacities (\(G<0\)), system failure \(P_{f}\) is achieved:  
\[P_{f}=P(G<0)=P(R-S<0) \tag{2}\]  
The reliability index can be defined as the inverse of the coefficient of variation of the Cumulative Distribution Function of the limit state function [31; 32]. In Equation (3), \(\beta\) is the reliability index, \(\Phi\) is the normal distribution and \(P_{f}\) is the probability of failure of the structure.  
\[\beta=-\Phi^{-1}\bigg{(}P_{f}\bigg{)} \tag{3}\]  
Next, fragility analysis is performed using the discharge as the intensity measure. It should be noted that for increasing values of discharge, the probabilistic distribution of the scour depth, as well as the scour geometry, is recalculated. Subsequently, a lognormal adjustment is used to better represent and ease the implementation of the flood fragility curve in the decision-making process related to bridge performance [13].  
Lastly, the procedure may be implemented for different flow angles of attack of the case study, which allows defining a fragility surface. Furthermore, geotechnical properties related to the soil materials of the case study may be changed, which allows a parametric analysis for assessing and identifying the influence of geotechnical parameters on the in-plane bridge fragility under local scouring from a case study.