## 3 Probabilistic Analysis  
### Scour Analysis  
Scouring can be formed around the pier, i.e., at the laterals of the pier, due to the bifurcation of the flow after impacting with the frontal (upstream) face of the pier [39]. Different scour profiles representing the incremental nature of the scouring process, both symmetrical and asymmetrical, have been used in the failure analysis of MABs, with shapes that are based on the soil's friction angle and the increasing water flow values [5; 18]. Moreover, pier geometry, and more specifically, pier shape, directly affects the development of the scour profile around the pier. For instance, it was found that for piers with sharp triangular noses, lateral scour occurs symmetrically and overall produces less turbulence strength and consequently less scour [39].  
In this study, different scour profile shapes are used to represent the propagation of each level of scour depth (see Figure 4); Stage 0 shows the case with no scour, and stages 1 to 3 show values of scour depth smaller, equal, and larger than the footing thickness (\(F_{t}\)), respectively. It was found that for scour profiles where the scour depth does not erode the soil beneath the foundation base, its influence on the structural response of the MABs is neglectable [5; 6; 7]. Nevertheless, when that limit is surpassed, settlement increases, leading to cracking of the arch barrel.  
The variables involved in the computation of scour depth are gathered; then, the ones considered random are introduced in the UQlab software, which was implemented using a script with its probabilistic distribution [27]. Next, the Monte Carlo sampling method is used for its ability to factor the full range of values of the uncertainties of the parameters to obtain a 1000 value sample, which will be used to introduce randomness to the computation of the scour depth using the HEC-18 formulation [40]. When an angle of attack is defined, asymmetric scour is being considered, therefore creating a dragging force on the pier surface, which is assumed to be rectangular due to flow velocity being constant along the flow depth [41]. Finally, model uncertainties are introduced to account for the reliability of the HEC-18 formulation [26; 42]. Parameter uncertainties were considered in the assessment of the scour depth. These parameters are related to geometric parameters, such as riverbed conditions, width, longitudinal slope, pier width and length, and Manning coefficient [40]. Flow angle, which is formed between the main direction of the flow and pier alignment, was defined as follows: 0\({}^{\circ}\); 5\({}^{\circ}\); 10\({}^{\circ}\); 20\({}^{\circ}\); 30\({}^{\circ}\); and 45\({}^{\circ}\)[43], whereas the following values for the flow discharge in m\({}^{3}\)/s were used: 1; 5; 10; 25; 50; 100; 150; and 200, being addressed as Q1; Q2; Q3; Q4; Q5; Q6; Q7; and Q8, respectively. In Table 1, the probabilistic distributions and parameters of each random variable are presented.  
Figure 4: Stages of the scour depth when compared to the footing thickness (\(F_{t}\)) for the different scour scenarios: (**a**) symmetric; (**b**) asymmetric.  
A probabilistic distribution for the flow depth was obtained for each one of the considered discharge values, and it was observed that when the discharge value increases, the mean value of the flow depth also increases. Nevertheless, the value corresponding with the 95% quantile was assumed to consider the most unfavorable scenario [8]. Finally, failure for both profile types (symmetrical and asymmetrical) was achieved due to equilibrium loss. As expected, the behavior of the scour depth is like the one observed for the flow depth in which the mean value of the scour depth increases when the discharge increases, as can be seen from Figure 5.