## 2 Numerical Methods
### Numerical Procedure to Conservatively Assess the Reserve Capacity in Existing Masonry Arches  
For the generic masonry arch geometry presented in Figure 1, the method for the evaluation of the Load Rating Factor \(V\) consists in two subsequential analyses, summarized hereafter and outlined in Figure 2:  
1. Capacity analysis--Minimum depth and horizontal thrust definition
2. Being the span of the arch \(L\) and rise-to-span ratio \(f/L\) the only geometrical information available, a preliminary analysis for the minimum depth and horizontal thrust definition is necessary to univocally define the arch. These data are determined for a given design load, and the arch therefore requires such a depth to be verified (Section 2.2.1) for the given design load \(q_{d}\). The procedure is conducted as follows: 1. Evaluation of the minimum depth of the arch \(s_{min}\), for which the arch is verified for the design code, and for which there can be no further load increases, i.e., with the hypothesis of critical load multiplier \(\alpha_{cr}=1\). 2. Evaluation of the design horizontal thrust \(H_{MAX,d}\) as the maximum of all the minimum horizontal forces related to the lines of thrust, for which the equilibrium is satisfied, for every position of the load on the arch of depth \(s_{min}\), defined as the previous bullet point. Since \(s=s_{min}\) does not imply that the most restrictive value of \(H_{MAX,d}\) is found as well, a certain number of depths \(s>s_{min}\) can be stored with their associated \(H_{MAX,d}\) values, to define more configurations to employ in the following demand analysis. Small variations of \(H_{MAX,d}\) are found when the value of \(s_{min}\) isincreased, and the most severe situation should be found for values of \(s\) directly next to \(s_{min}\). 2. Demand analysis--Load Rating Factor V definition Once the capacity analysis is concluded, the demand analysis is to be conducted for each \(s\) and \(H_{MAX,d}\) configuration. For such configurations, the Load Rating Factor is evaluated for the railway line load \(q_{v}\), for which the reserve capacity should be investigated. The procedure is the following: 1. Evaluation of the critical load multiplier \(\alpha_{cr}\) of the line load, for which the arch is verified (Section 2.2.1) and the horizontal thrust \(H\) does not exceed \(H_{MAX,d}\). In the case of moving loads, the critical load multiplier \(\alpha_{cr,k}\) is evaluated for every position of the load moving on the deck, as indicated in Section 2.1. Assessment of the Load Rating Factor \(V\) as the minimum load multiplier, or rather the minimum of the multiplier \(\alpha_{cr,k}\), i.e., \(\alpha_{cr}=\min(\alpha_{cr,k})\) for moving loads or \(\alpha_{cr}\) for stationary loads. The Load Rating Factor \(V\) of the whole structure is the minimum of the factors found for every \(s\) and \(H_{MAX,d}\) configuration. It can be observed that additional inputs, i.e., the density of the materials \(\gamma\), the thickness of the arch \(B\), and the depths of the ballast \(b\) and filling layer \(c\), are necessary to evaluate the exact critical load multiplier of a single arch configuration. However, for the application of the full method (capacity analysis and demand analysis) to a stock of arches, conservative assumptions can be made on the additional data, e.g., no filling layer  
Figure 1: Geometry of a generic arch. Figure 2: Scheme of the procedure. \(c=0\), which reduces the size of the input information, yet allows the obtainment of valid estimates of the Load Rating Factor \(V\). Indeed, considering that the minimum geometry of the arch is governed by the shape of the bending moment, rather than by its magnitude, assumptions on additional data have a limited effect compared to the definition of the transiting loads. In fact, the shape of the bending moment, induced by the ballast and filling layer, is less critical than the one caused by the transiting load, since the latter is more concentrated.