## 4 Application to a Stock of Bridges with Current Verification Loads  
An application of the developed method is proposed in this section. An extensive parametric analysis was performed to assess the reserve capacity of a stock of masonry arches of the railway system (Figure 7). Figure 6: Lines of thrust obtained with the numerical algorithm for each case study, in comparison with the plastic hinges position obtained with the software Ring: (**a**) case study 1, (**b**) case study 2, (**c**) case study 3, (**d**) case study 4. To obtain results on the safe side, conservative assumptions were made on the geometrical quantities, and neither the fill layer above the crown of the arch, nor the ballast, were considered (\(c=0\) m, \(b=0\) m). The arch thickness was set as \(B=3.5\) m. The unit weights \(\gamma_{AR}=20\) kN/m\({}^{3}\), \(\gamma_{BALL}=18\) kN/m\({}^{3}\), and \(\gamma_{FILL}=18\) kN/m\({}^{3}\) were considered for the masonry, ballast, and fill layer, respectively, and the train-track load was assumed \(q_{TT}=2\) kN/m\({}^{2}\). The arches were each discretized in 40 blocks (\(n_{blocks}=40\)) and the assumptions on the load distribution were the ones described in 2.4. Arches with eight different spans \(L\) (between 5 and 50 m) and seven \(f/L\) ratios were investigated. Four design codes, from 1925 to 1945, were considered, resulting in different design loads for the existing arches, namely 1925 [60], 1926 [61], and 1945A and 1945B [62]. The reserve capacity of the arches was evaluated with respect to three actual loads acting on the infrastructures in accordance with the classification of lines, or sections of lines, as reported in [54,55]. The B2, C3, and D4 line loads shown in Figure 8 were used in this study. The load configurations employed for all these design and line loads were single or multiple concentrated forces, with the only exception of the 1926 code, which recommends the only use of distributed loads, as shown in Figure 8.  
In Figure 8 the loads are shown in tons (t) and tons per square meter (t/m\({}^{2}\)), whereas the interaxle spacing is shown in centimeters (cm). The stability of the arches is governed by the shape, rather than the magnitude of the bending moment. Therefore, only load configurations consisting of single wagons were considered in the analyses, since they lead to more uneven distributions of the bending moment and thus were judged to be more critical for stability. Nevertheless, the developed method allows for the use of different load layouts, if more extensive parametric analyses are intended. Figure 7: Inputs of the parametric analysis. Figure 8: Design and verification loads used for parametric analysis (load weights in tons or tons per square meter). For exemplificative purposes, the procedure is illustrated for the 1945A design load and the B2 line. Firstly, the design arch depth \(s\) was sought for each one of the different load configurations associated with the 1945A trains. Three load configurations were employed, consisting of a single load and the loads representing two single train components, namely a locomotive and a wagon. The minimum depth \(s_{min}\), together with the associated value of horizontal thrust \(H_{MAX,d}\), is reported for arches with \(L=5\) m in Table 1. The maximum among the \(s_{min}\) of each load configuration was selected to define the arch geometry to be verified, since it represents the minimum design depth to comply with the 1945A code. However, choosing an arch with the minimum depth \(s_{min}\) did not imply that the lowest value of \(H_{MAX,d}\) was found as well. For this reason, 15 depths from \(s_{min}\) to \(s_{min}+0.15\) m, with the associated \(H_{MAX,d}\) values, were stored, defining a list of \(s\)-\(H_{MAX,d}\) configurations to be verified. In detail, for each configuration, the B2 load was moved along the span, and for every \(k\)-th load position the value of \(\alpha_{cr,k}\) was determined.