# Safety Assessment of Masonry Arch Bridges Considering the Fracturing Benefit
## 3 Masonry Arch Bridge Assessment: Elastic-Fracture-Plastic Transitions
The first case study refers to the Mosca bridge, located in Turin, Italy (Figure 7). The Mosca bridge is the most daring construction built in Turin in the first half of the XIXth century. The total length of the arch bridge is 129 m, and its width is 13.7 m.  
The principal structure, entirely cut in stone, is a single shallow arch of 45 m in length, with a rise of 5.5 m, and a depth of 1.5 m. The impressive arch structure shallowness ratio results to be 1/8. The structural material of the arch structure is Malanaggio stone, a greenish-gray gneiss with mechanical characteristics similar to those of granite. It was extracted from the main quarry of Malanaggio, near Turin [32]. The main geometrical and mechanical characteristics of the Mosca bridge are reported in Table 1.  
The purpose of the comparison between the three above mentioned analyses is to obtain the elastic-fracture-plastic transitions in terms of maximum live load, as uniformly distributed over the whole bridge deck, and the position of the critical sections representing both the brittle hinges and cracks within the fracture analysis, or the plastic hinges within the limit analysis. In Figure 8, the elastic-fracture-plastic transitions are shown. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Arch** \\ **Bridge** \\ \end{tabular} } & \begin{tabular}{c} **Arch** \\ **Span** \\ **(m)** \\ \end{tabular} & \begin{tabular}{c} **Shallownes** \\ **Ratio** \\ **(-)** \\ \end{tabular} & \begin{tabular}{c} **Arch** \\ **Depth** \\ **(cm)** \\ \end{tabular} & \begin{tabular}{c} **Slandeness** \\ **Ratio** \\ **(-)** \\ \end{tabular} & \begin{tabular}{c} **Young** \\ **Moulus** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Masony** \\ **Density** \\ **(kN/m\({}^{3}\))** \\ \end{tabular} & \begin{tabular}{c} **Compressive** \\ **Strength** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Tensile** \\ **Strength** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Fracture** \\ **Toughness** \\ **(daN/cm\({}^{3/2}\))** \\ \end{tabular} \\ \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} _Mosca_ \\ _Limyna_ \\ _Crimario_ \\ \end{tabular} } & 45 & 1/8 & 150 & 1/30 & 50,000 & 25 & 50 & 1.5 & 100 \\ \hline \hline \multirow{2}{*}{
\begin{tabular}{c} _Vitorio_ \\ _Emanuele 1_ \\ \end{tabular} } & 10 & 1/5 & 100 & 1/10 & 30,000 & 18 & 30 & 1.0 & 70 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical and mechanical parameters of the analyzed arch bridges. Figure 7: Mosca Bridge, Turin (Italy): Overview (**a**); Arch sketch by [PERSON], 1879 [31] (**b**). A Mosca bridge elastic analysis (Figure 8a) presents a line of thrust outside the central kern (middle third) for a uniformly distributed live load equal to zero, since tensile stresses in the masonry arise immediately. The elastic limit is overtaken in the two arch section at the springings. In Figure 8a, red circles represent the points where the line of thrust exceeds the middle third of the arch cross section. Limit analysis (Figure 8b) shows that the uniformly distributed maximum live load affordable by the arch structure is 1190 kN/m, a value for which the masonry compressive strength in four points is reached. In these positions, although the line of thrust results to be within the arch volume, plastic hinges (red circles in Figure 8b) due to compressive failure take places triggering the kinematic collapse of the arch bridge. On the other hand, fracture analysis (Figure 8c) allows the uniformly distributed live load to reach the maximum value of \(q_{line}\) = 870 kN/m. Note that the positions of the cracked sections, the so-called brittle hinges [21; 30], are identified by means of the abovementioned fracture mechanics approach, considering the damaging mechanisms (Figures 4 and 5) in a step-by-step loading process. In particular, for \(q_{line}\) = 5 kN/m, two cracks of length \(\xi\) = 0.2, i.e., \(a\) = 30 cm, appear at the springings.