# Safety Assessment of Masonry Arch Bridges Considering the Fracturing Benefit
## 2 Masonry Arch Off-Center Compression: LEFM Based Approach
Moreover, a LEFM-based approach to the analysis of masonry arches can take into account scale effects [12; 25; 26; 27], that are always present in structural engineering due to the different physical dimensions of two mechanical parameters of the materials: strength, \(\sigma_{\rm u}\), and fracture toughness, \(K_{\rm IC}\). The former is related to the plastic or ultimate strength collapse, whereas the latter is related to the brittle crack propagation. The competition between the two collapses of different nature in masonry arches can easily be shown by considering Equation (5), for the critical condition \(K_{\rm I}=K_{\rm IC}\)[20]. Considering the arch cross section subjected to off-center compression, if both members of Equation (5) are divided by \(\sigma_{\rm u}\ h^{1/2}\), where \(\sigma_{\rm u}\) is the masonry compressive strength, we obtain:  
\[\sigma=\frac{K_{\rm IC}}{\sqrt{\pi a}} \tag{2}\]0  
where \(s\) is a dimensionless parameter, the so-called brittleness number [13], which allows one to measure the scale-dependent susceptibility of fractures to propagate in unstable condition, \(s=\frac{K_{\rm IC}}{\sigma_{\rm u}h^{1/2}}\). On the other hand, it is possible to consider the non-dimensional load of ultimate strength in the arch cross-section of depth \(h\)-\(\xi\):  
\[\sigma=\frac{K_{\rm IC}}{\sqrt{\pi a}} \tag{2}\]1  
Equations (8) and (9) are plotted in Figure 6 as functions of the normalized crack depth, \(\xi\), and different values of relative eccentricity of the axial force, \(e/h\). While the former produces a set of curves by varying brittleness number, \(s\), the latter is represented by a unique curve [20]. \[\sigma=\frac{K_{\rm IC}}{\sqrt{\pi a}} \tag{2}\]2 Crack closing condition for off-center compression (arch-structure crack closure): case of an arch cross-section with a crack depth equal to \(\xi\) = 0.40, and subjected to an off-center compression with \(e/h\) = 0.23, which returns a crack closure corresponding to \(\xi^{*}\) = 0.25. \[\sigma=\frac{K_{\rm IC}}{\sqrt{\pi a}} \tag{2}\]3 Scale effects in masonry arch structures: Dimensionless load of brittle crack propagation versus normalized crack depth, \(\xi\), for \(e/h=0.15\) (**a**); \(e/h=0.20\) (**b**); \(e/h=0.35\) (**c**). Note that, in the arch cross-section, crack propagation collapse is possible only if the value \(\xi\) = 0.7 is reached, i.e., the crack depth damages at least 70% of section height [28]: graphically, this means that fracturing collapse occurs only if the curve plotted from Equation (8) reaches \(\xi\) = 0.7 before intercepting the curve of ultimate strength (Equation (9)). In Figure 6a, with \(e/h\) = 0.15, it is evident that for low values of relative eccentricity of the axial force, i.e., for small bending moments, the ultimate strength collapse at the arch ligament precedes and obscures crack propagation collapse for each brittleness number \(s\): all the curves representing the arch structural behaviours intercept the ultimate strength limit before reaching the fracturing collapse condition \(\xi\) = 0.7. In Figure 6c, with \(e/h\) = 0.35, it is evident that for high values of relative eccentricity of the axial force, i.e., for large bending moments, the crack propagation collapse precedes ultimate strength collapse at the arch ligament for each brittleness number \(s\): all the curves representing the structural behaviours of the arch intercept the fracturing collapse, \(\xi\) = 0.7, before reaching the ultimate strength limit (thick curve). In the intermediate cases (Figure 6b), \(0.15\leq e/h\leq 0.35\), the ultimate strength collapse at the arch ligament precedes crack propagation collapse when the brittleness number, \(s\), shows higher values, i.e., when the arch structure presents high fracture toughness, low ultimate strength, and/or small structural size.