# Safety Assessment of Masonry Arch Bridges Considering the Fracturing Benefit
## 2 Masonry Arch Off-Center Compression: LEFM Based Approach  
In order to highlight the LEFM-based approach to the dominant problem concerning the off-center compression in masonry arches, the definition of the concept of stress concentration has to be compared with the more conceptual one of stress intensification. Whereas the former concept represents the amplification of the stress applied to a body, which occurs at the tip of a notch, the latter measures the severity of this amplification in the case where, instead of the notch, there is a sharp crack, as usually in the case of a masonry structure, and the amplification itself tends to infinity at the crack tip [12; 13]. For the sake of clarity, let us consider the following cases in terms of stress concentration vs. stress intensification (Figure 2): (i) structural plate weakened by circular hole; (ii) structural plate notched by elliptical hole; (iii) structural plate having a rectilinear crack. Let us consider that each structure is subjected to a tensile stress, \(\sigma\), applied from outside, orthogonally to the straight line to which the crack or the notch belongs. As it is well known from elasticity theory, if the notches are relatively small in comparison with the size of the corresponding structural plate, the failure point is reached for external stresses, \(\sigma\), that have the value of \(\sigma=\sigma_{u}/3\), where \(\sigma_{u}\) is the material strength, in the case of circular holes [14], and the value of \(\sigma=\frac{\sigma_{u}}{1+2\frac{\pi}{b}}\) in the case of elliptical holes, with the major semi-axis, \(a\), and the minor semi-axis, \(b\) (Figure 2). Following the elastic formulation of Inglis [15], when the elliptical hole tends to represent a crack, i.e., when the stress concentration factor \(1+2a/b\) tends to infinity, the failure point becomes \(\sigma=0\), then no stresses can be applied to the damaged structure. For this reason, the concept of stress concentration must be overcome in order to take into account the structural damage, and, in the case of a plate structure weakened by an internal crack (Figure 2), the fracture point is a function of an amplification factor, the so-called stress-intensity factor, \(K_{\rm I}\), which is a real measure of crack opening severity. When \(K_{\rm I}\) reaches its critical value, \(K_{\rm IC}\), the crack propagates. Considering the plate structure weakened by an internal crack [16], the stress intensification following LEFM [17; 18] is represented by:  
\[K_{\rm I}=\sigma\sqrt{\pi a}, \tag{1}\]  
and, therefore, the critical condition becomes:  
\[\sigma=\frac{K_{\rm IC}}{\sqrt{\pi a}} \tag{2}\]  
The possibility of generalizing this reasoning to geometries different from the abovementioned academic ones is very important, as in the case of an arch subjected to off-center compression (Figure 3). Note that, for certain structural geometries and loading conditions, the process of crack propagation is unstable, in the sense that once a certain critical load is reached, the crack propagates spontaneously and without further increments in load, as at first considered by [PERSON] [16]. Figure 2: Transition from stress concentration (circular and elliptical hole) to stress intensification (crack). On the other hand, there are cases in which crack propagation is first stable and then unstable, or, vice versa, first unstable and then stable. Among the latter there is the case of the eccentric compression of a cracked beam of rectangular cross section (Figure 3; [19]). This case, explained below, can represent in an effective way the static regime of an arch segment subjected to axial force and bending moment [20; 21; 22].