# Safety Assessment of Masonry Arch Bridges Considering the Fracturing Benefit
## 2 Masonry Arch Off-Center Compression: LEFM Based Approach
As it is well known from linear elastic fracture mechanics [23], a bending moment, \(M\) (Figure 3), induces a stress-intensity factor at the crack tip equal to:  
\[K_{\rm IM}=\frac{M}{h^{3/2}t}Y_{\rm M}(\xi), \tag{3}\]  
while a tensile axial force, \(F\), induces the factor:  
\[K_{\rm IF}=\frac{F}{h^{1/2}t}Y_{\rm F}(\xi), \tag{4}\]  
where \(\xi=a/h\) is the relative crack depth, and the so-called weight functions, \(Y_{\rm M}\) and \(Y_{F}\), for a relative crack depth \(0\leq\xi\leq 0.7\), are [23]: \(Y_{\rm M}(\xi)=6\) (\(1.99\xi^{1/2}-2.47\xi^{2/32}+12.97\xi^{5/2}-23.17\xi^{7/2}+24.80\xi^{9/2}\)) and \(Y_{\rm F}(\xi)=1.99\xi^{1/2}-0.41\xi^{23/2}+18.70\xi^{5/2}-38.48\xi^{7/2}+53. 85\xi^{9/2}\). When the axial force, \(F\), is a compression and the bending moment, \(M\), tends to open the crack, as generally happens at the cross sections of arch-structures, the total stress-intensity factor representing the loading parameter can be obtained by applying the superposition principle:  
\[K_{\rm I}=K_{\rm IM}-K_{\rm IF}=\frac{F}{h^{1/2}t}\bigg{[}\frac{e}{h}Y_{\rm M} (\xi)-Y_{\rm F}(\xi)\bigg{]}, \tag{5}\]  
where \(e\) denotes the eccentricity of the equivalent eccentric axial force. From the critical condition of crack propagation, i.e., when the loading parameter reaches the fracture toughness, \(K_{\rm I}=K_{\rm IC}\), the dimensionless axial force of crack extension can be obtained, as a function of the relative crack depth, \(\xi\), and as the relative eccentricity \(e/h\) of the load varies:  
\[\overline{F}_{\rm C}=\frac{F_{\rm C}}{h^{1/2}t_{\rm IC}}=\frac{1}{\frac{e}{h} Y_{\rm M}(\xi)-Y_{\rm F}(\xi)}. \tag{6}\]  
Figure 3: Arch segment subjected to off-center compression. The curves of Figure 4 graphically represent the arch-structure fracturing process (Equation (6)), and show how, with the eccentricity \(e/h\) fixed, fracturing reaches a stable stage only after presenting an unstable one. If the load \(F\) does not have the possibility of following the unstable descending branch of the curve \(e/h\) = _constant_, fracturing will display a catastrophic behavior and the representative point in the diagram of Figure 4 will advance horizontally to meet up again with the curve \(e/h\) = _constant_ on its stable branch. In Figure 4, an example of mechanism of crack opening for off-center compression is shown: starting from a dimensionless axial force, \(\widetilde{F}_{\mathrm{C}}\), equal to 16, and a relative eccentricity of the axial force, \(e/h\), equal to 0.25, the corresponding relative crack depth in the cross-section is found as \(\xi\) = 0.28. It is moreover important to consider that, for each relative crack depth, \(\xi\), a relative eccentricity, \(e/h\), exists, under which the crack closes again, at least partially [19; 24]. From the closing condition, \(K_{\mathrm{I}}\) = 0, we have:  
\[\frac{e}{h}=Y_{\mathrm{F}}(\xi)/Y_{\mathrm{M}}(\xi). \tag{7}\]  
The curve of Figure 5 graphically represents Equation (7). The points under the curve represent arch-structure cracks and loading conditions for which \(K_{\mathrm{I}}\) < 0, i.e., the condition of crack closure. In Figure 5, an example of mechanism of crack closure for off-center compression is shown: starting from an arch cross-section with a crack depth equal to \(\xi\) = 0.40, and subjected to an off-center compression with a relative eccentricity of the axial force \(e/h\) = 0.23, it returns a crack closure corresponding to \(\xi^{*}\) = 0.25. Figure 4: Mechanism of crack opening for off-center compression (arch-structure fracturing process): case of an axial force corresponding to \(\widetilde{F}_{\mathrm{C}}\) = 16, with a relative eccentricity \(e/h\) = 0.25, leading to a relative crack depth in the cross-section \(\xi\) = 0.28 (dotted line).