## 4 Results and Discussion  
As indicated previously, according to the Griffith criterion, the value of maximum and minimum principal stresses (at the center) are respectively obtained using Equations (3) and (4). The numerical ratio \(\sigma_{c}^{G}/\sigma_{t}^{G}\) between the tensile and compressive stress at the specimen center was presented for all simulated concretes and mortars in Tables 3 and 5, respectively. It should be pointed out that the ratio \(\sigma_{c}^{G}/\sigma_{t}^{G}\) calculated numerically for concretes and mortars respects the Griffith failure criterion confirming for another time the validity of our numerical model.  
The initial crack location observed numerically was presented for each case studied in Tables 4 and 5. For some samples, the fractures are not at the center [23]. This last remark was indeed approved by the results obtained for the different concretes and mortars.  
Figures 6 and 7 respectively give the crack initiation positions for all simulated concretes and mortars under different arch loading angles studied as indicated in Tables 4 and 5, respectively.  
Figure 6: _Cont._  
Figure 6: Distribution of maximum shear strains. Crack initiation point positions in the disk of concretes tested by [8].  
Figure 7: _Cont._  
In the case of concrete, it is clear that the initiation of cracks starts from the center of the disks if the loading angle is between 20-45\({}^{\circ}\), thus allowing the validity of the test (Figure 6). From Figure 6, angles that equal to 5\({}^{\circ}\) and 10\({}^{\circ}\), the shear failure is recorded at the loaded points. For 2\(\alpha=15^{\circ}\), a tension failure near the loaded points is recorded. So, we can say that the tensile strength values for 2\(\alpha\), which varies from 5-15\({}^{\circ}\), were insignificant (unsuccessful test).  
Figure 7 shows the initiation of crack point positions in the case of mortar disks under a various arch loading angle. In contrast with concrete, it was observed that the initiation of cracks starts in the center of the mortar disks if the loading angle 2\(\alpha\) is higher than 10\({}^{\circ}\).  
In the case of 2\(\alpha=10^{\circ}\), the crack initiation starts from the loading points (tension failure), but spreads rapidly towards the center, thus giving significant tensile strength value.  
The distribution of horizontal and vertical stress along the loaded vertical diameter for the concrete disks tested by [PERSON] et al. [8] in two cases of arch loading angles 15\({}^{\circ}\) and 40\({}^{\circ}\), respectively, is shown in Figure 8. Numerical and analytical predictions are in good agreement with a maximum relative difference of 8.54%.  
A maximum horizontal tensile stress \(\sigma_{\mathrm{x}}\) along the loaded vertical diameter is localized at the center of the disk, which decreases when moving away from the center, becoming horizontal compressive stress at the loaded disk ends [30].  
It is noted that the compressive stresses and the stress concentration degree near the loaded points of the disk can be dramatically decreased with increasing the loading angle. Therefore, we can say that the possibility of compressive failure near the specimens loaded points is reduced considerably if the loading angle is increased. Consequently, the possibility of a central crack initiating will increase, thus giving a validated and a successful response [31].  
Figure 7: Distribution of maximum shear strains. Crack initiation point positions in the disk of mortar tested by [8].