# Analysis of Optimal Loading Angle in Dynamic Flattened Brazilian Disc Splitting Test for Concrete
## 3 Theoretical Analysis  
It is known from the literature [37] that under the same loading rate the stress at any point in the BD specimen on the loading under the action of horizontal radial concentrated force can be expressed as Appendix A, as shown in Figure 5.  
The horizontal stress \(\sigma_{x}\) in the loading BD specimen decreases from the A\({}_{1}\) and the A\({}_{2}\) to the \(O\), as shown in Figure 6. Moreover, with the increase in the loading angle, the vertical compressive stress decreases significantly, from 18.11 when the loading angle is 20\({}^{\circ}\) to 12.21 when the loading angle is 30\({}^{\circ}\), so its change rate with the loading angle is 0.59. The compression-tension stress ratio in the loading BD decreases with the increase in loading angle, as shown in Table 2.  
Figure 4: Measurement point compression and tensile stress distribution in the radial loading direction. It can be seen from Figure 7 that the maximum value of vertical tensile stress \(\sigma_{y}\) appears at the \(O\) point of the disc and gradually decreases from the \(O\) point to both ends A\({}_{1}\) and A\({}_{2}\) and finally becomes vertical compressive stress. Moreover, with the increase in loading angle, the vertical tensile stress in the BD specimen of the loading decreases slowly, from 1.88 at 20\({}^{\circ}\) to 1.74 at 30\({}^{\circ}\), and its change rate with loading angle is 0.014, which can be considered as a constant value. The maximum value of vertical compressive stress is reduced from 12.87 to 9.86. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Loading Angle (\({}^{\circ}\))**} & **Compressive Stress** & **Compressive Stress** & **Tensile Stress \(\sigma_{\text{xtrans}}\)** & \(\sigma_{\text{xmax}}\)/\(\sigma_{\text{x2 max}}\) & \(\sigma_{\text{ymax}}\)/\(\sigma_{\text{xmax}}\) \\  & \(\sigma_{\text{max}}\) & \(\sigma_{\text{ymax}}\) & & & \\ \hline
20 & 18.11 & 12.87 & 1.88 & 9.63 & 6.84 \\
22 & 16.52 & 12.13 & 1.86 & 8.88 & 6.52 \\
24 & 15.15 & 11.48 & 1.83 & 8.27 & 6.27 \\
26 & 13.96 & 10.90 & 1.81 & 7.71 & 6.02 \\
28 & 13.08 & 10.36 & 1.78 & 7.34 & 5.82 \\
30 & 12.21 & 9.86 & 1.74 & 7.01 & 5.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ratios of non-dimensional compressive stress to tensile stress inside discs. Figure 5: The flattened Brazilian test. Figure 6: Non-dimensional horizontal stresses along loaded diameter. Meanwhile, it is found that the tensile stress zone in the loading BD specimen gradually decreases towards the center of the disc with the increase in the loading angle; that is, increasing the loading angle reduces the tensile stress zone in the loading direction of the disc and increases the possibility of cracking at the center. It is found that increasing the loading angle can obviously reduce the compressive stress value and stress concentration at the loading place, thus reducing the possibility of compression failure of the BD specimen near the loading and creating conditions for the specimen to take the lead in central tensile splitting failure. According to [PERSON] strength criterion [38; 39] to judge the damage of SCC, the specific form is:  
\[\left\{ \begin{array}{c}\sigma_{G}=-\frac{\left(\sigma_{1}-\sigma_{3}\right) ^{2}}{8\left(\sigma_{1}+\sigma_{3}\right)}\geq\sigma_{\tau},\sigma_{1}+3 \sigma_{3}\leq 0\\ \sigma_{G}=\left|\sigma_{3}\right|\geq\sigma_{\tau},\sigma_{1}+3\sigma_{3}>0 \end{array} \right. \tag{1}\]  
where \(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\) are three principal stresses, which satisfy this relationship in turn. In the loading diameter direction (\(x=0\)), it can be known from Formula (1) that \(\tau_{xy}\) is 0, and it is easy to judge that \(\sigma_{y}\geq\sigma_{x}\) corresponds to \(\sigma_{1}\geq\sigma_{3}\).