## 3 Gradation Design
### Gradation Synthesis
#### 3.2.2 Aggregate Grading Optimization  
The coarse aggregate ratio \(R_{\text{CA}}\) was used to evaluate the rationality of the composition between different parts of the aggregate gradation. The calculation formula is shown in Equation (5). The nominal maximum particle size of the aggregate was 13.2 mm, and the standard \(P_{C_{P}}=13.2\times 0.22=2.9\) mm was determined by the Bailey method, which was between 2.36 mm and 4.75 mm. Therefore, 2.9 mm was used as the virtual control sieve hole. When calculating \(P_{\text{S/2}}\), \(P_{\text{S/2}}=13.2\times 0.5=6.6\) mm was obtained, which was between 4.75 mm and 9.5 mm. When the aggregate quantity was sufficient, the particle size of the mineral aggregate was uniformly and continuously changed. The pass rate of the virtual sieve hole calculated by the interpolation method was in line with that of the actual situation, and the theoretical formula of the interpolation is shown in Equations (6) and (7). From this, the \(R_{\text{CA}}\) values of design gradations 1 and 2 in different filling states could be obtained, as shown in Table 11:  
\[R_{\text{CA}}\ =\ \frac{P_{\text{S/2}}-P_{C_{P}}}{100\ -\ P_{\text{S/2}}}, \tag{5}\]  
where \(P_{\text{S/2}}\ =\) the half-sieve mesh pass rate (%), and \(P_{C_{P}}\) = the basic control screen pass rate (%). Equations (6) and (7) are  
\[P_{2.9}\ =\ \frac{(4.75\ -\ 2.9)}{(4.75\ -\ 2.36)}P_{2.36}+\frac{(2.9\ -\ 2.36)}{(4.75\ -\ 2.36)}P_{4.75}=0.77P_{2.36}+0.23P_{4.75}, \tag{6}\]  
\[P_{6.6}\ =\ \frac{(9.5\ -\ 6.6)}{(9.5\ -\ 4.75)}P_{4.75}+\frac{(6.6\ -\ 4.75)}{(9.5\ -\ 4.75)}P_{9.5}=0.61P_{4.75}+0.39P_{9.5}. \tag{7}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Gradation**} & \multicolumn{2}{c}{**Particle Size**} & \multirow{2}{*}{**16**} & \multirow{2}{*}{**13.2**} & \multirow{2}{*}{**9.5**} & \multirow{2}{*}{**4.75**} & \multirow{2}{*}{**2.36**} & \multirow{2}{*}{**1.18**} & \multirow{2}{*}{**0.6**} & \multirow{2}{*}{**0.3**} & \multirow{2}{*}{**0.15**} & \multirow{2}{*}{**0.075**} \\  & **(mm)** & & & & & & & & & \\ \hline \multirow{4}{*}{1} & Tamping state & 100.00 & 92.42 & 72.34 & 38.28 & 29.20 & 24.45 & 20.56 & 14.80 & 8.63 & 5.00 \\  & Select tight & 100.00 & 92.92 & 74.18 & 42.38 & 33.90 & 28.23 & 23.58 & 16.70 & 9.34 & 5.00 \\  & loose state & 100.00 & 93.13 & 74.92 & 44.03 & 35.80 & 29.75 & 24.80 & 17.47 & 9.62 & 5.00 \\ \hline \multirow{4}{*}{2} & Tamping state & 100.00 & 88.30 & 60.75 & 28.90 & 24.47 & 20.85 & 16.52 & 9.85 & 6.49 & 4.00 \\  & Select tight & 100.00 & 88.95 & 62.90 & 32.80 & 27.68 & 23.49 & 18.48 & 10.77 & 6.88 & 4.00 \\ \cline{1-1}  & loose state & 100.00 & 89.27 & 64.01 & 34.80 & 29.32 & 24.85 & 19.48 & 11.24 & 7.08 & 4.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Gradation under different filling methods. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Gradation** & **State** & **X (\%)** & **Y (\%)** & **Z (\%)** \\ \hline \multirow{4}{*}{1} & Tamping state & 70.8 & 24.2 & 4.58 \\  & Select tight state & 66.1 & 28.9 & 4.58 \\  & Loose state & 64.2 & 30.8 & 4.58 \\ \hline \multirow{4}{*}{2} & Tamping state & 71.1 & 24.9 & 4.56 \\  & Select tight state & 67.2 & 28.8 & 4.56 \\  & Loose state & 65.2 & 30.8 & 4.56 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Calculation results under different filling methods. It can be seen from Table 10 that the \(R_{\rm CA}\) of gradation 1 exceeded the range required by the Bailey method, so it was discarded. The \(R_{\rm CA}\) of gradation 2 was within the range required by the Bailey method. The aforementioned design aggregate gradation was compared to the commonly used SMA-13 specification gradation as a test grading, as shown in Figure 3.  
It can be seen from Figure 5 that the 13.2-mm mesh pass rate was less than 90%, and the content of the nominal maximum particle size aggregate was increased relative to the Stone Matrix Asphalt (SMA) grade. This was advantageous for improving the compressive strength and stability of the skeleton structure.