## 3 Shear Fatigue Characteristics of Waterproof Cohesive Layer Materials
### Equivalent Temperature Calculation  
According to the fatigue equations shown in Table 11, the relationship between the interlaminar shear fatigue life of the waterproof cohesive layer, applied stress, and its temperature can be expressed as the following function of shear stress and temperature:  
\[N_{f}=X_{N}\bigg{(}\frac{1}{\tau}\bigg{)}^{Y_{N}}\bigg{(}\frac{1}{T}\bigg{)}^{ Z_{N}} \tag{1}\]  
where \(X_{N}\), \(Y_{N}\), and \(Z_{N}\) are regression coefficients. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Test Temperature (\({}^{\circ}\)C)**} & \multirow{2}{*}{_T/T\({}_{0}\)_} & \multicolumn{2}{c}{\(N_{fT}/N_{fT0}\)} \\ \cline{3-4}  & & **SBS-Modified** & **90 Ordinary Hot** & **SBS-Modified** \\  & & **Emulsified Asphalt** & **Asphalt** & **Asphalt** \\ \hline
15 & 1.00 & 1.000 & 1.000 & 1.000 \\
20 & 1.33 & 0.732 & 0.707 & 0.739 \\
25 & 1.67 & 0.581 & 0.545 & 0.576 \\
30 & 2.00 & 0.471 & 0.448 & 0.479 \\
35 & 2.33 & 0.400 & 0.360 & 0.406 \\
40 & 2.67 & 0.339 & 0.305 & 0.353 \\ \(\log_{(T/T_{0}}\big{(}N_{fT}/N_{fT0}\big{)}\) & \(-\)1.10 & \(-\)1.20 & \(-\)1.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Temperature correction coefficient for the fatigue life model. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Material**} & \multicolumn{1}{c}{**Prediction Model**} \\ \hline SBS-modified emulsified asphalt & \(N_{f}=34.38\tau^{-6.704}\big{(}\frac{T}{T_{0}}\big{)}^{-1.10}\) \\
90 ordinary hot asphalt & \(N_{f}=18.42\tau^{-6.675}\big{(}\frac{T}{T_{0}}\big{)}^{-1.20}\) \\ SBS-modified asphalt & \(N_{f}=41.97\tau^{-6.719}\big{(}\frac{T}{T_{0}}\big{)}^{-1.06}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Shear fatigue life prediction model for waterproof cohesive layer materials with temperature correction. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Material**} & \multicolumn{1}{c}{**Shear Fatigue Equation**} & \multicolumn{1}{c}{**Prediction Model**} & \multicolumn{1}{c}{
\begin{tabular}{c} **Coefficient of** \\ **Determination** \\ \end{tabular} } \\ \hline SBS-modified emulsified asphalt & \(\lg N_{f}=1.536-6.704\lg\tau\) & \(N_{f}=34.38\tau^{-6.704}\) & 0.982 \\
90 ordinary hot asphalt & \(\lg N_{f}=1.265-6.675\lg\tau\) & \(N_{f}=18.42\tau^{-6.675}\) & 0.980 \\ SBS-modified asphalt & \(\lg N_{f}=1.623-6.719\lg\tau\) & \(N_{f}=41.97\tau^{-6.719}\) & 0.977 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Shear fatigue equations for different waterproof cohesive layer materials. [PERSON]'s linear cumulative damage theory [28; 29] can be used to express the accumulated fatigue damage, as shown in Equation (2):  
\[D=\sum_{i=1}^{m}\frac{n_{i}}{N_{i}} \tag{2}\]  
where \(D\) is the cumulative fatigue damage rate, \(Ni\) is the number of repetitions corresponding to loads at various loads during material failure, and \(n_{i}\) is the number of loads at various levels in practice. The shear stress and shear strength will change due to variations in the temperature distribution \(T(z,t)\) of the bridge deck [30]. According to the integral mean value theorem, there must be an equivalent temperature \(T_{eq}\) in the temperature field. At this temperature, the shear fatigue life of the bridge deck interlayer material is \(N_{eq}\), and there exists \(D=D_{eq}=\sum_{i=1}^{m}\frac{n_{i}}{N_{eq}}\), i.e., \(N_{eq}=\frac{\sum_{i=1}^{m}n_{i}}{D}\). It is difficult to accurately predict the vehicle load distribution at a certain temperature \(T_{i}\). To simplify the calculation, it is assumed that the axle loads are uniformly distributed at all time periods, and the horizontal shear stress on the waterproof cohesive layer material does not change with temperature [31], as defined in Equation (3):  
\[n_{i}=N\times f_{i} \tag{3}\]  
where \(n_{i}\) is the number of standard axle loads at temperature \(T_{i}\), \(f_{i}\) is the distribution frequency at the tack coat temperature \(T_{i}\) during the design period, and \(N\) is the cumulative frequency of standard axle loads on the asphalt pavement during the design period.