## 3 Consideration of Dynamic Loads in Axle Load Spectra
### Dynamic Axle Load Spectra  
Dynamic load distribution should consider the impact of deviation of axle loads from static magnitudes. The discrete distribution of axle loads is determined by the percentage of axle passes falling into a set of axle load intervals. The Dynamic Axle Loads Spectrum includes both static axle load spectra and distribution of dynamic loads coming from different axle load intervals [13]. The percentage of dynamic loads \(d_{j}\) in the interval \(j\) comes from dynamic loads induced by axles with static loads of \(Q_{k}\). The dynamic axle load spectra, the total percentage \(d_{j}\) of all axle loads in a given load interval j, is calculated as a sum of dynamic loads from all the static axle load spectra intervals k. The percentage of dynamic loads \(d_{j}\) is computed using Equation (2).  
\[d_{i,j}=\sum_{k=1}^{n}s_{k}\frac{1}{DLC_{i,k}Q_{k}\sqrt{2\pi}}exp\left(-\frac {\left(Q_{k}-Q_{j}\right)^{2}}{2 DIC_{i,k}^{2}Q_{k}^{2}}\right) \tag{2}\]  
where, \(d_{i,j}\) are the percentage of dynamic loads; \(Q_{j}\) are dynamic loads (kips); and \(s_{k}\) are the percentage of number of axles with static load \(Q_{k}\) determined from WIM data.  
It was assumed that dynamic loads can be characterized as normal distribution with the average value in the middle of a given interval and standard deviation, which has been proved and applied in previous works [6; 13]. The DLC values in the formula would change with pavement roughness levels and axle load intervals designated by MEPDG. Figure 4 illustrates the comparison of static axle load spectra and dynamic axle load spectra on major and minor highways. The axle load spectra were extracted from measured data in the long-term pavement performance (LTPP) database at two WIM sites on one major and one minor highway in NJ, respectively, which were considered as the reference loads on smooth pavement. The dynamic loads at three different IRI values of 1 m/km (63 in/mile), 2 m/km (126 in/mile) and 3 m/km (190 in/mile) were compared with the static loads on smooth pavement surface. The spikes in the distribution of axle loads were flatten after transformation because of the greater deviation of dynamic loads of vehicles. The increase in DLCs led to a reduction in the percentage of lighter loads but increase in the percentage of heavier loads. The heavier axle loads were expected to accelerate pavement deterioration in the M-E pavement design. The distinction of load distribution on tandem and tridem axles were more obvious, especially for the IRI value of 3 m/km.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Route Type** & **Axle Type** & \(a\) & \(b\) & \(c\) & **R-Squared Value** \\ \hline \multirow{2}{*}{Major highway (104 km/h)} & Single & 0.067 & 0.030 & 0.232 & 0.89 \\  & Tandem & 0.072 & 0.042 & 0.277 & 0.87 \\  & Tridem & 0.068 & 0.042 & 0.265 & 0.86 \\ \hline \multirow{2}{*}{Minor highway (80 km/h)} & Single & 0.096 & 0.025 & 0.308 & 0.85 \\  & Tandem & 0.089 & 0.033 & 0.317 & 0.88 \\ \cline{1-1}  & Tridem & 0.081 & 0.041 & 0.277 & 0.82 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of regression model of DLC with respect to IRI and axle load.  
Figure 4: _Cont._  
Figure 4: Static and dynamic axle load spectra: (**a**) single axle (major); (**b**) tandem axle (major); (**c**) tridem axle (major); (**d**) single axle (minor); (**e**) tandem axle (minor); and (**f**) tridem axle (minor).