# Automated Trucks' Impact on pavement Fatigue Damage
## 2 Materials and Methods
### Computation of the Frequency Matrices (ND) for HDTs and ATs  
This study considers a total load repetition of 30 million for a 20-year design period. This value is derived from typical traffic data, as referenced in [PERSON] et al. [17], and serves as a benchmark for assessing the pavement's fatigue life. The assumption of 30 million loading cycles is only valid if the pavement does not fail before reaching this period. The reason for considering this value is to provide a standardised and reasonable basis for comparing the damage index between normal traffic loading and AT loading behaviours. Using a consistent benchmark, the study aims to identify how the unique loading patterns of ATs might influence pavement fatigue differently from conventional traffic. This comparison is crucial for understanding potential risks and developing strategies to mitigate additional stresses caused by ATs. The 30 million vehicle passes are distributed within the lane considering the wheel wander scenarios calculated based on the market penetration rate and lane width to construct the frequency matrices (\(ND\)). We used field data from a previous study by [PERSON] et al. [34] to simulate the frequency distributions of HDTs. They measured the standard deviations associated with the normal distribution of vehicle displacement across a traffic lane. Based on the findings of their study, we assumed standard deviations (\(SD\)) of 0.24 for lane widths of 3 m and 0.29 m for 3.5 m lane widths. It is worth noting that the frequency of HDT passes was assumed to follow a normal distribution, with a probability density function expressed by Equation (7). In this equation, \(x\) represents the distance from a reference point, \(\mu\) is the mean value, and \(SD\) is the standard deviation.  
\[f(x)=\frac{1}{SD\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^ {2}} \tag{7}\]  
The total number of AT passes is applied to the centre point (i.e., the centre of the dual tyre), regardless of the lane width, to simulate the zero-wander distribution for ATs. Moreover, there are no AT passes on other points.  
For the uniform-wander distribution simulation, the total number of vehicles is distributed uniformly within the available width of the road lane. The truck's width is subtracted from the lane width, using the centre of a dual tyre as the reference point for a vehicle pass to calculate the wandering space. For example, a lane width of 3 m with a truck width of 2.59 m would result in a wandering space of 0.41 m, which means the centre of the dual tyre can move 0.205 m (i.e., 0.41 m/2) to the left or right.  
To calculate the vehicle passes at each transverse point for mixed traffic scenarios, we added the AT and HDT passes at each transverse point, considering the ATs' market penetration rate. This allows us to obtain a comprehensive picture of the vehicle frequency distribution across the traffic lane for different scenarios involving mixed traffic of ATs with zero- or uniform-wander mode and HDTs with normal-wander mode.  
This study considered AT market penetrations of 0%, 20%, 40%, 60%, 80%, and 100% with zero- and uniform-wander modes for lane widths of 3 m and 3.5 m. This created 24 different wander mode distribution scenarios, combined with the 36 flexible pavement cross-sections discussed above, resulting in 864 unique scenarios.  
The research methodology is systematically outlined in the flowchart (Figure 1).  
Figure 1: Flowchart depicting the sequential steps undertaken in the research methodology.