# Fatigue Assessment of Prestressed Concrete Slab-Between-Girder Bridges
## 3 Results
### Resulting Wohler Curve  
To find the Wohler curve of the fatigue experiments, the relation between the logarithm of the number of cycles \(N\) and the applied load ratio \(F/P_{\text{max}}\) was plotted, see Figure 6. For this curve, we interpreted the variable amplitude loading tests as follows: if \(N_{1}\) cycles at load level \(F_{1}\) are applied, followed by \(N_{2}\) cycles at load level \(F_{2}\), and then \(N_{3}\) cycles to failure at \(F_{3}\), with increasing load levels \(F_{1}\)\(<F_{2}<F_{3}\), it is conservative to assume that the slab can withstand \(N_{1}+N_{2}+N_{3}\) cycles at the load level \(F_{1}\), \(N_{2}+N_{3}\) cycles at load level \(F_{2}\), and \(N_{3}\) cycles at load level \(F_{3}\). This approach led to three datapoints for one variable amplitude fatigue test. As a result of this approach, we obtained 16 datapoints on the first setup and 28 datapoints on the second setup, resulting in 44 datapoints in Figure 6. The average  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Test** & **Setup** & \begin{tabular}{c} **Size Load** \\ **(mm \(\times\) mm)** \\ \end{tabular}  & **Wheel** & \(F/P_{\text{max}}\) & \(N\) & \begin{tabular}{c} **Age** \\ **(days)** \\ \end{tabular}  & \begin{tabular}{c} _fon,cube_ \\ **(MPa)** \\ \end{tabular}  \\ \hline BB30 & 1, new & \(150\times 115\) & D & \ \begin{tabular}{c} 0.58 \\ 0.67 \\ 0.58 \\ 0.75 \\ 0.81 \\ 0.81 \\ 0.75 \\ 0.81 \\ 0.81 \\ \end{tabular}  &
\begin{tabular}{c} 100,000 \\ 140,000 \\ 150 \\ 15  
value of the Wohler curve is shown as \"mean\" in Figure 6, and it is described with the following expression, using \(S\) for the load ratio and \(N\) for the number of cycles to failure:  
\[S=-0.062\log N+0.969 \tag{1}\]  
The current approach was based on a linear fit for the Wohler curve. For an improved approach, two- and three-parameter Weibull distribution models could be used as in Reference [38]. In [39; 40; 41], the methodology for selecting the Weibull distribution models and the compatibility requirements over the whole \(S\)-\(N\) field are given. This approach may be suitable in ascertaining the predictive fatigue life assessment. For the current approach, the goodness-of-fit was calculated using the chi-squared test. For all the fatigue tests (datapoints in Figure 6), the value equals 1. As such, the approach was considered satisfactory for our purposes. Since the assessment was carried out separately for one and two wheel prints, it was interesting to examine the difference in the Wohler curve for the experiments with one and two wheel prints. Figure 7 gives these results, with Figure 7a showing the datapoints from the FAT series for the single wheel print, and Figure 7b showing the datapoints for the double wheel print. The markers in Figure 7 are different for the datapoints obtained at a number of cycles that resulted in failure and a number of cycles that were calculated using the previously mentioned conservative assumption. The Wohler curve for the datapoints with a single wheel load is:  
\[S=-0.066\log N+1.026 \tag{2}\]  
The 5% lower bound (characteristic value) of this expression, which can be used for the assessment, is:  
\[S_{char}=-0.066\log N+0.922 \tag{3}\]  
The Wohler curve for the datapoints with a double wheel load is:  
\[S=-0.045\log N+0.885 \tag{4}\]  
Figure 6: Relation between the number of cycles \(N\) and the applied load ratio \(F/P_{\text{max}}\) in all the fatigue experiments, adapted from Reference [37]. Reprinted with permission. This figure was originally published in Vol. 116 of the ACI Structural Journal. The 5% lower bound of this expression is:  
\[S_{char}=-0.045\log N+0.825 \tag{5}\]  
The slope of the Wohler curve for the case with two wheel loads is lower compared to the case with a single wheel load. However, for the case with a double wheel load, no low-cycle fatigue experimental results are available. For one load cycle Equation (2) gives a load ratio of 1.026, and Equation (4) gives a load ratio of 0.885.