## 4 Probabilistic Fatigue Reliability Assessment
### Probabilistic Fatigue Action  
Given the fact that the stress response of a suspender mainly depends on the dynamic input of the coupled traffic-bridge-wind system, i.e., the wind and traffic loads, it is necessary to consider all the possible wind and traffic loading scenarios for accurately estimating the fatigue reliability of the suspenders. As already discussed in Section 3, the key parameters of wind load, i.e., the distribution of the mean cross-wind speed, can be obtained from the statistical analysis on the meteorological data. Accordingly, the full range of the mean cross-wind speed at the bridge site can be represented by 8 representative mean cross-wind speeds with the probability of occurrence listed in Table 2. Similarly, the key parameters of traffic loads included the vehicle category, gross vehicle weight (GVW), hourly traffic volume, and vehicle traffic lane occupancy ratios, which can be obtained through the traffic condition assessment. The traffic condition assessment revealed that the traffic conditions at the bridge site fall into two categories, i.e., free-flow condition and busy-flow condition, which can be properly simulated with the CA-based stochastic traffic model, as shown in Figure 4.  
Based on the wind and traffic information given above, the wind load is estimated with 8 reprehensive mean cross-wind speeds, while the traffic load is represented by free-flow and busy-flow conditions. As a result, a total of 16 wind and traffic loading scenarios can be used to reflect all possible wind and traffic loading conditions. Furthermore, by assuming the wind load and the traffic load as two independent variables, the daily effective stress range accounting for all possible combinations of wind and traffic loads can be expressed as follows:  
\[S_{re}^{*}=\left(\sum\
olimits_{i=1}^{2}\sum\
olimits_{j=1}^{8}P_{T,i}\cdot P _{W,j}\cdot S_{n,ij}^{m}\right)^{1/m}, \tag{13}\]  
where \(S_{re}^{*}\) = the daily effective stress range; subscript \(T\) denotes the traffic condition, and subscript \(W\) denotes the mean cross-wind speed; \(P_{T,\,i}\) = the probability occurrence of the \(i\)th (\(i\) = 1, 2) traffic condition in which \(P_{T,\,1}\) = 0.583 represents the probability occurrence of free-flow traffic and \(P_{T,\,2}\) = 0.417 represents the probability occurrence busy-flow traffic; \(P_{W,j}\) = the probability occurrence of the \(j\)th (\(j\) = 1, 2,..., 8) mean cross-wind speed, which is tabulated in Table 2; and \(S_{re,\,ij}\) = the equivalent fatigue stress range under combination of \(i\)th traffic condition and \(j\)th wind speed. The procedures for computing \(S_{re}^{*}\) are as followings. The first step is to perform the traffic-bridge-wind analysis to extract the stress response of the suspender under one wind and traffic loading scenario, which is further used to obtain the stress bin histogram through the rain-flow counting method. Meanwhile, the effective stress range can also be computed using Equation (12). Subsequently, the procedures in the first step are conduced repeatedly to obtain all 16 stress bin histograms and the associated effective stress range, each of which corresponds to one wind and traffic loading scenario. Finally, a new effective stress range accounting for all wind and traffic loading scenarios can be established by individual effective stress ranges from each stress bin histogram and the occurrence probability associated with wind and wave loading scenarios. Similarly, the predicted effective average daily number of cycles, \(N_{avg}^{*}\), can be derived using the occurrence probabilities associated with all possible wind and traffic loading scenarios. \[N_{avg}^{*}=\sum\
olimits_{i=1}^{2}\sum\
olimits_{j=1}^{8}P_{T,i}P_{W,j}N_{avg,j}, \tag{14}\]  
where \(N_{avg,\,i\!\!\!/}\) = average daily number of cycles under combination of \(i\)th traffic condition and \(j\)th wind speed.