# Residual Stress Relaxation of DRWDs in OSDs under Constant/Variable Amplitude Cyclic Loading
## 4 Residual Stress Relaxation of DRWDs by External Cyclic Loading
### Welding Residual Stress Release of DRWDs under Variable Amplitude Cyclic Loading  
During the service period of the bridge, the steel deck bears the repeated action of the vehicle loading directly, and the wheel loading is usually the variable amplitude loading. For the sake of convenience, 12 variable-amplitude loading cases are designated, as shown in Table 3, and their influence on the residual stress release in DRWs is analyzed. The symbols \(\left(\sigma_{app}\right)_{n}(n=1,2,\ldots 5)\) denote the stress of the \(n\)th cyclic loading. According to the study of 25 constant amplitude loading cases in the previous section, the residual stress relaxation mostly occurred in the first loading cycle, and tended to be stable after five cycles. Therefore, the number of cycles in this section is set to five. Taking C1 and C12 as examples, the stress time-history curves are shown in Figure 20. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Cases** & \(\left(\sigma_{app}\right)_{1}\) & \(\left(\sigma_{app}\right)_{2}\) & \(\left(\sigma_{app}\right)_{3}\) & \(\left(\sigma_{app}\right)_{4}\) & \(\left(\sigma_{app}\right)_{5}\) \\ \hline C1 & 130 & 40 & 40 & 40 & 40 \\ C2 & 40 & 130 & 40 & 40 & 40 \\ C3 & 40 & 40 & 130 & 40 & 40 \\ C4 & 40 & 40 & 40 & 40 & 130 \\ C6 & 32.5 & 65 & 65 & 65 & 65 \\ C7 & 65 & 65 & 65 & 65 & 65 \\ C8 & 97.5 & 65 & 65 & 65 & 65 \\ C9 & 130 & 65 & 65 & 65 & 65 \\ C10 & 130 & 100 & 65 & 65 & 65 \\ C11 & 130 & 65 & 100 & 65 & 65 \\ C12 & 65 & 130 & 100 & 65 & 65 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Twelve variable-amplitude cyclic loading cases (unit: MPa). Figure 19: Comparison between numerical simulation and formula prediction. For convenience, the initial transverse residual stress (transverse residual stress at \(n=0\)) is denoted as \((\sigma_{x})_{ini}\); the initial longitudinal residual stress is denoted as \((\sigma_{z})_{ini}\). According to the simulation results in Section 2.3, \((\sigma_{x})_{ini}\) was 146.54 MPa, and \((\sigma_{x})_{ini}\) was calculated to be 147.32 MPa based on the extracted FE data. In C1-C5, the cyclic order of the maximum loading is 1 to 5, respectively, and their common feature is that they have the same maximum loading (130 MPa). As shown in Figure 21a, the final \(\sigma_{x}\) at \(n=5\) are the same in C1-C5, which indicates that the residual stress releases in those five cases is the same. The reason is that the maximum value of the variable amplitude cyclic loading they are subjected to is equal. Moreover, the residual stress release only occurs in the first cycle in C1. However, residual stress release occurred in the first and second cycles, the first and third cycles, the first and fourth cycles, and the first and fifth cycles in C2-C5, respectively. This is due to the fact that the magnitude of residual stress relaxation is controlled by the maximum stress in the variable amplitude cyclic loading. Finally, the stress will not continue to be released if the DRWDs have the same or smaller amplitude loading, after the residual stress is released. For instance, if the maximum stress of cyclic loading appears in the first cycle, then thereafter the loading stress will be smaller than that of the first cycle in C1. The corresponding residual stress release only appears in the first cycle, and there is almost no relaxation of residual stress in the second to fifth stress loading processes. In C3, although the absolute maximum stress appears in the third cycle, the relative maximum stress is \((\sigma_{app})_{1}\) in the first cycle, and the first stress relaxation occurs in this cycle. In the second cycle, \((\sigma_{app})_{2}=(\sigma_{app})_{1}\), the residual stress of the cycle has little relaxation. In the third cycle, \((\sigma_{app})_{3}>(\sigma_{app})_{1}=(\sigma_{app})_{2}\), the residual stress of the cycle is released a second time. After that \((\sigma_{app})_{3}>(\sigma_{app})_{1}=(\sigma_{app})_{2}=(\sigma_{app})_{4}=( \sigma_{app})_{5}\), there is little relaxation of residual stress in the fourth and fifth cycles.