## 5 Influence Surface-Based Stress Spectra Analysis
### Fatigue Life Assessment  
In the Eurocode 3 (EC3) [23], the three-stages stress-life (S-N) model is proposed, including two particular points, i.e., the transition point at five million cycles and the cut-out limit at 10 million cycles. Between the transition point and the cut-out limit, the power index \(m\) will change from three to five. Beyond the cut-out limit, infinite fatigue life could be expected for welding joints, indicating that the power index \(m\) is infinite. However, this only happens in the ideal situation when the pure constant amplitude loads are applied [29]. According to the fracture mechanics [44], the fatigue limit depends on both the material properties and the size of cracks, as shown in Equation (2):  
\[\Delta K_{th}=B\Delta\sigma_{L}\sqrt{\pi a_{i}} \tag{2}\]  
where \(\Delta K_{th}\) is the fatigue threshold of the material, below which the crack will not grow; \(B\) is a constant related to the fatigue strength; \(\Delta\sigma_{L}\) stands for the fatigue limit; \(a_{i}\) is the crack size.  
According to Equation (2), the fatigue limit will decrease with the increase in the crack size under the fixed fatigue threshold. Based on this assumption, the high-level stress ranges induced by heavy trucks will degrade the fatigue limit of joints with time. As a result, the stress ranges lower than the original limit will gradually contribute to fatigue damage [29]. On this end, the S-N should be expanded beyond 10 million cycles using the power index \(m=5\). Based on the above analysis, the S-N curves of RD joints have been modified, as shown in Figure 14a,b.  
Figure 13: Derivation of stress spectra using Monte Carlo simulations.  
In this study, the linear damage accumulation rule is considered [45], as shown in Equation (3):  
\[D=\sum\
olimits_{i=1}^{n}\left(\frac{n_{i}}{N_{i}}\right) \tag{3}\]  
where \(D\) is the damage factor that failure; \(n_{i}\) stands for the number of cycles corresponding to the stress range \(\Delta\sigma_{i}\); \(N_{i}\) is the allowable fatigue life under the range \(\Delta\sigma_{i}\).  
Generally, the fatigue failure of welding joints is assumed when the damage factor \(D\) equals to 1.0 [20; 22; 23]. However, such consideration may lead to non-conservative results for OSDs, which is under the variable amplitude loading by different vehicles. Alternatively, the allowance of the damage factor \(D\) has been lowered to 0.5, as suggested by [PERSON] [29]. This factor is common in construction sector [46; 47; 48].  
The fatigue life under the specific stress range could be calculated using the S-N curves, as shown in Equation (4):  
\[N_{i}=\left\{\begin{array}{cc}(\Delta\sigma_{C}/\Delta\sigma_{i})^{3}\cdot 2 \times 10^{6}&\forall\Delta\sigma_{i}>\Delta\sigma_{D}\\ (\Delta\sigma_{D}/\Delta\sigma_{i})^{5}\cdot 5\times 10^{6}&\forall\Delta\sigma_{i} \leq\Delta\sigma_{D}\end{array}\right. \tag{4}\]  
where \(\Delta\sigma_{c}\) and \(\Delta\sigma_{D}\) are the design fatigue strength at 2 million and 5 million cycles, respectively.  
By setting the maximum damage factor as 0.5, the allowance of lifetime traffic volume could be solved through Equations (3) and (4).