## 5 Influence Surface-Based Stress Spectra Analysis
### Influence Surface-Based Solution  
In the fatigue evaluation of welding joints, an essential step is to solve the full stress history under the vehicle model. Generally, the history induced by one vehicle should be solved with many solution steps, in which the vehicle is loaded at a series of adjacent locations in the longitudinal direction. Further, a large-size database of vehicles (i.e., \(10^{5}\)) is often needed in deriving the stress spectra, even if the advanced Latino hyper sampling [42] is employed, which further escalates the computational complexity. Thus, the influence surface method has been employed to reduce the solving cost [1,25]. According to studies [1,33], the local stress in RD joints is highly sensitive to the location of wheel loads, and the influence of the loads is limited within one segment in length and two U-ribs in width. Based on this assumption, the loading area is selected, as shown in Figure 9.  
Figure 8: Multiple-scale finite element model. The procedures of stress analysis are demonstrated in Figure 10, including the FE analysis, influence surface generation and stress history derivation. In the FE analysis, discrete stress responses have been solved by applying the unit nodal force on each node inside the loading area. Then, quadratic interpolation is used to construct the matrix of influence surface under the specific density, i.e., 2 mm in this study. After that, the influence surface of the unit wheel load could be easily generated through matrix operations on the influence surface of unit loads. Given the vehicle configuration and lateral position, the stress history induced by each wheel could be directly solved by vector operations. Finally, the full stress history of the vehicle could be calculated by the linear superposition of the wheel-based history over the time domain. The approach shown in Figure 10 has been coded in the MATLAB [43], in combination with the stress result solved from the FE model. Figure 11a,b shows the influence surface of the unit nodal load in terms of the nominal stress and hot spot stress, respectively. Figure 11: Unitised influence surface of joint U12\({}_{L}\): (**a**) Nominal stress; (**b**) Hot spot stress. Figure 10: Flowchart of the influence surface-based stress analysis. Figure 9: Loading area selected for analysis. For a better illustration, the influence surface is utilised in Figure 10. The two results are highly similar with notable features: (1) a positive peak could be found in the vicinity of RD joints, which is extremely narrow; (2) near the positive peak there are two negative peaks, which are lower in the amplitude but much wider. Based on the above analysis, the stress history could be solved under the vehicle passing through different lateral positions. For illustration, the stress history with the highest response has been employed as the most critical history to compare the four traffic models. It can be found in Figure 4 that the joint U12\({}_{\text{L}}\) is prone to fatigue cracking since it is in the slow lane and close to the frequent loading position of wheels. Thus, the critical stress history of the U12\({}_{\text{L}}\) has been selected for the comparison, as shown in Figure 12a-d. It is worth stating that the stress history of one vehicle could be generated in less than 1/160 s by performing the influence surface-based analysis on a four-cores workstation, which may take more than 300 s using the same device (i.e., increased by almost 50,000 times in efficiency). Unlike the standard truck models, the observation model takes into consideration the difference between vehicle groups as well as the uncertainties in axle weights of the vehicles within the same group. Thus, In the case of the observation model, the calculation is made using the most critical six-axles truck with the upper 97.7%th percentile value of axle weights [37]. The result shows that the stress history in RD joints is mainly determined by the axle number and axle weight.