## 2 Theoretical Background
### Stress Calculation from Hull Girder Loads  
The global hull girder loads are defined by VBM (vertical bending moment), HBM (horizontal bending moment), and TM (torsional moment). These hull girder moments can be calculated by superposition of the hull girder mode. Under the assumption that the springing vibration response caused by the hull behavior has little effect on changes in local loads (such as external and internal pressure), it was applied to calculate the local stress response using only the hull girder global loads. The target ship was divided into several strips in the x-direction (longitudinal direction), and the loads distributed in the longitudinal direction could be defined as loads on each specific strip. In addition, the vertical and horizontal bending moments could be easily calculated using beam theory. The longitudinal distribution of the torsional load (\(M_{T}\)) components acting on each section can be calculated as in Equation (20). \[M_{T}(x)=-\int_{0}^{x}m(x)dx \tag{20}\]  
As the HSF (horizontal shear force) does not act at the center of the shear, the HBM always interacts with the TM simultaneously. In general, it is assumed that the TM acts at the shear center of the transverse section. In this study, it was assumed that the horizontal shear force was applied at a position of approximately 65% of the scantling draught based on the bottom line (in the case of maximum hogging loading in trim and stability, which is generally applied for fatigue strength evaluation). Owing to this assumption, one more constraint introduced by horizontal shear forces was created. The load distribution applied to each node is shown in Figure 6. And is defined as from (a) to (d). Assuming that the number of nodes in the strip section is \(n\), the sum of the nodal forces in the transverse direction \((f_{H})\) is expressed as Equation (21). \[f_{H}=\sum_{n}(\gamma_{1}+\gamma_{2}y) \tag{21}\]  
Figure 6: Nodal force distribution. The moment at the shear center \((SC)\) by the previously assumed horizontal force is expressed by Equation (22). \[f_{H}(0.65T+SC)=\sum_{n}(\gamma_{1}+\gamma_{2}z)(z+SC) \tag{22}\]  
Similarly, the sum of the nodal forces in the vertical direction \((f_{v})\) is expressed as Equation (23). \[f_{v}=\sum_{n}(\gamma_{3}+\gamma_{4}y) \tag{23}\]  
The torsional loads at the shear center \((SC)\) by the sum of nodal forces in the vertical direction are expressed by Equation (24). \[m_{T}+f_{H}(0.65T+SC)=\sum_{n}(\gamma_{3}+\gamma_{4}y)y \tag{24}\]  
Here, \(m_{T}\) is the torsional load generated by the horizontal and vertical loads acting on the nodes of the strip section. Given four equations for the unknown factors \((\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})\), the determinant can be constructed as in Equation (25), and the node load factor can be obtained. \[\left[ \begin{array}{cccc}n&\sum\limits_{i=1}^{n}z_{i}&0&0\\ \sum\limits_{i=1}^{n}z_{i}&\sum\limits_{i=1}^{n}z_{i}^{2}+sc\sum\limits_{i=1}^ {n}z_{i}&0&0\\ 0&0&n&\sum\limits_{i=1}^{n}y\\ 0&0&\sum\limits_{i=1}^{n}y_{i}&\sum\limits_{i=1}^{n}y_{i}^{2}\end{array}  \right]\left[ \begin{array}{c}\gamma_{1}\\ \gamma_{2}\\ \gamma_{3}\\ \gamma_{4}\end{array} \right]=\left[ \begin{array}{c}f_{H}\\ f_{H}(0.65T+SC)\\ f_{v}\\ m_{T}+f_{H}(0.65T+SC)\end{array} \right] \tag{25}\]  
To examine the correlation between the calculated stress and load distribution calculated from the above equations, the following can be done: first, the three hull girder load components (VBM, HBM, and TM) can be uniformly combined in (a) the case of positive direction \((+)\), (b) the case of negative direction \((-)\), and (c) the case of loads that do not act. Next, by analyzing the stress on the hot spot position through structural analysis, the correlation between the load distribution and the hot spot stress can be analyzed. The stress acting on any position of the hull can be obtained from Equation (26). A more detailed theoretical explanation has been provided by [PERSON] et al. (2020).