## 2 Existing Procedures for Hull Girder Load Adjustment
### Calculation of Reaction Force and Local Hull Grider Load  
The local hull girder loads can be calculated by integrating the local loads shown in Figure 6 and the reaction forces due to the boundary conditions. For the local load without reaction forces, as shown in Figure 7, the hull girder load at \(j^{\text{th}}\) station is calculated by integrating the local load acting from the stern to \(j^{\text{th}}\) station with respect to \(j^{\text{th}}\) station. The same method is used for all defined stations. The integral of force and moment over the local loads can be calculated using the load vector \(F_{i}\) and the distance vector \(r_{i}\), as shown in Equation (1). Since the local loads are located in the negative direction of \(j\)th station, they are signed minus according to the sign convention defined earlier. The subscript \(i\) indicates the point where the load is applied. \[F=-\underset{i}{\sum}F_{i},\quad M=-\underset{i}{\sum}r_{i}\times F_{i} \tag{1}\]  
Using Equation (1), the hull girder load at \(j\)th station can be calculated as Equations (2)-(5). \[AF_{j}=-\underset{i}{\sum}f_{xi},\quad HSF_{j}=-\underset{i}{\sum}f_{yi}, \quad VSF_{j}=-\underset{i}{\sum}f_{zi} \tag{2}\]  
\[TM_{j}=-\underset{i}{\sum}\{(y_{i}-y_{0})f_{zi}-(z_{i}-z_{s})f_{yi}\} \tag{3}\]  
\[VBM_{j}=-\underset{i}{\sum}\{(z_{i}-z_{0})f_{xi}-(x_{i}-x_{0})f_{zi}\} \tag{4}\]  
\[HBM_{j}=-\underset{i}{\sum}\{(x_{i}-x_{0})f_{yi}-(y_{i}-y_{0})f_{xi}\} \tag{5}\]  
where  
\(AF_{j}=\) the axial force summation from aft end to \(j\)th station;  
\(HSF_{j}=\) the horizontal shear force summation from aft end to \(j\)th station;  
\(VSF_{j}=\) the vertical shear force summation from aft end to \(j\)th station;  
\(TM_{j}=\) the torsional moment summation from aft end to \(j\)th station;  
\(VBM_{j}=\) the vertical bending moment summation from aft end to \(j\)th station;  
\(HBM_{j}=\) the horizontal bending moment summation from aft end to \(j\)th station;  
\(F_{i}\big{(}f_{xi},f_{yi},f_{zi}\big{)}=\) the applied force at \(i\)th point;  
\(X_{i}(x_{i},y_{i},z_{i})=\) the coordinate of the applied force point;  
\(X_{0}(x_{0},y_{0},z_{0})=\) the coordinate of the reference point at \(i\)th station;  
\(z_{s}=\) the vertical coordinate of the shear centre. Figure 7: Description of applied forces. In a structural analysis where no reaction forces are generated at the boundary conditions, Equation (2) can be used to calculate the hull girder load. However, in a cargo hold structural analysis where reaction forces are generated, the hull girder load is calculated by considering the reaction forces. The reaction forces at the boundary condition can be obtained from Equations (6)-(8) with reference to DNV-CG-0127 [25] or DNV [25] and Figure 8.  
\[R_{xf}=-\underset{i}{\sum}f_{xi} \tag{6}\]  
\[R_{yf}=-\frac{\underset{i}{\sum}(x_{i}-x_{a})f_{yi}}{\left(x_{f}-x_{a}\right)},\ \ R_{ya}=-\underset{i}{\sum}f_{yi}-R_{yf} \tag{7}\]  
\[R_{zf}=-\frac{\underset{i}{\sum}(x_{i}-x_{a})f_{zi}}{\left(x_{f}-x_{a}\right)},\ \ R_{za}=-\underset{i}{\sum}f_{zi}-R_{zf} \tag{8}\]  
Using Equations (6)-(8), the local hull girder load from Equations (2)-(5) can be recalculated as Equations (9)-(12), including the reaction forces that will be applied in the structural analysis.