## 3 Numerical Validation and Experiments
### Deformation of Stiffened Panel with Joints and Mixed Boundary Conditions  
Due to the mechanical interaction of the joined parts, the stringer-skin panel will suffer the secondary deformation in the joining process. Semi-analytical calculations and FE simulation of panel substructure model with different joining points are performed to analyze the deformation propagation of stiffened panel. Measurement experiments are conducted to verify the accuracy of the proposed method, as shown in Figure 3b. The intermediate quantity, concentrated force \(f_{z}{}^{st}\) is calculated by semi-analytical solutions, as listed in Table 5. The displacements of joining points on the stringers and skin are predicted by solving unknown coefficients \({}^{m}J_{i}\) and \(C_{i}\), as shown in Figure 7.  
\begin{table}
\begin{tabular}{c c} \hline
**Point** & **Concentrated Forces/N** \\ \hline \(\mathrm{P_{9}}^{\mathrm{I}}\) & 95.1882 \\ \(\mathrm{P_{9}}^{\mathrm{II}}\) & 104.734 \\ \(\mathrm{P^{I}}_{2n+1},\mathrm{P^{II}}_{2n+1}(n=1\cdots 8)\) & 103.89, 73.42, 37.73, 20.8, 15.24, 21.8, 33.15, 79.55, 61.45, 113.55, \\  & 92.09, 55.49, 30.64, 17.9, 14.57, 32.85, 28.99, 152.88 \\ \hline \end{tabular}
\end{table}
Table 5: Calculated intermediate joining forces.  
Figure 7: Deformation of stiffened panel with joints at different positions. (**a**) One joining point P9\({}^{\text{I}}\) on stringer I; (**b**) one joining point P9\({}^{\text{II}}\) on stringer II; (**c**) nine joining points on stringer I and nine joining points on stringer II; (**d**) magnitude of displacement given by FEM and proposed method and the deviation between both  
From Figure 7, it can be seen the joining operation causes dramatical deformation of shell and relatively slighter deformation of stringers. Figure 7a,b show that the absolute maximum deviations between the predicted and actual clearances are 0.180 mm and 0.193 mm when the point P9I and P9 II are joined respectively. Displacements in the direction of surface normal provided by the proposed method and FEM are consistent with each other. Figure 7c,d indicates that the final positions of the stiffened panel with joints are similar to the stringers, which have higher stiffness than skin. Maximum deviation of shell displacement between FEM and the proposed results are 0.198 mm which can verify the accuracy of the semi-analytical solutions. Compared with the skin deformation shown in Figure 4a, the area of panel between two stiffeners deforms obviously with the joining operation conducted. Moreover, Table 5 shows the values of concentrated forces imposed on the same joining points, P9I and P9 II, change in different situations. Figure 7 illustrates that the clearances \(\Delta\)w have similar values when the points on the stringer I and II are joined simultaneously or separately, but the joining forces change a lot.  
It can be concluded that the concentrated force on the joining point differs by the change of joining sequences. The relationship between the changing force and joining point position cannot be expressed by the discrete stiffness model which considers stiffeners just as immovable boundaries. Besides, discrete integration models need more zero and first spatial derivatives at the ends of individual segments to be re-entered into the deformation calculations when the joining force and clearances change. Moreover, compared with the curve interpolating algorithm, the proposed semi-analytical substructure model can provide the values of changing force with no more supplementary parameters introduced when the positions of joining points are different.