## 2 Governing Equations of Stiffened Panel and Its Solution Procedure
### Shell Subjected to Concentrated Force at Joining Interface and Functional Boundary Conditions  
Kirchhoff-Love shell theory is utilized to establish the deformation model of single curvature skin, regarded as a thin cylindrical shell. The curvilinear coordinate system of the undeformed shell is denoted as \(\mathcal{O}\alpha\beta\gamma\). Curve edges of the panel are located on fixturing boards, which induce the position variations on boundary conditions, as shown in Figure 1.  
The equilibrium equations of cylindrical shell [11] along axes of local coordinate system \(\mathcal{O}\tau_{1}\tau_{2}\tau_{3}\) are established by  
\[\left\{ \begin{array}{c}\frac{\partial N_{1}}{\partial\alpha}+\frac{ \partial N_{21}}{\partial\beta}+q_{1}=0\\ \frac{\partial N_{12}}{\partial\alpha}+\frac{\partial N_{2}}{\partial\beta}+ \frac{1}{R}\Big{(}\frac{\partial M_{12}}{\partial\alpha}+\frac{\partial M_{ 2}}{\partial\beta}\Big{)}+q_{2}=0\\ -\frac{N_{2}}{R}+\frac{\partial}{\partial\alpha}\Big{(}\frac{\partial M_{1}} {\partial\alpha}+\frac{\partial M_{21}}{\partial\beta}\Big{)}+\frac{\partial }{\partial\beta}\Big{(}\frac{\partial M_{12}}{\partial\alpha}+\frac{\partial M _{2}}{\partial\beta}\Big{)}+q_{3}=0\end{array} \right., \tag{1}\]  
where \(N_{12}=N_{21}\), \(M_{12}=M_{21}\). Moreover, the third terms of second equation in Equation (1) are small enough quantities to be eliminated for thin shell [19]. The stress components are expressed in terms of the middle surface stain components, as follows:  
\[\left[ \begin{array}{ccc}\sigma_{1}&\sigma_{2}&\sigma_{12}\end{array} \right] ^{T}=\left[ \begin{array}{cc}\mathbf{Q}_{3\times 3}^{\mathbf{I}}&\mathbf{Q}_{3 \times 3}^{\mathbf{I}}\end{array} \right]\left[ \begin{array}{ccc}\mathbf{E}_{3 \times 3}&0\\ 0&\gamma\mathbf{E}_{3\times 3}\end{array} \right]\left[ \begin{array}{ccc} \epsilon_{1}^{0}&\epsilon_{2}^{0}&\epsilon_{12}^{0}&\chi_{1}&\chi_{2}&\chi_{ 12}\end{array} \right]^{T}, \tag{2}\]  
Figure 1: Assembled aircraft stiffened panel structure. where \([\mathbf{E}_{3\times 3}]\) is a unit matrix, elements of matrix \([\mathbf{Q^{I}}]\) are \(Q_{11}=Q_{22}=\frac{E}{1-\mu^{2}}\), \(Q_{12}=Q_{21}=\frac{E\mu}{1-\mu^{2}}\), \(Q_{33}=\frac{E}{2(1+\mu)}\), other elements of \([\mathbf{Q^{I}}]\) are zeroes, strains are \(\varepsilon_{1}^{0}=\frac{\partial u}{\partial\alpha}\), \(\varepsilon_{2}^{0}=\frac{\partial v}{\partial\beta}+\frac{w}{R}\) and \(\varepsilon_{12}^{0}=\frac{\partial v}{\partial\alpha}+\frac{\partial u}{ \partial\beta}\), changes in curvature and twist are \(\chi_{1}=-\frac{\partial^{2}w}{\partial\alpha^{2}}\), \(\chi_{2}=-\frac{\partial^{2}w}{\partial\beta^{2}}\) and \(\chi_{12}=-2\frac{\partial^{2}w}{\partial\alpha\partial\beta}\). The stress resultants and stress couples of a thin cylindrical shell are given by:  
\[\left[ \begin{array}{cccccc}N_{1}&N_{2}&N_{12}&M_{1}&M_{2}&M_{12} \end{array} \right]^{T}=\int_{-h/2}^{h/2}\!\!\left[ \begin{array}{c}\mathbf{E}_ {3\times 3}\\ \gamma\mathbf{E}_{3\times 3}\end{array} \right]\left[ \begin{array}{c} \mathbf{\alpha}_{1}\\ \gamma\mathbf{E}_{3\times 3}\end{array} \right]\left[ \begin{array}{c} \mathbf{\alpha}_{1}\\ \mathbf{\alpha}_{2}\\ \mathbf{\alpha}_{12}\end{array} \right]\!\!\!dy=[\mathbf{Q^{r}}]\!\!\left[ \begin{array}{cccccc}\varepsilon_{1}^{0}&\varepsilon_{2}^{0}&\varepsilon_{ 12}^{0}&\chi_{1}&\chi_{2}&\chi_{12}\end{array} \right]^{T} \tag{3}\]  
where \([\mathbf{Q^{*}}]=\int_{-h/2}^{h/2}\!\!\left[ \begin{array}{c}\mathbf{E}_{3 \times 3}\\ \gamma\mathbf{E}_{3\times 3}\end{array} \right]\left[ \begin{array}{c} \mathbf{Q^{I}}_{3\times 3}&\mathbf{Q^{I}}_{3\times 3}\end{array} \right]\left[ \begin{array}{c}\mathbf{E}_{3\times 3}\\ 0&\gamma\mathbf{E}_{3\times 3}\end{array} \right]\!\!\!dy=\left[ \begin{array}{cc}h\mathbf{Q^{I}}&0\\ 0&(h^{3}/12)\mathbf{Q^{I}}\end{array} \right]\!\!\). Substituting Equation (3) into Equation (1), governing equations of the deformation of a thin cylindrical shell can be derived as  
\[[\mathbf{D}]\left[ \begin{array}{c}u\\ v\\ w\end{array} \right]+\frac{1-\mu^{2}}{Eh}\!\!\left[