## 2 Governing Equations of Stiffened Panel and Its Solution Procedure
### Shell Subjected to Concentrated Force at Joining Interface and Functional Boundary Conditions
\begin{array}{c}q_{1}\\ q_{2}\\ -q_{3}\end{array} \right]\!\!=\!\left[ \begin{array}{c}0\\ 0\\ 0\end{array} \right]\!\!. \tag{4}\]  
Differential operator matrix of the above equation is denoted by:  
\[D=\left[\begin{array}{ccc}\frac{\partial^{2}}{\partial\alpha^{2}}+\frac{1- \mu}{2}\frac{\partial^{2}}{\partial\beta^{2}}&\frac{1+\mu}{2}\frac{\partial^{2 }}{\partial\alpha\partial\beta}&\frac{\mu}{R}\frac{\partial}{\partial\alpha} \\ \frac{1+\mu}{2}\frac{\partial^{2}}{\partial\alpha\partial\beta}&\frac{\partial ^{2}}{\partial\beta^{2}}+\frac{1-\mu}{2}\frac{\partial^{2}}{\partial\alpha^{2}}  
where \([\mathbf{T_{m}}(\beta)]=\left[ \begin{array}{cccc}\sin\overline{m}\beta&0&0\\ 0&\cos\overline{m}\beta&0\\ 0&0&\sin\overline{m}\beta\end{array} \right],\overline{m}=m\pi/b\). The displacements and stress acting on the straight edges of shell, \(\beta_{1}=0\) and \(\beta_{2}=b\), have the following relationships:  
\[ \begin{array}{c}u\big{|}_{\beta=\beta_{i}}=w\big{|}_{\beta=\beta_{i}}=0\,\ N_{2}\bigg{|}_{\beta=\beta_{i}}=\frac{E^{k}h_{i}}{1-\mu^{2}}\left(\frac{ \partial x}{\partial\beta}+\frac{w}{K}+\mu\frac{\partial u}{\partial\alpha} \right)\bigg{|}_{\beta=\beta_{i}}=0\,,\\ M_{2}\bigg{|}_{\beta=\beta_{i}}=-\frac{E^{k}h^{3}}{12[1-\mu^{2}]}\left( \frac{\partial^{2}w}{\partial\beta^{2}}+\mu\frac{\partial^{2}w}{\partial \alpha^{2}}\right)\bigg{|}_{\beta=\beta_{i}}=0\,,\ (i=1,2),\end{array}  \tag{9}\]  
which are also the boundary conditions on straight edges of panel. Therefore, the variables in potential functions and the loads are decoupled by expanding the functions into the form of Fourier series as Equation (8):  
\[\Big{[}\ \Phi_{1}(\alpha,\beta)\quad\Phi_{2}(\alpha,\beta)\quad\Phi_{3}( \alpha,\beta)\ \ \Big{]}^{T}=\sum_{m=0}^{\infty}\ [\mathbf{T_{m}}(\beta)]\Big{[}\ \Phi_{1}^{*}(\alpha)\quad\Phi_{2}^{*}( \alpha)\quad\Phi_{3}^{*}(\alpha)\ \ \Big{]}^{T}, \tag{10}\]  
and  
\[\Big{[}\ q_{1}(\alpha,\beta)\quad q_{2}(\alpha,\beta)\quad q_{3}(\alpha,\beta )\ \ \Big{]}^{T}=\sum_{m=0}^{\infty}\ [\mathbf{T_{m}}(\beta)]\Big{[}\ q_{1m}(\alpha)\quad q_{2m}( \alpha)\quad q_{3m}(\alpha)\ \ \Big{]}^{T}, \tag{11}\]  
where  
\[ \begin{array}{c}\Big{[}\ q_{1m}\quad q_{2m}\quad q_{3m}\ \Big{]}^{T}=\frac{c_{m}}{b}\int_{0}^{b} \big{[}\mathbf{T_{m}}(\beta)\big{]}\Big{[}\ q_{1}(\alpha,\beta)\quad q_{2}( \alpha,\beta)\quad q_{3}(\alpha,\beta)\ \ \Big{]}^{T}d\beta,\\ c_{m}=\left\{\begin{array}{cc}1&m=0\\ 2&m\
eq 0\end{array}\right.\end{array} . \tag{12}\]  
Substituting Equations (8) and (11) into Equation (4), the displacements and loads with \(m=0\) can be rewritten as:  
\[u=0,\ w=0,\ q_{1}=0,\ q_{3}=0,\ q_{2}=q_{2m}(\alpha),\ D_{22}V(\alpha)+q_{2m}( \alpha)=0. \tag{13}\]  
where \(v=V(\alpha)=-\frac{1}{Q_{33}}\iint q_{2m}(\alpha)d\alpha^{2}+{}^{0}J_{1}\alpha +{}^{0}J_{2},\ \ {}^{0}J_{i}\ \ \ (i=1,2)\) is the coefficient to be determined. When \(m\
eq 0\), substituting the expanded Equations (10)-(12) into Equation (7) and eliminating the trigonometric terms, the following equations can be obtained:  
\[\begin{array}{c}G\Big{[}\ \ 1\quad-4\overline{m}^{2}\quad 6\overline{m}^{4}+ \frac{12(1-\mu^{2})}{h^{2}R^{2}}\quad-4\overline{m}^{6}\quad\overline{m}^{8} \ \ \Big{]}\ \All roots of Equation (15) are obtained by:  
\[ \begin{array}{cccccccc}\left[\begin{array}{cccccccc}\eta_{1}&\eta_{2}&\eta_{3}& \eta_{4}&\eta_{5}&\eta_{6}&\eta_{7}&\eta_{8}\end{array}\right]^{T}=\left[ \begin{array}{cccccccc}\mathbf{Y}_{4\times 2}&0&\\ 0&\mathbf{Y}_{4\times 2}&\end{array}\right]\left[\begin{array}{cccccccc} \xi_{1}&\xi_{2}&\xi_{3}&\xi_{4}\end{array}\right],\\ \mathbf{Y}_{4\times 2}=\left[\begin{array}{cccccccc}1&-1&1&-1\\ i&-i&-i&i\end{array}\right]^{T}\end{array}  \tag{16}\]  
where real and imaginary parts \(\xi_{j}\)\((j=1,2,3,4)\) of complex roots \(\eta_{j}\)\((j=1,2,\ldots 8)\) are listed as Equation (A1) in Appendix A. Solutions of the homogeneous equation corresponding to Equation (14) can be expressed by:  
\[{}^{h}\Phi_{i}^{*}(\alpha)=\left[ \begin{array}{cccccccc}e^{\alpha\xi_{1}}&e ^{\alpha\xi_{3}}&e^{-\alpha\xi_{3}}&e^{-\alpha\xi_{1}}\end{array}