## 2 Governing Equations of Stiffened Panel and Its Solution Procedure
### Shell Subjected to Concentrated Force at Joining Interface and Functional Boundary Conditions
\right][ \mathbf{S}][\mathbf{m}\mathbf{J}] \tag{17}\]  
where elements of \([\mathbf{S}]\) are \(S_{11}=S_{47}=\cos(\alpha\xi_{2})\), \(S_{12}=-S_{48}=\sin(\alpha\xi_{2})\), \(S_{23}=S_{35}=\cos(\alpha\xi_{4})\), \(S_{24}=-S_{36}=\sin(\alpha\xi_{4})\) and other elements of \([\mathbf{S}]\) are zeroes, \([\mathbf{m}\mathbf{J}]=\left[ \begin{array}{cccccccc}m\mathbf{J}_{1}&m \mathbf{J}_{2}&m\mathbf{J}_{3}&m\mathbf{J}_{4}&m\mathbf{J}_{5}&m\mathbf{J}_{6} &m\mathbf{J}_{7}&m\mathbf{J}_{8}\end{array} \right]^{T}\) is a set of real coefficients in displacement expressions to be determined. Secondly, the particular solutions of Equation (14) are derived by using the Wronskian determinant. The Wronskian determinant of the functions \(e^{\eta_{0}\alpha}\)\((n=1,2,\ldots 8)\), a fundamental set of solutions to the associated homogeneous equation to Equation (14), is defined by:  
\[|W|=e^{\alpha\sum\limits_{n=1}^{8}\eta_{n}}\prod\limits_{n=1}^{7}\prod\limits _{j=n+1}^{8}\left(\eta_{n}-\eta_{j}\right). \tag{18}\]  
The determinant \(\left|W_{j}\right|\) is denoted by replacing the \(j\)-th column of the Wronskian with the column (0,0,...,0,1)\({}^{T}\). Particular solutions of Equation (14) can be derived from the following expression:  
\[{}^{p}\Phi_{i}^{*}(\alpha)=\sum\limits_{j=1}^{8}\left\{e^{\eta_{i}/\alpha} \int\left[-\frac{t}{G}q_{im}\right]\frac{\left|W_{j}\right|}{\left|W\right|}d \alpha\right\}. \tag{19}\]  
The load functions of concentrated force and uniform load encountered in the practical joining process of aircraft panel assembly are introduced to derive the explicit expressions of Equation (19). When the concentrated force \(f\) is imposed on the joining interface of shell at point \((\alpha_{p},\beta_{p})\), the associated distributed load \(q\) introduced into the deformation equation is expressed as the Dirac delta function \(f\delta(\alpha-\alpha_{p})\delta(\beta-\beta_{p})\). Using Equation (12), the following expression can be derived:  
\[\left[ \begin{array}{cccccccc}q_{1m}(\alpha)&q_{2m}(\alpha)&q_{3m}(\alpha) \end{array} \right]^{T}=\frac{c_{m}}{b}\delta(\alpha-\alpha_{p})\int_{0}^{b} \delta(\beta-\beta_{p})\left[\mathbf{T_{m}}(\beta_{p})\right]\left[ \begin{array} []{cccccccc}f_{1}&f_{2}&f_{3}\end{array} \right]^{T} \tag{20}\]  
Then, the particular solution of Equation (14) with concentrated force applied on shell when \(m\
eq 0\) is transformed as:  
\[\left[ \begin{array}{cccccccc}p&\[PERSON]_{1}^{*}&p&\[PERSON]_{2}^{*}&p&\[PERSON]_{3}^{*} \end{array} \right]^{T}=\frac{2}{Gb}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\! where \(q_{im}\) is a constant or linear function of \(\alpha\). Since the distributed load is a combination of concentrated force and uniform load, the complete solution of Equation (14) with \(m+0\) should be the superposition of Equations (17), (21) and (22) as \(\[PERSON]_{i}^{*}={}^{h}\[PERSON]_{i}^{*}+{}^{P}\[PERSON]_{i}^{*}={}^{h}\[PERSON]_{i}^{*}+{}^{P}_{uni }\[PERSON]_{i}^{*}+{}^{P}_{con}\[PERSON]_{i}^{*}\). Subsequently, substituting \(\Phi_{i}^{*}\) into Equations (10) and (6), the displacements expressions with coefficients \({}^{m}J_{i}\) (\(i=1,2...8\)) can be obtained as Equation (A2).  
Moreover, the presented displacement functions should satisfy the boundary conditions prescribed on shell edges. The initial variations on the skin curve edges are regarded as the non-zero kinematic boundary conditions imposed on edges \(\alpha=\alpha_{i}\) (\(i=1,2\)), as follows:  
\[u\big{|}_{\alpha=\alpha_{i}}=U_{i}(\beta)\,\ v\big{|}_{\alpha=\alpha_{i}}=V_{i}( \beta)\,\ w\big{|}_{\alpha=\alpha_{i}}=W_{i}(\beta)\,\ \varphi_{v}\big{|}_{ \alpha=\alpha_{i}}=\Psi V_{i}(\beta), \tag{23}\]  
where \(\varphi_{v}=-\partial w/\partial\alpha\) is the rotation around the axis \(\beta\). Expanding \(U_{i}(\beta)\), \(V_{i}(\beta)\), \(W_{i}(\beta)\) and \(\Psi V_{i}(\beta)\) into Fourier series as the form of displacements expressions in Equations (A3) and (A4), a set of linear algebraic equations specific to coefficients are generated.