## 2 Governing Equations of Stiffened Panel and Its Solution Procedure
### Fundamental Equation of Spatial Stiffer with Arbitrary Boundaries  
Deformation of the stiffener is analyzed by the modified EBB model. Modified EBB approach introduces the transformation relations of displacements and rotations between arbitrary points on the same cross-section of the stringent entity. Cartesian coordinate system \(O^{b}xyz\) of beam is shown in Figure 1. \(\varphi_{x}\), \(\varphi_{y}=-dw^{b}/dx\) and \(\varphi_{z}=dv^{b}/dx\) are the rotations around three coordinate axes, respectively. With the assumptions imposed in EBB theory, the stress resultants can be described as [21; 22]:  
\[\Big{[} \begin{array}{cccc}N_{x}&M_{y}&M_{z}&M_{yz}\end{array} \Big{]}^{ \mathrm{T}}=\big{[} \begin{array}{cccc}\mathbf{B}\end{array} \Big{]}\Big{[} \begin{array}{cccc}\varepsilon_{xx}&K_{y}&K_{z}&K_{x}\end{array} \Big{]}^{ \mathrm{T}}, \tag{24}\]  
where elements of matrix **B** are \(B_{11}=E^{b}A\), \(B_{22}=E^{b}I_{yy}\), \(B_{23}=B_{32}=E^{b}I_{yz}\), \(B_{33}=E^{b}I_{zz}\), \(B_{44}=\frac{E^{b}I_{xx}}{2(1+\mu^{b})}\), other elements of **B** are zeroes, \(\varepsilon_{xx}=du^{b}/dx\) is the axial strain, \(K_{x}=d\varphi_{x}/dx\), \(K_{y}=-d^{2}w^{b}/dx^{2}\) and \(K_{z}=d^{2}\varphi^{b}/dx^{2}\) denote components of the curvature vector of axis \(x\). The governing differential equations of EBB are derived based on the principle of minimum total potential energy, which implies that the total potential energy of the system must be stationary and the stationary value is always a minimum. The total potential energy of the system is defined by:  
\[\Gamma(u^{b},v^{b},w^{b})=\frac{1}{2}S_{\Sigma}(u^{b},v^{b},w^{b})-\Theta_{ \Sigma}(u^{b},v^{b},w^{b}), \tag{25}\]  
where \(\Gamma(u^{b},v^{b},w^{b})\) is a functional, \(\frac{1}{2}S_{\Sigma}(u^{b},v^{b},w^{b})\) is total strain energy, \(\Theta_{\Sigma}(u^{b},v^{b},w^{b})\) is the work done by external forces. To achieve the stationary total potential energy, the variation of Equation (25) should satisfy the following relations:  
\[\delta\Gamma=\delta S_{\Sigma}-\delta\Theta_{\Sigma}=0. \tag{26}\]  
The strain energy of a beam is defined in terms of six components: one axial force, two bending moments, two shear forces and one torsional moment [23]. Since the shear deformation is neglected in the EBB theory, the total strain energy is expressed by the sum of the strain energy components due to tension or compression, bending and torsion force. The variation of the strain energy is described as:  
\[\delta S_{\Sigma}=\int\Big{(}N_{x}\delta\varepsilon_{xx}+M_{y}\delta K_{y}+M_ {z}\delta K_{z}+M_{yz}\delta K_{x}\Big{)}dx. \tag{27}\]The virtual work of external forces is given by:  
\[\delta\Theta_{\Sigma}=\int\Big{(}q_{x}\delta u^{b}+q_{rx}\delta\varphi_{x}\Big{)} dx+\int_{x_{1}}^{x_{2}}\Big{(}q_{y}\delta v^{b}+q_{z}\delta w^{b}\Big{)}dx+ \Big{[}q_{y}\delta v^{b}+q_{z}\delta w^{b}+q_{ry}\delta\varphi_{y}+q_{rz}\delta \varphi_{z}\Big{]}_{x_{1}}^{\mathbb{N}_{2}}, \tag{28}\]  
where the notation \([\ldots]_{x_{1}}^{x_{2}}\) means the minus of the function values with selected independent variable values \(x_{2}\) and \(x_{1}\). Using the [PERSON]'s theorem and method of integration by parts, Equation (26) is rewritten in the following form:  
\[