## 2 Stiffened Laminated Panel with Piezoelectric Materials
### Finite Element Formulations
\begin{array}{c}0\\ 0\\ 1/t_{a}\end{array} \right\}\phi_{a}=-\mathbf{B}_{\phi}\phi_{a} \tag{7}\]  
where \(t_{a}\) and \(\phi_{a}\) are the thickness of the actuator and the electric potential, respectively. The present element has elastic degrees of freedom \(u_{p}^{o}\), \(v_{p}^{o}\), \(\theta_{xp}\), \(\theta_{yp}\), and \(w_{p}^{o}\) per node and electrical degree of freedom \(\phi\) per piezoelectric layer. On the other hand, the strains of the \(x\)-stiffener are indicated as follows [17]:  
\[\varepsilon_{xs}=\left\{ \begin{array}{c}\varepsilon_{xx}\\ \gamma_{xz}\end{array} \right\}_{xs}=\left[ \begin{array}{cccc}1&z&0&0\\ 0&0&1&1\end{array} \right]\left\{ \begin{array}{c}u_{xs,x}^{o}\\ \theta_{xs,x}\\ \theta_{xs}\\ w_{xs,x}^{o}\end{array} \right\}=\mathbf{H}_{xs}\varepsilon_{xs}^{o} \tag{8}\]  
The stress resultant can be derived as follows:  
\[\left\{ \begin{array}{c}F_{xs}\\ M_{xs}\\ V_{xs}\end{array} \right\}=\left[ \begin{array}{cccc}A_{xs,11}&B_{xs,11}&0&0 \\ B_{xs,11}&D_{xs,11}&0&0\\ 0&0&S_{xs,55}&S_{xs,55}\end{array} \right]\varepsilon_{xs}^{o}=\mathbf{D}_{xs }\varepsilon_{xs}^{o} \tag{9}\]  
where \(A_{xs,ij},B_{xs,ij},D_{xs,ij},S_{xs,ij}=\sum\limits_{k=1}^{n}\int_{x_{k-1}}^{z _{k}}Q_{ij}\Big{(}1,\,z,\,z^{2},\,\kappa_{xs}\Big{)}dz\). To obtain the energy equations, the strain energy function is derived using the variational principles as follows:  
\[ \begin{array}{lcl}V^{e}&=&V_{p}^{e}+V_{xs}^{e}\\ &=&\frac{1}{2}\delta_{p}^{eT}\int_{A}\Big{(}\mathbf{B}_{m}{}^{T}\mathbf{A}_{ ij}\mathbf{B}_{m}+\mathbf{B}_{m}{}^{T}\mathbf{B}_{ij}\mathbf{B}_{b}+\mathbf{B}_{b}{}^{T} \mathbf{B}_{ij}\mathbf{B}_{m}+\mathbf{B}_{b}{}^{T}\mathbf{D}_{ij}\mathbf{B}_{ b}+\mathbf{B}_{s}{}^{T}\mathbf{S}_{ij}\mathbf{B}_{s}\Big{)}dA\delta_{p}^{e}\\ &&+\frac{1}{2}\delta_{p}^{eT}\int_{A}\Big{(}\mathbf{B}_{m}{}^{T}\mathbf{E}_{1} \mathbf{B}_{\phi}+\mathbf{B}_{b}{}^{T}\mathbf{E}_{2}\mathbf{B}_{\phi}\Big{)} dA\phi^{e}+\frac{1}{2}\delta_{p}^{eT}\int_{L}\mathbf{T}_{xs}{}^{T}\mathbf{B}_{xs}{}^{T} \mathbf{D}_{xs}\mathbf{B}_{xs}\mathbf{T}_{xs}dx\xi_{p}^{e}\\ &=&\frac{1}{2}\delta_{p}^{eT}\mathbf{K}_{p}^{e}\delta_{p}^{e}+\frac{1}{2}\delta_ {p}^{eT}\mathbf{K}_{u\phi}^{e}\phi^{e}+\frac{1}{2}\delta_{p}^{eT}\mathbf{K}_{xs }^{e}\delta_{p}^{e}\end{array}  \tag{10}\]  
where \(\mathbf{E}_{1},\mathbf{E}_{2}=\int_{x_{p}}\mathbf{e}_{j}{}^{T}(1,z)dz\), \(\mathbf{D}_{xs}=b_{xs}\int_{x_{S}}\mathbf{H}_{xs}{}^{T}\mathbf{Q}_{xs}\mathbf{ H}_{xs}dz\). \(\mathbf{K}_{p}^{e}\) is the element stiffness matrix of plate and \(\mathbf{K}_{u\phi}^{e}\) is the element elastic-electric coupling stiffness matrix. \(\mathbf{K}_{xs}^{e}\) is the element stiffness matrix of the \(x\)-stiffeners. The element kinetic energy can be written as  
\[ \begin{array}{lcl}T^{e}&=&T_{p}^{e}+T_{xs}^{e}\\ &=&\frac{1}{2}\dot{\delta}_{p}^{eT}\int_{A}\Big{[}\mathbf{P}(\overline{\mathbf{N} }_{pu}{}^{T}\overline{\mathbf{N}}_{pu}+\overline{\mathbf{N}}_{pv}{}^{T} \overline{\mathbf{N}}_{pv}+\overline{\mathbf{N}}_{pv}{}^{T}\overline{\mathbf{N}}_ {pv})+I\Big{(}\overline{\mathbf{N}}_{p\rho_{x}}{}^{T}\overline{\mathbf{N}}_{p \rho_{x}}+\overline{\mathbf{N}}_{p\rho_{y}}{}^{T}\overline{\mathbf{N}}_{p\rho_{y}} \Big{)}\Big{]}dA\xi_{p}^{e}\\ &&+\frac{1}{2}\dot{\delta}_{p}^{eT}\overline{\mathbf{N}}_{xp}{}^{T}\int_{L}\int_{L} \Big{[}\mathbf{P}(\overline{\mathbf{N}}_{xx}{}^{T}\overline{\mathbf{N}}_{xs}+ \overline{\mathbf{N}}_{xs}{}^{T}\overline{\mathbf{N}}_{xs}\overline{\mathbf{N}}_ {xs}\overline{\mathbf{N}}_{xs}\overline{\mathbf{N}}_{xs}\Big{)}+I\overline{ \mathbf{N}}_{xs\rho_{x}}{}^{T}\overline{\mathbf{N}}_{xs\rho_{y}}\Big{]}dx\Upsilon_{xs }\dot{\xi}_{p}^{e}\\ &=&\frac{1}{2}\dot{\delta}_{p}^{eT}\overline{\mathbf{M}}_{p}^{e}\delta_{p}^{e}+ \frac{1}{2}\dot{\delta}_{p}^{eT}\mathbf{M}_{xs}^{e}\dot{\xi}_{p}^{e}\end{array}  \tag{11}\]where \(P,I=\sum\limits_{k=1}^{n}\int_{z_{k-1}}^{z_{k}}\rho\Big{(}1,z^{2}\Big{)}dz\) and \(\overline{\mathbf{N}}\) is the shape function matrices relating the primary variables.