## 2 Stiffened Laminated Panel with Piezoelectric Materials
### Finite Element Formulations  
First of all, the \(k^{th}\) layer's piezoelectric coupling relations between the elastic and the electrical fields could be presented, and the equations can be written as follows [15]:  
\[\mathbf{D}_{k}=\mathbf{e}_{k}\varepsilon_{k}+\varepsilon_{k}\mathbf{E}_{k} \tag{1}\]  
\[\boldsymbol{\sigma}=\mathbf{Q}\boldsymbol{\varepsilon}-\mathbf{e}_{k}^{T} \mathbf{E}_{k} \tag{2}\]  
where \(\boldsymbol{\varepsilon}\), \(\boldsymbol{\sigma}\), \(\mathbf{D}\), and \(\mathbf{E}\) are the strain, stress component, electric displacement, and electrical field vectors, respectively. In addition, \(\boldsymbol{\varepsilon}\), \(\boldsymbol{\varepsilon}\), and \(\mathbf{Q}\) are piezoelectric constants, permittivity coefficients, and reduced elastic constants matrices, respectively. Equation (1) means the direct piezoelectric effect, and Equation (2) indicates the converse piezoelectric effect. The stiffened plate elements are derived with the plate element and stiffener elements. The first-order shear deformation theory (FSDT) applied, the displacement fields are defined as follows:  
\[ \begin{array}{l}u(x,y,z)=u_{p}^{o}(x,y)+z\theta_{xp}(x,y),\\ v(x,y,z)=v_{p}^{p}(x,y)+z\theta_{yp}(x,y),\\ w(x,y,z)=w_{p}^{o}(x,y,z).\end{array}  \tag{3}\]  
where \(u_{p}^{o}\), \(v_{p}^{o}\), and \(w_{p}^{o}\) are the mid-plane displacements, and \(\theta_{xp}\) and \(\theta_{yp}\) are the rotations of transverse normal about \(x\) and \(y\), respectively. The displacement fields of the \(x\)-directional stiffener and \(y\)-directional stiffener are expressed as follows:  
\[ \begin{array}{l}u_{xs}(x,z)=u_{xs}^{o}(x)+z\theta_{xs}(x),\ \ w_{xs}(x,z)=w_{xs}^{o}(x)\\ v_{ys}(y,z)=v_{ys}^{o}(y)+z\theta_{ys}(y),\ \ w_{ys}(y,z)=w_{ys}^{o}(y)\end{array}  \tag{4}\]  
where \(x\)-axis is taken along the \(x\)-directional stiffener center line and \(z\)-axis is its upward normal line. The detailed formulations can be found in reference [16]. The strains are presented as a function of the nodal displacement variables. \[ \begin{array}{l}\boldsymbol{\varepsilon}_{p}=\left\{\begin{array}{l} \varepsilon_{xx}\\ \varepsilon_{yy}\\ \gamma_{xy}\end{array}\right\}_{p}=\left\{\begin{array}{l}u_{p,x}^{o}\\ v_{p,y}^{o}\\ u_{p,y}^{o}+v_{p,x}^{o}\end{array}\right\}+z\left\{\begin{array}{l}\theta_{ xp,x}\\ \theta_{yp,y}\\ \theta_{xp,y}+\theta_{yp,x}\end{array}\right\}=\boldsymbol{\varepsilon}_{p}^{o} +z\boldsymbol{\kappa}_{p}=\mathbf{B}_{m}\boldsymbol{\kappa}_{p}+z\mathbf{B}_{ b}\boldsymbol{\delta}_{p}\\ \boldsymbol{\gamma}_{p}=\left\{\begin{array}{l}\gamma_{xz}\\ \gamma_{yz}\end{array}\right\}_{p}=\left\{\begin{array}{l}w_{p,x}^{o}-\theta _{p,x}\\ w_{p,y}^{o}-\theta_{p,y}\end{array}\right\}=\mathbf{B}_{s}\boldsymbol{\delta} _{p}\end{array}  \tag{5}\]  
where \(\mathbf{B}_{m}\), \(\mathbf{B}_{b}\), and \(\mathbf{B}_{s}\) are the derivative operators between the strain and nodal displacements. Figure 1: Stiffened model with piezoelectric sensors and actuators. The stress resultant can be written as  
\[ \begin{array}{l}\mathbf{F}=\mathbf{A}_{ij}\varepsilon_{p}^{o}+\mathbf{B}_{ij} \kappa_{p}-\int_{t_{p}}\mathbf{e}^{T}\mathbf{E}dz\\ \mathbf{M}=\mathbf{B}_{ij}\varepsilon_{p}^{o}+\mathbf{D}_{ij}\kappa_{p}-\int_ {t_{p}}z\mathbf{e}^{T}\mathbf{E}dz\\ \mathbf{V}_{p}=\mathbf{S}\gamma_{p}\end{array}  \tag{6}\]  
where \(\left(\mathbf{A}_{ij},\mathbf{B}_{ij},\mathbf{D}_{ij}\right)=\sum\limits_{k=1 }^{n}\int_{x_{k-1}}^{z_{k}}\mathbf{Q}_{ij}\Big{(}1,z,z^{2}\Big{)}dz\) and \(\mathbf{S}_{ij}=\sum\limits_{k=1}^{n}\kappa_{p}\int_{x_{k-1}}^{z_{k}}\mathbf{ Q}_{ij}dz\) with \((i,j=4,\,5)\). \(\kappa_{p}=5/6\) is shear correction factor. \(\mathbf{A}_{ij}\), \(\mathbf{B}_{ij}\), \(\mathbf{D}_{ij}\), and \(\mathbf{S}_{ij}\) are extensional stiffness, bending-extension coupling stiffness, bending stiffness, and shear stiffness matrix, respectively. Then, the electric field vector considering the piezoelectric actuator layers can be derived as follows:  
\[\mathbf{E}=\left\{ \begin{array}{c}E_{x}\\ E_{y}\\ E_{z}\end{array} \right\}=-\left\{