## 2 Materials and Methods
### Displacements and Strain Assumption  
A higher-order shear deformation theory was developed by [PERSON] [30]. It exhibits a higher accuracy in describing the displacement field in laminate elements, and it is employed to model the laminates. The displacements are assumed as follows:  
\[u(x,y,z)=u(x,y,0)+z\varphi_{y}(x,y,0)-\frac{4}{3h^{2}}z^{3}\bigg{(}\varphi_{y} +\frac{\partial w_{0}}{\partial x}\bigg{)}, \tag{1}\]  
\[v(x,y,z)=v(x,y,0)+z\varphi_{x}(x,y,0)-\frac{4}{3h^{2}}z^{3}\bigg{(}\varphi_{x} +\frac{\partial w_{0}}{\partial y}\bigg{)}, \tag{2}\]  
\[w(x,y,z)=w(x,y,0)=w_{0}(x,y), \tag{3}\]  
Figure 1: Active CFRC rib reflector with PZT actuators.  
where \(h\) is thickness of the laminates; \(u\), \(v\), \(w\) are displacements in the \(x\), \(y\), and \(z\) directions; and \(\varphi_{x}\) and \(\varphi_{y}\) are rotation angles in the Y-Z and X-Z planes, respectively. The plane \(z=0\) is a neutral plane of the laminates along the thickness direction. Geometry equations are denoted in Equation (4). \(\varepsilon_{xx}\), \(\varepsilon_{yy}\), and \(\gamma_{xy}\) are the plane strains of the plate; \(\gamma_{yz}\) and \(\gamma_{xz}\) are the shear strains along the thickness direction.  
\[\begin{bmatrix}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \gamma_{xy}\\ \gamma_{yz}\\ \gamma_{yz}\\ \gamma_{xzz}\end{bmatrix}=\begin{bmatrix}\frac{\partial}{\partial x}&0&0\\ 0&\frac{\partial}{\partial y}&0\\ \frac{\partial}{\partial y}&\frac{\partial}{\partial x}&0\\ 0&\frac{\partial}{\partial z}&\frac{\partial}{\partial y}\\ \end{bmatrix}\begin{bmatrix}u\\ v\\ w\end{bmatrix}=\
abla\mathbf{d}, \tag{4}\]  
\(\mathbf{\varepsilon}=\begin{bmatrix}\varepsilon_{xx}&\varepsilon_{yy}&\gamma_{xy }&\gamma_{yz}&\gamma_{xz}\end{bmatrix}^{T}\) is a plate strain vector, \(\mathbf{d}=[u~{}v~{}w]^{T}\) is a plate displacement vector, and \(\
abla\) is a differential operator for the plate. For a three-dimensional (3-D) piezoelectric beam element, which can be used to model PZT actuators, the displacement assumption can be written as  
\[u^{\prime}=u_{0}^{\prime}(x,0,0)+y\varphi_{z}^{\prime}(x,0,0)+z\varphi_{y}^{ \prime}(x,0,0), \tag{5}\]  
\[v^{\prime}=v_{0}^{\prime}(x,0,0), \tag{6}\]  
\[w^{\prime}=w_{0}^{\prime}(x,0,0), \tag{7}\]  
\(u^{\prime}\),\(v^{\prime}\), and \(w^{\prime}\) denote displacements along the \(x\)-, \(y\)-, and \(z\)-directions, respectively. \(\varphi_{y}^{\prime}\) and \(\varphi_{z}^{\prime}\) are rotation angles in the X-Z and X-Y planes, respectively. Geometry equations of a 3-D beam are written as  
\[\begin{bmatrix}\varepsilon_{x}^{\prime}\\ \gamma_{xy}^{\prime}\\ \gamma_{xz}^{\prime}\\ \kappa_{x}^{\prime}\end{bmatrix}=\begin{bmatrix}\frac{\partial}{\partial x} &0&0&0\\ \frac{\partial}{\partial y}&\frac{\partial}{\partial x}&0&0\\ \frac{\partial}{\partial z}&0&0&\frac{\partial}{\partial x}&0\\ 0&0&0&\frac{\partial}{\partial x}\end{bmatrix}\begin{bmatrix}u^{\prime}\\ v^{\prime}\\ w^{\prime}\\ \varphi_{x}^{\prime}\end{bmatrix}=\
abla^{\prime}\mathbf{d}^{\prime}, \tag{8}\]  
where \(\varepsilon_{x}^{\prime}\) denotes the axial strain; \(\gamma_{xy}^{\prime}\) and \(\gamma_{xz}^{\prime}\) denote the shear strains; \(\varphi_{x}^{\prime}\) is a twist angle of the beam in the X-Y plane; \(\kappa_{x}^{\prime}\) is a twist angle per length along the \(x\)-direction; \(\mathbf{\varepsilon}^{\prime}=\begin{bmatrix}\varepsilon_{x}^{\prime}&\gamma_{xy }^{\prime}&\gamma_{xy}^{\prime}&\kappa_{x}^{\prime}\end{bmatrix}^{T}\) and \(\mathbf{d}^{\prime}=\begin{bmatrix}u^{\prime}&v^{\prime}&w^{\prime}&\varphi_{x}^{ \prime}\end{bmatrix}^{T}\) are a strain vector and a displacement vector of the beam, respectively; \(\
abla^{\prime}\) is a differential operator for the beam.