# Influence of Applied Loads on Free Vibrations of Functionally Graded Material Plate-Shell Panels
## 4 Virtual Work Principle  
### Static Analysis  
The governing equations of the nonlinear problem are obtained from the virtual work principle in conjugation with an updated Lagrangian formulation [16; 17]. A reference configuration is associated with time \(t\), and the updated configuration is associated with the current time \(t^{\prime}=t+\Delta t\). \[ \begin{split}\sum_{k=1}^{N}\left\{\int_{t_{\mathbf{A}^{e}}}\int_{ h_{\mathbf{R}-1}}^{h_{\mathbf{R}}}\delta\big{(}\epsilon_{k}^{\mathrm{L}} \big{)}\mathrm{Q}_{kt}\epsilon_{k}^{\mathrm{L}}\mathrm{d}z^{\mathrm{t}} \mathrm{d}A^{\mathrm{e}}+\int_{t_{\mathbf{A}^{e}}}\int_{h_{\mathbf{R}-1}}^{h_ {\mathbf{R}}}\delta\big{(}\epsilon_{k}^{\mathrm{NL}}\big{)}^{t}\sigma_{k}^{ \mathrm{L}}\mathrm{d}z^{\mathrm{t}}\mathrm{d}A^{\mathrm{e}}\right\}=\mathrm{t} ^{+\Delta t}\Pi^{e}-\\ \sum_{k=1}^{N}\int_{t_{\mathbf{A}^{e}}}\int_{h_{\mathbf{R}-1}}^{h_ {\mathbf{R}}}\delta\big{(}\epsilon_{k}^{\mathrm{L}}\big{)}^{t}\sigma_{k}^{ \mathrm{L}}\mathrm{d}z^{\mathrm{t}}\mathrm{d}A^{\mathrm{e}}\end{split}  \tag{15}\]  
where \(t^{+\Delta t}_{t}\Pi^{e}\) is the element external virtual work. The membrane, bending, and shear strains, as well the higher order bending and shear strains can be represented by the following:  
\[\epsilon_{\mathbf{m}}=\mathbf{B}^{m}\mathbf{a}^{e};\ \epsilon_{\mathbf{b}}= \mathbf{B}^{b}\mathbf{z}\mathbf{a}^{e};\ \epsilon_{\mathbf{s}}=\mathbf{B}^{s}\mathbf{a}^{e};\ \epsilon_{\mathbf{b}}^{ \ast}=\mathbf{B}^{\ast b}\mathbf{z}^{3}\mathbf{a}^{e};\ \epsilon_{\mathbf{s}}^{ \ast}=\mathbf{B}^{\ast s}\mathbf{z}^{2}\mathbf{a}^{e} \tag{16}\]  
where the \(\mathbf{B}^{m}\), \(\mathbf{B}^{b}\), \(\mathbf{B}^{\ast b}\), \(\mathbf{B}^{s}\), and \(\mathbf{B}^{\ast s}\) are components of the strain-displacement matrix \(\mathbf{B}\) and are given explicitly in [PERSON] et al. [18]. The virtual work principle applied to a finite element can be written in the following matrix form:  
\[\int_{t_{\mathbf{A}^{e}}}\mathbf{B}^{\mathsf{T}}\mathbf{\hat{D}}\mathbf{B} \Delta a^{\mathrm{ct}}\mathrm{d}A^{\mathrm{e}}+\int_{t_{\mathbf{A}^{e}}} \mathbf{G}^{\mathsf{T}}\mathbf{\hat{\tau}}\mathbf{G}\ \Delta a^{\mathrm{ct}}\mathrm{d}A^{\mathrm{e}}=\int_{A^{e}}\mathbf{N}^{ \mathsf{T}}\mathbf{p}dA^{e}-\int_{t_{\mathbf{A}^{e}}}\mathbf{B}^{\mathsf{T}t} \mathbf{\hat{\sigma}}^{\ \mathrm{t}}\mathrm{d}A^{\mathrm{e}} \tag{17}\]  
From this equation, the element linear stiffness matrix \(\mathbf{K}_{L}^{e}\), the element geometric stiffness matrix \(\mathbf{K}_{\sigma}^{e}\), the element external load vector \(\mathbf{F}_{ext}^{e}\), and the element internal force vector \(\mathbf{F}_{int}^{e}\) can be obtained [18]. The element geometric stiffness matrix \(\mathbf{K}_{\sigma}^{e}\) is given by the following:  
\[\mathbf{K}_{\sigma}^{e}=\int_{t_{\mathbf{A}^{e}}}\Big{(}\mathbf{G}^{\mathsf{ T}}\mathbf{\hat{\tau}}\mathbf{G}\Big{)}^{\mathrm{t}}\mathrm{d}A^{\mathrm{e}} \tag{18}\]  
\[\mathrm{G}_{\mathrm{i}}=\begin{bmatrix}\frac{\partial_{\mathrm{L}}}{\partial x }&0&0&0&0&0&0&0\\ \frac{\partial_{\mathrm{L}}}{\partial y}&0&0&0&0&0&0&0\\ 0&\frac{\partial_{\mathrm{L}}}{\partial x}&0&0&0&0&0&0\\ 0&\frac{\partial_{\mathrm{L}}}{\partial y}&0&0&0&0&0&0\\ 0&0&\frac{\partial_{\mathrm{L}}N_{i}}{\partial x}&\frac{\partial_{2}N_{i}}{ \partial x}&\frac{\partial_\[\hat{\tau}= \begin{bmatrix}N_{xx}&N_{xy}&0&0&0&0\\ N_{xy}&N_{yy}&0&0&0&0\\ 0&0&N_{xx}&N_{xy}&0&0\\ 0&0&N_{xy}&N_{yy}&0&0\\ 0&0&0&0&N_{xx}&N_{xy}\\ 0&0&0&0&N_{xy}&N_{yy}\end{bmatrix}  \tag{20}\]  
To solve for general structures, these matrices and vectors, initially computed in the local coordinate system attached to the element, are transformed to a global coordinate system in the usual way [15]. Afterwards, by adding the contributions of all the elements in the domain, the system equilibrium equations can be obtained as follows:  
\[