in which $\mathbf{u}=[u,v,w{]}^{\mathrm{t}}$, ${\mathbf{u}}_0=[u_0,v_0,w{]}^{\mathrm{t}},\mathsf{\varphi }=[{\psi }_x,{\psi }_y,0{]}^{\mathrm{t}}$ for the plate, $\mathsf{\varphi }=[{\varphi }_x,{\varphi }_{\theta },0{]}^{\mathrm{t}}$ for the shell, $\partial \mathbf{x}=[\partial x,\partial y,\partial z{]}^{\mathrm{t}}$ for the plate, and $\partial \mathbf{x}=[\partial x,R\partial \theta ,\partial z{]}^{\mathrm{t}}$ for the shell, where u and v are the tangential displacements. $u_0$ and $v_0$ are the tangential displacements in the in-surface coordinate direction, $w$ is the transverse displacement in the out-of-surface coordinate z axis-direction of the middle-plane of plate-shells. ${\psi }_x$ and ${\psi }_y$ are shear rotations for the plates. ${\varphi }_x$ and ${\varphi }_{\theta }$ are shear rotations for the shells. R is the middle-surface radius of the shells. ${\omega }_{mn}$ is the natural frequency with respect to mode shapes m and n in subscripts. The superscript t is operating the transpose in the bold-form vector.

in which $\mathsf{\sigma }=[{\sigma }_x,{\sigma }_y,{\sigma }_{xy},{\sigma }_{yz}{{,\sigma }_{xz}]}^{\mathrm{t}}$ for the plates, $\mathsf{\sigma }=[{\sigma }_x,{\sigma }_{\theta },{\sigma }_{x\theta },{\sigma }_{\theta z}{{,\sigma }_{xz}]}^{\mathrm{t}}$ for the shells, $\mathit{\epsilon }=[{\epsilon }_{\mathit{x}},{\epsilon }_y,{\epsilon }_{xy},{\epsilon }_{yz}{{,\epsilon }_{xz}]}^{\mathrm{t}}$ for the plates, $\mathsf{\epsilon }=[{\epsilon }_{\mathit{x}},{\epsilon }_{\theta },{\epsilon }_{x\theta },{\epsilon }_{\theta z}{{,\epsilon }_{xz}]}^{\mathrm{t}}$ for the shells, $\mathsf{\alpha }=[{\alpha }_{\mathit{x}},{\alpha }_y,{\alpha }_{xy},0,{0]}^{\mathrm{t}}$ for the plates, $\mathsf{\alpha }=[{\alpha }_{\mathit{x}},{\alpha }_{\theta },{\alpha }_{x\theta },0,{0]}^{\mathrm{t}}$ for the shells, and

where ${\sigma }_x$ and ${\sigma }_y$ are the normal stresses in the plates, ${\sigma }_x$ and ${\sigma }_{\theta }$ are the normal stresses in the shells,${\sigma }_{xy},{\sigma }_{yz}$, and ${\sigma }_{xz}$ are the shear stresses in the plates, ${\sigma }_{x\theta },{\sigma }_{\theta z}$, and ${\sigma }_{xz}$ are the shear stresses in the shells. ${\epsilon }_{\mathit{x}},{\epsilon }_y$, and ${\epsilon }_{xy}$ are in-plane strains in the plates, ${\epsilon }_{\mathit{x}},{\epsilon }_{\theta }$, and ${\epsilon }_{x\theta }$ are in-plane strains in the shells, ${\epsilon }_{yz}$ and ${\epsilon }_{xz}$ are shear strains that cannot be negligible for the thick plates, ${\epsilon }_{\theta z}$ and ${\epsilon }_{xz}$ are shear strains that cannot be negligible for the thick shells, as demonstrated using strain–displacement correlations for displacement derivatives with respect to coordinates. ${\alpha }_{\mathit{x}}$ and ${\alpha }_y$ are the coefficients of thermal expansion for the plates, ${\alpha }_{\mathit{x}}$ and ${\alpha }_{\theta }$ are the coefficients of thermal expansion for the shells,${\alpha }_{xy}$ is the coefficient of thermal shear for the plates, and ${\alpha }_{x\theta }$ is the coefficient of thermal shear for the shells. ${\overline{Q}}_{ij}$ with subscripts i, j = 1,2,4,5, and 6 are the stiffness values of the FGMs, e.g., ${\overline{Q}}_{11}={\overline{Q}}_{22}={\displaystyle \frac{E_{fgm}}{1-{{\nu }_{fgm}}^2}}$, ${\overline{Q}}_{44}={\displaystyle \frac{E_{fgm}}{2 (1+{\nu }_{fgm})}}$ are used for the plates, ${\overline{Q}}_{12}={\overline{Q}}_{21}={\displaystyle \frac{{\nu }_{fgm}{E}_{fgm}}{\left(1+\frac{z}{R}\right) (1-{{\nu }_{fgm}}^2)}}$, ${\overline{Q}}_{55}={\overline{Q}}_{66}={\displaystyle \frac{E_{fgm}}{2\left(1+\frac{z}{R}\right) (1+{\nu }_{fgm})}}$ are used for the shells, and ${\overline{Q}}_{16}={\overline{Q}}_{26}={\overline{Q}}_{45}=0$. When the values of time sinusoidal displacements are obtained, then the stress values can be calculated under the time sinusoidal ∆T. The distribution of the four-layer FGMs, e.g., Poisson’s ratio ${\nu }_{\mathrm{f}\mathrm{g}\mathrm{m}}$ and Young’s modulus $E_{fgm}$, influences the dynamic behavior of the coupled plates and cylindrical shells seen from the stiffness value of ${\overline{Q}}_{ij}$ in (18) and stiffness integrals $A_{11}$, …, $H_{11}$, …, $H_{44}$ with respect to ${\nu }_{\mathrm{f}\mathrm{g}\mathrm{m}}$, $k_{\alpha }$, $E_1$, $E_2$, $E_3$, $E_4$, $h^{*}$, $R_n$ of the plates can be seen in (5)–(15) and $A_{12}$, …, $H_{12}$, …, $H_{55}$ with respect to ${\nu }_{\mathrm{f}\mathrm{g}\mathrm{m}}$, $k_{\alpha }$, $E_1$, $E_2$, $E_3$, $E_4$, $R$, $h^{*}$, $R_n$ of the shells can be seen in (A10)–(A16). When the values of ${\nu }_{\mathrm{f}\mathrm{g}\mathrm{m}}$, $k_{\alpha }$, and $E_{fgm}$ are obtained, then the displacement and stress dynamic response values can be calculated under the time sinusoidal ∆T.

3.3. Responses of $w$(a/2, b/2) and ${\sigma }_x$ Versus T