Usually the shear effect on the displacements and stresses were more considered in the thick thickness materials than in the thin one under every external loads. Some papers of functionally graded materials (FGMs) were investigated usually including the shear effect. Abouelregal et al. [1] in 2023 applied the Laplace transform and decoupling techniques to study the thermal vibration of spinning FGM isotropic piezoelectric rod. The effecting numerical results for non-homogeneous index in the displacement, temperature, electric potential and thermal stress were presented. The thermal stress variation was found in one of the most significant results want to be considered. In 2023, Tang et al. [2] used the radial integral boundary element method (RIBEM) to study the FGM plate under thermal shock loading. The numerical results of displacement, temperature and thermal stress were presented in the thick FGM plate. In 2022, Chen et al. [3] used the Rayleih–Ritz energy method with the first-order shear deformation theory (FSDT) of displacement models to investigate the vibration of FGM stepped cylindrical shell coupled with annular plate under thermal environment. The temperature-dependent constituent materials SUS304 and Si₃N₄ were used to study the transient responses of displacements. In 2022, Ramteke et al. [4] presented a nonlinear eigen- frequency responses with the higher-order shear deformation theory (HSDT) of displacement models for porous FGM shell panel under thermal environment by using the finite element method (FEM). In 2021, Saeedi et al. [5] applied differential quadrature (DQ) method to study the stresses distributions of thick FGM cylindrical shell with aluminum and silicon carbide under internal pressure and thermal loadings. In 2022, Gee and Hashemi [6] used the dynamic finite element (DFE) method to investigate the numerical results of natural frequencies for the FGM beams. In 2019, Khoshgoftar [7] used the virtual work principle and second-order shear deformation theory (SSDT) of displacement models to obtain the numerical displacement results of axis-symmetric thick FGM shells under non-uniform pressure. In 2019, Trinh and Kim [8] presented the nonlinear stability of moderately thick FGM sandwich shells supported by elastic foundations under thermo-mechanical loadings by using the Bubnov–Galerkin procedure, harmonic balance principle and FSDT of displacement models. The numerical solutions for thermal buckling load are obtained. In 2017, Nejad et al. [9] presented a review for some models used in the solving methods, elasticity theories and shear deformation theories used for the thick FGM cylindrical and conical shells.

Some generalized differential quadrature (GDQ) computational results with nonlinear displacement third-order shear deformation theory (TSDT), thermal temperature of environment and external heating loads were presented for the thermal vibrations of FGMs. Hong [10] in 2023 studied the GDQ solutions of thick FGM plates with the effects of TSDT model but not in function of nonlinear coefficient $c_1$ term of TSDT model. In 2022, Hong [11] presented the dynamic advanced GDQ solutions of thick FGM plates under thermal vibration with the effects of TSDT and advanced shear factor that was in function of $c_1$ term of TSDT model, FGM power law index and environment temperature, but not in function of total thickness. Hong [12] in 2022 presented the GDQ solutions of thick FGM circular cylindrical shells under thermal vibration with the effects of TSDT and varied shear factor that was not in function of $c_1$ term of TSDT model. Hong [13] in 2022 studied the advanced GDQ results of thick FGM plates-cylindrical shells under thermal vibration with the effects of TSDT and advanced shear factor that was in function of $c_1$ term of TSDT model, FGM power law index and environment temperature, but not in function of total thickness of plates-cylindrical shells. The further study now providing the GDQ vibration results of stresses and displacements with the effects of TSDT, advanced varied shear correction coefficient, environment temperature and power law index.

Two constituent material FGM circular cylindrical shell is studied in Figure 1, in which the parameter of $h_1$ denotes the inner layer thickness of constituent material 1,$h_2$ denotes the outer layer thickness of constituent material 2, L denotes the axial length, and $h^{*}$ denotes the total thickness. The material properties in power-law function are studied and parameter of $E_{fgm}$ denotes Young’s modulus in standard variation form of FGMs, $R_n$ denotes power law index, and ${\nu }_{fgm}$ denotes Poisson’s ratios of FGMs that is assumed in the simple average form as follows [10], where $E_1$ and $E_2$ are the Young’s modulus, ${\nu }_1$ and ${\nu }_2$ are the Poisson’s ratios of the constituent material 1 and material 2, respectively in the FGMs.

The properties $P_i$ of individual constituent material of FGMs, e.g., $E_1$, $E_2$, ${\nu }_1$ and ${\nu }_2$ are in functions of environment temperature T can be calculated by the following form, in which $P_0$, $P_{-1}$, $P_1$, $P_2$ and $P_3$ are the temperature coefficients.

The time dependent displacement vector $\stackrel{⃑}{\mathbf{u}}$ of thick FGM circular cylindrical shells are studied in the nonlinear of $z^3$ form with nonlinear coefficient $c_1$ term of TSDT equations [12,14] as follows, where $\mathbf{u}=[u,v,w{]}^{\mathrm{t}}$, ${\mathbf{u}}_0=[u_0\left(x,\theta ,t\right),v_0\left(x,\theta ,t\right),w\left(x,\theta ,t\right){]}^{\mathrm{t}}$, $\mathsf{\varphi }=[{\varphi }_x(x,\theta ,t),{\varphi }_{\theta }(x,\theta ,t),0{]}^{\mathrm{t}}$, $c_1=4/(3{h^{\mathrm{*}}}^2)$ and $\partial \mathbf{x}=[\partial x,R\partial \theta ,\partial z{]}^{\mathrm{t}}$, in which $t$ denotes the time, u and v denote respectively the tangential displacements in the direction of x and $\theta$ axes. $u_0$ and $v_0$ denote respectively the tangential displacements in the direction of x and $\theta$ axes, w denotes the transverse displacement in the direction of z axis in the middle-plane of FGM shells. ${\varphi }_x$ and ${\varphi }_{\theta }$ denote the shear rotations. R denotes the radius of middle-surface in FGM shells. The superscript t is operating the transpose of vector.

The stress vector $\mathsf{\sigma }$ in the thick FGM circular cylindrical shells under external heating loads with temperature difference $∆T$ for the kth constituent layer are studied in the following equations [15], where $\mathsf{\sigma }=[{\sigma }_x,{\sigma }_{\theta },{\sigma }_{x\theta },{\sigma }_{\theta z}{{,\sigma }_{xz}]}^{\mathrm{t}}$, $\mathsf{\epsilon }=[{\epsilon }_{\mathit{x}},{\epsilon }_{\theta },{\epsilon }_{x\theta },{\epsilon }_{\theta z}{{,\epsilon }_{xz}]}^{\mathrm{t}}$, $\mathsf{\alpha }=[{\alpha }_{\mathit{x}},{\alpha }_{\theta },{\alpha }_{x\theta },0,{0]}^{\mathrm{t}}$ and in which ${\sigma }_x$ and ${\sigma }_{\theta }$ are the normal stresses, ${\sigma }_{x\theta },{\sigma }_{\theta z}$ and ${\sigma }_{xz}$ are the shear stresses. ${\epsilon }_{\mathit{x}},{\epsilon }_{\theta }$ and ${\epsilon }_{x\theta }$ are in-plane strains, ${\epsilon }_{\theta z}$ and ${\epsilon }_{xz}$ are shear strains can't negligible for thick shells. ${\alpha }_{\mathit{x}}$ and ${\alpha }_{\theta }$ are thermal expansion coefficients, ${\alpha }_{x\theta }$ is the thermal shear coefficient. ${\overline{Q}}_{ij}$ with subscripts i, j=1,2,4,5 and 6 are the stiffness of FGM shells.

The advanced varied shear correction coefficient $k_{\mathit{\alpha }}$ can be used as a coefficient in the following stiffness ${\overline{Q}}_{ij}$ integral equations $A_{i^s{j}^s}$, $B_{i^s{j}^s}$, $D_{i^s{j}^s}$,$E_{i^s{j}^s}$, $F_{i^s{j}^s}$, $H_{i^s{j}^s}$ and $A_{i^{*}{j}^{*}}$, $B_{i^{*}{j}^{*}}$, $D_{i^{*}{j}^{*}}$,$E_{i^{*}{j}^{*}}$, $F_{i^{*}{j}^{*}}$, $H_{i^{*}{j}^{*}}$ with subscripts $i^{*},j^{*}$=4,5 when ${\epsilon }_{\theta z}$ and ${\epsilon }_{xz}$ can't negligible in thick shells, also for subscripts $i^s,j^s$=1,2,6 when ${\epsilon }_{\mathit{x}},{\epsilon }_{\theta }$ and ${\epsilon }_{x\theta }$ are applied. where simple forms of ${\overline{Q}}_{i^s{j}^s}$ and ${\overline{Q}}_{i^{*}{j}^{*}}$ are used in the following with z/R term cannot be neglected [16], ${\overline{Q}}_{11}={\overline{Q}}_{22}=\frac{E_{fgm}}{1-{{\nu }_{fgm}}^2}$,${\overline{Q}}_{12}={\overline{Q}}_{21}=\frac{{\nu }_{fgm}{E}_{fgm}}{\left(1+\frac{z}{R}\right)(1-{{\nu }_{fgm}}^2)}$, ${\overline{Q}}_{44}=\frac{E_{fgm}}{2(1+{\nu }_{fgm})}$,${\overline{Q}}_{55}={\overline{Q}}_{66}=\frac{E_{fgm}}{2\left(1+\frac{z}{R}\right)(1+{\nu }_{fgm})}$ and ${\overline{Q}}_{16}={\overline{Q}}_{26}={\overline{Q}}_{45}=0$.

And the advanced $k_{\mathit{\alpha }}$ expression can be derived and expressed in the following equation by using the total strain energy principle to apply [11], in which $FGMZSV={(FGMZS-c_{1}FGMZSN)}^2$, $FGMZIV=FGMZI-2c_{1}FGMZIV1+{c_1}^{2}FGMZIV2$ with parameters FGMZS, FGMZSN, FGMZI, FGMZIV1 and FGMZIV2 are in functions of $E_1$, $E_2$, $h^{*}$ and $R_n$. The calculated values of nonlinear advanced $k_{\mathit{\alpha }}$ are usually functions of $c_1$, $R_n$ and T.

The $∆T$ time dependent expression can be studied between the FGM shell and curing area under external heating loads by using the following equation in linear of z, where temperature parameter ${T_1=T}_1(x,\theta ,t)$, also the simple form heat conduction equation in cylindrical axes is used in the following [10], in which $K={\kappa }_{fgm}/({\rho }_{fgm}{C_v}_{fgm})$, ${\kappa }_{fgm}$ denotes the thermal conductivity of FGMs, ${\rho }_{fgm}$ denotes the density of FGMs, ${C_v}_{fgm}$ denotes the specific heat of FGMs, they are all assumed in the simple average property form of constituent material 1 and 2, e.g., ${\rho }_{fgm}=({\rho }_1+{\rho }_2)/2$ with ${\rho }_1,{\rho }_2$ is density of FGM constituent material 1 and 2, respectively.

The dynamic equations of motion in the thick FGM cylindrical shells with TSDT are given in the following that can be derived by using the generalized Hamilton total energy principle [13], where ${\overline{M}}_{\alpha \beta }=M_{\alpha \beta }-c_1{P}_{\alpha \beta }$, ${\overline{Q}}_{\alpha }=Q_{\alpha }-3c_1{R}_{\alpha }$, subscripts$(\alpha ,\beta =x,\theta )$,$\left\{\begin{array}{c}{N}_{xx}\\ N_{\theta \theta }\\ N_{x\theta }\end{array}\right\}={\int }_{-\frac{h^{\mathrm{*}}}{2}}^{\frac{h^{\mathrm{*}}}{2}}\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_{\theta }\\ {\sigma }_{x\theta }\end{array}\right\}\mathrm{d}z$, $\left\{\begin{array}{c}{M}_{xx}\\ M_{\theta \theta }\\ M_{x\theta }\end{array}\right\}={\int }_{-\frac{h^{\mathrm{*}}}{2}}^{\frac{h^{\mathrm{*}}}{2}}\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_{\theta }\\ {\sigma }_{x\theta }\end{array}\right\}z\mathrm{d}z$, $\left\{\begin{array}{c}{P}_{xx}\\ P_{\theta \theta }\\ P_{x\theta }\end{array}\right\}={\int }_{-\frac{h^{\mathrm{*}}}{2}}^{\frac{h^{\mathrm{*}}}{2}}\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_{\theta }\\ {\sigma }_{x\theta }\end{array}\right\}{z}^3\mathrm{d}z$, $\left\{\begin{array}{c}{R}_{\theta }\\ R_x\end{array}\right\}={\int }_{-\frac{h^{\mathrm{*}}}{2}}^{\frac{h^{\mathrm{*}}}{2}}\left\{\begin{array}{c}{\sigma }_{\theta z}\\ {\sigma }_{xz}\end{array}\right\}{z}^2\mathrm{d}z$, $\left\{\begin{array}{c}{Q}_{\theta }\\ Q_x\end{array}\right\}={\int }_{-\frac{h^{\mathrm{*}}}{2}}^{\frac{h^{\mathrm{*}}}{2}}\left\{\begin{array}{c}{\sigma }_{\theta z}\\ {\sigma }_{xz}\end{array}\right\}\mathrm{d}z$, q is external pressure load, $I_i={\displaystyle \sum _{k=1}^{N^{*}}{\displaystyle {\int }_k^{k+1}{\rho }^{(k)}}}{z}^{i}dz$, (i= 0, 1, 2,…,6), with ${\mathrm{N}}^{*}$ denotes the constituent layers total number, ${\rho }^{(k)}$ denotes the density of superscript k th constituent layer, $J_i=I_i-c_1{I}_{i+2}$, in which subscript i=1, 4 and $K_2=I_2-2c_1{I}_4+{c_1}^2{I}_6$. And assumed strain-displacement relations with $\frac{\partial v_0}{\partial z}=\frac{-v_0}{R}$, $\frac{\partial u_0}{\partial z}=\frac{-u_0}{R}$and $\frac{\partial w}{\partial z}=\frac{\partial {\varphi }_x}{\partial z}=\frac{\partial {\varphi }_{\theta }}{\partial z}=0$ are used as follows,

Substitute eqs. (3) and (7) into eq. (6), the dynamic equilibrium differential equations in the cylindrical coordinates of thick FGM circular cylindrical shells in terms of partial derivatives of $u_0,v_0,w,{\varphi }_x \mathrm{a}\mathrm{n}\mathrm{d} {\varphi }_{\theta }$ with TSDT under partial derivatives of external loads can be written in matrix forms as follows [13]. where $\mathbf{H}\mathbf{M}=[{\mathrm{H}\mathrm{M}}_{ij}]$ is a 5 by 5 matrix, $\mathbf{E}\mathbf{M}=[{\mathrm{E}\mathrm{M}}_{ij}]$ is a 5 by 12 matrix, $\mathbf{F}\mathbf{M}=[{\mathrm{F}\mathrm{M}}_{ij}]$ is a 5 by 8 matrix, $\mathbf{A}\mathbf{M}=[{\mathrm{A}\mathrm{M}}_{ij}]$ is a 5 by 9 matrix, $\mathbf{B}\mathbf{M}=[{\mathrm{B}\mathrm{M}}_{ij}]$ is a 5 by 6 matrix, $\mathbf{K}\mathbf{E}=[{\mathrm{K}\mathrm{E}}_{ij}]$ is a 5 by 10 matrix, $\mathbf{F}\mathbf{Q}=[{\mathrm{F}\mathrm{Q}}_{ij}]$ is a 5 by 5 matrix, for more details of these matrices are expressed in the section Appendix A. with in which $p_1$ and $p_2$ are external in-plane distributed forces in x and $\theta$ direction respectively.

By using the GDQ numerical method to solve the equation (8) for the thick FGM circular cylindrical shells under time sinusoidal vibrations of displacement, shear rotations and thermal temperature difference of external heating loads only, i.e., $p_1=p_2=q=0$ are studied as follows, where ${\omega }_{mn}$ denotes the natural frequency with subscripts m, n denote the mode shape numbers of FGMs, $\gamma$ denotes the applied heat flux frequency, ${\overline{T}}_1$ denotes the temperature amplitude.

For more detail procedures used in GDQ numerical method can be referred [10], the GDQ dynamic discrete equations could be obtained and expressed into the matrix equation as follows for a interior grid point at subscripts (i, j) in the computation, in which $\left[A\right]$ denotes the coefficient matrix with dimension of ${\mathrm{N}}^{**}$ by ${\mathrm{N}}^{**}$ (${\mathrm{N}}^{**}=5\left(N-2\right)(\mathrm{M}-2)$) contains ($A_{i^s{j}^s}$, $B_{i^s{j}^s}$, $D_{i^s{j}^s}$,$E_{i^s{j}^s}$, $F_{i^s{j}^s}$, $H_{i^s{j}^s}$ and $A_{i^{*}{j}^{*}}$, $B_{i^{*}{j}^{*}}$, $D_{i^{*}{j}^{*}}$,$E_{i^{*}{j}^{*}}$, $F_{i^{*}{j}^{*}}$, $H_{i^{*}{j}^{*}}$) and weighting parameters ($A_{i,l}^{(m)},B_{j,m}^{(m)}$) for the superscript m th-order derivative of the functions, e.g., $u_0(x,\theta )$, $v_0(x,\theta )$, $w(x,\theta )$, ${\varphi }_x(x,\theta )$, ${\varphi }_{\theta }(x,\theta )$, ${\overline{N}}_{xx}$, ${\overline{N}}_{x\theta }$, ${\overline{N}}_{\theta \theta }$, ${\overline{P}}_{xx}$, ${\overline{P}}_{x\theta }$, ${\overline{P}}_{\theta \theta }$, ${\tilde{\overline{M}}}_{xx}$,${\tilde{\overline{M}}}_{x\theta }$ and${\tilde{\overline{M}}}_{\theta \theta }$ with respect to the x and $\theta$ directions. is a ${\mathrm{N}}^{**}$th-order unknown column vector, in which non-dimensional parameters ${U=u}_0/L$, ${V=v}_0/R$ and $W=w/h^{*}$ are used for column vector $\left[W^{\mathrm{*}}\right]$ and $\left[B\right]=[{F_1\cdots F_1,F_2\cdots F_2,F_3\cdots F_3,F_4\cdots F_4,F_5\cdots F_5]}^{\mathrm{t}}$ is a ${\mathrm{N}}^{**}$th-order row external loads vector, in which