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Thermal buckling analysis of functionally graded annular plate with variable thickness

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Abstract

This paper reports on the thermal buckling characteristics of the functionally graded (FG) annular plate with variable thickness. The differential quadrature finite element method is used to derive the control differential equation of the FG annular plate with variable thickness in the thermal field. The numerical model is established on the basis of the first-order shear deformation theory. The reliability and accuracy of the proposed model are verified through a series of comparative studies. The influences of various parameters on the natural frequency, the critical buckling temperature difference and the associated buckling mode shapes are also analyzed and discussed in detail. In addition, the influences of key parameters such as power-law exponent, geometric structure and temperature change on thermal buckling characteristics are investigated under different boundary conditions, cross-sectional shapes and temperature distributions in the thermal field. The results show that the boundary conditions on the outer ring exert a more enormous function on the natural frequency and the critical buckling temperature difference. Frequency parameters of the annular plates with different cross-sectional shapes affect the temperature changes under totally different levels.

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This manuscript has associated data in a data repository. [Authors’ comment: All data included in this manuscript are available upon request by contacting with the corresponding author.]

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Acknowledgements

The authors gratefully acknowledge the financial supports from the National Natural Science Foundation of China (Grant No. 51705537), Key Laboratory of Vibration and Control of Aero-Propulsion System, Ministry of Education, Northeastern University (VCAME202006). The authors also gratefully acknowledge the supports from State Key Laboratory of High Performance Complex Manufacturing, Central South University, China (Grant No. ZZYJKT2019-08).

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Authors and Affiliations

  1. College of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, People’s Republic of China

    Qingshan Wang, Ruihua Wang, Xianlei Guan & Wen Shao

  2. State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, 410083, People’s Republic of China

    Qingshan Wang, Ruihua Wang, Xianlei Guan & Wen Shao

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  1. Qingshan Wang
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  2. Ruihua Wang
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  3. Xianlei Guan
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  4. Wen Shao
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Correspondence to Wen Shao.

Appendix 1

Appendix 1

$$ \begin{aligned} {\mathbf{K}}_{11}^{e} & = \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} A_{11} + \frac{1}{{\mathbf{r}}}{\mathbf{E}}^{{\text{T}}} A_{12} } \right){\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} A_{66} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} A_{12} + \frac{1}{{\mathbf{r}}}{\mathbf{E}}^{{\text{T}}} A_{11} } \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{12}^{e} & = \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} A_{12} + \frac{1}{{\mathbf{r}}}{\mathbf{E}}^{{\text{T}}} A_{11} } \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} \frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} \left( {A_{66} {\mathbf{A}}_{02}^{\left( 1 \right)} - A_{66} \frac{1}{{\mathbf{r}}}{\mathbf{E}}} \right) \\ {\mathbf{K}}_{13}^{e} & = \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} B_{11} + {\mathbf{E}}^{{\text{T}}} B_{12} \frac{1}{{\mathbf{r}}}} \right){\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} B_{12} + {\mathbf{E}}^{{\text{T}}} B_{11} \frac{1}{{\mathbf{r}}}} \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{14}^{e} & = \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} B_{12} + {\mathbf{E}}^{{\text{T}}} B_{11} \frac{1}{{\mathbf{r}}}} \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} \frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} - {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{15}^{e} & = {\mathbf{0}} \\ {\mathbf{K}}_{21}^{e} & = {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} A_{12} \frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} A_{11} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} A_{66} - {\mathbf{E}}^{{\text{T}}} A_{66} \frac{1}{{\mathbf{r}}}} \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ {\mathbf{K}}_{22}^{e} & = {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} A_{11} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} A_{66} - {\mathbf{E}}^{{\text{T}}} A_{66} \frac{1}{{\mathbf{r}}}} \right){\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + \left( {{\mathbf{E}}^{{\text{T}}} A_{66} \frac{1}{{\mathbf{r}}} - {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} A_{66} } \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{23}^{e} & = {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{12} \frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{11} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} - {\mathbf{E}}^{{\text{T}}} B_{66} \frac{1}{{\mathbf{r}}}} \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ {\mathbf{K}}_{24}^{e} & = {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{11} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} - {\mathbf{E}}^{{\text{T}}} B_{66} \frac{1}{{\mathbf{r}}}} \right){\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + \left( {{\mathbf{E}}^{{\text{T}}} B_{66} \frac{1}{{\mathbf{r}}} - {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} } \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{25}^{e} & = {\mathbf{0}} \\ \end{aligned} $$
$$ \begin{aligned} {\mathbf{K}}_{31}^{e} & = \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} {\mathbf{B}}_{11} + {\mathbf{E}}^{{\text{T}}} {\mathbf{B}}_{12} \frac{1}{{\mathbf{r}}}} \right){\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} {\mathbf{B}}_{12} + {\mathbf{E}}^{{\text{T}}} {\mathbf{B}}_{11} \frac{1}{{\mathbf{r}}}} \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} {\mathbf{B}}_{66} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ {\mathbf{K}}_{32}^{e} & = \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} B_{12} + {\mathbf{E}}^{{\text{T}}} B_{11} \frac{1}{{\mathbf{r}}}} \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} \frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} - {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{33}^{e} & = \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} D_{11} + {\mathbf{E}}^{{\text{T}}} D_{12} \frac{1}{{\mathbf{r}}}} \right){\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} D_{66} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ & \quad + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} D_{12} + {\mathbf{E}}^{{\text{T}}} D_{11} \frac{1}{{\mathbf{r}}}} \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} + {\mathbf{E}}^{{\text{T}}} A_{55} {\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{34}^{e} & = \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} D_{12} + {\mathbf{E}}^{{\text{T}}} D_{11} \frac{1}{{\mathbf{r}}}} \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} D_{66} \frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} - {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} D_{66} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{35}^{e} & = {\mathbf{E}}^{{\text{T}}} A_{55} {\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} \\ \end{aligned} $$
$$ \begin{aligned} {\mathbf{K}}_{41}^{e} & = {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{12} \frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{11} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} - {\mathbf{E}}^{{\text{T}}} B_{66} \frac{1}{{\mathbf{r}}}} \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ {\mathbf{K}}_{42}^{e} & = {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} B_{11} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} - {\mathbf{E}}^{{\text{T}}} B_{66} \frac{1}{{\mathbf{r}}}} \right){\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + \left( {{\mathbf{E}}^{{\text{T}}} B_{66} \frac{1}{{\mathbf{r}}} - {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} B_{66} } \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{43}^{e} & = {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} D_{12} \frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} D_{11} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} D_{66} - \frac{1}{{\mathbf{r}}}{\mathbf{E}}^{{\text{T}}} D_{66} } \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ {\mathbf{K}}_{44}^{e} & = {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} D_{11} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} + \left( {{\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} D_{66} - {\mathbf{E}}^{{\text{T}}} D_{66} \frac{1}{{\mathbf{r}}}} \right){\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} \\ & \quad + \left( {{\mathbf{E}}^{{\text{T}}} D_{66} \frac{1}{{\mathbf{r}}} - {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} D_{66} } \right)\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{E}} + {\mathbf{E}}^{{\text{T}}} A_{44} {\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{45}^{e} & = {\mathbf{E}}^{{\text{T}}} A_{44} \frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ \end{aligned} $$
$$ \begin{aligned} {\mathbf{K}}_{51}^{e} & = {\mathbf{0}} \\ {\mathbf{K}}_{52}^{e} & = {\mathbf{0}} \\ {\mathbf{K}}_{53}^{e} & = {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} A_{55} {\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{54}^{e} & = \frac{1}{{\mathbf{r}}}{\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} A_{44} {\mathbf{rC}}_{02} {\mathbf{E}} \\ {\mathbf{K}}_{55}^{e} & = {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} A_{55} {\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} A_{44} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ \end{aligned} $$
$$ \begin{aligned} {\mathbf{K}}_{{11{\text{T}}}}^{e} & = {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} N_{{r{\text{T}}}}^{{}} {\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} N_{{\theta {\text{T}}}}^{{}} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ {\mathbf{K}}_{{22{\text{T}}}}^{e} & = {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} N_{{r{\text{T}}}}^{{}} {\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} N_{{\theta {\text{T}}}}^{{}} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ {\mathbf{K}}_{{33{\text{T}}}}^{e} & = {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} M_{{r{\text{T}}}}^{{}} {\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} M_{{\theta {\text{T}}}}^{{}} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ {\mathbf{K}}_{{44{\text{T}}}}^{e} & = {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} M_{{r{\text{T}}}}^{{}} {\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} M_{{\theta {\text{T}}}}^{{}} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ {\mathbf{K}}_{{55{\text{T}}}}^{e} & = {\mathbf{A}}_{02}^{{\left( 1 \right){\text{T}}}} N_{{r{\text{T}}}}^{{}} {\mathbf{rC}}_{02} {\mathbf{A}}_{02}^{\left( 1 \right)} + {\mathbf{B}}_{02}^{{\left( 1 \right){\text{T}}}} N_{{\theta {\text{T}}}}^{{}} \frac{1}{{\mathbf{r}}}\frac{1}{{\mathbf{r}}}{\mathbf{rC}}_{02} {\mathbf{B}}_{02}^{\left( 1 \right)} \\ \end{aligned} $$

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Wang, Q., Wang, R., Guan, X. et al. Thermal buckling analysis of functionally graded annular plate with variable thickness. Eur. Phys. J. Plus 136, 1218 (2021). https://doi.org/10.1140/epjp/s13360-021-02107-2

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