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Nonlinear Buckling and Postbuckling of FGM Shear-Deformable Truncated Conical Shells Reinforced by FGM Stiffeners

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Mechanics of Composite Materials Aims and scope

In this study, the nonlinear buckling of stiffened FGM truncated conical shells resting on an elastic foundation and subjected to a uniform axial compressive load is considered. The shells are reinforced by FGM stringers and rings. Using the analytical approach, FSDT, Galerkin method, geometrical nonlinearity in the von Karman–Donnell sense, and Leknitskii smeared stiffener technique, the governing equations are derived. Closed-form expressions for determining the critical buckling load and for analyzing the postbuckling load–deflection curves are obtained. Finally, the effect of stiffeners, dimensional parameters, semivertex angle, material properties, and foundations on the nonlinear response of FGM truncated conical shells are analyzed and discussed in detail.

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Acknowledgement

This research was funded by the Vietnam National University, Hanoi, under grant QG.17.45. The authors are grateful for this support.

Author information

Authors and Affiliations

  1. University of Transport Technology, 54 Trieu Khuc, Hanoi, Thanh Xuan, Vietnam

    D. Q. Chan

  2. VNU, Hanoi, University of science, 334 Nguyen Trai, Hanoi, Thanh Xuan, Vietnam

    D. Q. Chan & V. D. Long

  3. Advanced Materials and Structures Laboratory, VNU, Hanoi University of Engineering and Technology, 144 Xuan Thuy, Hanoi, Cau Giay, Vietnam

    N. D. Duc

  4. Infrastructure Engineering Program, VNU, Hanoi - Vietnam-Japan University, My Dinh 1, Hanoi, Tu Liem, Vietnam

    N. D. Duc

  5. National Research Laboratory, Department of Civil and Environmental Engineering, Sejong University, 209 Neungdong-ro, Seoul, Gwangjin-gu, 05006, Korea

    N. D. Duc

Authors
  1. D. Q. Chan
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  2. V. D. Long
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  3. N. D. Duc
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Corresponding author

Correspondence to N. D. Duc.

Additional information

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 54, No. 6, pp. 1079-1104, November-December, 2018.

Appendices

Appendix A

$$ {\displaystyle \begin{array}{c}{E}_1{E}_m+\left(\frac{E_c-{E}_m}{k+1}\right)h,\kern1em {E}_2=\frac{\left({E}_c-{E}_m\right)k{h}^2}{2\left(k+1\right)\left(k+2\right)},\\ {}{E}_3=\left\{\frac{E}{12}+\left({E}_c-{E}_m\right)\left[\frac{1}{k+3}-\frac{1}{k+2}+\frac{1}{4k+4}\right]\right\}{h}^3,\kern1em {\uplambda}_0=\frac{2\pi \sin \beta }{n_s},{A}_s={b}_s{h}_s,{A}_r={b}_r{h}_r,\\ {}{z}_s=\frac{h+{h}_s}{2},{z}_r=\frac{h+{h}_r}{2},{d}_s(x)={\uplambda}_0x,\Big[{d}_r=\frac{L}{n_r},{c}_1^0=\pm \frac{E_s{A}_s{z}_z}{\uplambda_0},{c}_1(x)=\frac{c_1^0}{x},\\ {}c=\pm \frac{E_r{A}_r{z}_r}{d_r},{I}_s=\frac{b_s{h}_s^3}{12}+{A}_s{z}_s^2,\kern1em {I}_r=\frac{b_r{h}_r^3}{12}+{A}_r{z}_r^2,{A}_{11}={A}_{22}=\frac{E_1}{1-{v}^2},\\ {}{A}_{12}=\frac{v{E}_1}{1-{v}^2},\Big[{A}_{44}={A}_{55}=\frac{5{E}_1}{12\left(1+v\right)},{A}_{66}=\frac{E_1}{2\left(1+v\right)},{B}_{11}={B}_{22}=\frac{E_2}{1-{v}^2},\\ {}{B}_{12}=\frac{v{E}_2}{1\kern1em {v}^2},{B}_{66}=\frac{E_2}{2\left(1+v\right)},{D}_{11}={D}_{22}=\frac{E_3}{1\kern1em {v}^2},{D}_{12}=\frac{v{E}_3}{1\kern1em {v}^2},\\ {}{D}_{66}=\frac{E_3}{2\left(1+v\right)},\end{array}} $$

Appendix B

$$ {\displaystyle \begin{array}{c}{H}_1={A}_{22}+\frac{E_r{A}_r}{d_r};{H}_2={B}_{22}+{c}_2;{H}_3={A}_{11}x+\frac{E_s{A}_s}{\uplambda_0};\\ {}{H}_4={D}_{11}x+\frac{E_s{I}_s}{\uplambda_0};{H}_s={D}_{22}+\frac{E_r{I}_r}{d_r}\end{array}} $$

In Eqs. (11-15),

$$ {S}_{11}={H}_3\frac{\partial^2}{\partial {x}^2}+\frac{1}{x\sin^2\beta }{A}_{66}\frac{\partial^2}{\partial {\theta}^2}+{A}_{11}\frac{\partial }{\partial x}-{H}_1\frac{1}{x}, $$
$$ {S}_{12}=\frac{1}{\sin \beta}\left({A}_{12}+{A}_{66}\right)\frac{\partial^2}{\partial x\partial \theta }-\frac{1}{x\sin \beta}\left({H}_1+{A}_{66}\right)\frac{\partial }{\partial \theta },{S}_{13}=\cot \beta {A}_{12}\frac{\partial }{\partial x}-\cot \beta \frac{1}{x}{H}_1, $$
$$ {S}_{14}=\left({B}_{11}x+{c}_1^0\right)\frac{\partial^2}{\partial {x}^2}+\frac{1}{x\sin^2\beta }{B}_{66}\frac{\partial^2}{\partial {\theta}^2}+{B}_{11}\frac{\partial }{\partial x}-{H}_2\frac{1}{x}, $$
$$ {S}_{15}=\frac{1}{\sin \beta}\left({B}_{12}+{B}_{66}\right)\frac{\partial^2}{\partial x\partial \theta }-\frac{1}{x\sin \beta}\left({H}_2+{B}_{66}\right)\frac{\partial }{\partial \theta }; $$
$$ {\displaystyle \begin{array}{c}{S}_{16}={H}_3\frac{\partial }{\partial x}\frac{\partial^2}{\partial {x}^2}+\frac{1}{2}\left({A}_{11}-{A}_{12}\right){\left(\frac{\partial }{\partial x}\right)}^2\\ {}+\frac{1}{x\sin^2\beta}\left({A}_{12}+{A}_{66}\right)\frac{\partial }{\partial \theta}\frac{\partial^2}{\partial x\partial \theta }+\frac{1}{x\sin^2\beta }{A}_{66}\frac{\partial }{\partial x}\frac{\partial^2}{\partial {\theta}^2}-\left({A}_{12}+{H}_1\right)\frac{1}{2{x}^2}{\left(\frac{\partial }{\partial \theta}\right)}^2,\end{array}} $$
$$ {S}_{21}=\frac{1}{\sin \beta}\left({A}_{12}+{A}_{66}\right)\frac{\partial^2}{\partial x\partial \theta }+\frac{1}{x\sin \beta}\left({H}_1+{A}_{66}\right)\frac{\partial }{\partial \theta }; $$
$$ {S}_{22}={A}_{66}\left(x\frac{\partial^2}{\partial {x}^2}+\frac{\partial }{\partial x}-\frac{1}{x}\right)+\frac{1}{x\sin^2\beta }{H}_1\frac{\partial^2}{\partial {\theta}^2},\kern1em {S}_{23}={H}_1\cot \beta \frac{1}{x\sin \beta}\frac{\partial }{\partial \theta }, $$
$$ {S}_{24}=\frac{1}{\sin \beta}\left({B}_{12}+{B}_{66}\right)\frac{\partial^2}{\partial x\partial \theta }+\frac{1}{x\sin \beta}\left({H}_2+{B}_{66}\right)\frac{\partial }{\partial \theta }, $$
$$ {S}_{25}={B}_{66}\left(x\frac{\partial^2}{\partial {x}^2}+\frac{\partial }{\partial x}-\frac{1}{x}\right)+\frac{1}{x\sin^2\beta }{H}_2\frac{\partial^2}{\partial {\theta}^2}, $$
$$ {\displaystyle \begin{array}{c}{S}_{26}=\frac{1}{\sin \beta }{A}_{66}\frac{\partial }{\partial \theta}\frac{\partial^2}{\partial {x}^2}+\frac{1}{\sin \beta}\left({A}_{12}+{A}_{66}\right)\frac{\partial }{\partial x}\frac{\partial^2}{\partial x\partial \theta}\\ {}+\frac{1}{x\sin \beta }{A}_{66}\frac{\partial }{\partial x}\frac{\partial }{\partial \theta }+{H}_1\frac{1}{\sin^3\beta {x}^2}\frac{\partial }{\partial \theta}\frac{\partial^2}{\partial {\theta}^2},\end{array}} $$
$$ {\displaystyle \begin{array}{c}{S}_{31}=\left({B}_{11}x+{c}_1^0\right)\frac{\partial^3}{\partial {x}^3}+\frac{1}{x\sin^2\beta}\left({B}_{12}+2{B}_{66}\right)\frac{\partial^3}{\partial x\partial {\theta}^2}+2{B}_{11}\frac{\partial^2}{\partial {x}^2}\\ {}+\frac{1}{x^2{\sin}^2\beta }{H}_2\frac{\partial^2}{\partial {\theta}^2}-\left({A}_{12}\cot \beta +{H}_2\frac{1}{x}\right)\frac{\partial }{\partial x}-\left({H}_1\cot \beta -{H}_2\frac{1}{x}\right)\frac{1}{x},\end{array}} $$
$$ {\displaystyle \begin{array}{c}{S}_{32}=\left({B}_{12}+2{B}_{66}\right)\frac{1}{\sin \beta}\frac{\partial^3}{\partial {x}^2\partial \theta }+{H}_2\left(\frac{1}{x^2\sin \beta}\frac{\partial^3}{\partial {\theta}^3}-\frac{1}{x\sin \beta}\frac{\partial^2}{\partial x\partial \theta}\right)\\ {}-\left({H}_1\cot \beta -{H}_2\frac{1}{x}\right)\frac{1}{x\sin \beta}\frac{\partial }{\partial \theta },\end{array}} $$
$$ {\displaystyle \begin{array}{c}{S}_{33}={B}_{12}\cot \beta \frac{\partial^2}{\partial {x}^2}+{H}_2\cot \beta \frac{1}{x^2{\sin}^2\beta}\frac{\partial^2}{\partial {\theta}^2}-{H}_2\cot \beta \frac{1}{x}\frac{\partial }{\partial x}\\ {}-\left({H}_1\cot \beta -{H}_2\frac{1}{x}\right)\cot \beta \frac{1}{x}-x{K}_1+{K}_2\left(x\frac{\partial^2}{\partial {x}^2}+\frac{1}{x\sin^2\beta}\frac{\partial^2}{\partial {\theta}^2}+\frac{\partial }{\partial x}\right),\end{array}} $$
$$ {\displaystyle \begin{array}{c}{S}_{34}={H}_4\frac{\partial^3}{\partial {x}^3}+\frac{1}{x\sin^2\beta}\left({D}_{12}+2{D}_{66}\right)\frac{\partial^3}{\partial x\partial {\theta}^2}+2{D}_{11}\frac{\partial^2}{\partial {x}^2}\\ {}+{H}_5\frac{1}{x^2{\sin}^2\beta}\frac{\partial^2}{\partial {\theta}^2}-\left({B}_{12}\cot \beta +{H}_5\frac{1}{x}\right)\frac{\partial }{\partial x}-\left({H}_2\cot \beta -{H}_5\frac{1}{x}\right)\frac{1}{x},\end{array}} $$
$$ {\displaystyle \begin{array}{c}{S}_{35}=\left({D}_{12}+2{D}_{66}\right)\frac{1}{\sin \beta}\frac{\partial^3}{\partial {x}^2\partial \theta }+{H}_5\left(\frac{1}{x^2{\sin}^3\beta}\frac{\partial^3}{\partial {\theta}^3}-\frac{1}{x\sin \beta}\frac{\partial^2}{\partial x\partial \theta}\right)\\ {}-\left({H}_2\cot \beta -{H}_5\frac{1}{x}\right)\frac{1}{x\sin \beta}\frac{\partial }{\partial \theta },\end{array}} $$
$$ {\displaystyle \begin{array}{c}{S}_{36}=\frac{1}{x\sin^2\beta}\left(\frac{1}{x}{H}_1u+{A}_{12}\frac{\partial u}{\partial x}\right)\frac{\partial^2}{\partial {\theta}^2}+\left({H}_3\frac{\partial u}{\partial x}+{A}_{12}u\right)\frac{\partial^2}{\partial {x}^2}\\ {}+\frac{1}{x\sin^2\beta}\left[\frac{1}{x}\left({H}_1-{A}_{66}\right)\frac{\partial u}{\partial \theta }+\left({A}_{12}+{A}_{66}\right)\frac{\partial^2u}{\partial x\partial \theta}\right]\frac{\partial }{\partial \theta}\\ {}+{H}_3\frac{\partial^2u}{\partial {x}^2}\frac{\partial }{\partial x}+\frac{2}{x\sin^2\beta }{A}_{66}\frac{\partial u}{\partial \theta}\frac{\partial^2}{\partial x\partial \theta }+\frac{1}{x\sin^2\beta }{A}_{66}\frac{\partial^2u}{\partial {\theta}^2}\frac{\partial }{\partial x}+\left(2{A}_{11}+{A}_{12}\right)\frac{\partial u}{\partial x}\frac{\partial }{\partial x}\\ {}+\frac{1}{x^2{\sin}^3\beta}\left({H}_1+{A}_{12}\cot \beta \frac{E_r{A}_r}{d_r}\frac{\partial^2}{\partial {x}^2}.w\right)\frac{\partial v}{\partial \theta}\frac{\partial^2}{\partial {\theta}^2}+\frac{1}{x^2{\sin}^3\beta }{H}_1\frac{\partial^2}{\partial {\theta}^2}\frac{\partial^2}{\partial {\theta}^2}+\frac{1}{x^2{\sin}^3\beta }{H}_1\frac{\partial^2v}{\partial {\theta}^2}\frac{\partial }{\partial \theta}\\ {}+\frac{1}{\sin \beta}\left[\left({A}_{12}+{A}_{66}\right)\frac{\partial^2v}{\partial x\partial \theta }-\frac{1}{x}{A}_{66}\frac{\partial v}{\partial \theta}\right]\frac{\partial }{\partial x}+\frac{2}{\sin \beta }{A}_{66}\left(\frac{\partial \mid }{\partial x}-\frac{1}{x}v\right)\frac{\partial^2}{\partial x\partial \theta}\\ {}+\frac{1}{\sin \beta }{A}_{66}\left(\frac{\partial^2v}{\partial {x}^2}-\frac{1}{x}\frac{\partial v}{\partial x}+\frac{1}{x^2}v\right)\frac{\partial }{\partial \theta }+\frac{1}{x\sin^2\beta}\left({B}_{12}\frac{\partial {\phi}_x}{\partial x}+\frac{1}{x}{H}_2{\phi}_x\right)\frac{\partial^2}{\partial {\theta}^2}\\ {}+\left[\left({B}_{11}x+{c}_1^0\right)\frac{\partial {\phi}_x}{\partial x}+{B}_{12}{\phi}_x\right]\frac{\partial^2}{\partial {x}^2}+\frac{1}{x\sin^2\beta }{B}_{66}\frac{\partial^2{\phi}_x}{\partial {\theta}^2}\frac{\partial }{\partial x}\\ {}+\left[\left({B}_{11}x+{c}_1^0\right)\frac{\partial^2{\phi}_x}{\partial {x}^2}+\left({B}_{11}+{B}_{12}\right)\frac{\partial {\phi}_x}{\partial x}\right]\frac{\partial }{\partial x}+\frac{2}{x\sin^2\beta }{B}_{66}\frac{\partial {\phi}_x}{\partial \theta}\frac{\partial^2}{\partial x\partial \theta}\\ {}+\frac{1}{x\sin^2\beta}\left({B}_{12}+{B}_{66}\right)\frac{\partial^2{\phi}_x}{\partial x\partial \theta}\frac{\partial }{\partial \theta }+\frac{1}{x^2{\sin}^2\beta}\left({H}_2-{B}_{66}\right)\frac{\partial {\phi}_x}{\partial \theta}\frac{\partial }{\partial \theta}\\ {}+\frac{1}{\sin \beta }{B}_{66}\frac{\partial^2{\phi}_{\theta }}{\partial {x}^2}\frac{\partial }{\partial \theta }+\frac{1}{\sin \beta}\left({B}_{12}+{B}_{66}\right)\frac{\partial^2{\phi}_{\theta }}{\partial x\partial \theta}\frac{\partial }{\partial x}+\frac{1}{x^2{\sin}^3\beta }{H}_2\frac{\partial^2{\phi}_{\theta }}{\partial {\theta}^2}\frac{\partial }{\partial \theta}\\ {}+\frac{1}{\sin \beta}\left({B}_{12}\frac{\partial^2}{\partial {x}^2}+\frac{1}{x^2{\sin}^2\beta }{H}_2\frac{\partial^2}{\partial {\theta}^2}-\frac{1}{x}{B}_{66}\frac{\partial }{\partial x}\right)\frac{\partial {\phi}_{\theta }}{\partial \theta }+\frac{1}{\sin \beta }{B}_{66}\left(2\frac{\partial^2}{\partial x\partial \theta }-\frac{1}{x}\frac{\partial }{\partial \theta}\right)\frac{\partial {\phi}_{\theta }}{\partial x}\\ {}+\frac{1}{x\sin \beta }{B}_{66}\left(\frac{1}{x}\frac{\partial }{\partial \theta }-2\frac{\partial^2}{\partial x\partial \theta}\right){\phi}_{\theta }+\frac{1}{2}{A}_{12}\cot \beta {\left(\frac{\partial }{\partial x}\right)}^{\mid 2}+\frac{2}{x\sin^2\beta }{B}_{12}{\left(\frac{\partial^2}{\partial x\partial \theta}\right)}^2\\ {}+\frac{2}{x\sin^2\beta }{B}_{66}{\left(\frac{\partial^2}{\partial x\partial \theta}\right)}^2+\left({B}_{11}x+{c}_1^0\right)\frac{\partial }{\partial x}\frac{\partial^3}{\partial {x}^3}\\ {}+\frac{1}{x\sin^2\beta}\left({B}_{12}+2{B}_{66}\right)\left(\frac{\partial }{\partial \theta}\frac{\partial^3}{\partial {x}^2\partial \theta }+\frac{\partial }{\partial x}\frac{\partial^3}{\partial x\partial {\theta}^2}\right)+\frac{1}{x^3{\sin}^4\beta }{H}_2\frac{\partial }{\partial \theta}\frac{\partial^3}{\partial {\theta}^3}\\ {}+\frac{1}{2x\sin^2\beta}\left[\begin{array}{l}\left({A}_{12}+2{A}_{66}\right)\frac{\partial^2}{\partial {x}^2}+\frac{1}{x^2}\left(x{A}_{22}\cot \beta +2{H}_2\right)-\frac{1}{x}\left({A}_{12}+2{A}_{66}\right)\frac{\partial }{\partial x}\\ {}+\frac{3}{x^2{\sin}^2\beta }{H}_1\frac{\partial^2}{\partial {\theta}^2}+\frac{1}{x}\left(\frac{1}{x}{B}_{12}+\frac{E_r{A}_r}{d_r}\right)\end{array}\right]{\left(\frac{\partial }{\partial \theta}\right)}^2\\ {}+\left({B}_{11}x+{c}_1^0\right){\left(\frac{\partial^2}{\partial {x}^2}\right)}^2+\frac{1}{x^3{\sin}^4\beta }{H}_2{\left(\frac{\partial^2}{\partial {\theta}^2}\right)}^2\\ {}+\frac{1}{x\sin^2\beta}\left[2\left({A}_{12}+2{A}_{66}\right)\frac{\partial^2}{\partial x\partial \theta}\frac{\partial }{\partial x}-\frac{1}{x}\left(2{B}_{12}+{H}_2\right)\frac{\partial^2}{\partial x\partial \theta}\right]\frac{\partial }{\partial \theta}\\ {}+\frac{1}{x\sin^2\beta}\left[x{A}_{22}\cot \beta .w+\frac{1}{2}\left({A}_{12}+2{A}_{66}\right){\left(\frac{\partial }{\partial x}\right)}^2\right]\frac{\partial^2}{\partial {\theta}^2}+\frac{1}{2}{A}_{11}{\left(\frac{\partial }{\partial x}\right)}^3\\ {}+\left[{A}_{12}\cot \beta .w+\frac{2}{x\sin^2\beta }{B}_{66}\frac{\partial }{\partial {\theta}^2}+\left(2{B}_{11}-{B}_{12}\right)\frac{\partial }{\partial x}+\frac{3}{2}{H}_3{\left(\frac{\partial }{\partial x}\right)}^2\right]\frac{\partial^2}{\partial {x}^2},\end{array}} $$
$$ {S}_{41}=\left({B}_{11}x+{c}_1^0\right)\sin \beta \frac{\partial^2}{\partial {x}^2}+{B}_{66}\frac{1}{x\sin \beta}\frac{\partial^2}{\partial {\theta}^2}+{B}_{11}\sin \beta \frac{\partial }{\partial x}-{H}_2\sin \beta \frac{1}{x}, $$
$$ {S}_{42}=\left({B}_{12}+{B}_{66}\right)\frac{\partial^2}{\partial x\partial \theta }-\left({H}_2+{B}_{66}\right)\frac{1}{x}\frac{\partial }{\partial \theta }, $$
$$ {S}_{43}=\left({B}_{12}\cos \beta -{A}_{44}x\sin \beta \right)\frac{\partial }{\partial x}-{H}_2\cos \beta \frac{1}{x}, $$
$$ {S}_{44}={H}_4\sin \beta \frac{\partial^2}{\partial {x}^2}+\frac{1}{x\sin \beta }{D}_{66}\frac{\partial^2}{\partial {\theta}^2}+{D}_{11}\sin \beta \frac{\partial }{\partial x}-\left({A}_{44}x+{H}_5\frac{1}{x}\right)\sin \beta, $$
$$ {S}_{45}=\left({D}_{12}+{D}_{66}\right)\frac{\partial^2}{\partial x\partial \theta }-\left({H}_5+{D}_{66}\right)\frac{1}{x}\frac{\partial }{\partial \theta }, $$
$$ {\displaystyle \begin{array}{c}{S}_{46}=-\frac{1}{2{x}^2\sin \beta}\left({B}_{12}+{H}_2\right){\left(\frac{\partial }{\partial \theta}\right)}^2+\frac{1}{x\sin \beta}\left({B}_{12}+{B}_{66}\right)\frac{\partial^2}{\partial x\partial \theta}\frac{\partial }{\partial \theta}\\ {}+\sin \beta \left({B}_{11}x+{c}_1^0\right)\frac{\partial }{\partial x}\frac{\partial^2}{\partial {x}^2}+\frac{1}{x\sin \beta }{B}_{66}\frac{\partial }{\partial \theta}\frac{\partial^2}{\partial {\theta}^2}+\frac{1}{2\sin \beta}\left({B}_{11}-{B}_{12}\right){\left(\frac{\partial }{\partial x}\right)}^2,\end{array}} $$
$$ {S}_{51}=\left({B}_{12}+{B}_{66}\right)\frac{\partial^2}{\partial x\partial \theta }+\left({H}_2+{B}_{66}\right)\frac{1}{x}\frac{\partial }{\partial \theta }, $$
$$ {S}_{52}={B}_{66}x\sin \beta \frac{\partial^2}{\partial {x}^2}+\frac{1}{x\sin \beta }{H}_2\frac{\partial^2}{\partial {\theta}^2}+{B}_{66}\left(\sin \beta \frac{\partial }{\partial x}-\sin \beta \frac{1}{x}\right), $$
$$ {S}_{53}=-\left({A}_{55}-{H}_2\cot \beta \frac{1}{x}\right)\frac{\partial }{\partial \theta },{S}_{54}=\left({D}_{12}+{D}_{66}\right)\frac{\partial^2}{\partial x\partial \theta }+\left({H}_5+{D}_{66}\right)\frac{1}{x}\frac{\partial }{\partial \theta }, $$
$$ {S}_{55}={D}_{66}x\sin \beta \frac{\partial^2}{\partial {x}^2}+{H}_5\frac{1}{x\sin \beta}\frac{\partial^2}{\partial {\theta}^2}+{D}_{66}\sin \beta \frac{\partial }{\partial x}-\left({A}_{55}x+{D}_{66}\frac{1}{x}\right)\sin \beta, $$
$$ {S}_{56}=\left[\frac{1}{x^2{\sin}^2\beta }{H}_2\frac{\partial^2}{\partial {\theta}^2}+{B}_{66}\left(\frac{1}{x}\frac{\partial }{\partial x}+\frac{\partial^2}{\partial {x}^2}\right)\right]\frac{\partial }{\partial \theta }+\left({B}_{12}+{B}_{66}\right)\frac{\partial^2}{\partial x\partial \theta}\frac{\partial }{\partial x}. $$

Appendix C

$$ {\displaystyle \begin{array}{c}{H}_6=\frac{{\left({x}_0+L\right)}^3-{x}_0^3}{6}+\frac{L^3}{4{m}^2{\pi}^2},\kern1em {H}_7=\frac{{\left({x}_0+L\right)}^3-{x}_0^3}{6}-\frac{L^3}{4{m}^2{\pi}^2},\\ {}{H}_9=\frac{{\left({x}_0+L\right)}^4-{x}_0^4}{8}-\frac{3{L}^3\left(2{x}_0+L\right)}{8{m}^2{\pi}^2},\\ {}{H}_{10}=\frac{{\left({x}_0+L\right)}^5-{x}_0^5}{10}+\frac{L^2\left[{x}_0^3-{\left({x}_0+L\right)}^3\right]}{2{m}^2{\pi}^2}+\frac{3{L}^5}{4{m}^4{\pi}^4},\end{array}} $$

In Eqs. (17a-17e),

$$ {\displaystyle \begin{array}{c}{G}_{11}=-\frac{m^2{\pi}^3}{L^3}{A}_{11}\sin \beta {H}_8-\frac{\pi }{4}L\left(2{x}_0+L\right)\sin \beta \left({H}_1+\frac{n^2}{\sin^2\beta }{A}_{66}\right)\\ {}-\frac{m^2{\pi}^3}{L^2}\frac{E_s{A}_s}{\uplambda_0}\sin \beta {H}_6+\frac{\pi }{4}L\left(2{x}_0+L\right){A}_{11}\sin \beta, \end{array}} $$
$$ {G}_{12}=-\frac{mn{\pi}^2}{L}\left({A}_{12}+{A}_{66}\right){H}_6-\frac{n{L}^2}{4m}\left({H}_1+{A}_{66}\right), $$
$$ {G}_{13}=\frac{m{\pi}^2}{L}\cot \beta \sin \beta {A}_{12}{H}_6+\cot \beta \sin \beta \frac{L^2}{4m}{H}_1, $$
$$ {\displaystyle \begin{array}{c}{G}_{14}=-\frac{m^2{\pi}^3}{L^2}{B}_{11}\sin \beta {H}_8-\frac{\pi }{4}L\left(2{x}_0+L\right)\sin \beta \left({H}_2+|\frac{n^2}{\sin^2\beta }{B}_{66}\right)\\ {}-\frac{m^2{\pi}^3}{L^2}{c}_1^0\sin \beta {H}_6+\frac{\pi }{4}L\left(2{x}_0+L\right){B}_{11}\sin \beta, \end{array}} $$
$$ {G}_{15}=-\frac{mn{\pi}^2}{4}\frac{1}{\sin \beta}\left(2{x}_0+L\right)\left({B}_{12}+{B}_{66}\right), $$
$$ {G}_{21}=-\frac{mn{\pi}^2}{L}\left({A}_{12}+{A}_{66}\right){H}_7-\frac{n{L}^2}{4m}\left({H}_2+{A}_{66}\right), $$
$$ {\displaystyle \begin{array}{c}{G}_{22}=-\frac{m^2{\pi}^3}{L^2}{A}_{66}\sin \beta {H}_9-\frac{\pi }{4}L\left(2{x}_0+L\right){A}_{66}\sin \beta \\ {}-\frac{\pi }{4}L\left(2{x}_0+L\right)\sin \beta \left(\frac{n^2}{\sin^2\beta }{H}_1+{A}_{66}\right),\end{array}} $$
$$ {G}_{23}=\frac{n\pi}{4}L\left(2{x}_0+L\right)\cot \beta {H}_1,\kern1em {G}_{24}=-\frac{mn{\pi}^2}{L}\left({B}_{12}+{B}_{66}\right){H}_7-\frac{n{L}^2}{4m}\left({H}_2+{B}_{66}\right), $$
$$ {G}_{25}=-\frac{m^2{\pi}^3}{L^2}{B}_{66}{H}_7-\frac{n^2\pi L}{2\sin^2\beta }{H}_2+\frac{\pi }{4}L{B}_{66}, $$
$$ {\displaystyle \begin{array}{c}{G}_{31}=\frac{m^3{\pi}^4}{L^3}{B}_{11}\sin \beta {H}_{10}+\left(\frac{m^3{\pi}^4}{L^3}{c}_1^0\sin \beta +\frac{m{\pi}^2}{L}\cot \beta \sin \beta {A}_{12}\right){H}_9\\ {}+\left[\frac{m{\pi}^2}{L}\sin \beta \left(3{B}_{11}+{H}_2\right)+\frac{m{n}^2{\pi}^2}{L\sin \beta}\left({B}_{12}+2{B}_{66}\right)\right]{H}_7\\ {}+\cot \beta \sin \beta {H}_1\frac{L^2\left(2{x}_0+L\right)}{4m}+\frac{L^2}{4m}\sin \beta {H}_2\left(\frac{n^2}{\sin^2\beta }-1\right),\end{array}} $$
$$ {G}_{32}=\frac{m^2n{\pi}^3}{L^2}\left({B}_{12}+2{B}_{66}\right){H}_9+ n\pi \cot \beta {H}_1{H}_7+\frac{n\pi L\left(2{x}_0+L\right)}{4}{H}_2\left(\frac{n^2}{\sin^2\beta }-2\right), $$
$$ {\displaystyle \begin{array}{c}{G}_{33}=-\frac{m^2{\pi}^3}{L^2}\cot \beta \sin \beta {B}_{12}{H}_9-\pi {\cot}^2\beta \sin \beta {H}_2{H}_7\\ {}-\frac{\pi }{4}L\left(2{x}_0+L\right)\cot \beta \sin \beta {H}_2\left(\frac{n^2}{\sin^2\beta }-2\right),\end{array}} $$
$$ {\displaystyle \begin{array}{c}{G}_{34}=\frac{m^3{\pi}^4}{L^3}{D}_{11}\sin \beta {H}_{10}+\left(\frac{m^3{\pi}^4}{L^3}\frac{E_s{I}_s}{\uplambda_0}\sin \beta +\frac{m{\pi}^2}{L}\cot \beta \sin \beta {B}_{12}\right){H}_9\\ {}+\left[\frac{m{\pi}^2}{L}\sin \beta \left(3{D}_{11}+{H}_5\right)+\frac{m{n}^2{\pi}^2}{L\sin \beta}\left({D}_{12}+2{D}_{66}\right)\right]{H}_7\\ {}+\frac{L^2\left(2{x}_0+L\right)}{4m}\cot \beta \sin \beta {H}_2+\frac{L^2}{4m}\sin \beta {H}_5\left(\frac{n^2}{\sin^2\beta }-1\right),\end{array}} $$
$$ {\displaystyle \begin{array}{c}{G}_{35}=\frac{m^2n{\pi}^3}{L^2}\frac{1}{\sin \beta}\left({D}_{12}+2{D}_{66}\right){H}_7\\ {}+\frac{n\pi}{4\sin \beta }L\left(2{x}_0+L\right)\cot \beta {H}_2-\frac{n\pi}{\sin \beta }L\left({D}_{12}+2{D}_{66}\right)\\ {}+\frac{n\pi}{2\sin \beta }L{H}_5\left(\frac{n^2}{\sin^2\beta }-2\right)-\frac{n\pi}{4\sin \beta }L\left({H}_5+2{D}_{12}+4{D}_{66}\right),\end{array}} $$
$$ {\displaystyle \begin{array}{c}{G}_{36}=-\frac{9}{320}{m}^4{\pi}^5{A}_{11}L\sin \beta -\frac{1}{32}{m}^2{\pi}^3L\sin \beta \left[\frac{3}{4}{A}_{11}+\frac{1}{3}\frac{n^2}{\sin^2\beta}\left({A}_{12}+2{A}_{66}\right)\right]\\ {}+\frac{1}{64}\pi L\sin \beta \left[\frac{63}{16}{A}_{11}+\frac{5}{2}\frac{n^2}{\sin^2\beta}\left({A}_{12}+2{A}_{66}\right)-9\frac{n^4}{\sin^4\beta }{A}_{22}\right]\\ {}-\frac{9}{256}{m}^4{\pi}^5\sin \beta \left(4{x}_0{A}_{11}+\frac{E_s{A}_s}{\uplambda_0}\right)+\frac{27}{1024}{m}^2{\pi}^3\sin \beta \frac{E_s{A}_s}{\uplambda_0}\\ {}-\frac{1}{32}{m}^2{\pi}^3{x}_0\sin \beta \left[\frac{9}{4}{A}_{11}+\frac{n^2}{\sin^2\beta}\left({A}_{12}+{A}_{66}\right)\right]-\frac{9}{64L}{m}^4{\pi}^5{x}_0\sin \beta \left(2{x}_0{A}_{11}+\frac{E_s{A}_s}{\uplambda_0}\right)\\ {}+\frac{27}{512L}{m}^2{\pi}^3{x}_0\sin \beta \frac{E_s{A}_s}{\uplambda_0}-\frac{1}{32}{m}^2{\pi}^3{x}_0^2\sin \beta \left[\frac{9}{4}{A}_{11}+\frac{n^2}{\mathrm{L}\;{\sin}^2\beta}\left({A}_{12}+2{A}_{66}\right)\right]\\ {}-\frac{9}{64{L}^2}{m}^4{\pi}^5{x}_0^3\sin \beta \left({x}_0{A}_{11}+\frac{E_s{A}_s}{\uplambda_0}\right)\left(1+\frac{1}{L}\right)-\frac{9}{64}\frac{n^4\pi L}{\sin^3\beta}\frac{E_r{A}_r}{d_r},\end{array}} $$
$$ {G}_{37}=-\pi \sin \beta {H}_{10},\kern1em {G}_{38}=-\frac{m^2{\pi}^3}{L^2}\sin \beta {H}_{10}-\frac{\pi }{2}\left(3\sin \beta +\frac{2{n}^2}{\sin \beta}\right){H}_6, $$
$$ {\displaystyle \begin{array}{c}{G}_{39}=-\frac{m^2{\pi}^3}{L^2}\sin \beta {H}_9,\kern1em {G}_{41}=-\frac{m^2{\pi}^3}{L^2}{B}_{11}{\sin}^2\beta {H}_8-\frac{m^2{\pi}^3}{L^2}{c}_1^0{\sin}^2\beta {H}_6\\ {}+\frac{\pi }{4}L\left(2{x}_0+L\right)\left[{\sin}^2\beta \left({B}_{11}-{H}_2\right)-{n}^2{B}_{66}\right],\end{array}} $$
$$ {G}_{42}=-\frac{mn{\pi}^2}{L}\sin \beta \left({B}_{12}+{B}_{66}\right){H}_6-\frac{n{L}^2}{4m}\sin \beta \left({H}_2+{B}_{66}\right), $$
$$ {G}_{43}=-\frac{m{\pi}^2}{L}{A}_{44}{\sin}^2\beta {H}_8+\frac{m{\pi}^2}{L}\sin \beta \cos \beta {B}_{12}{H}_6+\frac{L^2}{4m}\sin \beta \cos \beta {H}_2, $$
$$ {\displaystyle \begin{array}{c}{G}_{44}=-\left(\frac{m^2{\pi}^3}{L^2}{D}_{11}+\pi {A}_{44}\right){\sin}^2\beta {H}_8-\frac{m^2{\pi}^3}{L^2}\frac{E_s{I}_s}{\uplambda_0}{\sin}^2\beta {H}_6\\ {}+\frac{\pi }{4}L\left(2{x}_0+L\right)\left[{\sin}^2\beta \left({D}_{11}-{H}_5\right)-{n}^2{D}_{66}\right],\kern1em {G}_{45}=-\frac{mn{\pi}^2}{4}\left(2{x}_0+L\right)\left({D}_{12}+{D}_{66}\right),\end{array}} $$
$$ {G}_{51}=-\frac{mn{\pi}^2}{L}\left({B}_{12}+{B}_{66}\right){H}_7-\frac{n{L}^2}{4m}\left({H}_2+{B}_{66}\right), $$
$$ {G}_{52}=-\frac{m^2{\pi}^3}{L^3}{B}_{66}\sin \beta {H}_9-\frac{\pi }{2}L\left(2{x}_0+L\right)\left(\frac{n^2}{2\sin \beta }{H}_2+{B}_{66}\sin \beta \right), $$
$$ {G}_{53}=- n\pi {A}_{55}{H}_7+\frac{n\pi}{4}L\left(2{x}_0+L\right)\cot \beta {H}_2, $$
$$ {G}_{54}=-\frac{mn{\pi}^2}{L}\left({D}_{12}+{D}_{66}\right){H}_7-\frac{n{L}^2}{4m}\left({H}_5+{D}_{66}\right), $$
$$ {G}_{55}=-\pi \left({A}_{55}+\frac{m^2{\pi}^2}{L^2}{D}_{66}\right){H}_7+\frac{\pi }{4}L\left({D}_{66}-\frac{2{n}^2}{\sin^2\beta }{H}_5\right). $$

Appendix D

$$ {\displaystyle \begin{array}{c}{X}_{22}=\frac{G_{22}{G}_{11}-{G}_{12}{G}_{21}}{G_{11}},{X}_{23}=\frac{G_{13}{G}_{21}-{G}_{23}{G}_{11}}{G_{11}},\\ {}{X}_{24}=\frac{G_{24}{G}_{11}-{G}_{14}{G}_{21}}{G_{11}},{X}_{25}=\frac{G_{25}{G}_{11}-{G}_{15}{G}_{21}}{G_{11}},\\ {}{X}_{42}=\frac{G_{42}{G}_{11}-{G}_{12}{G}_{41}}{G_{11}},{X}_{43}=\frac{G_{13}{G}_{41}-{G}_{11}{G}_{43}}{G_{11}},\\ {}{X}_{44}=\frac{G_{44}{G}_{11}-{G}_{14}{G}_{41}}{G_{11}},{X}_{45}=\frac{G_{45}{G}_{11}-{G}_{15}{G}_{41}}{G_{11}},\\ {}{X}_{52}=\frac{G_{52}{G}_{11}-{G}_{12}{G}_{51}}{G_{11}}{X}_{53}=\frac{G_{13}{G}_{51}-{G}_{11}{G}_{53}}{G_{11}},\\ {}{X}_{54}=\frac{G_{54}{G}_{11}-{G}_{14}{G}_{51}}{G_{11}}{X}_{55}=\frac{G_{55}{G}_{11}-{G}_{15}{G}_{51}}{G_{11}},\\ {}{T}_{43}=\frac{X_{43}{X}_{22}-{X}_{42}{X}_{23}}{X_{22}},{T}_{44}=\frac{X_{44}{X}_{22}-{X}_{42}{X}_{24}}{X_{22}},{T}_{45}=\frac{X_{45}{X}_{22}-{X}_{42}{X}_{25}}{X_{22}},\\ {}{T}_{53}=\frac{X_{53}{X}_{22}-{X}_{52}{X}_{23}}{X_{22}},{T}_{54}=\frac{X_{54}{X}_{22}-{X}_{52}{X}_{24}}{X_{22}},\\ {}{T}_{55}=\frac{X_{55}{X}_{22}-{X}_{52}{X}_{25}}{X_{22}},{R}_{53}=\frac{T_{53}{T}_{44}-{T}_{54}{T}_{43}}{T_{44}},{R}_{55}=\frac{T_{55}{T}_{44}-{T}_{54}{T}_{45}}{T_{44}},\\ {}{L}_{35}=\frac{R_{53}}{R_{55}},{L}_{34}=\frac{T_{43}-{L}_{35}{T}_{45}}{T_{44}},{L}_{32}=\frac{X_{23}-{X}_{25}{L}_{35}-{X}_{24}{L}_{34}}{X_{22}},\\ {}{L}_{31}=-\frac{G_{13}+{G}_{12}{L}_{32}+{G}_{14}{L}_{34}+{G}_{15}{L}_{35}}{G_{11}}.\end{array}} $$

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Chan, D.Q., Long, V.D. & Duc, N.D. Nonlinear Buckling and Postbuckling of FGM Shear-Deformable Truncated Conical Shells Reinforced by FGM Stiffeners. Mech Compos Mater 54, 745–764 (2019). https://doi.org/10.1007/s11029-019-9780-x

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