Reddy’s third-order shear deformation shell theory for free vibration analysis of rotating stiffened advanced nanocomposite toroidal shell segments in thermal environments

Abstract

This study investigates the free vibration of rotating stiffened toroidal shell segments in thermal environments. The shell segments are made of a functionally graded graphene platelet reinforced composite (FG-GPLRC)—advanced nanocomposite. Functionally graded (FG)-X, FG-O, and uniform distribution-type graphene platelet distribution patterns are considered. The equations of motion of rotating stiffened FG-GPLRC toroidal shell segments are derived based on variants of Reddy’s third-order shear deformation shell theory (TSDT) and the smeared-stiffener technique. Then, solutions are obtained using the Rayleigh–Ritz procedure. The effects of centrifugal and Coriolis forces and the initial hoop tension resulting from rotation are considered. Various numerical examples are produced to verify the implemented scheme and demonstrate the effects of material properties, rotating speed, temperature increment, boundary conditions, geometric parameters, and stiffeners on the natural frequencies of the shell. In addition, while an incomplete version of Reddy’s TSDT has been used widely in many recent studies, the present study points out significant differences between that version and the complete version of Reddy’s TSDT by numerical results and discussions.

Appendix

The entries of ${\mathbf{K}},{\mathbf{M,}}\overline{{\mathbf{M}}} ,{\tilde{\mathbf{M}}}$ in Eq. (31) are given by

$$\begin{aligned} K_{ij}^{11} & = \pi R_{2} \int\limits_{0}^{L} {\left( {A_{11}^{{}} \phi_{i,x}^{u} \phi_{j,x}^{u} + A_{66}^{{}} \frac{{n^{2} }}{{R_{2}^{2} }}\phi_{i}^{u} \phi_{j}^{u} + k_{s} A_{55} \frac{1}{{R_{1}^{2} }}\phi_{i}^{u} \phi_{j}^{u} } \right){\text{d}}x} {;} \\ K_{ij}^{12} & = \pi R_{2} \int\limits_{0}^{L} {\left( {A_{12}^{{}} \frac{n}{{R_{2} }}\phi_{i,x}^{u} \phi_{j}^{v} - A_{66}^{{}} \frac{n}{{R_{2} }}\phi_{i}^{u} \phi_{j,x}^{v} } \right){\text{d}}x} {;} \\ K_{ij}^{13} & = \pi R_{2} \int\limits_{0}^{L} {\left( {A_{11}^{{}} \frac{1}{{R_{1} }}\phi_{i,x}^{u} \phi_{j}^{w} + A_{12}^{{}} \frac{1}{{R_{2} }}\phi_{i,x}^{u} \phi_{j}^{w} - k_{s} A_{55} \frac{1}{{R_{1} }}\phi_{i}^{u} \phi_{j,x}^{w} \, } \right){\text{d}}x} {;} \\ K_{ij}^{14} & = \pi R_{2} \int\limits_{0}^{L} {\left( {B_{11}^{{}} \phi_{i,x}^{u} \phi_{j,x}^{x} + B_{66}^{{}} \frac{{n^{2} }}{{R_{2}^{2} }}\phi_{i}^{u} \phi_{j}^{x} - k_{s} A_{55} \frac{1}{{R_{1} }}\phi_{i}^{u} \phi_{j}^{x} } \right){\text{d}}x} {;} \\ K_{ij}^{15} & = \pi R_{2} \int\limits_{0}^{L} {\left( {B_{12}^{{}} \frac{n}{{R_{2} }}\phi_{i,x}^{u} \phi_{j}^{y} - B_{66}^{{}} \frac{n}{{R_{2} }}\phi_{i}^{u} \phi_{j,x}^{y} } \right){\text{d}}x} {;} \\ K_{ij}^{22} & = \pi R_{2} \int\limits_{0}^{L} {\left( {A_{22}^{{}} \frac{{n^{2} }}{{R_{2}^{2} }}\phi_{i}^{v} \phi_{j}^{v} + A_{66}^{{}} \phi_{i,x}^{v} \phi_{j,x}^{v} + k_{s} A_{44} \frac{1}{{R_{2}^{2} }}\phi_{i}^{v} \phi_{j}^{v} - N_{x}^{T} \phi_{i,x}^{v} \phi_{i,x}^{v} } \right){\text{d}}x} {;} \\ K_{ij}^{23} & = \pi R_{2} \int\limits_{0}^{L} {\left( {A_{12}^{{}} \frac{n}{{R_{1} R_{2} }} + A_{22}^{{}} \frac{n}{{R_{2}^{2} }} + k_{s} A_{44} \frac{n}{{R_{2}^{2} }}} \right)\phi_{i}^{v} \phi_{j}^{w} {\text{d}}x} {;} \\ K_{ij}^{24} & = \pi R_{2} \int\limits_{0}^{L} {\left( {B_{12}^{{}} \frac{n}{{R_{2} }}\phi_{i}^{v} \phi_{j,x}^{x} - B_{66}^{{}} \frac{n}{{R_{2} }}\phi_{i,x}^{v} \phi_{j}^{x} } \right){\text{d}}x} {;} \\ K_{ij}^{25} & = \pi R_{2} \int\limits_{0}^{L} {\left( {B_{22}^{{}} \frac{{n^{2} }}{{R_{2}^{2} }}\phi_{i}^{v} \phi_{j}^{y} + B_{66}^{{}} \phi_{i,x}^{v} \phi_{j,x}^{y} - k_{s} A_{44} \frac{1}{{R_{2} }}\phi_{i}^{v} \phi_{j}^{y} } \right){\text{d}}x} {;} \\ K_{ij}^{33} & = \pi R_{2} \int\limits_{0}^{L} {\left[ {\left( {\frac{{A_{11}^{{}} }}{{R_{1}^{2} }} + 2\frac{{A_{12}^{{}} }}{{R_{1} R_{2} }} + \frac{{A_{22}^{{}} }}{{R_{2}^{2} }}} \right)\phi_{i}^{w} \phi_{j}^{w} + k_{s} A_{44}^{{}} \frac{{n^{2} }}{{R_{2}^{2} }}\phi_{i}^{w} \phi_{j}^{w} + k_{s} A_{55}^{{}} \phi_{i,x}^{w} \phi_{j,x}^{w} - N_{x}^{T} \phi_{i,x}^{w} \phi_{j,x}^{w} } \right]{\text{d}}x} {;} \\ K_{ij}^{34} & = \pi R_{2} \int\limits_{0}^{L} {\left( {\frac{{B_{11}^{{}} }}{{R_{1} }}\phi_{i}^{w} \phi_{j,x}^{x} + \frac{{B_{12}^{{}} }}{{R_{2} }}\phi_{i}^{w} \phi_{j,x}^{x} + k_{s} A_{55}^{{}} \phi_{i,x}^{w} \phi_{j}^{x} } \right){\text{d}}x{;}} \\ K_{ij}^{35} & = \pi R_{2} \int\limits_{0}^{L} {\left( {B_{12}^{{}} \frac{n}{{R_{1} R_{2} }} + B_{22}^{{}} \frac{n}{{R_{2}^{2} }} - k_{s} A_{44}^{{}} \frac{n}{{R_{2} }}} \right)\phi_{i}^{w} \phi_{j}^{y} {\text{d}}x} {;} \\ K_{ij}^{44} & = \pi R_{2} \int\limits_{0}^{L} {\left( {D_{11}^{{}} \phi_{i,x}^{x} \phi_{j,x}^{x} + D_{66}^{{}} \frac{{n^{2} }}{{R_{2}^{2} }}\phi_{i}^{x} \phi_{j}^{x} + k_{s} A_{55}^{{}} \phi_{i}^{x} \phi_{j}^{x} } \right){\text{d}}x} {;} \\ K_{ij}^{45} & = \pi R_{2} \int\limits_{0}^{L} {\left( {D_{12}^{{}} \frac{n}{{R_{2} }}\phi_{i,x}^{x} \phi_{j}^{y} - D_{66}^{{}} \frac{n}{{R_{2} }}\phi_{i}^{x} \phi_{j,x}^{y} } \right){\text{d}}x} {;} \\ K_{ij}^{55} & = \pi R_{2} \int\limits_{0}^{L} {\left( {D_{22}^{{}} \frac{{n^{2} }}{{R_{2}^{2} }}\phi_{i}^{y} \phi_{j}^{y} + D_{66}^{{}} \phi_{i,x}^{y} \phi_{i,x}^{y} + k_{s} A_{44}^{{}} \phi_{i}^{y} \phi_{j}^{y} } \right){\text{d}}x{;}} \\ \end{aligned}$$

(A.1)

$$\begin{aligned} M_{ij}^{11} & = \pi R_{2} J_{0} \int\limits_{0}^{L} {\phi_{i}^{u} \phi_{j}^{u} {\text{d}}x} {;}\;M_{ij}^{22} = \pi R_{2} J_{0} \int\limits_{0}^{L} {\phi_{i}^{v} \phi_{j}^{v} {\text{d}}x} {;}\;M_{ij}^{33} = \pi R_{2} J_{0} \int\limits_{0}^{L} {\phi_{i}^{w} \phi_{j}^{w} {\text{d}}x} {;} \\ M_{ij}^{44} & = \pi R_{2} J_{2} \int\limits_{0}^{L} {\phi_{i}^{x} \phi_{j}^{x} {\text{d}}x} {;}\;M_{ij}^{55} = \pi R_{2} J_{2} \int\limits_{0}^{L} {\phi_{i}^{y} \phi_{j}^{y} {\text{d}}x} {;} \\ M_{ij}^{14} & = \pi R_{2} J_{1} \int\limits_{0}^{L} {\phi_{i}^{u} \phi_{j}^{x} {\text{d}}x} {;}\;M_{ij}^{25} = \pi R_{2} J_{1} \int\limits_{0}^{L} {\phi_{i}^{v} \phi_{j}^{y} {\text{d}}x}; \\ \overline{M}_{ij}^{23} & = \pi R_{2} J_{0} \int\limits_{0}^{L} {\phi_{i}^{v} \phi_{j}^{w} {\text{d}}x} {;}\;\overline{M}_{ij}^{35} = \pi R_{2} J_{1} \int\limits_{0}^{L} {\phi_{i}^{w} \phi_{j}^{y} {\text{d}}x}; \\ \tilde{M}_{ij}^{11} & = n^{2} \pi R_{2}^{{}} J_{0} \int\limits_{0}^{L} {\phi_{i}^{u} \phi_{j}^{u} {\text{d}}x} {;}\;\tilde{M}_{ij}^{22} = n^{2} \pi R_{2} J_{0} \int\limits_{0}^{L} {\phi_{i}^{v} \phi_{j}^{v} {\text{d}}x} ;\\ \tilde{M}_{ij}^{33} & = n^{2} \pi R_{2}^{{}} J_{0} \int\limits_{0}^{L} {\phi_{i}^{w} \phi_{j}^{w} {\text{d}}x} {;}\;\tilde{M}_{ij}^{44} = n^{2} \pi R_{2} J_{2} \int\limits_{0}^{L} {\phi_{i}^{x} \phi_{j}^{x} {\text{d}}x}; \\ \tilde{M}_{ij}^{55} & = n^{2} \pi R_{2}^{{}} J_{2} \int\limits_{0}^{L} {\phi_{i}^{y} \phi_{j}^{y} {\text{d}}x} {;}\;\tilde{M}_{ij}^{14} = n^{2} \pi R_{2}^{{}} J_{1} \int\limits_{\Omega }^{{}} {\phi_{i}^{u} \phi_{j}^{x} {\text{d}}x}; \\ \tilde{M}_{ij}^{25} & = n^{2} \pi R_{2}^{{}} J_{1} \int\limits_{0}^{L} {\phi_{i}^{v} \phi_{j}^{y} {\text{d}}x} {;}\;\tilde{M}_{ij}^{23} = 2n\pi R_{2}^{{}} J_{0} \int\limits_{0}^{L} {\phi_{i}^{v} \phi_{j}^{w} {\text{d}}x}; \\ \tilde{M}_{ij}^{35} & = 2n\pi R_{2}^{{}} J_{1} \int\limits_{0}^{L} {\phi_{i}^{w} \phi_{j}^{y} {\text{d}}x} \\ \end{aligned}$$

(A.2)

where

$$\begin{gathered} J_{0} = \sum\limits_{k = 1}^{NL} {\int\limits_{{z_{k - 1} }}^{{z_{k} }} {\rho_{{{\text{eq}}}}^{\left( k \right)} {(}z{\text{)d}}z} } {;}\;\;\;J_{1} = \sum\limits_{k = 1}^{NL} {\int\limits_{{z_{k - 1} }}^{{z_{k} }} {z\rho_{{{\text{eq}}}}^{\left( k \right)} {(}z{\text{)d}}z} } {;}\;\;\;J_{2} = \sum\limits_{k = 1}^{NL} {\int\limits_{{z_{k - 1} }}^{{z_{k} }} {z^{2} \rho_{{{\text{eq}}}}^{\left( k \right)} {(}z{\text{)d}}z} } ; \hfill \\ \rho_{{{\text{eq}}}}^{{{(}k{)}}} = \rho^{{{(}k{)}}} + \frac{{\rho_{x} }}{{s_{x} }}\frac{{A_{x} }}{h} + \frac{{\rho_{y} }}{{s_{y} }}\frac{{A_{y} }}{h} .\hfill \\ \end{gathered}$$

(A.3)