Nonlinear dynamic modeling and experimental study of full-composite cylindrical shells with a foam-filled cavity lattice core

Abstract

A new full-composite cylindrical shell (FCCS) with a foam-filled cavity lattice core (FFCLC) is developed and prepared, and a nonlinear dynamic model considering the amplitude-dependent property of composite materials is proposed. Compared to traditional linear dynamic models, the lower frequencies and higher resonant responses of structures subjected to base harmonic excitations can be obtained in the proposed nonlinear dynamic model. The nonlinear dynamic behaviors of FFCLC-FCCSs are investigated theoretically and experimentally, in which the fabrication and assembly procedures of FFCLC-FCCS specimens are first provided, and vibration measurements are performed on those specimens subjected to different excitation amplitudes, wherein the soft nonlinear vibration phenomenon characterized by the amplitude-dependent property is discovered. Subsequently, in the framework of the first-order shear deformation theory based on the layerwise principle, the mode superposition approach and the Rayleigh–Ritz method are utilized to obtain the nonlinear frequencies, mode shapes, and resonant responses of the structure subjected to different excitation amplitudes. Therein, the equivalent material parameters of the core part are determined using the modified cross and fill equivalent principle, and the nonlinear elastic modulus with amplitude-dependent fitting coefficients of the skins and core are assumed by the Jones-Nelson nonlinear theory, and those coefficients are determined by using an inverse parameter identification and fitting technique based on experimental test data. Then, the validation work on the developed model is performed by comparing the calculated results of the model with those of the tests. Finally, the impacts of several critical parameters on the nonlinear dynamic behaviors of the structure are estimated, with some suggestions in favor of reducing the nonlinear resonant responses of FFCLC-FCCSs being clarified.

Appendices

Appendix A

The volumes $V_{ * }$, $V_{1}$, $V_{2}$ and $V_{3}$ in Eqs. (1a), (1b), and (1c) can be represented as:

$$ V_{ * } = {\uppi }L\left[ {\left( {R^{{\text{C}}} + \frac{{h^{{\text{C}}} }}{2}} \right)^{2} - \left( {R^{{\text{C}}} - \frac{{h^{{\text{C}}} }}{2}} \right)^{2} } \right] $$

(A-1)

$$ V_{1} = \left[ {n_{\varphi } \left( {n_{\theta } + 1} \right)} \right]\left( {\frac{{\phi_{{\text{C}}} }}{{360^{ \circ } }}} \right){\uppi }\left[ {\left( {R^{{\text{C}}} + \frac{{h^{{\text{C}}} }}{2}} \right)^{2} - \left( {R^{{\text{C}}} - \frac{{h^{{\text{C}}} }}{2}} \right)^{2} } \right]L_{\varphi } $$

(A-2)

$$ V_{2} = {\uppi }\left( {L_{\theta } - 2t_{{\text{C}}} } \right)\left[ {\left( {R^{{\text{C}}} + \frac{{h^{{\text{C}}} }}{2}} \right)^{2} - \left( {R^{{\text{C}}} - \frac{{h^{{\text{C}}} }}{2}} \right)^{2} } \right] $$

(A-3)

$$ V_{3} = N_{\varphi } \left[ {\frac{{1}}{2}\left( {L_{\theta } - 2t_{{\text{C}}} } \right)\left( {R^{{\text{C}}} + {{h^{{\text{C}}} } \mathord{\left/ {\vphantom {{h^{{\text{C}}} } {2 - t_{{\text{C}}} }}} \right. \kern-0pt} {2 - t_{{\text{C}}} }}} \right) - \frac{{\left( {L_{\theta } - 2t_{{\text{C}}} } \right)}}{{2\left( {R^{{\text{C}}} + {{h^{{\text{C}}} } \mathord{\left/ {\vphantom {{h^{{\text{C}}} } {2 - t_{{\text{C}}} }}} \right. \kern-0pt} {2 - t_{{\text{C}}} }}} \right)}}\left( {R^{{\text{C}}} - {{h^{{\text{C}}} } \mathord{\left/ {\vphantom {{h^{{\text{C}}} } {2 + t_{{\text{C}}} }}} \right. \kern-0pt} {2 + t_{{\text{C}}} }}} \right)^{2} } \right]L $$

(A-4)

$$ V_{4} = V_{ * } - \left( {V_{1} + V_{2} + V_{3} } \right) $$

(A-5)

where

$$ \phi_{{\text{C}}} = \frac{{360^{ \circ } }}{{n_{\varphi } }} - 360^{ \circ } \frac{{L_{\theta } }}{{2{\uppi }\left( {R^{{\text{C}}} + {{h^{{\text{C}}} } \mathord{\left/ {\vphantom {{h^{{\text{C}}} } 2}} \right. \kern-0pt} 2}} \right)}},\;L_{\varphi } = \frac{{\left( {L - n_{\theta } L_{\theta } } \right)}}{{\left( {n_{\theta } + 1} \right)}} $$

(A-6)

Appendix B

The mass matrix M in Eq. (23) can be expressed as:

$$ {\varvec{M}} = \int_{0}^{{2{\uppi }}} {\int_{0}^{l} {\left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{\text{I}}} } & {{\varvec{M}}_{{{\text{IO}}}} } & {\mathbf{0}} \\ {{\varvec{M}}_{{{\text{OI}}}} } & {{\varvec{M}}_{{\text{O}}} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\varvec{M}}_{{\text{W}}} } \\ \end{array} } \right]} } {\text{d}}\varphi {\text{d}}\theta $$

(B-1)

${\varvec{M}}_{{\text{I}}} = \left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{\text{I}}}^{uu} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{u\alpha } } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{vv} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{v\beta } } \\ {{\varvec{M}}_{{\text{I}}}^{\alpha u} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{\alpha \alpha } } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{\beta v} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{\beta \beta } } \\ \end{array} } \right]$

$$ {\varvec{M}}_{{\text{I}}} = \left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{\text{I}}}^{uu} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{u\alpha } } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{vv} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{v\beta } } \\ {{\varvec{M}}_{{\text{I}}}^{\alpha u} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{\alpha \alpha } } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{\beta v} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{I}}}^{\beta \beta } } \\ \end{array} } \right],\;{\varvec{M}}_{{\text{O}}} = \left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{\text{O}}}^{uu} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{O}}}^{u\alpha } } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{{\text{O}}}^{vv} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{O}}}^{v\beta } } \\ {{\varvec{M}}_{{\text{O}}}^{\alpha u} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{O}}}^{\alpha \alpha } } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{{\text{O}}}^{\beta v} } & {\mathbf{0}} & {{\varvec{M}}_{{\text{O}}}^{\beta \beta } } \\ \end{array} } \right] $$

(B-2)

$$ {\varvec{M}}_{{{\text{IO}}}} = \left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{{\text{IO}}}}^{uu} } & {\mathbf{0}} & {{\varvec{M}}_{{{\text{IO}}}}^{u\alpha } } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{{{\text{IO}}}}^{vv} } & {\mathbf{0}} & {{\varvec{M}}_{{{\text{IO}}}}^{v\beta } } \\ {{\varvec{M}}_{{{\text{IO}}}}^{\alpha u} } & {\mathbf{0}} & {{\varvec{M}}_{{{\text{IO}}}}^{\alpha \alpha } } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{{{\text{IO}}}}^{\beta v} } & {\mathbf{0}} & {{\varvec{M}}_{{{\text{IO}}}}^{\beta \beta } } \\ \end{array} } \right],\;{\varvec{M}}_{{{\text{OI}}}} = \left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{{\text{OI}}}}^{uu} } & {\mathbf{0}} & {{\varvec{M}}_{{{\text{OI}}}}^{u\alpha } } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{{{\text{OI}}}}^{vv} } & {\mathbf{0}} & {{\varvec{M}}_{{{\text{OI}}}}^{v\beta } } \\ {{\varvec{M}}_{{{\text{OI}}}}^{\alpha u} } & {\mathbf{0}} & {{\varvec{M}}_{{{\text{OI}}}}^{\alpha \alpha } } & {\mathbf{0}} \\ {\mathbf{0}} & {{\varvec{M}}_{{{\text{OI}}}}^{\beta v} } & {\mathbf{0}} & {{\varvec{M}}_{{{\text{OI}}}}^{\beta \beta } } \\ \end{array} } \right] $$

(B-3)

$$ {\varvec{M}}_{{\text{I}}}^{uu} = \left[ {I_{0}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} - \frac{1}{{h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{U}}_{{\text{I}}}^{{\text{T}}} {\varvec{U}}_{{\text{I}}} $$

(B-4)

$$ {\varvec{M}}_{{\text{I}}}^{vv} = \left[ {I_{0}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} - \frac{1}{{h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{V}}_{{\text{I}}}^{{\text{T}}} {\varvec{V}}_{{\text{I}}} $$

(B-5)

$$ {\varvec{M}}_{{\text{I}}}^{\alpha \alpha } = \left[ {I_{2}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{4}\left( {\frac{{h^{{\text{I}}} }}{2}} \right)^{2} I_{0}^{{\text{C}}} R^{{\text{C}}} - \left( {\frac{{h^{{\text{I}}} }}{2}} \right)^{2} \frac{1}{{h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \left( {\frac{{h^{{\text{I}}} }}{2}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} $$

(B-6)

$$ {\varvec{M}}_{{\text{I}}}^{\beta \beta } = \left[ {I_{2}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{4}\left( {\frac{{h^{{\text{I}}} }}{2}} \right)^{2} I_{0}^{{\text{C}}} R^{{\text{C}}} - \left( {\frac{{h^{{\text{I}}} }}{2}} \right)^{2} \frac{1}{{h^{{\text{c}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \left( {\frac{{h^{{\text{I}}} }}{2}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} $$

(B-7)

$$ {\varvec{M}}_{{\text{I}}}^{u\alpha } = {\varvec{M}}_{{\text{I}}}^{\alpha u} = \left[ {2I_{1}^{{\text{I}}} R^{{\text{I}}} + \frac{{h^{{\text{I}}} }}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} - \frac{{h^{{\text{I}}} }}{{2h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{U}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} $$

(B-8)

$$ {\varvec{M}}_{{\text{I}}}^{v\beta } = {\varvec{M}}_{{\text{I}}}^{\beta v} = \left[ {2I_{1}^{{\text{I}}} R^{{\text{I}}} + \frac{{h^{{\text{I}}} }}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} - \frac{{h^{{\text{I}}} }}{{2h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} $$

(B-9)

$$ {\varvec{M}}_{{\text{O}}}^{uu} = \left[ {I_{0}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} + \frac{1}{{h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{U}}_{{\text{O}}}^{{\text{T}}} {\varvec{U}}_{{\text{O}}} $$

(B-10)

$$ {\varvec{M}}_{{\text{O}}}^{vv} = \left[ {I_{0}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} + \frac{1}{{h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{V}}_{{\text{O}}}^{{\text{T}}} {\varvec{V}}_{{\text{O}}} $$

(B-11)

$$ {\varvec{M}}_{{\text{O}}}^{\alpha \alpha } = \left[ {I_{2}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{4}\left( {\frac{{h^{{\text{O}}} }}{2}} \right)^{2} I_{0}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{{h^{{\text{O}}} }}{2}} \right)^{2} \frac{1}{{h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \left( {\frac{{h^{{\text{O}}} }}{2}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} $$

(B-12)

$$ {\varvec{M}}_{{\text{O}}}^{\beta \beta } = \left[ {I_{2}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{4}\left( {\frac{{h^{{\text{O}}} }}{2}} \right)^{2} I_{0}^{{\text{C}}} R^{{\text{C}}} - \left( {\frac{{h^{{\text{O}}} }}{2}} \right)^{2} \frac{1}{{h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} + \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \left( {\frac{{h^{{\text{O}}} }}{2}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{\varPsi}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} $$

(B-13)

$$ {\varvec{M}}_{{\text{O}}}^{u\alpha } = {\varvec{M}}_{{\text{O}}}^{\alpha u} = \left[ {2I_{1}^{{\text{O}}} R^{{\text{O}}} - \frac{{h^{{\text{O}}} }}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} - \frac{{h^{{\text{O}}} }}{{2h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} - \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{U}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} $$

(B-14)

$$ {\varvec{M}}_{{\text{O}}}^{v\beta } = {\varvec{M}}_{{\text{O}}}^{\beta v} = \left[ {2I_{1}^{{\text{O}}} R^{{\text{O}}} - \frac{{h^{{\text{O}}} }}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} - \frac{{h^{{\text{O}}} }}{{2h^{{\text{C}}} }}I_{1}^{{\text{C}}} R^{{\text{C}}} - \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} $$

(B-15)

$$ {\varvec{M}}_{{{\text{IO}}}}^{uu} = {\varvec{M}}_{{{\text{OI}}}}^{uu} = \left[ {\frac{1}{2}I_{0}^{{\text{C}}} R^{{\text{C}}} - 2\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{U}}_{{\text{I}}}^{{\text{T}}} {\varvec{U}}_{{\text{O}}} ,\;{\varvec{M}}_{{{\text{IO}}}}^{vv} = {\varvec{M}}_{{{\text{OI}}}}^{vv} = \left[ {\frac{1}{2}I_{0}^{{\text{C}}} R^{{\text{C}}} - 2\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{V}}_{{\text{I}}}^{{\text{T}}} {\varvec{V}}_{{\text{O}}} $$

(B-16)

$$ {\varvec{M}}_{{{\text{IO}}}}^{\alpha \alpha } = {\varvec{M}}_{{{\text{OI}}}}^{\alpha \alpha } = \left[ {\frac{{h^{{\text{O}}} h^{{\text{I}}} }}{8}I_{0}^{{\text{C}}} R^{{\text{C}}} + \frac{{h^{{\text{O}}} h^{{\text{I}}} }}{2}\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} $$

(B-17)

$$ {\varvec{M}}_{{{\text{IO}}}}^{\beta \beta } = {\varvec{M}}_{{{\text{OI}}}}^{\beta \beta } = \left[ { - \frac{{h^{{\text{O}}} h^{{\text{I}}} }}{8}I_{0}^{{\text{C}}} R^{{\text{C}}} + \frac{{h^{{\text{O}}} h^{{\text{I}}} }}{2}\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} $$

(B-18)

$$ {\varvec{M}}_{{{\text{IO}}}}^{u\alpha } = {\varvec{M}}_{{{\text{OI}}}}^{\alpha u} = \left[ { - \frac{{h^{{\text{O}}} }}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} + 2\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{U}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} $$

(B-19)

$$ {\varvec{M}}_{{{\text{OI}}}}^{u\alpha } = {\varvec{M}}_{{{\text{IO}}}}^{\alpha u} = \left[ {\frac{{h^{{\text{I}}} }}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} - 2\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{U}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} ,\;{\varvec{M}}_{{\text{W}}} = \left( {I_{0}^{{\text{O}}} R^{{\text{O}}} + I_{0}^{{\text{C}}} R^{{\text{c}}} + I_{0}^{{\text{I}}} R^{{\text{I}}} } \right){\varvec{W}}^{{\text{T}}} {\varvec{W}} $$

(B-20)

$$ {\varvec{M}}_{{{\text{IO}}}}^{v\beta } = {\varvec{M}}_{{{\text{OI}}}}^{\beta v} = \left[ { - \frac{{h^{{\text{I}}} }}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} + 2\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} $$

(B-21)

$$ {\varvec{M}}_{{{\text{OI}}}}^{v\beta } = {\varvec{M}}_{{{\text{IO}}}}^{\beta v} = \left[ {\frac{{h^{{\text{I}}} }}{4}I_{0}^{{\text{C}}} R^{{\text{C}}} - 2\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} I_{2}^{{\text{C}}} R^{{\text{C}}} } \right]{\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} $$

(B-22)

$$ {\varvec{M}}_{{\text{W}}} = \left( {I_{0}^{{\text{O}}} R^{{\text{O}}} + I_{0}^{{\text{C}}} R^{{\text{C}}} + I_{0}^{{\text{I}}} R^{{\text{I}}} } \right){\varvec{W}}^{{\text{T}}} {\varvec{W}} $$

(B-23)

where

$$ {\varvec{U}}_{{\text{b}}} = {\varvec{U}}_{{\text{t}}} = {\varvec{C}}_{\theta } \otimes {\varvec{J}}_{\varphi } , \, {\varvec{V}}_{{\text{b}}} = {\varvec{V}}_{{\text{t}}} = {\varvec{S}}_{\theta } \otimes {\varvec{J}}_{\varphi } , \,{\varvec{\varPhi}}_{{\text{b}}} ={\varvec{\varPhi}}_{{\text{t}}} = {\varvec{C}}_{\theta } \otimes {\varvec{J}}_{\varphi } , \,{\varvec{\varPsi}}_{{\text{b}}} ={\varvec{\varPsi}}_{{\text{t}}} = {\varvec{S}}_{\theta } \otimes {\varvec{J}}_{\varphi } , \, {\varvec{W}} = {\varvec{C}}_{\theta } \otimes {\varvec{J}}_{\varphi } $$

(B-24)

$$ {\varvec{J}}_{\varphi } = \left[ {J_{0} , \, J_{1} , \cdots {, }J_{m} , \cdots {, }J_{M} } \right] $$

(B-25)

$$ {\varvec{C}}_{\theta } = \left[ {{\text{cos}}\left( {0\theta } \right),{\text{ cos}}\left( {1\theta } \right), \, \cdots {\text{, cos}}\left( {n\theta } \right), \, \cdots {\text{, cos}}\left( {N\theta } \right) \, } \right]^{{\text{T}}} $$

(B-26)

$$ {\varvec{S}}_{\theta } = \left[ {\sin \left( {0\theta } \right), \, \sin \left( {1\theta } \right), \, \cdots {, }\sin \left( {n\theta } \right), \, \cdots {, }\sin \left( {N\theta } \right) \, } \right]^{{\text{T}}} $$

(B-27)

The stiffness matrix K in Eq. (23) can be indicated as:

$$ {\varvec{K}} = \int_{0}^{{2{\uppi }}} {\int_{0}^{l} {\left[ {\begin{array}{*{20}c} {{\varvec{K}}_{{\text{I}}} } & {{\varvec{K}}_{{{\text{IO}}}} } & {{\varvec{K}}_{{{\text{IW}}}} } \\ {{\varvec{K}}_{{{\text{OI}}}} } & {{\varvec{K}}_{{\text{O}}} } & {{\varvec{K}}_{{{\text{OW}}}} } \\ {{\varvec{K}}_{{{\text{WI}}}} } & {{\varvec{K}}_{{{\text{WO}}}} } & {{\varvec{K}}_{{\text{W}}} } \\ \end{array} } \right]} } {\text{d}}\varphi {\text{d}}\theta $$

(B-28)

$$ {\varvec{K}}_{i} = \left[ {\begin{array}{*{20}c} {{\varvec{K}}_{i}^{uu} + \overline{\user2{K}}_{i}^{uu} } & {{\varvec{K}}_{i}^{uv} + \overline{\user2{K}}_{i}^{uv} } & {{\varvec{K}}_{i}^{u\alpha } + \overline{\user2{K}}_{i}^{u\alpha } } & {{\varvec{K}}_{i}^{u\beta } + \overline{\user2{K}}_{i}^{u\beta } } \\ {{\varvec{K}}_{i}^{vu} + \overline{\user2{K}}_{i}^{vu} } & {{\varvec{K}}_{i}^{vv} + \overline{\user2{K}}_{i}^{vv} } & {{\varvec{K}}_{i}^{v\alpha } + \overline{\user2{K}}_{i}^{v\alpha } } & {{\varvec{K}}_{i}^{v\beta } + \overline{\user2{K}}_{i}^{v\beta } } \\ {{\varvec{K}}_{i}^{\alpha u} + \overline{\user2{K}}_{i}^{\alpha u} } & {{\varvec{K}}_{i}^{\alpha v} + \overline{\user2{K}}_{i}^{\alpha v} } & {{\varvec{K}}_{i}^{\alpha \alpha } + \overline{\user2{K}}_{i}^{\alpha \alpha } } & {{\varvec{K}}_{i}^{\alpha \beta } + \overline{\user2{K}}_{i}^{\alpha \beta } } \\ {{\varvec{K}}_{i}^{\beta u} + \overline{\user2{K}}_{i}^{\beta u} } & {{\varvec{K}}_{i}^{\beta v} + \overline{\user2{K}}_{i}^{\beta v} } & {{\varvec{K}}_{i}^{\beta \phi } + \overline{\user2{K}}_{i}^{\beta \phi } } & {{\varvec{K}}_{i}^{\beta \beta } + \overline{\user2{K}}_{i}^{\beta \beta } } \\ \end{array} } \right]\left( {i = {\text{I}},{\text{ O}},{\text{ IO}},{\text{ OI}}} \right) $$

(B-29)

$$\begin{aligned}&{\varvec{K}}_{{j{\text{W}}}} = \left[ {\begin{array}{*{20}c} {{\varvec{K}}_{{j{\text{W}}}}^{uw} } & {{\varvec{K}}_{{j{\text{W}}}}^{vw} } & {{\varvec{K}}_{{j{\text{W}}}}^{\alpha w} } & {{\varvec{K}}_{{j{\text{W}}}}^{\beta w} } \\ \end{array} } \right]^{{\text{T}}} \left( {j = {\text{I}},{\text{ O}}} \right),\\ &{\varvec{K}}_{{{\text{W}}j}} = \left[ {\begin{array}{*{20}c} {{\varvec{K}}_{{{\text{W}}j}}^{wu} } & {{\varvec{K}}_{{{\text{W}}j}}^{wv} } & {{\varvec{K}}_{{{\text{W}}j}}^{w\alpha } } & {{\varvec{K}}_{{{\text{W}}j}}^{w\beta } } \\ \end{array} } \right]^{{\text{T}}} \left( {j = {\text{I}},{\text{ O}}} \right)\end{aligned}$$

(B-30)

$$ \begin{gathered} {\varvec{K}}_{{\text{I}}}^{uu} = \left[ {T_{11}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{4}T_{11}^{{\text{C}}} R^{{\text{C}}} - C_{11}^{{\text{C}}} \frac{{R^{{\text{C}}} }}{{h^{{\text{C}}} }} + F_{11}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{b}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{U}}_{{\text{b}}} }}{\partial \varphi } + \left[ {2T_{16}^{{\text{I}}} + \frac{1}{2}T_{16}^{{\text{C}}} - 2C_{16}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + 2F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right] \\ \times \frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{U}}_{{\text{I}}} }}{\partial \varphi } + \left[ {T_{66}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{4}T_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} - C_{66}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} R^{{\text{C}}} }} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{1}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{U}}_{{\text{I}}} }}{\partial \theta } + S_{r} A_{55}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} {\varvec{U}}_{{\text{I}}}^{{\text{T}}} {\varvec{U}}_{{\text{I}}} \\ \end{gathered} $$

(B-31)

$$ \begin{gathered} {\varvec{K}}_{{\text{I}}}^{vv} = \left[ {T_{22}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{4}A_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} - C_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} h^{{\text{C}}} }} + F_{22}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{1}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \theta } \\ + \left[ {2T_{26}^{{\text{I}}} + \frac{1}{2}T_{26}^{{\text{C}}} - 2C_{26}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + 2F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right] \times \frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \varphi } \\ + \left[ {T_{66}^{{\text{I}}} R^{{\text{I}}} + T_{66}^{{\text{C}}} \frac{{R^{{\text{C}}} }}{4} - C_{66}^{{\text{C}}} \frac{{r^{{\text{C}}} }}{{h^{{\text{C}}} }} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \varphi } \\ + \left[ {S_{{\text{r}}} T_{44}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + S_{r} T_{44}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + \frac{1}{2}S_{r} T_{44}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + 2S_{r} T_{44}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]{\varvec{V}}_{{\text{I}}}^{{\text{T}}} {\varvec{V}}_{{\text{I}}} \\ \end{gathered} $$

(B-32)

$$ \begin{gathered} {\varvec{K}}_{{\text{I}}}^{\alpha \alpha } = \left[ {F_{11}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{16}T_{11}^{{\text{C}}} R^{{\text{C}}} \left( {h^{{\text{I}}} } \right)^{2} - \frac{1}{4}C_{11}^{{\text{C}}} \frac{{R^{{\text{C}}} }}{{h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} + \frac{1}{4}F_{11}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi } + S_{{\text{r}}} T_{55}^{{\text{I}}} R^{{\text{I}}}{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} \\ + \left[ {2F_{16}^{{\text{I}}} + \frac{1}{8}T_{16}^{{\text{C}}} \left( {h^{{\text{I}}} } \right)^{2} - \frac{1}{2}C_{16}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} + \frac{1}{2}F_{16}^{{\text{C}}} \left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi } + \frac{1}{4}S_{{\text{r}}} T_{55}^{{\text{C}}} \left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}}{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} \\ + \left[ {F_{66}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{16}T_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} - \frac{1}{4}C_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} + \frac{1}{4}F_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \theta } \\ \end{gathered} $$

(B-33)

$$ \begin{gathered} {\varvec{K}}_{{\text{I}}}^{\beta \beta } = \left[ {F_{22}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{16}T_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} - \frac{1}{4}C_{22}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} R^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} + \frac{1}{4}F_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } \\ + \left[ {2F_{26}^{{\text{I}}} + \frac{1}{8}T_{26}^{{\text{C}}} \left( {h^{{\text{I}}} } \right)^{2} - \frac{1}{2}C_{26}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} + \frac{1}{2}F_{26}^{{\text{C}}} \left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } \\ + \left[ {F_{66}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{16}T_{66}^{{\text{C}}} \left( {h^{{\text{I}}} } \right)^{2} R^{{\text{C}}} - \frac{1}{4}C_{66}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} R^{{\text{C}}} + \frac{1}{4}F_{66}^{{\text{C}}} \left( {\frac{{h^{{\text{C}}} }}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } \\ + \left[ {S_{{\text{r}}} T_{44}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{8}S_{{\text{r}}} \frac{1}{{R^{{\text{C}}} }}T_{44}^{{\text{C}}} \left( {h^{{\text{I}}} } \right)^{2} + \frac{1}{2}S_{{\text{r}}} T_{44}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} + \frac{1}{4}S_{{\text{r}}} \frac{1}{{h^{{\text{C}}} }}T_{44}^{{\text{C}}} \left( {h^{{\text{I}}} } \right)^{2} } \right]{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ \end{gathered} $$

(B-34)

$$ \begin{gathered} {\varvec{K}}_{{\text{I}}}^{uv} = {\varvec{K}}_{{\text{I}}}^{vu} = \left[ {2T_{12}^{{\text{I}}} + \frac{1}{2}T_{12}^{{\text{C}}} - 2C_{12}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + 2F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \theta } + \left[ {S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]{\varvec{U}}_{{\text{I}}}^{{\text{T}}} {\varvec{V}}_{{\text{I}}} \\ + \left[ {2T_{16}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{2}T_{16}^{{\text{C}}} R^{{\text{C}}} - 2C_{16}^{{\text{C}}} \frac{{R^{{\text{C}}} }}{{h^{{\text{C}}} }} + 2F_{16}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \varphi } \\ + \left[ {2T_{26}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{2}T_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} - 2C_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} h^{{\text{C}}} }} + 2F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{1}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \theta } \\ + \left[ {2T_{66}^{{\text{I}}} + \frac{1}{2}T_{66}^{{\text{C}}} - 2C_{66}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + 2F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \varphi } \\ \end{gathered} $$

(B-35)

$$ \begin{gathered} {\varvec{K}}_{{\text{I}}}^{u\alpha } = {\varvec{K}}_{{\text{I}}}^{\alpha u} = \left[ {2C_{11}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{4}A_{11}^{{\text{C}}} h^{{\text{I}}} R^{{\text{C}}} - C_{11}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} + F_{11}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi } + S_{{\text{r}}} T_{55}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} T^{{\text{C}}} {\varvec{U}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} \\ + \left[ {2C_{16}^{{\text{I}}} + \frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{I}}} - \frac{1}{2}C_{16}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }} + F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\left( {\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \theta } + \frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi }} \right) \\ + \left[ {2C_{66}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{4}T_{66}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }} - C_{66}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \theta } \\ \end{gathered} $$

(B-36)

$$ \begin{gathered} {\varvec{K}}_{{\text{I}}}^{u\beta } = {\varvec{K}}_{{\text{I}}}^{\beta u} = \left[ {2C_{12}^{{\text{I}}} + \frac{1}{4}T_{12}^{{\text{C}}} h^{{\text{I}}} - C_{12}^{{\text{c}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} + F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } + \frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}{\varvec{U}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ + \left[ {2C_{16}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{I}}} R^{{\text{C}}} - C_{16}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} + F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } - S_{{\text{r}}} T_{45}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} {\varvec{U}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ + \left[ {2C_{26}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{4}T_{26}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }} - C_{26}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} + F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } \\ + \left[ {2C_{66}^{{\text{I}}} + \frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{I}}} - C_{66}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } \\ \end{gathered} $$

(B-37)

$$ \begin{gathered} {\varvec{K}}_{{\text{I}}}^{v\alpha } = {\varvec{K}}_{{\text{I}}}^{\alpha v} = \left[ {2C_{12}^{{\text{I}}} + \frac{1}{4}T_{12}^{{\text{C}}} h^{{\text{I}}} - C_{12}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }} + F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi } - 2S_{{\text{r}}} T_{45}^{{\text{I}}} {\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} \\ + \left[ {2C_{16}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{I}}} R^{{\text{C}}} - C_{16}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} + F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi } + \frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}{\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} \\ + \left[ {2C_{26}^{{\text{I}}} \frac{1}{{R^{{\text{b}}} }} + \frac{1}{4}T_{26}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }} - C_{26}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} + F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \theta } + S_{{\text{r}}} A_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} {\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} \\ + \left[ {2C_{66}^{{\text{I}}} + \frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{I}}} - C_{66}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \theta } \\ \end{gathered} $$

(B-38)

$$ \begin{gathered} {\varvec{K}}_{{\text{I}}}^{v\beta } = {\varvec{K}}_{{\text{I}}}^{\beta v} = \left[ {2C_{22}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{4}T_{22}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }} - C_{22}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} + F_{22}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } - 2S_{{\text{r}}} T_{44}^{{\text{I}}} {\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ + \left[ {C_{26}^{{\text{I}}} + \frac{1}{4}T_{26}^{{\text{C}}} h^{{\text{I}}} - C_{26}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }} + F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\left( {\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } + \frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi }} \right) + S_{{\text{r}}} T_{44}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{I}}} {\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ + \left[ {2C_{66}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{I}}} R^{{\text{C}}} - C_{66}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} + D_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } + \left( {\frac{1}{4}S_{{\text{r}}} T_{44}^{{\text{c}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }} + S_{{\text{r}}} T_{44}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right) \\ \end{gathered} $$

(B-39)

$$ \begin{gathered} {\varvec{K}}_{{\text{I}}}^{\alpha \beta } = {\varvec{K}}_{{\text{I}}}^{\alpha \beta } = \left[ {2F_{12}^{{\text{I}}} + \frac{1}{8}T_{12}^{{\text{C}}} \left( {h^{{\text{I}}} } \right)^{2} - \frac{1}{2}C_{12}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} + \frac{1}{2}F_{12}^{{\text{C}}} \left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } - \frac{1}{4}S_{{\text{r}}} A_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2}{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ + \left[ {2F_{16}^{{\text{I}}} R^{{\text{I}}} + \frac{1}{8}T_{16}^{{\text{C}}} R^{{\text{C}}} \left( {h^{{\text{I}}} } \right)^{2} - \frac{1}{2}C_{16}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} R^{{\text{C}}} + \frac{1}{2}F_{16}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } \\ + \left[ {2F_{26}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{8}T_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} - \frac{1}{2}C_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} + \frac{1}{2}F_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } \\ + \left[ {2F_{66}^{{\text{I}}} + \frac{1}{8}T_{66}^{{\text{C}}} \left( {h^{{\text{I}}} } \right)^{2} - \frac{1}{2}C_{66}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} + \frac{1}{2}F_{66}^{{\text{C}}} \left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } + \frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}} \right)^{2}{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ \end{gathered} $$

(B-40)

$$ \begin{gathered} {\varvec{K}}_{{\text{O}}}^{uu} = \left[ {T_{11}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{4}T_{11}^{{\text{C}}} R^{{\text{C}}} + C_{11}^{{\text{C}}} \frac{{R^{{\text{C}}} }}{{h^{{\text{C}}} }} + F_{11}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{U}}_{{\text{O}}} }}{\partial \varphi } \\ + \left[ {2T_{16}^{{\text{O}}} + \frac{1}{2}T_{16}^{{\text{C}}} + 2C_{16}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + 2F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right] \times \frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{U}}_{{\text{O}}} }}{\partial \varphi } + S_{r} T_{55}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} {\varvec{U}}_{{\text{O}}}^{{\text{T}}} {\varvec{U}}_{{\text{O}}} \\ + \left[ {T_{66}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} + \frac{1}{4}T_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + B_{66}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} R^{{\text{C}}} }} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{1}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{U}}_{{\text{O}}} }}{\partial \theta } \\ \end{gathered} $$

(B-41)

$$ \begin{gathered} {\varvec{K}}_{{\text{O}}}^{vv} = \left[ {T_{22}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} + \frac{1}{4}T_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + C_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} h^{{\text{C}}} }} + F_{22}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{1}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \theta } \\ + \left[ {2T_{26}^{{\text{O}}} + \frac{1}{2}T_{26}^{{\text{C}}} + 2C_{26}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + 2F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right] \times \frac{{\partial {\varvec{V}}_{{\text{t}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{t}}} }}{\partial \varphi } \\ + \left[ {T_{66}^{{\text{O}}} R^{{\text{O}}} + T_{66}^{{\text{C}}} \frac{{R^{{\text{C}}} }}{4} + C_{66}^{{\text{C}}} \frac{{R^{{\text{C}}} }}{{h^{{\text{C}}} }} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \varphi } \\ + \left[ {S_{{\text{r}}} T_{44}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} - S_{r} T_{44}^{{\text{C}}} \frac{{R^{{\text{C}}} }}{{h^{{\text{C}}} }} + \frac{1}{2}S_{r} T_{44}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + 2S_{r} T_{44}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]{\varvec{V}}_{{\text{O}}}^{{\text{T}}} {\varvec{V}}_{{\text{O}}} \\ \end{gathered} $$

(B-42)

$$ \begin{gathered} {\varvec{K}}_{{\text{O}}}^{\alpha \alpha } = \left[ {F_{11}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{16}T_{11}^{{\text{C}}} R^{{\text{C}}} \left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{4}C_{11}^{{\text{C}}} \frac{{R^{{\text{C}}} }}{{h^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{4}F_{11}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi } + S_{{\text{r}}} T_{55}^{{\text{O}}} R^{{\text{O}}}{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} \\ + \left[ {2F_{16}^{{\text{O}}} + \frac{1}{8}T_{16}^{{\text{C}}} \left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}C_{16}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{I}}} } \right)^{2} + \frac{1}{2}F_{16}^{{\text{C}}} \left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi } + \frac{1}{4}S_{{\text{r}}} T_{55}^{{\text{C}}} \left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}}{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} \\ + \left[ {F_{66}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} + \frac{1}{16}T_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{4}C_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} h^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{4}F_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \theta } \\ \end{gathered} $$

(B-43)

$$ \begin{gathered} {\varvec{K}}_{{\text{O}}}^{\beta \beta } = \left[ {F_{22}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} + \frac{1}{16}T_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{4}C_{22}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} R^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{4}F_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } \\ + \left[ {2D_{26}^{{\text{O}}} + \frac{1}{8}A_{26}^{{\text{C}}} \left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}T_{26}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}F_{26}^{{\text{C}}} \left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } \\ + \left[ {F_{66}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{16}T_{66}^{{\text{C}}} \left( {h^{{\text{O}}} } \right)^{2} R^{{\text{C}}} + \frac{1}{4}C_{66}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} R^{{\text{C}}} + \frac{1}{4}F_{66}^{{\text{C}}} \left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } \\ + \left[ {S_{{\text{r}}} T_{44}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{8}S_{{\text{r}}} \frac{1}{{R^{{\text{C}}} }}T_{44}^{{\text{C}}} \left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}S_{{\text{r}}} T_{44}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} - \frac{1}{4}S_{{\text{r}}} \frac{1}{{h^{{\text{C}}} }}T_{44}^{{\text{C}}} \left( {h^{{\text{O}}} } \right)^{2} } \right]{\varvec{\varPsi}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} \\ \end{gathered} $$

(B-44)

$$ \begin{gathered} {\varvec{K}}_{{\text{O}}}^{uv} = {\varvec{K}}_{{\text{O}}}^{vu} = \left[ {2T_{12}^{{\text{O}}} + \frac{1}{2}T_{12}^{{\text{C}}} + 2C_{12}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + 2F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \theta } + \left[ {S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} - S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]{\varvec{U}}_{{\text{O}}}^{{\text{T}}} {\varvec{V}}_{{\text{O}}} \\ + \left[ {2T_{16}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{2}T_{16}^{{\text{C}}} R^{{\text{C}}} + 2C_{16}^{{\text{C}}} \frac{{R^{{\text{C}}} }}{{h^{{\text{C}}} }} + 2F_{16}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \varphi } \\ + \left[ {2T_{26}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} + \frac{1}{2}T_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + 2C_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} h^{{\text{C}}} }} + 2F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{1}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \theta } \\ + \left[ {2T_{66}^{{\text{O}}} + \frac{1}{2}T_{66}^{{\text{C}}} + 2C_{66}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + 2F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \varphi } \\ \end{gathered} $$

(B-45)

$$ \begin{gathered} {\varvec{K}}_{{\text{O}}}^{u\alpha } = {\varvec{K}}_{{\text{O}}}^{\alpha u} = \left[ {2C_{11}^{{\text{O}}} R^{{\text{O}}} - \frac{1}{4}T_{11}^{{\text{C}}} h^{{\text{O}}} R^{{\text{C}}} - C_{11}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} - F_{11}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi } \\ + \left[ {2C_{16}^{{\text{O}}} - \frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{O}}} - \frac{1}{2}C_{16}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }} - F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\left( {\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \theta } + \frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi }} \right) \\ + \left[ {2C_{66}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} - \frac{1}{4}T_{66}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }} - C_{66}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} - F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{c}}} }}} \right)^{2} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \theta } - S_{{\text{r}}} A_{55}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} {\varvec{U}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} \\ \end{gathered} $$

(B-46)

$$ \begin{gathered} {\varvec{K}}_{{\text{O}}}^{u\beta } = {\varvec{K}}_{{\text{O}}}^{\beta u} = \left[ {2C_{12}^{{\text{O}}} - \frac{1}{4}T_{12}^{{\text{C}}} h^{{\text{O}}} - C_{12}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} - F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } - \frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}{\varvec{U}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} \\ + \left[ {2C_{16}^{{\text{O}}} R^{{\text{O}}} - \frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{O}}} R^{{\text{C}}} - C_{16}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} - F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } \\ - S_{{\text{r}}} A_{45}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} {\varvec{U}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} + \left[ {2C_{26}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} - \frac{1}{4}F_{26}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }} - C_{26}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} - F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } \\ + \left[ {2C_{66}^{{\text{O}}} - \frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{O}}} - C_{66}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }} - F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } \\ \end{gathered} $$

(B-47)

$$ \begin{gathered} {\varvec{K}}_{{\text{O}}}^{v\alpha } = {\varvec{K}}_{{\text{O}}}^{\alpha v} = \left[ {2C_{12}^{{\text{O}}} - \frac{1}{4}T_{12}^{{\text{C}}} h^{{\text{O}}} - C_{12}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }} - F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi } - 2S_{{\text{r}}} T_{45}^{{\text{O}}} {\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} \\ + \left[ {2C_{16}^{{\text{O}}} R^{{\text{O}}} - \frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{O}}} R^{{\text{C}}} - C_{16}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} - F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} r^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi } + \frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}{\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} \\ + \left[ {2C_{26}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} - \frac{1}{4}T_{26}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }} - C_{26}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} - F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \theta } - S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} {\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} \\ + \left[ {2C_{66}^{{\text{O}}} - \frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{O}}} - C_{66}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }} - F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \theta } \\ \end{gathered} $$

(B-48)

$$ \begin{gathered} {\varvec{K}}_{{\text{O}}}^{v\beta } = {\varvec{K}}_{{\text{O}}}^{\beta v} = \left[ {2C_{22}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} - \frac{1}{4}T_{22}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }} - C_{22}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} - F_{22}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } - 2S_{{\text{r}}} T_{44}^{{\text{O}}} {\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} \\ + \left[ {C_{26}^{{\text{O}}} - \frac{1}{4}T_{26}^{{\text{C}}} h^{{\text{O}}} - C_{26}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }} - D_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\left( {\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } + \frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi }} \right) \\ + \left[ {2C_{66}^{{\text{O}}} R^{{\text{O}}} - \frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{O}}} R^{{\text{C}}} - C_{66}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} - F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } \\ + \left( { - \frac{1}{4}S_{{\text{r}}} T_{44}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }} + S_{{\text{r}}} T_{44}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right){\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} - S_{{\text{r}}} T_{44}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} {\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} \\ \end{gathered} $$

(B-49)

$$ \begin{gathered} {\varvec{K}}_{{\text{O}}}^{\beta \alpha } = \left[ {2F_{12}^{{\text{O}}} + \frac{1}{8}T_{12}^{{\text{C}}} \left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}C_{12}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}F_{12}^{{\text{C}}} \left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } - \frac{1}{4}S_{{\text{r}}} A_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2}{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} \\ + \left[ {2F_{16}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{8}T_{16}^{{\text{C}}} R^{{\text{C}}} \left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}C_{16}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} R^{{\text{C}}} + \frac{1}{2}F_{16}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } \\ + \left[ {2F_{26}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} + \frac{1}{8}T_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}C_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} h^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}F_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } \\ + \left[ {2F_{66}^{{\text{O}}} + \frac{1}{8}T_{66}^{{\text{C}}} \left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}C_{66}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}\left( {h^{{\text{O}}} } \right)^{2} + \frac{1}{2}F_{66}^{{\text{C}}} \left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } + \frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}} \right)^{2}{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} \\ \end{gathered} $$

(B-50)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{uu} = {\varvec{K}}_{{{\text{OI}}}}^{uu} = \left[ {\frac{1}{2}T_{11}^{{\text{C}}} R^{{\text{C}}} - F_{11}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{U}}_{{\text{O}}} }}{\partial \varphi } + \left[ {T_{16}^{{\text{C}}} - 4F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right] \times \frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{U}}_{{\text{O}}} }}{\partial \varphi } \\ + \left[ {\frac{1}{2}T_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} - 2F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{1}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{U}}_{{\text{O}}} }}{\partial \theta } - 2S_{r} T_{55}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} {\varvec{U}}_{{\text{O}}}^{{\text{T}}} {\varvec{U}}_{{\text{I}}} \\ \end{gathered} $$

(B-51)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{vv} = {\varvec{K}}_{{{\text{OI}}}}^{vv} = \left[ {\frac{1}{2}T_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} - 2F_{22}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{1}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \theta } + \left[ {T_{26}^{{\text{C}}} - 4T_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right] \times \frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \varphi } \\ + \left[ {\frac{1}{2}T_{66}^{{\text{C}}} R^{{\text{C}}} - 2F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \varphi } + \left[ {\frac{1}{2}S_{r} T_{44}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} - 2S_{r} T_{44}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} R^{{\text{C}}} } \right]{\varvec{V}}_{{\text{I}}}^{{\text{T}}} {\varvec{V}}_{{\text{O}}} \\ \end{gathered} $$

(B-52)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{\alpha \alpha } = {\varvec{K}}_{{{\text{OI}}}}^{\beta \beta } = \left[ { - \frac{1}{8}T_{11}^{{\text{C}}} R^{{\text{C}}} h^{{\text{I}}} h^{{\text{O}}} + \frac{1}{2}F_{11}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} h^{{\text{O}}} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi } + \left[ { - \frac{1}{4}A_{16}^{{\text{C}}} h^{{\text{I}}} h^{{\text{O}}} + F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} h^{{\text{O}}} } \right] \\ \times \frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi } + \left[ { - \frac{1}{8}T_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}h^{{\text{I}}} h^{{\text{O}}} + \frac{1}{2}F_{66}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} h^{{\text{O}}} } \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \theta } + \frac{1}{2}S_{{\text{r}}} T_{55}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} h^{{\text{O}}}{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} \\ \end{gathered} $$

(B-53)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{\beta \beta } = {\varvec{K}}_{{{\text{OI}}}}^{\beta \beta } = \left[ { - \frac{1}{8}T_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + \frac{1}{2}F_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]h^{{\text{I}}} h^{{\text{O}}} \frac{{\partial{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } + \left[ { - \frac{1}{4}T_{26}^{{\text{C}}} + F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]h^{{\text{I}}} h^{{\text{O}}} \frac{{\partial{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } \hfill \\ + \left[ { - \frac{1}{8}T_{66}^{{\text{C}}} R^{{\text{C}}} + \frac{1}{2}F_{66}^{{\text{C}}} R^{{\text{C}}} } \right]h^{{\text{I}}} h^{{\text{O}}} \frac{{\partial{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } + \left[ { - \frac{1}{8}S_{{\text{r}}} A_{44}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + \frac{1}{2}S_{{\text{r}}} T_{44}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]h^{{\text{I}}} h^{{\text{O}}}{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} \hfill \\ \end{gathered} $$

(B-54)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{uv} = {\varvec{K}}_{{{\text{OI}}}}^{vu} = \left[ {\frac{1}{2}T_{12}^{{\text{C}}} - 2F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \theta } + \left[ {S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} - S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]{\varvec{U}}_{{\text{I}}}^{{\text{T}}} {\varvec{V}}_{{\text{O}}} \\ + \left[ {\frac{1}{2}T_{16}^{{\text{C}}} R^{{\text{C}}} - 2F_{16}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \varphi } + \left[ {\frac{1}{2}T_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} - 2F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{1}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \theta } \\ + \left[ {\frac{1}{2}T_{66}^{{\text{C}}} - 2F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{O}}} }}{\partial \varphi } \\ \end{gathered} $$

(B-55)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{vu} = {\varvec{K}}_{{{\text{OI}}}}^{uv} = \left[ {\frac{1}{2}T_{12}^{{\text{C}}} - 2F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \theta } + \left[ { - S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} - S_{{\text{r}}} A_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]{\varvec{U}}_{{\text{O}}}^{{\text{T}}} {\varvec{V}}_{{\text{I}}} \\ + \left[ {\frac{1}{2}T_{16}^{{\text{C}}} R^{{\text{C}}} - 2F_{16}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \varphi } + \left[ {\frac{1}{2}T_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} - 2F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{1}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \theta } \\ + \left[ {\frac{1}{2}T_{66}^{{\text{C}}} - 2F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{V}}_{{\text{I}}} }}{\partial \varphi } \\ \end{gathered} $$

(B-56)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{u\alpha } = {\varvec{K}}_{{{\text{OI}}}}^{\alpha u} = \left[ { - \frac{1}{4}T_{11}^{{\text{C}}} h^{{\text{O}}} R^{{\text{C}}} - \frac{1}{2}C_{11}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} + F_{11}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi } + S_{{\text{r}}} T_{55}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} {\varvec{U}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} \\ + \left[ { - \frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{O}}} - \frac{1}{2}C_{16}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }} + F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\left( {\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \theta } + \frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi }} \right) \\ + \left[ { - \frac{1}{4}T_{66}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }} - C_{66}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \theta } \\ \end{gathered} $$

(B-57)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{\alpha u} = {\varvec{K}}_{{{\text{OI}}}}^{u\alpha } = \left[ {\frac{1}{4}T_{11}^{{\text{C}}} h^{{\text{I}}} R^{{\text{C}}} + \frac{1}{2}C_{11}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} - F_{11}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi } - S_{{\text{r}}} T_{55}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} {\varvec{U}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} \\ + \left[ {\frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{O}}} + \frac{1}{2}C_{16}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }} - F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\left( {\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \theta } + \frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi }} \right) \\ + \left[ { - \frac{1}{4}T_{66}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }} + C_{66}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} - F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \theta } \\ \end{gathered} $$

(B-58)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{u\beta } = {\varvec{K}}_{{{\text{OI}}}}^{\beta u} = \left[ { - \frac{1}{4}T_{12}^{{\text{C}}} h^{{\text{O}}} + F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } - \frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}{\varvec{U}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} \\ + \left[ { - \frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{O}}} R^{{\text{C}}} + F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } + S_{{\text{r}}} A_{45}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} {\varvec{U}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} \\ + \left[ { - \frac{1}{4}T_{26}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }} + F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } + \left[ { - \frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{O}}} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } \\ \end{gathered} $$

(B-59)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{\beta u} = {\varvec{K}}_{{{\text{OI}}}}^{u\beta } = \left[ {\frac{1}{4}T_{12}^{{\text{C}}} h^{{\text{I}}} - F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } + \frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}{\varvec{U}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ + \left[ {\frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{I}}} R^{{\text{C}}} - F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } - S_{{\text{r}}} T_{45}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{O}}} {\varvec{U}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ + \left[ {\frac{1}{4}T_{26}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }} - F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } + \left[ { - \frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{I}}} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } \\ \end{gathered} $$

(B-60)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{v\alpha } = {\varvec{K}}_{{{\text{OI}}}}^{\alpha v} = \left[ { - \frac{1}{4}T_{12}^{{\text{C}}} h^{{\text{O}}} + F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \varphi } + \left[ { - \frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{O}}} R^{{\text{C}}} + F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi } \\ + \left[ { - \frac{1}{4}T_{26}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }} + F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \theta } + \left[ {S_{{\text{r}}} A_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} + \frac{1}{2}S_{{\text{r}}} A_{45}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}{\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} } \right]{\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} \\ + \left[ { - \frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{O}}} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}} }}{\partial \theta } \\ \end{gathered} $$

(B-61)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{\alpha v} = {\varvec{K}}_{{{\text{OI}}}}^{v\alpha } = \left[ {\frac{1}{4}T_{12}^{{\text{C}}} h^{{\text{I}}} - F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi } + \left[ { - \frac{1}{4}T_{16}^{{\text{C}}} h^{{\text{I}}} R^{{\text{C}}} + F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \varphi } \\ + \left[ {\frac{1}{4}T_{26}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }} - F_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \theta } + \left[ {S_{{\text{r}}} A_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} + \frac{1}{2}S_{{\text{r}}} A_{45}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}{\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{O}}} } \right]{\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPhi}}_{{\text{I}}} \\ + \left[ {\frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{I}}} - F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}} }}{\partial \theta } \\ \end{gathered} $$

(B-62)

$$ \begin{gathered} {\varvec{K}}_{{{\text{OI}}}}^{v\beta } = {\varvec{K}}_{{{\text{IO}}}}^{\beta v} = \left[ { - \frac{1}{4}T_{22}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }} + C_{22}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} + F_{22}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } + S_{{\text{r}}} A_{44}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} {\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ - \frac{1}{4}S_{{\text{r}}} T_{44}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{R^{{\text{C}}} }}{\varvec{V}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} + \left[ { - \frac{1}{4}A_{26}^{{\text{C}}} h^{{\text{O}}} + D_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} } \right]\left( {\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } + \frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi }} \right) \\ + \left[ { - \frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{O}}} R^{{\text{C}}} + C_{66}^{{\text{C}}} \frac{{h^{{\text{O}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} + F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } \\ \end{gathered} $$

(B-63)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{\beta v} = {\varvec{K}}_{{{\text{OI}}}}^{v\beta } = \left[ {\frac{1}{4}T_{22}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }} + C_{22}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} h^{{\text{C}}} }} - F_{22}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } + \left( { - \frac{1}{4}S_{{\text{r}}} T_{44}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{R^{{\text{C}}} }} - S_{{\text{r}}} T_{44}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} } \right) \\ \times {\varvec{V}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} + \left[ {\frac{1}{4}T_{26}^{{\text{C}}} h^{{\text{I}}} - D_{26}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} } \right]\left( {\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } + \frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi }} \right) \\ + \left[ {\frac{1}{4}T_{66}^{{\text{C}}} h^{{\text{I}}} R^{{\text{C}}} + C_{66}^{{\text{C}}} \frac{{h^{{\text{I}}} }}{{h^{{\text{C}}} }}R^{{\text{C}}} - F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{I}}} R^{{\text{C}}} } \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } \\ \end{gathered} $$

(B-64)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{\alpha \beta } = {\varvec{K}}_{{{\text{OI}}}}^{\beta \alpha } = \left[ { - \frac{1}{8}T_{12}^{{\text{C}}} + \frac{1}{2}F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]h^{{\text{O}}} h^{{\text{I}}} \frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } + \left[ {\frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} - \frac{1}{4}S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}} \right]h^{{\text{O}}} h^{{\text{I}}}{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{O}}} \\ + \left[ { - \frac{1}{8}T_{16}^{{\text{C}}} + \frac{1}{2}F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]R^{{\text{C}}} h^{{\text{O}}} h^{{\text{I}}} \frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \varphi } + \left[ { - \frac{1}{8}T_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}h^{{\text{O}}} h^{{\text{I}}} + \frac{1}{2}F_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} h^{{\text{I}}} } \right] \\ \times \frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } + \left[ { - \frac{1}{8}T_{66}^{{\text{C}}} + \frac{1}{2}F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]h^{{\text{O}}} h^{{\text{I}}} \frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{O}}} }}{\partial \theta } \\ \end{gathered} $$

(B-65)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IO}}}}^{\beta \alpha } = {\varvec{K}}_{{{\text{OI}}}}^{\alpha \beta } = \left[ { - \frac{1}{8}T_{12}^{{\text{C}}} + \frac{1}{2}F_{12}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]h^{{\text{O}}} h^{{\text{I}}} \frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } + \left[ {\frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} - \frac{1}{4}S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}} \right]h^{{\text{O}}} h^{{\text{I}}}{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}}{\varvec{\varPsi}}_{{\text{I}}} \\ + \left[ { - \frac{1}{8}T_{16}^{{\text{C}}} + \frac{1}{2}F_{16}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]R^{{\text{C}}} h^{{\text{O}}} h^{{\text{I}}} \frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \varphi } + \left[ { - \frac{1}{8}T_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}h^{{\text{O}}} h^{{\text{I}}} + \frac{1}{2}F_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} h^{{\text{O}}} h^{{\text{I}}} } \right] \\ \times \frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } + \left[ { - \frac{1}{8}T_{66}^{{\text{C}}} + \frac{1}{2}F_{66}^{{\text{C}}} \left( {\frac{1}{{h^{{\text{C}}} }}} \right)^{2} } \right]h^{{\text{O}}} h^{{\text{I}}} \frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial{\varvec{\varPsi}}_{{\text{I}}} }}{\partial \theta } \\ \end{gathered} $$

(B-66)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IW}}}}^{uw} = {\varvec{K}}_{{{\text{WI}}}}^{wu} = \left[ {2T_{12}^{{\text{I}}} \user2{ + }T_{12}^{{\text{C}}} - 2C_{12}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }{\varvec{W}} - 2S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}{\varvec{U}}_{{\text{I}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \varphi } \\ \user2{ + }\left[ {2T_{26}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }}\user2{ + }T_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} - 2C_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }{\varvec{W}} - 2S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}{\varvec{U}}_{{\text{I}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \theta } \\ \end{gathered} $$

(B-67)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IW}}}}^{vw} = {\varvec{K}}_{{{\text{WI}}}}^{wv} = \left[ {2T_{22}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }}\user2{ + }T_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} - 2C_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }{\varvec{W}} + \left[ { - 2S_{{\text{r}}} A_{44}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} - S_{{\text{r}}} T_{44}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}} \right]{\varvec{V}}_{{\text{O}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \theta } \\ - 2S_{{\text{r}}} T_{44}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}{\varvec{V}}_{{\text{O}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \theta }\user2{ + }\left[ {2T_{26}^{{\text{I}}} \user2{ + }T_{26}^{{\text{C}}} - S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}R^{{\text{C}}} - 2C_{26}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }{\varvec{W}} \\ + \left[ { - 2S_{{\text{r}}} T_{45}^{{\text{O}}} - S_{{\text{r}}} T_{45}^{{\text{C}}} - 2S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}} \right]{\varvec{V}}_{{\text{O}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \varphi } \\ \end{gathered} $$

(B-68)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IW}}}}^{\alpha w} = {\varvec{K}}_{{{\text{WI}}}}^{w\beta } = \left[ {2S_{{\text{r}}} T_{45}^{{\text{I}}} - S_{{\text{r}}} T_{45}^{{\text{C}}} h^{{\text{I}}} \frac{1}{{h^{{\text{C}}} }}} \right]{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \theta } + \left[ {2S_{{\text{r}}} T_{55}^{{\text{I}}} R^{{\text{I}}} - S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} h^{{\text{I}}} \frac{1}{{h^{{\text{C}}} }}} \right]{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \varphi } \\ + \left[ {2C_{12}^{{\text{C}}} + \frac{1}{2}T_{12}^{{\text{C}}} h^{{\text{I}}} - C_{12}^{{\text{C}}} h^{{\text{I}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }{\varvec{W}} + \left[ {2C_{26}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{2}T_{26}^{{\text{C}}} h^{{\text{I}}} \frac{1}{{R^{{\text{C}}} }} - C_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}h^{{\text{I}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }{\varvec{W}} \\ \end{gathered} $$

(B-69)

$$ \begin{gathered} {\varvec{K}}_{{{\text{IW}}}}^{\beta w} = {\varvec{K}}_{{{\text{WI}}}}^{w\beta } = \left[ {2S_{{\text{r}}} T_{44}^{{\text{I}}} - \frac{1}{2}S_{{\text{r}}} T_{44}^{{\text{C}}} h^{{\text{I}}} \frac{1}{{R^{{\text{C}}} }} - S_{{\text{r}}} T_{44}^{{\text{C}}} h^{{\text{I}}} \frac{1}{{h^{{\text{C}}} }}} \right]{\varvec{\varPsi}}_{{\text{I}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \theta } + \left[ {2S_{{\text{r}}} T_{45}^{{\text{I}}} R^{{\text{I}}} - \frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} h^{{\text{I}}} - S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} h^{{\text{I}}} \frac{1}{{h^{{\text{C}}} }}} \right] \\ \times{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \varphi } + \left[ {2T_{22}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + \frac{1}{2}T_{22}^{{\text{C}}} h^{{\text{I}}} \frac{1}{{R^{{\text{C}}} }} - C_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}h^{{\text{I}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \theta }{\varvec{W}} \\ + \left[ {2C_{26}^{{\text{I}}} + \frac{1}{2}T_{26}^{{\text{C}}} h^{{\text{I}}} - 2C_{26}^{{\text{C}}} h^{{\text{I}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} }}{\partial \varphi }{\varvec{W}} \\ \end{gathered} $$

(B-70)

$$ \begin{gathered} {\varvec{K}}_{{{\text{OW}}}}^{uw} = {\varvec{K}}_{{{\text{WO}}}}^{wu} = \left[ {2T_{12}^{{\text{O}}} \user2{ + }T_{12}^{{\text{C}}} + 2S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + 2C_{12}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\user2{W + }2S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}{\varvec{U}}_{{\text{O}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \theta } \\ \user2{ + }\left[ {2T_{26}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }}\user2{ + }T_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + 2S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }} + 2C_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{U}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\user2{W + }2S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}{\varvec{U}}_{{\text{O}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \varphi } \\ \end{gathered} $$

(B-71)

$$ \begin{gathered} {\varvec{K}}_{{{\text{OW}}}}^{vw} = {\varvec{K}}_{{{\text{WO}}}}^{wv} = \left[ {2T_{22}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }}\user2{ + }T_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + 2C_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}\frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }\user2{W + }\left[ { - S_{{\text{r}}} T_{44}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} - S_{{\text{r}}} T_{44}^{{\text{C}}} + 2S_{{\text{r}}} T_{44}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}} \right]{\varvec{V}}_{{\text{O}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \theta } \\ \user2{ + }\left[ {2T_{26}^{{\text{O}}} \user2{ + }T_{26}^{{\text{C}}} + S_{{\text{r}}} T_{45}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}R^{{\text{C}}} + 2C_{26}^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial {\varvec{V}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }\user2{W + }\left[ { - 2S_{{\text{r}}} T_{45}^{{\text{O}}} - S_{{\text{r}}} T_{45}^{{\text{C}}} - 2S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} \frac{1}{{h^{{\text{C}}} }}} \right]{\varvec{V}}_{{\text{O}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \varphi } \\ \end{gathered} $$

(B-72)

$$ \begin{gathered} {\varvec{K}}_{{{\text{OW}}}}^{\alpha w} = {\varvec{K}}_{{{\text{WO}}}}^{w\alpha } = \left[ {2S_{{\text{r}}} T_{45}^{{\text{O}}} - S_{{\text{r}}} T_{45}^{{\text{C}}} h^{{\text{O}}} \frac{1}{{h^{{\text{C}}} }}} \right]{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \theta } + \left[ {2S_{{\text{r}}} T_{55}^{{\text{O}}} R^{{\text{O}}} - S_{{\text{r}}} T_{55}^{{\text{C}}} R^{{\text{C}}} h^{{\text{O}}} \frac{1}{{h^{{\text{C}}} }}} \right]{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \varphi } \\ + \left[ {2C_{12}^{{\text{O}}} - \frac{1}{2}T_{12}^{{\text{C}}} h^{{\text{O}}} - C_{12}^{{\text{C}}} h^{{\text{O}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }{\varvec{W}} + \left[ {2C_{26}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} - \frac{1}{2}T_{26}^{{\text{C}}} h^{{\text{O}}} \frac{1}{{R^{{\text{C}}} }} - C_{26}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}h^{{\text{O}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }{\varvec{W}} \\ \end{gathered} $$

(B-73)

$$ \begin{gathered} {\varvec{K}}_{{{\text{WC}}}}^{\beta w} = {\varvec{K}}_{{{\text{WC}}}}^{w\beta } = \left[ {2S_{{\text{r}}} T_{44}^{{\text{O}}} + \frac{1}{2}S_{{\text{r}}} T_{44}^{{\text{C}}} h^{{\text{O}}} \frac{1}{{R^{{\text{C}}} }} - S_{{\text{r}}} T_{44}^{{\text{C}}} h^{{\text{O}}} \frac{1}{{h^{{\text{C}}} }}} \right]{\varvec{\varPsi}}_{{\text{O}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \theta } \\ + \left[ {2S_{{\text{r}}} A_{45}^{{\text{O}}} R^{{\text{O}}} + \frac{1}{2}S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} h^{{\text{O}}} - S_{{\text{r}}} T_{45}^{{\text{C}}} R^{{\text{C}}} h^{{\text{O}}} \frac{1}{{h^{{\text{C}}} }}} \right] \times{\varvec{\varPhi}}_{{\text{I}}}^{{\text{T}}} \frac{{\partial {\varvec{W}}}}{\partial \varphi } \\ + \left[ {2C_{22}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }} - \frac{1}{2}T_{22}^{{\text{C}}} h^{{\text{O}}} \frac{1}{{R^{{\text{C}}} }} - C_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }}h^{{\text{O}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \theta }{\varvec{W}} + \left[ {2C_{26}^{{\text{O}}} - \frac{1}{2}T_{26}^{{\text{C}}} h^{{\text{O}}} - 2C_{26}^{{\text{C}}} h^{{\text{O}}} \frac{1}{{h^{{\text{C}}} }}} \right]\frac{{\partial{\varvec{\varPhi}}_{{\text{O}}}^{{\text{T}}} }}{\partial \varphi }{\varvec{W}} \\ \end{gathered} $$

(B-74)

$$ \begin{gathered} {\varvec{K}}_{{\text{W}}} = \left[ {T_{22}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + T_{22}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + T_{22}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }}} \right]{\varvec{W}}^{{\text{T}}} \user2{W + }\left[ {S_{{\text{r}}} T_{44}^{{\text{I}}} \frac{1}{{R^{{\text{I}}} }} + S_{{\text{r}}} T_{44}^{{\text{C}}} \frac{1}{{R^{{\text{C}}} }} + S_{{\text{r}}} T_{44}^{{\text{O}}} \frac{1}{{R^{{\text{O}}} }}} \right]\frac{{\partial {\varvec{W}}^{{\text{T}}} }}{\partial \theta }\frac{{\partial {\varvec{W}}}}{\partial \theta } \\ \left[ {S_{{\text{r}}} T_{55}^{{\text{I}}} R^{{\text{I}}} + S_{{\text{r}}} T_{55}^{{\text{C}}} R^{{\text{C}}} + S_{{\text{r}}} T_{55}^{{\text{O}}} R^{{\text{O}}} } \right]\frac{{\partial {\varvec{W}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{W}}}}{\partial \varphi } + \left[ {2S_{{\text{r}}} T_{45}^{{\text{I}}} + S_{{\text{r}}} T_{45}^{{\text{C}}} + S_{{\text{r}}} T_{45}^{{\text{O}}} } \right]\frac{{\partial {\varvec{W}}^{{\text{T}}} }}{\partial \varphi }\frac{{\partial {\varvec{W}}}}{\partial \theta } \\ \end{gathered} $$

(B-75)

The mode shape eigenvector q in Eq. (23) can be stated as:

$$ \begin{gathered} {\varvec{q}} = [U_{00}^{{\text{O}}} , \cdots ,U_{mn}^{{\text{O}}} , \cdots ,U_{MN}^{{\text{O}}} ,V_{00}^{{\text{O}}} , \cdots ,V_{mn}^{{\text{O}}} , \cdots ,V_{MN}^{{\text{O}}} ,a_{00}^{{\text{O}}} , \cdots ,a_{mn}^{{\text{O}}} , \cdots ,a_{MN}^{{\text{O}}} ,b_{00}^{{\text{O}}} , \cdots ,b_{mn}^{{\text{O}}} , \cdots ,b_{MN}^{{\text{O}}} , \\ \, U_{00}^{{\text{I}}} , \cdots ,U_{mn}^{{\text{I}}} , \cdots ,U_{MN}^{{\text{I}}} ,V_{00}^{{\text{I}}} , \cdots ,V_{mn}^{{\text{I}}} , \cdots ,V_{MN}^{{\text{I}}} ,a_{00}^{{\text{I}}} , \cdots ,a_{mn}^{{\text{I}}} , \cdots ,a_{MN}^{{\text{I}}} ,b_{00}^{{\text{I}}} , \cdots ,b_{mn}^{{\text{I}}} , \cdots ,b_{MN}^{{\text{I}}} , \\ \, W_{00} , \cdots ,W_{mn} , \cdots ,W_{MN} ]^{{\text{T}}} \\ \end{gathered} $$

(B-76)