An efficient nine-node quadrilateral element for free vibration analysis of deep doubly curved soft-core sandwich shells

Abstract

This paper presents a free vibration analysis of deep doubly curved laminated composite sandwich shells with soft core under various boundary conditions, using a novel ${C}^{0}$ efficient nine-node quadrilateral element. The proposed model is formulated based on the extended higher-order sandwich panel theory (EHSAPT) in which the compressibility as well as in-plane stresses of the core are considered. Accordingly, the first-order shear deformation theory is used to model the face sheets, while third- and second-order functions are assumed to model the in-plane and transverse displacements of the core, respectively, maintaining an interlaminar displacement continuity. The important feature of this model in comparison with other layerwise models is that by increasing the number of lamina layers, the number of variables is fixed and does not increase, which makes this model an appropriate choice for facilitating engineering analysis. The equations of motion and relevant boundary conditions for sandwich structure are derived using Hamilton’s principle. Convergence and comparison studies are carried out to demonstrate the efficiency and accuracy of the solution. The method is validated against the results found in the literature based on various analytical and numerical methods. The comparison studies indicate that the developed finite element model is more accurate, fast convergence and simpler in comparison with results found in the literature and it is valid for both thin and thick sandwich plates and shells. Finally, a parametric study is performed to illustrate the effect of various parameters on vibration characteristics of cylindrical, spherical and hyperbolic paraboloid sandwich panels along with sandwich plates with different boundary conditions which may be used as benchmark results for future studies.

Appendices

Appendix 1

The strain–displacement relations for the face sheets can be written in compact form in terms of mid-surface generalized strains as follows:

$$\begin{gathered} \varepsilon_{\alpha }^{i} = \frac{1}{{\left( {1 + z_{i} /R_{\alpha }^{i} } \right)}}\left( {\varepsilon_{0\alpha }^{i} + z_{i} \kappa_{\alpha }^{i} } \right) \hfill \\ \varepsilon_{\beta }^{i} = \frac{1}{{\left( {1 + z_{i} /R_{\beta }^{i} } \right)}}\left( {\varepsilon_{0\beta }^{i} + z_{i} \kappa_{\beta }^{i} } \right) \hfill \\ \varepsilon_{z}^{i} = \varepsilon_{0z}^{i} \hfill \\ \gamma_{\alpha z}^{i} = \frac{{\gamma_{0\alpha z}^{i} }}{{\left( {1 + z_{i} /R_{\alpha }^{i} } \right)}} \hfill \\ \gamma_{\beta z}^{i} = \frac{{\gamma_{0\beta z}^{i} }}{{\left( {1 + z_{i} /R_{\beta }^{i} } \right)}} \hfill \\ \gamma_{\alpha \beta }^{i} = \frac{1}{{\left( {1 + z_{i} /R_{\alpha }^{i} } \right)}}\left( {\varepsilon_{0\alpha \beta }^{i} + z_{i} \kappa_{\alpha \beta }^{i} } \right) + \frac{1}{{\left( {1 + z_{i} /R_{\beta }^{i} } \right)}}\left( {\varepsilon_{0\beta \alpha }^{i} + z_{i} \kappa_{\beta \alpha }^{i} } \right) \hfill \\ \end{gathered}$$

(37)

And for the core layer,

$$\begin{aligned} \varepsilon_{\alpha }^{c} &= \frac{1}{{A_{c} \left( {1 + z_{c} /R_{\alpha }^{c} } \right)}}\left( {\varepsilon_{0\alpha }^{c} + z_{c} \varepsilon_{1\alpha }^{c} + z_{c}^{2} \varepsilon_{2\alpha }^{c} + z_{c}^{3} \varepsilon_{3\alpha }^{c} } \right) \\ \varepsilon_{\beta }^{c} &= \frac{1}{{B_{c} \left( {1 + z_{c} /R_{\beta }^{c} } \right)}}\left( {\varepsilon_{0\beta }^{c} + z_{c} \varepsilon_{1\beta }^{c} + z_{c}^{2} \varepsilon_{2\beta }^{c} + z_{c}^{3} \varepsilon_{3\beta }^{c} } \right) \\ \varepsilon_{z}^{c} &= \varepsilon_{0z}^{c} + z_{c} \varepsilon_{1z}^{c} \\ \gamma_{\alpha z}^{c} &= \frac{1}{{A_{c} \left( {1 + z_{c} /R_{\alpha }^{c} } \right)}}\left( {\gamma_{0\alpha z}^{c} + z_{c} \gamma_{1\alpha z}^{c} + z_{c}^{2} \gamma_{2\alpha z}^{c} + z_{c}^{3} \gamma_{3\alpha z}^{c} } \right) \\ \gamma_{\beta z}^{c} &= \frac{1}{{B_{c} \left( {1 + z_{c} /R_{\beta }^{c} } \right)}}\left( {\gamma_{0\beta z}^{c} + z_{c} \gamma_{1\beta z}^{c} + z_{c}^{2} \gamma_{2\beta z}^{c} + z_{c}^{3} \gamma_{3\beta z}^{c} } \right) \\ \gamma_{\alpha \beta }^{c} &= \frac{1}{{A_{c} \left( {1 + z_{c} /R_{\alpha }^{c} } \right)}}\left( {\gamma_{0\alpha \beta }^{c} + z_{c} \gamma_{1\alpha \beta }^{c} + z_{c}^{2} \gamma_{2\alpha \beta }^{c} + z_{c}^{3} \gamma_{3\alpha \beta }^{c} } \right) + \frac{1}{{B_{c} \left( {1 + z_{c} /R_{\beta }^{c} } \right)}}\left( {\gamma_{0\alpha z}^{c} + z_{c} \gamma_{1\alpha z}^{c} + z_{c}^{2} \gamma_{2\alpha z}^{c} + z_{c}^{3} \gamma_{3\alpha z}^{c} } \right) \end{aligned}$$

(38)

in which

$$\begin{gathered} \varepsilon_{0\alpha }^{i} = \frac{1}{{A_{i} }}\frac{{\partial u_{0}^{i} }}{\partial \alpha } + \frac{{v_{0}^{i} }}{{A_{i} B_{i} }}\frac{{\partial A_{i} }}{\partial \beta } + \frac{{w_{0}^{i} }}{{R_{\alpha }^{i} }} \hfill \\ \varepsilon_{0\beta }^{i} = \frac{1}{{B_{i} }}\frac{{\partial v_{0}^{i} }}{\partial \beta } + \frac{{u_{0}^{i} }}{{A_{i} B_{i} }}\frac{{\partial B_{i} }}{\partial \alpha } + \frac{{w_{0}^{i} }}{{R_{\beta }^{i} }} \hfill \\ \varepsilon_{0\alpha \beta }^{i} = \frac{1}{{A_{i} }}\frac{{\partial v_{0}^{i} }}{\partial \alpha } - \frac{{u_{0}^{i} }}{{A_{i} B_{i} }}\frac{{\partial A_{i} }}{\partial \beta } \hfill \\ \varepsilon_{0\beta \alpha }^{i} = \frac{1}{{B_{i} }}\frac{{\partial u_{0}^{i} }}{\partial \beta } - \frac{{v_{0}^{i} }}{{A_{i} B_{i} }}\frac{{\partial B_{i} }}{\partial \alpha } \hfill \\ \gamma_{0\alpha z}^{i} = \frac{1}{{A_{i} }}\frac{{\partial w_{0}^{i} }}{\partial \alpha } - \frac{{u_{0}^{i} }}{{R_{\alpha }^{i} }} + \psi_{\alpha }^{i} \hfill \\ \gamma_{0\beta z}^{i} = \frac{1}{{B_{i} }}\frac{{\partial w_{0}^{i} }}{\partial \beta } - \frac{{v_{0}^{i} }}{{R_{\beta }^{i} }} + \psi_{\beta }^{i} \hfill \\ \kappa_{\alpha }^{i} = \frac{1}{{A_{i} }}\frac{{\partial \psi_{\alpha }^{i} }}{\partial \alpha } + \frac{{\psi_{\beta }^{i} }}{{A_{i} B_{i} }}\frac{{\partial A_{i} }}{\partial \beta } \hfill \\ \kappa_{\beta }^{i} = \frac{1}{{B_{i} }}\frac{{\partial \psi_{\beta }^{i} }}{\partial \beta } + \frac{{\psi_{\alpha }^{i} }}{{A_{i} B_{i} }}\frac{{\partial B_{i} }}{\partial \alpha } \hfill \\ \kappa_{\alpha \beta }^{i} = \frac{1}{{A_{i} }}\frac{{\partial \psi_{\beta }^{i} }}{\partial \alpha } - \frac{{\psi_{\alpha }^{i} }}{{A_{i} B_{i} }}\frac{{\partial A_{i} }}{\partial \beta } \hfill \\ \kappa_{\beta \alpha }^{i} = \frac{1}{{B_{i} }}\frac{{\partial \psi_{\alpha }^{i} }}{\partial \beta } - \frac{{\psi_{\beta }^{i} }}{{A_{i} B_{i} }}\frac{{\partial B_{i} }}{\partial \alpha } \hfill \\ \end{gathered}$$

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$$\begin{aligned} \varepsilon_{0\alpha }^{c} &= \frac{{\partial u_{0} }}{\partial \alpha } + \frac{{v_{0} }}{{B_{c} }}\frac{{\partial A_{c} }}{\partial \beta } + \frac{{A_{c} w_{0} }}{{R_{\alpha } }}\quad \varepsilon_{0\beta }^{c} = \frac{{\partial v_{0} }}{\partial \beta } + \frac{{u_{0} }}{{A_{c} }}\frac{{\partial B_{c} }}{\partial \alpha } + \frac{{B_{c} w_{0} }}{{R_{\beta } }} \\ \varepsilon_{1\alpha }^{c} &= \frac{{\partial u_{1} }}{\partial \alpha } + \frac{{v_{1} }}{{B_{c} }}\frac{{\partial A_{c} }}{\partial \beta } + \frac{{A_{c} w_{1} }}{{R_{\alpha } }}\quad \varepsilon_{1\beta }^{c} = \frac{{\partial v_{1} }}{\partial \beta } + \frac{{u_{1} }}{{A_{c} }}\frac{{\partial B_{c} }}{\partial \alpha } + \frac{{B_{c} w_{1} }}{{R_{\beta } }} \\ \varepsilon_{2\alpha }^{c} &= \frac{{\partial u_{2} }}{\partial \alpha } + \frac{{v_{2} }}{{B_{c} }}\frac{{\partial A_{c} }}{\partial \beta } + \frac{{A_{c} w_{2} }}{{R_{\alpha } }}\quad \varepsilon_{2\beta }^{c} = \frac{{\partial v_{2} }}{\partial \beta } + \frac{{u_{2} }}{{A_{c} }}\frac{{\partial B_{c} }}{\partial \alpha } + \frac{{B_{c} w_{2} }}{{R_{\beta } }} \\ \varepsilon_{3\alpha }^{c} &= \frac{{\partial u_{3} }}{\partial \alpha } + \frac{{v_{3} }}{{B_{c} }}\frac{{\partial A_{c} }}{\partial \beta }\quad \varepsilon_{3\beta }^{c} = \frac{{\partial v_{3} }}{\partial \beta } + \frac{{u_{3} }}{{A_{c} }}\frac{{\partial B_{c} }}{\partial \alpha } \\ \varepsilon_{0z}^{c} &= w_{1}\varepsilon_{1z}^{c} = 2w_{2} \\ \gamma_{0\alpha z}^{c} &= A_{c} u_{1} + \frac{{\partial w_{0} }}{\partial \alpha } - \frac{{A_{c} u_{0} }}{{R_{\alpha } }}\quad \gamma_{0\beta z}^{c} = B_{c} v_{1} + \frac{{\partial w_{0} }}{\partial \beta } - B_{c} \frac{{v_{0} }}{{R_{\beta } }} \\ \gamma_{1\alpha z}^{c} &= 2A_{c} u_{2} + \frac{{\partial w_{1} }}{\partial \alpha }\quad \gamma_{1\beta z}^{c} = 2B_{c} v_{2} + \frac{{\partial w_{1} }}{\partial \beta } \\ \gamma_{2\alpha z}^{c} &= 3A_{c} u_{3} + \frac{{\partial w_{2} }}{\partial \alpha } + \frac{{A_{c} u_{2} }}{{R_{\alpha } }}\quad \gamma_{2\beta z}^{c} = 3B_{c} v_{3} + \frac{{\partial w_{2} }}{\partial \beta } + B_{c} \frac{{v_{2} }}{{R_{\beta } }} \\ \gamma_{3\alpha z}^{c} &= \frac{{2A_{c} u_{3} }}{{R_{\alpha } }} \gamma_{3\beta z}^{c} = 2B_{c} \frac{{v_{3} }}{{R_{\beta } }} \\ \gamma_{01\alpha \beta }^{c} &= \frac{1}{{R_{\alpha } }}\frac{{\partial v_{0} }}{\partial \alpha } - \frac{{u_{0} }}{{B_{c} }}\frac{{\partial A_{c} }}{\partial \beta }\quad \gamma_{02\alpha \beta }^{c} = \frac{1}{{R_{\beta } }}\frac{{\partial u_{0} }}{\partial \beta } - \frac{{v_{0} }}{{A_{c} }}\frac{{\partial B_{c} }}{\partial \alpha } \\ \gamma_{11\alpha \beta }^{c} &= \frac{1}{{R_{\alpha } }}\frac{{\partial v_{1} }}{\partial \alpha } - \frac{{u_{1} }}{{B_{c} }}\frac{{\partial A_{c} }}{\partial \beta }\quad \gamma_{12\alpha \beta }^{c} = \frac{1}{{R_{\beta } }}\frac{{\partial u_{1} }}{\partial \beta } - \frac{{v_{1} }}{{A_{c} }}\frac{{\partial B_{c} }}{\partial \alpha } \\ \gamma_{21\alpha \beta }^{c} &= \frac{1}{{R_{\alpha } }}\frac{{\partial v_{2} }}{\partial \alpha } - \frac{{u_{2} }}{{B_{c} }}\frac{{\partial A_{c} }}{\partial \beta }\quad \gamma_{22\alpha \beta }^{c} = \frac{1}{{R_{\beta } }}\frac{{\partial u_{2} }}{\partial \beta } - \frac{{v_{2} }}{{A_{c} }}\frac{{\partial B_{c} }}{\partial \alpha } \\ \gamma_{31\alpha \beta }^{c} &= \frac{1}{{R_{\alpha } }}\frac{{\partial v_{3} }}{\partial \alpha } - \frac{{u_{3} }}{{B_{c} }}\frac{{\partial A_{c} }}{\partial \beta }\quad \gamma_{32\alpha \beta }^{c} = \frac{1}{{R_{\beta } }}\frac{{\partial u_{3} }}{\partial \beta } - \frac{{v_{3} }}{{A_{c} }}\frac{{\partial B_{c} }}{\partial \alpha } \end{aligned}$$

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Appendix 2

The stress and couple resultants for the core can be expressed, as follows:

$$\begin{aligned}& \left[ {\begin{array}{*{20}l} {N_{\alpha } } \hfill \\ {N_{\beta } } \hfill \\ {N_{{\alpha \beta }} } \hfill \\ {N_{{\beta \alpha }} } \hfill \\ {M_{{1\alpha }} } \hfill \\ {M_{{1\beta }} } \hfill \\ {M_{{1\alpha \beta }} } \hfill \\ {M_{{1\beta \alpha }} } \hfill \\ {M_{{2\alpha }} } \hfill \\ {M_{{2\beta }} } \hfill \\ {M_{{2\alpha \beta }} } \hfill \\ {M_{{2\beta \alpha }} } \hfill \\ {M_{{3\alpha }} } \hfill \\ {M_{{3\beta }} } \hfill \\ {M_{{3\alpha \beta }} } \hfill \\ {M_{{3\beta \alpha }} } \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\bar{A}_{{11}} } \hfill & {A_{{12}} } \hfill & {\bar{A}_{{16}} } \hfill & {A_{{16}} } \hfill & {\bar{B}_{{11}} } \hfill & {B_{{12}} } \hfill & {\bar{B}_{{16}} } \hfill & {B_{{16}} } \hfill \\ {A_{{12}} } \hfill & {\tilde{A}_{{22}} } \hfill & {A_{{26}} } \hfill & {\tilde{A}_{{26}} } \hfill & {B_{{12}} } \hfill & {\tilde{B}_{{22}} } \hfill & {B_{{26}} } \hfill & {\tilde{B}_{{26}} } \hfill \\ {\bar{A}_{{16}} } \hfill & {A_{{26}} } \hfill & {\bar{A}_{{66}} } \hfill & {A_{{66}} } \hfill & {\bar{B}_{{16}} } \hfill & {B_{{26}} } \hfill & {\bar{B}_{{66}} } \hfill & {B_{{66}} } \hfill \\ {A_{{16}} } \hfill & {\tilde{A}_{{26}} } \hfill & {A_{{66}} } \hfill & {\tilde{A}_{{66}} } \hfill & {B_{{16}} } \hfill & {\tilde{B}_{{26}} } \hfill & {B_{{66}} } \hfill & {\tilde{B}_{{66}} } \hfill \\ {\bar{B}_{{11}} } \hfill & {B_{{12}} } \hfill & {\bar{B}_{{16}} } \hfill & {B_{{16}} } \hfill & {\bar{C}_{{11}} } \hfill & {C_{{12}} } \hfill & {\bar{C}_{{16}} } \hfill & {C_{{16}} } \hfill \\ {B_{{12}} } \hfill & {\tilde{B}_{{22}} } \hfill & {B_{{26}} } \hfill & {\tilde{B}_{{26}} } \hfill & {C_{{12}} } \hfill & {\tilde{C}_{{22}} } \hfill & {C_{{26}} } \hfill & {\tilde{C}_{{26}} } \hfill \\ {\bar{B}_{{16}} } \hfill & {B_{{26}} } \hfill & {\bar{B}_{{66}} } \hfill & {B_{{66}} } \hfill & {\bar{C}_{{16}} } \hfill & {C_{{26}} } \hfill & {\bar{C}_{{66}} } \hfill & {C_{{66}} } \hfill \\ {B_{{16}} } \hfill & {\tilde{B}_{{26}} } \hfill & {B_{{66}} } \hfill & {\tilde{B}_{{66}} } \hfill & {C_{{16}} } \hfill & {\tilde{C}_{{26}} } \hfill & {C_{{66}} } \hfill & {\tilde{C}_{{66}} } \hfill \\ {\bar{C}_{{11}} } \hfill & {C_{{12}} } \hfill & {\bar{C}_{{16}} } \hfill & {C_{{16}} } \hfill & {\bar{D}_{{11}} } \hfill & {D_{{12}} } \hfill & {\bar{D}_{{16}} } \hfill & {D_{{16}} } \hfill \\ {C_{{12}} } \hfill & {\tilde{C}_{{22}} } \hfill & {C_{{26}} } \hfill & {\tilde{C}_{{26}} } \hfill & {D_{{12}} } \hfill & {\tilde{D}_{{22}} } \hfill & {D_{{26}} } \hfill & {\tilde{D}_{{26}} } \hfill \\ {\bar{C}_{{16}} } \hfill & {C_{{26}} } \hfill & {\bar{C}_{{66}} } \hfill & {C_{{66}} } \hfill & {\bar{D}_{{16}} } \hfill & {D_{{26}} } \hfill & {\bar{D}_{{66}} } \hfill & {D_{{66}} } \hfill \\ {C_{{16}} } \hfill & {\tilde{C}_{{26}} } \hfill & {C_{{66}} } \hfill & {\tilde{C}_{{66}} } \hfill & {D_{{16}} } \hfill & {\tilde{D}_{{26}} } \hfill & {D_{{66}} } \hfill & {\tilde{D}_{{66}} } \hfill \\ {\bar{D}_{{11}} } \hfill & {D_{{12}} } \hfill & {\bar{D}_{{16}} } \hfill & {D_{{16}} } \hfill & {\bar{E}_{{11}} } \hfill & {E_{{12}} } \hfill & {\bar{E}_{{16}} } \hfill & {E_{{16}} } \hfill \\ {D_{{12}} } \hfill & {\tilde{D}_{{22}} } \hfill & {D_{{26}} } \hfill & {\tilde{D}_{{26}} } \hfill & {E_{{12}} } \hfill & {\tilde{E}_{{22}} } \hfill & {E_{{26}} } \hfill & {\tilde{E}_{{26}} } \hfill \\ {\bar{D}_{{16}} } \hfill & {D_{{26}} } \hfill & {\bar{D}_{{66}} } \hfill & {D_{{66}} } \hfill & {\bar{E}_{{16}} } \hfill & {E_{{26}} } \hfill & {\bar{E}_{{66}} } \hfill & {E_{{66}} } \hfill \\ {D_{{16}} } \hfill & {\tilde{D}_{{26}} } \hfill & {D_{{66}} } \hfill & {\tilde{D}_{{66}} } \hfill & {E_{{16}} } \hfill & {\tilde{E}_{{26}} } \hfill & {E_{{66}} } \hfill & {\tilde{E}_{{66}} } \hfill \\ \end{array} } \right. \\ & \quad \quad \left. {\begin{array}{*{20}l} {\bar{C}_{{11}} } \hfill & {C_{{12}} } \hfill & {\bar{C}_{{16}} } \hfill & {C_{{16}} } \hfill & {\bar{D}_{{11}} } \hfill & {D_{{12}} } \hfill & {\bar{D}_{{16}} } \hfill & {D_{{16}} } \hfill \\ {C_{{12}} } \hfill & {\tilde{C}_{{22}} } \hfill & {C_{{26}} } \hfill & {\tilde{C}_{{26}} } \hfill & {D_{{12}} } \hfill & {\tilde{D}_{{22}} } \hfill & {D_{{26}} } \hfill & {\tilde{D}_{{26}} } \hfill \\ {\bar{C}_{{16}} } \hfill & {C_{{26}} } \hfill & {\bar{C}_{{66}} } \hfill & {C_{{66}} } \hfill & {\bar{D}_{{16}} } \hfill & {D_{{26}} } \hfill & {\bar{D}_{{66}} } \hfill & {D_{{66}} } \hfill \\ {C_{{16}} } \hfill & {\tilde{C}_{{26}} } \hfill & {C_{{66}} } \hfill & {\tilde{C}_{{66}} } \hfill & {D_{{16}} } \hfill & {\tilde{D}_{{26}} } \hfill & {D_{{66}} } \hfill & {\tilde{D}_{{66}} } \hfill \\ {\bar{D}_{{11}} } \hfill & {D_{{12}} } \hfill & {\bar{D}_{{16}} } \hfill & {D_{{16}} } \hfill & {\bar{E}_{{11}} } \hfill & {E_{{12}} } \hfill & {\bar{E}_{{16}} } \hfill & {E_{{16}} } \hfill \\ {D_{{12}} } \hfill & {\tilde{D}_{{22}} } \hfill & {D_{{26}} } \hfill & {\tilde{D}_{{26}} } \hfill & {E_{{12}} } \hfill & {\tilde{E}_{{22}} } \hfill & {E_{{26}} } \hfill & {\tilde{E}_{{26}} } \hfill \\ {\bar{D}_{{16}} } \hfill & {D_{{26}} } \hfill & {\bar{D}_{{66}} } \hfill & {D_{{66}} } \hfill & {\bar{E}_{{16}} } \hfill & {E_{{26}} } \hfill & {\bar{E}_{{66}} } \hfill & {E_{{66}} } \hfill \\ {D_{{16}} } \hfill & {\tilde{D}_{{26}} } \hfill & {D_{{66}} } \hfill & {\tilde{D}_{{66}} } \hfill & {E_{{16}} } \hfill & {\tilde{E}_{{26}} } \hfill & {E_{{66}} } \hfill & {\tilde{E}_{{66}} } \hfill \\ {\bar{E}_{{11}} } \hfill & {E_{{12}} } \hfill & {\bar{E}_{{16}} } \hfill & {E_{{16}} } \hfill & {\bar{F}_{{11}} } \hfill & {F_{{12}} } \hfill & {\bar{F}_{{16}} } \hfill & {F_{{16}} } \hfill \\ {E_{{12}} } \hfill & {\tilde{E}_{{22}} } \hfill & {E_{{26}} } \hfill & {\tilde{E}_{{26}} } \hfill & {F_{{12}} } \hfill & {\tilde{F}_{{22}} } \hfill & {F_{{26}} } \hfill & {\tilde{F}_{{26}} } \hfill \\ {\bar{E}_{{16}} } \hfill & {E_{{26}} } \hfill & {\bar{E}_{{66}} } \hfill & {E_{{66}} } \hfill & {\bar{F}_{{16}} } \hfill & {F_{{26}} } \hfill & {\bar{F}_{{66}} } \hfill & {F_{{66}} } \hfill \\ {E_{{16}} } \hfill & {\tilde{E}_{{26}} } \hfill & {E_{{66}} } \hfill & {\tilde{E}_{{66}} } \hfill & {F_{{16}} } \hfill & {\tilde{F}_{{26}} } \hfill & {F_{{66}} } \hfill & {\tilde{F}_{{66}} } \hfill \\ {\bar{F}_{{11}} } \hfill & {F_{{12}} } \hfill & {\bar{F}_{{16}} } \hfill & {F_{{16}} } \hfill & {\bar{G}_{{11}} } \hfill & {G_{{12}} } \hfill & {\bar{G}_{{16}} } \hfill & {G_{{16}} } \hfill \\ {F_{{12}} } \hfill & {\tilde{F}_{{22}} } \hfill & {F_{{26}} } \hfill & {\tilde{F}_{{26}} } \hfill & {G_{{12}} } \hfill & {\tilde{G}_{{22}} } \hfill & {G_{{26}} } \hfill & {\tilde{G}_{{26}} } \hfill \\ {\bar{F}_{{16}} } \hfill & {F_{{26}} } \hfill & {\bar{F}_{{66}} } \hfill & {F_{{66}} } \hfill & {\bar{G}_{{16}} } \hfill & {G_{{26}} } \hfill & {\bar{G}_{{66}} } \hfill & {G_{{66}} } \hfill \\ {F_{{16}} } \hfill & {\tilde{F}_{{26}} } \hfill & {F_{{66}} } \hfill & {\tilde{F}_{{66}} } \hfill & {G_{{16}} } \hfill & {\tilde{G}_{{26}} } \hfill & {G_{{66}} } \hfill & {\tilde{G}_{{66}} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {\varepsilon _{{0\alpha }} } \hfill \\ {\varepsilon _{{0\beta }} } \hfill \\ {\varepsilon _{{0\alpha \beta }} } \hfill \\ {\varepsilon _{{0\beta \alpha }} } \hfill \\ {\varepsilon _{{1\alpha }} } \hfill \\ {\varepsilon _{{1\beta }} } \hfill \\ {\varepsilon _{{1\alpha \beta }} } \hfill \\ {\varepsilon _{{1\beta \alpha }} } \hfill \\ {\varepsilon _{{2\alpha }} } \hfill \\ {\varepsilon _{{2\beta }} } \hfill \\ {\varepsilon _{{2\alpha \beta }} } \hfill \\ {\varepsilon _{{2\beta \alpha }} } \hfill \\ {\varepsilon _{{3\alpha }} } \hfill \\ {\varepsilon _{{3\beta }} } \hfill \\ {\varepsilon _{{3\alpha \beta }} } \hfill \\ {\varepsilon _{{3\beta \alpha }} } \hfill \\ \end{array} } \right] \\ \end{aligned}$$

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where:

$$\begin{gathered} \overline{A}_{mn} = A_{mn} - c_{0} B_{mn} \quad \tilde{A}_{mn} = A_{mn} + c_{0} B_{mn} \hfill \\ \overline{B}_{mn} = B_{mn} - c_{0} C_{mn} \quad \tilde{B}_{mn} = B_{mn} + c_{0} C_{mn} \hfill \\ \overline{C}_{mn} = C_{mn} - c_{0} D_{mn} \quad \tilde{C}_{mn} = C_{mn} + c_{0} D_{mn} \hfill \\ \overline{D}_{mn} = D_{mn} - c_{0} E_{mn} \quad \tilde{D}_{mn} = D_{mn} + c_{0} E_{mn} \hfill \\ \overline{E}_{mn} = E_{mn} - c_{0} F_{mn} \quad \tilde{E}_{mn} = E_{mn} + c_{0} F_{mn} \hfill \\ \overline{F}_{mn} = F_{mn} - c_{0} G_{mn} \quad \tilde{F}_{mn} = F_{mn} + c_{0} G_{mn} \hfill \\ \overline{G}_{mn} = G_{mn} - c_{0} H_{mn} \quad \tilde{G}_{mn} = G_{mn} + c_{0} H_{mn} \hfill \\ \end{gathered}$$

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And the coefficients ${A}_{mn}$, ${B}_{mn},$ …, ${G}_{mn}$ of rigidity matrix are:

$$\left({A}_{mn}, {B}_{mn}, {C}_{mn}, {D}_{mn}, {E}_{mn}, {F}_{mn}, {G}_{mn}, {H}_{mn}\right)=\sum_{i=1}^{N}{\overline{Q} }_{mn}^{(k)}\int \limits_{{{z}_{c}}_{k-1}}^{{{{z}_{c}}_{k-1}}}(1,{z}_{c},{{z}_{c}}^{2},{{z}_{c}}^{3},{{z}_{c}}^{4},{{z}_{c}}^{5},{{z}_{c}}^{6},{{z}_{c}}^{7})\mathrm{d}{z}_{c}$$

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Appendix 3

The governing equations of motion are derived as:

$$\delta {u}_{0}^{t}:{\left({B}_{t}{N}_{\alpha }^{t}\right)}_{,\alpha }+{\left({A}_{t}{N}_{\beta \alpha }^{t}\right)}_{,\beta }-{B}_{t,\alpha }{N}_{\beta }^{t}+{A}_{t,\beta }{N}_{\alpha \beta }^{t}-\frac{2{B}_{c,\alpha }{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}+\frac{4{A}_{c,\beta }{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}-\frac{4{B}_{c,\alpha }{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}+\frac{2{A}_{c,\beta }{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}+\frac{{A}_{t}{B}_{t}{Q}_{\alpha z}^{t}}{{R}_{\alpha }^{t}}-\frac{4{A}_{c}{B}_{c}{M}_{Q1\alpha z}^{c}}{{{h}_{c}}^{2}}-\frac{8{A}_{c}{B}_{c}{M}_{Q3\alpha z}^{c}}{{{h}_{c}}^{3}{R}_{\alpha }^{c}}-\frac{2{A}_{c}{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}{R}_{\alpha }^{c}}-\frac{12{A}_{c}{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{3}}+{\left(\frac{4{B}_{c}{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}\right)}_{,\alpha }+{\left(\frac{2{B}_{c}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }+{\left(\frac{2{A}_{c}{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }+{\left(\frac{4{A}_{c}{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}\right)}_{,\beta }+\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{6}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{1,tt}}{{{h}_{c}}^{3}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{1,tt}}{{{h}_{c}}^{4}}+\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{0,tt}}{{{h}_{c}}^{5}}+\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{u}_{1,tt}}{{{h}_{c}}^{5}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{u}_{0,tt}}{{{h}_{c}}^{2}}-\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{u}_{0,tt}}{{{h}_{c}}^{3}}-\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{u}_{1,tt}}{{{h}_{c}}^{2}}-2{A}_{t}{B}_{t}{I}_{1}^{t}{\psi }_{\alpha ,tt}^{t}-\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{4}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{5}}+\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{6}}+\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{6}}-2{A}_{t}{B}_{t}{I}_{0}^{t}{u}_{0,tt}^{t}-\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{5}}-\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{6}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{4}}=0$$

(44)

$$\delta {u}_{0}:{\left({B}_{c}{N}_{\alpha }^{c}\right)}_{,\alpha }+{\left({A}_{c}{N}_{\beta \alpha }^{c}\right)}_{,\beta }-{B}_{c,\alpha }{N}_{\beta }^{c}-\frac{4{A}_{c,\beta }{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}+{A}_{c,\beta }{N}_{\alpha \beta }^{c}+\frac{4{B}_{c,\alpha }{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}+\frac{8{A}_{c}{B}_{c}{M}_{Q1\alpha z}^{c}}{{{h}_{c}}^{2}}+\frac{{A}_{c}{B}_{c}{Q}_{\alpha z}^{c}}{{R}_{\alpha }^{c}}+\frac{4{A}_{c}{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}{R}_{\alpha }^{c}}-{\left(\frac{4{B}_{c}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }-{\left(\frac{4{A}_{c}{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{4}}-\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{1,tt}}{{{h}_{c}}^{4}}-\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{5}}+\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{5}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{2}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{2}}+\frac{16{A}_{c}{B}_{c}{I}_{3}^{c}{u}_{1,tt}}{{{h}_{c}}^{2}}-\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{3}}-2{A}_{c}{B}_{c}{I}_{1}^{c}{u}_{1,tt}-\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{2}}+\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{2}}+\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{4}}-\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{5}}+\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{3}}+\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{3}}-2{A}_{c}{B}_{c}{I}_{0}^{c}{u}_{0,tt}-\frac{32{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}}{{{h}_{c}}^{4}}+\frac{32{A}_{c}{B}_{c}{I}_{2}^{c}{u}_{0,tt}}{{{h}_{c}}^{2}}=0$$

(45)

$${\delta {u}_{0}^{b}:\left({B}_{b}{N}_{\alpha }^{b}\right)}_{,\alpha }+{\left({A}_{b}{N}_{\beta \alpha }^{b}\right)}_{,\beta }-\frac{2{B}_{c,\alpha }{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}-\frac{4{A}_{c,\beta }{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}+\frac{4{B}_{c,\alpha }{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}+\frac{2{A}_{c,\beta }{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}+\frac{{A}_{b}{B}_{b}{Q}_{\alpha z}^{b}}{{R}_{\alpha }^{b}}-{B}_{b,\alpha }{N}_{\beta }^{b}+{A}_{b,\beta }{N}_{\alpha \beta }^{b}-\frac{4{A}_{c}{B}_{c}{M}_{Q1\alpha z}^{c}}{{{h}_{c}}^{2}}+\frac{8{A}_{c}{B}_{c}{M}_{Q3\alpha z}^{c}}{{{h}_{c}}^{3}{R}_{\alpha }^{c}}-\frac{2{A}_{c}{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}{R}_{\alpha }^{c}}+\frac{12{A}_{c}{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{3}}-{\left(\frac{4{B}_{c}{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}\right)}_{,\alpha }+{\left(\frac{2{B}_{c}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }+{\left({B}_{b}{N}_{\alpha }^{b}\right)}_{,\alpha }+{\left(\frac{2{A}_{c}{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }-{\left(\frac{4{A}_{c}{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}\right)}_{,\beta }+{\left({A}_{b}{N}_{\beta \alpha }^{b}\right)}_{,\beta }+\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{6}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{4}}+\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{1,tt}}{{{h}_{c}}^{3}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{1,tt}}{{{h}_{c}}^{4}}-\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{0,tt}}{{{h}_{c}}^{5}}-\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{u}_{1,tt}}{{{h}_{c}}^{5}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{u}_{0,tt}}{{{h}_{c}}^{2}}+\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{u}_{0,tt}}{{{h}_{c}}^{3}}-\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{u}_{1,tt}}{{{h}_{c}}^{2}}-2{A}_{b}{B}_{b}{I}_{1}^{b}{\psi }_{\alpha ,tt}^{b}-\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{4}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{6}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{6}}-2{A}_{b}{B}_{b}{I}_{0}^{b}{u}_{0,tt}^{b}+\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{5}}-\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{6}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{4}}=0$$

(46)

$${\delta {v}_{0}^{t}:\left({B}_{t}{N}_{\alpha \beta }^{t}\right)}_{,\alpha }+{\left({A}_{t}{N}_{\beta }^{t}\right)}_{,\beta }-{A}_{t,\beta }{N}_{\alpha }^{t}+{B}_{t,\alpha }{N}_{\beta \alpha }^{t}+\frac{2{B}_{c,\alpha }{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}-\frac{4{A}_{c,\beta }{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}+\frac{4{B}_{c,\alpha }{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}+\frac{{A}_{t}{B}_{t}{Q}_{\beta z}^{t}}{{R}_{\beta }^{t}}-\frac{2{A}_{c,\beta }{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}-\frac{4{A}_{c}{B}_{c}{M}_{Q1\beta z}^{c}}{{{h}_{c}}^{2}}-\frac{2{A}_{c}{B}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}{R}_{\beta }^{c}}-\frac{8{A}_{c}{B}_{c}{M}_{Q3\beta z}^{c}}{{{h}_{c}}^{3}{R}_{\beta }^{c}}-\frac{12{A}_{c}{B}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{3}}+{\left(\frac{4{B}_{c}{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}\right)}_{,\alpha }+{\left(\frac{2{B}_{c}{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }+{\left(\frac{4{A}_{c}{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}\right)}_{,\beta }+{\left(\frac{2{A}_{c}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }-2{A}_{t}{B}_{t}{I}_{1}^{t}{\psi }_{\beta ,tt}^{t}+\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{v}_{1,tt}}{{h}_{c}^{5}}+\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{v}_{0,tt}^{b}}{{h}_{c}^{6}}-\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{v}_{0,tt}}{{{h}_{c}}^{3}}-\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{v}_{1,tt}}{{{h}_{c}}^{2}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{1,tt}}{{{h}_{c}}^{3}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{1,tt}}{{{h}_{c}}^{4}}+\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{0,tt}}{{{h}_{c}}^{5}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{v}_{0,tt}}{{{h}_{c}}^{2}}-\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{4}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{5}}+\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{6}}+\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{6}}-2{A}_{t}{B}_{t}{I}_{0}^{t}{v}_{0,tt}^{t}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{4}}-\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{5}}-\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{6}}=0$$

(47)

$${\delta {v}_{0}^{b}:\left({B}_{b}{N}_{\alpha \beta }^{b}\right)}_{,\alpha }+{\left({A}_{b}{N}_{\beta }^{b}\right)}_{,\beta }-{\left(\frac{4{B}_{c}{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}\right)}_{,\alpha }+{\left(\frac{2{B}_{c}{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }+{\left(\frac{2{A}_{c}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }-{\left(\frac{4{A}_{c}{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}\right)}_{,\beta }+\frac{4{A}_{c,\beta }{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}-\frac{4{B}_{c,\alpha }{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}+\frac{2{B}_{c,\alpha }{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}-\frac{2{A}_{c,\beta }{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}+\frac{{A}_{b}{B}_{b}{Q}_{\beta z}^{b}}{{R}_{\beta }^{b}}-{A}_{b,\beta }{N}_{\alpha }^{b}+{B}_{b,\alpha }{N}_{\beta \alpha }^{b}-\frac{4{A}_{c}{B}_{c}{M}_{Q1\beta z}^{c}}{{{h}_{c}}^{2}}+\frac{8{A}_{c}{B}_{c}{M}_{Q3\beta z}^{c}}{{{h}_{c}}^{3}{R}_{\beta }^{c}}-\frac{2{A}_{c}{B}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}{R}_{\beta }^{c}}+\frac{12{A}_{c}{B}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{3}}-\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{v}_{1,tt}}{{{h}_{c}}^{5}}+\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{6}}+\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{v}_{0,tt}}{{{h}_{c}}^{3}}-\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{v}_{1,tt}}{{{h}_{c}}^{2}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{4}}+\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{1,tt}}{{{h}_{c}}^{3}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{1,tt}}{{{h}_{c}}^{4}}-\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{0,tt}}{{{h}_{c}}^{5}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{v}_{0,tt}}{{{h}_{c}}^{2}}-2{A}_{b}{B}_{b}{I}_{1}^{b}{\psi }_{\beta ,tt}^{b}-\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{4}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{6}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{6}}-2{A}_{b}{B}_{b}{I}_{0}^{b}{v}_{0,tt}^{b}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{4}}+\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{5}}-\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{6}}=0$$

(48)

$${\delta {v}_{0}:\left({B}_{c}{N}_{\alpha \beta }^{c}\right)}_{,\alpha }-{\left(\frac{4{B}_{c}{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }+{\left({A}_{c}{N}_{\beta }^{c}\right)}_{,\beta }-{\left(\frac{4{A}_{c}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }+{B}_{c,\alpha }{N}_{\beta \alpha }^{c}-{A}_{c,\beta }{N}_{\alpha }^{c}-\frac{4{B}_{c,\alpha }{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}+\frac{4{A}_{c,\beta }{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}+\frac{8{A}_{c}{B}_{c}{M}_{Q1\beta z}^{c}}{{{h}_{c}}^{2}}+\frac{{A}_{c}{B}_{c}{Q}_{\beta z}^{c}}{{R}_{\beta }^{c}}+\frac{4{A}_{c}{B}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}{R}_{\beta }^{c}}-2{A}_{c}{B}_{c}{I}_{1}^{c}{v}_{1,tt}+\frac{16{A}_{c}{B}_{c}{I}_{3}^{c}{v}_{1,tt}}{{{h}_{c}}^{2}}+\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{3}}-\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{3}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{4}}-\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{1,tt}}{{{h}_{c}}^{4}}-\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{5}}+\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{5}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{2}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{2}}-\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{2}}+\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{2}}+\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{3}}+\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{3}}+\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{4}}-\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{5}}-2{A}_{c}{B}_{c}{I}_{0}^{c}{v}_{0,tt}+\frac{16{A}_{c}{B}_{c}{I}_{2}^{c}{v}_{0,tt}}{{{h}_{c}}^{2}}-\frac{32{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}}{{{h}_{c}}^{4}}=0$$

(49)

$${\delta {w}_{0}^{t}:\left({B}_{t}{Q}_{\alpha z}^{t}\right)}_{,\alpha }+{\left(\frac{{B}_{c}{M}_{Q1\alpha z}^{c}}{{h}_{c}}\right)}_{,\alpha }+{\left(\frac{2{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }+{\left({A}_{t}{Q}_{\beta z}^{t}\right)}_{,\beta }+{\left(\frac{{A}_{c}{M}_{Q1\beta z}^{c}}{{h}_{c}}\right)}_{,\beta }+{\left(\frac{2{A}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }-\frac{{A}_{t}{B}_{t}{N}_{\alpha }^{t}}{{R}_{\alpha }^{t}}-\frac{{A}_{t}{B}_{t}{N}_{\beta }^{t}}{{R}_{\beta }^{t}}-\frac{4{M}_{z}^{c}{A}_{c}{B}_{c}}{{{h}_{c}}^{2}}-\frac{{R}_{z}^{c}{A}_{c}{B}_{c}}{{h}_{c}}-\frac{{A}_{c}{B}_{c}{M}_{1\alpha }^{c}}{{h}_{c}{R}_{\alpha }^{c}}-\frac{2{A}_{c}{B}_{c}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}{R}_{\alpha }^{c}}-\frac{{A}_{c}{B}_{c}{M}_{1\beta }^{c}}{{h}_{c}{R}_{\beta }^{c}}-\frac{2{A}_{c}{B}_{c}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}{R}_{\beta }^{c}}+\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{w}_{0,tt}}{{{h}_{c}}^{3}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{w}_{0,tt}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{w}_{0,tt}^{b}}{{{h}_{c}}^{4}}-\frac{2{A}_{c}{B}_{c}{I}_{1}^{c}{w}_{0,tt}}{{h}_{c}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{w}_{0,tt}}{{{h}_{c}}^{2}}+\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{w}_{0,tt}^{b}}{{{h}_{c}}^{2}}-2{A}_{t}{B}_{t}{I}_{0}^{t}-\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{w}_{0,tt}^{t}}{{{h}_{c}}^{2}}-\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{w}_{0,tt}^{t}}{{{h}_{c}}^{3}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{w}_{0,tt}^{t}}{{{h}_{c}}^{4}}=0$$

(50)

$${\delta {w}_{0}:\left({B}_{c}{Q}_{\alpha z}^{c}\right)}_{,\alpha }-{\left(\frac{4{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }+{\left({A}_{c}{Q}_{\beta z}^{c}\right)}_{,\beta }-{\left(\frac{4{A}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }+\frac{4{A}_{c}{B}_{c}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}{R}_{\beta }^{c}}+\frac{4{A}_{c}{B}_{c}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}{R}_{\alpha }^{c}}+\frac{8{A}_{c}{B}_{c}{M}_{z}^{c}}{{{h}_{c}}^{2}}-\frac{{A}_{c}{B}_{c}{N}_{\alpha }^{c}}{{R}_{\alpha }^{c}}-\frac{{A}_{c}{B}_{c}{N}_{\beta }^{c}}{{R}_{\beta }^{c}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{w}_{0,tt}^{t}}{{{h}_{c}}^{2}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{w}_{0,tt}^{b}}{{{h}_{c}}^{2}}-\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{w}_{0,tt}^{b}}{{{h}_{c}}^{3}}+\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{w}_{0,tt}^{t}}{{{h}_{c}}^{3}}-\frac{2{A}_{c}{B}_{c}{I}_{1}^{c}{w}_{0,tt}^{t}}{{h}_{c}}+\frac{2{A}_{c}{B}_{c}{I}_{1}^{c}{w}_{0,tt}^{b}}{{h}_{c}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{w}_{0,tt}^{b}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{w}_{0,tt}^{t}}{{{h}_{c}}^{4}}-2{A}_{c}{B}_{c}{I}_{0}^{c}{w}_{0,tt}-\frac{32{A}_{c}{B}_{c}{I}_{4}^{c}{w}_{0,tt}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{2}^{c}{w}_{0,tt}}{{{h}_{c}}^{4}}=0$$

(51)

$$\delta {w}_{0}^{b}:{\left({B}_{b}{Q}_{\alpha z}^{b}\right)}_{,\alpha }+{\left({A}_{b}{Q}_{\beta z}^{b}\right)}_{,\beta }+{\left(\frac{2{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }-{\left(\frac{{B}_{c}{M}_{Q1\alpha z}^{c}}{{h}_{c}}\right)}_{,\alpha }+{\left(\frac{2{A}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }-{\left(\frac{{A}_{c}{M}_{Q1\beta z}^{c}}{{h}_{c}}\right)}_{,\beta }-\frac{{A}_{b}{B}_{b}{N}_{\alpha }^{b}}{{R}_{\alpha }^{b}}-\frac{{A}_{b}{B}_{b}{N}_{\beta }^{b}}{{R}_{\beta }^{b}}-\frac{4{A}_{c}{B}_{c}{M}_{z}^{c}}{{{h}_{c}}^{2}}+\frac{{A}_{c}{B}_{c}{R}_{z}^{c}}{{h}_{c}}-\frac{2{A}_{c}{B}_{c}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}{R}_{\alpha }^{c}}+\frac{{A}_{c}{B}_{c}{M}_{1\beta }^{c}}{{h}_{c}{R}_{\beta }^{c}}-\frac{2{A}_{c}{B}_{c}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}{R}_{\beta }^{c}}+\frac{{A}_{c}{B}_{c}{M}_{1\alpha }^{c}}{{h}_{c}{R}_{\alpha }^{c}}-\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{w}_{0,tt}}{{{h}_{c}}^{3}}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{w}_{0,tt}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{w}_{0,tt}^{t}}{{{h}_{c}}^{4}}+\frac{2{A}_{c}{B}_{c}{I}_{1}^{c}{w}_{0,tt}}{{h}_{c}}-\frac{4{A}_{c}{B}_{c}{I}_{2}^{c}{w}_{0,tt}}{{{h}_{c}}^{2}}+\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{w}_{0,tt}^{t}}{{{h}_{c}}^{2}}-2{A}_{b}{B}_{b}{I}_{0}^{b}{w}_{0}^{b}-\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{w}_{0}^{b}}{{{h}_{c}}^{2}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{w}_{0}^{b}}{{{h}_{c}}^{4}}+\frac{8{A}_{c}{B}_{c}{I}_{3}^{c}{w}_{0}^{b}}{{{h}_{c}}^{3}}=0$$

(52)

$${\delta {u}_{1}:\left({B}_{c}{M}_{1\alpha }^{c}\right)}_{,\alpha }-{\left(\frac{4{B}_{c}{M}_{3\alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }+{\left({A}_{c}{M}_{1\beta \alpha }^{c}\right)}_{,\beta }-{\left(\frac{4{A}_{c}{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }+\frac{4{B}_{c,\alpha }{M}_{3\beta }^{c}}{{{h}_{c}}^{2}}-\frac{4{A}_{c,\beta }{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{2}}+{A}_{c,\beta }{M}_{1\alpha \beta }^{c}-{B}_{c,\alpha }{M}_{1\beta }^{c}+\frac{12{A}_{c}{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}}-{A}_{c}{B}_{c}{Q}_{\alpha z}^{c}+\frac{8{A}_{c}{B}_{c}{M}_{Q3\alpha z}^{c}}{{{h}_{c}}^{2}{R}_{\alpha }^{c}}-2{A}_{c}{B}_{c}{I}_{1}^{c}{u}_{0,tt}-\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{5}}+\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{5}}-\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{2}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{3}}+\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{3}}-\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{0,tt}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{u}_{0,tt}^{t}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{3}^{c}{u}_{0,tt}}{{{h}_{c}}^{2}}-\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{u}_{0,tt}^{b}}{{{h}_{c}}^{2}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{3}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{3}}+\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{4}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{5}}-\frac{2{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{b}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{2}}+\frac{2{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{2}}-2{A}_{c}{B}_{c}{I}_{2}^{c}{u}_{1,tt}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{u}_{1,tt}}{{{h}_{c}}^{2}}-\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{u}_{1,tt}}{{{h}_{c}}^{4}}=0$$

(53)

$${\delta {v}_{1}:\left({B}_{c}{M}_{1\alpha \beta }^{c}\right)}_{,\alpha }-{\left(\frac{4{B}_{c}{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }+{\left({A}_{c}{M}_{1\beta }^{c}\right)}_{,\beta }-{\left(\frac{4{A}_{c}{M}_{3\beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }+{B}_{c,\alpha }{M}_{1\beta \alpha }^{c}+\frac{4{A}_{c,\beta }{M}_{3\alpha }^{c}}{{{h}_{c}}^{2}}-\frac{4{B}_{c,\alpha }{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{2}}-{A}_{c,\beta }{M}_{1\alpha }^{c}+\frac{12{A}_{c}{B}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}}-{A}_{c}{B}_{c}{Q}_{\beta z}^{c}+\frac{8{A}_{c}{B}_{c}{M}_{Q3\beta z}^{c}}{{{h}_{c}}^{2}{R}_{\beta }^{c}}-2{A}_{c}{B}_{c}{I}_{1}^{c}{v}_{0,tt}-\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{5}}+\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{5}}+\frac{16{A}_{c}{B}_{c}{I}_{3}^{c}{v}_{0,tt}}{{{h}_{c}}^{2}}-\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{2}}-\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{2}}+\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{3}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{3}}-\frac{32{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{0,tt}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{0,tt}^{b}}{{{h}_{c}}^{4}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{v}_{0,tt}^{t}}{{{h}_{c}}^{4}}-\frac{2{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{2}}+\frac{2{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{2}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{3}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{3}}+\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{4}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{5}}-2{A}_{c}{B}_{c}{I}_{2}^{c}{v}_{1,tt}+\frac{16{A}_{c}{B}_{c}{I}_{4}^{c}{v}_{1,tt}}{{{h}_{c}}^{2}}-\frac{32{A}_{c}{B}_{c}{I}_{6}^{c}{v}_{1,tt}}{{{h}_{c}}^{4}}=0$$

(54)

$${\delta {\psi }_{\alpha }^{t}:\left({B}_{t}{M}_{\alpha }^{t}\right)}_{,\alpha }-{\left(\frac{{B}_{c}{h}_{t}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }-{\left(\frac{2{B}_{c}{h}_{t}{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}\right)}_{,\alpha }+{\left({A}_{t}{M}_{\beta \alpha }^{t}\right)}_{,\beta }-{\left(\frac{{A}_{c}{h}_{t}{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }-{\left(\frac{2{A}_{c}{h}_{t}{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}\right)}_{,\beta }+\frac{6{A}_{c}{B}_{c}{h}_{t}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{3}}+\frac{4{A}_{c}{B}_{c}{h}_{t}{M}_{Q3\alpha z}^{c}}{{{h}_{c}}^{3}{R}_{\alpha }^{c}}+\frac{2{A}_{c}{B}_{c}{h}_{t}{M}_{Q1\alpha z}^{c}}{{{h}_{c}}^{2}}+\frac{{A}_{c}{B}_{c}{h}_{t}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}{R}_{\alpha }^{c}}+{A}_{t,\beta }{M}_{\alpha \beta }^{t}-{A}_{t}{B}_{t}{Q}_{\alpha z}^{t}-\frac{{h}_{t}{A}_{c,\beta }{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}+\frac{{h}_{t}{B}_{c,\alpha }{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}-\frac{2{h}_{t}{A}_{c,\beta }{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}+\frac{2{h}_{t}{B}_{c,\alpha }{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}-\frac{8{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{h}_{t}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{6}}+\frac{2{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{h}_{t}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{4}}-2{A}_{t}{B}_{t}{I}_{1}^{c}{u}_{0,tt}^{t}+\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{t}{u}_{0,tt}}{{{h}_{c}}^{3}}+\frac{2{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{t}{u}_{1,tt}}{{{h}_{c}}^{2}}-\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{u}_{1,tt}}{{{h}_{c}}^{4}}-\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{u}_{0,tt}}{{{h}_{c}}^{5}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{u}_{0,tt}^{t}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{u}_{1,tt}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{u}_{0,tt}^{b}}{{{h}_{c}}^{6}}+\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{u}_{0,tt}^{t}}{{{h}_{c}}^{6}}+\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{h}_{t}{u}_{0,tt}}{{{h}_{c}}^{2}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{u}_{1,tt}}{{{h}_{c}}^{3}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{u}_{0,tt}}{{{h}_{c}}^{4}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{u}_{0,tt}^{b}}{{{h}_{c}}^{4}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{u}_{0,tt}^{t}}{{{h}_{c}}^{4}}-2{A}_{t}{B}_{t}{I}_{2}^{t}{\psi }_{\alpha ,tt}^{t}-\frac{8{A}_{c}{B}_{c}{I}_{6}^{c}{{h}_{t}}^{2}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{6}}-\frac{2{A}_{c}{B}_{c}{I}_{4}^{c}{{h}_{t}}^{2}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{{h}_{t}}^{2}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{5}}=0$$

(55)

$${\delta {\psi }_{\alpha }^{b}:\left({B}_{b}{M}_{\alpha }^{b}\right)}_{,\alpha }+{\left(\frac{{B}_{c}{h}_{b}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }-{\left(\frac{2{B}_{c}{h}_{b}{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}\right)}_{,\alpha }+{\left({A}_{b}{M}_{\beta \alpha }^{b}\right)}_{,\beta }+{\left(\frac{{A}_{c}{h}_{b}{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }-{\left(\frac{2{A}_{c}{h}_{b}{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}\right)}_{,\beta }+\frac{4{A}_{c}{B}_{c}{h}_{b}{M}_{Q3\alpha z}^{c}}{{{h}_{c}}^{3}{R}_{\alpha }^{c}}+\frac{6{A}_{c}{B}_{c}{h}_{b}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{3}}-\frac{2{A}_{c}{B}_{c}{h}_{b}{M}_{Q1\alpha z}^{c}}{{{h}_{c}}^{2}}-\frac{{A}_{c}{B}_{c}{h}_{b}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}{R}_{\alpha }^{c}}-{A}_{b}{B}_{b}{Q}_{\alpha z}^{b}+{A}_{b,\beta }{M}_{\alpha \beta }^{b}+\frac{{h}_{b}{A}_{c,\beta }{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}-\frac{{h}_{b}{B}_{c,\alpha }{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}-\frac{2{h}_{b}{A}_{c,\beta }{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}+\frac{2{h}_{b}{B}_{c,\beta }{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}-\frac{8{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{6}}+\frac{2{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{h}_{t}{\psi }_{\alpha ,tt}^{t}}{{{h}_{c}}^{4}}-2{A}_{b}{B}_{b}{I}_{1}^{b}{u}_{0,tt}^{b}-\frac{2{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{b}{u}_{1,tt}}{{{h}_{c}}^{2}}+\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{u}_{1,tt}}{{{h}_{c}}^{4}}-\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{u}_{0,tt}}{{{h}_{c}}^{5}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{u}_{0,tt}^{b}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{u}_{1,tt}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{u}_{0,tt}^{b}}{{{h}_{c}}^{6}}+\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{u}_{0,tt}^{t}}{{{h}_{c}}^{6}}-\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{h}_{b}{u}_{0,tt}}{{{h}_{c}}^{2}}+\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{b}{u}_{0,tt}}{{{h}_{c}}^{3}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{u}_{1,tt}}{{{h}_{c}}^{3}}+\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{u}_{0,tt}}{{{h}_{c}}^{4}}-\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{u}_{0,tt}^{b}}{{{h}_{c}}^{4}}-\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{u}_{0,tt}^{t}}{{{h}_{c}}^{4}}-2{A}_{b}{B}_{b}{I}_{2}^{b}{\psi }_{\alpha ,tt}^{b}-\frac{8{A}_{c}{B}_{c}{I}_{6}^{c}{{h}_{b}}^{2}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{6}}-\frac{2{A}_{c}{B}_{c}{I}_{4}^{c}{{h}_{b}}^{2}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{4}}+\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{{h}_{b}}^{2}{\psi }_{\alpha ,tt}^{b}}{{{h}_{c}}^{5}}=0$$

(56)

$${\delta {\psi }_{\beta }^{t}:\left({B}_{t}{M}_{\alpha \beta }^{t}\right)}_{,\alpha }-{\left(\frac{{B}_{c}{h}_{t}{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }-{\left(\frac{{2B}_{c}{h}_{t}{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}\right)}_{,\alpha }+{\left({A}_{t}{M}_{\beta }^{t}\right)}_{,\beta }-{\left(\frac{{A}_{c}{h}_{t}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }-{\left(\frac{2{A}_{c}{h}_{t}{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}\right)}_{,\beta }+\frac{2{A}_{c}{B}_{c}{h}_{t}{M}_{Q1\beta z}^{c}}{{{h}_{c}}^{2}}+\frac{{A}_{c}{B}_{c}{h}_{t}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}{R}_{\beta }^{c}}+\frac{4{A}_{c}{B}_{c}{h}_{t}{M}_{Q3\beta z}^{c}}{{{h}_{c}}^{3}{R}_{\beta }^{c}}+\frac{6{A}_{c}{B}_{c}{h}_{t}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{3}}-{A}_{t,\beta }{M}_{\alpha }^{t}-{B}_{t,\alpha }{M}_{\beta }^{t}+{B}_{t,\alpha }{M}_{\beta \alpha }^{t}-{A}_{t}{B}_{t}{Q}_{\beta z}^{t}-\frac{{h}_{t}{B}_{c,\alpha }{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}+\frac{{h}_{t}{A}_{c,\beta }{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}-\frac{2{h}_{t}{B}_{c,\alpha }{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}+\frac{2{h}_{t}{A}_{c,\beta }{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}+\frac{2{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{h}_{t}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{h}_{t}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{6}}-2{A}_{t}{B}_{t}{I}_{1}^{t}{v}_{0,tt}^{t}+\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{t}{v}_{0,tt}}{{{h}_{c}}^{3}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{v}_{1,tt}}{{{h}_{c}}^{3}}-\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{v}_{0,tt}}{{{h}_{c}}^{4}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{v}_{0,tt}^{b}}{{{h}_{c}}^{4}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{t}{v}_{0,tt}^{t}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{v}_{1,tt}}{{{h}_{c}}^{4}}-\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{v}_{0,tt}}{{{h}_{c}}^{5}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{t}{v}_{0,tt}^{t}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{v}_{1,tt}}{{{h}_{c}}^{5}}+\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{h}_{t}{v}_{0,tt}}{{{h}_{c}}^{2}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{v}_{0,tt}^{b}}{{{h}_{c}}^{6}}+\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{t}{v}_{0,tt}^{t}}{{{h}_{c}}^{6}}+\frac{2{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{t}{v}_{1,tt}}{{{h}_{c}}^{2}}-2{A}_{t}{B}_{t}{I}_{2}^{t}{\psi }_{\beta ,tt}^{t}-\frac{8{A}_{c}{B}_{c}{I}_{6}^{c}{{h}_{t}}^{2}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{6}}-\frac{2{A}_{c}{B}_{c}{I}_{4}^{c}{{h}_{t}}^{2}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{{h}_{t}}^{2}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{5}}=0$$

(57)

$${\delta {\psi }_{\beta }^{b}:\left({B}_{b}{M}_{\alpha \beta }^{b}\right)}_{,\alpha }+{\left(\frac{{B}_{c}{h}_{b}{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\alpha }-{\left(\frac{{2B}_{c}{h}_{b}{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}\right)}_{,\alpha }+{\left({A}_{b}{M}_{\beta }^{b}\right)}_{,\beta }+{\left(\frac{{A}_{c}{h}_{b}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}\right)}_{,\beta }-{\left(\frac{2{A}_{c}{h}_{b}{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}\right)}_{,\beta }-\frac{2{A}_{c}{B}_{c}{h}_{b}{M}_{Q1\beta z}^{c}}{{{h}_{c}}^{2}}-\frac{{A}_{c}{B}_{c}{h}_{b}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}{R}_{\beta }^{c}}+\frac{4{A}_{c}{B}_{c}{h}_{b}{M}_{Q3\beta z}^{c}}{{{h}_{c}}^{3}{R}_{\beta }^{c}}+\frac{6{A}_{c}{B}_{c}{h}_{b}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{3}}-{A}_{b,\beta }{M}_{\alpha }^{b}-{B}_{b,\alpha }{M}_{\beta }^{b}+{B}_{b,\alpha }{M}_{\beta \alpha }^{b}-{A}_{b}{B}_{b}{Q}_{\beta z}^{b}+\frac{{h}_{b}{B}_{c,\alpha }{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}-\frac{{h}_{b}{A}_{c,\beta }{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}-\frac{2{h}_{b}{B}_{c,\alpha }{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}+\frac{2{h}_{b}{A}_{c,\beta }{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}+\frac{2{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{4}}-\frac{8{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{h}_{t}{\psi }_{\beta ,tt}^{t}}{{{h}_{c}}^{6}}-2{A}_{b}{B}_{b}{I}_{1}^{b}{v}_{0,tt}^{b}+\frac{4{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{b}{v}_{0,tt}}{{{h}_{c}}^{3}}+\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{v}_{1,tt}}{{{h}_{c}}^{3}}+\frac{8{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{v}_{0,tt}}{{{h}_{c}}^{4}}-\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{v}_{0,tt}^{b}}{{{h}_{c}}^{4}}-\frac{4{A}_{c}{B}_{c}{I}_{4}^{c}{h}_{b}{v}_{0,tt}^{t}}{{{h}_{c}}^{4}}+\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{v}_{1,tt}}{{{h}_{c}}^{4}}-\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{v}_{0,tt}}{{{h}_{c}}^{5}}+\frac{16{A}_{c}{B}_{c}{I}_{5}^{c}{h}_{b}{v}_{0,tt}^{b}}{{{h}_{c}}^{5}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{v}_{1,tt}}{{{h}_{c}}^{5}}-\frac{2{A}_{c}{B}_{c}{I}_{2}^{c}{h}_{b}{v}_{0,tt}}{{{h}_{c}}^{2}}-\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{v}_{0,tt}^{b}}{{{h}_{c}}^{6}}+\frac{16{A}_{c}{B}_{c}{I}_{6}^{c}{h}_{b}{v}_{0,tt}^{t}}{{{h}_{c}}^{6}}-\frac{2{A}_{c}{B}_{c}{I}_{3}^{c}{h}_{b}{v}_{1,tt}}{{{h}_{c}}^{2}}-2{A}_{b}{B}_{b}{I}_{2}^{b}{\psi }_{\beta ,tt}^{b}-\frac{8{A}_{c}{B}_{c}{I}_{6}^{c}{{h}_{b}}^{2}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{6}}-\frac{2{A}_{c}{B}_{c}{I}_{4}^{c}{{h}_{b}}^{2}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{4}}+\frac{8{A}_{c}{B}_{c}{I}_{5}^{c}{{h}_{b}}^{2}{\psi }_{\beta ,tt}^{b}}{{{h}_{c}}^{5}}=0$$

(58)

Boundary conditions at $\alpha ={\alpha }_{i}$ and ${\alpha }_{i}=0, a$:

$${B}_{t}{N}_{\alpha }^{t}+\frac{4{B}_{c}{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}+\frac{2{B}_{c}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{u}^{t}}^{\alpha ={\alpha }_{i}}{u}_{0}^{t}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {u}_{0}^{t}=0$$

(59)

$${B}_{c}{N}_{\alpha }^{c}-\frac{4{B}_{c}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{u}_{0}}^{\alpha ={\alpha }_{i}}{u}_{0}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {u}_{0}=0$$

(60)

$${B}_{b}{N}_{\alpha }^{b}-\frac{4{B}_{c}{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}+\frac{2{B}_{c}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{u}^{b}}^{\alpha ={\alpha }_{i}}{u}_{0}^{b}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {u}_{0}^{b}=0$$

(61)

$${B}_{t}{N}_{\alpha \beta }^{t}+\frac{4{B}_{c}{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}+\frac{2{B}_{c}{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{v}^{t}}^{\alpha ={\alpha }_{i}}{v}_{0}^{t}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {v}_{0}^{t}=0$$

(62)

$${B}_{b}{N}_{\alpha \beta }^{b}-\frac{4{B}_{c}{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}+\frac{2{B}_{c}{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{v}^{b}}^{\alpha ={\alpha }_{i}}{v}_{0}^{b}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {v}_{0}^{b}=0$$

(63)

$${B}_{c}{N}_{\alpha \beta }^{c}-\frac{4{B}_{c}{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{v}_{0}}^{\alpha ={\alpha }_{i}}{v}_{0}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {v}_{0}=0$$

(64)

$${B}_{t}{Q}_{\alpha z}^{t}+\frac{{B}_{c}{M}_{Q1\alpha z}^{c}}{{h}_{c}}+\frac{2{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}}\pm {K}_{{w}_{0}^{t}}^{\alpha ={\alpha }_{i}}{w}_{0}^{t}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {w}_{0}^{t}=0$$

(65)

$${B}_{b}{Q}_{\alpha z}^{b}+\frac{2{B}_{c}{M}_{Q2\alpha z}^{c}}{{{h}_{c}}^{2}}-\frac{{B}_{c}{M}_{Q1\alpha z}^{c}}{{h}_{c}}\pm {K}_{{w}_{0}^{b}}^{\alpha ={\alpha }_{i}}{w}_{0}^{b}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {w}_{0}^{b}=0$$

(66)

$${B}_{c}{M}_{1\alpha }^{c}-\frac{4{B}_{c}{M}_{3\alpha }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{u}_{1}}^{\alpha ={\alpha }_{i}}{u}_{1}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {u}_{1}=0$$

(67)

$${B}_{c}{M}_{1\alpha \beta }^{c}-\frac{4{B}_{c}{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{v}_{1}}^{\alpha ={\alpha }_{i}}{v}_{1}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {v}_{1}=0$$

(68)

$${B}_{t}{M}_{\alpha }^{t}-\frac{{B}_{c}{h}_{t}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}-\frac{2{B}_{c}{h}_{t}{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{{\psi }_{\alpha }}^{t}}^{\alpha ={\alpha }_{i}}{{\psi }_{\alpha }}^{t}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or } \; \delta {\psi }_{\alpha }^{t}=0$$

(69)

$${B}_{b}{M}_{\alpha }^{b}+\frac{{B}_{c}{h}_{b}{M}_{2\alpha }^{c}}{{{h}_{c}}^{2}}-\frac{2{B}_{c}{h}_{b}{M}_{3\alpha }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{{\psi }_{\alpha }}^{b}}^{\alpha ={\alpha }_{i}}{{\psi }_{\alpha }}^{b}\left({\alpha }_{i},\beta \right)=0 \; \mathrm{ or }\;\delta {\psi }_{\alpha }^{b}=0$$

(70)

$${B}_{t}{M}_{\alpha \beta }^{t}-\frac{{B}_{c}{h}_{t}{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}-\frac{{2B}_{c}{h}_{t}{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{{\psi }_{\beta }}^{t}}^{\alpha ={\alpha }_{i}}{{\psi }_{\beta }}^{t}\left({\alpha }_{i},\beta \right)=0\; \mathrm{ or } \; \delta {\psi }_{\beta }^{t}=0$$

(71)

$${B}_{b}{M}_{\alpha \beta }^{b}+\frac{{B}_{c}{h}_{b}{M}_{2\alpha \beta }^{c}}{{{h}_{c}}^{2}}-\frac{{2B}_{c}{h}_{b}{M}_{3\alpha \beta }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{{\psi }_{\beta }}^{b}}^{\alpha ={\alpha }_{i}}{{\psi }_{\beta }}^{b}\left({\alpha }_{i},\beta \right)=0 \;\mathrm{or } \; \delta {\psi }_{\beta }^{b}=0$$

(72)

Boundary conditions at $\beta ={\beta }_{i}$ and ${\beta }_{i}=0,b$:

$${A}_{t}{N}_{\beta \alpha }^{t}+\frac{2{A}_{c}{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}+\frac{4{A}_{c}{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{u}^{t}}^{\beta ={\beta }_{i}}{u}_{0}^{t}\left(\alpha ,{\beta }_{i}\right)=0\; \mathrm{or }\; \delta {u}_{0}^{t}=0$$

(73)

$${A}_{c}{N}_{\beta \alpha }^{c}-\frac{4{A}_{c}{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{u}_{0}}^{\beta ={\beta }_{i}}{u}_{0}\left(\alpha ,{\beta }_{i}\right)=0 \; \mathrm{or } \; \delta {u}_{0}=0$$

(74)

$${A}_{b}{N}_{\beta \alpha }^{b}+\frac{2{A}_{c}{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}-\frac{4{A}_{c}{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{u}^{b}}^{\beta ={\beta }_{i}}{u}_{0}^{b}\left(\alpha ,{\beta }_{i}\right)=0 \; \mathrm{or } \; \delta {u}_{0}^{b}=0$$

(75)

$${A}_{t}{N}_{\beta }^{t}+\frac{4{A}_{c}{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}+\frac{2{A}_{c}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{v}^{t}}^{\beta ={\beta }_{i}}{v}_{0}^{t}\left(\alpha ,{\beta }_{i}\right)=0\; \mathrm{or } \; \delta {v}_{0}^{t}=0$$

(76)

$${A}_{b}{N}_{\beta }^{b}+\frac{2{A}_{c}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}-\frac{4{A}_{c}{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{v}^{b}}^{\beta ={\beta }_{i}}{v}_{0}^{b}\left(\alpha ,{\beta }_{i}\right)=0 \; \mathrm{or } \; \delta {v}_{0}^{b}=0$$

(77)

$${A}_{c}{N}_{\beta }^{c}-\frac{4{A}_{c}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{v}_{0}}^{\beta ={\beta }_{i}}{v}_{0}\left(\alpha ,{\beta }_{i}\right)=0 \; \mathrm{or } \;\delta {v}_{0}=0$$

(78)

$${A}_{t}{Q}_{\beta z}^{t}+\frac{{A}_{c}{M}_{Q1\beta z}^{c}}{{h}_{c}}+\frac{2{A}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}}\pm {K}_{{w}^{t}}^{\beta ={\beta }_{i}}{w}_{0}^{t}\left(\alpha ,{\beta }_{i}\right)=0 \; \mathrm{or } \; \delta {w}_{0}^{t}=0$$

(79)

$${A}_{b}{Q}_{\beta z}^{b}+\frac{2{A}_{c}{M}_{Q2\beta z}^{c}}{{{h}_{c}}^{2}}-\frac{{A}_{c}{M}_{Q1\beta z}^{c}}{{h}_{c}}\pm {K}_{{w}^{b}}^{\beta ={\beta }_{i}}{w}_{0}^{b}\left(\alpha ,{\beta }_{i}\right)=0\; \mathrm{or }\;\delta {w}_{0}^{b}=0$$

(80)

$${A}_{c}{M}_{1\beta \alpha }^{c}-\frac{4{A}_{c}{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{u}_{1}}^{\beta ={\beta }_{i}}{u}_{1}\left(\alpha ,{\beta }_{i}\right)=0 \; \mathrm{or } \; \delta {u}_{1}=0$$

(81)

$${M}_{1\beta }^{c}-\frac{4{A}_{c}{M}_{3\beta }^{c}}{{{h}_{c}}^{2}}\pm {K}_{{v}_{1}}^{\beta ={\beta }_{i}}{v}_{1}\left(\alpha ,{\beta }_{i}\right)=0 \; \mathrm{or }\; \delta {v}_{1}=0$$

(82)

$${A}_{t}{M}_{\beta \alpha }^{t}-\frac{{A}_{c}{h}_{t}{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}-\frac{2{A}_{c}{h}_{t}{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{\psi }_{\alpha }^{t}}^{\beta ={\beta }_{i}}{\psi }_{\alpha }^{t}\left(\alpha ,{\beta }_{i}\right)=0\; \mathrm{or }\;\delta {\psi }_{\alpha }^{t}=0$$

(83)

$${A}_{b}{M}_{\beta \alpha }^{b}+\frac{{A}_{c}{h}_{b}{M}_{2\beta \alpha }^{c}}{{{h}_{c}}^{2}}-\frac{2{A}_{c}{h}_{b}{M}_{3\beta \alpha }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{\psi }_{\alpha }^{b}}^{\beta ={\beta }_{i}}{\psi }_{\alpha }^{b}\left(\alpha ,{\beta }_{i}\right)=0 \mathrm{or }\delta {\psi }_{\alpha }^{b}=0$$

(84)

$${A}_{t}{M}_{\beta }^{t}-\frac{{A}_{c}{h}_{t}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}-\frac{2{A}_{c}{h}_{t}{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{\psi }_{\beta }^{t}}^{\beta ={\beta }_{i}}{\psi }_{\beta }^{t}\left(\alpha ,{\beta }_{i}\right)=0 \delta {\psi }_{\beta }^{t}=0$$

(85)

$${A}_{b}{M}_{\beta }^{b}+\frac{{A}_{c}{h}_{b}{M}_{2\beta }^{c}}{{{h}_{c}}^{2}}-\frac{2{A}_{c}{h}_{b}{M}_{3\beta }^{c}}{{{h}_{c}}^{3}}\pm {K}_{{\psi }_{\beta }^{b}}^{\beta ={\beta }_{i}}{\psi }_{\beta }^{b}\left(\alpha ,{\beta }_{i}\right)=0 \delta {\psi }_{\beta }^{b}=0$$

(86)

Appendix 4

The relation between the displacements and rotations with their nodal values is as follows:

$$\begin{aligned} & u_{0}^{t} = \left[ {N_{{u_{0}^{t} }} } \right]\left\{ \delta \right\} = \left[ {N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{{14}} } \right]\left\{ \delta \right\} \\ & \psi _{\alpha }^{t} = \left[ {N_{{\psi _{\alpha }^{t} }} } \right]\left\{ \delta \right\} = \left[ {0,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{{13}} } \right]\left\{ \delta \right\} \\ & v_{0}^{t} = \left[ {N_{{v_{0}^{t} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{2} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{5} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{{12}} } \right]\left\{ \delta \right\} \\ & \psi _{\beta }^{t} = \left[ {N_{{\psi _{\beta }^{t} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{3} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{{11}} } \right]\left\{ \delta \right\} \\ & w_{0}^{t} = \left[ {N_{{w_{0}^{t} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{4} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{{10}} } \right]\left\{ \delta \right\} \\ & u_{0}^{b} = \left[ {N_{{u_{0}^{b} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{5} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{9} } \right]\left\{ \delta \right\} \\ & \psi _{\alpha }^{t} = \left[ {N_{{\psi _{\alpha }^{b} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{6} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{8} } \right]\left\{ \delta \right\} \\ & v_{0}^{t} = \left[ {N_{{v_{0}^{b} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{7} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{7} } \right]\left\{ \delta \right\} \\ & \psi _{\beta }^{t} = \left[ {N_{{\psi _{\beta }^{b} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{8} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{6} } \right]\left\{ \delta \right\} \\ & w_{0}^{t} = \left[ {N_{{w_{0}^{b} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{9} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{5} } \right]\left\{ \delta \right\} \\ & u_{0} = \left[ {N_{{u_{0} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{{10}} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{4} } \right]\left\{ \delta \right\} \\ & u_{1} = \left[ {N_{{u_{1} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{{11}} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{3} } \right]\left\{ \delta \right\} \\ & v_{0} = \left[ {N_{{v_{0} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{{12}} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,\left\{ 0 \right\}_{2} } \right]\left\{ \delta \right\} \\ & ~v_{1} = \left[ {N_{{v_{1} }} } \right]\left\{ \delta \right\} = \left[ {\left\{ 0 \right\}_{{13}} ,N_{1} ,\left\{ 0 \right\}_{{14}} ,N_{2} ,\left\{ 0 \right\}_{{14}} ,N_{3} ,\left\{ 0 \right\}_{{14}} ,N_{4} ,\left\{ 0 \right\}_{{14}} ,N_{5} ,\left\{ 0 \right\}_{{14}} ,N_{6} ,\left\{ 0 \right\}_{{14}} ,N_{7} ,\left\{ 0 \right\}_{{14}} ,N_{8} ,\left\{ 0 \right\}_{{14}} ,N_{9} ,0} \right]\left\{ \delta \right\} \\ \end{aligned}$$

(87)