Appendix
Governing equations of motion can be presented in terms of displacement components as:
$$ \begin{aligned} & \delta u: - Q_{110} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }}u} \right) + Q_{111} \left( {\frac{{\partial^{3} }}{{\partial x^{3} }}w_{\text{b}} } \right) + Q_{113} \left( {\frac{{\partial^{3} }}{{\partial x^{3} }}w_{\text{s}} } \right) - Q_{120} \left( {\frac{{\partial^{2} }}{\partial x\partial y}v} \right) + Q_{121} \left( {\frac{\partial 3}{{\partial x\partial y^{2} }}w_{\text{b}} } \right) + Q_{123} \left( {\frac{{\partial^{3} }}{{\partial x\partial y^{2} }}w_{\text{s}} } \right) \\ & \quad - Q_{136} \left( {\frac{\partial }{\partial x}\lambda } \right) - E_{310} \left( {\frac{\partial }{\partial x}\Psi } \right) - Q_{660} \left( {\frac{{\partial^{2} }}{{\partial y^{2} }}u} \right) + 2Q_{661} \left( {\frac{{\partial^{3} }}{{\partial x\partial y^{2} }}w_{\text{b}} } \right) + 2Q_{663} \left( {\frac{{\partial^{3} }}{{\partial x\partial y^{2} }}w_{\text{s}} } \right) - Q_{660} \left( {\frac{{\partial^{2} }}{\partial x\partial y}v} \right) \\ & \quad - \frac{1}{4}l^{2} T_{0} \left( {\frac{{\partial^{4} }}{{\partial x\partial y^{3} }}v} \right) + \frac{1}{4}l^{2} T_{0} \left( {\frac{{\partial^{4} }}{{\partial y^{4} }}u} \right) - \frac{1}{4}l^{2} T_{0} \left( {\frac{{\partial^{4} }}{{\partial x^{3} \partial y}}v} \right) + \frac{1}{4}l^{2} T_{0} \left( {\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}u} \right) - I_{0} \left( {\frac{{\partial^{2} }}{{\partial t^{2} }}u} \right) + I_{1} \left( {\frac{{\partial^{3} }}{{\partial t^{2} \partial x}}w_{\text{b}} } \right) \\ & \quad + I_{3} \left( {\frac{{\partial^{3} }}{{\partial t^{2} \partial x}}w_{\text{s}} } \right) = 0. \\ \end{aligned} $$
(40)
$$ \begin{aligned} & \delta v: - Q_{120} \left( {\frac{{\partial^{2} }}{\partial x\partial y}u} \right) + Q_{121} \left( {\frac{{\partial^{3} }}{{\partial x^{2} \partial y}}w_{\text{b}} } \right) + Q_{123} \left( {\frac{{\partial^{3} }}{{\partial x^{2} \partial y}}w_{\text{s}} } \right) - Q_{220} \left( {\frac{{\partial^{2} }}{{\partial y^{2} }}v} \right) + Q_{221} \left( {\frac{{\partial^{3} }}{{\partial y^{3} }}w_{\text{b}} } \right) + Q_{223} \left( {\frac{{\partial^{3} }}{{\partial y^{3} }}w_{\text{s}} } \right) \\ & \quad - Q_{236} \left( {\frac{\partial }{\partial y}\lambda } \right) - E_{320} \left( {\frac{\partial }{\partial y}\Psi } \right) - Q_{660} \left( {\frac{{\partial^{2} }}{\partial y\partial x}u} \right) + 2Q_{661} \left( {\frac{{\partial^{3} }}{{\partial x^{2} \partial y}}w_{\text{b}} } \right) + 2Q_{663} \left( {\frac{{\partial^{3} }}{{\partial x^{2} \partial y}}w_{\text{s}} } \right) - Q_{660} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }}v} \right) \\ & \quad + \frac{1}{4}l^{2} T_{0} \left( {\frac{{\partial^{4} }}{{\partial x^{4} }}v} \right) - \frac{1}{4}l^{2} T_{0} \left( {\frac{{\partial^{4} }}{{\partial x^{3} \partial y}}u} \right) + \frac{1}{4}l^{2} T_{0} \left( {\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}v} \right) - \frac{1}{4}l^{2} T_{0} \left( {\frac{{\partial^{4} }}{{\partial x\partial y^{3} }}u} \right) - I_{0} \left( {\frac{{\partial^{2} }}{{\partial t^{2} }}v} \right) + I_{1} \left( {\frac{{\partial^{3} }}{{\partial t^{2} \partial y}}w_{\text{b}} } \right) \\ & \quad + I_{3} \left( {\frac{{\partial^{3} }}{{\partial t^{2} \partial y}}w_{\text{s}} } \right) = 0. \\ \end{aligned} $$
(41)
$$ \begin{aligned} & \delta w_{\text{b}} : - Q_{121} (\frac{{\partial^{3} }}{{\partial x\partial y^{2} }}u) + 2Q_{122} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{b}} ) + 2Q_{124} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{s}} ) - Q_{221} (\frac{{\partial^{3} }}{{\partial y^{3} }}v) + Q_{222} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{b}} ) + Q_{224} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{s}} ) \\ & \quad - Q_{237} (\frac{{\partial^{2} }}{{\partial y^{2} }}\lambda ) - E_{321} (\frac{{\partial^{2} }}{{\partial y^{2} }}\Psi ) + 4Q_{662} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{b}} ) + 4Q_{664} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{s}} ) - 2Q_{661} (\frac{{\partial^{3} }}{{\partial y^{2} \partial x}}u) - Q_{111} (\frac{{\partial^{3} }}{{\partial x^{3} }}u) \\ & \quad + Q_{112} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{b}} ) + Q_{114} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{s}} ) - Q_{121} (\frac{{\partial^{3} }}{{\partial x^{2} \partial y}}v) + 2l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{b}} ) + l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{s}} ) - 2Q_{661} (\frac{{\partial^{3} }}{{\partial x^{2} \partial y}}v) \\ & \quad + l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}\lambda ) - l^{2} T_{2} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{s}} ) + l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{b}} ) + \frac{1}{2}l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{s}} ) + \frac{1}{2}l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial y^{4} }}\lambda ) + \frac{1}{2}l^{2} T_{2} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{s}} ) \\ & \quad + l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{b}} ) + \frac{1}{2}l^{2} T_{2} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{s}} ) + \frac{1}{2}l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{s}} ) + \frac{1}{2}l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial x^{4} }}\lambda ) - Q_{137} (\frac{{\partial^{4} }}{{\partial x^{2} }}\lambda ) - E_{311} (\frac{{\partial^{4} }}{{\partial x^{2} }}\Psi ) \\ & \quad - I_{1} (\frac{{\partial^{3} }}{{\partial t^{2} \partial x}}u) + I_{2} (\frac{{\partial^{4} }}{{\partial t^{2} \partial x^{2} }}w_{\text{b}} ) + I_{5} (\frac{{\partial^{4} }}{{\partial t^{2} \partial x^{2} }}w_{\text{s}} ) - I_{1} (\frac{{\partial^{3} }}{{\partial t^{2} \partial y}}v) + I_{2} (\frac{{\partial^{4} }}{{\partial t^{2} \partial y^{2} }}w_{\text{b}} ) + I_{5} (\frac{{\partial^{4} }}{{\partial t^{2} \partial y^{2} }}w_{\text{s}} ) \\ & \quad - I_{0} (\frac{{\partial^{2} }}{{\partial t^{2} }}w_{\text{b}} ) - I_{0} (\frac{{\partial^{2} }}{{\partial t^{2} }}w_{\text{s}} ) - I_{4} (\frac{{\partial^{2} }}{{\partial t^{2} }}\lambda ) = 0. \\ \end{aligned} $$
(42)
$$ \begin{aligned} & \delta w_{\text{s}} :\frac{1}{2}l^{2} T_{2} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{b}} ) + \frac{1}{4}l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{s}} ) + \frac{1}{4}l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial x^{4} }}\lambda ) + Q_{114} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{b}} ) + \frac{1}{2}l^{2} T_{2} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{b}} ) + \frac{1}{4}l^{2} T_{5} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{s}} ) \\ & \quad + \frac{1}{4}l^{2} T_{4} (\frac{{\partial^{4} }}{{\partial x^{4} }}\lambda ) - \frac{1}{4}l^{2} T_{6} (\frac{{\partial^{2} }}{{\partial x^{2} }}w_{\text{s}} ) - \frac{1}{4}l^{2} T_{6} (\frac{{\partial^{2} }}{{\partial y^{2} }}w_{\text{s}} ) - \frac{1}{4}l^{2} T_{7} (\frac{{\partial^{2} }}{{\partial x^{2} }}\lambda ) - \frac{1}{4}l^{2} T_{7} (\frac{{\partial^{2} }}{{\partial y^{2} }}\lambda ) - Q_{113} (\frac{{\partial^{3} }}{{\partial x^{3} }}u) \\ & \quad - Q_{123} (\frac{{\partial^{3} }}{{\partial x^{2} \partial y}}v) + l^{2} T_{2} (\frac{{\partial^{4} }}{{\partial y^{2} \partial x^{2} }}w_{\text{b}} ) + l^{2} T_{2} (\frac{{\partial^{4} }}{{\partial y^{2} \partial x^{2} }}w_{\text{s}} ) - E_{132} (\frac{{\partial^{2} }}{{\partial x^{2} }}\lambda ) - Q_{123} (\frac{{\partial^{3} }}{{\partial y^{2} \partial x}}u) + Q_{115} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{s}} ) \\ & \quad + \frac{1}{2}l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{s}} ) + E_{2410} (\frac{{\partial^{2} }}{{\partial y^{2} }}\Psi ) - Q_{223} (\frac{{\partial^{3} }}{{\partial y^{3} }}v) + Q_{224} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{b}} ) + Q_{225} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{s}} ) - E_{232} (\frac{{\partial^{2} }}{{\partial y^{2} }}\lambda ) \\ & \quad - Q_{138} (\frac{{\partial^{2} }}{{\partial x^{2} }}\lambda ) - Q_{4410} (\frac{{\partial^{2} }}{{\partial y^{2} }}\Psi ) - E_{323} (\frac{{\partial^{2} }}{{\partial y^{2} }}\Psi ) - 2Q_{663} (\frac{{\partial^{3} }}{{\partial x\partial y^{2} }}u) - Q_{4410} (\frac{{\partial^{2} }}{{\partial y^{2} }}w_{\text{s}} ) - Q_{238} (\frac{{\partial^{2} }}{{\partial y^{2} }}\lambda ) \\ & \quad + l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{b}} ) + \frac{1}{2}l^{2} T_{4} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}\lambda ) + \frac{1}{2}l^{2} T_{5} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{s}} ) + \frac{1}{2}l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{b}} ) + \frac{1}{2}l^{2} T_{2} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{s}} ) \\ & \quad + \frac{1}{4}l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{s}} ) + \frac{1}{4}l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial y^{4} }}\lambda ) + 4Q_{664} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{b}} ) - Q_{5510} (\frac{{\partial^{2} }}{{\partial x^{2} }}w_{\text{s}} ) + 4Q_{665} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{s}} ) \\ & \quad + \frac{1}{2}l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}\lambda ) + \frac{1}{2}l^{2} T_{2} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{s}} ) + \frac{1}{4}l^{2} T_{4} (\frac{{\partial^{4} }}{{\partial y^{4} }}\lambda ) - 2Q_{663} (\frac{{\partial^{3} }}{{\partial x^{2} \partial y}}v) + 2Q_{125} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{s}} ) \\ & \quad - E_{313} (\frac{{\partial^{2} }}{{\partial x^{2} }}\Psi ) + E_{1510} (\frac{{\partial^{2} }}{{\partial x^{2} }}\Psi ) - Q_{5510} (\frac{{\partial^{2} }}{{\partial x^{2} }}\Psi ) + \frac{1}{4}l^{2} T_{5} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{s}} ) + 2Q_{124} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{b}} ) + \frac{1}{2}l^{2} T_{0} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{b}} ) \\ & \quad - I_{3} (\frac{{\partial^{3} }}{{\partial t^{2} \partial x}}u) + I_{6} (\frac{{\partial^{4} }}{{\partial t^{2} \partial x^{2} }}w_{\text{s}} ) - I_{3} (\frac{{\partial^{3} }}{{\partial t^{2} \partial y}}v) + I_{5} (\frac{{\partial^{4} }}{{\partial t^{2} \partial y^{2} }}w_{\text{b}} ) + I_{6} (\frac{{\partial^{4} }}{{\partial t^{2} \partial y^{2} }}w_{\text{s}} ) - I_{0} (\frac{{\partial^{2} }}{{\partial t^{2} }}w_{\text{b}} ) \\ & \quad - I_{0} (\frac{{\partial^{2} }}{{\partial t^{2} }}w_{\text{s}} ) - I_{4} (\frac{{\partial^{2} }}{{\partial t^{2} }}\lambda ) + I_{5} (\frac{{\partial^{4} }}{{\partial t^{2} \partial x^{2} }}w_{\text{b}} ) = 0. \\ \end{aligned} $$
(43)
$$ \begin{aligned} & \delta \lambda :\frac{1}{4}l^{2} T_{4} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{s}} ) + \frac{1}{2}l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{b}} ) - \frac{1}{4}l^{2} T_{7} (\frac{{\partial^{2} }}{{\partial x^{2} }}w_{\text{s}} ) + \frac{1}{4}l^{2} T_{4} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{s}} ) - \frac{1}{4}l^{2} T_{9} (\frac{{\partial^{2} }}{{\partial y^{2} }}\lambda ) - \frac{1}{4}l^{2} T_{7} (\frac{{\partial^{2} }}{{\partial y^{2} }}w_{\text{s}} ) \\ & \quad - \frac{1}{4}l^{2} T_{9} (\frac{{\partial^{2} }}{{\partial x^{2} }}\lambda ) - Q_{5510} (\frac{{\partial^{2} }}{{\partial x^{2} }}w_{\text{s}} ) - Q_{5510} (\frac{{\partial^{2} }}{{\partial x^{2} }}\Psi ) + E_{1510} (\frac{{\partial^{2} }}{{\partial x^{2} }}\Psi ) + l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial x^{2} \partial y^{2} }}w_{\text{b}} ) - Q_{4410} (\frac{{\partial^{2} }}{{\partial y^{2} }}\Psi ) \\ & \quad + \frac{1}{2}l^{2} T_{3} (\frac{{\partial^{4} }}{{\partial y^{2} \partial x^{2} }}\lambda ) + \frac{1}{2}l^{2} T_{4} (\frac{{\partial^{4} }}{{\partial y^{2} \partial x^{2} }}w_{\text{s}} ) + \frac{1}{4}l^{2} T_{3} (\frac{{\partial^{4} }}{{\partial x^{4} }}\lambda ) + E_{2410} (\frac{{\partial^{2} }}{{\partial y^{2} }}\Psi ) + \frac{1}{4}l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial x^{4} }}w_{\text{s}} ) \\ & \quad + \frac{1}{4}l^{2} T_{3} (\frac{{\partial^{4} }}{{\partial y^{4} }}\lambda ) + \frac{1}{2}l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial y^{2} \partial x^{2} }}w_{\text{s}} ) + \frac{1}{2}l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{b}} ) - Q_{138} (\frac{{\partial^{2} }}{{\partial x^{2} }}w_{\text{s}} ) + \frac{1}{4}l^{2} T_{1} (\frac{{\partial^{4} }}{{\partial y^{4} }}w_{\text{s}} ) + Q_{339} (\lambda ) \\ & \quad + E_{334} (\Psi ) + Q_{236} \left( {\frac{\partial }{\partial y}v} \right) + Q_{136} \left( {\frac{\partial }{\partial x}u} \right) - Q_{137} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }}w_{\text{b}} } \right) - Q_{4410} \left( {\frac{{\partial^{2} }}{{\partial y^{2} }}w_{\text{s}} } \right) - Q_{237} \left( {\frac{{\partial^{2} }}{{\partial y^{2} }}w_{\text{b}} } \right) - Q_{238} \left( {\frac{{\partial^{2} }}{{\partial y^{2} }}w_{\text{s}} } \right) \\ & \quad - I_{4} \left( {\frac{{\partial^{2} }}{{\partial t^{2} }}w_{\text{b}} } \right) + I_{4} \left( {\frac{{\partial^{2} }}{{\partial t^{2} }}w_{\text{s}} } \right) + I_{4} \left( {\frac{{\partial^{2} }}{{\partial t^{2} }}\lambda } \right) = 0. \\ \end{aligned} $$
(44)
$$ \begin{aligned} & \delta \Psi :E_{1510} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }}\Psi } \right) + E_{5510} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }}w_{\text{s}} } \right) + D_{111} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }}\Psi } \right) + E_{2410} \left( {\frac{{\partial^{2} }}{{\partial y^{2} }}\Psi } \right) + E_{2410} \left( {\frac{{\partial^{2} }}{{\partial y^{2} }}w_{\text{s}} } \right) + D_{222} \left( {\frac{{\partial^{2} }}{{\partial y^{2} }}\Psi } \right) \\ & \quad + E_{320} \left( {\frac{\partial }{\partial y}v} \right) + E_{310} \left( {\frac{\partial }{\partial x}u} \right) - E_{311} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }}w_{\text{b}} } \right) - E_{321} \left( {\frac{{\partial^{2} }}{{\partial y^{2} }}w_{\text{b}} } \right) - E_{323} \left( {\frac{{\partial^{2} }}{{\partial y^{2} }}w_{\text{s}} } \right) - E_{313} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }}w_{\text{s}} } \right) \\ & \quad + E_{334} \lambda - 2D_{331} V_{0} - D_{333} \Psi = 0. \\ \end{aligned} $$
(45)
The used coefficients in governing equations of (40)–(45) are introduced as below:
$$ \begin{aligned} & \left\{ {\begin{array}{*{20}c} {I_{7} ,} & {I_{6} ,} & {I_{5} ,} & {I_{4} ,} & {I_{3} ,} & {I_{2} ,} & {I_{1} ,} & {I_{0} } \\ \end{array} } \right\} \\ & \quad = \int_{{ - \frac{h}{2}}}^{\frac{h}{2}} \,\rho_{{\text{c,f}}} \left\{ {\begin{array}{*{20}c} {g(z)^{2} ,} & {f(z)^{2} ,} & {zf(z),} & {g(z),} & {f(z),} & {z^{2} ,} & {z,} & 1 \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \left\{ {\begin{array}{*{20}c} {Q_{110} ,} & {Q_{111} ,} & {Q_{112} } \\ \end{array} } \right\} = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{11} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{11} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{11} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z, $$
$$ \left\{ {\begin{array}{*{20}c} {Q_{120} ,} & {Q_{121} ,} & {Q_{122} } \\ \end{array} } \right\} = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{12} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{12} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{12} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z, $$
$$ \left\{ {\begin{array}{*{20}c} {Q_{130} ,} & {Q_{131} ,} & {Q_{132} } \\ \end{array} } \right\} = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{13} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{13} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{13} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z, $$
$$ \left\{ {\begin{array}{*{20}c} {Q_{230} ,} & {Q_{231} ,} & {Q_{232} } \\ \end{array} } \right\} = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{23} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{23} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{23} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z, $$
$$ \begin{aligned}& \left\{ {\begin{array}{*{20}c} {Q_{i\,i0} ,} & {Q_{i\,i1} ,} & {Q_{i\,i2} } \\ \end{array} } \right\} & \\ & \quad= \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{ii} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{ii} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z \\ & \qquad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{ii} \left\{ {\begin{array}{*{20}c} {1,} & {z,} & {z^{2} } \\ \end{array} } \right\}{\text{d}}z,\quad (i = 2,3,4,5,6) \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{113} ,} & {Q_{114} ,} & {Q_{115} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{11} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{11} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{11} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{123} ,} & {Q_{124} ,} & {Q_{125} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{12} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{12} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{12} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{133} ,} & {Q_{134} ,} & {Q_{135} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{13} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{13} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{13} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{233} ,} & {Q_{234} ,} & {Q_{235} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{23} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{23} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{23} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{i\,i3} ,} & {Q_{i\,i4} ,} & {Q_{i\,i5} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{i\,i} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{i\,i} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{i\,i} f(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z,\quad (i = 2,3,4,5,6) \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{116} ,} & {Q_{117} ,} & {Q_{118} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{11} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{11} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{11} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{126} ,} & {Q_{127} ,} & {Q_{128} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{12} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{12} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{12} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{116} ,} & {Q_{117} ,} & {Q_{118} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{11} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{11} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{11} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{126} ,} & {Q_{127} ,} & {Q_{128} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{12} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{12} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{12} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{136} ,} & {Q_{137} ,} & {Q_{138} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{13} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{13} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{13} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{236} ,} & {Q_{237} ,} & {Q_{238} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{23} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{23} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{23} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{i\,i6} ,} & {Q_{i\,i7} ,} & {Q_{i\,i8} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{i\,i} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{i\,i} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{i\,i} g^{\prime}(z)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z)} \\ \end{array} } \right\}{\text{d}}z,\quad (i = 2,3,4,5,6) \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{119} ,} & {Q_{1110} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{11} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{11} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{11} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{129} ,} & {Q_{1210} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{12} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{12} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{12} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{139} ,} & {Q_{1310} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{13} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{13} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{13} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{239} ,} & {Q_{2310} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{23} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{23} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{23} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {Q_{i\,i9} ,} & {Q_{i\,i10} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} Q_{i\,i} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2}}} C_{i\,i} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} Q_{i\,i} \left\{ {\begin{array}{*{20}c} {\left[ {\,g^{\prime}(z)} \right]^{2} ,} & {g^{2} (z)} \\ \end{array} } \right\}{\text{d}}z,\quad (i = 2,3,4,5,6) \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {E_{310} ,} & {E_{311} ,} & {E_{313} ,} & {E_{314} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} \frac{\pi }{{h_{\text{f}} }}e_{31} \sin (\frac{\pi z}{{h_{\text{f}} }})\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z),} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} \frac{\pi }{{h_{\text{f}} }}e_{31} \sin (\frac{\pi z}{{h_{\text{f}} }})\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z),} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {E_{320} ,} & {E_{321} ,} & {E_{323} ,} & {E_{324} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} \frac{\pi }{{h_{\text{f}} }}e_{32} \sin (\frac{\pi z}{{h_{\text{f}} }})\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z),} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} \frac{\pi }{{h_{\text{f}} }}e_{32} \sin (\frac{\pi z}{{h_{\text{f}} }})\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z),} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {E_{330} ,} & {E_{331} ,} & {E_{333} ,} & {E_{334} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} \frac{\pi }{{h_{\text{f}} }}e_{33} \sin (\frac{\pi z}{{h_{\text{f}} }})\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z),} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} \frac{\pi }{{h_{\text{f}} }}e_{33} \sin (\frac{\pi z}{{h_{\text{f}} }})\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z),} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {E_{150} ,} & {E_{151} ,} & {E_{153} ,} & {E_{154} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} \frac{\pi }{{h_{\text{f}} }}e_{15} \sin \left( {\frac{\pi z}{{h_{\text{f}} }}} \right)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z),} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} \frac{\pi }{{h_{\text{f}} }}e_{15} \sin \left( {\frac{\pi z}{{h_{\text{f}} }}} \right)\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z),} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {E_{240} ,} & {E_{241} ,} & {E_{243} ,} & {E_{244} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} \frac{\pi }{{h_{\text{f}} }}e_{24} \sin (\frac{\pi z}{{h_{\text{f}} }})\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z),} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} \frac{\pi }{{h_{\text{f}} }}e_{24} \sin (\frac{\pi z}{{h_{\text{f}} }})\left\{ {\begin{array}{*{20}c} {1,} & {z,} & {f(z),} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \left\{ {\begin{array}{*{20}c} {E_{1510} ,} & {E_{2410} } \\ \end{array} } \right\} = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} \sin \left( {\frac{\pi z}{{h_{\text{f}} }}} \right)g(z)\left\{ {\begin{array}{*{20}c} {e_{15} ,} & {e_{24} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} \sin \left( {\frac{\pi z}{{h_{\text{f}} }}} \right)g(z)\left\{ {\begin{array}{*{20}c} {e_{15} ,} & {e_{24} } \\ \end{array} } \right\}{\text{d}}z, $$
$$ \left\{ {\begin{array}{*{20}c} {D_{111} ,} & {D_{222} } \\ \end{array} } \right\} = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} \left[ {\sin \left( {\frac{\pi z}{{h_{\text{f}} }}} \right)} \right]^{2} \left\{ {\begin{array}{*{20}c} {d_{11} ,} & {d_{22} } \\ \end{array} } \right\}{\text{d}}z + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} \left[ {\sin \left( {\frac{\pi z}{{h_{\text{f}} }}} \right)} \right]^{2} \left\{ {\begin{array}{*{20}c} {d_{11} ,} & {d_{22} } \\ \end{array} } \right\}{\text{d}}z, $$
$$ \begin{aligned} \left\{ {\begin{array}{*{20}c} {D_{333} ,} & {D_{331} } \\ \end{array} } \right\} & = \int_{{\frac{{h_{\text{c}} }}{2}}}^{{\frac{{h_{\text{c}} }}{2} + h_{\text{t}} }} d_{33} \left\{ {\begin{array}{*{20}c} {\left[ {\frac{\pi }{{h_{\text{f}} }}\sin \left( {\frac{\pi z}{{h_{\text{f}} }}} \right)} \right]^{2} ,} & {\frac{\pi }{{h^{2}_{f} }}\sin \left( {\frac{\pi z}{{h_{\text{f}} }}} \right)} \\ \end{array} } \right\}{\text{d}}z \\ & \quad + \int_{{ - \frac{{h_{\text{c}} }}{2} - h_{\text{b}} }}^{{ - \frac{{h_{\text{c}} }}{2}}} d_{33} \left\{ {\begin{array}{*{20}c} {\left[ {\frac{\pi }{{h_{\text{f}} }}\sin \left( {\frac{\pi z}{{h_{\text{f}} }}} \right)} \right]^{2} ,} & {\frac{\pi }{{h^{2}_{f} }}\sin \left( {\frac{\pi z}{{h_{\text{f}} }}} \right)} \\ \end{array} } \right\}{\text{d}}z, \\ \end{aligned} $$
$$ \left\{ {\begin{array}{*{20}c} {T_{1} ,} & {T_{2} ,} & {T_{3} ,} & {T_{4} ,} & {T_{5} } \\ \end{array} } \right\} = \int_{{ - \frac{h}{2}}}^{\frac{h}{2}} \,\mu \left\{ {\begin{array}{*{20}c} {g(z),} & {f^{\prime}(z),} & {g^{2} (z),} & {g(z)f^{\prime}(z),} & {\,f^{\prime}(z)^{2} } \\ \end{array} } \right\}{\text{d}}z, $$
$$ \left\{ {\begin{array}{*{20}c} {T_{6} ,} & {T_{7} ,} & {T_{8} ,} & {T_{9} ,} & {T_{10} } \\ \end{array} } \right\} = \int_{{ - \frac{h}{2}}}^{\frac{h}{2}} \,\mu \left\{ {\begin{array}{*{20}c} {f^{\prime\prime}(z)^{2} ,} & {g^{\prime}(z)f^{\prime\prime}(z),} & {f^{\prime\prime}(z),} & {g^{\prime}(z)^{2} ,} & {g^{\prime}(z)} \\ \end{array} } \right\}{\text{d}}z. $$
