Analytical solution for static and dynamic analysis of FGP cylinders integrated with FG-GPLs patches exposed to longitudinal magnetic field

Abstract

Motivated by progressive sandwich structures’ usage due to their high strength to weight ratio in different industries, the current paper aimed at evaluating static and dynamic behaviors of three-layered cylinders including FG porous core and two graphene nanoplatelet (GPLs)-reinforced composite as face sheets. The whole sandwich cylinder rests on Pasternak substrate and it is also exposed to a longitudinal magnetic field. Epoxy has used as matrix and GPLs as the reinforcing phases for top and bottom face sheets and based on the rule of mixture and Halpin–Tsai micromechanical models, effective values for mechanical properties of skins are gained. Besides, regarding the integrity of current research, all layers of model are assumed to be FG, which means for the porous core the placement of the pores is considered and for the faces, the GPLs dispersion patterns are regarded. Among different shell theories, sinusoidal shear deformation shells theory (SSDST) is utilized to define displacement components along with the major axes. Hamilton’s principle is hired to attain governing equations for vibrational and buckling analyses. In the end, the effects of different variables’ alternation as the model’s geometry, foundation moduli, mode number, and mid-radius on vibrational and buckling behaviors are interpreted in type of different plots and tables. Pores’ placement and GPLs dispersion patterns play important roles in static and dynamic responses of the under-consideration cylinder. The outcomes of this study may help to create more efficient engineering structures such as pressure vessels.

Appendices

Appendix A

The used stress resultants in Eqs. (31)–(35) are defined as follows:

$$\begin{gathered} \left\{ {\begin{array}{*{20}c} {N_{ii} } \\ {F_{ii} } \\ {G_{ii} } \\ \end{array} } \right\} = \int\limits_{z} {} \left( {C_{jj} \left[ {\begin{array}{*{20}c} 1 & {f(z)} & { - g(z)} \\ {f(z)} & {f(z)^{2} } & { - f(z)g(z)} \\ {g(z)} & {f(z)g(z)} & { - g(z)^{2} } \\ \end{array} } \right] + \frac{1}{R}C_{jj} \left[ {\begin{array}{*{20}c} 1 & {f(z)} & { - g(z)} \\ {f(z)} & {f(z)^{2} } & { - f(z)g(z)} \\ {g(z)} & {g(z)f(z)} & { - g(z)^{2} } \\ \end{array} \,\,\,\,\begin{array}{*{20}c} 1 \\ {f(z)} \\ {g(z)} \\ \end{array} } \right]} \right) \hfill \\ \left\{ {\begin{array}{*{20}c} {\frac{{\partial u_{0} }}{\partial x}} & {\frac{{\partial \phi_{x} }}{\partial x}} & {\frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} & {\frac{{\partial v_{0} }}{\partial \theta }} & {\frac{{\partial \phi_{\theta } }}{\partial \theta }} & {\frac{{\partial^{2} w_{0} }}{{\partial \theta^{2} }}} & {w_{0} } \\ \end{array} } \right\}^{{\text{T}}} {\text{d}}z,\;i = x,\theta ;\,j = 1,2. \hfill \\ \end{gathered}$$

$$\left\{ {\begin{array}{*{20}c} {\alpha_{1} } \\ {\alpha_{2} } \\ {\alpha_{3} } \\ \end{array} } \right\} = \int\limits_{z} {} C_{55} \,\left[ {\begin{array}{*{20}c} {1 - g^{\prime}(z)} & {f^{\prime}(z)} \\ {f^{\prime}(z)(1 - g^{\prime}(z))} & {f^{\prime}(z)^{2} } \\ {g^{\prime}(z)(1 - g^{\prime}(z))} & {g^{\prime}(z)f^{\prime}(z)} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\frac{{\partial w_{0} }}{\partial x}} & {\phi_{x} } \\ \end{array} } \right\}^{{\text{T}}} \,{\text{d}}z,$$

$$\left\{ {\begin{array}{*{20}c} {\alpha_{4} } \\ {\alpha_{5} } \\ {\alpha_{6} } \\ {\alpha_{7} } \\ {\alpha_{8} } \\ \end{array} } \right\} = \int\limits_{z} {} C_{44} \,\,\left[ {\begin{array}{*{20}c} {f^{\prime}(z) - f(z)} & {\frac{1}{R}(g(z) + \frac{1}{R}g(z) + 1)} & { - \frac{1}{R}} \\ {f(z)(f^{\prime}(z) - f(z))} & {\frac{f(z)}{R}(g(z) + \frac{1}{R}g(z) + 1)} & { - \frac{f(z)}{R}} \\ {g(z)(f^{\prime}(z) - f(z))} & {\frac{g(z)}{R}(g(z) + \frac{1}{R}g(z) + 1)} & { - \frac{g(z)}{R}} \\ {f^{\prime}(z)(f^{\prime}(z) - f(z))} & {\frac{{f^{\prime}(z)}}{R}(g(z) + \frac{1}{R}g(z) + 1)} & { - \frac{{f^{\prime}(z)}}{R}} \\ {g^{\prime}(z)(f^{\prime}(z) - f(z))} & {\frac{{g^{\prime}(z)}}{R}(g(z) + \frac{1}{R}g(z) + 1)} & { - \frac{{g^{\prime}(z)}}{R}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\phi_{\theta } } & {\frac{{\partial w_{0} }}{\partial \theta }} & {v_{0} } \\ \end{array} } \right\}^{{\text{T}}} \,{\text{d}}z,$$

$$\left\{ {\begin{array}{*{20}c} {\alpha_{9} } \\ {\alpha_{10} } \\ {\alpha_{11} } \\ \end{array} } \right\} = \int\limits_{z} {} C_{66} \left[ {\begin{array}{*{20}c} 1 & {f(z)} & { - \frac{2}{R}g(z)} & \frac{1}{R} & {\frac{1}{R}f(z)} \\ {f(z)} & {f(z)^{2} } & { - \frac{2}{R}f(z)g(z)} & {\frac{f(z)}{R}} & {\frac{1}{R}f(z)^{2} } \\ {g(z)} & {g(z)f(z)} & { - \frac{2}{R}g(z)^{2} } & {\frac{g(z)}{R}} & {\frac{1}{R}f(z)g(z)} \\ \end{array} } \right]\,\,\,\left\{ {\begin{array}{*{20}c} {\frac{{\partial v_{0} }}{\partial x}} & {\frac{{\partial \phi_{\theta } }}{\partial x}} & {\frac{{\partial^{2} w_{0} }}{\partial x\partial \theta }} & {\frac{{\partial u_{0} }}{\partial \theta }} & {\frac{{\partial \phi_{x} }}{\partial \theta }} \\ \end{array} } \right\}^{{\text{T}}} \,{\text{d}}z,$$

$$\left\{ {\begin{array}{*{20}c} {\alpha_{12} } \\ {\alpha_{13} } \\ \end{array} } \right\} = \int\limits_{z} {} \eta H_{x}^{2} \left\{ {\begin{array}{*{20}c} 1 & {,\,\,\,\,f^{\prime}(z)} \\ \end{array} } \right\}\,{\text{d}}z,$$

$$\begin{gathered} J_{0} ,J_{1} ,J_{2} ,J_{3} ,J_{4} ,J_{5} = \int_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2} + h_{{\text{t}}} }} \rho_{{\text{f}}} \left( z \right)(1,f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} )dz \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \int_{{ - \frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} \rho_{{\text{c}}} \left( z \right)(1,f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} )dz \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \int_{{ - \frac{{h_{{\text{c}}} }}{2} - h_{{\text{b}}} }}^{{ - \frac{{h_{{\text{c}}} }}{2}}} \rho_{{\text{f}}} \left( z \right)(1,f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} )dz. \hfill \\ \end{gathered}$$

Appendix B

The arrays of matrices of Eq. (38a, 38b) are as follows:

$$K_{11} = C_{110} \Xi_{1}^{2} + \frac{{C_{660} \Xi_{2}^{2} }}{{R^{2} }},$$

$$K_{12} = \frac{1}{R}(C_{120} \Xi_{1} {\mkern 1mu} \Xi_{2} + C_{660} \Xi_{1} {\mkern 1mu} \Xi_{2} ),$$

$$K_{13} = - C_{112} \Xi_{1}^{3} - \frac{{C_{122} \Xi_{1} {\mkern 1mu} \Xi_{2}^{2} }}{{R^{2} }} - \frac{{C_{120} \Xi_{1} }}{R} - 2{\mkern 1mu} \frac{{C_{662} \Xi_{1} {\mkern 1mu} \Xi_{2}^{2} }}{{R^{2} }},$$

$$K_{14} = C_{111} \Xi_{1}^{2} + \frac{{C_{661} \Xi_{2}^{2} }}{{R^{2} }},$$

$$K_{15} = \frac{{C_{121} \Xi_{1} \Xi_{2} }}{R} + \frac{{C_{661} \Xi_{1} {\mkern 1mu} \Xi_{2} }}{R},$$

$$K_{21} = \frac{{C_{120} \Xi_{1} \beta }}{R} + \frac{{C_{660} \Xi_{1} {\mkern 1mu} \beta }}{R},$$

$$K_{22} = \frac{{C_{220} \Xi_{2}^{2} }}{{R^{2} }} + Q_{660} \Xi_{1}^{2} ,$$

$$K_{23} = - \frac{{C_{122} \Xi_{1}^{2} \Xi_{2} }}{R} - \frac{{C_{222} \Xi_{2}^{3} }}{{R^{3} }} - \frac{{C_{220} \Xi_{2} }}{{R^{2} }} - 2{\mkern 1mu} \frac{{C_{662} \Xi_{1}^{2} \Xi_{2} }}{R},$$

$$K_{24} = \frac{{C_{121} \Xi_{1} {\mkern 1mu} \Xi_{2} }}{R} + \frac{{C_{661} \Xi_{1} \Xi_{2} }}{R},$$

$$K_{25} = \frac{{C_{221} \Xi_{2}^{2} }}{{R^{2} }} + C_{661} \Xi_{1}^{2} ,$$

$$K_{31} = - C_{112} \Xi_{1}^{3} - \frac{{C_{122} \Xi_{1} \Xi_{2}^{2} }}{{R^{2} }} - \frac{{C_{120} \Xi_{1} }}{R} - 2{\mkern 1mu} \frac{{C_{662} \Xi_{1} {\mkern 1mu} \Xi_{2}^{2} }}{{R^{2} }},$$

$$K_{32} = - \frac{{C_{122} \Xi_{1}^{2} \Xi_{2} }}{R} - \frac{{C_{222} \Xi_{2}^{3} }}{{R^{3} }} - \frac{{C_{220} \Xi_{2} }}{{R^{2} }} - 2{\mkern 1mu} \frac{{C_{662} \Xi_{1}^{2} \Xi_{2} }}{R},$$

$$\begin{gathered} K_{33} = 2{\mkern 1mu} \frac{{C_{122} \Xi_{1}^{2} }}{R} + 4{\mkern 1mu} \frac{{C_{665} \Xi_{1}^{2} \Xi_{2}^{2} }}{{R^{2} }} + 2{\mkern 1mu} \frac{{C_{125} \Xi_{1}^{2} \Xi_{2}^{2} }}{{R^{2} }} + \frac{{C_{220} }}{{R^{2} }} + C_{115} \Xi_{1}^{4} + C_{550} \Xi_{1}^{2} - 2{\mkern 1mu} C_{559} \Xi_{1}^{2} \hfill \\ + C_{5511} \Xi_{1}^{2} + \frac{{C_{225} \Xi_{2}^{4} }}{{R^{4} }} + 2{\mkern 1mu} \frac{{C_{222} \Xi_{2}^{2} }}{{R^{3} }} + \frac{{C_{440} \Xi_{2}^{2} }}{{R^{2} }} - 2{\mkern 1mu} \frac{{C_{449} \Xi_{2}^{2} }}{{R^{2} }} + \frac{{C_{4411} \Xi_{2}^{2} }}{{R^{2} }} \hfill \\ + k_{2} \Xi_{1}^{2} + \frac{1}{{R^{2} }}k_{2} \Xi_{2}^{2} + k_{1} + N_{x}^{0} \Xi_{1}^{2} {}_{0} - \alpha_{12} \Xi_{1}^{2} - \frac{{\alpha_{12} \Xi_{2}^{2} }}{{R^{2} }} - \frac{{\alpha_{13} \Xi_{2}^{2} }}{R}, \hfill \\ \end{gathered}$$

$$K_{34} = - 2{\mkern 1mu} \frac{{C_{664} \Xi_{1} {\mkern 1mu} \Xi_{2}^{2} }}{{R^{2} }} - \frac{{C_{124} \Xi_{1} {\mkern 1mu} \Xi_{2}^{2} }}{{R^{2} }} - C_{114} \Xi_{1}^{3} + C_{556} \Xi_{1} - C_{5510} \Xi_{1} - \frac{{C_{121} \Xi_{1} }}{R},$$

$$K_{35} = - \frac{{C_{124} \Xi_{1}^{2} \Xi_{2} }}{R} - 2{\mkern 1mu} \frac{{C_{664} \Xi_{1}^{2} \Xi_{2} }}{R} - \frac{{C_{224} \Xi_{2}^{3} }}{{R^{3} }} - \frac{{C_{221} \Xi_{2} }}{{R^{2} }} + \frac{{C_{446} \Xi_{2} }}{R} - \frac{{C_{4410} \Xi_{2} }}{R}{ - }\frac{{\alpha_{13} \Xi_{2} }}{R},$$

$$K_{41} = C_{111} \Xi_{1}^{2} + \frac{{C_{661} \Xi_{2}^{2} }}{{R^{2} }},$$

$$K_{42} = \frac{{C_{121} \Xi_{1} {\mkern 1mu} \Xi_{2} }}{R} + \frac{{C_{661} \Xi_{1} {\mkern 1mu} \Xi_{2} }}{R},$$

$$K_{43} = - 2{\mkern 1mu} \frac{{C_{664} \Xi_{1} {\mkern 1mu} \Xi_{2}^{2} }}{{R^{2} }} - \frac{{C_{124} \Xi_{1} {\mkern 1mu} \Xi_{2}^{2} }}{{R^{2} }} - C_{114} \Xi_{1}^{3} + C_{556} \Xi_{1} - C_{5510} \Xi_{1} - \frac{{C_{121} \Xi_{1} }}{R},$$

$$K_{44} = - 2{\mkern 1mu} \frac{{C_{664} \Xi_{1} {\mkern 1mu} \Xi_{2}^{2} }}{{R^{2} }} - \frac{{C_{124} \Xi_{1} {\mkern 1mu} \Xi_{2}^{2} }}{{R^{2} }} - C_{114} \Xi_{1}^{3} + C_{556} \Xi_{1} - C_{5510} \Xi_{1} - \frac{{C_{121} \Xi_{1} }}{R},$$

$$K_{45} = \frac{{C_{123} \Xi_{1} {\mkern 1mu} \Xi_{2} }}{R} + \frac{{C_{663} \Xi_{1} {\mkern 1mu} \Xi_{2} }}{R},$$

$$K_{51} = \frac{{C_{121} \Xi_{1} {\mkern 1mu} \Xi_{2} }}{R} + \frac{{C_{661} \Xi_{1} \Xi_{2} }}{R},$$

$$K_{52} = \frac{{C_{221} \Xi_{2}^{2} }}{{R^{2} }} + C_{661} \Xi_{1}^{2} ,$$

$$K_{53} = \frac{1}{R}\left( { - C_{124} \Xi_{1}^{2} \Xi_{2} - 2{\mkern 1mu} C_{664} \Xi_{1}^{2} \Xi_{2} - \frac{{C_{224} \Xi_{2}^{3} }}{{R^{2} }} - \frac{{C_{221} \Xi_{2} }}{R} + C_{446} \Xi_{2} - C_{4410} \Xi_{2} } \right),$$

$$K_{54} = \frac{{C_{123} \Xi_{1} {\mkern 1mu} \Xi_{2} }}{R} + \frac{{C_{663} \Xi_{1} {\mkern 1mu} \Xi_{2} }}{R},$$

$$K_{55} = \frac{{C_{223} \Xi_{2}^{2} }}{{R^{2} }} + C_{663} \Xi_{1}^{2} + C_{448} ,$$

$$\begin{gathered} M_{11} = J_{0} , \hfill \\ M_{12} = 0, \hfill \\ M_{13} = 0, \hfill \\ M_{14} = J_{1} , \hfill \\ M_{15} = 0, \hfill \\ \end{gathered}$$

$$\begin{gathered} M_{21} = 0, \hfill \\ M_{22} = J_{0} , \hfill \\ M_{23} = - J_{1} \Xi_{2} /R, \hfill \\ M_{24} = J_{1} , \hfill \\ M_{25} = 0, \hfill \\ \end{gathered}$$

$$\begin{gathered} M_{31} = - J_{2} \Xi_{1} , \hfill \\ M_{32} = - J_{1} \Xi_{2} /R, \hfill \\ M_{33} = J_{5} \Xi_{1}^{2} + J_{3} \Xi_{2}^{2} + J_{0} , \hfill \\ M_{34} = - J_{4} \Xi_{1} , \hfill \\ M_{35} = - J_{3} \Xi_{2} /R, \hfill \\ \end{gathered}$$

$$\begin{gathered} M_{41} = J_{1} , \hfill \\ M_{42} = 0, \hfill \\ M_{43} = - J_{4} \Xi_{1} , \hfill \\ M_{44} = J_{3} , \hfill \\ M_{45} = 0, \hfill \\ \end{gathered}$$

$$\begin{gathered} M_{51} = 0, \hfill \\ M_{52} = J_{1} , \hfill \\ M_{53} = - J_{3} \Xi_{2} /R, \hfill \\ M_{54} = 0, \hfill \\ M_{55} = J_{3} , \hfill \\ \end{gathered}$$

where

$$\begin{gathered} C_{110} ,C_{111} ,C_{112} ,C_{113} ,C_{114} ,C_{115} = \int_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2} + h_{{\text{t}}} }} C_{11}^{f} \left( z \right)(\,1,{\mkern 1mu} f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} ){\text{d}}z \hfill \\ + \int_{{ - \frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{11}^{c} \left( z \right)(1,{\mkern 1mu} f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} ){\text{d}}z \hfill \\ \, + \int_{{ - \frac{{h_{{\text{c}}} }}{2} - h_{{\text{b}}} }}^{{ - \frac{{h_{{\text{c}}} }}{2}}} C_{11}^{f} \left( z \right)(\,1,{\mkern 1mu} f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} ){\text{d}}z,\,\,\, \hfill \\ \end{gathered}$$

$$\begin{gathered} C_{120} ,C_{121} ,C_{122} ,C_{123} ,C_{124} = \int_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2} + h_{{\text{t}}} }} C_{12}^{f} \left( z \right)(\,1,{\mkern 1mu} f(z),g(z),f(z)^{2} ,f(z)g(z)){\text{d}}z \hfill \\ + \int_{{ - \frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{12}^{c} \left( z \right)(1,{\mkern 1mu} f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} ){\text{d}}z \hfill \\ \, + \int_{{ - \frac{{h_{{\text{c}}} }}{2} - h_{{\text{b}}} }}^{{ - \frac{{h_{{\text{c}}} }}{2}}} C_{12}^{f} \left( z \right)(\,1,{\mkern 1mu} f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} ){\text{d}}z, \hfill \\ \end{gathered}$$

$$\begin{gathered} C_{220} ,C_{221} ,C_{223} = \int_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2} + h_{{\text{t}}} }} C_{22}^{f} \left( z \right)(\,1,{\mkern 1mu} f(z),f(z)^{2} ){\text{d}}z\, + \int_{{ - \frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{22}^{c} \left( z \right)(1,{\mkern 1mu} f(z),f(z)^{2} ){\text{d}}z \hfill \\ + \int_{{ - \frac{{h_{{\text{c}}} }}{2} - h_{{\text{b}}} }}^{{ - \frac{{h_{{\text{c}}} }}{2}}} C_{22}^{f} \left( z \right)(\,1,{\mkern 1mu} f(z),f(z)^{2} ){\text{d}}z, \hfill \\ \end{gathered}$$

$$\begin{gathered} C_{440} ,C_{441} ,C_{443} ,C_{446} ,C_{447} ,C_{448} = \int_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2} + h_{{\text{t}}} }} C_{44}^{f} \left( z \right)(\,1,{\mkern 1mu} f(z),f(z)^{2} ,f(z)^{\prime},f(z)^{\prime}f(z),f(z)^{{\prime}{2}} ){\text{d}}z \hfill \\ + \int_{{ - \frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{44}^{c} \left( z \right)(\,1,{\mkern 1mu} f(z),f(z)^{2} ,f(z)^{\prime},f(z)^{\prime}f(z),f(z)^{{\prime}{2}} ){\text{d}}z \hfill \\ + \int_{{ - \frac{{h_{{\text{c}}} }}{2} - h_{{\text{b}}} }}^{{ - \frac{{h_{{\text{c}}} }}{2}}} C_{44}^{f} \left( z \right)(\,\,1,{\mkern 1mu} f(z),f(z)^{2} ,f(z)^{\prime},f(z)^{\prime}f(z),f(z)^{{\prime}{2}} ){\text{d}}z,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ \end{gathered}$$

$$\begin{gathered} C_{550} ,C_{556} ,C_{558} ,C_{559} ,C_{5510} ,C_{5511} = \int_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2} + h_{{\text{t}}} }} C_{55}^{f} \left( z \right)(\,1,{\mkern 1mu} f(z)^{\prime},f(z)^{{\prime}{2}} ,g(z)^{\prime},f(z)^{\prime}g(z)^{\prime},g(z)^{{\prime}{2}} ){\text{d}}z \hfill \\ + \int_{{ - \frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{55}^{c} \left( z \right)(\,\,1,{\mkern 1mu} f(z)^{\prime},f(z)^{{\prime}{2}} ,g(z)^{\prime},f(z)^{\prime}g(z)^{\prime},g(z)^{{\prime}{2}} ){\text{d}}z \hfill \\ + \int_{{ - \frac{{h_{{\text{c}}} }}{2} - h_{{\text{b}}} }}^{{ - \frac{{h_{{\text{c}}} }}{2}}} C_{55}^{f} \left( z \right)(\,\,\,1,{\mkern 1mu} f(z)^{\prime},f(z)^{{\prime}{2}} ,g(z)^{\prime},f(z)^{\prime}g(z)^{\prime},g(z)^{{\prime}{2}} ){\text{d}}z, \hfill \\ \end{gathered}$$

$$\begin{gathered} C_{660} ,C_{661} ,C_{662} ,C_{663} ,C_{664} ,C_{665} = \int_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2} + h_{{\text{t}}} }} C_{66}^{f} \left( z \right)(\,1,{\mkern 1mu} f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} ){\text{d}}z \hfill \\ + \int_{{ - \frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{66}^{c} \left( z \right)(\,\,\,1,{\mkern 1mu} f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} ){\text{d}}z \hfill \\ + \int_{{ - \frac{{h_{{\text{c}}} }}{2} - h_{{\text{b}}} }}^{{ - \frac{{h_{{\text{c}}} }}{2}}} C_{66}^{f} \left( z \right)(\,\,1,{\mkern 1mu} f(z),g(z),f(z)^{2} ,f(z)g(z),g(z)^{2} ){\text{d}}z. \hfill \\ \end{gathered}$$