Theoretical analysis and experimental measurement of coupling dynamic characteristics for transversely isotropic rectangular plate based on modified FSDT assumption

Abstract

The theoretical analysis of thick plate vibration behavior has been investigated in the literature, but most of the studies were focused on flexural dynamic characteristics and lack experimental verification. In this study, the analytical solutions based on the superposition method for both the flexural and extensional vibrations are presented to obtain resonant frequencies and associated mode shapes for a transversely isotropic thick rectangular plate. The displacement equilibrium equations and boundary conditions of the modified first-order shear deformation theory (FSDT) are derived by utilizing Hamilton’s principle and the variation method. To verify the validity of the theoretical model, the finite-element method (FEM) and impact experiment results for thick plates are employed in this work. Excellent agreement of resonant frequencies and associated mode shapes is obtained for FEM calculation and theoretical analysis. To excite vibrations of a thick rectangular plate, a steel ball controlled by an electromagnet is utilized. A steel ball is dropped freely from a height of 231 mm on the top of the plate surface, and a transient wave will be generated after the ball impact on the specimen. The frequency spectrum of the thick rectangular plate is constructed by using the fast Fourier transform of the time-domain transient response. The excited resonant frequencies obtained from experimental measurement are compared with theoretical results. The comparisons show that the modified FSDT provides an excellent prediction of the resonant frequencies and mode shapes for the dynamic characteristics of thick rectangular plates.

Appendices

Appendix A: Constants in the flexural type displacement solutions

The constants in the flexural type displacement solutions are listed as follows:

As $m=0$, the coefficients of the first and third building block are

$$\begin{aligned} \left[ {{\begin{array}{llll} {\theta X_{10k1}^{F} } \\ {\theta X_{10k2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {1 \big / {\left( {S_{10k1}^{F} +X_{101}^{F} S_{10k2}^{F} } \right) }} \\ {X_{101}^{F} \theta X_{10k1}^{F} } \\ \end{array} }} \right] , \ \left[ {{\begin{array}{l} {\theta Z_{10k1}^{F} } \\ {\theta Z_{10k2}^{F} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {S_{10k1}^{F} \theta X_{10k1}^{F} } \\ {S_{10k2}^{F} \theta X_{10k2}^{F} } \\ \end{array} }} \right] ,\hbox { and }k=1,2,3, \end{aligned}$$

(A.1)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{30k1}^{F} } \\ {\theta X_{30k2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {1 \big / {\left( {S_{30k1}^{F} +X_{301}^{F} S_{30k2}^{F} } \right) }} \\ {X_{301}^{F} \theta X_{30k1}^{F} } \\ \end{array} }} \right] ,\ \left[ {{\begin{array}{l} {\theta Z_{30k1}^{F} } \\ {\theta Z_{30k2}^{F} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {S_{30k1}^{F} \theta X_{30k1}^{F} } \\ {S_{30k2}^{F} \theta X_{30k2}^{F} } \\ \end{array} }} \right] ,\hbox {and }k=1,2,3, \end{aligned}$$

(A.2)

$$\begin{aligned} X_{101}^{F}= & {} -{BQ_{10k1}^{F} } \big / {BQ_{10k2}^{F} }, \ X_{301}^{F} =-{BQ_{30k1}^{F} } \big / {BQ_{30k2}^{F} }, \end{aligned}$$

(A.3)

$$\begin{aligned} BQ_{10k1}^{F}= & {} \left\{ {{\begin{array}{l} {S_{10k1}^{F} -{\alpha _{m}^{F} } \big / \phi \hbox { as }R_{m1}^{F}<0}, \\ {S_{10k1}^{F} +{\alpha _{m}^{F} } \big / \phi \hbox { as }R_{m1}^{F}>0}, \\ \end{array} }} \right. BQ_{10k2}^{F} =\left\{ {{\begin{array}{l} {S_{10k2}^{F} -{\beta _{m}^{F} } \big / \phi \hbox { as }R_{m2}^{F} <0}, \\ {S_{10k2}^{F} +{\beta _{m}^{F} } \big / \phi \hbox { as }R_{m2}^{F} >0}, \\ \end{array} }} \right. \end{aligned}$$

(A.4)

$$\begin{aligned} BQ_{30k1}^{F}= & {} \left\{ {{\begin{array}{l} {S_{30k1}^{F} +{\alpha _{m}^{F} } \big / \phi \hbox { as }R_{m1}^{F}<0}, \\ {S_{30k1}^{F} -{\alpha _{m}^{F} } \big / \phi \hbox { as }R_{m1}^{F}>0}, \\ \end{array} }} \right. BQ_{30k2}^{F} =\left\{ {{\begin{array}{l} {S_{30k2}^{F} +{\beta _{m}^{F} } \big / \phi \hbox { as }R_{m2}^{F} <0}, \\ {S_{30k2}^{F} -{\beta _{m}^{F} } \big / \phi \hbox { as }R_{m2}^{F} >0}, \\ \end{array} }} \right. \end{aligned}$$

(A.5)

$$\begin{aligned} S_{10k1}^{F}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-a_{m13}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+a_{m14}^{F} } \right] } \big / {\left( {c_{m14}^{F} \alpha _{m}^{F} } \right) } \text{ as } R_{m1}^{F}<0}, \\ {-{\left[ {a_{m13}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+a_{m14}^{F} } \right] } \big / {\left( {c_{m14}^{F} \alpha _{m}^{F} } \right) } \text{ as } R_{m1}^{F} >0}, \\ \end{array} }} \right. \end{aligned}$$

(A.6)

$$\begin{aligned} S_{10k2}^{F}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-a_{m13}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+a_{m14}^{F} } \right] } \big / {\left( {c_{m14}^{F} \beta _{m}^{F} } \right) } \text{ as } R_{m2}^{F}<0},\\ {-{\left[ {a_{m13}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+a_{m14}^{F} } \right] } \big / {\left( {c_{m14}^{F} \beta _{m}^{F} } \right) } \text{ as } R_{m2}^{F} >0}, \\ \end{array} }} \right. \end{aligned}$$

(A.7)

$$\begin{aligned} S_{30k1}^{F}= & {} \left\{ {{\begin{array}{l} {{\left[ {-a_{m13}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+a_{m14}^{F} } \right] } \big / {\left( {c_{m14}^{F} \alpha _{m}^{F} } \right) } \text{ as } R_{m1}^{F}<0}, \\ {{\left[ {a_{m13}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+a_{m14}^{F} } \right] } \big / {\left( {c_{m14}^{F} \alpha _{m}^{F} } \right) } \text{ as } R_{m1}^{F} >0,} \\ \end{array} }} \right. \end{aligned}$$

(A.8)

$$\begin{aligned} S_{30k2}^{F}= & {} \left\{ {{\begin{array}{l} {{\left[ {-a_{m13}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+a_{m14}^{F} } \right] } \big / {\left( {c_{m14}^{F} \beta _{m}^{F} } \right) } \text{ as } R_{m2}^{F}<0}, \\ {{\left[ {a_{m13}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+a_{m14}^{F} } \right] } \big / {\left( {c_{m14}^{F} \beta _{m}^{F} } \right) } \text{ as } R_{m2}^{F} >0,} \\ \end{array} }} \right. \end{aligned}$$

(A.9)

As $m\ge 1$, the coefficients of the first and third building block are

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{1mk1}^{F} } \\ {\theta X_{1mk2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {1 \big / {\left( {S_{1mk1}^{F} +X_{1m1}^{F} S_{1mk2}^{F} +X_{1m2}^{F} S_{1mk3}^{F} } \right) }} \\ {X_{1m1}^{F} \theta X_{1mk1}^{F} } \\ \end{array} }} \right] ,\hbox { and }k=1,2,3, \end{aligned}$$

(A.10)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta Y_{1mk1}^{F} } \\ {\theta Y_{1mk2}^{F} } \\ {\theta Y_{1mk3}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {R_{1mk1}^{F} \theta X_{1mk1}^{F} } \\ {R_{1mk2}^{F} \theta X_{1mk2}^{F} } \\ {X_{1m2}^{F} \theta X_{1mk1}^{F} } \\ \end{array} }} \right] , \ \left[ {{\begin{array}{l} {\theta Z_{1mk1}^{F} } \\ {\theta Z_{1mk2}^{F} } \\ {\theta Z_{1mk3}^{F} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {S_{1mk1}^{F} \theta X_{1mk1}^{F} } \\ {S_{1mk2}^{F} \theta X_{1mk2}^{F} } \\ {S_{1mk3}^{F} \theta Y_{1mk3}^{F} } \\ \end{array} }} \right] , \end{aligned}$$

(A.11)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{3mk1}^{F} } \\ {\theta X_{3mk2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {1 \big / {\left( {S_{3mk1}^{F} +X_{3m1}^{F} S_{3mk2}^{F} +X_{3m2}^{F} S_{3mk3}^{F} } \right) }} \\ {X_{3m1}^{F} \theta X_{3mk1}^{F} } \\ \end{array} }} \right] ,\hbox {and }k=1,2,3, \end{aligned}$$

(A.12)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta Y_{3mk1}^{F} } \\ {\theta Y_{3mk2}^{F} } \\ {\theta Y_{3mk3}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {R_{3mk1}^{F} \theta X_{3mk1}^{F} } \\ {R_{3mk2}^{F} \theta X_{3mk2}^{F} } \\ {X_{3m2}^{F} \theta X_{3mk1}^{F} } \\ \end{array} }} \right] , \ \left[ {{\begin{array}{l} {\theta Z_{3mk1}^{F} } \\ {\theta Z_{3mk2}^{F} } \\ {\theta Z_{3mk3}^{F} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {S_{3mk1}^{F} \theta X_{3mk1}^{F} } \\ {S_{3mk2}^{F} \theta X_{3mk2}^{F} } \\ {S_{3mk3}^{F} \theta Y_{3mk3}^{F} } \\ \end{array} }} \right] , \end{aligned}$$

(A.13)

$$\begin{aligned} \left[ {{\begin{array}{l} {X_{1m1}^{F} } \\ {X_{1m2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {{-\left( {BM_{1mk1}^{F} BQ_{1mk3}^{F} -BQ_{1mk1}^{F} BM_{1mk3}^{F} } \right) } \big / {\left( {BM_{1mk2}^{F} BQ_{1mk3}^{F} -BQ_{1mk2}^{F} BM_{1mk3}^{F} } \right) }} \\ {{-\left( {BM_{1mk1}^{F} +BM_{1mk2}^{F} X_{1m1}^{F} } \right) } \big / {BM_{1mk3}^{F} }} \\ \end{array} }} \right] , \end{aligned}$$

(A.14)

$$\begin{aligned} \left[ {{\begin{array}{l} {X_{3m1}^{F} } \\ {X_{3m2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {{-\left( {BM_{3mk1}^{F} BQ_{3mk3}^{F} -BQ_{3mk1}^{F} BM_{3mk3}^{F} } \right) } \big / {\left( {BM_{3mk2}^{F} BQ_{3mk3}^{F} -BQ_{3mk2}^{F} BM_{3mk3}^{F} } \right) }} \\ {{-\left( {BM_{3mk1}^{F} +BM_{3mk2}^{F} X_{3m1}^{F} } \right) } \big / {BM_{3mk3}^{F} }} \\ \end{array} }} \right] , \end{aligned}$$

(A.15)

$$\begin{aligned} BM_{1mk1}^{F}= & {} \left\{ {{\begin{array}{l} {{-R_{1mk1}^{F} \alpha _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{1mk1}^{F} \hbox { as }R_{m1}^{F}<0,} \\ {{R_{1mk1}^{F} \alpha _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{1mk1}^{F} \hbox { as }R_{m1}^{F}>0,} \\ \end{array} }} \right. \nonumber \\ BM_{1mk2}^{F}= & {} \left\{ {{\begin{array}{l} {{-R_{1mk2}^{F} \beta _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{1mk2}^{F} \hbox { as }R_{m2}^{F}<0,} \\ {{R_{1mk2}^{F} \beta _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{1mk2}^{F} \hbox { as }R_{m2}^{F}>0,} \\ \end{array} }} \right. \nonumber \\ BM_{1mk3}^{F}= & {} \left\{ {{\begin{array}{l} {{-\gamma _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{1mk3}^{F} \hbox { as }R_{m3}^{F} <0,} \\ {{\gamma _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{1mk3}^{F} \hbox { as }R_{m3}^{F} >0,} \\ \end{array} }} \right. \end{aligned}$$

(A.16)

$$\begin{aligned} BQ_{1mk1}^{F}= & {} \left\{ {{\begin{array}{l} {S_{1mk1}^{F} -{\alpha _{m}^{F} } \big / \phi \hbox { as }R_{m1}^{F}<0,} \\ {S_{1mk1}^{F} +{\alpha _{m}^{F} } \big / \phi \hbox { as }R_{m1}^{F}>0,} \\ \end{array} }} \right. \nonumber \\ BQ_{1mk2}^{F}= & {} \left\{ {{\begin{array}{l} {S_{1mk2}^{F} -{\beta _{m}^{F} } \big / \phi \hbox { as }R_{m2}^{F} <0,} \\ {S_{1mk2}^{F} +{\beta _{m}^{F} } \big / \phi \hbox { as }R_{m2}^{F} >0,} \\ \end{array} }} \right. \nonumber \\ BQ_{1mk3}^{F}= & {} S_{1mk3}^{F} , \end{aligned}$$

(A.17)

$$\begin{aligned} BM_{3mk1}^{F}= & {} \left\{ {{\begin{array}{l} {{R_{3mk1}^{F} \alpha _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{3mk1}^{F} \hbox { as }R_{m1}^{F}<0,} \\ {{-R_{3mk1}^{F} \alpha _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{3mk1}^{F} \hbox { as }R_{m1}^{F}>0,} \\ \end{array} }} \right. \nonumber \\ BM_{3mk2}^{F}= & {} \left\{ {{\begin{array}{l} {{R_{3mk2}^{F} \beta _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{3mk2}^{F} \hbox { as }R_{m2}^{F}<0}, \\ {{-R_{3mk2}^{F} \beta _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{3mk2}^{F} \hbox { as }R_{m2}^{F}>0}, \\ \end{array} }} \right. \nonumber \\ BM_{3mk3}^{F}= & {} \left\{ {{\begin{array}{l} {{\gamma _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{3mk3}^{F} \hbox { as }R_{m3}^{F} <0}, \\ {-{\gamma _{m}^{F} } \big / \phi -\left( {m\pi } \right) S_{3mk3}^{F} \hbox { as }R_{m3}^{F} >0}, \\ \end{array} }} \right. \end{aligned}$$

(A.18)

$$\begin{aligned} BQ_{3mk1}^{F}= & {} \left\{ {{\begin{array}{l} {S_{3mk1}^{F} +{\alpha _{m}^{F} } \big / \phi \hbox { as }R_{m1}^{F}<0,} \\ {S_{3mk1}^{F} -{\alpha _{m}^{F} } \big / \phi \hbox { as }R_{m1}^{F}>0} ,\\ \end{array} }} \right. \nonumber \\ BQ_{3mk2}^{F}= & {} \left\{ {{\begin{array}{l} {S_{3mk2}^{F} +{\beta _{m}^{F} } \big / \phi \hbox { as }R_{m2}^{F} <0}, \\ {S_{3mk2}^{F} -{\beta _{m}^{F} } \big / \phi \hbox { as }R_{m2}^{F} >0}, \\ \end{array} }} \right. \nonumber \\ BQ_{3mk3}^{F}= & {} S_{3mk3}^{F} , \end{aligned}$$

(A.19)

$$\begin{aligned} R_{1mk1}^{F}= & {} R_{3mk1}^{F} =\left\{ {{\begin{array}{l} {-{\left[ {-\alpha _{m11}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+\alpha _{m12}^{F} } \right] } \big / {\left[ {-\beta _{m11}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+\beta _{m12}^{F} } \right] }\hbox { as }R_{m1}^{F}<0}, \\ {-{\left[ {\alpha _{m11}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+\alpha _{m12}^{F} } \right] } \big / {\left[ {\beta _{m11}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+\beta _{m12}^{F} } \right] }\hbox { as }R_{m1}^{F}>0}, \\ \end{array} }} \right. \nonumber \\ R_{1mk2}^{F}= & {} R_{3mk2}^{F} =\left\{ {{\begin{array}{l} {-{\left[ {-\alpha _{m11}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+\alpha _{m12}^{F} } \right] } \big / {\left[ {-\beta _{m11}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+\beta _{m12}^{F} } \right] }\hbox { as }R_{m2}^{F} <0}, \\ {-{\left[ {\alpha _{m11}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+\alpha _{m12}^{F} } \right] } \big / {\left[ {\beta _{m11}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+\beta _{m12}^{F} } \right] }\hbox { as }R_{m2}^{F} >0}, \\ \end{array} }} \right. \end{aligned}$$

(A.20)

$$\begin{aligned} S_{1mk1}^{F}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-a_{m13}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+a_{m14}^{F} +b_{m14}^{F} R_{1mk1}^{F} } \right] } \big / {\left( {c_{m14}^{F} \alpha _{m}^{F} } \right) }\hbox { as }R_{m1}^{F}<0}, \\ {-{\left[ {a_{m13}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+a_{m14}^{F} +b_{m14}^{F} R_{1mk1}^{F} } \right] } \big / {\left( {c_{m14}^{F} \alpha _{m}^{F} } \right) }\hbox { as }R_{m1}^{F}>0}, \\ \end{array} }} \right. \nonumber \\ S_{1mk2}^{F}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-a_{m13}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+a_{m14}^{F} +b_{m14}^{F} R_{1mk2}^{F} } \right] } \big / {\left( {c_{m14}^{F} \beta _{m}^{F} } \right) }\hbox { as }R_{m2}^{F} <0}, \\ {-{\left[ {a_{m13}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+a_{m14}^{F} +b_{m14}^{F} R_{1mk2}^{F} } \right] } \big / {\left( {c_{m14}^{F} \beta _{m}^{F} } \right) }\hbox { as }R_{m2}^{F} >0} ,\\ \end{array} }} \right. \nonumber \\ S_{1mk3}^{F}= & {} -{b_{m14}^{F} } \big / {\left( {c_{m14}^{F} \gamma _{m}^{F} } \right) }, \end{aligned}$$

(A.21)

$$\begin{aligned} S_{3mk1}^{F}= & {} \left\{ {{\begin{array}{l} {{\left[ {-a_{m13}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+a_{m14}^{F} +b_{m14}^{F} R_{3mk1}^{F} } \right] } \big / {\left( {c_{m14}^{F} \alpha _{m}^{F} } \right) }\hbox { as }R_{m1}^{F}<0}, \\ {{\left[ {a_{m13}^{F} \left( {\alpha _{m}^{F} } \right) ^{2}+a_{m14}^{F} +b_{m14}^{F} R_{3mk1}^{F} } \right] } \big / {\left( {c_{m14}^{F} \alpha _{m}^{F} } \right) }\hbox { as }R_{m1}^{F}>0}, \\ \end{array} }} \right. \nonumber \\ S_{3mk2}^{F}= & {} \left\{ {{\begin{array}{l} {{\left[ {-a_{m13}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+a_{m14}^{F} +b_{m14}^{F} R_{3mk2}^{F} } \right] } \big / {\left( {c_{m14}^{F} \beta _{m}^{F} } \right) }\hbox { as }R_{m2}^{F} <0}, \\ {{\left[ {a_{m13}^{F} \left( {\beta _{m}^{F} } \right) ^{2}+a_{m14}^{F} +b_{m14}^{F} R_{3mk2}^{F} } \right] } \big / {\left( {c_{m14}^{F} \beta _{m}^{F} } \right) }\hbox { as }R_{m2}^{F} >0}, \\ \end{array} }} \right. \nonumber \\ S_{3mk3}^{F}= & {} {b_{m14}^{F} } \big / {\left( {c_{m14}^{F} \gamma _{m}^{F} } \right) }. \end{aligned}$$

(A.22)

As $n=0$, the coefficients of the second and fourth building block are

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{20k1}^{F} } \\ {\theta X_{20k2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {1 \big / {\left( {R_{20k1}^{F} +X_{201}^{F} R_{20k2}^{F} } \right) }} \\ {X_{201}^{F} \theta X_{20k1}^{F} } \\ \end{array} }} \right] , \ \left[ {{\begin{array}{l} {\theta Y_{20k1}^{F} } \\ {\theta Y_{20k2}^{F} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {R_{20k1}^{F} \theta X_{20k1}^{F} } \\ {R_{20k2}^{F} \theta X_{20k2}^{F} } \\ \end{array} }} \right] ,\hbox { and }k=1,2,3, \end{aligned}$$

(A.23)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{40k1}^{F} } \\ {\theta X_{40k2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {1 \big / {\left( {R_{40k1}^{F} +X_{401}^{F} R_{40k2}^{F} } \right) }} \\ {X_{401}^{F} \theta X_{40k1}^{F} } \\ \end{array} }} \right] , \ \left[ {{\begin{array}{l} {\theta Y_{40k1}^{F} } \\ {\theta Y_{40k2}^{F} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {R_{40k1}^{F} \theta X_{40k1}^{F} } \\ {R_{40k2}^{F} \theta X_{40k2}^{F} } \\ \end{array} }} \right] ,\hbox { and }k=1,2,3, \end{aligned}$$

(A.24)

$$\begin{aligned} X_{201}^{F}= & {} -{BQ_{20k1}^{F} } \big / {BQ_{20k2}^{F} }, \ X_{401}^{F} =-{BQ_{40k1}^{F} } \big / {BQ_{40k2}^{F} }, \end{aligned}$$

(A.25)

$$\begin{aligned} BQ_{20k1}^{F}= & {} \left\{ {{\begin{array}{l} {R_{20k1}^{F} -{\alpha }'{_{n}^{F}} \hbox { as }R_{n1}^{F}<0}, \\ {R_{20k1}^{F} +{\alpha }'{_{n}^{F}} \hbox { as }R_{n1}^{F}>0}, \\ \end{array} }} \right. \ BQ_{20k2}^{F} =\left\{ {{\begin{array}{l} {R_{20k2}^{F} -{\beta }'{_{n}^{F}} \hbox { as }R_{n2}^{F} <0}, \\ {R_{20k2}^{F} +{\beta }'{_{n}^{F}} \hbox { as }R_{n2}^{F} >0}, \\ \end{array} }} \right. \end{aligned}$$

(A.26)

$$\begin{aligned} BQ_{40k1}^{F}= & {} \left\{ {{\begin{array}{l} {R_{40k1}^{F} +{\alpha }'{_{n}^{F}} \hbox { as }R_{n1}^{F}<0,} \\ {R_{40k1}^{F} -{\alpha }'{_{n}^{F}} \hbox { as }R_{n1}^{F}>0}, \\ \end{array} }} \right. \ BQ_{40k2}^{F} =\left\{ {{\begin{array}{l} {R_{40k2}^{F} +{\beta }'{_{n}^{F}} \hbox { as }R_{n2}^{F} <0,} \\ {R_{40k2}^{F} -{\beta }'{_{n}^{F}} \hbox { as }R_{n2}^{F} >0}, \\ \end{array} }} \right. \end{aligned}$$

(A.27)

$$\begin{aligned} R_{20k1}^{F}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-{a}'{_{n13}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\alpha }'{_{n}^{F}} } \right) } \text{ as } R_{n1}^{F}<0,} \\ {-{\left[ {{a}'{_{n13}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\alpha }'{_{n}^{F}} } \right) } \text{ as } R_{n1}^{F} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(A.28)

$$\begin{aligned} R_{20k2}^{F}= & {} {} \left\{ {{\begin{array}{l} {-{\left[ {-{a}'{_{n13}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\beta }'{_{n}^{F}} } \right) } \text{ as } R_{n2}^{F}<0}, \\ {-{\left[ {{a}'{_{n13}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\beta }'{_{n}^{F}} } \right) } \text{ as } R_{n2}^{F} >0}, \\ \end{array} }} \right. \end{aligned}$$

(A.29)

$$\begin{aligned} R_{40k1}^{F}= & {} {} \left\{ {{\begin{array}{l} {{\left[ {-{a}'{_{n13}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\alpha }'{_{n}^{F}} } \right) } \text{ as } R_{n1}^{F}<0} ,\\ {{\left[ {{a}'{_{n13}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\alpha }'{_{n}^{F}} } \right) } \text{ as } R_{n1}^{F} >0}, \\ \end{array} }} \right. \end{aligned}$$

(A.30)

$$\begin{aligned} R_{40k2}^{F}= & {} {} \left\{ {{\begin{array}{l} {{\left[ {-{a}'{_{n13}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\beta }'{_{n}^{F}} } \right) } \text{ as } R_{n2}^{F}<0}, \\ {{\left[ {{a}'{_{n13}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\beta }'{_{n}^{F}} } \right) } \text{ as } R_{n2}^{F} >0.} \\ \end{array} }} \right. \end{aligned}$$

(A.31)

As $n\ge 1$, the coefficients of the first and third building block are

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{2nk1}^{F} } \\ {\theta X_{2nk2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {1 \big / {\left( {R_{2nk1}^{F} +X_{2n1}^{F} R_{2nk2}^{F} +X_{2n2}^{F} R_{2nk3}^{F} } \right) }} \\ {X_{2n1}^{F} \theta X_{2nk1}^{F} } \\ \end{array} }} \right] \hbox {, and }k=1,2,3, \end{aligned}$$

(A.32)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta Z_{2nk1}^{F} } \\ {\theta Z_{2nk2}^{F} } \\ {\theta Z_{2nk3}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {S_{2nk1}^{F} \theta X_{2nk1}^{F} } \\ {S_{2nk2}^{F} \theta X_{2nk2}^{F} } \\ {X_{2n2}^{F} \theta X_{2nk1}^{F} } \\ \end{array} }} \right] ,\ \left[ {{\begin{array}{l} {\theta Y_{2nk1}^{F} } \\ {\theta Y_{2nk2}^{F} } \\ {\theta Y_{2nk3}^{F} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {R_{2nk1}^{F} \theta X_{2nk1}^{F} } \\ {R_{2nk2}^{F} \theta X_{2nk2}^{F} } \\ {R_{2nk3}^{F} \theta Z_{2nk3}^{F} } \\ \end{array} }} \right] , \end{aligned}$$

(A.33)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{4nk1}^{F} } \\ {\theta X_{4nk2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {1 \big / {\left( {R_{4nk1}^{F} +X_{4n1}^{F} R_{4nk2}^{F} +X_{4n2}^{F} R_{4nk3}^{F} } \right) }} \\ {X_{4n1}^{F} \theta X_{4nk1}^{F} } \\ \end{array} }} \right] \hbox {, and }k=1,2,3, \end{aligned}$$

(A.34)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta Z_{4nk1}^{F} } \\ {\theta Z_{4nk2}^{F} } \\ {\theta Z_{4nk3}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {S_{4nk1}^{F} \theta X_{4nk1}^{F} } \\ {S_{4nk2}^{F} \theta X_{4nk2}^{F} } \\ {X_{4n2}^{F} \theta X_{4nk1}^{F} } \\ \end{array} }} \right] ,\ \left[ {{\begin{array}{l} {\theta Y_{4nk1}^{F} } \\ {\theta Y_{4nk2}^{F} } \\ {\theta Y_{4nk3}^{F} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {R_{4nk1}^{F} \theta X_{4nk1}^{F} } \\ {R_{4nk2}^{F} \theta X_{4nk2}^{F} } \\ {R_{4nk3}^{F} \theta Z_{4nk3}^{F} } \\ \end{array} }} \right] , \end{aligned}$$

(A.35)

$$\begin{aligned} \left[ {{\begin{array}{l} {X_{2n1}^{F} } \\ {X_{2n2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {{-\left( {BM_{2nk1}^{F} BQ_{2nk3}^{F} -BQ_{2nk1}^{F} BM_{2nk3}^{F} } \right) } \big / {\left( {BM_{2nk2}^{F} BQ_{2nk3}^{F} -BQ_{2nk2}^{F} BM_{2nk3}^{F} } \right) }} \\ {{-\left( {BM_{2nk1}^{F} +BM_{2nk2}^{F} X_{2n1}^{F} } \right) } \big / {BM_{2nk3}^{F} }} \\ \end{array} }} \right] ,\nonumber \\ \end{aligned}$$

(A.36)

$$\begin{aligned} \left[ {{\begin{array}{c} {X_{4n1}^{F} } \\ {X_{4n2}^{F} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {{-\left( {BM_{4nk1}^{F} BQ_{4nk3}^{F} -BQ_{4nk1}^{F} BM_{4nk3}^{F} } \right) } \big / {\left( {BM_{4nk2}^{F} BQ_{4nk3}^{F} -BQ_{4nk2}^{F} BM_{4nk3}^{F} } \right) }} \\ {{-\left( {BM_{4nk1}^{F} +BM_{4nk2}^{F} X_{4n1}^{F} } \right) } \big / {BM_{4nk3}^{F} }} \\ \end{array} }} \right] ,\nonumber \\ \end{aligned}$$

(A.37)

$$\begin{aligned} BM_{2nk1}^{F}= & {} \left\{ {{\begin{array}{l} {-S_{2nk1}^{F} {\alpha }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{2nk1}^{F} \hbox { as }R_{n1}^{F}<0} ,\\ {S_{2nk1}^{F} {\alpha }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{2nk1}^{F} \hbox { as }R_{n1}^{F}>0} ,\\ \end{array} }} \right. \nonumber \\ BM_{2nk2}^{F}= & {} \left\{ {{\begin{array}{l} {-S_{2nk2}^{F} {\beta }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{2nk2}^{F} \hbox { as }R_{n2}^{F}<0} ,\\ {S_{2nk2}^{F} {\beta }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{2nk2}^{F} \hbox { as }R_{n2}^{F}>0} ,\\ \end{array} }} \right. \nonumber \\ BM_{2nk3}^{F}= & {} \left\{ {{\begin{array}{l} {-{\gamma }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{2nk3}^{F} \hbox { as }R_{n3}^{F} <0} ,\\ {{\gamma }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{2nk3}^{F} \hbox { as }R_{n3}^{F} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(A.38)

$$\begin{aligned} BQ_{2nk1}^{F}= & {} \left\{ {{\begin{array}{l} {R_{2nk1}^{F} -{\alpha }'{_{n}^{F}} \hbox { as }R_{n1}^{F}<0} ,\\ {R_{2nk1}^{F} +{\alpha }'{_{n}^{F}} \hbox { as }R_{n1}^{F}>0} ,\\ \end{array} }} \right. \nonumber \\ BQ_{2nk2}^{F}= & {} \left\{ {{\begin{array}{l} {R_{2nk2}^{F} -{\beta }'{_{n}^{F}} \hbox { as }R_{n2}^{F} <0} ,\\ {R_{2nk2}^{F} +{\beta }'{_{n}^{F}} \hbox { as }R_{n2}^{F} >0} ,\\ \end{array} }} \right. \nonumber \\ BQ_{2nk3}^{F}= & {} R_{2nk3}^{F} , \end{aligned}$$

(A.39)

$$\begin{aligned} BM_{4nk1}^{F}= & {} \left\{ {{\begin{array}{l} {S_{4nk1}^{F} {\alpha }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{4nk1}^{F} \hbox { as }R_{n1}^{F}<0} ,\\ {-S_{4nk1}^{F} {\alpha }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{4nk1}^{F} \hbox { as }R_{n1}^{F}>0} ,\\ \end{array} }} \right. \nonumber \\ BM_{4nk2}^{F}= & {} \left\{ {{\begin{array}{l} {S_{4nk2}^{F} {\beta }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{4nk2}^{F} \hbox { as }R_{n2}^{F}<0} ,\\ {-S_{4nk2}^{F} {\beta }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{4nk2}^{F} \hbox { as }R_{n2}^{F}>0} ,\\ \end{array} }} \right. \nonumber \\ BM_{4nk3}^{F}= & {} \left\{ {{\begin{array}{l} {{\gamma }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{4nk3}^{F} \hbox { as }R_{n3}^{F} <0} ,\\ {-{\gamma }'{_{n}^{F}} -\left( {{n\pi } \big / \phi } \right) R_{4nk3}^{F} \hbox { as }R_{n3}^{F} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(A.40)

$$\begin{aligned} BQ_{4nk1}^{F}= & {} \left\{ {{\begin{array}{l} {R_{4nk1}^{F} +{\alpha }'{_{n}^{F}} \hbox { as }R_{n1}^{F}<0} ,\\ {R_{4nk1}^{F} -{\alpha }'{_{n}^{F}} \hbox { as }R_{n1}^{F}>0} ,\\ \end{array} }} \right. \nonumber \\ BQ_{4nk2}^{F}= & {} \left\{ {{\begin{array}{l} {R_{4nk2}^{F} +{\beta }'{_{n}^{F}} \hbox { as }R_{n2}^{F} <0} ,\\ {R_{4nk2}^{F} -{\beta }'{_{n}^{F}} \hbox { as }R_{n2}^{F} >0} ,\\ \end{array} }} \right. \nonumber \\ BQ_{4nk3}^{F}= & {} R_{4nk3}^{F} , \end{aligned}$$

(A.41)

$$\begin{aligned} S_{2nk1}^{F}= & {} S_{4nk1}^{F} =\left\{ {{\begin{array}{l} {{-\left[ {-{\alpha }'{_{n11}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{\alpha }'{_{n12}^{F}} } \right] } \big / {\left[ {-{\beta }'{_{n11}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{\beta }'{_{n12}^{F}} } \right] }\hbox { as }R_{n1}^{F}<0} ,\\ {{-\left[ {{\alpha }'{_{n11}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{\alpha }'{_{n12}^{F}} } \right] } \big / {\left[ {{\beta }'{_{n11}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{\beta }'{_{n12}^{F}} } \right] }\hbox { as }R_{n1}^{F}>0} ,\\ \end{array} }} \right. \nonumber \\ S_{2nk2}^{F}= & {} S_{4nk2}^{F} =\left\{ {{\begin{array}{l} {{-\left[ {-{\alpha }'{_{n11}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{\alpha }'{_{n12}^{F}} } \right] } \big / {\left[ {-{\beta }'{_{n11}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{\beta }'{_{n12}^{F}} } \right] }\hbox { as }R_{n2}^{F} <0} ,\\ {{-\left[ {{\alpha }'{_{n11}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{\alpha }'{_{n12}^{F}} } \right] } \big / {\left[ {{\beta }'{_{n11}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{\beta }'{_{n12}^{F}} } \right] }\hbox { as }R_{n2}^{F} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(A.42)

$$\begin{aligned} R_{2nk1}^{F}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-{a}'{_{n13}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} +{b}'{_{n14}^{F}} S_{2nk1}^{F} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\alpha }'{_{n}^{F}} } \right) }\hbox { as }R_{n1}^{F}<0} ,\\ {-{\left[ {{a}'{_{n13}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} +{b}'{_{n14}^{F}} S_{2nk1}^{F} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\alpha }'{_{n}^{F}} } \right) }\hbox { as }R_{n1}^{F}>0} ,\\ \end{array} }} \right. \nonumber \\ R_{2nk2}^{F}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-{a}'{_{n13}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} +{b}'{_{n14}^{F}} S_{2nk2}^{F} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\beta }'{_{n}^{F}} } \right) }\hbox { as }R_{n2}^{F} <0} ,\\ {-{\left[ {{a}'{_{n13}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} +{b}'{_{n14}^{F}} S_{2nk2}^{F} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\beta }'{_{n}^{F}} } \right) }\hbox { as }R_{n2}^{F} >0} ,\\ \end{array} }} \right. \nonumber \\ R_{2nk3}^{F}= & {} -{{b}'{_{n14}^{F}} } \big / {\left( {{c}'{_{n14}^{F}} {\gamma }'{_{n}^{F}} } \right) } , \end{aligned}$$

(A.43)

$$\begin{aligned} R_{4nk1}^{F}= & {} \left\{ {{\begin{array}{l} {{\left[ {-{a}'{_{n13}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} +{b}'{_{n14}^{F}} S_{4nk1}^{F} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\alpha }'{_{n}^{F}} } \right) }\hbox { as }R_{n1}^{F}<0} ,\\ {{\left[ {{a}'{_{n13}^{F}} \left( {{\alpha }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} +{b}'{_{n14}^{F}} S_{4nk1}^{F} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\alpha }'{_{n}^{F}} } \right) }\hbox { as }R_{n1}^{F}>0} ,\\ \end{array} }} \right. \nonumber \\ R_{4nk2}^{F}= & {} \left\{ {{\begin{array}{l} {{\left[ {-{a}'{_{n13}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} +{b}'{_{n14}^{F}} S_{4nk2}^{F} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\beta }'{_{n}^{F}} } \right) }\hbox { as }R_{n2}^{F} <0} ,\\ {{\left[ {{a}'{_{n13}^{F}} \left( {{\beta }'{_{n}^{F}} } \right) ^{2}+{a}'{_{n14}^{F}} +{b}'{_{n14}^{F}} S_{4nk2}^{F} } \right] } \big / {\left( {{c}'{_{n14}^{F}} {\beta }'{_{n}^{F}} } \right) }\hbox { as }R_{n2}^{F} >0,} \\ \end{array} }} \right. \nonumber \\ R_{4nk3}^{F}= & {} {{b}'{_{n14}^{F}} } \big / {\left( {{c}'{_{n14}^{F}} {\gamma }'{_{n}^{F}} } \right) } . \end{aligned}$$

(A.44)

Appendix B: Constants in the extensional type displacement solution

The constants in the extensional type displacement solutions are listed as follows:

As $m=0$, the coefficients of the first and third building blocks are

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{10k1}^{E} } \\ {\theta X_{10k2}^{E} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {1 \big / {\left( {S_{10k1}^{E} +X_{101}^{E} S_{10k2}^{E} } \right) }} \\ {X_{101}^{E} \theta X_{10k1}^{E} } \\ \end{array} }} \right] , \left[ {{\begin{array}{l} {\theta Z_{10k1}^{E} } \\ {\theta Z_{10k2}^{E} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {S_{10k1}^{E} \theta X_{10k1}^{E} } \\ {S_{10k2}^{E} \theta X_{10k2}^{E} } \\ \end{array} }} \right] \hbox {, and }k=1,2,3, \end{aligned}$$

(B.1)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{30k1}^{E} } \\ {\theta X_{30k2}^{E} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {1 \big / {\left( {S_{30k1}^{E} +X_{301}^{E} S_{30k2}^{E} } \right) }} \\ {X_{301}^{E} \theta X_{30k1}^{E} } \\ \end{array} }} \right] , \left[ {{\begin{array}{l} {\theta Z_{30k1}^{E} } \\ {\theta Z_{30k2}^{E} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {S_{30k1}^{E} \theta X_{30k1}^{E} } \\ {S_{30k2}^{E} \theta X_{30k2}^{E} } \\ \end{array} }} \right] \hbox {, and }k=1,2,3, \end{aligned}$$

(B.2)

$$\begin{aligned} X_{101}^{E}= & {} -{BQ_{10k1}^{E} } \big / {BQ_{10k2}^{E} }, X_{301}^{E} =-{BQ_{30k1}^{E} } \big / {BQ_{30k2}^{E} }, \end{aligned}$$

(B.3)

$$\begin{aligned} BQ_{10k1}^{E}= & {} \left\{ {{\begin{array}{l} {-\alpha _{m}^{E} \hbox { as }R_{m1}^{E}<0}, \\ {\alpha _{m}^{E} \hbox { as }R_{m1}^{E}>0}, \\ \end{array} }} \right. BQ_{10k2}^{E} =\left\{ {{\begin{array}{l} {-\beta _{m}^{E} \hbox { as }R_{m2}^{E} <0}, \\ {\beta _{m}^{E} \hbox { as }R_{m2}^{E} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(B.4)

$$\begin{aligned} BQ_{30k1}^{E}= & {} \left\{ {{\begin{array}{l} {\alpha _{m}^{E} \hbox { as, }R_{m1}^{E}<0},\\ {-\alpha _{m}^{E} \hbox { as }R_{m1}^{E}>0} ,\\ \end{array} }} \right. BQ_{30k2}^{E} =\left\{ {{\begin{array}{l} {\beta _{m}^{E} \hbox { as }R_{m2}^{E} <0} ,\\ {-\beta _{m}^{E} \hbox { as }R_{m2}^{E} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(B.5)

$$\begin{aligned} S_{10k1}^{E}= & {} {} \left\{ {{\begin{array}{l} {-{\left[ {-a_{m13}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+a_{m14}^{E} } \right] } \big / {\left( {c_{m14}^{E} \alpha _{m}^{E} } \right) } \text{ as } R_{m1}^{E}<0} ,\\ {-{\left[ {a_{m13}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+a_{m14}^{E} } \right] } \big / {\left( {c_{m14}^{E} \alpha _{m}^{E} } \right) } \text{ as } R_{m1}^{E} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(B.6)

$$\begin{aligned} S_{10k2}^{E}= & {} {} \left\{ {{\begin{array}{l} {-{\left[ {-a_{m13}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+a_{m14}^{E} } \right] } \big / {\left( {c_{m14}^{E} \beta _{m}^{E} } \right) } \text{ as } R_{m2}^{E}<0} ,\\ {-{\left[ {a_{m13}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+a_{m14}^{E} } \right] } \big / {\left( {c_{m14}^{E} \beta _{m}^{E} } \right) } \text{ as } R_{m2}^{E} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(B.7)

$$\begin{aligned} S_{30k1}^{E}= & {} {} \left\{ {{\begin{array}{l} {{\left[ {-a_{m13}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+a_{m14}^{E} } \right] } \big / {\left( {c_{m14}^{E} \alpha _{m}^{E} } \right) } \text{ as } R_{m1}^{E}<0} ,\\ {{\left[ {a_{m13}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+a_{m14}^{E} } \right] } \big / {\left( {c_{m14}^{E} \alpha _{m}^{E} } \right) } \text{ as } R_{m1}^{E} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(B.8)

$$\begin{aligned} S_{30k2}^{E}= & {} {} \left\{ {{\begin{array}{l} {{\left[ {-a_{m13}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+a_{m14}^{E} } \right] } \big / {\left( {c_{m14}^{E} \beta _{m}^{E} } \right) } \text{ as } R_{m2}^{E}<0} ,\\ {{\left[ {a_{m13}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+a_{m14}^{E} } \right] } \big / {\left( {c_{m14}^{E} \beta _{m}^{E} } \right) } \text{ as } R_{m2}^{E} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(B.9)

As $m\ge 1$, the coefficients of the first and third building blocks are

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{1mk1}^{E} } \\ {\theta X_{1mk2}^{E} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {1 \big / {\left( {S_{1mk1}^{E} +X_{1m1}^{E} S_{1mk2}^{E} +X_{1m2}^{E} S_{1mk3}^{E} } \right) }} \\ {X_{1m1}^{E} \theta X_{1mk1}^{E} } \\ \end{array} }} \right] \hbox {, and }k=1,2,3, \end{aligned}$$

(B.10)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta Y_{1mk1}^{E} } \\ {\theta Y_{1mk2}^{E} } \\ {\theta Y_{1mk3}^{E} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {R_{1mk1}^{E} \theta X_{1mk1}^{E} } \\ {R_{1mk2}^{E} \theta X_{1mk2}^{E} } \\ {X_{1m2}^{E} \theta X_{1mk1}^{E} } \\ \end{array} }} \right] , \left[ {{\begin{array}{l} {\theta Z_{1mk1}^{E} } \\ {\theta Z_{1mk2}^{E} } \\ {\theta Z_{1mk3}^{E} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {S_{1mk1}^{E} \theta X_{1mk1}^{E} } \\ {S_{1mk2}^{E} \theta X_{1mk2}^{E} } \\ {S_{1mk3}^{E} \theta Y_{1mk3}^{E} } \\ \end{array} }} \right] , \end{aligned}$$

(B.11)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{3mk1}^{E} } \\ {\theta X_{3mk2}^{E} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {1 \big / {\left( {S_{3mk1}^{E} +X_{3m1}^{E} S_{3mk2}^{E} +X_{3m2}^{E} S_{3mk3}^{E} } \right) }} \\ {X_{3m1}^{E} \theta X_{3mk1}^{E} } \\ \end{array} }} \right] \hbox {, and }k=1,2,3, \end{aligned}$$

(B.12)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta Y_{3mk1}^{E} } \\ {\theta Y_{3mk2}^{E} } \\ {\theta Y_{3mk3}^{E} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {R_{3mk1}^{E} \theta X_{3mk1}^{E} } \\ {R_{3mk2}^{E} \theta X_{3mk2}^{E} } \\ {X_{3m2}^{E} \theta X_{3mk1}^{E} } \\ \end{array} }} \right] , \left[ {{\begin{array}{l} {\theta Z_{3mk1}^{E} } \\ {\theta Z_{3mk2}^{E} } \\ {\theta Z_{3mk3}^{E} } \\ \end{array} }} \right] =\left[ {{\begin{array}{l} {S_{3mk1}^{E} \theta X_{3mk1}^{E} } \\ {S_{3mk2}^{E} \theta X_{3mk2}^{E} } \\ {S_{3mk3}^{E} \theta Y_{3mk3}^{E} } \\ \end{array} }} \right] , \end{aligned}$$

(B.13)

$$\begin{aligned} \left[ {{\begin{array}{l} {X_{1m1}^{E} } \\ {X_{1m2}^{E} } \\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {{-\left( {BM_{1mk1}^{E} BQ_{1mk3}^{E} -BQ_{1mk1}^{E} BM_{1mk3}^{E} } \right) } \big / {\left( {BM_{1mk2}^{E} BQ_{1mk3}^{E} -BQ_{1mk2}^{E} BM_{1mk3}^{E} } \right) }} \\ {{-\left( {BM_{1mk1}^{E} +BM_{1mk2}^{E} X_{1m1}^{E} } \right) } \big / {BM_{1mk3}^{E} }} \\ \end{array} }} \right] , \end{aligned}$$

(B.14)

$$\begin{aligned} \left[ {{\begin{array}{l} {X_{3m1}^{E} } ,\\ {X_{3m2}^{E} } ,\\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {{-\left( {BM_{3mk1}^{E} BQ_{3mk3}^{E} -BQ_{3mk1}^{E} BM_{3mk3}^{E} } \right) } \big / {\left( {BM_{3mk2}^{E} BQ_{3mk3}^{E} -BQ_{3mk2}^{E} BM_{3mk3}^{E} } \right) }} ,\\ {{-\left( {BM_{3mk1}^{E} +BM_{3mk2}^{E} X_{3m1}^{E} } \right) } \big / {BM_{3mk3}^{E} }} ,\\ \end{array} }} \right] , \end{aligned}$$

(B.15)

$$\begin{aligned} BM_{1mk1}^{E}= & {} \left\{ {{\begin{array}{l} {{-R_{1mk1}^{E} \alpha _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{1mk1}^{E} \hbox { as }R_{m1}^{E}<0},\\ {{R_{1mk1}^{E} \alpha _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{1mk1}^{E} \hbox { as }R_{m1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BM_{1mk2}^{E}= & {} \left\{ {{\begin{array}{l} {{-R_{1mk2}^{E} \beta _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{1mk2}^{E} \hbox { as }R_{m2}^{E}<0},\\ {{R_{1mk2}^{E} \beta _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{1mk2}^{E} \hbox { as }R_{m2}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BM_{1mk3}^{E}= & {} \left\{ {{\begin{array}{l} {{-\gamma _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{1mk3}^{E} \hbox { as }R_{m3}^{E}<0},\\ {{\gamma _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{1mk3}^{E} \hbox { as }R_{m3}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BQ_{1mk1}^{E}= & {} \left\{ {{\begin{array}{l} {-\alpha _{m}^{E} \hbox { as }R_{m1}^{E}<0},\\ {\alpha _{m}^{E} \hbox { as }R_{m1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BQ_{1mk2}^{E}= & {} \left\{ {{\begin{array}{l} {-\beta _{m}^{E} \hbox { as }R_{m2}^{E} <0,} \\ {\beta _{m}^{E} \hbox { as }R_{m2}^{E} >0},\\ \end{array} }} \right. \nonumber \\ BQ_{1mk3}^{E}= & {} 0 \end{aligned}$$

(B.16)

$$\begin{aligned} BM_{3mk1}^{E}= & {} \left\{ {{\begin{array}{l} {{R_{3mk1}^{E} \alpha _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{3mk1}^{E} \hbox { as }R_{m1}^{E}<0},\\ {{-R_{3mk1}^{E} \alpha _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{3mk1}^{E} \hbox { as }R_{m1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BM_{3mk2}^{E}= & {} \left\{ {{\begin{array}{l} {{R_{3mk2}^{E} \beta _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{3mk2}^{E} \hbox { as }R_{m2}^{E}<0},\\ {-{R_{3mk2}^{E} \beta _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{3mk2}^{E} \hbox { as }R_{m2}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BM_{3mk3}^{E}= & {} \left\{ {{\begin{array}{l} {{\gamma _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{3mk3}^{E} \hbox { as }R_{m3}^{E}<0},\\ {{-\gamma _{m}^{E} } \big / \phi -\left( {m\pi } \right) S_{3mk3}^{E} \hbox { as }R_{m3}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BQ_{3mk1}^{E}= & {} \left\{ {{\begin{array}{l} {\alpha _{m}^{E} \hbox { as }R_{m1}^{E}<0},\\ {-\alpha _{m}^{E} \hbox { as }R_{m1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BQ_{3mk2}^{E}= & {} \left\{ {{\begin{array}{l} {\beta _{m}^{E} \hbox { as }R_{m2}^{E} <0},\\ {-\beta _{m}^{E} \hbox { as }R_{m2}^{E} >0} \\ \end{array} }} \right. \nonumber \\ BQ_{3mk3}^{E}= & {} 0; \end{aligned}$$

(B.17)

$$\begin{aligned} R_{1mk1}^{E}= & {} R_{3mk1}^{E} =\left\{ {{\begin{array}{l} {-{\left[ {-\alpha _{m11}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+\alpha _{m12}^{E} } \right] } \big / {\left[ {-\beta _{m11}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+\beta _{m12}^{E} } \right] }\hbox { as }R_{m1}^{E}<0},\\ {-{\left[ {\alpha _{m11}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+\alpha _{m12}^{E} } \right] } \big / {\left[ {\beta _{m11}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+\beta _{m12}^{E} } \right] }\hbox { as }R_{m1}^{E}>0,} \\ \end{array} }} \right. \nonumber \\ R_{1mk2}^{E}= & {} R_{3mk2}^{E} =\left\{ {{\begin{array}{l} {-{\left[ {-\alpha _{m11}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+\alpha _{m12}^{E} } \right] } \big / {\left[ {-\beta _{m11}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+\beta _{m12}^{E} } \right] }\hbox { as }R_{m2}^{E} <0,} \\ {-{\left[ {\alpha _{m11}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+\alpha _{m12}^{E} } \right] } \big / {\left[ {\beta _{m11}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+\beta _{m12}^{E} } \right] }\hbox { as }R_{m2}^{E} >0},\\ \end{array} }} \right. \end{aligned}$$

(B.18)

$$\begin{aligned} S_{1mk1}^{E}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-a_{m13}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+a_{m14}^{E} +b_{m14}^{E} R_{1mk1}^{E} } \right] } \big / {\left( {c_{m14}^{E} \alpha _{m}^{E} } \right) }\hbox { as }R_{m1}^{E}<0},\\ {-{\left[ {a_{m13}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+a_{m14}^{E} +b_{m14}^{E} R_{1mk1}^{E} } \right] } \big / {\left( {c_{m14}^{E} \alpha _{m}^{E} } \right) }\hbox { as }R_{m1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ S_{1mk2}^{E}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-a_{m13}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+a_{m14}^{E} +b_{m14}^{E} R_{1mk2}^{E} } \right] } \big / {\left( {c_{m14}^{E} \beta _{m}^{E} } \right) }\hbox { as }R_{m2}^{E} <0,} \\ {-{\left[ {a_{m13}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+a_{m14}^{E} +b_{m14}^{E} R_{1mk2}^{E} } \right] } \big / {\left( {c_{m14}^{E} \beta _{m}^{E} } \right) }\hbox { as }R_{m2}^{E} >0},\\ \end{array} }} \right. \nonumber \\ S_{1mk3}^{E}= & {} -{b_{m14}^{E} } \big / {\left( {c_{m14}^{E} \gamma _{m}^{E} } \right) } , \end{aligned}$$

(B.19)

$$\begin{aligned} S_{3mk1}^{E}= & {} \left\{ {{\begin{array}{l} {{\left[ {-a_{m13}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+a_{m14}^{E} +b_{m14}^{E} R_{3mk1}^{E} } \right] } \big / {\left( {c_{m14}^{E} \alpha _{m}^{E} } \right) }\hbox { as }R_{m1}^{E}<0},\\ {{\left[ {a_{m13}^{E} \left( {\alpha _{m}^{E} } \right) ^{2}+a_{m14}^{E} +b_{m14}^{E} R_{3m11}^{E} } \right] } \big / {\left( {c_{m14}^{E} \alpha _{m}^{E} } \right) }\hbox { as }R_{m1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ S_{3mk2}^{E}= & {} \left\{ {{\begin{array}{l} {{\left[ {-a_{m13}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+a_{m14}^{E} +b_{m14}^{E} R_{3mk2}^{E} } \right] } \big / {\left( {c_{m14}^{E} \beta _{m}^{E} } \right) }\hbox { as }R_{m2}^{E} <0},\\ {{\left[ {a_{m13}^{E} \left( {\beta _{m}^{E} } \right) ^{2}+a_{m14}^{E} +b_{m14}^{E} R_{3mk2}^{E} } \right] } \big / {\left( {c_{m14}^{E} \beta _{m}^{E} } \right) }\hbox { as }R_{m2}^{E} >0},\\ \end{array} }} \right. \nonumber \\ S_{3mk3}^{E}= & {} {b_{m14}^{E} } \big / {\left( {c_{m14}^{E} \gamma _{m}^{E} } \right) } . \end{aligned}$$

(B.20)

As $n=0$, the coefficients of the second and fourth building blocks are

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{40k1}^{E} }\\ {\theta X_{40k2}^{E} }\\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {1 \big / {\left( {R_{40k1}^{E} +X_{401}^{E} R_{40k2}^{E} } \right) }}\\ {X_{401}^{E} \theta X_{40k1}^{E} }\\ \end{array} }} \right] , \ \left[ {{\begin{array}{l} {\theta Y_{40k1}^{E} }\\ {\theta Y_{40k2}^{E} }\\ \end{array} }} \right] =\left[ {{\begin{array}{l} {R_{40k1}^{E} \theta X_{40k1}^{E} }\\ {R_{40k2}^{E} \theta X_{40k2}^{E} }\\ \end{array} }} \right] \hbox {, and }k=1,2,3, \end{aligned}$$

(B.21)

$$\begin{aligned} X_{201}^{E}= & {} -{BQ_{20k1}^{E} } \big / {BQ_{20k2}^{E} },\ X_{401}^{E} =-{BQ_{40k1}^{E} } \big / {BQ_{40k2}^{E} }, \end{aligned}$$

(B.22)

$$\begin{aligned} BQ_{20k1}^{E}= & {} \left\{ {{\begin{array}{l} {-{\alpha }'{_{n}^{F}} \hbox { as }R_{n1}^{E}<0},\\ {{\alpha }'{_{n}^{F}} \hbox { as }R_{n1}^{E}>0},\\ \end{array} }} \right. BQ_{20k2}^{E} =\left\{ {{\begin{array}{l} {-{\beta }'{_{n}^{E}} \hbox { as }R_{n2}^{E} <0},\\ {{\beta }'{_{n}^{E}} \hbox { as }R_{n2}^{E} >0},\\ \end{array} }} \right. \end{aligned}$$

(B.23)

$$\begin{aligned} BQ_{40k1}^{E}= & {} \left\{ {{\begin{array}{l} {{\alpha }'{_{n}^{E}} \hbox { as }R_{n1}^{E}<0},\\ {-{\alpha }'{_{n}^{E}} \hbox { as }R_{n1}^{E}>0},\\ \end{array} }} \right. BQ_{40k2}^{E} =\left\{ {{\begin{array}{l} {{\beta }'{_{n}^{E}} \hbox { as }R_{n2}^{E} <0},\\ {-{\beta }'{_{n}^{E}} \hbox { as }R_{n2}^{E} >0},\\ \end{array} }} \right. \end{aligned}$$

(B.24)

$$\begin{aligned} R_{20k1}^{E}= & {} {} \left\{ {{\begin{array}{l} {-{\left[ {-{a}'{_{n13}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\alpha }'{_{n}^{E}} } \right) } \text{ as } R_{n1}^{E}<0},\\ {-{\left[ {{a}'{_{n13}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\alpha }'{_{n}^{E}} } \right) } \text{ as } R_{n1}^{E} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(B.25)

$$\begin{aligned} R_{20k2}^{E}= & {} {} \left\{ {{\begin{array}{l} {-{\left[ {-{a}'{_{n13}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\beta }'{_{n}^{E}} } \right) } \text{ as } R_{n2}^{E}<0} ,\\ {-{\left[ {{a}'{_{n13}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\beta }'{_{n}^{E}} } \right) } \text{ as } R_{n2}^{E} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(B.26)

$$\begin{aligned} R_{40k1}^{E}= & {} {} \left\{ {{\begin{array}{l} {{\left[ {-{a}'{_{n13}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\alpha }'{_{n}^{E}} } \right) } \text{ as } R_{n1}^{E}<0} ,\\ {{\left[ {{a}'{_{n13}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\alpha }'{_{n}^{E}} } \right) } \text{ as } R_{n1}^{E} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(B.27)

$$\begin{aligned} R_{40k2}^{E}= & {} {} \left\{ {{\begin{array}{l} {{\left[ {-{a}'{_{n13}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\beta }'{_{n}^{E}} } \right) } \text{ as } R_{n2}^{E}<0} ,\\ {{\left[ {{a}'{_{n13}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\beta }'{_{n}^{E}} } \right) } \text{ as } R_{n2}^{E} >0} ,\\ \end{array} }} \right. \end{aligned}$$

(B.28)

As $n\ge 1$, the coefficients of the first and third building blocks are

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{2nk1}^{E} }\\ {\theta X_{2nk2}^{E} }\\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {1 \big / {\left( {R_{2nk1}^{E} +X_{2n1}^{E} R_{2nk2}^{E} +X_{2n2}^{E} R_{2nk3}^{E} } \right) }}\\ {X_{2n1}^{E} \theta X_{2nk1}^{E} }\\ \end{array} }} \right] \hbox {, and }k=1,2,3, \end{aligned}$$

(B.29)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta Z_{2nk1}^{E} }\\ {\theta Z_{2nk2}^{E} }\\ {\theta Z_{2nk3}^{E} }\\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {S_{2nk1}^{E} \theta X_{2nk1}^{E} }\\ {S_{2nk2}^{E} \theta X_{2nk2}^{E} }\\ {X_{2n2}^{E} \theta X_{2nk1}^{E} }\\ \end{array} }} \right] , \ \left[ {{\begin{array}{l} {\theta Y_{2nk1}^{E} }\\ {\theta Y_{2nk2}^{E} }\\ {\theta Y_{2nk3}^{E} }\\ \end{array} }} \right] =\left[ {{\begin{array}{l} {R_{2nk1}^{E} \theta X_{2nk1}^{E} }\\ {R_{2nk2}^{E} \theta X_{2nk2}^{E} }\\ {R_{2nk3}^{E} \theta Z_{2nk3}^{E} }\\ \end{array} }} \right] , \end{aligned}$$

(B.30)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta X_{4nk1}^{E} }\\ {\theta X_{4nk2}^{E} }\\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {1 \big / {\left( {R_{4nk1}^{E} +X_{4n1}^{E} R_{4nk2}^{E} +X_{4n2}^{E} R_{4nk3}^{E} } \right) }}\\ {X_{4n1}^{E} \theta X_{4nk1}^{E} }\\ \end{array} }} \right] \hbox {, and }k=1,2,3, \end{aligned}$$

(B.31)

$$\begin{aligned} \left[ {{\begin{array}{l} {\theta Z_{4nk1}^{E} }\\ {\theta Z_{4nk2}^{E} }\\ {\theta Z_{4nk3}^{E} }\\ \end{array} }} \right]= & {} \left[ {{\begin{array}{l} {S_{4nk1}^{E} \theta X_{4nk1}^{E} }\\ {S_{4nk2}^{E} \theta X_{4nk2}^{E} }\\ {X_{4n2}^{E} \theta X_{4nk1}^{E} }\\ \end{array} }} \right] , \ \left[ {{\begin{array}{l} {\theta Y_{4nk1}^{E} }\\ {\theta Y_{4nk2}^{E} }\\ {\theta Y_{4nk3}^{E} }\\ \end{array} }} \right] =\left[ {{\begin{array}{l} {R_{4nk1}^{E} \theta X_{4nk1}^{E} }\\ {R_{4nk2}^{E} \theta X_{4nk2}^{E} }\\ {R_{4nk3}^{E} \theta Z_{4nk3}^{E} }\\ \end{array} }} \right] , \end{aligned}$$

(B.32)

$$\begin{aligned} \left[ {{\begin{array}{l} {X_{2n1}^{E} }\\ {X_{2n2}^{E} }\\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {{-\left( {BM_{2nk1}^{E} BQ_{2nk3}^{E} -BQ_{2nk1}^{E} BM_{2nk3}^{E} } \right) } \big / {\left( {BM_{2nk2}^{E} BQ_{2nk3}^{E} -BQ_{2nk2}^{E} BM_{2nk3}^{E} } \right) }}\\ {{-\left( {BM_{2nk1}^{E} +BM_{2nk2}^{E} X_{2n1}^{E} } \right) } \big / {BM_{2nk3}^{E} }}\\ \end{array} }} \right] , \end{aligned}$$

(B.33)

$$\begin{aligned} \left[ {{\begin{array}{l} {X_{4n1}^{E} }\\ {X_{4n2}^{E} }\\ \end{array} }} \right]= & {} \left[ {{\begin{array}{c} {{-\left( {BM_{4nk1}^{E} BQ_{4nk3}^{E} -BQ_{4nk1}^{E} BM_{4nk3}^{E} } \right) } \big / {\left( {BM_{4nk2}^{E} BQ_{4nk3}^{E} -BQ_{4nk2}^{E} BM_{4nk3}^{E} } \right) }}\\ {{-\left( {BM_{4nk1}^{E} +BM_{4nk2}^{E} X_{4n1}^{E} } \right) } \big / {BM_{4nk3}^{E} }},\\ \end{array} }} \right] , \end{aligned}$$

(B.34)

$$\begin{aligned} BM_{2nk1}^{E}= & {} \left\{ {{\begin{array}{l} {-S_{2nk1}^{E} {\alpha }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{2nk1}^{E} \hbox { as }R_{n1}^{E}<0},\\ {S_{2nk1}^{E} {\alpha }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{2nk1}^{E} \hbox { as }R_{n1}^{E}>0,} \nonumber \\ \end{array} }} \right. \nonumber \\ BM_{2nk2}^{E}= & {} \left\{ {{\begin{array}{l} {-S_{2nk2}^{E} {\beta }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{2nk2}^{E} \hbox { as }R_{n2}^{E}<0},\\ {S_{2nk2}^{E} {\beta }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{2nk2}^{E} \hbox { as }R_{n2}^{E}>0},\\ \end{array}}} \right. \nonumber \\ BM_{2nk3}^{E}= & {} \left\{ {{\begin{array}{l} {-{\gamma }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{2nk3}^{E} \hbox { as }R_{n3}^{E}<0},\\ {{\gamma }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{2nk3}^{E} \hbox { as }R_{n3}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BQ_{2nk1}^{E}= & {} \left\{ {{\begin{array}{l} {-{\alpha }'{_{n}^{E}} \hbox { as }R_{n1}^{E}<0},\\ {{\alpha }'{_{n}^{E}} \hbox { as }R_{n1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BQ_{2nk2}^{E}= & {} \left\{ {{\begin{array}{l} {-\beta _{n}^{E} \hbox { as }R_{n2}^{E} <0},\\ {\beta _{n}^{E} \hbox { as }R_{n2}^{E} >0},\\ \end{array} }} \right. \nonumber \\ BQ_{2nk3}^{E}= & {} 0 \end{aligned}$$

(B.35)

$$\begin{aligned} BM_{4nk1}^{E}= & {} \left\{ {{\begin{array}{l} {S_{4nk1}^{E} {\alpha }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{4nk1}^{E} \hbox { as }R_{n1}^{E}<0},\\ {-S_{4nk1}^{E} {\alpha }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{4nk1}^{E} \hbox { as }R_{n1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BM_{4nk2}^{E}= & {} \left\{ {{\begin{array}{l} {S_{4nk2}^{E} {\beta }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{4nk2}^{E} \hbox { as }R_{n2}^{E}<0},\\ {-S_{4nk2}^{E} {\beta }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{4nk2}^{E} \hbox { as }R_{n2}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BM_{4nk3}^{E}= & {} \left\{ {{\begin{array}{l} {{\gamma }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{4nk3}^{E} \hbox { as }R_{n3}^{E}<0},\\ {-{\gamma }'{_{n}^{E}} -\left( {{n\pi } \big / \phi } \right) R_{4nk3}^{E} \hbox { as }R_{n3}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BQ_{4nk1}^{E}= & {} \left\{ {{\begin{array}{l} {{\alpha }'{_{n}^{E}} \hbox { as }R_{n1}^{E}<0},\\ {-{\alpha }'{_{n}^{E}} \hbox { as }R_{n1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ BQ_{4nk2}^{E}= & {} \left\{ {{\begin{array}{l} {\beta _{n}^{E} \hbox { as }R_{n2}^{E} <0},\\ {-\beta _{n}^{E} \hbox { as }R_{n2}^{E} >0},\\ \end{array} }} \right. \nonumber \\ BQ_{4nk3}^{E}= & {} 0; \end{aligned}$$

(B.36)

$$\begin{aligned} S_{2nk1}^{E}= & {} S_{4nk1}^{E} =\left\{ {{\begin{array}{l} {-{\left[ {-{\alpha }'{_{n11}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{\alpha }'{_{n12}^{E}} } \right] } \big / {\left[ {-{\beta }'{_{n11}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{\beta }'{_{n12}^{E}} } \right] }\hbox { as }R_{n1}^{E}<0},\\ {-{\left[ {{\alpha }'{_{n11}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{\alpha }'{_{n12}^{E}} } \right] } \big / {\left[ {{\beta }'{_{n11}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{\beta }'{_{n12}^{E}} } \right] }\hbox { as }R_{n1}^{E}>0} \\ \end{array} }} \right. \nonumber \\ S_{2nk2}^{E}= & {} S_{4nk2}^{E} =\left\{ {{\begin{array}{l} {{-\left[ {-{\alpha }'{_{n11}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{\alpha }'{_{n12}^{E}} } \right] } \big / {\left[ {-{\beta }'{_{n11}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{\beta }'{_{n12}^{E}} } \right] }\hbox { as }R_{n2}^{E} <0,} \\ {{-\left[ {{\alpha }'{_{n11}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{\alpha }'{_{n12}^{E}} } \right] } \big / {\left[ {{\beta }'{_{n11}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{\beta }'{_{n12}^{E}} } \right] }\hbox { as }R_{n2}^{E} >0},\\ \end{array} }} \right. \end{aligned}$$

(B.37)

$$\begin{aligned} R_{2nk1}^{E}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-{a}'{_{n13}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} +{b}'{_{n14}^{E}} S_{2nk1}^{E} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\alpha }'{_{n}^{E}} } \right) }\hbox { as }R_{n1}^{E}<0},\\ {-{\left[ {{a}'{_{n13}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} +{b}'{_{n14}^{E}} S_{2nk1}^{E} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\alpha }'{_{n}^{E}} } \right) }\hbox { as }R_{n1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ R_{2nk2}^{E}= & {} \left\{ {{\begin{array}{l} {-{\left[ {-{a}'{_{n13}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} +{b}'{_{n14}^{E}} S_{2nk2}^{E} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\beta }'{_{n}^{E}} } \right) }\hbox { as }R_{n2}^{E} <0,} \\ {-{\left[ {{a}'{_{n13}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} +{b}'{_{n14}^{E}} S_{2nk2}^{E} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\beta }'{_{n}^{E}} } \right) }\hbox { as }R_{n2}^{E} >0},\\ \end{array} }} \right. \nonumber \\ R_{2nk3}^{E}= & {} -{{b}'{_{n14}^{E}} } \big / {\left( {{c}'{_{n14}^{E}} {\gamma }'{_{n}^{E}} } \right) } , \end{aligned}$$

(B.38)

$$\begin{aligned} R_{4nk1}^{E}= & {} \left\{ {{\begin{array}{l} {{\left[ {-{a}'{_{n13}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} +{b}'{_{n14}^{E}} S_{4nk1}^{E} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\alpha }'{_{n}^{E}} } \right) }\hbox { as }R_{n1}^{E}<0},\\ {{\left[ {{a}'{_{n13}^{E}} \left( {{\alpha }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} +{b}'{_{n14}^{E}} S_{4nk1}^{E} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\alpha }'{_{n}^{E}} } \right) }\hbox { as }R_{n1}^{E}>0},\\ \end{array} }} \right. \nonumber \\ R_{4nk2}^{E}= & {} \left\{ {{\begin{array}{l} {{\left[ {-{a}'{_{n13}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} +{b}'{_{n14}^{E}} S_{4nk2}^{E} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\beta }'{_{n}^{E}} } \right) }\hbox { as }R_{n2}^{E} <0,} \\ {{\left[ {{a}'{_{n13}^{E}} \left( {{\beta }'{_{n}^{E}} } \right) ^{2}+{a}'{_{n14}^{E}} +{b}'{_{n14}^{E}} S_{4nk2}^{E} } \right] } \big / {\left( {{c}'{_{n14}^{E}} {\beta }'{_{n}^{E}} } \right) }\hbox { as }R_{n2}^{E} >0},\\ \end{array} }} \right. \nonumber \\ R_{4nk3}^{E}= & {} {{b}'{_{n14}^{E}} } \big / {\left( {{c}'{_{n14}^{E}} {\gamma }'{_{n}^{E}} } \right) } . \end{aligned}$$

(B.39)

Appendix C: Constants of stress components

The constants associated with flexural type stress terms are

$$\begin{aligned} TEX_{1mkj}^{F}= & {} \left( {m\pi } \right) {\bar{r}}_{11} \theta Y_{1mkj}^{F} +\frac{{\bar{r}}_{12} }{\phi }\alpha _{mj}^{F} \theta Z_{1mkj}^{F} , \hbox { as }m=0, TEX_{10k3}^{F} =0, \end{aligned}$$

(C.1)

$$\begin{aligned} TEX_{2nkj}^{F}= & {} {\bar{r}}_{11} {\alpha }'{_{nj}^{F}} \theta Y_{2nkj}^{F} +\frac{{\bar{r}}_{12} }{\phi }\left( {n\pi } \right) \theta Z_{2nkj}^{F} \hbox {, as }n=0, TEX_{20k3}^{F} =0, \end{aligned}$$

(C.2)

$$\begin{aligned} TEX_{3mkj}^{F}= & {} \left( {m\pi } \right) {\bar{r}}_{11} \theta Y_{3mkj}^{F} -\frac{{\bar{r}}_{12} }{\phi }\alpha _{mj}^{F} \theta Z_{3mkj}^{F} ,\hbox { as }m=0, TEX_{30k3}^{F} =0, \end{aligned}$$

(C.3)

$$\begin{aligned} TEX_{4nkj}^{F}= & {} -{\bar{r}}_{11} {\alpha }'{_{nj}^{F}} \theta Y_{4nkj}^{F} +\frac{{\bar{r}}_{12} }{\phi }\left( {n\pi } \right) \theta Z_{4nkj}^{F} \hbox {, as }m=0, TEX_{40k3}^{F} =0, \end{aligned}$$

(C.4)

$$\begin{aligned} TEY_{1mkj}^{F}= & {} \left( {m\pi } \right) {\bar{r}}_{12} \theta Y_{1mkj}^{F} +\frac{{\bar{r}}_{11} }{\phi }\alpha _{mj}^{F} \theta Z_{1mkj}^{F} \hbox {, as }m=0, TEY_{10k3}^{F} =0 \end{aligned}$$

(C.5)

$$\begin{aligned} TEY_{2nkj}^{F}= & {} {\bar{r}}_{12} {\alpha }'{_{nj}^{F}} \theta Y_{2nkj}^{F} +\frac{{\bar{r}}_{11} }{\phi }\left( {n\pi } \right) \theta Z_{2nkj}^{F} \hbox {, as }n=0, TEY_{20k3}^{F} =0, \end{aligned}$$

(C.6)

$$\begin{aligned} TEY_{3mkj}^{F}= & {} \left( {m\pi } \right) {\bar{r}}_{12} \theta Y_{3mkj}^{F} -\frac{{\bar{r}}_{11} }{\phi }\alpha _{mj}^{F} \theta Z_{3mkj}^{F} , \hbox { as }m=0, TEY_{30k3}^{F} =0, \end{aligned}$$

(C.7)

$$\begin{aligned} TEY_{4nkj}^{F}= & {} -{\bar{r}}_{12} {\alpha }'{_{nj}^{F}} \theta Y_{4nkj}^{F} +\frac{{\bar{r}}_{11} }{\phi }\left( {n\pi } \right) \theta Z_{4nkj}^{F} \hbox {, as }m=0, TEY_{40k3}^{F} =0. \end{aligned}$$

(C.8)

The constants associated with extensional type stress terms are

$$\begin{aligned}&TEX_{1mkj}^{E} =\left\{ {{\begin{array}{l} {\left( {m\pi } \right) \theta Y_{1mkj}^{E} +\frac{r_{12} }{\phi }\alpha _{mj}^{E} \theta Z_{1mkj}^{E} +r_{13} \theta X_{1mkj}^{E} \hbox { as }j=1,2,}\\ {\left( {m\pi } \right) \theta Y_{1mkj}^{E} +\frac{r_{12} }{\phi }\alpha _{mj}^{E} \theta Z_{1mkj}^{E} \hbox { as }j=3,}\\ \end{array} }} \right. \hbox { as }m=0 \quad TEX_{10k3}^{F} =0,\nonumber \\ \end{aligned}$$

(C.9)

$$\begin{aligned}&TEX_{2nkj}^{E} =\left\{ {{\begin{array}{l} {{\alpha }'{_{nj}^{E}} \theta Y_{2nkj}^{E} +\frac{r_{12} }{\phi }\left( {n\pi } \right) \theta Z_{2nkj}^{E} +r_{13} \theta X_{2nkj}^{E} \hbox { as }j=1,2,}\\ {{\alpha }'{_{nj}^{E}} \theta Y_{2nkj}^{E} +\frac{r_{12} }{\phi }\left( {n\pi } \right) \theta Z_{2nkj}^{E} \hbox { as }j=3,}\\ \end{array} }} \right. \hbox { as }n=0 \quad TEX_{20k3}^{E} =0,\nonumber \\ \end{aligned}$$

(C.10)

$$\begin{aligned}&TEX_{3mkj}^{E} =\left\{ {{\begin{array}{l} {\left( {m\pi } \right) \theta Y_{3mkj}^{E} -\frac{r_{12} }{\phi }\alpha _{mj}^{E} \theta Z_{3mkj}^{E} +r_{13} \theta X_{3mkj}^{E} \hbox { as }j=1,2},\\ {\left( {m\pi } \right) \theta Y_{3mkj}^{E} -\frac{r_{12} }{\phi }\alpha _{mj}^{E} \theta Z_{3mkj}^{E} \hbox { as }j=3,}\\ \end{array} }} \right. \hbox { as }m=0 \quad TEX_{30k3}^{F} =0,\nonumber \\ \end{aligned}$$

(C.11)

$$\begin{aligned}&TEX_{4nkj}^{E} =\left\{ {{\begin{array}{l} {-{\alpha }'{_{nj}^{E}} \theta Y_{4nkj}^{E} +\frac{r_{12} }{\phi }\left( {n\pi } \right) \theta Z_{4nkj}^{E} +r_{13} \theta X_{4nkj}^{E} \hbox { as }j=1,2,}\\ {-{\alpha }'{_{nj}^{E}} \theta Y_{4nkj}^{E} +\frac{r_{12} }{\phi }\left( {n\pi } \right) \theta Z_{4nkj}^{E} \hbox { as }j=3,}\\ \end{array} }} \right. \hbox { as }m=0 \quad TEX_{40k3}^{E} =0,\nonumber \\ \end{aligned}$$

(C.12)

$$\begin{aligned}&TEY_{1mkj}^{E} =\left\{ {{\begin{array}{l} {\left( {m\pi } \right) r_{12} \theta Y_{1mkj}^{E} +\frac{\alpha _{mj}^{E} }{\phi }\theta Z_{1mkj}^{E} +r_{13} \theta X_{1mkj}^{E} \hbox { as }j=1,2,}\\ {\left( {m\pi } \right) r_{12} \theta Y_{1mkj}^{E} +\frac{\alpha _{mj}^{E} }{\phi }\theta Z_{1mkj}^{E} \hbox { as }j=3,}\\ \end{array} }} \right. \hbox { as }m=0 \quad TEY_{10k3}^{E} =0,\nonumber \\ \end{aligned}$$

(C.13)

$$\begin{aligned}&TEY_{2nkj}^{E} =\left\{ {{\begin{array}{l} {r_{12} {\alpha }'{_{nj}^{E}} \theta Y_{2nkj}^{E} +\frac{\left( {n\pi } \right) }{\phi }\theta Z_{2nkj}^{E} +r_{13} \theta X_{2nkj}^{E} \hbox { as }j=1,2,}\\ {r_{12} {\alpha }'{_{nj}^{E}} \theta Y_{2nkj}^{E} +\frac{\left( {n\pi } \right) }{\phi }\theta Z_{2nkj}^{E} \hbox { as }j=3},\\ \end{array} }} \right. \hbox { as }n=0 \quad TEY_{20k3}^{E} =0,\nonumber \\ \end{aligned}$$

(C.14)

$$\begin{aligned}&TEY_{3mkj}^{E} =\left\{ {{\begin{array}{l} {\left( {m\pi } \right) r_{12} \theta Y_{3mkj}^{E} -\frac{\alpha _{mj}^{E} }{\phi }\theta Z_{3mkj}^{E} +r_{13} \theta X_{3mkj}^{E} \hbox { as }j=1,2,}\\ {\left( {m\pi } \right) r_{12} \theta Y_{3mkj}^{E} -\frac{\alpha _{mj}^{E} }{\phi }\theta Z_{3mkj}^{E} \hbox { as }j=3,}\\ \end{array} }} \right. \hbox { as }m=0 \quad TEY_{30k3}^{E} =0,\nonumber \\ \end{aligned}$$

(C.15)

$$\begin{aligned}&TEY_{4nkj}^{E} =\left\{ {{\begin{array}{l} {-r_{12} {\alpha }'{_{nj}^{E}} \theta Y_{4nkj}^{E} +\frac{\left( {n\pi } \right) }{\phi }\theta Z_{4nkj}^{E} +r_{13} \theta X_{4nkj}^{E} \hbox { as }j=1,2}\\ {-r_{12} {\alpha }'{_{nj}^{E}} \theta Y_{4nkj}^{E} +\frac{\left( {n\pi } \right) }{\phi }\theta Z_{4nkj}^{E} \hbox { as }j=3},\\ \end{array} }} \right. \hbox {, as }m=0, TEY_{40k3}^{E} =0.\nonumber \\ \end{aligned}$$

(C.16)