Dynamic damage analysis of a ten-layer circular composite plate subjected to low-velocity impact

Abstract

A new theoretical solution is presented to determine the stress distribution in a ten-layer simply-supported circular composite plate subjected to the low-velocity impact. The aim of the current study is the investigation of the dynamic analysis of the composite plate when a cylindrical impactor hits the top layer of the plate with an initial velocity of 1 m/s. The plate is made of two adhesive layers adhere two aluminum layers to a six-layer carbon-epoxy laminated plate. The classical non-adhesive elastic contact theory and Hunter's relationship are used to simulate the contact behavior in terms of time and contact radius. By using Hamilton's principle and Layerwise theory, thirty-two equations of motion are derived. Moreover, Johnson–Cook’s criteria, the plastic simulation model, the normal stress–strain failure criterion theory were used for failure analysis of the aluminum, adhesive, and carbon-epoxy layers, respectively. The numerical method was used to solve the thirty-two differential equations of motion based on the finite difference method. Moreover, the relationship between stress and strain is re-written in the numerical code so that the failure criterion theories are satisfied. Moreover, according to the defined failure criterion for each layer, the damage is checked at the end of every time step. In addition, the damping behavior of the composite plate after applying the contact pressure caused by the impact was also investigated. The results showed that the impact resulted in residual stress in the plate.

Appendices

Appendix A

Expressions for the displacement components in the ten-layer circular composite plate based on LT along with HSDT.

$$u_{1}^{\left( 1 \right)} = \left\{ \begin{gathered} u_{0} + \left[ {\xi^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}\xi^{{\left( 1 \right)^{3} }} } \right]\psi_{1}^{\left( 1 \right)} - \frac{1}{2}\xi^{{\left( 1 \right)^{2} }} \alpha_{1}^{\left( 1 \right)} - \frac{4}{3}\left[ \begin{gathered} \frac{1}{{h_{1}^{2} }}\beta_{1}^{\left( 1 \right)} + \frac{1}{{h_{2}^{2} }}\beta_{1}^{\left( 2 \right)} + \frac{1}{{h_{3}^{2} }}\beta_{1}^{\left( 3 \right)} \hfill \\ + \frac{1}{{h_{4}^{2} }}\beta_{1}^{\left( 4 \right)} + \frac{1}{{h_{5}^{2} }}\beta_{1}^{\left( 5 \right)} \hfill \\ \end{gathered} \right] \hfill \\ - \frac{1}{3}\xi^{{\left( 1 \right)^{3} }} \lambda_{1}^{\left( 1 \right)} - \frac{2}{3}\left[ {h_{2} \psi_{1}^{\left( 2 \right)} + h_{3} \psi_{1}^{\left( 3 \right)} + h_{4} \psi_{1}^{\left( 4 \right)} + h_{5} \psi_{1}^{\left( 5 \right)} } \right] \hfill \\ + \frac{1}{12}\left[ {h_{2}^{3} \lambda_{1}^{\left( 2 \right)} + h_{3}^{3} \lambda_{1}^{\left( 3 \right)} + h_{4}^{3} \lambda_{1}^{\left( 4 \right)} + h_{5}^{3} \lambda_{1}^{\left( 5 \right)} } \right] - \frac{{h_{1} }}{3}\psi_{1}^{\left( 1 \right)} + \frac{{h_{1}^{2} }}{8}\alpha_{1}^{\left( 1 \right)} + \frac{{h_{1}^{3} }}{24}\lambda_{1}^{\left( 1 \right)} \hfill \\ \end{gathered} \right\}$$

(A1)

$$u_{3}^{\left( 1 \right)} = w_{0} + \xi^{\left( 1 \right)} \psi_{3}^{\left( 1 \right)} + \xi^{{\left( 1 \right)^{2} }} \alpha_{3}^{\left( 1 \right)} - h_{2} \psi_{3}^{\left( 2 \right)} - h_{3} \psi_{3}^{\left( 3 \right)} - h_{4} \psi_{3}^{\left( 4 \right)} - h_{5} \psi_{3}^{\left( 5 \right)} - \frac{{h_{1} }}{2}\psi_{3}^{\left( 1 \right)} - \frac{{h_{1}^{2} }}{4}\alpha_{3}^{\left( 1 \right)}$$

(A2)

$$u_{1}^{\left( 2 \right)} = \left\{ \begin{gathered} u_{0} + \left[ {\xi^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}\xi^{{\left( 2 \right)^{3} }} } \right]\psi_{1}^{\left( 2 \right)} - \frac{1}{2}\xi^{{\left( 2 \right)^{2} }} \alpha_{1}^{\left( 2 \right)} - \frac{4}{3}\left[ \begin{gathered} \frac{1}{{h_{2}^{2} }}\beta_{1}^{\left( 2 \right)} + \frac{1}{{h_{3}^{2} }}\beta_{1}^{\left( 3 \right)} \hfill \\ + \frac{1}{{h_{4}^{2} }}\beta_{1}^{\left( 4 \right)} + \frac{1}{{h_{5}^{2} }}\beta_{1}^{\left( 5 \right)} \hfill \\ \end{gathered} \right] \hfill \\ - \frac{1}{3}\xi^{{\left( 2 \right)^{3} }} \lambda_{1}^{\left( 2 \right)} - \frac{2}{3}\left[ {h_{3} \psi_{1}^{\left( 3 \right)} + h_{4} \psi_{1}^{\left( 4 \right)} + h_{5} \psi_{1}^{\left( 5 \right)} } \right] + \frac{1}{12}\left[ {h_{3}^{3} \lambda_{1}^{\left( 3 \right)} + h_{4}^{3} \lambda_{1}^{\left( 4 \right)} + h_{5}^{3} \lambda_{1}^{\left( 5 \right)} } \right] \hfill \\ - \frac{{h_{2} }}{3}\psi_{1}^{\left( 2 \right)} + \frac{{h_{2}^{2} }}{8}\alpha_{1}^{\left( 2 \right)} + \frac{{h_{2}^{3} }}{24}\lambda_{1}^{\left( 2 \right)} \hfill \\ \end{gathered} \right\}$$

(A3)

$$u_{3}^{\left( 2 \right)} = w_{0} + \xi^{\left( 2 \right)} \psi_{3}^{\left( 2 \right)} + \xi^{{\left( 2 \right)^{2} }} \alpha_{3}^{\left( 2 \right)} - h_{3} \psi_{3}^{\left( 3 \right)} - h_{4} \psi_{3}^{\left( 4 \right)} - h_{5} \psi_{3}^{\left( 5 \right)} - \frac{{h_{2} }}{2}\psi_{3}^{\left( 2 \right)} - \frac{{h_{2}^{2} }}{4}\alpha_{3}^{\left( 2 \right)}$$

(A4)

$$u_{1}^{\left( 3 \right)} = \left\{ \begin{gathered} u_{0} + \left[ {\xi^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}\xi^{{\left( 3 \right)^{3} }} } \right]\psi_{1}^{\left( 3 \right)} - \frac{1}{2}\xi^{{\left( 3 \right)^{2} }} \alpha_{1}^{\left( 3 \right)} - \frac{4}{3}\left[ {\frac{1}{{h_{3}^{2} }}\beta_{1}^{\left( 3 \right)} + \frac{1}{{h_{4}^{2} }}\beta_{1}^{\left( 4 \right)} + \frac{1}{{h_{5}^{2} }}\beta_{1}^{\left( 5 \right)} } \right] \hfill \\ - \frac{1}{3}\xi^{{\left( 3 \right)^{3} }} \lambda_{1}^{\left( 3 \right)} - \frac{2}{3}\left[ {h_{4} \psi_{1}^{\left( 4 \right)} + h_{5} \psi_{1}^{\left( 5 \right)} } \right] + \frac{1}{12}\left[ {h_{4}^{3} \lambda_{1}^{\left( 4 \right)} + h_{5}^{3} \lambda_{1}^{\left( 5 \right)} } \right] \hfill \\ - \frac{{h_{3} }}{3}\psi_{1}^{\left( 3 \right)} + \frac{{h_{3}^{2} }}{8}\alpha_{1}^{\left( 3 \right)} + \frac{{h_{3}^{3} }}{24}\lambda_{1}^{\left( 3 \right)} \hfill \\ \end{gathered} \right\}$$

(A5)

$$u_{3}^{\left( 3 \right)} = w_{0} + \xi^{\left( 3 \right)} \psi_{3}^{\left( 3 \right)} + \xi^{{\left( 3 \right)^{2} }} \alpha_{3}^{\left( 3 \right)} - h_{4} \psi_{3}^{\left( 4 \right)} - h_{5} \psi_{3}^{\left( 5 \right)} - \frac{{h_{3} }}{2}\psi_{3}^{\left( 3 \right)} - \frac{{h_{3}^{2} }}{4}\alpha_{3}^{\left( 3 \right)}$$

(A6)

$$u_{1}^{\left( 4 \right)} = \left\{ \begin{gathered} u_{0} + \left[ {\xi^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}\xi^{{\left( 4 \right)^{3} }} } \right]\psi_{1}^{\left( 4 \right)} - \frac{1}{2}\xi^{{\left( 4 \right)^{2} }} \alpha_{1}^{\left( 4 \right)} - \frac{4}{3}\left[ {\frac{1}{{h_{4}^{2} }}\beta_{1}^{\left( 4 \right)} + \frac{1}{{h_{5}^{2} }}\beta_{1}^{\left( 5 \right)} } \right] \hfill \\ - \frac{1}{3}\xi^{{\left( 4 \right)^{3} }} \lambda_{1}^{\left( 4 \right)} - \frac{{2h_{5} }}{3}\psi_{1}^{\left( 5 \right)} + \frac{{h_{5}^{3} }}{12}\lambda_{1}^{\left( 5 \right)} - \frac{{h_{4} }}{3}\psi_{1}^{\left( 4 \right)} + \frac{{h_{4}^{2} }}{8}\alpha_{1}^{\left( 4 \right)} + \frac{{h_{7}^{3} }}{24}\lambda_{1}^{\left( 4 \right)} \hfill \\ \end{gathered} \right\}$$

(A7)

$$u_{3}^{\left( 4 \right)} = w_{0} + \xi^{\left( 4 \right)} \psi_{3}^{\left( 4 \right)} + \xi^{{\left( 4 \right)^{2} }} \alpha_{3}^{\left( 4 \right)} - h_{5} \psi_{3}^{\left( 5 \right)} - \frac{{h_{4} }}{2}\psi_{3}^{\left( 4 \right)} - \frac{{h_{4}^{2} }}{4}\alpha_{3}^{\left( 4 \right)}$$

(A8)

$$u_{1}^{\left( 5 \right)} = \left\{ \begin{gathered} u_{0} + \left[ {\xi^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}\xi^{{\left( 5 \right)^{3} }} } \right]\psi_{1}^{\left( 5 \right)} - \frac{1}{2}\xi^{{\left( 5 \right)^{2} }} \alpha_{1}^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}\beta_{1}^{\left( 5 \right)} \hfill \\ - \frac{1}{3}\xi^{{\left( 5 \right)^{3} }} \lambda_{1}^{\left( 5 \right)} - \frac{{h_{5} }}{3}\psi_{1}^{\left( 5 \right)} + \frac{{h_{5}^{2} }}{8}\alpha_{1}^{\left( 5 \right)} + \frac{{h_{5}^{3} }}{24}\lambda_{1}^{\left( 5 \right)} \hfill \\ \end{gathered} \right\}$$

(A9)

$$u_{3}^{\left( 5 \right)} = w_{0} + \xi^{\left( 5 \right)} \psi_{3}^{\left( 5 \right)} + \xi^{{\left( 5 \right)^{2} }} \alpha_{3}^{\left( 5 \right)} - \frac{{h_{5} }}{2}\psi_{3}^{\left( 5 \right)} - \frac{{h_{5}^{2} }}{4}\alpha_{3}^{\left( 5 \right)}$$

(A10)

$$u_{1}^{\left( 6 \right)} = \left\{ \begin{gathered} u_{0} + \left[ {\xi^{\left( 6 \right)} - \frac{4}{{3h_{6}^{2} }}\xi^{{\left( 6 \right)^{3} }} } \right]\psi_{1}^{\left( 6 \right)} - \frac{1}{2}\xi^{{\left( 6 \right)^{2} }} \alpha_{1}^{\left( 6 \right)} - \frac{4}{{3h_{6}^{2} }}\beta_{1}^{\left( 6 \right)} \hfill \\ - \frac{1}{3}\xi^{{\left( 6 \right)^{3} }} \lambda_{1}^{\left( 6 \right)} + \frac{{h_{6} }}{3}\psi_{1}^{\left( 6 \right)} + \frac{{h_{6}^{2} }}{8}\alpha_{1}^{\left( 6 \right)} - \frac{{h_{6}^{3} }}{24}\lambda_{1}^{\left( 6 \right)} \hfill \\ \end{gathered} \right\}$$

(A11)

$$u_{3}^{\left( 6 \right)} = w_{0} + \xi^{\left( 6 \right)} \psi_{3}^{\left( 6 \right)} + \xi^{{\left( 6 \right)^{2} }} \alpha_{3}^{\left( 6 \right)} + \frac{{h_{6} }}{2}\psi_{3}^{\left( 6 \right)} - \frac{{h_{6}^{2} }}{4}\alpha_{3}^{\left( 6 \right)}$$

(A12)

$$u_{1}^{\left( 7 \right)} = \left\{ \begin{gathered} u_{0} + \left[ {\xi^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}\xi^{{\left( 7 \right)^{3} }} } \right]\psi_{1}^{\left( 7 \right)} - \frac{1}{2}\xi^{{\left( 7 \right)^{2} }} \alpha_{1}^{\left( 7 \right)} - \frac{4}{3}\left[ {\frac{1}{{h_{6}^{2} }}\beta_{1}^{\left( 6 \right)} + \frac{1}{{h_{7}^{2} }}\beta_{1}^{\left( 7 \right)} } \right] \hfill \\ - \frac{1}{3}\xi^{{\left( 7 \right)^{3} }} \lambda_{1}^{\left( 7 \right)} + \frac{{2h_{6} }}{3}\psi_{1}^{\left( 6 \right)} - \frac{{h_{6}^{3} }}{12}\lambda_{1}^{\left( 6 \right)} + \frac{{h_{7} }}{3}\psi_{1}^{\left( 7 \right)} + \frac{{h_{7}^{2} }}{8}\alpha_{1}^{\left( 7 \right)} - \frac{{h_{7}^{3} }}{24}\lambda_{1}^{\left( 7 \right)} \hfill \\ \end{gathered} \right\}$$

(A13)

$$u_{3}^{\left( 7 \right)} = w_{0} + \xi^{\left( 7 \right)} \psi_{3}^{\left( 7 \right)} + \xi^{{\left( 7 \right)^{2} }} \alpha_{3}^{\left( 7 \right)} + h_{6} \psi_{3}^{\left( 6 \right)} + \frac{{h_{7} }}{2}\psi_{3}^{\left( 7 \right)} - \frac{{h_{7}^{2} }}{4}\alpha_{3}^{\left( 7 \right)}$$

(A14)

$$u_{1}^{\left( 8 \right)} = \left\{ \begin{gathered} u_{0} + \left[ {\xi^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}\xi^{{\left( 8 \right)^{3} }} } \right]\psi_{1}^{\left( 8 \right)} - \frac{1}{2}\xi^{{\left( 8 \right)^{2} }} \alpha_{1}^{\left( 8 \right)} - \frac{4}{3}\left[ {\frac{1}{{h_{6}^{2} }}\beta_{1}^{\left( 6 \right)} + \frac{1}{{h_{7}^{2} }}\beta_{1}^{\left( 7 \right)} + \frac{1}{{h_{8}^{2} }}\beta_{1}^{\left( 8 \right)} } \right] \hfill \\ - \frac{1}{3}\xi^{{\left( 8 \right)^{3} }} \lambda_{1}^{\left( 8 \right)} + \frac{2}{3}\left[ {h_{6} \psi_{1}^{\left( 6 \right)} + h_{7} \psi_{1}^{\left( 7 \right)} } \right] - \frac{1}{12}\left[ {h_{6}^{3} \lambda_{1}^{\left( 6 \right)} + h_{7}^{3} \lambda_{1}^{\left( 7 \right)} } \right] \hfill \\ + \frac{{h_{8} }}{3}\psi_{1}^{\left( 8 \right)} + \frac{{h_{8}^{2} }}{8}\alpha_{1}^{\left( 8 \right)} - \frac{{h_{8}^{3} }}{24}\lambda_{1}^{\left( 8 \right)} \hfill \\ \end{gathered} \right\}$$

(A15)

$$u_{3}^{\left( 8 \right)} = w_{0} + \xi^{\left( 8 \right)} \psi_{3}^{\left( 8 \right)} + \xi^{{\left( 8 \right)^{2} }} \alpha_{3}^{\left( 8 \right)} + h_{6} \psi_{3}^{\left( 6 \right)} + h_{7} \psi_{3}^{\left( 7 \right)} + \frac{{h_{8} }}{2}\psi_{3}^{\left( 8 \right)} - \frac{{h_{8}^{2} }}{4}\alpha_{3}^{\left( 8 \right)}$$

(A16)

$$u_{1}^{\left( 9 \right)} = \left\{ \begin{gathered} u_{0} + \left[ {\xi^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}\xi^{{\left( 9 \right)^{3} }} } \right]\psi_{1}^{\left( 9 \right)} - \frac{1}{2}\xi^{{\left( 9 \right)^{2} }} \alpha_{1}^{\left( 9 \right)} - \frac{4}{3}\left[ \begin{gathered} \frac{1}{{h_{6}^{2} }}\beta_{1}^{\left( 6 \right)} + \frac{1}{{h_{7}^{2} }}\beta_{1}^{\left( 7 \right)} \hfill \\ + \frac{1}{{h_{8}^{2} }}\beta_{1}^{\left( 8 \right)} + \frac{1}{{h_{9}^{2} }}\beta_{1}^{\left( 9 \right)} \hfill \\ \end{gathered} \right] \hfill \\ - \frac{1}{3}\xi^{{\left( 9 \right)^{3} }} \lambda_{1}^{\left( 9 \right)} + \frac{2}{3}\left[ {h_{6} \psi_{1}^{\left( 6 \right)} + h_{7} \psi_{1}^{\left( 7 \right)} + h_{8} \psi_{1}^{\left( 8 \right)} } \right] - \frac{1}{12}\left[ {h_{6}^{3} \lambda_{1}^{\left( 6 \right)} + h_{7}^{3} \lambda_{1}^{\left( 7 \right)} + h_{8}^{3} \lambda_{1}^{\left( 8 \right)} } \right] \hfill \\ + \frac{{h_{9} }}{3}\psi_{1}^{\left( 9 \right)} + \frac{{h_{9}^{2} }}{8}\alpha_{1}^{\left( 9 \right)} - \frac{{h_{9}^{3} }}{24}\lambda_{1}^{\left( 9 \right)} \hfill \\ \end{gathered} \right\}$$

(A17)

$$u_{3}^{\left( 9 \right)} = w_{0} + \xi^{\left( 9 \right)} \psi_{3}^{\left( 9 \right)} + \xi^{{\left( 9 \right)^{2} }} \alpha_{3}^{\left( 9 \right)} + h_{6} \psi_{3}^{\left( 6 \right)} + h_{7} \psi_{3}^{\left( 7 \right)} + h_{8} \psi_{3}^{\left( 8 \right)} + \frac{{h_{9} }}{2}\psi_{3}^{\left( 9 \right)} - \frac{{h_{9}^{2} }}{4}\alpha_{3}^{\left( 9 \right)}$$

(A18)

$$u_{1}^{{\left( {10} \right)}} = \left\{ \begin{gathered} u_{0} + \left[ {\xi^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}\xi^{{\left( {10} \right)^{3} }} } \right]\psi_{1}^{{\left( {10} \right)}} - \frac{1}{2}\xi^{{\left( {10} \right)^{2} }} \alpha_{1}^{{\left( {10} \right)}} - \frac{4}{3}\left[ \begin{gathered} \frac{1}{{h_{6}^{2} }}\beta_{1}^{\left( 6 \right)} + \frac{1}{{h_{7}^{2} }}\beta_{1}^{\left( 7 \right)} + \frac{1}{{h_{8}^{2} }}\beta_{1}^{\left( 8 \right)} \hfill \\ + \frac{1}{{h_{9}^{2} }}\beta_{1}^{\left( 9 \right)} + \frac{1}{{h_{10}^{2} }}\beta_{1}^{{\left( {10} \right)}} \hfill \\ \end{gathered} \right] \hfill \\ - \frac{1}{3}\xi^{{\left( {10} \right)^{3} }} \lambda_{1}^{{\left( {10} \right)}} + \frac{2}{3}\left[ {h_{6} \psi_{1}^{\left( 6 \right)} + h_{7} \psi_{1}^{\left( 7 \right)} + h_{8} \psi_{1}^{\left( 8 \right)} + h_{9} \psi_{1}^{\left( 9 \right)} } \right] \hfill \\ - \frac{1}{12}\left[ {h_{6}^{3} \lambda_{1}^{\left( 6 \right)} + h_{7}^{3} \lambda_{1}^{\left( 7 \right)} + h_{8}^{3} \lambda_{1}^{\left( 8 \right)} + h_{9}^{3} \lambda_{1}^{\left( 9 \right)} } \right] + \frac{{h_{10} }}{3}\psi_{1}^{{\left( {10} \right)}} + \frac{{h_{10}^{2} }}{8}\alpha_{1}^{{\left( {10} \right)}} - \frac{{h_{10}^{3} }}{24}\lambda_{1}^{{\left( {10} \right)}} \hfill \\ \end{gathered} \right\}$$

(A19)

$$u_{3}^{{\left( {10} \right)}} = w_{0} + \xi^{{\left( {10} \right)}} \psi_{3}^{{\left( {10} \right)}} + \xi^{{\left( {10} \right)^{2} }} \alpha_{3}^{{\left( {10} \right)}} + h_{6} \psi_{3}^{\left( 6 \right)} + h_{7} \psi_{3}^{\left( 7 \right)} + h_{8} \psi_{3}^{\left( 8 \right)} + h_{9} \psi_{3}^{\left( 9 \right)} + \frac{{h_{10} }}{2}\psi_{3}^{{\left( {10} \right)}} - \frac{{h_{10}^{2} }}{4}\alpha_{3}^{{\left( {10} \right)}}$$

(A20)

Where;

$$\alpha_{1}^{\left( k \right)} = \frac{{\partial \psi_{3}^{\left( k \right)} }}{{\partial \xi_{1} }};\,\beta_{1}^{\left( k \right)} = \frac{{\partial w_{0} }}{{\partial \xi_{1} }};\,\lambda_{1}^{\left( k \right)} = \frac{{\partial \alpha_{3}^{\left( k \right)} }}{{\partial \xi_{1} }};\,k = 1,...,10.$$

(A21)

In Eqs. (A1)–(A20), u₀ and w₀ stand for the displacement of the mid-surface along ξ₁ and ξ, respectively. In addition, ξ^(k) (k = 1, 2, …, 10) is measured from the k^th layer mid-surface (see Fig. 1). Furthermore, it is assumed that $u_{3}^{\left( k \right)}$ (k = 1, 2, …, 10) is considered a function in terms of ξ₁ and ξ^(k).

Appendix B

Expressions for the coefficient of A_i (i = 1, …, 32) defined in Eq. (4b) for the virtual kinetic energy of the ten-layer circular composite plate.

$$A_{1} = \left[ \begin{gathered} \left( {I_{0}^{{\left( 1 \right)}} + I_{0}^{{\left( 2 \right)}} + I_{0}^{{\left( 3 \right)}} + I_{0}^{{\left( 4 \right)}} + I_{0}^{{\left( 5 \right)}} + I_{0}^{{\left( 6 \right)}} + I_{0}^{{\left( 7 \right)}} + I_{0}^{{\left( 8 \right)}} + I_{0}^{{\left( 9 \right)}} + I_{0}^{{\left( {10} \right)}} } \right)\dot{u}_{0} + \left( { - \frac{{h_{1} }}{3}I_{0}^{{\left( 1 \right)}} + I_{1}^{{\left( 1 \right)}} - \frac{4}{{3h_{1}^{2} }}I_{3}^{{\left( 1 \right)}} } \right)\dot{\psi }_{1}^{{\left( 1 \right)}} \hfill \\ + \left( { - \frac{{2h_{2} }}{3}I_{0}^{{\left( 1 \right)}} - \frac{{h_{2} }}{3}I_{0}^{{\left( 2 \right)}} + I_{1}^{{\left( 2 \right)}} - \frac{4}{{3h_{2}^{2} }}I_{3}^{{\left( 2 \right)}} } \right)\dot{\psi }_{1}^{{\left( 2 \right)}} + \left( { - \frac{{2h_{3} }}{3}I_{0}^{{\left( 1 \right)}} - \frac{{2h_{3} }}{3}I_{0}^{{\left( 2 \right)}} - \frac{{h_{3} }}{3}I_{0}^{{\left( 3 \right)}} + I_{1}^{{\left( 3 \right)}} - \frac{4}{{3h_{3}^{2} }}I_{3}^{{\left( 3 \right)}} } \right)\dot{\psi }_{1}^{{\left( 3 \right)}} \hfill \\ + \left( { - \frac{{2h_{4} }}{3}I_{0}^{{\left( 1 \right)}} - \frac{{2h_{4} }}{3}I_{0}^{{\left( 2 \right)}} - \frac{2}{3}I_{0}^{{\left( 3 \right)}} - \frac{{h_{4} }}{3}I_{0}^{{\left( 4 \right)}} + I_{1}^{{\left( 4 \right)}} - \frac{4}{{3h_{4}^{2} }}I_{3}^{{\left( 4 \right)}} } \right)\dot{\psi }_{1}^{{\left( 4 \right)}} \hfill \\ + \left( { - \frac{{2h_{5} }}{3}I_{0}^{{\left( 1 \right)}} - \frac{{2h_{5} }}{3}I_{0}^{{\left( 2 \right)}} - \frac{{2h_{5} }}{3}I_{0}^{{\left( 3 \right)}} - \frac{{2h_{5} }}{3}I_{0}^{{\left( 4 \right)}} - \frac{{h_{5} }}{3}I_{0}^{{\left( 5 \right)}} + I_{1}^{{\left( 5 \right)}} - \frac{4}{{3h_{5}^{2} }}I_{3}^{{\left( 5 \right)}} } \right)\dot{\psi }_{1}^{{\left( 5 \right)}} \hfill \\ + \left( {\frac{{h_{6} }}{3}I_{0}^{{\left( 6 \right)}} + I_{1}^{{\left( 6 \right)}} - \frac{4}{{3h_{6}^{2} }}I_{3}^{{\left( 6 \right)}} + \frac{{2h_{6} }}{3}I_{0}^{{\left( 7 \right)}} + \frac{{2h_{6} }}{3}I_{0}^{{\left( 8 \right)}} + \frac{{2h_{6} }}{3}I_{0}^{{\left( 9 \right)}} + \frac{{2h_{6} }}{3}I_{0}^{{\left( {10} \right)}} } \right)\dot{\psi }_{1}^{{\left( 6 \right)}} \hfill \\ + \left( {\frac{{h_{7} }}{3}I_{0}^{{\left( 7 \right)}} + I_{1}^{{\left( 7 \right)}} - \frac{4}{{3h_{7}^{2} }}I_{3}^{{\left( 7 \right)}} + \frac{{2h_{7} }}{3}I_{0}^{{\left( 8 \right)}} + \frac{{2h_{7} }}{3}I_{0}^{{\left( 9 \right)}} + \frac{{2h_{7} }}{3}I_{0}^{{\left( {10} \right)}} } \right)\dot{\psi }_{1}^{{\left( 7 \right)}} \hfill \\ + \left( {\frac{{h_{8} }}{3}I_{0}^{{\left( 8 \right)}} + I_{1}^{{\left( 8 \right)}} - \frac{4}{{3h_{8}^{2} }}I_{3}^{{\left( 8 \right)}} + \frac{{2h_{8} }}{3}I_{0}^{{\left( 9 \right)}} + \frac{{2h_{8} }}{3}I_{0}^{{\left( {10} \right)}} } \right)\dot{\psi }_{1}^{{\left( 8 \right)}} + \left( {\frac{{h_{9} }}{3}I_{0}^{{\left( 9 \right)}} + I_{1}^{{\left( 9 \right)}} - \frac{4}{{3h_{9}^{2} }}I_{3}^{{\left( 9 \right)}} + \frac{{2h_{9} }}{3}I_{0}^{{\left( {10} \right)}} } \right)\dot{\psi }_{1}^{{\left( 9 \right)}} \hfill \\ + \left( {\frac{{h_{{10}} }}{3} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{{10}}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\psi }_{1}^{{\left( {10} \right)}} + \left( {\frac{{h_{1}^{3} }}{{24}}I_{0}^{{\left( 1 \right)}} - \frac{1}{3}I_{3}^{{\left( 1 \right)}} } \right)\dot{\lambda }_{1}^{{\left( 1 \right)}} + \left( {\frac{{h_{2}^{3} }}{{12}}I_{0}^{{\left( 1 \right)}} + \frac{{h_{2}^{3} }}{{24}}I_{0}^{{\left( 2 \right)}} - \frac{1}{3}I_{3}^{{\left( 2 \right)}} } \right)\dot{\lambda }_{1}^{{\left( 2 \right)}} \hfill \\ + \left( {\frac{{h_{3}^{3} }}{{12}}I_{0}^{{\left( 1 \right)}} + \frac{{h_{3}^{3} }}{{12}}I_{0}^{{\left( 2 \right)}} + \frac{{h_{3}^{3} }}{{24}}I_{0}^{{\left( 3 \right)}} - \frac{1}{3}I_{3}^{{\left( 3 \right)}} } \right)\dot{\lambda }_{1}^{{\left( 3 \right)}} + \left( {\frac{{h_{4}^{3} }}{{12}}I_{0}^{{\left( 1 \right)}} + \frac{{h_{4}^{3} }}{{12}}I_{0}^{{\left( 2 \right)}} + \frac{{h_{4}^{3} }}{{12}}I_{0}^{{\left( 3 \right)}} + \frac{{h_{4}^{3} }}{{24}}I_{0}^{{\left( 4 \right)}} - \frac{1}{3}I_{3}^{{\left( 4 \right)}} } \right)\dot{\lambda }_{1}^{{\left( 4 \right)}} \hfill \\ + \left( {\frac{{h_{5}^{3} }}{{12}}I_{0}^{{\left( 1 \right)}} + \frac{{h_{5}^{3} }}{{12}}I_{0}^{{\left( 2 \right)}} + \frac{{h_{5}^{3} }}{{12}}I_{0}^{{\left( 3 \right)}} + \frac{{h_{5}^{3} }}{{12}}I_{0}^{{\left( 4 \right)}} + \frac{{h_{5}^{3} }}{{24}}I_{0}^{{\left( 5 \right)}} - \frac{1}{3}I_{3}^{{\left( 5 \right)}} } \right)\dot{\lambda }_{1}^{{\left( 5 \right)}} \hfill \\ - \left( {\frac{{h_{6}^{3} }}{{24}}I_{0}^{{\left( 6 \right)}} + \frac{1}{3}I_{3}^{{\left( 6 \right)}} + \frac{{h_{6}^{3} }}{{12}}I_{0}^{{\left( 7 \right)}} + \frac{{h_{6}^{3} }}{{12}}I_{0}^{{\left( 8 \right)}} + \frac{{h_{6}^{3} }}{{12}}I_{0}^{{\left( 9 \right)}} + \frac{{h_{6}^{3} }}{{12}}I_{0}^{{\left( {10} \right)}} } \right)\dot{\lambda }_{1}^{{\left( 6 \right)}} - \left( {\frac{{h_{7}^{3} }}{{24}}I_{0}^{{\left( 7 \right)}} + \frac{1}{3}I_{3}^{{\left( 7 \right)}} + \frac{{h_{7}^{3} }}{{12}}I_{0}^{{\left( 8 \right)}} + \frac{{h_{7}^{3} }}{{12}}I_{0}^{{\left( 9 \right)}} + \frac{{h_{7}^{3} }}{{12}}I_{0}^{{\left( {10} \right)}} } \right)\dot{\lambda }_{1}^{{\left( 7 \right)}} \hfill \\ - \left( {\frac{{h_{8}^{3} }}{{24}}I_{0}^{{\left( 8 \right)}} + \frac{1}{3}I_{3}^{{\left( 8 \right)}} + \frac{{h_{8}^{3} }}{{12}}I_{0}^{{\left( 9 \right)}} + \frac{{h_{8}^{3} }}{{12}}I_{0}^{{\left( {10} \right)}} } \right)\dot{\lambda }_{1}^{{\left( 8 \right)}} - \left( {\frac{{h_{9}^{3} }}{{24}}I_{0}^{{\left( 9 \right)}} + \frac{1}{3}I_{3}^{{\left( 9 \right)}} + \frac{{h_{9}^{3} }}{{12}}I_{0}^{{\left( {10} \right)}} } \right)\dot{\lambda }_{1}^{{\left( 9 \right)}} \hfill \\ - \left( {\frac{{h_{{10}}^{3} }}{{24}}I_{0}^{{\left( {10} \right)}} + \frac{1}{3}I_{3}^{{\left( {10} \right)}} } \right)\dot{\lambda }_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{1}^{2} }}I_{0}^{{\left( 1 \right)}} \dot{\beta }_{1}^{{\left( 1 \right)}} - \frac{4}{{3h_{2}^{2} }}\left( {I_{0}^{{\left( 1 \right)}} + I_{0}^{{\left( 2 \right)}} } \right)\dot{\beta }_{1}^{{\left( 2 \right)}} - \frac{4}{{3h_{3}^{2} }}\left( {I_{0}^{{\left( 1 \right)}} + I_{0}^{{\left( 2 \right)}} + I_{0}^{{\left( 3 \right)}} } \right)\dot{\beta }_{1}^{{\left( 3 \right)}} \hfill \\ - \frac{4}{{3h_{4}^{2} }}\left( {I_{0}^{{\left( 1 \right)}} + I_{0}^{{\left( 2 \right)}} + I_{0}^{{\left( 3 \right)}} + I_{0}^{{\left( 4 \right)}} } \right)\dot{\beta }_{1}^{{\left( 4 \right)}} - \frac{4}{{3h_{5}^{2} }}\left( {I_{0}^{{\left( 1 \right)}} + I_{0}^{{\left( 2 \right)}} + I_{0}^{{\left( 3 \right)}} + I_{0}^{{\left( 4 \right)}} + I_{0}^{{\left( 5 \right)}} } \right)\dot{\beta }_{1}^{{\left( 5 \right)}} \hfill \\ - \frac{4}{{3h_{6}^{2} }}\left( {I_{0}^{{\left( 6 \right)}} + I_{0}^{{\left( 7 \right)}} + I_{0}^{{\left( 8 \right)}} + I_{0}^{{\left( 9 \right)}} + I_{0}^{{\left( {10} \right)}} } \right)\dot{\beta }_{1}^{{\left( 6 \right)}} - \frac{4}{{3h_{7}^{2} }}\left( {I_{0}^{{\left( 7 \right)}} + I_{0}^{{\left( 8 \right)}} + I_{0}^{{\left( 9 \right)}} + I_{0}^{{\left( {10} \right)}} } \right)\dot{\beta }_{1}^{{\left( 7 \right)}} \hfill \\ - \frac{4}{{3h_{8}^{2} }}\left( {I_{0}^{{\left( 8 \right)}} + I_{0}^{{\left( 9 \right)}} + I_{0}^{{\left( {10} \right)}} } \right)\dot{\beta }_{1}^{{\left( 8 \right)}} - \frac{4}{{3h_{9}^{2} }}\left( {I_{0}^{{\left( 9 \right)}} + I_{0}^{{\left( {10} \right)}} } \right)\dot{\beta }_{1}^{{\left( 9 \right)}} - \frac{4}{{3h_{{10}}^{2} }}I_{0}^{{\left( {10} \right)}} \dot{\beta }_{1}^{{\left( {10} \right)}} \hfill \\ + \left( {\frac{{h_{1}^{2} }}{8}I_{0}^{{\left( 1 \right)}} - \frac{1}{2}I_{2}^{{\left( 1 \right)}} } \right)\dot{\alpha }_{1}^{{\left( 1 \right)}} + \left( {\frac{{h_{2}^{2} }}{8}I_{0}^{{\left( 2 \right)}} - \frac{1}{2}I_{2}^{{\left( 2 \right)}} } \right)\dot{\alpha }_{1}^{{\left( 2 \right)}} + \left( {\frac{{h_{3}^{2} }}{8}I_{0}^{{\left( 3 \right)}} - \frac{1}{2}I_{2}^{{\left( 3 \right)}} } \right)\dot{\alpha }_{1}^{{\left( 3 \right)}} + \left( {\frac{{h_{4}^{2} }}{8}I_{0}^{{\left( 4 \right)}} - \frac{1}{2}I_{2}^{{\left( 4 \right)}} } \right)\dot{\alpha }_{1}^{{\left( 4 \right)}} \hfill \\ + \left( {\frac{{h_{5}^{2} }}{8}I_{0}^{{\left( 5 \right)}} - \frac{1}{2}I_{2}^{{\left( 5 \right)}} } \right)\dot{\alpha }_{1}^{{\left( 5 \right)}} + \left( {\frac{{h_{6}^{2} }}{8}I_{0}^{{\left( 6 \right)}} - \frac{1}{2}I_{2}^{{\left( 6 \right)}} } \right)\dot{\alpha }_{1}^{{\left( 6 \right)}} + \left( {\frac{{h_{7}^{2} }}{8}I_{0}^{{\left( 7 \right)}} - \frac{1}{2}I_{2}^{{\left( 7 \right)}} } \right)\dot{\alpha }_{1}^{{\left( 7 \right)}} + \left( {\frac{{h_{8}^{2} }}{8}I_{0}^{{\left( 8 \right)}} - \frac{1}{2}I_{2}^{{\left( 8 \right)}} } \right)\dot{\alpha }_{1}^{{\left( 8 \right)}} \hfill \\ + \left( {\frac{{h_{9}^{2} }}{8}I_{0}^{{\left( 9 \right)}} - \frac{1}{2}I_{2}^{{\left( 9 \right)}} } \right)\dot{\alpha }_{1}^{{\left( 9 \right)}} + \left( {\frac{{h_{{10}}^{2} }}{8}I_{0}^{{\left( {10} \right)}} - \frac{1}{2}I_{2}^{{\left( {10} \right)}} } \right)\dot{\alpha }_{1}^{{\left( {10} \right)}} \hfill \\ \end{gathered} \right]$$

(B1)

$$A_{2} = \left[ \begin{gathered} \left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{u}_{0} + \left( { - \frac{{h_{1} }}{3}I_{1}^{\left( 1 \right)} + I_{2}^{\left( 1 \right)} + \frac{4}{{9h_{1} }}I_{3}^{\left( 1 \right)} - \frac{8}{{3h_{1}^{2} }}I_{4}^{\left( 1 \right)} + \frac{16}{{9h_{1}^{4} }}I_{6}^{\left( 1 \right)} - \frac{{h_{1} }}{3}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)} \right)\dot{\psi }_{1}^{\left( 1 \right)} \hfill \\ - \frac{1}{2}\left( { - \frac{{h_{1} }}{3}I_{2}^{\left( 1 \right)} + I_{3}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{5}^{\left( 1 \right)} - \frac{{h_{1}^{2} }}{4}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)} \right)\dot{\alpha }_{1}^{\left( 1 \right)} - \frac{1}{3}\left( { - \frac{{h_{1} }}{3}I_{3}^{\left( 1 \right)} + I_{4}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{6}^{\left( 1 \right)} } \right)\dot{\lambda }_{1}^{\left( 1 \right)} \hfill \\ - \frac{2}{3}h_{2} \left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\psi }_{1}^{\left( 2 \right)} - \frac{2}{3}h_{3} \left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\psi }_{1}^{\left( 3 \right)} - \frac{2}{3}h_{4} \left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\psi }_{1}^{\left( 4 \right)} \hfill \\ - \frac{2}{3}h_{5} \left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\psi }_{1}^{\left( 5 \right)} + \frac{{h_{1}^{3} }}{24}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\lambda }_{1}^{\left( 1 \right)} + \frac{{h_{2}^{3} }}{12}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\lambda }_{1}^{\left( 2 \right)} \hfill \\ + \frac{{h_{3}^{3} }}{12}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\lambda }_{1}^{\left( 3 \right)} + \frac{{h_{4}^{3} }}{12}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\lambda }_{1}^{\left( 4 \right)} + \frac{{h_{5}^{3} }}{12}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\lambda }_{1}^{\left( 5 \right)} \hfill \\ - \frac{4}{{3h_{1}^{2} }}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\beta }_{1}^{\left( 1 \right)} - \frac{4}{{3h_{2}^{2} }}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\beta }_{1}^{\left( 2 \right)} - \frac{4}{{3h_{3}^{2} }}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\beta }_{1}^{\left( 3 \right)} \hfill \\ - \frac{4}{{3h_{4}^{2} }}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\beta }_{1}^{\left( 4 \right)} - \frac{4}{{3h_{5}^{2} }}\left( { - \frac{{h_{1} }}{3}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}I_{3}^{\left( 1 \right)} } \right)\dot{\beta }_{1}^{\left( 5 \right)} \hfill \\ \end{gathered} \right]$$

(B2)

$$A_{3} = \left[ \begin{gathered} \left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{u}_{0} + \left( { - \frac{{h_{2} }}{3}I_{1}^{\left( 2 \right)} + I_{2}^{\left( 2 \right)} + \frac{4}{{9h_{2} }}I_{3}^{\left( 2 \right)} - \frac{8}{{3h_{2}^{2} }}I_{4}^{\left( 2 \right)} + \frac{16}{{9h_{2}^{4} }}I_{6}^{\left( 2 \right)} - \frac{{h_{2} }}{3}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)} \right)\dot{\psi }_{1}^{\left( 2 \right)} \hfill \\ - \frac{1}{2}\left( { - \frac{{h_{2} }}{3}I_{2}^{\left( 2 \right)} + I_{3}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{5}^{\left( 2 \right)} - \frac{{h_{2}^{2} }}{4}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)} \right)\dot{\alpha }_{1}^{\left( 2 \right)} - \frac{1}{3}\left( \begin{subarray}{l} - \frac{{h_{2} }}{3}I_{3}^{\left( 2 \right)} + I_{4}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{6}^{\left( 2 \right)} \\ - \frac{{h_{2}^{3} }}{8}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right) \end{subarray} \right)\dot{\lambda }_{1}^{\left( 2 \right)} \hfill \\ - \frac{{2h_{3} }}{3}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{\psi }_{1}^{\left( 3 \right)} - \frac{{2h_{4} }}{3}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{\psi }_{1}^{\left( 4 \right)} - \frac{{2h_{5} }}{3}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{\psi }_{1}^{\left( 5 \right)} \hfill \\ - \frac{4}{{3h_{2}^{2} }}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{\beta }_{1}^{\left( 2 \right)} - \frac{4}{{3h_{3}^{2} }}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{\beta }_{1}^{\left( 3 \right)} - \frac{4}{{3h_{4}^{2} }}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{\beta }_{1}^{\left( 4 \right)} \hfill \\ - \frac{4}{{3h_{5}^{2} }}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{\beta }_{1}^{\left( 5 \right)} + \frac{{h_{3}^{3} }}{12}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{\lambda }_{1}^{\left( 3 \right)} + \frac{{h_{4}^{3} }}{12}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{\lambda }_{1}^{\left( 4 \right)} \hfill \\ + \frac{{h_{5}^{3} }}{12}\left( { - \frac{{h_{2} }}{3}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}I_{3}^{\left( 2 \right)} } \right)\dot{\lambda }_{1}^{\left( 5 \right)} \hfill \\ \end{gathered} \right]$$

(B3)

$$A_{4} = \left[ \begin{gathered} \left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right)\dot{u}_{0} + \left( { - \frac{{h_{3} }}{3}I_{1}^{\left( 3 \right)} + I_{2}^{\left( 3 \right)} + \frac{4}{{9h_{3} }}I_{3}^{\left( 3 \right)} - \frac{8}{{3h_{3}^{2} }}I_{4}^{\left( 3 \right)} + \frac{16}{{9h_{3}^{4} }}I_{6}^{\left( 3 \right)} - \frac{{h_{3} }}{3}\left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right)} \right)\dot{\psi }_{1}^{\left( 3 \right)} \hfill \\ - \frac{1}{2}\left( { - \frac{{h_{3} }}{3}I_{2}^{\left( 3 \right)} + I_{3}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{5}^{\left( 3 \right)} - \frac{{h_{3}^{2} }}{4}\left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right)} \right)\dot{\alpha }_{1}^{\left( 3 \right)} - \frac{1}{3}\left( \begin{subarray}{l} - \frac{{h_{3} }}{3}I_{3}^{\left( 3 \right)} + I_{4}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{6}^{\left( 3 \right)} \\ - \frac{{h_{3}^{3} }}{8}\left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right) \end{subarray} \right)\dot{\lambda }_{1}^{\left( 3 \right)} \hfill \\ - \frac{{2h_{4} }}{3}\left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right)\dot{\psi }_{1}^{\left( 4 \right)} - \frac{{2h_{5} }}{3}\left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right)\dot{\psi }_{1}^{\left( 5 \right)} + \frac{{h_{4}^{3} }}{12}\left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right)\dot{\lambda }_{1}^{\left( 4 \right)} \hfill \\ + \frac{{h_{5}^{3} }}{12}\left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right)\dot{\lambda }_{1}^{\left( 5 \right)} - \frac{4}{{3h_{3}^{2} }}\left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right)\dot{\beta }_{1}^{\left( 3 \right)} - \frac{4}{{3h_{4}^{2} }}\left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right)\dot{\beta }_{1}^{\left( 4 \right)} \hfill \\ - \frac{4}{{3h_{5}^{2} }}\left( { - \frac{{h_{3} }}{3}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}I_{3}^{\left( 3 \right)} } \right)\dot{\beta }_{1}^{\left( 5 \right)} \hfill \\ \end{gathered} \right]$$

(B4)

$$A_{5} = \left[ \begin{gathered} \left( { - \frac{{h_{4} }}{3}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}I_{3}^{\left( 4 \right)} } \right)\dot{u}_{0} + \left( { - \frac{{h_{4} }}{3}I_{1}^{\left( 4 \right)} + I_{2}^{\left( 4 \right)} + \frac{4}{{9h_{4} }}I_{3}^{\left( 4 \right)} - \frac{8}{{3h_{4}^{2} }}I_{4}^{\left( 4 \right)} + \frac{16}{{9h_{4}^{4} }}I_{6}^{\left( 4 \right)} - \frac{{h_{4} }}{3}\left( { - \frac{{h_{4} }}{3}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}I_{3}^{\left( 4 \right)} } \right)} \right)\dot{\psi }_{1}^{\left( 4 \right)} \hfill \\ - \frac{1}{2}\left( { - \frac{{h_{4} }}{3}I_{2}^{\left( 4 \right)} + I_{3}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}I_{5}^{\left( 4 \right)} - \frac{{h_{4}^{2} }}{4}\left( { - \frac{{h_{4} }}{3}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}I_{3}^{\left( 4 \right)} } \right)} \right)\dot{\alpha }_{1}^{\left( 3 \right)} - \frac{1}{3}\left( \begin{subarray}{l} - \frac{{h_{4} }}{3}I_{3}^{\left( 4 \right)} + I_{4}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}I_{6}^{\left( 4 \right)} \\ - \frac{{h_{4}^{3} }}{8}\left( { - \frac{{h_{4} }}{3}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}I_{3}^{\left( 4 \right)} } \right) \end{subarray} \right)\dot{\lambda }_{1}^{\left( 4 \right)} \hfill \\ - \frac{{2h_{5} }}{3}\left( { - \frac{{h_{4} }}{3}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}I_{3}^{\left( 4 \right)} } \right)\dot{\psi }_{1}^{\left( 5 \right)} + \frac{{h_{5}^{3} }}{12}\left( { - \frac{{h_{4} }}{3}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}I_{3}^{\left( 4 \right)} } \right)\dot{\lambda }_{1}^{\left( 5 \right)} - \frac{4}{{3h_{4}^{2} }}\left( { - \frac{{h_{4} }}{3}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}I_{3}^{\left( 4 \right)} } \right)\dot{\beta }_{1}^{\left( 4 \right)} \hfill \\ - \frac{4}{{3h_{5}^{2} }}\left( { - \frac{{h_{4} }}{3}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}I_{3}^{\left( 4 \right)} } \right)\dot{\beta }_{1}^{\left( 5 \right)} \hfill \\ \end{gathered} \right]$$

(B5)

$$A_{6} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{5} }}{3}I_{0}^{\left( 5 \right)} + I_{1}^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}I_{3}^{\left( 5 \right)} } \right)\dot{u}_{0} + \left( { - \frac{{h_{5} }}{3}I_{1}^{\left( 5 \right)} + I_{2}^{\left( 5 \right)} + \frac{4}{{9h_{5} }}I_{3}^{\left( 5 \right)} - \frac{8}{{3h_{5}^{2} }}I_{4}^{\left( 5 \right)} + \frac{16}{{9h_{5}^{4} }}I_{6}^{\left( 5 \right)} - \frac{{h_{5} }}{3}\left( { - \frac{{h_{5} }}{3}I_{0}^{\left( 5 \right)} + I_{1}^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}I_{3}^{\left( 5 \right)} } \right)} \right)\dot{\psi }_{1}^{\left( 5 \right)} \\ - \frac{1}{2}\left( { - \frac{{h_{5} }}{3}I_{2}^{\left( 5 \right)} + I_{3}^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}I_{5}^{\left( 5 \right)} - \frac{{h_{5}^{2} }}{4}\left( { - \frac{{h_{5} }}{3}I_{0}^{\left( 5 \right)} + I_{1}^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}I_{3}^{\left( 5 \right)} } \right)} \right)\dot{\alpha }_{1}^{\left( 5 \right)} \\ - \frac{1}{3}\left( { - \frac{{h_{5} }}{3}I_{3}^{\left( 5 \right)} + I_{4}^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}I_{6}^{\left( 5 \right)} - \frac{{h_{5}^{3} }}{8}\left( { - \frac{{h_{5} }}{3}I_{0}^{\left( 5 \right)} + I_{1}^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}I_{3}^{\left( 5 \right)} } \right)} \right)\dot{\lambda }_{1}^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}\left( { - \frac{{h_{5} }}{3}I_{0}^{\left( 5 \right)} + I_{1}^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}I_{3}^{\left( 5 \right)} } \right)\dot{\beta }_{1}^{\left( 5 \right)} \end{subarray} \right]$$

(B6)

$$A_{7} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{6} }}{3}I_{0}^{\left( 6 \right)} + I_{1}^{\left( 6 \right)} - \frac{4}{{3h_{6}^{2} }}I_{3}^{\left( 6 \right)} } \right)\dot{u}_{0} + \left( { - \frac{{h_{6} }}{3}I_{1}^{\left( 6 \right)} + I_{2}^{\left( 6 \right)} + \frac{4}{{9h_{6} }}I_{3}^{\left( 6 \right)} - \frac{8}{{3h_{6}^{2} }}I_{4}^{\left( 6 \right)} + \frac{16}{{9h_{6}^{4} }}I_{6}^{\left( 6 \right)} + \frac{{h_{6} }}{3}\left( { - \frac{{h_{6} }}{3}I_{0}^{\left( 6 \right)} + I_{1}^{\left( 6 \right)} - \frac{4}{{3h_{6}^{2} }}I_{3}^{\left( 6 \right)} } \right)} \right)\dot{\psi }_{1}^{\left( 6 \right)} \\ - \frac{1}{2}\left( { - \frac{{h_{6} }}{3}I_{2}^{\left( 6 \right)} + I_{3}^{\left( 6 \right)} - \frac{4}{{3h_{6}^{2} }}I_{5}^{\left( 6 \right)} - \frac{{h_{6}^{2} }}{4}\left( { - \frac{{h_{6} }}{3}I_{0}^{\left( 6 \right)} + I_{1}^{\left( 6 \right)} - \frac{4}{{3h_{6}^{2} }}I_{3}^{\left( 6 \right)} } \right)} \right)\dot{\alpha }_{1}^{\left( 6 \right)} - \frac{1}{3}\left( \begin{subarray}{l} - \frac{{h_{6} }}{3}I_{3}^{\left( 6 \right)} + I_{4}^{\left( 6 \right)} - \frac{4}{{3h_{6}^{2} }}I_{6}^{\left( 6 \right)} \\ + \frac{{h_{6}^{3} }}{8}\left( { - \frac{{h_{6} }}{3}I_{0}^{\left( 6 \right)} + I_{1}^{\left( 6 \right)} - \frac{4}{{3h_{6}^{2} }}I_{3}^{\left( 6 \right)} } \right) \end{subarray} \right)\dot{\lambda }_{1}^{\left( 6 \right)} \\ - \frac{4}{{3h_{6}^{2} }}\left( { - \frac{{h_{6} }}{3}I_{0}^{\left( 6 \right)} + I_{1}^{\left( 6 \right)} - \frac{4}{{3h_{6}^{2} }}I_{3}^{\left( 6 \right)} } \right)\dot{\beta }_{1}^{\left( 6 \right)} \end{subarray} \right]$$

(B7)

$$A_{8} = \left[ \begin{gathered} \left( { - \frac{{h_{7} }}{3}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}I_{3}^{\left( 7 \right)} } \right)\dot{u}_{0} + \frac{{2h_{6} }}{3}\left( { - \frac{{h_{7} }}{3}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}I_{3}^{\left( 7 \right)} } \right)\dot{\psi }_{1}^{\left( 6 \right)} - \frac{{h_{6}^{3} }}{12}\left( { - \frac{{h_{7} }}{3}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}I_{3}^{\left( 7 \right)} } \right)\dot{\lambda }_{1}^{\left( 6 \right)} \hfill \\ + \left( { - \frac{{h_{7} }}{3}I_{1}^{\left( 7 \right)} + I_{2}^{\left( 7 \right)} + \frac{4}{{9h_{7} }}I_{3}^{\left( 7 \right)} - \frac{8}{{3h_{7}^{2} }}I_{4}^{\left( 7 \right)} + \frac{16}{{9h_{7}^{4} }}I_{6}^{\left( 7 \right)} + \frac{{h_{7} }}{3}\left( { - \frac{{h_{7} }}{3}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}I_{3}^{\left( 7 \right)} } \right)} \right)\dot{\psi }_{1}^{\left( 7 \right)} \hfill \\ - \frac{1}{2}\left( { - \frac{{h_{7} }}{3}I_{2}^{\left( 7 \right)} + I_{3}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}I_{5}^{\left( 7 \right)} - \frac{{h_{7}^{2} }}{4}\left( { - \frac{{h_{7} }}{3}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}I_{3}^{\left( 7 \right)} } \right)} \right)\dot{\alpha }_{1}^{\left( 7 \right)} - \frac{1}{3}\left( \begin{subarray}{l} - \frac{{h_{7} }}{3}I_{3}^{\left( 7 \right)} + I_{4}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}I_{6}^{\left( 7 \right)} \\ + \frac{{h_{7}^{3} }}{8}\left( { - \frac{{h_{7} }}{3}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}I_{3}^{\left( 7 \right)} } \right)\dot{\lambda }_{1}^{\left( 7 \right)} \end{subarray} \right)\dot{\lambda }_{1}^{\left( 7 \right)} \hfill \\ - \frac{4}{{3h_{6}^{2} }}\left( { - \frac{{h_{7} }}{3}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}I_{3}^{\left( 7 \right)} } \right)\dot{\beta }_{1}^{\left( 6 \right)} - \frac{4}{{3h_{7}^{2} }}\left( { - \frac{{h_{7} }}{3}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}I_{3}^{\left( 7 \right)} } \right)\dot{\beta }_{1}^{\left( 7 \right)} \hfill \\ \end{gathered} \right]$$

(B8)

$$A_{9} = \left[ \begin{gathered} \left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right)\dot{u}_{0} + \left( { - \frac{{h_{8} }}{3}I_{1}^{\left( 8 \right)} + I_{2}^{\left( 8 \right)} + \frac{4}{{9h_{8} }}I_{3}^{\left( 8 \right)} - \frac{8}{{3h_{8}^{2} }}I_{4}^{\left( 8 \right)} + \frac{16}{{9h_{8}^{4} }}I_{6}^{\left( 8 \right)} + \frac{{h_{8} }}{3}\left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right)} \right)\dot{\psi }_{1}^{\left( 8 \right)} \hfill \\ - \frac{1}{2}\left( { - \frac{{h_{8} }}{3}I_{2}^{\left( 8 \right)} + I_{3}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{5}^{\left( 8 \right)} - \frac{{h_{8}^{2} }}{4}\left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right)} \right)\dot{\alpha }_{1}^{\left( 8 \right)} - \frac{1}{3}\left( \begin{subarray}{l} - \frac{{h_{8} }}{3}I_{3}^{\left( 8 \right)} + I_{4}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{6}^{\left( 8 \right)} \\ + \frac{{h_{8}^{3} }}{8}\left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right) \end{subarray} \right)\dot{\lambda }_{1}^{\left( 8 \right)} \hfill \\ + \frac{{2h_{6} }}{3}\left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right)\dot{\psi }_{1}^{\left( 6 \right)} + \frac{{2h_{7} }}{3}\left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right)\dot{\psi }_{1}^{\left( 7 \right)} - \frac{{h_{6}^{3} }}{12}\left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right)\dot{\lambda }_{1}^{\left( 6 \right)} \hfill \\ - \frac{{h_{7}^{3} }}{12}\left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right)\dot{\lambda }_{1}^{\left( 7 \right)} - \frac{4}{{3h_{6}^{2} }}\left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right)\dot{\beta }_{1}^{\left( 6 \right)} - \frac{4}{{3h_{7}^{2} }}\left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right)\dot{\beta }_{1}^{\left( 7 \right)} \hfill \\ - \frac{4}{{3h_{8}^{2} }}\left( { - \frac{{h_{8} }}{3}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}I_{3}^{\left( 8 \right)} } \right)\dot{\beta }_{1}^{\left( 8 \right)} \hfill \\ \end{gathered} \right]$$

(B9)

$$A_{10} = \left[ \begin{gathered} \left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{u}_{0} + \left( { - \frac{{h_{9} }}{3}I_{1}^{\left( 9 \right)} + I_{2}^{\left( 9 \right)} + \frac{4}{{9h_{9} }}I_{3}^{\left( 9 \right)} - \frac{8}{{3h_{9}^{2} }}I_{4}^{\left( 9 \right)} + \frac{16}{{9h_{9}^{4} }}I_{6}^{\left( 9 \right)} + \frac{{h_{9} }}{3}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)} \right)\dot{\psi }_{1}^{\left( 9 \right)} \hfill \\ - \frac{1}{2}\left( { - \frac{{h_{9} }}{3}I_{2}^{\left( 9 \right)} + I_{3}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{5}^{\left( 9 \right)} - \frac{{h_{9}^{2} }}{4}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)} \right)\dot{\alpha }_{1}^{\left( 9 \right)} - \frac{1}{3}\left( \begin{subarray}{l} - \frac{{h_{9} }}{3}I_{3}^{\left( 9 \right)} + I_{4}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{6}^{\left( 9 \right)} \\ + \frac{{h_{9}^{3} }}{8}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right) \end{subarray} \right)\dot{\lambda }_{1}^{\left( 9 \right)} \hfill \\ + \frac{{2h_{6} }}{3}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{\psi }_{1}^{\left( 6 \right)} + \frac{{2h_{7} }}{3}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{\psi }_{1}^{\left( 7 \right)} + \frac{{2h_{8} }}{3}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{\psi }_{1}^{\left( 8 \right)} \hfill \\ - \frac{{h_{6}^{3} }}{12}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{\lambda }_{1}^{\left( 6 \right)} - \frac{{h_{7}^{3} }}{12}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{\lambda }_{1}^{\left( 7 \right)} - \frac{{h_{8}^{3} }}{12}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{\lambda }_{1}^{\left( 8 \right)} \hfill \\ - \frac{4}{{3h_{6}^{2} }}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{\beta }_{1}^{\left( 6 \right)} - \frac{4}{{3h_{7}^{2} }}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{\beta }_{1}^{\left( 7 \right)} - \frac{4}{{3h_{8}^{2} }}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{\beta }_{1}^{\left( 8 \right)} \hfill \\ - \frac{4}{{3h_{9}^{2} }}\left( { - \frac{{h_{9} }}{3}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}I_{3}^{\left( 9 \right)} } \right)\dot{\beta }_{1}^{\left( 9 \right)} \hfill \\ \end{gathered} \right]$$

(B10)

$$A_{11} = \left[ \begin{gathered} \left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{u}_{0} + \frac{{2h_{6} }}{3}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\psi }_{1}^{\left( 6 \right)} \hfill \\ + \frac{{2h_{7} }}{3}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\psi }_{1}^{\left( 7 \right)} + \frac{{2h_{8} }}{3}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\psi }_{1}^{\left( 8 \right)} \hfill \\ + \frac{{2h_{9} }}{3}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\psi }_{1}^{\left( 9 \right)} + \left( \begin{subarray}{l} - \frac{{h_{10} }}{3}I_{1}^{{\left( {10} \right)}} + I_{2}^{{\left( {10} \right)}} + \frac{4}{{9h_{10} }}I_{3}^{{\left( {10} \right)}} - \frac{8}{{3h_{10}^{2} }}I_{4}^{{\left( {10} \right)}} \\ + \frac{16}{{9h_{10}^{4} }}I_{6}^{{\left( {10} \right)}} + \frac{{h_{10} }}{3}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right) \end{subarray} \right)\dot{\psi }_{1}^{{\left( {10} \right)}} \hfill \\ - \frac{1}{2}\left( { - \frac{{h_{10} }}{3}I_{2}^{{\left( {10} \right)}} + I_{3}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{5}^{{\left( {10} \right)}} - \frac{{h_{10}^{2} }}{4}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)} \right)\dot{\alpha }_{1}^{{\left( {10} \right)}} \hfill \\ - \frac{1}{3}\left( { - \frac{{h_{10} }}{3}I_{3}^{{\left( {10} \right)}} + I_{4}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{6}^{{\left( {10} \right)}} - \frac{{h_{10}^{3} }}{8}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)} \right)\dot{\lambda }_{1}^{{\left( {10} \right)}} \hfill \\ - \frac{{h_{6}^{3} }}{12}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\lambda }_{1}^{\left( 6 \right)} - \frac{{h_{7}^{3} }}{12}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\lambda }_{1}^{\left( 7 \right)} \hfill \\ - \frac{{h_{8}^{3} }}{12}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\lambda }_{1}^{\left( 8 \right)} - \frac{{h_{9}^{3} }}{12}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\lambda }_{1}^{\left( 9 \right)} \hfill \\ - \frac{4}{{3h_{6}^{2} }}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\beta }_{1}^{\left( 6 \right)} - \frac{4}{{3h_{7}^{2} }}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\beta }_{1}^{\left( 7 \right)} \hfill \\ - \frac{4}{{3h_{8}^{2} }}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\beta }_{1}^{\left( 8 \right)} - \frac{4}{{3h_{9}^{2} }}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\beta }_{1}^{\left( 9 \right)} \hfill \\ - \frac{4}{{3h_{10}^{2} }}\left( { - \frac{{h_{10} }}{3}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}I_{3}^{{\left( {10} \right)}} } \right)\dot{\beta }_{1}^{{\left( {10} \right)}} \hfill \\ \end{gathered} \right]$$

(B11)

$$A_{12} = \left[ \begin{gathered} \left( {I_{0}^{\left( 1 \right)} + I_{0}^{\left( 2 \right)} + I_{0}^{\left( 3 \right)} + I_{0}^{\left( 4 \right)} + I_{0}^{\left( 5 \right)} + I_{0}^{\left( 6 \right)} + I_{0}^{\left( 7 \right)} + I_{0}^{\left( 8 \right)} + I_{0}^{\left( 9 \right)} + I_{0}^{{\left( {10} \right)}} } \right)\dot{w}_{0} + \left( { - \frac{{h_{1} }}{2}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} } \right)\dot{\psi }_{3}^{\left( 1 \right)} \hfill \\ + \left( { - h_{2} I_{0}^{\left( 1 \right)} - \frac{{h_{2} }}{2}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} } \right)\dot{\psi }_{3}^{\left( 2 \right)} + \left( { - h_{3} I_{0}^{\left( 1 \right)} - h_{3} I_{0}^{\left( 2 \right)} - \frac{{h_{3} }}{2}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} } \right)\dot{\psi }_{3}^{\left( 3 \right)} \hfill \\ + \left( { - h_{4} I_{0}^{\left( 1 \right)} - h_{4} I_{0}^{\left( 2 \right)} - h_{4} I_{0}^{\left( 3 \right)} - \frac{{h_{4} }}{2}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} } \right)\dot{\psi }_{3}^{\left( 4 \right)} + \left( { - h_{5} I_{0}^{\left( 1 \right)} - h_{5} I_{0}^{\left( 2 \right)} - h_{5} I_{0}^{\left( 3 \right)} - h_{5} I_{0}^{\left( 4 \right)} - \frac{{h_{5} }}{2}I_{0}^{\left( 5 \right)} + I_{1}^{\left( 5 \right)} } \right)\dot{\psi }_{3}^{\left( 5 \right)} \hfill \\ + \left( {\frac{{h_{6} }}{2}I_{0}^{\left( 6 \right)} + h_{6} I_{0}^{\left( 7 \right)} + h_{6} I_{0}^{\left( 8 \right)} + h_{6} I_{0}^{\left( 9 \right)} + h_{6} I_{0}^{{\left( {10} \right)}} + I_{1}^{\left( 6 \right)} } \right)\dot{\psi }_{3}^{\left( 6 \right)} + \left( {\frac{{h_{7} }}{2}I_{0}^{\left( 7 \right)} + h_{7} I_{0}^{\left( 8 \right)} + h_{7} I_{0}^{\left( 9 \right)} + h_{7} I_{0}^{{\left( {10} \right)}} + I_{1}^{\left( 7 \right)} } \right)\dot{\psi }_{3}^{\left( 7 \right)} \hfill \\ + \left( {\frac{{h_{8} }}{2}I_{0}^{\left( 8 \right)} + h_{8} I_{0}^{\left( 9 \right)} + h_{8} I_{0}^{{\left( {10} \right)}} + I_{1}^{\left( 8 \right)} } \right)\dot{\psi }_{3}^{\left( 8 \right)} + \left( {\frac{{h_{9} }}{2}I_{0}^{\left( 9 \right)} + h_{9} I_{0}^{{\left( {10} \right)}} + I_{1}^{\left( 9 \right)} } \right)\dot{\psi }_{3}^{\left( 9 \right)} + \left( {\frac{{h_{10} }}{2}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} } \right)\dot{\psi }_{3}^{{\left( {10} \right)}} \hfill \\ + \left( { - \frac{{h_{1}^{2} }}{4}I_{0}^{\left( 1 \right)} + I_{2}^{\left( 1 \right)} } \right)\dot{\alpha }_{3}^{\left( 1 \right)} + \left( { - \frac{{h_{2}^{2} }}{4}I_{0}^{\left( 2 \right)} + I_{2}^{\left( 2 \right)} } \right)\dot{\alpha }_{3}^{\left( 2 \right)} + \left( { - \frac{{h_{3}^{2} }}{4}I_{0}^{\left( 3 \right)} + I_{2}^{\left( 3 \right)} } \right)\dot{\alpha }_{3}^{\left( 3 \right)} + \left( { - \frac{{h_{4}^{2} }}{4}I_{0}^{\left( 4 \right)} + I_{2}^{\left( 4 \right)} } \right)\dot{\alpha }_{3}^{\left( 4 \right)} \hfill \\ + \left( { - \frac{{h_{5}^{2} }}{4}I_{0}^{\left( 5 \right)} + I_{2}^{\left( 5 \right)} } \right)\dot{\alpha }_{3}^{\left( 5 \right)} + \left( { - \frac{{h_{6}^{2} }}{4}I_{0}^{\left( 6 \right)} + I_{2}^{\left( 6 \right)} } \right)\dot{\alpha }_{3}^{\left( 6 \right)} + \left( { - \frac{{h_{7}^{2} }}{4}I_{0}^{\left( 7 \right)} + I_{2}^{\left( 7 \right)} } \right)\dot{\alpha }_{3}^{\left( 7 \right)} + \left( { - \frac{{h_{8}^{2} }}{4}I_{0}^{\left( 8 \right)} + I_{2}^{\left( 8 \right)} } \right)\dot{\alpha }_{3}^{\left( 8 \right)} \hfill \\ + \left( { - \frac{{h_{9}^{2} }}{4}I_{0}^{\left( 9 \right)} + I_{2}^{\left( 9 \right)} } \right)\dot{\alpha }_{3}^{\left( 9 \right)} + \left( { - \frac{{h_{10}^{2} }}{4}I_{0}^{{\left( {10} \right)}} + I_{2}^{{\left( {10} \right)}} } \right)\dot{\alpha }_{3}^{{\left( {10} \right)}} \hfill \\ \end{gathered} \right]$$

(B12)

$$A_{13} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{1} }}{2}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} } \right)\dot{w}_{0} + \left( {\frac{{h_{1}^{2} }}{4}I_{0}^{\left( 1 \right)} - h_{1} I_{1}^{\left( 1 \right)} + I_{2}^{\left( 1 \right)} } \right)\dot{\psi }_{3}^{\left( 1 \right)} - h_{2} \left( { - \frac{{h_{1} }}{2}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} } \right)\dot{\psi }_{3}^{\left( 2 \right)} - h_{3} \left( { - \frac{{h_{1} }}{2}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} } \right)\dot{\psi }_{3}^{\left( 3 \right)} \\ - h_{4} \left( { - \frac{{h_{1} }}{2}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} } \right)\dot{\psi }_{3}^{\left( 4 \right)} - h_{5} \left( { - \frac{{h_{1} }}{2}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} } \right)\dot{\psi }_{3}^{\left( 5 \right)} + \left( { - \frac{{h_{1} }}{2}I_{2}^{\left( 1 \right)} + I_{3}^{\left( 1 \right)} - \frac{{h_{1}^{2} }}{4}\left( { - \frac{{h_{1} }}{2}I_{0}^{\left( 1 \right)} + I_{1}^{\left( 1 \right)} } \right)} \right)\dot{\alpha }_{3}^{\left( 1 \right)} \end{subarray} \right]$$

(B13)

$$A_{14} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{1}^{2} }}{4}I_{0}^{\left( 1 \right)} + I_{2}^{\left( 1 \right)} } \right)\dot{w}_{0} + \left( { - \frac{{h_{1}^{2} }}{4}I_{1}^{\left( 1 \right)} + I_{3}^{\left( 1 \right)} - \frac{{h_{1} }}{2}\left( { - \frac{{h_{1}^{2} }}{4}I_{0}^{\left( 1 \right)} + I_{2}^{\left( 1 \right)} } \right)} \right)\dot{\psi }_{3}^{\left( 1 \right)} - h_{2} \left( { - \frac{{h_{1}^{2} }}{4}I_{0}^{\left( 1 \right)} + I_{2}^{\left( 1 \right)} } \right)\dot{\psi }_{3}^{\left( 2 \right)} \\ - h_{3} \left( { - \frac{{h_{1}^{2} }}{4}I_{0}^{\left( 1 \right)} + I_{2}^{\left( 1 \right)} } \right)\dot{\psi }_{3}^{\left( 3 \right)} - h_{4} \left( { - \frac{{h_{1}^{2} }}{4}I_{0}^{\left( 1 \right)} + I_{2}^{\left( 1 \right)} } \right)\dot{\psi }_{3}^{\left( 4 \right)} - h_{5} \left( { - \frac{{h_{1}^{2} }}{4}I_{0}^{\left( 1 \right)} + I_{2}^{\left( 1 \right)} } \right)\dot{\psi }_{3}^{\left( 5 \right)} + \left( {\frac{{h_{1}^{4} }}{16}I_{0}^{\left( 1 \right)} - \frac{{h_{1}^{2} }}{2}I_{2}^{\left( 1 \right)} + I_{4}^{\left( 1 \right)} } \right)\dot{\alpha }_{3}^{\left( 1 \right)} \end{subarray} \right]$$

(B14)

$$A_{15} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{2} }}{2}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} } \right)\dot{w}_{0} + \left( {\frac{{h_{2}^{2} }}{4}I_{0}^{\left( 2 \right)} - h_{2} I_{1}^{\left( 2 \right)} + I_{2}^{\left( 2 \right)} } \right)\dot{\psi }_{3}^{\left( 2 \right)} - h_{3} \left( { - \frac{{h_{2} }}{2}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} } \right)\dot{\psi }_{3}^{\left( 3 \right)} - h_{4} \left( { - \frac{{h_{2} }}{2}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} } \right)\dot{\psi }_{3}^{\left( 4 \right)} \\ - h_{5} \left( { - \frac{{h_{2} }}{2}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} } \right)\dot{\psi }_{3}^{\left( 5 \right)} + \left( { - \frac{{h_{2} }}{2}I_{2}^{\left( 2 \right)} + I_{3}^{\left( 2 \right)} - \frac{{h_{2}^{2} }}{4}\left( { - \frac{{h_{2} }}{2}I_{0}^{\left( 2 \right)} + I_{1}^{\left( 2 \right)} } \right)} \right)\dot{\alpha }_{3}^{\left( 2 \right)} \end{subarray} \right]$$

(B15)

$$A_{16} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{2}^{2} }}{4}I_{0}^{\left( 2 \right)} + I_{2}^{\left( 2 \right)} } \right)\dot{w}_{0} + \left( { - \frac{{h_{2}^{2} }}{4}I_{1}^{\left( 2 \right)} + I_{3}^{\left( 2 \right)} - \frac{{h_{2} }}{2}\left( { - \frac{{h_{2}^{2} }}{4}I_{0}^{\left( 2 \right)} + I_{2}^{\left( 2 \right)} } \right)} \right)\dot{\psi }_{3}^{\left( 2 \right)} - h_{3} \left( { - \frac{{h_{2}^{2} }}{4}I_{0}^{\left( 2 \right)} + I_{2}^{\left( 2 \right)} } \right)\dot{\psi }_{3}^{\left( 3 \right)} - h_{4} \left( { - \frac{{h_{2}^{2} }}{4}I_{0}^{\left( 2 \right)} + I_{2}^{\left( 2 \right)} } \right)\dot{\psi }_{3}^{\left( 4 \right)} \\ - h_{5} \left( { - \frac{{h_{2}^{2} }}{4}I_{0}^{\left( 2 \right)} + I_{2}^{\left( 2 \right)} } \right)\dot{\psi }_{3}^{\left( 5 \right)} + \left( {\frac{{h_{2}^{4} }}{16}I_{0}^{\left( 2 \right)} - \frac{{h_{2}^{2} }}{2}I_{2}^{\left( 2 \right)} + I_{4}^{\left( 2 \right)} } \right)\dot{\alpha }_{3}^{\left( 2 \right)} \end{subarray} \right]$$

(B16)

$$A_{17} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{3} }}{2}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} } \right)\dot{w}_{0} + \left( {\frac{{h_{3}^{2} }}{4}I_{0}^{\left( 3 \right)} - h_{3} I_{1}^{\left( 3 \right)} + I_{2}^{\left( 3 \right)} } \right)\dot{\psi }_{3}^{\left( 3 \right)} - h_{4} \left( { - \frac{{h_{3} }}{2}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} } \right)\dot{\psi }_{3}^{\left( 4 \right)} - h_{5} \left( { - \frac{{h_{3} }}{2}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} } \right)\dot{\psi }_{3}^{\left( 5 \right)} \\ + \left( { - \frac{{h_{3} }}{2}I_{2}^{\left( 3 \right)} + I_{3}^{\left( 3 \right)} - \frac{{h_{3}^{2} }}{4}\left( { - \frac{{h_{3} }}{2}I_{0}^{\left( 3 \right)} + I_{1}^{\left( 3 \right)} } \right)} \right)\dot{\alpha }_{3}^{\left( 3 \right)} \end{subarray} \right]$$

(B17)

$$A_{18} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{3}^{2} }}{4}I_{0}^{\left( 3 \right)} + I_{2}^{\left( 3 \right)} } \right)\dot{w}_{0} + \left( { - \frac{{h_{3}^{2} }}{4}I_{1}^{\left( 3 \right)} + I_{3}^{\left( 3 \right)} - \frac{{h_{3} }}{2}\left( { - \frac{{h_{3}^{2} }}{4}I_{0}^{\left( 3 \right)} + I_{2}^{\left( 3 \right)} } \right)} \right)\dot{\psi }_{3}^{\left( 3 \right)} - h_{4} \left( { - \frac{{h_{3}^{2} }}{4}I_{0}^{\left( 3 \right)} + I_{2}^{\left( 3 \right)} } \right)\dot{\psi }_{3}^{\left( 4 \right)} \\ - h_{5} \left( { - \frac{{h_{3}^{2} }}{4}I_{0}^{\left( 3 \right)} + I_{2}^{\left( 3 \right)} } \right)\dot{\psi }_{3}^{\left( 5 \right)} + \left( {\frac{{h_{3}^{4} }}{16}I_{0}^{\left( 3 \right)} - \frac{{h_{3}^{2} }}{2}I_{2}^{\left( 3 \right)} + I_{4}^{\left( 3 \right)} } \right)\dot{\alpha }_{3}^{\left( 3 \right)} \end{subarray} \right]$$

(B18)

$$A_{19} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{4} }}{2}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} } \right)\dot{w}_{0} + \left( {\frac{{h_{4}^{2} }}{4}I_{0}^{\left( 4 \right)} - h_{4} I_{1}^{\left( 4 \right)} + I_{2}^{\left( 4 \right)} } \right)\dot{\psi }_{3}^{\left( 4 \right)} - h_{5} \left( { - \frac{{h_{4} }}{2}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} } \right)\dot{\psi }_{3}^{\left( 5 \right)} \\ + \left( { - \frac{{h_{4} }}{2}I_{2}^{\left( 4 \right)} + I_{3}^{\left( 4 \right)} - \frac{{h_{4}^{2} }}{4}\left( { - \frac{{h_{4} }}{2}I_{0}^{\left( 4 \right)} + I_{1}^{\left( 4 \right)} } \right)} \right)\dot{\alpha }_{3}^{\left( 4 \right)} \end{subarray} \right]$$

(B19)

$$A_{20} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{4}^{2} }}{4}I_{0}^{\left( 4 \right)} + I_{2}^{\left( 4 \right)} } \right)\dot{w}_{0} + \left( { - \frac{{h_{4}^{2} }}{4}I_{1}^{\left( 4 \right)} + I_{3}^{\left( 4 \right)} - \frac{{h_{4} }}{2}\left( { - \frac{{h_{4}^{2} }}{4}I_{0}^{\left( 4 \right)} + I_{2}^{\left( 4 \right)} } \right)} \right)\dot{\psi }_{3}^{\left( 4 \right)} \\ - h_{5} \left( { - \frac{{h_{4}^{2} }}{4}I_{0}^{\left( 4 \right)} + I_{2}^{\left( 4 \right)} } \right)\dot{\psi }_{3}^{\left( 5 \right)} + \left( {\frac{{h_{4}^{4} }}{16}I_{0}^{\left( 4 \right)} - \frac{{h_{4}^{2} }}{2}I_{2}^{\left( 4 \right)} + I_{4}^{\left( 4 \right)} } \right)\dot{\alpha }_{3}^{\left( 4 \right)} \end{subarray} \right]$$

(B20)

$$A_{21} = \left[ {\left( { - \frac{{h_{5} }}{2}I_{0}^{\left( 5 \right)} + I_{1}^{\left( 5 \right)} } \right)\dot{w}_{0} + \left( {\frac{{h_{5}^{2} }}{4}I_{0}^{\left( 5 \right)} - h_{5} I_{1}^{\left( 5 \right)} + I_{2}^{\left( 5 \right)} } \right)\dot{\psi }_{3}^{\left( 5 \right)} + \left( { - \frac{{h_{5} }}{2}I_{2}^{\left( 5 \right)} + I_{3}^{\left( 5 \right)} - \frac{{h_{5}^{2} }}{4}\left( { - \frac{{h_{5} }}{2}I_{0}^{\left( 5 \right)} + I_{1}^{\left( 5 \right)} } \right)} \right)\dot{\alpha }_{3}^{\left( 5 \right)} } \right]$$

(B21)

$$A_{22} = \left[ {\left( { - \frac{{h_{5}^{2} }}{4}I_{0}^{\left( 5 \right)} + I_{2}^{\left( 5 \right)} } \right)\dot{w}_{0} + \left( { - \frac{{h_{5}^{2} }}{4}I_{1}^{\left( 5 \right)} + I_{3}^{\left( 5 \right)} - \frac{{h_{5} }}{2}\left( { - \frac{{h_{5}^{2} }}{4}I_{0}^{\left( 5 \right)} + I_{2}^{\left( 5 \right)} } \right)} \right)\dot{\psi }_{3}^{\left( 5 \right)} + \left( {\frac{{h_{5}^{4} }}{16}I_{0}^{\left( 5 \right)} - \frac{{h_{5}^{2} }}{2}I_{2}^{\left( 5 \right)} + I_{4}^{\left( 5 \right)} } \right)\dot{\alpha }_{3}^{\left( 5 \right)} } \right]$$

(B22)

$$A_{23} = \left[ {\left( { - \frac{{h_{6} }}{2}I_{0}^{\left( 6 \right)} + I_{1}^{\left( 6 \right)} } \right)\dot{w}_{0} + \left( {\frac{{h_{6}^{2} }}{4}I_{0}^{\left( 6 \right)} - h_{6} I_{1}^{\left( 6 \right)} + I_{2}^{\left( 6 \right)} } \right)\dot{\psi }_{3}^{\left( 6 \right)} + \left( { - \frac{{h_{6} }}{2}I_{2}^{\left( 6 \right)} + I_{3}^{\left( 6 \right)} - \frac{{h_{6}^{2} }}{4}\left( { - \frac{{h_{6} }}{2}I_{0}^{\left( 6 \right)} + I_{1}^{\left( 6 \right)} } \right)} \right)\dot{\alpha }_{3}^{\left( 6 \right)} } \right]$$

(B23)

$$A_{24} = \left[ {\left( { - \frac{{h_{6}^{2} }}{4}I_{0}^{\left( 6 \right)} + I_{2}^{\left( 6 \right)} } \right)\dot{w}_{0} + \left( { - \frac{{h_{6}^{2} }}{4}I_{1}^{\left( 6 \right)} + I_{3}^{\left( 6 \right)} + \frac{{h_{6} }}{2}\left( { - \frac{{h_{6}^{2} }}{4}I_{0}^{\left( 6 \right)} + I_{2}^{\left( 6 \right)} } \right)} \right)\dot{\psi }_{3}^{\left( 6 \right)} + \left( {\frac{{h_{6}^{4} }}{16}I_{0}^{\left( 6 \right)} - \frac{{h_{6}^{2} }}{2}I_{2}^{\left( 6 \right)} + I_{4}^{\left( 6 \right)} } \right)\dot{\alpha }_{3}^{\left( 6 \right)} } \right]$$

(B24)

$$A_{25} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{7} }}{2}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} } \right)\dot{w}_{0} + h_{6} \left( { - \frac{{h_{7} }}{2}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} } \right)\dot{\psi }_{3}^{\left( 6 \right)} + \left( {\frac{{h_{7}^{2} }}{4}I_{0}^{\left( 7 \right)} - h_{7} I_{1}^{\left( 7 \right)} + I_{2}^{\left( 7 \right)} } \right)\dot{\psi }_{3}^{\left( 7 \right)} \\ + \left( { - \frac{{h_{7} }}{2}I_{2}^{\left( 7 \right)} + I_{3}^{\left( 7 \right)} - \frac{{h_{7}^{2} }}{4}\left( { - \frac{{h_{7} }}{2}I_{0}^{\left( 7 \right)} + I_{1}^{\left( 7 \right)} } \right)} \right)\dot{\alpha }_{3}^{\left( 7 \right)} \end{subarray} \right]$$

(B25)

$$A_{26} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{7}^{2} }}{4}I_{0}^{\left( 7 \right)} + I_{2}^{\left( 7 \right)} } \right)\dot{w}_{0} + h_{6} \left( { - \frac{{h_{7}^{2} }}{4}I_{0}^{\left( 7 \right)} + I_{2}^{\left( 7 \right)} } \right)\dot{\psi }_{3}^{\left( 6 \right)} + \left( { - \frac{{h_{7}^{2} }}{4}I_{1}^{\left( 7 \right)} + I_{3}^{\left( 7 \right)} + \frac{{h_{7} }}{2}\left( { - \frac{{h_{7}^{2} }}{4}I_{0}^{\left( 7 \right)} + I_{2}^{\left( 7 \right)} } \right)} \right)\dot{\psi }_{3}^{\left( 7 \right)} \\ + \left( {\frac{{h_{7}^{4} }}{16}I_{0}^{\left( 7 \right)} - \frac{{h_{7}^{2} }}{2}I_{2}^{\left( 7 \right)} + I_{4}^{\left( 7 \right)} } \right)\dot{\alpha }_{3}^{\left( 7 \right)} \end{subarray} \right]$$

(B26)

$$A_{27} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{8} }}{2}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} } \right)\dot{w}_{0} + h_{6} \left( { - \frac{{h_{8} }}{2}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} } \right)\dot{\psi }_{3}^{\left( 6 \right)} + h_{7} \left( { - \frac{{h_{8} }}{2}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} } \right)\dot{\psi }_{3}^{\left( 7 \right)} + \left( {\frac{{h_{8}^{2} }}{4}I_{0}^{\left( 8 \right)} - h_{8} I_{1}^{\left( 8 \right)} + I_{2}^{\left( 8 \right)} } \right)\dot{\psi }_{3}^{\left( 8 \right)} \\ + \left( { - \frac{{h_{8} }}{2}I_{2}^{\left( 8 \right)} + I_{3}^{\left( 8 \right)} - \frac{{h_{8}^{2} }}{4}\left( { - \frac{{h_{8} }}{2}I_{0}^{\left( 8 \right)} + I_{1}^{\left( 8 \right)} } \right)} \right)\dot{\alpha }_{3}^{\left( 8 \right)} \end{subarray} \right]$$

(B27)

$$A_{28} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{8}^{2} }}{4}I_{0}^{\left( 8 \right)} + I_{2}^{\left( 8 \right)} } \right)\dot{w}_{0} + h_{6} \left( { - \frac{{h_{8}^{2} }}{4}I_{0}^{\left( 8 \right)} + I_{2}^{\left( 8 \right)} } \right)\dot{\psi }_{3}^{\left( 6 \right)} + h_{7} \left( { - \frac{{h_{8}^{2} }}{4}I_{0}^{\left( 8 \right)} + I_{2}^{\left( 8 \right)} } \right)\dot{\psi }_{3}^{\left( 7 \right)} + \left( { - \frac{{h_{8}^{2} }}{4}I_{1}^{\left( 8 \right)} + I_{3}^{\left( 8 \right)} + \frac{{h_{8} }}{2}\left( { - \frac{{h_{8}^{2} }}{4}I_{0}^{\left( 8 \right)} + I_{2}^{\left( 8 \right)} } \right)} \right)\dot{\psi }_{3}^{\left( 8 \right)} \\ + \left( {\frac{{h_{8}^{4} }}{16}I_{0}^{\left( 8 \right)} - \frac{{h_{8}^{2} }}{2}I_{2}^{\left( 8 \right)} + I_{4}^{\left( 8 \right)} } \right)\dot{\alpha }_{3}^{\left( 8 \right)} \end{subarray} \right]$$

(B28)

$$A_{29} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{9} }}{2}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} } \right)\dot{w}_{0} + h_{6} \left( { - \frac{{h_{9} }}{2}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} } \right)\dot{\psi }_{3}^{\left( 6 \right)} + h_{7} \left( { - \frac{{h_{9} }}{2}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} } \right)\dot{\psi }_{3}^{\left( 7 \right)} + h_{8} \left( { - \frac{{h_{9} }}{2}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} } \right)\dot{\psi }_{3}^{\left( 8 \right)} \\ + \left( {\frac{{h_{9}^{2} }}{4}I_{0}^{\left( 9 \right)} - h_{9} I_{1}^{\left( 9 \right)} + I_{2}^{\left( 9 \right)} } \right)\dot{\psi }_{3}^{\left( 9 \right)} + \left( { - \frac{{h_{9} }}{2}I_{2}^{\left( 9 \right)} + I_{3}^{\left( 9 \right)} - \frac{{h_{9}^{2} }}{4}\left( { - \frac{{h_{9} }}{2}I_{0}^{\left( 9 \right)} + I_{1}^{\left( 9 \right)} } \right)} \right)\dot{\alpha }_{3}^{\left( 9 \right)} \end{subarray} \right]$$

(B29)

$$A_{30} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{9}^{2} }}{4}I_{0}^{\left( 9 \right)} + I_{2}^{\left( 9 \right)} } \right)\dot{w}_{0} + h_{6} \left( { - \frac{{h_{9}^{2} }}{4}I_{0}^{\left( 9 \right)} + I_{2}^{\left( 9 \right)} } \right)\dot{\psi }_{3}^{\left( 6 \right)} + h_{7} \left( { - \frac{{h_{9}^{2} }}{4}I_{0}^{\left( 9 \right)} + I_{2}^{\left( 9 \right)} } \right)\dot{\psi }_{3}^{\left( 7 \right)} + h_{8} \left( { - \frac{{h_{9}^{2} }}{4}I_{0}^{\left( 9 \right)} + I_{2}^{\left( 9 \right)} } \right)\dot{\psi }_{3}^{\left( 8 \right)} \\ + \left( { - \frac{{h_{9}^{2} }}{4}I_{1}^{\left( 9 \right)} + I_{3}^{\left( 9 \right)} + \frac{{h_{9} }}{2}\left( { - \frac{{h_{9}^{2} }}{4}I_{0}^{\left( 9 \right)} + I_{2}^{\left( 9 \right)} } \right)} \right)\dot{\psi }_{3}^{\left( 9 \right)} + \left( {\frac{{h_{9}^{4} }}{16}I_{0}^{\left( 9 \right)} - \frac{{h_{9}^{2} }}{2}I_{2}^{\left( 9 \right)} + I_{4}^{\left( 9 \right)} } \right)\dot{\alpha }_{3}^{\left( 9 \right)} \end{subarray} \right]$$

(B30)

$$A_{31} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{10} }}{2}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} } \right)\dot{w}_{0} + h_{6} \left( { - \frac{{h_{10} }}{2}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} } \right)\dot{\psi }_{3}^{\left( 6 \right)} + h_{7} \left( { - \frac{{h_{10} }}{2}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} } \right)\dot{\psi }_{3}^{\left( 7 \right)} + h_{8} \left( { - \frac{{h_{10} }}{2}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} } \right)\dot{\psi }_{3}^{\left( 8 \right)} \\ + h_{9} \left( { - \frac{{h_{10} }}{2}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} } \right)\dot{\psi }_{3}^{\left( 9 \right)} + \left( {\frac{{h_{10}^{2} }}{4}I_{0}^{{\left( {10} \right)}} - h_{10} I_{1}^{{\left( {10} \right)}} + I_{2}^{{\left( {10} \right)}} } \right)\dot{\psi }_{3}^{{\left( {10} \right)}} + \left( { - \frac{{h_{10} }}{2}I_{2}^{{\left( {10} \right)}} + I_{3}^{{\left( {10} \right)}} - \frac{{h_{10}^{2} }}{4}\left( { - \frac{{h_{10} }}{2}I_{0}^{{\left( {10} \right)}} + I_{1}^{{\left( {10} \right)}} } \right)} \right)\dot{\alpha }_{3}^{{\left( {10} \right)}} \end{subarray} \right]$$

(B31)

$$A_{32} = \left[ \begin{subarray}{l} \left( { - \frac{{h_{10}^{2} }}{4}I_{0}^{{\left( {10} \right)}} + I_{2}^{{\left( {10} \right)}} } \right)\dot{w}_{0} + h_{6} \left( { - \frac{{h_{10}^{2} }}{4}I_{0}^{{\left( {10} \right)}} + I_{2}^{{\left( {10} \right)}} } \right)\dot{\psi }_{3}^{\left( 6 \right)} + h_{7} \left( { - \frac{{h_{10}^{2} }}{4}I_{0}^{{\left( {10} \right)}} + I_{2}^{{\left( {10} \right)}} } \right)\dot{\psi }_{3}^{\left( 7 \right)} + h_{8} \left( { - \frac{{h_{10}^{2} }}{4}I_{0}^{{\left( {10} \right)}} + I_{2}^{{\left( {10} \right)}} } \right)\dot{\psi }_{3}^{\left( 8 \right)} \\ + h_{9} \left( { - \frac{{h_{10}^{2} }}{4}I_{0}^{{\left( {10} \right)}} + I_{2}^{{\left( {10} \right)}} } \right)\dot{\psi }_{3}^{\left( 9 \right)} + \left( { - \frac{{h_{10}^{2} }}{4}I_{1}^{{\left( {10} \right)}} + I_{3}^{{\left( {10} \right)}} + \frac{{h_{10} }}{2}\left( { - \frac{{h_{10}^{2} }}{4}I_{0}^{{\left( {10} \right)}} + I_{2}^{{\left( {10} \right)}} } \right)} \right)\dot{\psi }_{3}^{{\left( {10} \right)}} + \left( {\frac{{h_{10}^{4} }}{16}I_{0}^{{\left( {10} \right)}} - \frac{{h_{10}^{2} }}{2}I_{2}^{{\left( {10} \right)}} + I_{4}^{{\left( {10} \right)}} } \right)\dot{\alpha }_{3}^{{\left( {10} \right)}} \end{subarray} \right]$$

(B32)

where in Eqs. (B1)–(B32), $I_{q}^{\left( k \right)}$ (k = 1, …, 10 and q = 0, 1, 2, 3, 4, 5, 6) is defined as Eq. (B33).

$$I_{q}^{\left( k \right)} = \int\limits_{{ - \frac{{h_{k} }}{2}}}^{{\frac{{h_{k} }}{2}}} {\rho^{\left( k \right)} \xi^{{\left( k \right)^{q} }} d\xi^{\left( k \right)} } \,\,\,\,;\,{\text{Where}}\,\,k = 1,\,...,10\,\,\,\,{\text{and}}\,\,\,q = 0,1,2,3,4,5,6.$$

(B33)

Appendix C

Expressions for the coefficient of B_i (i = 1, …, 32) defined in Eq. (5b) for the virtual strain energy of the ten-layer circular composite plate.

$$B_{1} = \sum\limits_{k = 1}^{10} {\left( {\frac{{\partial N_{11}^{\left( k \right)} }}{{\partial \xi_{1} }} + \frac{{N_{22}^{\left( k \right)} }}{{\xi_{1} }}} \right)}$$

(C1)

$$B_{2} = N_{13}^{\left( 1 \right)} - \frac{4}{{h_{1}^{2} }}P_{13}^{\left( 1 \right)} - \frac{{h_{1} }}{3}\left[ {\frac{{\partial N_{11}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + N_{22}^{\left( 1 \right)} } \right] - \left[ {\frac{{\partial M_{11}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + M_{22}^{\left( 1 \right)} - \frac{4}{{3h_{1}^{2} }}\left( {\frac{{\partial q_{11}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + q_{22}^{\left( 1 \right)} } \right)} \right]$$

(C2)

$$B_{3} = N_{13}^{\left( 2 \right)} - \frac{4}{{h_{2}^{2} }}P_{13}^{\left( 2 \right)} - \frac{{h_{2} }}{3}\left[ { - 2\frac{{\partial N_{11}^{\left( 1 \right)} }}{{\partial \xi_{1} }} - 2N_{22}^{\left( 1 \right)} + \frac{{\partial N_{11}^{\left( 2 \right)} }}{{\partial \xi_{1} }} + N_{22}^{\left( 2 \right)} } \right] - \left[ {\frac{{\partial M_{11}^{\left( 2 \right)} }}{{\partial \xi_{1} }} + M_{22}^{\left( 2 \right)} - \frac{4}{{3h_{2}^{2} }}\left( {\frac{{\partial q_{11}^{\left( 2 \right)} }}{{\partial \xi_{1} }} + q_{22}^{\left( 2 \right)} } \right)} \right]$$

(C3)

$$B_{4} = N_{13}^{\left( 3 \right)} - \frac{4}{{h_{3}^{2} }}P_{13}^{\left( 3 \right)} - \frac{{h_{3} }}{3}\left[ { - 2\left[ {\frac{{\partial N_{11}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 2 \right)} }}{{\partial \xi_{1} }}} \right] - 2\left[ {N_{22}^{\left( 1 \right)} + N_{22}^{\left( 2 \right)} } \right] + \frac{{\partial N_{11}^{\left( 3 \right)} }}{{\partial \xi_{1} }} + N_{22}^{\left( 3 \right)} } \right] - \left[ {\frac{{\partial M_{11}^{\left( 3 \right)} }}{{\partial \xi_{1} }} + M_{22}^{\left( 3 \right)} - \frac{4}{{3h_{3}^{2} }}\left( {\frac{{\partial q_{11}^{\left( 3 \right)} }}{{\partial \xi_{1} }} + q_{22}^{\left( 3 \right)} } \right)} \right]$$

(C4)

$$B_{5} = \left[ {N_{13}^{\left( 4 \right)} - \frac{4}{{h_{4}^{2} }}P_{13}^{\left( 4 \right)} - \frac{{h_{4} }}{3}\left[ { - 2\left[ {\frac{{\partial N_{11}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 2 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 3 \right)} }}{{\partial \xi_{1} }}} \right] - 2\left[ {N_{22}^{\left( 1 \right)} + N_{22}^{\left( 2 \right)} + N_{22}^{\left( 3 \right)} } \right] + \frac{{\partial N_{11}^{\left( 4 \right)} }}{{\partial \xi_{1} }} + N_{22}^{\left( 4 \right)} } \right] - \left[ {\frac{{\partial M_{11}^{\left( 4 \right)} }}{{\partial \xi_{1} }} + M_{22}^{\left( 4 \right)} - \frac{4}{{3h_{4}^{2} }}\left( {\frac{{\partial q_{11}^{\left( 4 \right)} }}{{\partial \xi_{1} }} + q_{22}^{\left( 4 \right)} } \right)} \right]} \right]$$

(C5)

$$B_{6} = \left[ \begin{subarray}{l} N_{13}^{\left( 5 \right)} - \frac{4}{{h_{5}^{2} }}P_{13}^{\left( 5 \right)} - \frac{{h_{5} }}{3}\left[ { - 2\left[ {\frac{{\partial N_{11}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 2 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 3 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 4 \right)} }}{{\partial \xi_{1} }}} \right] - 2\left[ {N_{22}^{\left( 1 \right)} + N_{22}^{\left( 2 \right)} + N_{22}^{\left( 3 \right)} + N_{22}^{\left( 4 \right)} } \right] + \frac{{\partial N_{11}^{\left( 5 \right)} }}{{\partial \xi_{1} }} + N_{22}^{\left( 5 \right)} } \right] \\ - \left[ {\frac{{\partial M_{11}^{\left( 5 \right)} }}{{\partial \xi_{1} }} + M_{22}^{\left( 5 \right)} - \frac{4}{{3h_{5}^{2} }}\left( {\frac{{\partial q_{11}^{\left( 5 \right)} }}{{\partial \xi_{1} }} + q_{22}^{\left( 5 \right)} } \right)} \right] \end{subarray} \right]$$

(C6)

$$B_{7} = \left[ \begin{subarray}{l} N_{13}^{\left( 6 \right)} - \frac{4}{{h_{6}^{2} }}P_{13}^{\left( 6 \right)} + \frac{{h_{6} }}{3}\left[ { - 2\left[ {\frac{{\partial N_{11}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 9 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 8 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 7 \right)} }}{{\partial \xi_{1} }}} \right] - 2\left[ {N_{22}^{{\left( {10} \right)}} + N_{22}^{\left( 9 \right)} + N_{22}^{\left( 8 \right)} + N_{22}^{\left( 7 \right)} } \right] + \frac{{\partial N_{11}^{\left( 6 \right)} }}{{\partial \xi_{1} }} + N_{22}^{\left( 6 \right)} } \right] \\ - \left[ {\frac{{\partial M_{11}^{\left( 6 \right)} }}{{\partial \xi_{1} }} + M_{22}^{\left( 6 \right)} - \frac{4}{{3h_{6}^{2} }}\left( {\frac{{\partial q_{11}^{\left( 6 \right)} }}{{\partial \xi_{1} }} + q_{22}^{\left( 6 \right)} } \right)} \right] \end{subarray} \right]$$

(C7)

$$B_{8} = \left[ {N_{13}^{\left( 7 \right)} - \frac{4}{{h_{7}^{2} }}P_{13}^{\left( 7 \right)} + \frac{{h_{7} }}{3}\left[ { - 2\left[ {\frac{{\partial N_{11}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 9 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 8 \right)} }}{{\partial \xi_{1} }}} \right] - 2\left[ {N_{22}^{{\left( {10} \right)}} + N_{22}^{\left( 9 \right)} + N_{22}^{\left( 8 \right)} } \right] + \frac{{\partial N_{11}^{\left( 7 \right)} }}{{\partial \xi_{1} }} + N_{22}^{\left( 7 \right)} } \right] - \left[ {\frac{{\partial M_{11}^{\left( 7 \right)} }}{{\partial \xi_{1} }} + M_{22}^{\left( 7 \right)} - \frac{4}{{3h_{7}^{2} }}\left( {\frac{{\partial q_{11}^{\left( 7 \right)} }}{{\partial \xi_{1} }} + q_{22}^{\left( 7 \right)} } \right)} \right]} \right]$$

(C8)

$$B_{9} = N_{13}^{\left( 8 \right)} - \frac{4}{{h_{8}^{2} }}P_{13}^{\left( 8 \right)} + \frac{{h_{8} }}{3}\left[ { - 2\left[ {\frac{{\partial N_{11}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + \frac{{\partial N_{11}^{\left( 9 \right)} }}{{\partial \xi_{1} }}} \right] - 2\left[ {N_{22}^{{\left( {10} \right)}} + N_{22}^{\left( 9 \right)} } \right] + \frac{{\partial N_{11}^{\left( 8 \right)} }}{{\partial \xi_{1} }} + N_{22}^{\left( 8 \right)} } \right] - \left[ {\frac{{\partial M_{11}^{\left( 8 \right)} }}{{\partial \xi_{1} }} + M_{22}^{\left( 8 \right)} - \frac{4}{{3h_{8}^{2} }}\left( {\frac{{\partial q_{11}^{\left( 8 \right)} }}{{\partial \xi_{1} }} + q_{22}^{\left( 8 \right)} } \right)} \right]$$

(C9)

$$B_{10} = N_{13}^{\left( 9 \right)} - \frac{4}{{h_{9}^{2} }}P_{13}^{\left( 9 \right)} + \frac{{h_{9} }}{3}\left[ { - 2\frac{{\partial N_{11}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} - 2N_{22}^{{\left( {10} \right)}} + \frac{{\partial N_{11}^{\left( 9 \right)} }}{{\partial \xi_{1} }} + N_{22}^{\left( 9 \right)} } \right] - \left[ {\frac{{\partial M_{11}^{\left( 9 \right)} }}{{\partial \xi_{1} }} + M_{22}^{\left( 9 \right)} - \frac{4}{{3h_{9}^{2} }}\left( {\frac{{\partial q_{11}^{\left( 9 \right)} }}{{\partial \xi_{1} }} + q_{22}^{\left( 9 \right)} } \right)} \right]$$

(C10)

$$B_{11} = N_{13}^{{\left( {10} \right)}} - \frac{4}{{h_{10}^{2} }}P_{13}^{{\left( {10} \right)}} + \frac{{h_{10} }}{3}\left[ {\frac{{\partial N_{11}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + N_{22}^{{\left( {10} \right)}} } \right] - \left[ {\frac{{\partial M_{11}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + M_{22}^{{\left( {10} \right)}} - \frac{4}{{3h_{10}^{2} }}\left( {\frac{{\partial q_{11}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + q_{22}^{{\left( {10} \right)}} } \right)} \right]$$

(C11)

$$B_{12} = \sum\limits_{k = 1}^{10} {\left( {\frac{{\partial N_{13}^{\left( k \right)} }}{{\partial \xi_{1} }}} \right)}$$

(C12)

$$B_{13} = N_{33}^{\left( 1 \right)} - \frac{{h_{1} }}{2}\frac{{\partial N_{13}^{\left( 1 \right)} }}{{\partial \xi_{1} }} - \frac{{\partial M_{13}^{\left( 1 \right)} }}{{\partial \xi_{1} }}$$

(C13)

$$B_{14} = - 2M_{33}^{\left( 1 \right)} - \frac{{h_{1}^{2} }}{4}\frac{{\partial N_{13}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial P_{13}^{\left( 1 \right)} }}{{\partial \xi_{1} }}$$

(C14)

$$B_{15} = N_{33}^{\left( 2 \right)} - \frac{{h_{2} }}{2}\left[ { - 2\frac{{\partial N_{13}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 2 \right)} }}{{\partial \xi_{1} }}} \right] - \frac{{\partial M_{13}^{\left( 2 \right)} }}{{\partial \xi_{1} }}$$

(C15)

$$B_{16} = - 2M_{33}^{\left( 2 \right)} - \frac{{h_{2}^{2} }}{4}\frac{{\partial N_{13}^{\left( 2 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial P_{13}^{\left( 2 \right)} }}{{\partial \xi_{1} }}$$

(C16)

$$B_{17} = N_{33}^{\left( 3 \right)} - \frac{{h_{3} }}{2}\left[ { - 2\left[ {\frac{{\partial N_{13}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 2 \right)} }}{{\partial \xi_{1} }}} \right] + \frac{{\partial N_{13}^{\left( 3 \right)} }}{{\partial \xi_{1} }}} \right] - \frac{{\partial M_{13}^{\left( 3 \right)} }}{{\partial \xi_{1} }}$$

(C17)

$$B_{18} = - 2M_{33}^{\left( 3 \right)} - \frac{{h_{3}^{2} }}{4}\frac{{\partial N_{13}^{\left( 3 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial P_{13}^{\left( 3 \right)} }}{{\partial \xi_{1} }}$$

(C18)

$$B_{19} = N_{33}^{\left( 4 \right)} - \frac{{h_{4} }}{2}\left[ { - 2\left[ {\frac{{\partial N_{13}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 2 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 3 \right)} }}{{\partial \xi_{1} }}} \right] + \frac{{\partial N_{13}^{\left( 4 \right)} }}{{\partial \xi_{1} }}} \right] - \frac{{\partial M_{13}^{\left( 4 \right)} }}{{\partial \xi_{1} }}$$

(C19)

$$B_{20} = - 2M_{33}^{\left( 4 \right)} - \frac{{h_{4}^{2} }}{4}\frac{{\partial N_{13}^{\left( 4 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial P_{13}^{\left( 4 \right)} }}{{\partial \xi_{1} }}$$

(C20)

$$B_{21} = N_{33}^{\left( 5 \right)} - \frac{{h_{5} }}{2}\left[ { - 2\left[ {\frac{{\partial N_{13}^{\left( 1 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 2 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 3 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 4 \right)} }}{{\partial \xi_{1} }}} \right] + \frac{{\partial N_{13}^{\left( 5 \right)} }}{{\partial \xi_{1} }}} \right] - \frac{{\partial M_{13}^{\left( 5 \right)} }}{{\partial \xi_{1} }}$$

(C21)

$$B_{22} = - 2M_{33}^{\left( 5 \right)} - \frac{{h_{5}^{2} }}{4}\frac{{\partial N_{13}^{\left( 5 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial P_{13}^{\left( 5 \right)} }}{{\partial \xi_{1} }}$$

(C22)

$$B_{23} = N_{33}^{\left( 6 \right)} + \frac{{h_{6} }}{2}\left[ { - 2\left[ {\frac{{\partial N_{13}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 9 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 8 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 7 \right)} }}{{\partial \xi_{1} }}} \right] + \frac{{\partial N_{13}^{\left( 6 \right)} }}{{\partial \xi_{1} }}} \right] - \frac{{\partial M_{13}^{\left( 6 \right)} }}{{\partial \xi_{1} }}$$

(C23)

$$B_{24} = - 2M_{33}^{\left( 6 \right)} - \frac{{h_{6}^{2} }}{4}\frac{{\partial N_{13}^{\left( 6 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial P_{13}^{\left( 6 \right)} }}{{\partial \xi_{1} }}$$

(C24)

$$B_{25} = N_{33}^{\left( 7 \right)} + \frac{{h_{7} }}{2}\left[ { - 2\left[ {\frac{{\partial N_{13}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 9 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 8 \right)} }}{{\partial \xi_{1} }}} \right] + \frac{{\partial N_{13}^{\left( 7 \right)} }}{{\partial \xi_{1} }}} \right] - \frac{{\partial M_{13}^{\left( 7 \right)} }}{{\partial \xi_{1} }}$$

(C25)

$$B_{26} = - 2M_{33}^{\left( 7 \right)} - \frac{{h_{7}^{2} }}{4}\frac{{\partial N_{13}^{\left( 7 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial P_{13}^{\left( 7 \right)} }}{{\partial \xi_{1} }}$$

(C26)

$$B_{27} = N_{33}^{\left( 8 \right)} + \frac{{h_{8} }}{2}\left[ { - 2\left[ {\frac{{\partial N_{13}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 9 \right)} }}{{\partial \xi_{1} }}} \right] + \frac{{\partial N_{13}^{\left( 8 \right)} }}{{\partial \xi_{1} }}} \right] - \frac{{\partial M_{13}^{\left( 8 \right)} }}{{\partial \xi_{1} }}$$

(C27)

$$B_{28} = - 2M_{33}^{\left( 8 \right)} - \frac{{h_{8}^{2} }}{4}\frac{{\partial N_{13}^{\left( 8 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial P_{13}^{\left( 8 \right)} }}{{\partial \xi_{1} }}$$

(C28)

$$B_{29} = N_{33}^{\left( 9 \right)} + \frac{{h_{9} }}{2}\left[ { - 2\frac{{\partial N_{13}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + \frac{{\partial N_{13}^{\left( 9 \right)} }}{{\partial \xi_{1} }}} \right] - \frac{{\partial M_{13}^{\left( 9 \right)} }}{{\partial \xi_{1} }}$$

(C29)

$$B_{30} = - 2M_{33}^{\left( 9 \right)} - \frac{{h_{9}^{2} }}{4}\frac{{\partial N_{13}^{\left( 9 \right)} }}{{\partial \xi_{1} }} + \frac{{\partial P_{13}^{\left( 9 \right)} }}{{\partial \xi_{1} }}$$

(C30)

$$B_{31} = N_{33}^{{\left( {10} \right)}} + \frac{{h_{10} }}{2}\frac{{\partial N_{13}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} - \frac{{\partial M_{13}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }}$$

(C31)

$$B_{32} = - 2M_{33}^{{\left( {10} \right)}} - \frac{{h_{10}^{2} }}{4}\frac{{\partial N_{13}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }} + \frac{{\partial P_{13}^{{\left( {10} \right)}} }}{{\partial \xi_{1} }}$$

(C32)

where in Eqs. (C1)–(C32), $N_{ij}^{\left( k \right)}$, $M_{ij}^{\left( k \right)}$, $P_{ij}^{\left( k \right)}$ and $q_{ij}^{\left( k \right)}$(k = 1, …, 10 and i,j = 1, 2 and 3) are defined in Eqs. (C33)–(C35).

$$\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {N_{11}^{\left( k \right)} } \\ {N_{22}^{\left( k \right)} } \\ {N_{12}^{\left( k \right)} } \\ \end{array} } \\ {\begin{array}{*{20}c} {M_{11}^{\left( k \right)} } \\ {M_{22}^{\left( k \right)} } \\ {M_{12}^{\left( k \right)} } \\ \end{array} } \\ {\begin{array}{*{20}c} {P_{11}^{\left( k \right)} } \\ {P_{22}^{\left( k \right)} } \\ {P_{12}^{\left( k \right)} } \\ \end{array} } \\ {\begin{array}{*{20}c} {q_{11}^{\left( k \right)} } \\ {q_{22}^{\left( k \right)} } \\ {q_{12}^{\left( k \right)} } \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left[ A \right]^{\left( k \right)} } & {\left[ B \right]^{\left( k \right)} } & {\left[ D \right]^{\left( k \right)} } & {\left[ E \right]^{\left( k \right)} } \\ {\left[ B \right]^{\left( k \right)} } & {\left[ D \right]^{\left( k \right)} } & {\left[ E \right]^{\left( k \right)} } & {\left[ F \right]^{\left( k \right)} } \\ {\left[ D \right]^{\left( k \right)} } & {\left[ E \right]^{\left( k \right)} } & {\left[ F \right]^{\left( k \right)} } & {\left[ G \right]^{\left( k \right)} } \\ {\left[ E \right]^{\left( k \right)} } & {\left[ F \right]^{\left( k \right)} } & {\left[ G \right]^{\left( k \right)} } & {\left[ H \right]^{\left( k \right)} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\varepsilon_{11}^{{0^{\left( k \right)} }} } \\ {\varepsilon_{22}^{{0^{\left( k \right)} }} } \\ {\gamma_{12}^{{0^{\left( k \right)} }} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\varepsilon_{11}^{{1^{\left( k \right)} }} } \\ {\varepsilon_{22}^{{1^{\left( k \right)} }} } \\ {\gamma_{12}^{{1^{\left( k \right)} }} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\varepsilon_{11}^{{2^{\left( k \right)} }} } \\ {\varepsilon_{22}^{{2^{\left( k \right)} }} } \\ {\gamma_{12}^{{2^{\left( k \right)} }} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\varepsilon_{11}^{{3^{\left( k \right)} }} } \\ {\varepsilon_{22}^{{3^{\left( k \right)} }} } \\ {\gamma_{12}^{{3^{\left( k \right)} }} } \\ \end{array} } \\ \end{array} } \right]$$

(C33)

$$\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {N_{13}^{\left( k \right)} } \\ {N_{23}^{\left( k \right)} } \\ \end{array} } \\ {\begin{array}{*{20}c} {M_{13}^{\left( k \right)} } \\ {M_{23}^{\left( k \right)} } \\ \end{array} } \\ {\begin{array}{*{20}c} {P_{13}^{\left( k \right)} } \\ {P_{23}^{\left( k \right)} } \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {A_{55}^{\left( k \right)} } & 0 & {B_{55}^{\left( k \right)} } & 0 & {D_{55}^{\left( k \right)} } & 0 \\ 0 & {A_{44}^{\left( k \right)} } & 0 & {B_{44}^{\left( k \right)} } & 0 & {D_{44}^{\left( k \right)} } \\ {B_{55}^{\left( k \right)} } & 0 & {D_{55}^{\left( k \right)} } & 0 & {E_{55}^{\left( k \right)} } & 0 \\ 0 & {B_{44}^{\left( k \right)} } & 0 & {D_{44}^{\left( k \right)} } & 0 & {E_{44}^{\left( k \right)} } \\ {D_{55}^{\left( k \right)} } & 0 & {E_{55}^{\left( k \right)} } & 0 & {F_{55}^{\left( k \right)} } & 0 \\ 0 & {D_{44}^{\left( k \right)} } & 0 & {E_{44}^{\left( k \right)} } & 0 & {F_{44}^{\left( k \right)} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\gamma_{13}^{{0^{\left( k \right)} }} } \\ {\gamma_{23}^{{0^{\left( k \right)} }} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\gamma_{13}^{{1^{\left( k \right)} }} } \\ {\gamma_{23}^{{1^{\left( k \right)} }} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\gamma_{13}^{{2^{\left( k \right)} }} } \\ {\gamma_{23}^{{2^{\left( k \right)} }} } \\ \end{array} } \\ \end{array} } \right]$$

(C34)

$$\left[ {\begin{array}{*{20}c} {N_{33}^{\left( k \right)} } \\ {M_{33}^{\left( k \right)} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {A_{33}^{\left( k \right)} } & {B_{33}^{\left( k \right)} } \\ {B_{33}^{\left( k \right)} } & {D_{33}^{\left( k \right)} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varepsilon_{33}^{{0^{\left( k \right)} }} } \\ {\varepsilon_{33}^{{1^{\left( k \right)} }} } \\ \end{array} } \right]$$

(C35)

where the elements of matrices A, B, D, E, F, G and H (k = 1, …, 10) are given as;

$$\left( {A_{ij}^{\left( k \right)} ,B_{ij}^{\left( k \right)} ,D_{ij}^{\left( k \right)} ,E_{ij}^{\left( k \right)} ,F_{ij}^{\left( k \right)} ,G_{ij}^{\left( k \right)} ,H_{ij}^{\left( k \right)} } \right) = \int\limits_{{ - \frac{{h_{k} }}{2}}}^{{\frac{{h_{k} }}{2}}} {Q_{ij}^{\left( k \right)} \left( {1,\xi^{\left( k \right)} ,\xi^{{\left( k \right)^{2} }} ,\xi^{{\left( k \right)^{3} }} ,\xi^{{\left( k \right)^{4} }} ,\xi^{{\left( k \right)^{5} }} ,\xi^{{\left( k \right)^{6} }} } \right)d\xi^{\left( k \right)} \,\,\,for\,i,j = 1,2,6.}$$

(C36)

$$\left( {A_{ii}^{\left( k \right)} ,B_{ii}^{\left( k \right)} ,D_{ii}^{\left( k \right)} ,E_{ii}^{\left( k \right)} ,F_{ii}^{\left( k \right)} } \right) = \int\limits_{{ - \frac{{h_{k} }}{2}}}^{{\frac{{h_{k} }}{2}}} {Q_{ii}^{\left( k \right)} \left( {1,\xi^{\left( k \right)} ,\xi^{{\left( k \right)^{2} }} ,\xi^{{\left( k \right)^{3} }} ,\xi^{{\left( k \right)^{4} }} } \right)d\xi^{\left( k \right)} \,\,\,for\,i = 4,5.}$$

(C37)

$$\left( {A_{33}^{\left( k \right)} ,B_{33}^{\left( k \right)} ,D_{33}^{\left( k \right)} } \right) = \int\limits_{{ - \frac{{h_{k} }}{2}}}^{{\frac{{h_{k} }}{2}}} {Q_{33}^{\left( k \right)} \left( {1,\xi^{\left( k \right)} ,\xi^{{\left( k \right)^{2} }} } \right)d\xi^{\left( k \right)} }$$

(C38)