Nonlinear Damped Transient Vibrations of Carbon Nanotube-Reinforced Magneto-Electro-Elastic Shells with Different Electromagnetic Circuits

Abstract

Purpose

In this research work, the nonlinear damped transient response of functionally graded carbon nanotube (CNT)-reinforced magneto-electro-elastic (FG-CNTMEE) shells are investigated using finite element methods.

Method

The controlled response is obtained through active constrained layer damping (ACLD) treatment composed of a 1–3 piezoelectric (PZC) patch and the viscoelastic layer. The FG-CNTMEE shell subjected to different forms of load cases including mechanical and electro-magnetic loads are considered for evaluation. In addition, the influence of open circuit and closed circuit electro-magnetic boundary conditions on the damped transient response of the FG-CNTMEE shell is investigated for the first time in the literature. The equations of motion are derived using the principle of virtual work. The solutions are obtained through the condensation approach and the direct-iterative method.

Results

Several numerical examples are presented to assess the influence of parameters such as shell geometries, CNT distribution pattern, CNT volume fraction, and boundary conditions. Special attention has been paid to understand the effect of coupling fields on the damped response of the FG-CNTMEE shell.

Appendices

Appendix A

The expanded representation of various material property matrices of Eq. (2.a) can be shown as follows:

$$\begin{aligned} \left[ {\overline{C}_{b}^{h} } \right] & = \left[ {\begin{array}{*{20}c} {\overline{C}_{11}^{h} } & {\overline{C}_{12}^{h} } & {\overline{C}_{13}^{h} } & {\overline{C}_{16}^{h} } \\ {\overline{C}_{12}^{h} } & {\overline{C}_{22}^{h} } & {\overline{C}_{23}^{h} } & {\overline{C}_{26}^{h} } \\ {\overline{C}_{13}^{h} } & {\overline{C}_{32}^{h} } & {\overline{C}_{33}^{h} } & {\overline{C}_{36}^{h} } \\ {\overline{C}_{16}^{h} } & {\overline{C}_{26}^{h} } & {\overline{C}_{36}^{h} } & {\overline{C}_{66}^{h} } \\ \end{array} } \right];\;\left[ {\overline{C}_{s}^{h} } \right] = \left[ {\begin{array}{*{20}c} {\overline{C}_{55}^{h} } & {\overline{C}_{45}^{h} } \\ {\overline{C}_{45}^{h} } & {\overline{C}_{66}^{h} } \\ \end{array} } \right]; \, \\ \left\{ {\overline{e}^{h} } \right\} & = \left\{ {\begin{array}{*{20}c} {\overline{e}_{31}^{h} } \\ {\overline{e}_{32}^{h} } \\ {\overline{e}_{33}^{h} } \\ {\overline{e}_{36}^{h} } \\ \end{array} } \right\};\left\{ {\overline{q}^{h} } \right\} = \left\{ {\begin{array}{*{20}c} {\overline{q}_{31}^{h} } \\ {\overline{q}_{32}^{h} } \\ {\overline{q}_{33}^{h} } \\ {\overline{q}_{36}^{h} } \\ \end{array} } \right\}. \\ \end{aligned}$$

(27)

Also, the elastic stiffness coefficient matrices, piezoelectric coefficient matrices of 1–3 PZC (Eq. 3) can be given as follows:

$$\begin{aligned} \left[ {\overline{C}_{b}^{p} } \right] & = \left[ {\begin{array}{*{20}c} {\overline{C}_{11}^{p} } & {\overline{C}_{12}^{p} } & {\overline{C}_{13}^{p} } & {\overline{C}_{16}^{p} } \\ {\overline{C}_{12}^{p} } & {\overline{C}_{22}^{p} } & {\overline{C}_{23}^{p} } & {\overline{C}_{26}^{p} } \\ {\overline{C}_{13}^{p} } & {\overline{C}_{32}^{p} } & {\overline{C}_{33}^{p} } & {\overline{C}_{36}^{p} } \\ {\overline{C}_{16}^{p} } & {\overline{C}_{26}^{p} } & {\overline{C}_{36}^{p} } & {\overline{C}_{66}^{p} } \\ \end{array} } \right];\left[ {\overline{C}_{s}^{p} } \right] = \left[ {\begin{array}{*{20}c} {\overline{C}_{55}^{p} } & {\overline{C}_{45}^{p} } \\ {\overline{C}_{45}^{p} } & {\overline{C}_{66}^{p} } \\ \end{array} } \right];\;\left[ {\overline{C}_{bs}^{p} } \right] = \left[ {\begin{array}{*{20}c} {\overline{C}^{p}_{15} } & {\overline{C}^{p}_{25} } & {\overline{C}^{p}_{35} } & 0 \\ 0 & 0 & 0 & {\overline{C}^{p}_{46} } \\ \end{array} } \right]^{{\text{T}}} ; \\ \left[ {\overline{C}_{bs}^{p} } \right] & = \left[ {\begin{array}{*{20}c} {\overline{C}^{p}_{15} } & {\overline{C}^{p}_{25} } & {\overline{C}^{p}_{35} } & 0 \\ 0 & 0 & 0 & {\overline{C}^{p}_{46} } \\ \end{array} } \right]^{{\text{T}}} ; \\ \left\{ {\overline{e}_{b}^{p} } \right\} & = \left\{ {\begin{array}{*{20}c} {\overline{e}_{31}^{p} } \\ {\overline{e}_{32}^{p} } \\ {\overline{e}_{33}^{p} } \\ {\overline{e}_{36}^{p} } \\ \end{array} } \right\};\left\{ {\overline{e}_{s}^{p} } \right\} = \left\{ {\begin{array}{*{20}c} {\overline{e}_{35}^{p} } \\ {\overline{e}_{34}^{p} } \\ \end{array} } \right\}. \\ \end{aligned}$$

Appendix B

Equation (17) can be expanded using the FE parameters and can be re-written as follows:

$$\begin{gathered} \int\limits_{{\Omega ^{h} }} {\delta \left( {\left[ {B_{{tb}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{1} } \right]\left[ {B_{{rb}} } \right]\left\{ {d_{r}^{e} } \right\} + \frac{1}{2}\left[ {B_{1} } \right]\left[ {B_{2} } \right]\left\{ {d_{t}^{e} } \right\}} \right)^{{\text{T}}} \times \left[ {\left[ {\bar{C}_{b}^{h} } \right] \times \left[ {\left[ {B_{{tb}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{1} } \right]\left[ {B_{{rb}} } \right]\left\{ {d_{r}^{e} } \right\}} \right.} \right.} \hfill \\ \left. { + \frac{1}{2}\left[ {B_{1} } \right]\left[ {B_{2} } \right]\left\{ {d_{t}^{e} } \right\}} \right] + \left. {\left\{ {e_{b}^{h} } \right\}\frac{1}{h}\left[ {N_{\phi } } \right]\left\{ \phi \right\} + \left\{ {q_{b}^{h} } \right\}\frac{1}{H}\left[ {N_{\psi } } \right]\left\{ \psi \right\}} \right]{\text{d}}\Omega ^{h} \hfill \\ + \int\limits_{{\Omega ^{h} }} {\delta \left( {\left[ {B_{{ts}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{3} } \right]\left[ {B_{{rs}} } \right]\left\{ {d_{r}^{e} } \right\}} \right)^{{\text{T}}} \times \left[ {\left[ {\bar{C}_{s}^{h} } \right]\left( {\left[ {B_{{ts}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{3} } \right]\left[ {B_{{rs}} } \right]\left\{ {d_{r}^{e} } \right\}} \right)} \right]{\text{d}}\Omega ^{h} } \hfill \\ + \int\limits_{{\Omega ^{h} }} {\delta \left[ {\frac{1}{h}\left\{ {\phi ^{e} } \right\}^{{\text{T}}} \left[ {N_{\phi } } \right]^{{\text{T}}} } \right] \times \left[ {\left\{ {e_{b}^{h} } \right\}^{{\text{T}}} \left( {\left[ {B_{{tb}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{1} } \right]\left[ {B_{{rb}} } \right]\left\{ {d_{r}^{e} } \right\}} \right)} \right.} - \left. { \in _{{33}}^{h} \left( {\frac{1}{h}\left[ {N_{\phi } } \right]\left\{ {\phi ^{e} } \right\}} \right) - d_{{33}}^{h} \left( {\frac{1}{h}\left[ {N_{\psi } } \right]\left\{ {\psi ^{e} } \right\}} \right)} \right]{\text{d}}\Omega ^{h} \hfill \\ + \int\limits_{{\Omega ^{h} }} {\delta \left[ {\frac{1}{h}\left\{ {\psi ^{e} } \right\}^{{\text{T}}} \left[ {N_{\psi } } \right]^{{\text{T}}} } \right]} \times \left[ {\left\{ {q_{b}^{h} } \right\}^{{\text{T}}} \left( {\left[ {B_{{tb}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{1} } \right]\left[ {B_{{rb}} } \right]\left\{ {d_{r}^{e} } \right\}} \right)} \right. - \left. {d_{{33}}^{h} \left( {\frac{1}{h}\left[ {N_{\phi } } \right]\left\{ {\phi ^{e} } \right\}} \right) - \mu _{{33}}^{h} \left( {\frac{1}{h}\left[ {N_{\psi } } \right]\left\{ {\psi ^{e} } \right\}} \right)} \right]{\text{d}}\Omega ^{h} {\text{ }} \hfill \\ + \int\limits_{{\Omega ^{p} }} {\delta \left( {\left[ {B_{{tb}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{2} } \right]\left[ {B_{{rb}} } \right]\left\{ {d_{r}^{e} } \right\} + \frac{1}{2}\left[ {B_{1} } \right]\left[ {B_{2} } \right]\left\{ {d_{t}^{e} } \right\}} \right)^{{\text{T}}} \times \left[ {\left[ {\bar{C}_{b}^{p} } \right] \times \left[ {\left[ {B_{{tb}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{2} } \right]\left[ {B_{{rb}} } \right]\left\{ {d_{r}^{e} } \right\}} \right.} \right.} \hfill \\ \left. {\left. { + \frac{1}{2}\left[ {B_{1} } \right]\left[ {B_{2} } \right]\left\{ {d_{t}^{e} } \right\}} \right] + \left[ {\bar{C}_{{bs}}^{p} } \right]\left( {\left[ {B_{{ts}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{5} } \right]\left[ {B_{{rs}} } \right]\left\{ {d_{r}^{e} } \right\}} \right) + \frac{1}{{h_{p} }}\left\{ {e_{b}^{p} } \right\}V} \right]{\text{d}}\Omega ^{p} \hfill \\ \int\limits_{{\Omega ^{p} }} {\delta \left( {\left[ {B_{{ts}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{5} } \right]\left[ {B_{{rs}} } \right]\left\{ {d_{r}^{e} } \right\}} \right)^{{\text{T}}} \times \left[ {\left[ {\bar{C}_{{bs}}^{p} } \right]^{{\text{T}}} \times \left[ {\left[ {B_{{tb}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{2} } \right]\left[ {B_{{rb}} } \right]\left\{ {d_{r}^{e} } \right\}} \right.} \right.} \hfill \\ \left. {\left. { + \frac{1}{2}\left[ {B_{1} } \right]\left[ {B_{2} } \right]\left\{ {d_{t}^{e} } \right\}} \right] + \left[ {\bar{C}_{s}^{p} } \right]\left( {\left[ {B_{{ts}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{5} } \right]\left[ {B_{{rs}} } \right]\left\{ {d_{r}^{e} } \right\}} \right) + \frac{1}{{h_{p} }}\left\{ {e_{s}^{p} } \right\}V} \right]{\text{d}}\Omega ^{p} \hfill \\ + \int\limits_{{\Omega ^{v} }} {\delta \left( {\left[ {B_{{ts}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{4} } \right]\left[ {B_{{rs}} } \right]\left\{ {d_{r}^{e} } \right\}} \right)^{{\text{T}}} \left[ {\left[ {\overline{C} _{s}^{v} } \right]\left( {\left[ {B_{{ts}} } \right]\left\{ {d_{t}^{e} } \right\} + \left[ {Z_{4} } \right]\left[ {B_{{rs}} } \right]\left\{ {d_{r}^{e} } \right\}} \right)} \right]{\text{d}}\Omega ^{v} } \hfill \\ - \int\limits_{A} \delta \left\{ {d_{t} } \right\}^{{\text{T}}} \left\{ f \right\}dA + \int\limits_{A} {Q^{\phi } } \delta \left[ {N_{\phi } } \right]\left\{ \phi \right\}dA + \int\limits_{A} {Q^{\psi } } \delta \left[ {N_{\psi } } \right]\left\{ \psi \right\}{\text{d}}A \hfill \\ + \frac{1}{2}\left\{ {d_{t}^{e} } \right\}^{{\text{T}}} \left[ {M^{e} } \right]\left\{ {\ddot{d}_{t}^{e} } \right\}. \hfill \\ \end{gathered}$$

(29)

The different nonlinear and linear stiffness matrices, force vectors leading to Eq. (22) can be represented as follows:

$$\begin{aligned} \left[ {K_{{{\text{NL}}1}}^{e} } \right] & = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{2} } \right]^{{\text{T}}} \left[ {B_{1} } \right]^{{\text{T}}} \left[ {D_{1} } \right]\left[ {B_{{tb}} } \right]} } {\text{ d}}x{\text{d}}y;\;\;\left[ {K_{{{\text{NL}}2}}^{e} } \right] = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{2} } \right]^{{\text{T}}} \left[ {B_{1} } \right]^{{\text{T}}} \left[ {D_{2} } \right]\left[ {B_{{rb}} } \right]} } {\text{ d}}xdy; \\ \left[ {K_{{{\text{NL}}3}}^{e} } \right] & = \frac{1}{4}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{2} } \right]^{{\text{T}}} \left[ {B_{1} } \right]^{{\text{T}}} \left[ {D_{1} } \right]\left[ {B_{1} } \right]} } \left[ {B_{2} } \right]{\text{ d}}x{\text{d}}y;\;\left[ {K_{{{\text{NL}}4}}^{e} } \right] = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{2} } \right]^{{\text{T}}} \left[ {B_{1} } \right]^{{\text{T}}} \left[ {D_{3} } \right]} } {\text{ }}\left[ {N_{\phi } } \right]{\text{d}}x{\text{d}}y; \\ \left[ {K_{{{\text{NL}}5}}^{e} } \right] & = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{2} } \right]^{{\text{T}}} \left[ {B_{1} } \right]^{{\text{T}}} \left[ {D_{4} } \right]} } {\text{ }}\left[ {N_{\psi } } \right]{\text{d}}x{\text{d}}y;\;\left[ {K_{{{\text{NL}}6}}^{e} } \right] = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{T} \left[ {D_{5} } \right]} } {\text{ }}\left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x{\text{d}}y; \\ \left[ {K_{{{\text{NL}}7}}^{e} } \right] & = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{{\text{T}}} \left[ {D_{1} } \right]} } {\text{ }}\left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x{\text{d}}y;\left[ {K_{{{\text{NL}}8}}^{e} } \right] = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{2} } \right]^{T} \left[ {B_{1} } \right]^{T} \left[ {D_{{13}} } \right]} } {\text{ }}\left[ {B_{{tb}} } \right]{\text{d}}x{\text{d}}y; \\ \left[ {K_{{{\text{NL}}9}}^{e} } \right] & = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{2} } \right]^{{\text{T}}} \left[ {B_{1} } \right]^{{\text{T}}} \left[ {D_{{14}} } \right]} } {\text{ }}\left[ {B_{{rb}} } \right]{\text{d}}x{\text{d}}y;\left[ {K_{{{\text{NL}}10}}^{e} } \right] = \frac{1}{4}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{2} } \right]^{T} \left[ {B_{1} } \right]^{T} \left[ {D_{{13}} } \right]} } {\text{ }}\left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x{\text{d}}y; \\ \left[ {K_{{{\text{NL}}11}}^{e} } \right] & = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{2} } \right]^{{\text{T}}} \left[ {B_{1} } \right]^{{\text{T}}} \left[ {D_{{14}} } \right]} } {\text{ }}\left[ {B_{{ts}} } \right]{\text{d}}x{\text{d}}y;\left[ {K_{{{\text{NL}}12}}^{e} } \right] = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{2} } \right]^{T} \left[ {B_{1} } \right]^{T} \left[ {D_{{15}} } \right]} } {\text{ }}\left[ {B_{{rs}} } \right]{\text{d}}x{\text{d}}y; \\ \left[ {K_{{{\text{NL}}13}}^{e} } \right] & = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{{\text{T}}} \left[ {D_{{17}} } \right]} } {\text{ }}\left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x{\text{d}}y;\left[ {K_{{{\text{NL}}14}}^{e} } \right] = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{T} \left[ {D_{{13}} } \right]} } {\text{ }}\left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x{\text{d}}y; \\ \left[ {K_{{{\text{NL}}15}}^{e} } \right] & = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rs}} } \right]^{{\text{T}}} \left[ {D_{{24}} } \right]} } {\text{ }}\left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x{\text{d}}y;\left[ {K_{{{\text{NL}}16}}^{e} } \right] = \frac{1}{2}\int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{ts}} } \right]^{T} \left[ {D_{{24}} } \right]} } {\text{ }}\left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x{\text{d}}y; \\ \end{aligned}$$

$$\begin{aligned} \left[ {K_{1}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{{\text{T}}} \left[ {D_{5} } \right]\left[ {B_{{tb}} } \right]{\text{d}}x{\text{d}}y;} \;} \left[ {K_{2}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{{\text{T}}} \left[ {D_{6} } \right]\left[ {B_{{rb}} } \right]{\text{d}}x{\text{d}}y;} } \;\left[ {K_{3}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{{\text{T}}} \left[ {D_{7} } \right]\left[ {N_{\phi } } \right]{\text{d}}x{\text{d}}y;} } \;\left[ {K_{4}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{{\text{T}}} \left[ {D_{8} } \right]\left[ {N_{\psi } } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{5}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{{\text{T}}} \left[ {D_{1} } \right]} \left[ {B_{{tb}} } \right]{\text{d}}x{\text{d}}y;} \;\left[ {K_{6}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{{\text{T}}} \left[ {D_{2} } \right]\left[ {B_{{rb}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{7}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{{\text{T}}} \left[ {D_{3} } \right]\left[ {N_{\phi } } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{8}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{{\text{T}}} \left[ {D_{4} } \right]\left[ {N_{\psi } } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{9}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rs}} } \right]^{{\text{T}}} \left[ {D_{9} } \right]\left[ {B_{{ts}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{10}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rs}} } \right]^{T} \left[ {D_{{10}} } \right]\left[ {B_{{rs}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{11}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{ts}} } \right]^{{\text{T}}} \left[ {D_{{11}} } \right]\left[ {B_{{ts}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{12}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{ts}} } \right]^{{\text{T}}} \left[ {D_{{12}} } \right]\left[ {B_{{rs}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{13}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{{\text{T}}} \left[ {D_{{17}} } \right]\left[ {B_{{tb}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{14}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{{\text{T}}} \left[ {D_{{18}} } \right]\left[ {B_{{rb}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{15}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{{\text{T}}} \left[ {D_{{17s}} } \right]\left[ {B_{{ts}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{16}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{{\text{T}}} \left[ {D_{{19}} } \right]\left[ {B_{{rs}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{17}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{{\text{T}}} \left[ {D_{{13}} } \right]\left[ {B_{{tb}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{18}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{{\text{T}}} \left[ {D_{{14}} } \right]\left[ {B_{{rb}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{19}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{{\text{T}}} \left[ {D_{{21}} } \right]\left[ {B_{{ts}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{18}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{{\text{T}}} \left[ {D_{{14}} } \right]\left[ {B_{{rb}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{21}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rs}} } \right]^{{\text{T}}} \left[ {D_{{24}} } \right]\left[ {B_{{tb}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{22}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rs}} } \right]^{{\text{T}}} \left[ {D_{{25}} } \right]\left[ {B_{{rb}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{23}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rs}} } \right]^{{\text{T}}} \left[ {D_{{25}} } \right]\left[ {B_{{ts}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{24}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rs}} } \right]^{{\text{T}}} \left[ {D_{{26}} } \right]\left[ {B_{{rs}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{25}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{ts}} } \right]^{{\text{T}}} \left[ {D_{{24}} } \right]\left[ {B_{{tb}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{26}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{ts}} } \right]^{{\text{T}}} \left[ {D_{{28}} } \right]\left[ {B_{{rb}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{27}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{ts}} } \right]^{{\text{T}}} \left[ {D_{{29}} } \right]\left[ {B_{{ts}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{28}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{ts}} } \right]^{{\text{T}}} \left[ {D_{{30}} } \right]\left[ {B_{{rs}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{29}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rs}} } \right]^{{\text{T}}} \left[ {D_{{31}} } \right]\left[ {B_{{ts}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{30}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rs}} } \right]^{{\text{T}}} \left[ {D_{{32}} } \right]\left[ {B_{{rs}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{31}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{ts}} } \right]^{{\text{T}}} \left[ {D_{{33}} } \right]\left[ {B_{{ts}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{32}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{ts}} } \right]^{{\text{T}}} \left[ {D_{{34}} } \right]\left[ {B_{{rs}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{33}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {N_{\phi } } \right]^{{\text{T}}} \left[ {D_{{35}} } \right]\left[ {B_{{tb}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{34}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {N_{\phi } } \right]^{{\text{T}}} \left[ {D_{{36}} } \right]\left[ {B_{{rb}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{35}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {N_{\phi } } \right]^{{\text{T}}} \left[ {D_{{37}} } \right]\left[ {N_{\phi } } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{36}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {N_{\phi } } \right]^{{\text{T}}} \left[ {D_{{38}} } \right]\left[ {N_{\psi } } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{37}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {N_{\psi } } \right]^{{\text{T}}} \left[ {D_{{39}} } \right]\left[ {B_{{tb}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{38}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {N_{\psi } } \right]^{{\text{T}}} \left[ {D_{{40}} } \right]\left[ {B_{{rb}} } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {K_{{39}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {N_{\psi } } \right]^{{\text{T}}} \left[ {D_{{41}} } \right]\left[ {N_{\phi } } \right]{\text{d}}x{\text{d}}y;} } \left[ {K_{{40}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {N_{\psi } } \right]^{T} \left[ {D_{{42}} } \right]\left[ {N_{\psi } } \right]{\text{d}}x{\text{d}}y;} } \\ \left[ {F_{{{\text{tp}}N1}}^{e} } \right] & = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{1} } \right]^{{\text{T}}} \left[ {B_{2} } \right]\left[ {D_{{16}} } \right]{\text{d}}x{\text{d}}y;} } {\text{ }}\left[ {F_{{tp1}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{tb}} } \right]^{{\text{T}}} \left[ {D_{{23}} } \right]{\text{d}}x{\text{d}}y;} } \left[ {F_{{tp2}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{ts}} } \right]^{{\text{T}}} \left[ {D_{{27}} } \right]{\text{d}}x{\text{d}}y;} } {\text{ }}\left[ {F_{{rp1}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rb}} } \right]^{{\text{T}}} \left[ {D_{{20}} } \right]{\text{d}}x{\text{d}}y;} } {\text{ }}\left[ {F_{{rp2}}^{e} } \right] = \int\limits_{0}^{{a_{e} }} {\int\limits_{0}^{{b_{e} }} {\left[ {B_{{rs}} } \right]^{{\text{T}}} \left[ {D_{{27}} } \right]{\text{d}}x{\text{d}}y;} } \\ \end{aligned}$$

$$\begin{gathered} \left[ {K_{T1}^{e} } \right] = \left[ {K_{NL1} } \right] + \left[ {K_{NL3} } \right] + \left[ {K_{5} } \right] + \left[ {K_{NL17} } \right] + \left[ {K_{11} } \right];\left[ {K_{T2}^{e} } \right] = \left[ {K_{NL2} } \right] + \left[ {K_{6} } \right] + \left[ {K_{12} } \right]; \hfill \\ \left[ {K_{T3}^{e} } \right] = \left[ {K_{NL14} } \right] + \left[ {K_{7} } \right];\left[ {K_{T4}^{e} } \right] = \left[ {K_{NL15} } \right] + \left[ {K_{8} } \right];\left[ {K_{T5}^{e} } \right] = \left[ {K_{NL6} } \right] + \left[ {K_{1} } \right] + \left[ {K_{9} } \right]; \hfill \\ \left[ {K_{T6}^{e} } \right] = \left[ {K_{2} } \right] + \left[ {K_{10} } \right];\left[ {K_{T7}^{e} } \right] = \left[ {K_{NL8} } \right] + \left[ {K_{NL10} } \right] + \left[ {K_{17} } \right] + \left[ {K_{NL11} } \right] + \left[ {K_{19} } \right] + \left[ {K_{NL14} } \right]; \hfill \\ \left[ {K_{T8}^{e} } \right] = \left[ {K_{NL19} } \right] + \left[ {K_{NL12} } \right] + \left[ {K_{18} } \right] + \left[ {K_{20} } \right];\left[ {K_{T9}^{e} } \right] = \left[ {K_{NL13} } \right] + \left[ {K_{13} } \right] + \left[ {K_{15} } \right];\left[ {K_{T10}^{e} } \right] = \left[ {K_{14} } \right] + \left[ {K_{16} } \right];\left[ {K_{T11}^{e} } \right] = \left[ {K_{NL15} } \right] + \left[ {K_{21} } \right] + \left[ {K_{23} } \right];\left[ {K_{T12}^{e} } \right] = \left[ {K_{22} } \right] + \left[ {K_{24} } \right]; \hfill \\ \left[ {K_{T13}^{e} } \right] = \left[ {K_{NL16} } \right] + \left[ {K_{25} } \right] + \left[ {K_{27} } \right];\left[ {K_{T14}^{e} } \right] = \left[ {K_{26} } \right] + \left[ {K_{28} } \right] + \left[ {K_{32} } \right];\left[ {K_{T15}^{e} } \right] = \left[ {K_{T1} } \right] + \left[ {K_{T7} } \right] + \left[ {K_{T13} } \right];\left[ {K_{T16}^{e} } \right] = \left[ {K_{T2} } \right] + \left[ {K_{T8} } \right] + \left[ {K_{T14} } \right];\left[ {K_{T17}^{e} } \right] = \left[ {K_{T5} } \right] + \left[ {K_{T9} } \right] + \left[ {K_{T11} } \right] + \left[ {K_{T29} } \right]; \hfill \\ \left[ {K_{T18}^{e} } \right] = \left[ {K_{T6} } \right] + \left[ {K_{T10} } \right] + \left[ {K_{T12} } \right] + \left[ {K_{30} } \right]; \hfill \\ \left[ {K_{C1}^{{}} } \right] = - \left[ {K_{40} } \right]^{ - 1} \left[ {K_{37} } \right];\left[ {K_{C2}^{{}} } \right] = - \left[ {K_{40} } \right]^{ - 1} \left[ {K_{38} } \right];\left[ {K_{C3}^{{}} } \right] = - \left[ {K_{40} } \right]^{ - 1} \left[ {K_{39} } \right]; \hfill \\ \left[ {K_{C4}^{{}} } \right] = \left[ {K_{33} } \right] - \left[ {K_{36} } \right]\left[ {K_{C1} } \right];\left[ {K_{C5}^{{}} } \right] = \left[ {K_{34} } \right] - \left[ {K_{36} } \right]\left[ {K_{C2} } \right]; \hfill \\ \left[ {K_{C6}^{{}} } \right] = \left[ {K_{35} } \right] - \left[ {K_{36} } \right]\left[ {K_{C3} } \right];\left[ {K_{C7}^{{}} } \right] = - \left[ {K_{C6} } \right]^{ - 1} \left[ {K_{C4} } \right];\left[ {K_{C8}^{{}} } \right] = - \left[ {K_{C6} } \right]^{ - 1} \left[ {K_{C5} } \right]; \hfill \\ \left[ {K_{C9}^{{}} } \right] = \left[ {K_{T17} } \right] - \left[ {K_{4} } \right]\left[ {K_{C1} } \right];\left[ {K_{C10}^{{}} } \right] = \left[ {K_{T18} } \right] - \left[ {K_{4} } \right]\left[ {K_{C2} } \right];\left[ {K_{C11}^{{}} } \right] = \left[ {K_{3} } \right] - \left[ {K_{4} } \right]\left[ {K_{C3} } \right]; \hfill \\ \left[ {K_{C12}^{{}} } \right] = \left[ {K_{C9} } \right] - \left[ {K_{C11} } \right]\left[ {K_{C7} } \right];\left[ {K_{C13}^{{}} } \right] = \left[ {K_{C10} } \right] - \left[ {K_{C11} } \right]\left[ {K_{C8} } \right]; \hfill \\ \left[ {K_{C14}^{{}} } \right] = \left[ {K_{T15} } \right] - \left[ {K_{T4} } \right]\left[ {K_{C1} } \right];\left[ {K_{C15}^{{}} } \right] = \left[ {K_{T16} } \right] - \left[ {K_{T4} } \right]\left[ {K_{C2} } \right];\left[ {K_{C16}^{{}} } \right] = \left[ {K_{T3} } \right] - \left[ {K_{T4} } \right]\left[ {K_{C3} } \right]; \hfill \\ \left[ {K_{C17}^{{}} } \right] = \left[ {K_{C14} } \right] - \left[ {K_{C16} } \right]\left[ {K_{C7} } \right];\left[ {K_{C18}^{{}} } \right] = \left[ {K_{C15} } \right] - \left[ {K_{C16} } \right]\left[ {K_{C8} } \right]; \hfill \\ \left[ {K_{{}}^{*} } \right] = \left[ {K_{C17} } \right] - \left[ {K_{C18} } \right]\left[ {K_{C13} } \right]^{ - 1} \left[ {K_{C12} } \right]; \hfill \\ \left\{ {F_{{}}^{*} } \right\} = \left\{ F \right\} - \left( {\left\{ {F_{tp1} } \right\} + \left\{ {F_{tp2} } \right\} + \left\{ {F_{tpN1} } \right\}} \right)V + \left[ {K_{C18} } \right]\left[ {K_{C13} } \right]\left( {\left\{ {F_{rp1} } \right\} + \left\{ {F_{rp2} } \right\}} \right)V. \hfill \\ \end{gathered}$$

(30)

The various rigidity matrices contributing to the stiffness matrices and force vectors of Eq. (30) are shown as follows:

$$\begin{gathered} \left[ {D_{1} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {\bar{C}_{b}^{h} } \right]} {\text{ }}{\rm d}z;\left[ {D_{2} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {\bar{C}_{b}^{h} } \right]} \left[ {Z_{1} } \right]{\text{ }}{\rm d}z;\left[ {D_{3} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left\{ {\overline{e} _{b}^{h} } \right\}\frac{1}{h}} {\text{ }}{\rm d}z;\left[ {D_{4} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left\{ {\overline{q} _{b}^{h} } \right\}\frac{1}{h}{\text{ }}} {\rm d}z; \hfill \\ \left[ {D_{5} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{1} } \right]^{{\rm T}} \left[ {\bar{C}_{b}^{h} } \right]} {\text{ }}{\rm d}z;\left[ {D_{6} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{1} } \right]^{{\rm T}} \left[ {\bar{C}_{b}^{h} } \right]} \left[ {Z_{1} } \right]{\text{ }}{\rm d}z;\left[ {D_{7} } \right] = \int\limits_{{h1}}^{{h_{2} }} {\left[ {z_{1} } \right]^{{\rm T}} \left\{ {\overline{e} _{b}^{h} } \right\}\frac{1}{h}} {\text{ }}{\rm d}z; \hfill \\ \left[ {D_{8} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{1} } \right]^{{\rm T}} \left\{ {\overline{q} _{b}^{h} } \right\}} \frac{1}{h}{\rm d}z;\left[ {D_{9} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{3} } \right]^{{\rm T}} \left[ {\bar{C}_{s}^{h} } \right]} {\text{ }}{\rm d}z;\left[ {D_{{10}} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {Z_{3} } \right]^{{\rm T}} \left[ {\bar{C}_{s}^{h} } \right]} \left[ {Z_{3} } \right]{\text{ }}{\rm d}z; \hfill \\ \left[ {D_{{11}} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {\bar{C}_{s}^{h} } \right]} {\text{ }}{\rm d}z;\left[ {D_{{12}} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left[ {\bar{C}_{s}^{h} } \right]} \left[ {Z_{3} } \right]{\rm d}z;\left[ {D_{{13}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {\bar{C}_{b}^{p} } \right]} {\text{ }}{\rm d}z;\left[ {D_{{14}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {\bar{C}_{b}^{p} } \right]} \left[ {Z_{2} } \right]{\text{ }}{\rm d}z; \hfill \\ \left[ {D_{{14}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {\bar{C}_{{bs}}^{p} } \right]} {\text{ }}{\rm d}z;\left[ {D_{{15}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {\bar{C}_{{bs}}^{p} } \right]} \left[ {Z_{5} } \right]{\text{ }}{\rm d}z;\left[ {D_{{16}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left\{ {e_{b}^{p} } \right\}\frac{1}{{h_{p} }}} {\text{ }}{\rm d}z;\left[ {D_{{17}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{2} } \right]^{{\rm T}} {\text{ }}\left[ {\bar{C}_{b}^{p} } \right]} {\text{ }}{\rm d}z; \hfill \\ \left[ {D_{{17}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{2} } \right]^{{\rm T}} {\text{ }}\left[ {\bar{C}_{{bs}}^{p} } \right]} {\text{ }}{\rm d}z;\left[ {D_{{18}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{2} } \right]^{{\rm T}} {\text{ }}\left[ {\bar{C}_{b}^{p} } \right]} \left[ {Z_{2} } \right]{\text{ }}{\rm d}z;\left[ {D_{{19}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{2} } \right]^{{\rm T}} {\text{ }}\left[ {\bar{C}_{{bs}}^{p} } \right]} \left[ {Z_{5} } \right]{\text{ }}{\rm d}z; \hfill \\ \left[ {D_{{20}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {Z_{2} } \right]^{{\rm T}} \left\{ {e_{b}^{p} } \right\}\frac{1}{{h_{p} }}} {\text{ }}{\rm d}z;\left[ {D_{{21}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {{\text{ }}\left[ {\bar{C}_{{bs}}^{p} } \right]} {\text{ }}{\rm d}z;\left[ {D_{{22}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {{\text{ }}\left[ {\bar{C}_{{bs}}^{p} } \right]} \left[ {Z_{5} } \right]{\text{ }}{\rm d}z;\left[ {D_{{23}} } \right] = \left[ {D_{{16}} } \right]; \hfill \\ \left[ {D_{{24}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {{\text{ }}\left[ {Z_{5} } \right]^{{\rm T}} \left[ {\bar{C}_{{bs}}^{p} } \right]^{{\rm T}} } {\rm d}z;\left[ {D_{{25}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {{\text{ }}\left[ {Z_{5} } \right]^{{\rm T}} \left[ {\bar{C}_{{bs}}^{p} } \right]^{{\rm T}} } \left[ {Z_{2} } \right]{\rm d}z;\left[ {D_{{25\_s}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {{\text{ }}\left[ {Z_{5} } \right]^{{\rm T}} \left[ {\bar{C}_{s}^{p} } \right]} {\text{ }}{\rm d}z; \hfill \\ \left[ {D_{{26}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {{\text{ }}\left[ {Z_{5} } \right]^{{\rm T}} \left[ {\bar{C}_{s}^{p} } \right]^{{\rm T}} } \left[ {Z_{5} } \right]{\rm d}z;\left[ {D_{{27}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {\left[ {z_{5} } \right]^{{\rm T}} \left\{ {e_{s}^{p} } \right\}\frac{1}{{h_{p} }}} {\text{ }}{\rm d}z; \hfill \\ \left[ {D_{{28}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {{\text{ }}\left[ {\bar{C}_{{bs}}^{p} } \right]^{{\rm T}} } \left[ {Z_{2} } \right]{\rm d}z;\left[ {D_{{29}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {{\text{ }}\left[ {\bar{C}_{s}^{p} } \right]} {\text{ }}{\rm d}z;\left[ {D_{{30}} } \right] = \int\limits_{{h_{3} }}^{{h_{4} }} {{\text{ }}\left[ {\bar{C}_{s}^{p} } \right]} \left[ {Z_{5} } \right]{\rm d}z; \hfill \\ \left[ {D_{{31}} } \right] = \int\limits_{{h_{2} }}^{{h_{3} }} {{\text{ }}\left[ {Z_{4} } \right]^{{\rm T}} \left[ {\bar{C}_{s}^{v} } \right]{\text{ }}} {\rm d}z;\left[ {D_{{32}} } \right] = \int\limits_{{h_{2} }}^{{h_{3} }} {{\text{ }}\left[ {Z_{4} } \right]^{{\rm T}} \left[ {\bar{C}_{s}^{v} } \right]\left[ {Z_{4} } \right]{\text{ }}} {\rm d}z;\left[ {D_{{33}} } \right] = \int\limits_{{h_{2} }}^{{h_{3} }} {{\text{ }}\left[ {\bar{C}_{s}^{v} } \right]{\text{ }}} {\rm d}z; \hfill \\ \left[ {D_{{34}} } \right] = \int\limits_{{h_{2} }}^{{h_{3} }} {{\text{ }}\left[ {\bar{C}_{s}^{v} } \right]\left[ {Z_{4} } \right]{\text{ }}} {\rm d}z;\left[ {D_{{35}} } \right] = \left[ {D_{3} } \right]^{{\rm T}} ;\left[ {D_{{36}} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left\{ {e_{b}^{h} } \right\}\frac{1}{h}\left[ {Z_{1} } \right]} {\text{ }}{\rm d}z;\left[ {D_{{37}} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\frac{{\overline{ \in } _{{33}}^{h} }}{h}} {\text{ }}{\rm d}z; \hfill \\ \left[ {D_{{38}} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\frac{{\overline{d} _{{33}}^{h} }}{h}} {\text{ }}{\rm d}z;\left[ {D_{{39}} } \right] = \left[ {D_{4} } \right]^{{\rm T}} ;\left[ {D_{{40}} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\left\{ {q_{b}^{h} } \right\}\frac{1}{h}\left[ {Z_{1} } \right]} {\text{ }}{\rm d}z;\left[ {D_{{41}} } \right] = \left[ {D_{{38}} } \right]; \hfill \\ \left[ {D_{{42}} } \right] = \int\limits_{{h_{1} }}^{{h_{2} }} {\frac{{\overline{\mu } _{{33}}^{h} }}{h}{\text{ }}} {\rm d}z. \hfill \\ \end{gathered}$$

(31)