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Large deflection analysis of functionally graded magneto-electro-elastic porous flat panels

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Abstract

In this article, an attempt has been made on evaluating the large/nonlinear deflection of functionally graded magneto-electro-elastic porous (FG-MEEP) flat panels taking geometric skewness into consideration. Further, the flat panel is subjected to combined loads which include mechanical, electrical and magnetic loads. The mathematical formulation is derived through higher order shear deformation theory and von-Karman's geometric nonlinearity under the framework of finite element method (FEM). The effective material properties of FG-MEEP material are determined using modified power law. Two forms of material gradation such as ‘B’ rich bottom and ‘F’ rich bottom are modelled and implemented in the analysis. The numerical assessment is carried out to investigate the effect of prominent parameters such as skew angle, porosity distribution, gradient index, porosity volume, functionally graded pattern, electromagnetic loads on the nonlinear deflection of FG-MEEP flat panels. In addition, this study also makes an attempt to evaluate the degree of coupling associated with these parameters.

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Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.

Abbreviations

a, b, h :

Length, width and thickness of FG-MEEP flat panel

[\(\overline{C}^{{\rm fg}}\)(z)]:

Elastic stiffness coefficients of FG-MEEP composite

[\(\overline{e}^{{\rm fg}}\)(z)]:

Piezoelectric coefficients of FG-MEEP composite

[\(q^{{\rm fg}}\)(z)]:

Magnetostrictive coefficients of effective FG-MEEP composite

[\(\overline{m}^{{\rm fg}}\)(z)]:

Electromagnetic coefficients of effective FG-MEEP composite

[\(\overline{\eta }^{{\rm fg}}\)(z)]:

Dielectric coefficients of effective FG-MEEP composite

[\(\overline{\mu }^{{\rm fg}}\)(z)]:

Magnetic permeability coefficients of effective FG-MEEP composite

\(\left\{ {\varepsilon^{fg} } \right\}\) :

Strain tensor

\(\left\{ E \right\}\) :

Electric field vector

\(\left\{ H \right\}\) :

Magnetic field vector

\(\left\{ {\sigma^{{\rm fg}} \left( z \right)} \right\}\) :

Stress tensor

\(\left\{ {D^{{\rm fg}} \left( z \right)} \right\}\) :

Electric displacement vector

\(\left\{ {B^{{\rm fg}} \left( z \right)} \right\}\) :

Magnetic flux vector

u 0, v 0, and w 0 :

Midplane displacement along x-, y- and z-axes

\(\theta_{x} , \theta_{y}\) :

Normal transverse rotation about x- and y-axes

\(\left\{ {d_{{\rm t}} } \right\}\) :

Translational displacement

\(\left\{ {d_{{\rm r}} } \right\}\) :

Rotational displacement

\(\left\{ {d_{{\rm r}*} } \right\}\) :

Higher-order rotational displacement

\(\psi\) :

Magnetic potential

\(\phi\) :

Electric potential

λ :

Skew angle of the flat panel

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Acknowledgements

The author acknowledges the support of Indian Institute of Science, Bangalore, through C.V. Raman Post-doctoral fellowship R(IA)/CVR-PDF/2019/1630, under Institution of Eminence scheme.

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Authors and Affiliations

  1. Nonlinear Multifunctional Composites Analysis and Design (NMCAD) Laboratory, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, 560012, India

    Vinyas Mahesh & Dineshkumar Harursampath

  2. Composite Research: Experimentation, Analysis and Modelling (CREAM) Lab, Department of Mechanical Engineering, Nitte Meenakshi Institute of Technology, Bangalore, 560064, India

    Vinyas Mahesh

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  1. Vinyas Mahesh
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Correspondence to Vinyas Mahesh.

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Appendix

Appendix

The equivalent stiffness matrix and force vector appearing in Eq. (18) can be explicitly shown as follows:

$$\begin{aligned} & \left[ {K_{{{\text{eq}}}}^{{\text{e}}} } \right] = \left[ {K_{55}^{{\text{e}}} } \right] - \left[ {K_{56}^{{\text{e}}} } \right]\left[ {K_{47}^{{\text{e}}} } \right] \\ & \left\{ {F_{{{\text{eq}}}} } \right\} = \left[ {K_{56} } \right]\left[ {K_{46} } \right]^{ - 1} \left\{ {F{}_{4}} \right\} - \left\{ {F_{7} } \right\}. \\ \end{aligned}$$
(22)

The constituent stiffness matrices of Eq. (22) can be given as follows:

$$\begin{aligned} & \left[ {K_{1}^{{\text{e}}} } \right] = \left[ {K_{{{\text{tbNLbNL}}1\_{\text{tbtbNL}}1}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{ts}}1}}^{{\text{e}}} } \right];\quad \left[ {K_{2}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rbNL}}\_{\text{rtb}}24}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{rts}}13}}^{{\text{e}}} } \right]^{{\text{T}}} \\ & \left[ {K_{3}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rbNL}}\_{\text{rtb}}4}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{rts}}3}}^{{\text{e}}} } \right]^{{\text{T}}} ;\quad \left[ {K_{4}^{{\text{e}}} } \right] = \left[ {K_{{{\text{bNL}}\_{\text{tb}}\phi 1}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{ts}}\phi 1}}^{{\text{e}}} } \right]^{{\text{T}}} \\ & \left[ {K_{5}^{{\text{e}}} } \right] = \left[ {K_{{{\text{bNL}}\_{\text{tb}}\psi 1}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{ts}}\psi 1}}^{{\text{e}}} } \right]^{T} ;\quad \left[ {K_{6}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rtbrbNL}}24}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rts}}13}}^{{\text{e}}} } \right] \\ & \left[ {K_{7}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rrb}}3557}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rrs}}3513}}^{{\text{e}}} } \right];\quad \left[ {K_{8}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rrb}}57}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rrs}}35}}^{{\text{e}}} } \right] \\ & \left[ {K_{9}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rb}}\phi 24}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{r}}\phi {\text{s}}13}}^{{\text{e}}} } \right]^{{\text{T}}} ;\quad \left[ {K_{10}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rb}}\psi 24}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{r}}\psi {\text{s}}13}}^{{\text{e}}} } \right]^{{\text{T}}} \\ & \left[ {K_{11}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rtbrbNL}}4}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rts}}3}}^{{\text{e}}} } \right];\quad \left[ {K_{12}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rrb}}57}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rrs}}35}}^{{\text{e}}} } \right] \\ & \left[ {K_{13}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rrb}}7}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rrs}}5}}^{{\text{e}}} } \right];\quad \left[ {K_{14}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rb}}\phi 4}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{r}}\phi {\text{s}}3}}^{{\text{e}}} } \right]^{{\text{T}}} \\ & \left[ {K_{15}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rb}}\psi 4}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{r}}\psi {\text{s}}3}}^{{\text{e}}} } \right]^{{\text{T}}} ;\quad \left[ {K_{16}^{{\text{e}}} } \right] = \left[ {K_{{{\text{tb}}\phi 1}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{bNL}}\phi 1}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{ts}}\phi 1}}^{{\text{e}}} } \right]^{{\text{T}}} \\ & \left[ {K_{17}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rb}}\phi 2}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{rb}}\phi 4}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{r}}\phi {\text{s}}1}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{r}}\phi {\text{s}}3}}^{{\text{e}}} } \right]^{{\text{T}}} ; \\ & \left[ {K_{18}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rb}}\phi 4}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{r}}\phi {\text{s}}3}}^{{\text{e}}} } \right]^{{\text{T}}} ;\quad \left[ {K_{19}^{{\text{e}}} } \right] = \left[ {K_{{{\text{tb}}\psi 1}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{bN}}\psi 1}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{ts}}\psi 1}}^{{\text{e}}} } \right]^{{\text{T}}} \\ & \left[ {K_{20}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rb}}\psi 2}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{rb}}\psi 4}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{r}}\psi {\text{s}}1}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{r}}\psi {\text{s}}3}}^{{\text{e}}} } \right]^{{\text{T}}} ; \\ \end{aligned}$$
$$\begin{aligned} & \left[ {K_{21}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rb}}\psi 4}}^{{\text{e}}} } \right]^{{\text{T}}} + \left[ {K_{{{\text{r}}\psi {\text{s}}3}}^{{\text{e}}} } \right]^{{\text{T}}} ,\quad \left[ {K_{22}^{{\text{e}}} } \right] = \left[ {K_{16}^{{\text{e}}} } \right] - \left[ {K_{\phi \psi }^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{19}^{{\text{e}}} } \right] \\ & \left[ {K_{23}^{{\text{e}}} } \right] = \left[ {K_{17}^{{\text{e}}} } \right] - \left[ {K_{\phi \psi }^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{20}^{{\text{e}}} } \right],\quad \left[ {K_{24}^{{\text{e}}} } \right] = \left[ {K_{18}^{{\text{e}}} } \right] - \left[ {K_{\phi \psi }^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{21}^{{\text{e}}} } \right], \\ & \left[ {K_{25}^{{\text{e}}} } \right] = \left[ {K_{\phi \phi }^{{\text{e}}} } \right] - \left[ {K_{\phi \psi }^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{\phi \psi }^{{\text{e}}} } \right]^{{\text{T}}} \\ & \left[ {K_{26}^{{\text{e}}} } \right] = \left[ {K_{25}^{{\text{e}}} } \right]^{ - 1} \left[ {K_{22}^{{\text{e}}} } \right]^{{\text{T}}} ;\quad \left[ {K_{27}^{{\text{e}}} } \right] = \left[ {K_{25}^{{\text{e}}} } \right]^{ - 1} \left[ {K_{23}^{{\text{e}}} } \right]^{{\text{T}}} ;\quad \left[ {K_{28}^{{\text{e}}} } \right] = \left[ {K_{25}^{{\text{e}}} } \right]^{ - 1} \left[ {K_{24}^{{\text{e}}} } \right]^{{\text{T}}} \\ & \left[ {K_{29}^{{\text{e}}} } \right] = \left[ {K_{11}^{{\text{e}}} } \right] - \left[ {K_{15}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{19}^{{\text{e}}} } \right];\quad \left[ {K_{30}^{{\text{e}}} } \right] = \left[ {K_{12}^{{\text{e}}} } \right] - \left[ {K_{15}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{20}^{{\text{e}}} } \right] \\ & \left[ {K_{31}^{{\text{e}}} } \right] = \left[ {K_{13}^{{\text{e}}} } \right] - \left[ {K_{15}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{21}^{{\text{e}}} } \right];\left[ {K_{32}^{{\text{e}}} } \right] = \left[ {K_{14}^{{\text{e}}} } \right] - \left[ {K_{15}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{\phi \psi }^{{\text{e}}} } \right]^{{\text{T}}} \\ & \left[ {K_{33}^{{\text{e}}} } \right] = \left[ {K_{29}^{{\text{e}}} } \right] - \left[ {K_{32}^{{\text{e}}} } \right]\left[ {K_{26}^{{\text{e}}} } \right],\quad \left[ {K_{34}^{{\text{e}}} } \right] = \left[ {K_{30}^{{\text{e}}} } \right] - \left[ {K_{32}^{{\text{e}}} } \right]\left[ {K_{27}^{{\text{e}}} } \right] \\ & \left[ {K_{35}^{{\text{e}}} } \right] = \left[ {K_{31}^{{\text{e}}} } \right] - \left[ {K_{32}^{{\text{e}}} } \right]\left[ {K_{28}^{{\text{e}}} } \right],\quad \left[ {K_{36}^{{\text{e}}} } \right] = \left[ {K_{35}^{{\text{e}}} } \right]^{ - 1} \left[ {K_{33}^{{\text{e}}} } \right]^{{\text{T}}} ; \\ & \left[ {K_{37}^{{\text{e}}} } \right] = \left[ {K_{35}^{{\text{e}}} } \right]^{ - 1} \left[ {K_{34}^{{\text{e}}} } \right]^{T} \left[ {K_{38}^{{\text{e}}} } \right] = \left[ {K_{6}^{{\text{e}}} } \right] - \left[ {K_{10}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{19}^{{\text{e}}} } \right]; \\ & \left[ {K_{39}^{{\text{e}}} } \right] = \left[ {K_{7}^{{\text{e}}} } \right] - \left[ {K_{10}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{20}^{{\text{e}}} } \right];\quad \left[ {K_{40}^{{\text{e}}} } \right] = \left[ {K_{8}^{{\text{e}}} } \right] - \left[ {K_{10}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{21}^{{\text{e}}} } \right]; \\ \end{aligned}$$
$$\begin{aligned} & \left[ {K_{41}^{{\text{e}}} } \right] = \left[ {K_{9}^{{\text{e}}} } \right] - \left[ {K_{10}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{\phi \psi }^{{\text{e}}} } \right]^{{\text{T}}} ;\quad \left[ {K_{42}^{{\text{e}}} } \right] = \left[ {K_{38}^{{\text{e}}} } \right] - \left[ {K_{41}^{{\text{e}}} } \right]\left[ {K_{26}^{{\text{e}}} } \right] \\ & \left[ {K_{43}^{{\text{e}}} } \right] = \left[ {K_{39}^{{\text{e}}} } \right] - \left[ {K_{41}^{{\text{e}}} } \right]\left[ {K_{27}^{{\text{e}}} } \right];\quad \left[ {K_{44}^{{\text{e}}} } \right] = \left[ {K_{40}^{{\text{e}}} } \right] - \left[ {K_{41}^{{\text{e}}} } \right]\left[ {K_{28}^{{\text{e}}} } \right] \\ & \left[ {K_{45}^{{\text{e}}} } \right] = \left[ {K_{42}^{{\text{e}}} } \right] - \left[ {K_{44}^{{\text{e}}} } \right]\left[ {K_{36}^{{\text{e}}} } \right];\quad \left[ {K_{46}^{{\text{e}}} } \right] = \left[ {K_{43}^{{\text{e}}} } \right] - \left[ {K_{44}^{{\text{e}}} } \right]\left[ {K_{37}^{{\text{e}}} } \right] \\ & \left[ {K_{47}^{{\text{e}}} } \right] = \left[ {K_{46}^{{\text{e}}} } \right]^{ - 1} \left[ {K_{45}^{{\text{e}}} } \right];\quad \left[ {K_{48}^{{\text{e}}} } \right] = \left[ {K_{1}^{{\text{e}}} } \right] - \left[ {K_{5}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{19}^{{\text{e}}} } \right] \\ & \left[ {K_{49}^{{\text{e}}} } \right] = \left[ {K_{2}^{{\text{e}}} } \right] - \left[ {K_{5}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{20}^{{\text{e}}} } \right];\quad \left[ {K_{50}^{{\text{e}}} } \right] = \left[ {K_{3}^{{\text{e}}} } \right] - \left[ {K_{5}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{21}^{{\text{e}}} } \right] \\ & \left[ {K_{51}^{{\text{e}}} } \right] = \left[ {K_{4}^{{\text{e}}} } \right] - \left[ {K_{5}^{{\text{e}}} } \right]\left[ {K_{\psi \psi }^{{\text{e}}} } \right]^{ - 1} \left[ {K_{\phi \psi }^{{\text{e}}} } \right]^{{\text{T}}} ;\quad \left[ {K_{52}^{{\text{e}}} } \right] = \left[ {K_{48}^{{\text{e}}} } \right] - \left[ {K_{51}^{{\text{e}}} } \right]\left[ {K_{26}^{{\text{e}}} } \right] \\ & \left[ {K_{53}^{{\text{e}}} } \right] = \left[ {K_{49}^{{\text{e}}} } \right] - \left[ {K_{51}^{{\text{e}}} } \right]\left[ {K_{27}^{{\text{e}}} } \right];\quad \left[ {K_{54}^{{\text{e}}} } \right] = \left[ {K_{50}^{{\text{e}}} } \right] - \left[ {K_{51}^{{\text{e}}} } \right]\left[ {K_{28}^{{\text{e}}} } \right] \\ & \left[ {K_{55}^{{\text{e}}} } \right] = \left[ {K_{52}^{{\text{e}}} } \right] - \left[ {K_{54}^{{\text{e}}} } \right]\left[ {K_{36}^{{\text{e}}} } \right];\quad \left[ {K_{56}^{{\text{e}}} } \right] = \left[ {K_{53}^{{\text{e}}} } \right] - \left[ {K_{54}^{{\text{e}}} } \right]\left[ {K_{37}^{{\text{e}}} } \right] \\ \end{aligned}$$
(23)
$$\begin{aligned} & \left[ {K_{{{\text{rrs}}35}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rrs}}3}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rrs}}5}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{rrs}}13}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rrs}}1}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rrs}}3}}^{{\text{e}}} } \right], \\ & \left[ {K_{{{\text{rrs}}3513}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rrs}}35}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rrs}}13}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{rts}}13}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rts}}1}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rts}}3}}^{{\text{e}}} } \right] \\ & \left[ {K_{{{\text{r}}\psi {\text{s}}13}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{r}}\psi {\text{s}}1}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{r}}\psi {\text{s}}3}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{r}}\phi {\text{s}}13}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{r}}\phi {\text{s}}1}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{r}}\phi {\text{s}}3}}^{{\text{e}}} } \right] \\ & \left[ {K_{{{\text{tbtbNL}}1}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{tb}}1}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{tbNL}}1}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{rtb}}24}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rtb}}2}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rtb}}4}}^{{\text{e}}} } \right] \\ & \left[ {K_{{{\text{tbNLbNL}}1\_{\text{tbtbNL}}1}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{tbNLbNL}}1}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{tbtbNL}}1}}^{{\text{e}}} } \right], \\ & \left[ {K_{{{\text{rbNL}}\_{\text{rtb}}24}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rbNL}}24}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rtb}}24}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{rbNL}}\_{\text{rtb}}4}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rbNL}}4}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rtb}}4}}^{{\text{e}}} } \right] \\ & \left[ {K_{{{\text{bNL}}\_{\text{tb}}\phi 1}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{bNL}}\phi 1}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{tb}}\phi 1}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{bNL}}\_{\text{tb}}\psi 1}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{bNL}}\psi 1}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{tb}}\psi 1}}^{{\text{e}}} } \right] \\ & \left[ {K_{{{\text{rrb}}57}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rrb}}5}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rrb}}7}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{rtbrbNL}}4}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rtb}}4}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rbNL}}4}}^{{\text{e}}} } \right] \\ & \left[ {K_{{{\text{rtbrbNL}}2}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rtb}}2}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rbNL}}2}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{rtbrbNL}}24}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rtbrbNL}}2}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rtbrbNL}}4}}^{{\text{e}}} } \right] \\ & \left[ {K_{{{\text{rrb}}35}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rrb}}3}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rrb}}5}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{rrb}}5735}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rrb}}57}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rrb}}35}}^{{\text{e}}} } \right] \\ & \left[ {K_{{{\text{rb}}\phi 24}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rb}}\phi 2}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rb}}\phi 4}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{rb}}\psi 24}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rb}}\psi 2}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rb}}\psi 4}}^{{\text{e}}} } \right] \\ & \left[ {K_{{{\text{tbNLbNL}}1}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{tbNL}}1}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{bNL}}1}}^{{\text{e}}} } \right],\quad \left[ {K_{{{\text{rbNL}}24}}^{{\text{e}}} } \right] = \left[ {K_{{{\text{rbNL}}2}}^{{\text{e}}} } \right] + \left[ {K_{{{\text{rbNL}}4}}^{{\text{e}}} } \right]. \\ \end{aligned}$$
(24)

The force vectors and stiffness matrices contributing to \(\left\{{F}_{\mathrm{eq}}\right\}\) can be shown as follows:

$$\begin{aligned} & \left\{ {F{}_{1}} \right\} = - \left[ { - \left\{ {F_{\phi } } \right\} + \left[ {K_{\phi \psi } } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left\{ {F_{\psi } } \right\}} \right] \\ & \left\{ {F_{2} } \right\} = \left[ {K_{32} } \right]\left[ {K_{25} } \right]^{ - 1} \left\{ {F{}_{1}} \right\} + \left[ {K_{15} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left\{ {F{}_{\psi }} \right\} \\ & \left\{ {F_{3} } \right\} = \left[ {K_{41} } \right]\left[ {K_{25} } \right]^{ - 1} \left\{ {F{}_{1}} \right\} + \left[ {K_{10} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left\{ {F{}_{\psi }} \right\} \\ & \left\{ {F_{4} } \right\} = \left\{ {F_{3} } \right\} - \left[ {K_{44} } \right]\left[ {K_{35} } \right]^{ - 1} \left\{ {F{}_{2}} \right\} \\ & \left\{ {F_{5} } \right\} = \left\{ {F_{m} } \right\} - \left[ {K_{5} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left\{ {F{}_{\psi }} \right\} \\ & \left\{ {F_{6} } \right\} = \left[ {K_{51} } \right]\left[ {K_{25} } \right]^{ - 1} \left\{ {F{}_{1}} \right\} - \left\{ {F_{5} } \right\} \\ & \left\{ {F_{7} } \right\} = \left\{ {F_{6} } \right\} - \left[ {K_{54} } \right]\left[ {K_{35} } \right]^{ - 1} \left\{ {F{}_{2}} \right\}. \\ \end{aligned}$$
(25)

The expressions for stiffness matrices and force vectors are as follows:

$$\begin{aligned} & \left[ {K_{{{\text{rtb}}4}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}4}} } \right]} } \left[ {B_{{{\text{tb}}}} } \right]{\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{rbNL}}4}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{bNL}}4}} } \right]} } \left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{rrb}}5}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}5}} } \right]} } \left[ {B_{{{\text{rb}}}} } \right]{\text{d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{rrb}}7}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}7}} } \right]} } \left[ {B_{{{\text{rb}}}} } \right]{\text{d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{rb}}\phi 4}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}\phi 4}} } \right]} } \left[ {B_{\phi } } \right]{\text{d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{rb}}\psi 4}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}\psi 4}} } \right]} } \left[ {B_{\psi } } \right]\,{\text{d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{rt}}b4}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}4}} } \right]} } \left[ {B_{{{\text{tb}}}} } \right]\,{\text{d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{rtb}}2}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}2}} } \right]} } \left[ {B_{{{\text{tb}}}} } \right]\,{\text{d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{rbNL}}2}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{bNL}}2}} } \right]} } \left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{rrb}}3}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}3}} } \right]} } \left[ {B_{{{\text{rb}}}} } \right]{\text{ d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{rrb}}5}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}5}} } \right]} } \left[ {B_{{{\text{rb}}}} } \right]{\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{rb}}\phi 2}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}\phi 2}} } \right]} } \left[ {B_{\phi } } \right]{\text{d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{rb}}\psi 2}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}\psi 2}} } \right]} } \left[ {B_{\psi } } \right]{\text{d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{tbNL}}1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{tb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{bNL}}1}} } \right]} } \left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{ d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{bNL}}1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{2} } \right]^{{\text{T}}} \left[ {B_{1} } \right]^{{\text{T}}} \left[ {D_{{{\text{bbNL}}1}} } \right]} } \left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{ d}}x\,{\text{d}}y; \\ & \left[ {K_{{{\text{bNL}}\phi 1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\phi } } \right]^{{\text{T}}} \left[ {D_{{{\text{bNL}}\phi 1}} } \right]} } \left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{bNL}}\psi 1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\psi } } \right]^{{\text{T}}} \left[ {D_{{{\text{bNL}}\psi 1}} } \right]} } \left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{tb}}1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{tb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}1}} } \right]} } \left[ {B_{{{\text{tb}}}} } \right]{\text{d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{tb}}\phi 1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{tb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}\phi 1}} } \right]} } \left[ {B_{\phi } } \right]{\text{d}}x\,{\text{d}}y \\ \end{aligned}$$
$$\begin{aligned} & \left[ {K_{{{\text{tb}}\psi 1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{tb}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{b}}\psi 1}} } \right]} } \left[ {B_{\psi } } \right]{\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{rts}}3}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rs}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}3}} } \right]\left[ {B_{{{\text{ts}}}} } \right]} } {\text{ d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{rrs}}3}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rs}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}3}} } \right]\left[ {B_{{{\text{rs}}}} } \right]} } {\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{rrs}}5}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rs}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}5}} } \right]\left[ {B_{{{\text{rs}}}} } \right]} } {\text{ d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{r}}\psi {\text{s}}3}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rs}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}\psi 3}} } \right]\left[ {B_{\psi } } \right]} } {\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{r}}\phi {\text{s}}3}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rs}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}\phi 3}} } \right]\left[ {B_{\phi } } \right]} } {\text{ d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{ts}}\phi 1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{ts}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}\phi 1}} } \right]} } \left[ {B_{\phi } } \right]{\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{ts}}\psi 1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{ts}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}\psi 1}} } \right]} } \left[ {B_{\psi } } \right]{\text{ d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{r}}\phi {\text{s}}1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rs}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}\phi 1}} } \right]\left[ {B_{\phi } } \right]} } {\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{r}}\psi {\text{s}}1}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rs}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}\psi 1}} } \right]\left[ {B_{\psi } } \right]} } {\text{ d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{rts}}1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rs}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}1}} } \right]\left[ {B_{{{\text{ts}}}} } \right]} } {\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{rrs}}1}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rs}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}1}} } \right]\left[ {B_{{{\text{rs}}}} } \right]} } {\text{ d}}x\,{\text{d}}y \\ & \left[ {K_{{{\text{ts}}1}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{ts}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}1}} } \right]\left[ {B_{{{\text{ts}}}} } \right]} } {\text{ d}}x\,{\text{d}}y;\quad \left[ {K_{{{\text{rts}}3}}^{{\text{e}}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{{{\text{rs}}}} } \right]^{{\text{T}}} \left[ {D_{{{\text{s}}3}} } \right]\left[ {B_{{{\text{ts}}}} } \right]} } {\text{ d}}x\,{\text{d}}y \\ & \left\{ {F_{\psi } } \right\} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {N_{\psi } } \right]^{{\text{T}}} Q^{\psi } } } {\text{ d}}x\,{\text{d}}y;\left\{ {F_{\phi } } \right\} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {N_{\phi } } \right]^{{\text{T}}} Q^{\phi } } } {\text{ d}}x\,{\text{d}}y; \\ & \left\{ {F_{m} } \right\} = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {N_{{\text{t}}} } \right]^{{\text{T}}} \left[ {\begin{array}{*{20}c} {p_{x} } & {p_{y} } & {p_{z} } \\ \end{array} } \right]} } {\text{ d}}x\,{\text{d}}y. \\ \end{aligned}$$
(26)

The various rigidity matrices contributing to Eq. (26) can be denoted as follows:

$$\begin{aligned} & \left[ {D_{{{\text{b}}1}} } \right] = \int\limits_{ - h/2}^{h/2} {\left[ {\overline{{C_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{bbNL}}1}} } \right] = \frac{1}{4}\left[ {D_{{{\text{b}}1}} } \right];\quad \left[ {D_{{{\text{bNL}}1}} } \right] = \frac{1}{2}\left[ {D_{{{\text{b}}1}} } \right];\quad \left[ {D_{{{\text{b}}2}} } \right] = \int\limits_{ - h/2}^{h/2} {z\left[ {\overline{{C_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z \\ & \left[ {D_{{{\text{bNL}}2}} } \right] = \frac{1}{2}\left[ {D_{{{\text{b}}2}} } \right];\quad \left[ {D_{{{\text{b}}3}} } \right] = \int\limits_{ - h/2}^{h/2} {z^{2} \left[ {\overline{{C_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{b}}4}} } \right] = \int\limits_{ - h/2}^{h/2} {c_{1} z^{3} \left[ {\overline{{C_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{bNL}}4}} } \right] = \frac{1}{2}\left[ {D_{{{\text{b}}4}} } \right] \\ & \left[ {D_{{{\text{b}}5}} } \right] = \int\limits_{ - h/2}^{h/2} {c_{1} z^{4} \left[ {\overline{{C_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{b}}7}} } \right] = \int\limits_{ - h/2}^{h/2} {c_{1}^{2} z^{6} \left[ {\overline{{C_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{b}}\phi 1}} } \right] = \int\limits_{ - h/2}^{h/2} {\left[ {\overline{{e_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z; \\ & \left[ {D_{{{\text{bNL}}\phi 1}} } \right] = \left[ {D_{{{\text{b}}\phi 1}} } \right];\quad \left[ {D_{{{\text{b}}\phi 2}} } \right] = \int\limits_{ - h/2}^{h/2} {z\left[ {\overline{{e_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{b}}\phi 4}} } \right] = \int\limits_{ - h/2}^{h/2} {c_{1} z^{3} \left[ {\overline{{e_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z; \\ & \left[ {D_{{{\text{b}}\psi 1}} } \right] = \int\limits_{ - h/2}^{h/2} {\left[ {\overline{{q_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{bNL}}\psi 1}} } \right] = \left[ {D_{{{\text{b}}\psi 1}} } \right]; \\ & \left[ {D_{{{\text{b}}\psi 2}} } \right] = \int\limits_{ - h/2}^{h/2} {z\left[ {\overline{{q_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\left[ {D_{{{\text{s}}1}} } \right] = \int\limits_{ - h/2}^{h/2} {\left[ {\overline{{C_{{\text{s}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{b}}\psi 4}} } \right] = \int\limits_{ - h/2}^{h/2} {c_{1} z^{3} \left[ {\overline{{q_{{\text{b}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z; \\ & \left[ {D_{{{\text{s}}3}} } \right] = \int\limits_{ - h/2}^{h/2} {c_{2} z^{2} \left[ {\overline{{C_{{\text{s}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{s}}5}} } \right] = \int\limits_{ - h/2}^{h/2} {c_{2}^{2} z^{4} \left[ {\overline{{C_{{\text{s}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{s}}\phi 1}} } \right] = \int\limits_{ - h/2}^{h/2} {\left[ {\overline{{e_{{\text{s}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z; \\ & \left[ {D_{{{\text{s}}\phi 3}} } \right] = \int\limits_{ - h/2}^{h/2} {c_{2} z^{2} \left[ {\overline{{e_{{\text{s}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{s}}\psi 1}} } \right] = \int\limits_{ - h/2}^{h/2} {\left[ {\overline{{q_{{\text{s}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{{{\text{s}}\psi 3}} } \right] = \int\limits_{ - h/2}^{h/2} {c_{2} z^{2} \left[ {\overline{{q_{{\text{s}}} }}^{{{\text{fg}}}} } \right]} {\text{d}}z; \\ & \left[ {D_{\phi \phi } } \right] = \int\limits_{ - h/2}^{h/2} {\left[ {\overline{\eta }^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{\psi \psi } } \right] = \int\limits_{ - h/2}^{h/2} {\left[ {\overline{\mu }^{{{\text{fg}}}} } \right]} {\text{d}}z;\quad \left[ {D_{\phi \psi } } \right] = \int\limits_{ - h/2}^{h/2} {\left[ {\overline{m}^{{{\text{fg}}}} } \right]} {\text{d}}z. \\ \end{aligned}$$
(27)

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Mahesh, V., Harursampath, D. Large deflection analysis of functionally graded magneto-electro-elastic porous flat panels. Engineering with Computers 38 (Suppl 2), 1615–1634 (2022). https://doi.org/10.1007/s00366-020-01270-x

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