Appendix
Grouping the terms based on the degrees of freedom and assigning stiffness matrices, Eq. (15) can be expressed as follows:
$$ \delta \left\{ {d_{t} } \right\}^{\text{T}} \left[ \begin{aligned} & \left\{ {\left[ {K_{{{\text{tbNLBNL1}}\_{\text{tbtbNL1}}}} } \right] + \left[ {K_{\text{ts1}} } \right]} \right\}\left\{ {d_{\text{t}} } \right\} + \left\{ {\left[ {K_{{{\text{rbNl\_rtb24}}}} } \right]^{\text{T}} + \left[ {K_{\text{rts13}} } \right]^{\text{T}} } \right\}\left\{ {d_{\text{r}} } \right\} \hfill \\ & + \left\{ {\left[ {K_{{{\text{rbNL\_rtb4}}}} } \right] + \left[ {K_{\text{rts3}} } \right]} \right\}\left\{ {d_{\text{r*}} } \right\} + \left\{ {\left[ {K_{{{\text{bNL\_tb}}\phi 1}} } \right] + \left[ {K_{{{\text{ts}}\phi 1}} } \right]} \right\}\left\{ \phi \right\} \hfill \\ & + \left\{ {\left[ {K_{{{\text{bNL\_tb}}\psi 1}} } \right] + \left[ {K_{{{\text{ts}}\psi 1}} } \right]} \right\}\left\{ \psi \right\} \hfill \\ \end{aligned} \right] + $$
$$ \delta \left\{ {d_{\text{r}} } \right\}^{\text{T}} \left[ \begin{aligned} & \left\{ {\left[ {K_{\text{rtbrbNL24}} } \right] + \left[ {K_{\text{rts13}} } \right]} \right\}\left\{ {d_{\text{t}} } \right\} + \left\{ {\left[ {K_{\text{rrb3557}} } \right] + \left[ {K_{\text{rrs3513}} } \right]} \right\}\left\{ {d_{\text{r}} } \right\} \hfill \\ & + \left\{ {\left[ {K_{\text{rrb57}} } \right] + \left[ {K_{\text{rrs35}} } \right]} \right\}\left\{ {d_{\text{r*}} } \right\} + \left\{ {\left[ {K_{\text{rbf24}} } \right] + \left[ {K_{\text{rsf13}} } \right]} \right\}\left\{ \phi \right\} \hfill \\ & + \left\{ {\left[ {K_{{{\text{rb}}\psi 24}} } \right] + \left[ {K_{{{\text{r}}\psi 1 3}} } \right]} \right\}\left\{ \psi \right\} \hfill \\ \end{aligned} \right] + $$
$$ \delta \left\{ {d_{{{\text{r}}*}} } \right\}^{\text{T}} \left[ \begin{aligned} & \left\{ {\left[ {K_{\text{rtbrbNL4}} } \right] + \left[ {K_{\text{rts3}} } \right]} \right\}\left\{ {d_{\text{t}} } \right\} + \left\{ {\left[ {K_{\text{rrb57}} } \right] + \left[ {K_{\text{rrs35}} } \right]} \right\}\left\{ {d_{\text{r}} } \right\} \hfill \\ & + \left\{ {\left[ {K_{\text{rrb7}} } \right] + \left[ {K_{\text{rrs5}} } \right]} \right\}\left\{ {d_{\text{r*}} } \right\} + \left\{ {\left[ {K_{{{\text{rb}}\phi 4}} } \right] + \left[ {K_{{{\text{rs}}\phi 3}} } \right]} \right\}\left\{ \phi \right\} \hfill \\ & + \left\{ {\left[ {K_{{{\text{rb}}\psi 4}} } \right] + \left[ {K_{{{\text{r}}\psi {\text{s}}3}} } \right]} \right\}\left\{ \psi \right\} \hfill \\ \end{aligned} \right] + $$
$$ \delta \left\{ \phi \right\}^{\text{T}} \left[ \begin{aligned} & \left( {\left[ {K_{{{\text{tb}}\phi 1}} } \right]^{\text{T}} + \left[ {K_{{{\text{bNL}}\phi 1}} } \right]^{\text{T}} + \left[ {K_{{{\text{ts}}\phi 1}} } \right]^{\text{T}} } \right)\left\{ {d_{\text{t}} } \right\} \hfill \\ & + \left( {\left[ {K_{{{\text{rb}}\phi 2}} } \right]^{\text{T}} + \left[ {K_{{{\text{rb}}\phi 4}} } \right]^{\text{T}} + \left[ {K_{{{\text{rs}}\phi 1}} } \right]^{\text{T}} + \left[ {K_{{{\text{rs}}\phi 3}} } \right]^{\text{T}} } \right)\left\{ {d_{\text{r}} } \right\} \hfill \\ & + \left( {\left[ {K_{{{\text{rb}}\phi 4}} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\phi {\text{s}}3}} } \right]^{\text{T}} } \right)\left\{ {d_{\text{r*}} } \right\} + \left[ {K_{\phi \phi } } \right]^{\text{T}} \left\{ \phi \right\} + \left[ {K_{\phi \psi } } \right]^{\text{T}} \left\{ \psi \right\} \hfill \\ \end{aligned} \right] + $$
$$ \delta \left\{ \psi \right\}^{\text{T}} \left[ \begin{aligned} & \left( {\left[ {K_{{{\text{tb}}\psi 1}} } \right]^{\text{T}} + \left[ {K_{{{\text{bNL}}\psi 1}} } \right]^{\text{T}} + \left[ {K_{{{\text{ts}}\psi 1}} } \right]^{\text{T}} } \right)\left\{ {d_{\text{t}} } \right\} \hfill \\ & + \left( {\left[ {K_{{{\text{rb}}\psi 2}} } \right]^{\text{T}} + \left[ {K_{{{\text{rb}}\psi 4}} } \right]^{\text{T}} + \left[ {K_{{{\text{rs}}\psi 1}} } \right]^{\text{T}} + \left[ {K_{{{\text{rs}}\psi 3}} } \right]^{\text{T}} } \right)\left\{ {d_{\text{r}} } \right\} \hfill \\ & + \left( {\left[ {K_{{{\text{rb}}\psi 4}} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\psi {\text{s}}3}} } \right]^{\text{T}} } \right)\left\{ {d_{{{\text{r}}*}} } \right\} + \left[ {K_{\phi \psi } } \right]^{\text{T}} \left\{ \phi \right\} + \left[ {K_{\psi \psi } } \right]^{\text{T}} \left\{ \psi \right\} \hfill \\ \end{aligned} \right] + $$
$$ \delta \left\{ {d_{\text{t}} } \right\}^{\text{T}} \left[ {M_{\text{tt}} } \right]\left\{ {\ddot{d}_{\text{t}} } \right\} = 0. $$
(A-1)
The equations of motion can be obtained from Eq. (A-1) by separating out the coefficients of various degrees of freedom as follows:
$$ \left[ {M_{\text{tt}} } \right]\left\{ {\ddot{d}_{\text{t}} } \right\} + \left[ {K_{1} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{2} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{3} } \right]\left\{ {d_{\text{r*}} } \right\} + \left[ {K_{4} } \right]\left\{ \phi \right\} + \left[ {K_{5} } \right]\left\{ \psi \right\} = 0 $$
(A-2)
$$ \left[ {K_{6} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{7} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{8} } \right]\left\{ {d_{{{\text{r}}*}} } \right\} + \left[ {K_{9} } \right]\left\{ \phi \right\} + \left[ {K_{10} } \right]\left\{ \psi \right\} = 0 $$
(A-3)
$$ \left[ {K_{11} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{12} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{13} } \right]\left\{ {d_{\text{r*}} } \right\} + \left[ {K_{14} } \right]\left\{ \phi \right\} + \left[ {K_{15} } \right]\left\{ \psi \right\} = 0 $$
(A-4)
$$ \left[ {K_{16} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{17} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{18} } \right]\left\{ {d_{{{\text{r}}*}} } \right\} + \left[ {K_{\phi \phi } } \right]\left\{ \phi \right\} + \left[ {K_{\phi \psi } } \right]\left\{ \psi \right\} = 0 $$
(A-5)
$$ \left[ {K_{19} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{20} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{21} } \right]\left\{ {d_{{{\text{r}}*}} } \right\} + \left[ {K_{\psi \phi } } \right]\left\{ \phi \right\} + \left[ {K_{\psi \psi } } \right]\left\{ \psi \right\} = 0. $$
(A-6)
Solving for \( \left\{ \psi \right\} \) using Eq. (A-6), it can be shown that
$$ \left\{ \psi \right\} = - \left[ {K_{\psi \psi } } \right]^{ - 1} \left\{ {\left[ {K_{19} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{20} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{21} } \right]\left\{ {d_{{{\text{r}}*}} } \right\} + \left[ {K_{\phi \psi } } \right]^{\text{T}} \left\{ \phi \right\}} \right\}. $$
(A-7)
On substituting Eq. (A-7) in Eq. (A-5) and solving for \( \left\{ \phi \right\} \) yields
$$ \left[ {K_{16} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{17} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{18} } \right]\left\{ {d_{{{\text{r}}*}} } \right\} + \left[ {K_{\phi \phi } } \right]\left\{ \phi \right\} $$
$$ - \left[ {K_{\phi \psi } } \right]\left[ {\left[ {K_{\psi \psi } } \right]^{ - 1} \left( {\left[ {K_{19} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{20} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{21} } \right]\left\{ {d_{{{\text{r}}*}} } \right\} + \left[ {K_{\phi \psi } } \right]^{\text{T}} \left\{ \phi \right\}} \right)} \right] $$
$$ \left\{ {d_{\text{t}} } \right\}\left( {\left[ {K_{16} } \right] - \left[ {K_{\phi \psi } } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{19} } \right]} \right) + \left\{ {d_{\text{r}} } \right\}\left( {\left[ {K_{17} } \right] - \left[ {K_{\phi \psi } } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{20} } \right]} \right) $$
$$ + \left\{ {d_{{{\text{r}}*}} } \right\}\left( {\left[ {K_{18} } \right] - \left[ {K_{\phi \psi } } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{21} } \right]} \right) + \left\{ \phi \right\}\left( {\left[ {K_{\phi \phi } } \right] - \left[ {K_{\phi \psi } } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{\phi \psi } } \right]^{\text{T}} } \right) $$
$$ \left[ {K_{22} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{23} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{24} } \right]\left\{ {d_{\text{r*}} } \right\} + \left[ {K_{25} } \right]\left\{ \phi \right\} = 0 $$
$$ \left\{ \phi \right\} = - \left[ {K_{25} } \right]^{ - 1} \left\{ {\left[ {K_{22} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{23} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{24} } \right]\left\{ {d_{\text{r*}} } \right\}} \right\} $$
(A-8)
$$ \left\{ \phi \right\} = - \left[ {K_{26} } \right]\left\{ {d_{\text{t}} } \right\} - \left[ {K_{27} } \right]\left\{ {d_{\text{r}} } \right\} - \left[ {K_{28} } \right]\left\{ {d_{\text{r*}} } \right\}. $$
Similarly, substituting Eqs. (A-7) and (A-8) in Eq. (A-4) and solving for \( \left\{ {d_{\text{r*}} } \right\} \), we get
$$ \left[ {K_{11} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{12} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{13} } \right]\left\{ {d_{{{\text{r}}*}} } \right\} + \left[ {K_{14} } \right]\left\{ \phi \right\} $$
$$ - \left[ {K_{15} } \right]\left[ {\left[ {K_{\psi \psi } } \right]^{ - 1} \left( {\left[ {K_{19} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{20} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{21} } \right]\left\{ {d_{\text{r*}} } \right\} + \left[ {K_{\phi \psi } } \right]^{\text{T}} \left\{ \phi \right\}} \right)} \right] $$
$$ \left\{ {d_{\text{t}} } \right\}\left( {\left[ {K_{11} } \right] - \left[ {K_{15} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{19} } \right]} \right) + \left\{ {d_{\text{r}} } \right\}\left( {\left[ {K_{12} } \right] - \left[ {K_{15} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{20} } \right]} \right) $$
$$ + \left\{ {d_{\text{r*}} } \right\}\left( {\left[ {K_{13} } \right] - \left[ {K_{15} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{21} } \right]} \right) + \left\{ \phi \right\}\left( {\left[ {K_{14} } \right] - \left[ {K_{15} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{\phi \psi } } \right]^{\text{T}} } \right) $$
$$ \left[ {K_{29} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{30} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{31} } \right]\left\{ {d_{\text{r*}} } \right\} + \left[ {K_{32} } \right]\left\{ \phi \right\} $$
$$ = \left[ {K_{29} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{30} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{31} } \right]\left\{ {d_{\text{r*}} } \right\} - \left[ {K_{32} } \right]\left[ {K_{26} } \right]\left\{ {d_{\text{t}} } \right\} - \left[ {K_{32} } \right]\left[ {K_{27} } \right]\left\{ {d_{\text{r}} } \right\} - \left[ {K_{32} } \right]\left[ {K_{28} } \right]\left\{ {d_{\text{r*}} } \right\} $$
$$ = \left\{ {d_{\text{t}} } \right\}\left( {\left[ {K_{29} } \right] - \left[ {K_{32} } \right]\left[ {K_{26} } \right]} \right) + \left\{ {d_{\text{r}} } \right\}\left( {\left[ {K_{30} } \right] - \left[ {K_{32} } \right]\left[ {K_{27} } \right]} \right) + \left\{ {d_{\text{r*}} } \right\}\left( {\left[ {K_{31} } \right] - \left[ {K_{32} } \right]\left[ {K_{28} } \right]} \right) $$
$$ = \left[ {K_{33} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{34} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{35} } \right]\left\{ {d_{{{\text{r}}*}} } \right\} $$
$$ \left\{ {d_{\text{r*}} } \right\} = - \left[ {K_{35} } \right]^{ - 1} \left( {\left[ {K_{33} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{34} } \right]\left\{ {d_{\text{r}} } \right\}} \right) $$
$$ \left\{ {d_{{{\text{r}}*}} } \right\} = - \left[ {K_{36} } \right]\left\{ {d_{\text{t}} } \right\} - \left[ {K_{37} } \right]\left\{ {d_{\text{r}} } \right\}. $$
(A-9)
The expression for \( \left\{ {d_{\text{r}} } \right\} \) is obtained using Eqs. (A-7)–(A-9) in Eq. (A-3) as
$$ \left[ {K_{6} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{7} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{8} } \right]\left\{ {d_{\text{r*}} } \right\} + \left[ {K_{9} } \right]\left\{ \phi \right\} $$
$$ - \left[ {K_{10} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {\left[ {K_{19} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{20} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{21} } \right]\left\{ {d_{\text{r*}} } \right\} + \left[ {K_{\phi \psi } } \right]^{\text{T}} \left\{ \phi \right\}} \right] $$
$$ = \left\{ {d_{\text{t}} } \right\}\left( {\left[ {K_{6} } \right] - \left[ {K_{10} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{19} } \right]} \right) + \left\{ {d_{\text{r}} } \right\}\left( {\left[ {K_{7} } \right] - \left[ {K_{10} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{20} } \right]} \right) $$
$$ + \left\{ {d_{\text{r*}} } \right\}\left( {\left[ {K_{8} } \right] - \left[ {K_{10} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{21} } \right]} \right) + \left\{ \phi \right\}\left( {\left[ {K_{9} } \right] - \left[ {K_{10} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{\phi \psi } } \right]^{\text{T}} } \right) $$
$$ = \left[ {K_{38} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{39} } \right]\left\{ {d_{\text{r}} } \right\} - \left[ {K_{40} } \right]\left\{ {d_{\text{r*}} } \right\} - \left[ {K_{41} } \right]\left[ {K_{26} } \right]\left\{ {d_{\text{t}} } \right\} - \left[ {K_{41} } \right]\left[ {K_{27} } \right]\left\{ {d_{\text{r}} } \right\} - \left[ {K_{41} } \right]\left[ {K_{28} } \right]\left\{ {d_{\text{r*}} } \right\} $$
$$ = \left\{ {d_{\text{t}} } \right\}\left( {\left[ {K_{38} } \right] - \left[ {K_{41} } \right]\left[ {K_{26} } \right]} \right) + \left\{ {d_{\text{r}} } \right\}\left( {\left[ {K_{39} } \right] - \left[ {K_{41} } \right]\left[ {K_{27} } \right]} \right) + \left\{ {d_{{{\text{r}}*}} } \right\}\left( {\left[ {K_{40} } \right] - \left[ {K_{41} } \right]\left[ {K_{28} } \right]} \right) $$
$$ \left[ {K_{42} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{43} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{44} } \right]\left\{ {d_{{{\text{r}}*}} } \right\} = 0 $$
$$ = \left[ {K_{42} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{43} } \right]\left\{ {d_{\text{r}} } \right\} - \left[ {K_{44} } \right]\left[ {K_{36} } \right]\left\{ {d_{\text{t}} } \right\} - \left[ {K_{44} } \right]\left[ {K_{37} } \right]\left\{ {d_{\text{r}} } \right\} $$
$$ \left\{ {d_{\text{t}} } \right\}\left( {\left[ {K_{42} } \right] - \left[ {K_{44} } \right]\left[ {K_{36} } \right]} \right) + \left\{ {d_{\text{r}} } \right\}\left( {\left[ {K_{43} } \right] - \left[ {K_{44} } \right]\left[ {K_{37} } \right]} \right) $$
$$ \left[ {K_{45} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{46} } \right]\left\{ {d_{\text{r}} } \right\} = 0 $$
$$ \left\{ {d_{\text{r}} } \right\} = - \left[ {K_{46} } \right]^{ - 1} \left[ {\left[ {K_{45} } \right]\left\{ {d_{\text{t}} } \right\}} \right] $$
$$ \left\{ {d_{\text{r}} } \right\} = - \left[ {K_{47} } \right]\left\{ {d_{\text{t}} } \right\}. $$
(A-10)
Finally, \( \left[ {K_{\text{eq}} } \right] \) is obtained by making use of Eqs. (A-7)–(A-10) in Eq. (A-2) and simplifying as follows:
$$ \left[ {K_{1} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{2} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{3} } \right]\left\{ {d_{\text{r*}} } \right\} + \left[ {K_{4} } \right]\left\{ \phi \right\} $$
$$ - \left[ {K_{5} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {\left[ {K_{19} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{20} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{21} } \right]\left\{ {d_{\text{r*}} } \right\} + \left[ {K_{\phi \psi } } \right]^{\text{T}} \left\{ \phi \right\}} \right] $$
$$ = \left\{ {d_{\text{t}} } \right\}\left( {\left[ {K_{1} } \right] - \left[ {K_{5} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{19} } \right]} \right) + \left\{ {d_{\text{r}} } \right\}\left( {\left[ {K_{2} } \right] - \left[ {K_{5} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{20} } \right]} \right) $$
$$ + \left\{ {d_{\text{r*}} } \right\}\left( {\left[ {K_{3} } \right] - \left[ {K_{5} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{21} } \right]} \right) + \left\{ \phi \right\}\left( {\left[ {K_{4} } \right] - \left[ {K_{5} } \right]\left[ {K_{\psi \psi } } \right]^{ - 1} \left[ {K_{\phi \psi } } \right]^{\text{T}} } \right) $$
$$ = \left[ {K_{48} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{49} } \right]\left\{ {d_{\text{r}} } \right\} + \left[ {K_{50} } \right]\left\{ {d_{{{\text{r}}*}} } \right\} - \left[ {K_{51} } \right]\left[ {K_{26} } \right]\left\{ {d_{\text{t}} } \right\} - \left[ {K_{51} } \right]\left[ {K_{27} } \right]\left\{ {d_{\text{r}} } \right\} - \left[ {K_{51} } \right]\left[ {K_{28} } \right]\left\{ {d_{\text{r*}} } \right\} $$
$$ = \left\{ {d_{\text{t}} } \right\}\left( {\left[ {K_{48} } \right] - \left[ {K_{51} } \right]\left[ {K_{26} } \right]} \right) + \left\{ {d_{\text{r}} } \right\}\left( {\left[ {K_{49} } \right] - \left[ {K_{51} } \right]\left[ {K_{27} } \right]} \right) + \left\{ {d_{\text{r*}} } \right\}\left( {\left[ {K_{50} } \right] - \left[ {K_{51} } \right]\left[ {K_{28} } \right]} \right) $$
$$ = \left[ {K_{52} } \right]\left\{ {d_{\text{t}} } \right\} + \left[ {K_{53} } \right]\left\{ {d_{\text{r}} } \right\} - \left[ {K_{54} } \right]\left[ {K_{36} } \right]\left\{ {d_{\text{t}} } \right\} - \left[ {K_{54} } \right]\left[ {K_{37} } \right]\left\{ {d_{\text{r}} } \right\} $$
$$ = \left\{ {d_{\text{t}} } \right\}\left( {\left[ {K_{52} } \right] - \left[ {K_{54} } \right]\left[ {K_{36} } \right]} \right) + \left\{ {d_{\text{r}} } \right\}\left( {\left[ {K_{53} } \right] - \left[ {K_{54} } \right]\left[ {K_{37} } \right]} \right) $$
$$ = \left[ {K_{55} } \right]\left\{ {d_{\text{t}} } \right\} - \left[ {K_{56} } \right]\left[ {K_{47} } \right]\left\{ {d_{\text{t}} } \right\} $$
$$ \left[ {K_{\text{eq}} } \right]\left\{ {d_{\text{t}} } \right\} = 0 $$
$$ \left[ {K_{\text{eq}} } \right] = \left[ {K_{55} } \right] - \left[ {K_{56} } \right]\left[ {K_{47} } \right]. $$
(A-11)
The relation between various stiffness matrices contributing towards \( \left[ {K_{\text{eq}} } \right] \) are expressed as
$$ \left[ {K_{1}^{e} } \right] = \left[ {K_{{{\text{tbNLbNL1\_tbtbNL1}}}}^{e} } \right] + \left[ {K_{\text{ts1}}^{e} } \right] $$
$$ \left[ {K_{2}^{e} } \right] = \left[ {K_{{{\text{rbNL\_rtb}}24}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{rts}}13}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{3}^{e} } \right] = \left[ {K_{{{\text{rbNL\_rtb4}}}}^{e} } \right]^{\text{T}} + \left[ {K_{\text{rts3}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{4}^{e} } \right] = \left[ {K_{{{\text{bNL\_tbf1}}}}^{e} } \right]^{\text{T}} + \left[ {K_{\text{tsf1}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{5}^{e} } \right] = \left[ {K_{{{\text{bNL\_tb}}\psi 1}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{ts}}\psi 1}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{6}^{e} } \right] = \left[ {K_{\text{rtbrbNL24}}^{e} } \right] + \left[ {K_{{{\text{rts}}13}}^{e} } \right] $$
$$ \left[ {K_{7}^{e} } \right] = \left[ {K_{{{\text{rrb}}3557}}^{e} } \right] + \left[ {K_{{{\text{rrs}}3513}}^{e} } \right] $$
$$ \left[ {K_{8}^{e} } \right] = \left[ {K_{{{\text{rrb}}57}}^{e} } \right] + \left[ {K_{{{\text{rrs}}35}}^{e} } \right] $$
$$ \left[ {K_{9}^{e} } \right] = \left[ {K_{{{\text{rb}}\phi 24}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\phi {\text{s}}13}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{10}^{e} } \right] = \left[ {K_{{{\text{rb}}\psi 24}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\psi {\text{s}}13}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{11}^{e} } \right] = \left[ {K_{\text{rtbrbNL4}}^{e} } \right] + \left[ {K_{{{\text{rts}}3}}^{e} } \right] $$
$$ \left[ {K_{12}^{e} } \right] = \left[ {K_{{{\text{rrb}}57}}^{e} } \right] + \left[ {K_{{{\text{rrs}}35}}^{e} } \right] $$
$$ \left[ {K_{13}^{e} } \right] = \left[ {K_{{{\text{rrb}}7}}^{e} } \right] + \left[ {K_{{{\text{rrs}}5}}^{e} } \right] $$
$$ \left[ {K_{14}^{e} } \right] = \left[ {K_{{{\text{rb}}\phi 4}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\phi {\text{s}}3}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{15}^{e} } \right] = \left[ {K_{{{\text{rb}}\psi 4}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\psi {\text{s}}3}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{16}^{e} } \right] = \left[ {K_{{{\text{tb}}\phi 1}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{bNL}}\phi 1}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{ts}}\phi 1}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{17}^{e} } \right] = \left[ {K_{{{\text{rb}}\phi 2}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{rb}}\phi 4}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\phi {\text{s}}1}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\phi {\text{s}}3}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{18}^{e} } \right] = \left[ {K_{{{\text{rb}}\phi 4}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\phi {\text{s}}3}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{19}^{e} } \right] = \left[ {K_{{{\text{tb}}\psi 1}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{bN}}\psi 1}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{ts}}\psi 1}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{20}^{e} } \right] = \left[ {K_{{{\text{rb}}\psi 2}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{rb}}\psi 4}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\psi {\text{s}}1}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\psi {\text{s}}3}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{21}^{e} } \right] = \left[ {K_{{{\text{rb}}\psi 4}}^{e} } \right]^{\text{T}} + \left[ {K_{{{\text{r}}\psi {\text{s}}3}}^{e} } \right]^{\text{T}} $$
$$ \left[ {K_{22}^{e} } \right] = \left[ {K_{16}^{e} } \right] - \left[ {K_{\phi \psi }^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{19}^{e} } \right] $$
$$ \left[ {K_{23}^{e} } \right] = \left[ {K_{17}^{e} } \right] - \left[ {K_{\phi \psi }^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{20}^{e} } \right] $$
$$ \left[ {K_{24}^{e} } \right] = \left[ {K_{18}^{e} } \right] - \left[ {K_{\phi \psi }^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{21}^{e} } \right] $$
$$ \left[ {K_{25}^{e} } \right] = \left[ {K_{\phi \phi }^{e} } \right] - \left[ {K_{\phi \psi }^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{\phi \psi }^{e} } \right]^{\text{T}} $$
$$ \begin{aligned} \left[ {K_{26}^{e} } \right] = \left[ {K_{25}^{e} } \right]^{ - 1} \left[ {K_{22}^{e} } \right]^{\text{T}} ; \hfill \\ \left[ {K_{27}^{e} } \right] = \left[ {K_{25}^{e} } \right]^{ - 1} \left[ {K_{23}^{e} } \right]^{\text{T}} ; \hfill \\ \left[ {K_{28}^{e} } \right] = \left[ {K_{25}^{e} } \right]^{ - 1} \left[ {K_{24}^{e} } \right]^{\text{T}} \hfill \\ \end{aligned} $$
$$ \begin{aligned} \left[ {K_{29}^{e} } \right] = \left[ {K_{11}^{e} } \right] - \left[ {K_{15}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{19}^{e} } \right] \hfill \\ \left[ {K_{30}^{e} } \right] = \left[ {K_{12}^{e} } \right] - \left[ {K_{15}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{20}^{e} } \right] \hfill \\ \left[ {K_{31}^{e} } \right] = \left[ {K_{13}^{e} } \right] - \left[ {K_{15}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{21}^{e} } \right] \hfill \\ \left[ {K_{32}^{e} } \right] = \left[ {K_{14}^{e} } \right] - \left[ {K_{15}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{\phi \psi }^{e} } \right]^{\text{T}} \hfill \\ \end{aligned} $$
$$ \left[ {K_{33}^{e} } \right] = \left[ {K_{29}^{e} } \right] - \left[ {K_{32}^{e} } \right]\left[ {K_{26}^{e} } \right] $$
$$ \left[ {K_{34}^{e} } \right] = \left[ {K_{30}^{e} } \right] - \left[ {K_{32}^{e} } \right]\left[ {K_{27}^{e} } \right] $$
$$ \left[ {K_{35}^{e} } \right] = \left[ {K_{31}^{e} } \right] - \left[ {K_{32}^{e} } \right]\left[ {K_{28}^{e} } \right] $$
$$ \begin{aligned} \left[ {K_{36}^{e} } \right] = \left[ {K_{35}^{e} } \right]^{ - 1} \left[ {K_{33}^{e} } \right]^{\text{T}} ; \hfill \\ \left[ {K_{37}^{e} } \right] = \left[ {K_{35}^{e} } \right]^{ - 1} \left[ {K_{34}^{e} } \right]^{\text{T}} \hfill \\ \end{aligned} $$
$$ \begin{aligned} \left[ {K_{38}^{e} } \right] = \left[ {K_{6}^{e} } \right] - \left[ {K_{10}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{19}^{e} } \right]; \hfill \\ \left[ {K_{39}^{e} } \right] = \left[ {K_{7}^{e} } \right] - \left[ {K_{10}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{20}^{e} } \right] \hfill \\ \left[ {K_{40}^{e} } \right] = \left[ {K_{8}^{e} } \right] - \left[ {K_{10}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{21}^{e} } \right]; \hfill \\ \left[ {K_{41}^{e} } \right] = \left[ {K_{9}^{e} } \right] - \left[ {K_{10}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{\phi \psi }^{e} } \right]^{\text{T}} \hfill \\ \end{aligned} $$
$$ \left[ {K_{42}^{e} } \right] = \left[ {K_{38}^{e} } \right] - \left[ {K_{41}^{e} } \right]\left[ {K_{26}^{e} } \right] $$
$$ \left[ {K_{43}^{e} } \right] = \left[ {K_{39}^{e} } \right] - \left[ {K_{41}^{e} } \right]\left[ {K_{27}^{e} } \right] $$
$$ \left[ {K_{44}^{e} } \right] = \left[ {K_{40}^{e} } \right] - \left[ {K_{41}^{e} } \right]\left[ {K_{28}^{e} } \right] $$
$$ \left[ {K_{45}^{e} } \right] = \left[ {K_{42}^{e} } \right] - \left[ {K_{44}^{e} } \right]\left[ {K_{36}^{e} } \right] $$
$$ \left[ {K_{46}^{e} } \right] = \left[ {K_{43}^{e} } \right] - \left[ {K_{44}^{e} } \right]\left[ {K_{37}^{e} } \right] $$
$$ \left[ {K_{47}^{e} } \right] = \left[ {K_{46}^{e} } \right]^{ - 1} \left[ {K_{45}^{e} } \right] $$
$$ \begin{aligned} \left[ {K_{48}^{e} } \right] = \left[ {K_{1}^{e} } \right] - \left[ {K_{5}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{19}^{e} } \right] \hfill \\ \left[ {K_{49}^{e} } \right] = \left[ {K_{2}^{e} } \right] - \left[ {K_{5}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{20}^{e} } \right] \hfill \\ \left[ {K_{50}^{e} } \right] = \left[ {K_{3}^{e} } \right] - \left[ {K_{5}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{21}^{e} } \right] \hfill \\ \left[ {K_{51}^{e} } \right] = \left[ {K_{4}^{e} } \right] - \left[ {K_{5}^{e} } \right]\left[ {K_{\psi \psi }^{e} } \right]^{ - 1} \left[ {K_{\phi \psi }^{e} } \right]^{\text{T}} \hfill \\ \end{aligned} $$
$$ \left[ {K_{52}^{e} } \right] = \left[ {K_{48}^{e} } \right] - \left[ {K_{51}^{e} } \right]\left[ {K_{26}^{e} } \right] $$
$$ \left[ {K_{53}^{e} } \right] = \left[ {K_{49}^{e} } \right] - \left[ {K_{51}^{e} } \right]\left[ {K_{27}^{e} } \right] $$
$$ \left[ {K_{54}^{e} } \right] = \left[ {K_{50}^{e} } \right] - \left[ {K_{51}^{e} } \right]\left[ {K_{28}^{e} } \right] $$
$$ \left[ {K_{55}^{e} } \right] = \left[ {K_{52}^{e} } \right] - \left[ {K_{54}^{e} } \right]\left[ {K_{36}^{e} } \right] $$
$$ \left[ {K_{56}^{e} } \right] = \left[ {K_{53}^{e} } \right] - \left[ {K_{54}^{e} } \right]\left[ {K_{37}^{e} } \right] $$
$$ \left[ {K_{\text{eq}}^{e} } \right] = \left[ {K_{55}^{e} } \right] - \left[ {K_{56}^{e} } \right]\left[ {K_{47}^{e} } \right] $$
$$ \left[ {K_{{{\text{rrs}}35}}^{e} } \right] = \left[ {K_{{{\text{rrs}}3}}^{e} } \right] + \left[ {K_{{{\text{rrs}}5}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rrs}}13}}^{e} } \right] = \left[ {K_{{{\text{rrs}}1}}^{e} } \right] + \left[ {K_{{{\text{rrs}}3}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rrs}}3513}}^{e} } \right] = \left[ {K_{{{\text{rrs}}35}}^{e} } \right] + \left[ {K_{{{\text{rrs}}13}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rts}}13}}^{e} } \right] = \left[ {K_{{{\text{rts}}1}}^{e} } \right] + \left[ {K_{{{\text{rts}}3}}^{e} } \right] $$
$$ \left[ {K_{{_{{{\text{r}}\psi {\text{s}}13}} }}^{e} } \right] = \left[ {K_{{_{{{\text{r}}\psi {\text{s}}1}} }}^{e} } \right] + \left[ {K_{{_{{{\text{r}}\psi {\text{s}}3}} }}^{e} } \right] $$
$$ \left[ {K_{{_{{{\text{r}}\phi {\text{s}}13}} }}^{e} } \right] = \left[ {K_{{_{{{\text{r}}\phi {\text{s}}1}} }}^{e} } \right] + \left[ {K_{{_{{{\text{r}}\phi {\text{s}}3}} }}^{e} } \right] $$
$$ \left[ {K_{\text{tbtbNL1}}^{e} } \right] = \left[ {K_{\text{tb1}}^{e} } \right] + \left[ {K_{\text{tbNL1}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rtb}}24}}^{e} } \right] = \left[ {K_{{{\text{rtb}}2}}^{e} } \right] + \left[ {K_{{{\text{rtb}}4}}^{e} } \right] $$
$$ \left[ {K_{{{\text{tbNLbNL1\_tbtbNL1}}}}^{e} } \right] = \left[ {K_{\text{tbNLbNL1}}^{e} } \right] + \left[ {K_{\text{tbtbNL1}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rbNL\_rtb}}24}}^{e} } \right] = \left[ {K_{{{\text{rbNL}}24}}^{e} } \right] + \left[ {K_{{{\text{rtb}}24}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rbNL\_rtb}}4}}^{e} } \right] = \left[ {K_{{{\text{rbNL}}4}}^{e} } \right] + \left[ {K_{{{\text{rtb}}4}}^{e} } \right] $$
$$ \left[ {K_{{{\text{bNL\_tb}}\phi 1}}^{e} } \right] = \left[ {K_{{{\text{bNL}}\phi 1}}^{e} } \right] + \left[ {K_{{{\text{tb}}\phi 1}}^{e} } \right] $$
$$ \left[ {K_{{{\text{bNL\_tb}}\psi 1}}^{e} } \right] = \left[ {K_{{{\text{bNL}}\psi 1}}^{e} } \right] + \left[ {K_{{{\text{tb}}\psi 1}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rrb}}57}}^{e} } \right] = \left[ {K_{{{\text{rrb}}5}}^{e} } \right] + \left[ {K_{{{\text{rrb}}7}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rtbrbNL}}4}}^{e} } \right] = \left[ {K_{{{\text{rtb}}4}}^{e} } \right] + \left[ {K_{{{\text{rbNL}}4}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rtbrbNL}}2}}^{e} } \right] = \left[ {K_{{{\text{rtb}}2}}^{e} } \right] + \left[ {K_{{{\text{rbNL}}2}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rtbrbNL}}24}}^{e} } \right] = \left[ {K_{{{\text{rtbrbNL}}2}}^{e} } \right] + \left[ {K_{{{\text{rtbrbNL}}4}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rrb}}35}}^{e} } \right] = \left[ {K_{{{\text{rrb}}3}}^{e} } \right] + \left[ {K_{{{\text{rrb}}5}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rrb}}5735}}^{e} } \right] = \left[ {K_{{{\text{rrb}}57}}^{e} } \right] + \left[ {K_{{{\text{rrb}}35}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rb}}\phi 24}}^{e} } \right] = \left[ {K_{{{\text{rb}}\phi 2}}^{e} } \right] + \left[ {K_{{{\text{rb}}\phi 4}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rb}}\psi 24}}^{e} } \right] = \left[ {K_{{{\text{rb}}\psi 2}}^{e} } \right] + \left[ {K_{{{\text{rb}}\psi 4}}^{e} } \right] $$
$$ \left[ {K_{{{\text{tbNLbNL}}1}}^{e} } \right] = \left[ {K_{{{\text{tbNL}}1}}^{e} } \right] + \left[ {K_{{{\text{bNL}}1}}^{e} } \right] $$
$$ \left[ {K_{{{\text{rbNL}}24}}^{e} } \right] = \left[ {K_{{{\text{rbNL}}2}}^{e} } \right] + \left[ {K_{\text{rbNL4}}^{e} } \right]. $$
(A-12)
The expressions for stiffness matrices are as follows:
$$ \left[ {K_{{{\text{rtb}}4}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{\text{b4}} } \right]} } \left[ {B_{\text{tb}} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rbNL}}4}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{{{\text{bNL}}4}} } \right]} } \left[ {B_{1} } \right] \, \left[ {B_{2} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rrb}}5}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{\text{b5}} } \right]} } \left[ {B_{\text{rb}} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rrb}}7}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{\text{b7}} } \right]} } \left[ {B_{\text{rb}} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rb}}\phi 4}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{{{\text{b}}\phi 4}} } \right]} } \left[ {B_{\phi } } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rb}}\psi 4}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{{{\text{b}}\psi 4}} } \right]} } \left[ {B_{\psi } } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rtb}}4}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{{{\text{b}}4}} } \right]} } \left[ {B_{\text{tb}} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rtb}}2}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{{{\text{b}}2}} } \right]} } \left[ {B_{\text{tb}} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rbNL}}2}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{\text{bNL2}} } \right]} } \left[ {B_{1} } \right] \, \left[ {B_{2} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rrb}}3}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{\text{b3}} } \right]} } \left[ {B_{\text{rb}} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rrb}}5}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{\text{b5}} } \right]} } \left[ {B_{\text{rb}} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rb}}\phi 2}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{{{\text{b}}\phi 2}} } \right]} } \left[ {B_{\phi } } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rb}}\psi 2}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rb}} } \right]^{\text{T}} \left[ {D_{{{\text{b}}\psi 2}} } \right]} } \left[ {B_{\psi } } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{tbNL}}1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{tb}} } \right]^{\text{T}} \left[ {D_{{{\text{bNL}}1}} } \right]\left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x{\text{d}}y} } $$
$$ \left[ {K_{{{\text{bNL}}1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{2} } \right]^{\text{T}} \left[ {B_{1} } \right]^{\text{T}} \left[ {D_{\text{bbNL1}} } \right]\left[ {B_{1} } \right]\left[ {B_{2} } \right]{\text{d}}x{\text{d}}y} } $$
$$ \left[ {K_{{{\text{bNL}}\phi 1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\phi } } \right]^{\text{T}} \left[ {D_{{{\text{bNL}}\phi 1}} } \right]} } \left[ {B_{1} } \right] \, \left[ {B_{2} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{bNL}}\psi 1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\psi } } \right]^{\text{T}} \left[ {D_{{{\text{bNL}}\psi 1}} } \right]} } \left[ {B_{1} } \right] \, \left[ {B_{2} } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{tb}}1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{tb}} } \right]^{\text{T}} \left[ {D_{{{\text{b}}1}} } \right]\left[ {B_{\text{tb}} } \right]{\text{d}}x{\text{d}}y} } $$
$$ \left[ {K_{{{\text{tb}}\phi 1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{tb}} } \right]^{\text{T}} \left[ {D_{{{\text{b}}\phi 1}} } \right]\left[ {B_{\phi } } \right]{\text{d}}x{\text{d}}y} } $$
$$ \left[ {K_{{{\text{tb}}\psi 1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{tb}} } \right]^{\text{T}} \left[ {D_{{{\text{b}}\psi 1}} } \right]} } \left[ {B_{\psi } } \right]{\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{_{{{\text{rts}}3}} }}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rs}} } \right]^{\text{T}} \left[ {D_{{{\text{s}}3}} } \right]\left[ {B_{\text{ts}} } \right]} } {\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{rrs}}3}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rs}} } \right]^{\text{T}} \left[ {D_{{{\text{s}}3}} } \right]\left[ {B_{\text{rs}} } \right]} } {\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{_{\text{rrs5}} }}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rs}} } \right]^{\text{T}} \left[ {D_{{{\text{s}}5}} } \right]\left[ {B_{\text{rs}} } \right]} } {\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{r}}\psi {\text{s}}3}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rs}} } \right]^{\text{T}} \left[ {D_{{{\text{s}}\psi 3}} } \right]\left[ {B_{\psi } } \right]} } {\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{_{{{\text{r}}\phi {\text{s}}3}} }}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rs}} } \right]^{\text{T}} \left[ {D_{{{\text{s}}\phi 3}} } \right]\left[ {B_{\phi } } \right]} } {\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{ts}}\phi 1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{ts}} } \right]^{\text{T}} \left[ {D_{{{\text{s}}\phi 1}} } \right]} } \left[ {B_{\phi } } \right]{\text{d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{ts}}\psi 1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{ts}} } \right]^{\text{T}} \left[ {D_{{{\text{s}}\psi 1}} } \right]} } \left[ {B_{\psi } } \right]{\text{d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{r}}\phi {\text{s}}1}}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rs}} } \right]^{\text{T}} \left[ {D_{{{\text{s}}\phi 1}} } \right]\left[ {B_{\phi } } \right]} } {\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{r}}\psi {\text{s}}1}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rs}} } \right]^{\text{T}} \left[ {D_{s\psi 1} } \right]\left[ {B_{\psi } } \right]} } {\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{_{\text{rts1}} }}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rs}} } \right]^{\text{T}} \left[ {D_{\text{s1}} } \right]\left[ {B_{\text{ts}} } \right]} } {\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{_{\text{rrs1}} }}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rs}} } \right]^{\text{T}} \left[ {D_{\text{s1}} } \right]\left[ {B_{\text{rs}} } \right]} } {\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{{\text{ts}}1}} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{ts}} } \right]^{\text{T}} \left[ {D_{\text{s1}} } \right]\left[ {B_{\text{ts}} } \right]} } {\text{ d}}x{\text{d}}y $$
$$ \left[ {K_{{_{{{\text{rts}}3}} }}^{e} } \right] = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ {B_{\text{rs}} } \right]^{\text{T}} \left[ {D_{{{\text{s}}3}} } \right]\left[ {B_{\text{ts}} } \right]} } {\text{ d}}x{\text{d}}y. $$
(A-13)
The various rigidity matrices contributing to Eq. (A-13) can be denoted as follows:
$$ \left[ {D_{{{\text{b}}1}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {\left[ {C_{\text{b}} } \right]} {\text{d}}z $$
$$ \left[ {D_{{{\text{bbNL}}1}} } \right] = \frac{1}{4}\left[ {D_{\text{b1}} } \right] $$
$$ \left[ {D_{\text{bNL1}} } \right] = \frac{1}{2}\left[ {D_{\text{b1}} } \right] $$
$$ \left[ {D_{{{\text{b}}2}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {z\left[ {C_{\text{b}} } \right]} {\text{d}}z $$
$$ \left[ {D_{{{\text{bNL}}2}} } \right] = \frac{1}{2}\left[ {D_{\text{b2}} } \right] $$
$$ \left[ {D_{{{\text{b}}3}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {z^{2} \left[ {C_{\text{b}} } \right]} {\text{d}}z $$
$$ \left[ {D_{\text{b4}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {c_{1} z^{3} \left[ {C_{\text{b}} } \right]} {\text{d}}z $$
$$ \left[ {D_{{{\text{bNL}}4}} } \right] = \frac{1}{2}\left[ {D_{\text{b4}} } \right] $$
$$ \left[ {D_{{{\text{b}}5}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {c_{1} z^{4} \left[ {C_{\text{b}} } \right]} {\text{d}}z $$
$$ \left[ {D_{{{\text{b}}7}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {c_{1}^{2} z^{6} \left[ {C_{\text{b}} } \right]} {\text{d}}z $$
$$ \left[ {D_{{{\text{b}}\phi 1}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {\left[ {e_{\text{b}} } \right]} {\text{d}}z $$
$$ \left[ {D_{{{\text{bNL}}\phi 1}} } \right] = \left[ {D_{{{\text{b}}\phi 1}} } \right] $$
$$ \left[ {D_{{{\text{b}}\phi 2}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {z\left[ {e_{\text{b}} } \right]} {\text{d}}z $$
$$ \left[ {D_{{{\text{b}}\phi 4}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {c_{1} z^{3} \left[ {e_{\text{b}} } \right]} {\text{d}}z $$
$$ \left[ {D_{{{\text{b}}\psi 1}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {\left[ {q_{\text{b}} } \right]} {\text{d}}z, $$
$$ \left[ {D_{{{\text{bNL}}\psi 1}} } \right] = \left[ {D_{{{\text{b}}\psi 1}} } \right] $$
$$ \left[ {D_{{{\text{b}}\psi 2}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {z\left[ {q_{\text{b}} } \right]} {\text{d}}z, $$
$$ \left[ {D_{{{\text{b}}\psi 4}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {c_{1} z^{3} \left[ {q_{\text{b}} } \right]} {\text{d}}z $$
$$ \left[ {D_{\text{s1}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {\left[ {C_{\text{s}} } \right]} {\text{d}}z, $$
$$ \left[ {D_{{{\text{s}}3}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {c_{2} z^{2} \left[ {C_{\text{s}} } \right]} {\text{d}}z{ ,} $$
$$ \left[ {D_{{{\text{s}}5}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {c_{2}^{2} z^{4} \left[ {C_{\text{s}} } \right]} {\text{d}}z{ ,} $$
$$ \left[ {D_{{{\text{s}}\phi 1}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {\left[ {e_{\text{s}} } \right]} {\text{d}}z, $$
$$ \left[ {D_{{{\text{s}}\phi 3}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {c_{2} z^{2} \left[ {e_{\text{s}} } \right]} {\text{d}}z, $$
$$ \left[ {D_{{{\text{s}}\psi 1}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {\left[ {q_{\text{s}} } \right]} {\text{d}}z \, $$
$$ \left[ {D_{{{\text{s}}\psi 3}} } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {c_{2} z^{2} \left[ {q_{\text{s}} } \right]} {\text{d}}z $$
$$ \left[ {D_{\phi \phi } } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {\left[ \eta \right]} {\text{d}}z $$
$$ \left[ {D_{\psi \psi } } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {\left[ \mu \right]} {\text{d}}z{ ,} $$
$$ \left[ {D_{\phi \psi } } \right] = \int\limits_{{{{ - h} \mathord{\left/ {\vphantom {{ - h} 2}} \right. \kern-0pt} 2}}}^{{{h \mathord{\left/ {\vphantom {h 2}} \right. \kern-0pt} 2}}} {\left[ m \right]} {\text{d}}z. $$
(A-14)
The derivative of shape function matrices appearing in the FE formulation can be represented by
$$ \begin{aligned} \left[ {B_{\text{tb}} } \right] = \left[ {\begin{array}{*{20}c} {N_{i,x} } & 0 & 0 \\ 0 & {N_{i,y} } & 0 \\ {N_{i,y} } & {N_{i,x} } & 0 \\ \end{array} } \right], \, \left[ {B_{\text{rb}} } \right] = \left[ {\begin{array}{*{20}c} {N_{i,x} } & 0 \\ 0 & {N_{i,y} } \\ {N_{i,y} } & {N_{i,x} } \\ \end{array} } \right], \, \left[ {B_{\text{ts}} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & {N_{i,x} } \\ 0 & 0 & {N_{i,y} } \\ \end{array} } \right] , { }\left[ {B_{\text{rs}} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right] \hfill \\ \left[ {B_{1} } \right] = \left[ {\begin{array}{*{20}c} {w_{0,x} } & 0 & {w_{0,y} } \\ 0 & {w_{0,y} } & {w_{0,x} } \\ \end{array} } \right]^{T} ,\left[ {B_{2} } \right] = \left[ {\begin{array}{*{20}c} {B_{21} } & {B_{22} } & \cdots & {B_{28} } \\ \end{array} } \right]. \hfill \\ \end{aligned} $$
(A-15)
