Appendix 1: Coefficients of the full-order equations
The elements of the symmetric mass matrix are defined first:
$$\begin{aligned} \mathbf{M}^{11}= & {} \mathbf{M}^{22}=\rho h\int _\varOmega \mathbf{N}^{\mathbf{u}}{\mathbf{N}^\mathbf{u}}^\mathrm{T}\,\,\hbox {d}\varOmega ,\nonumber \\ \mathbf{M}^{33}= & {} \rho h\int _\varOmega \mathbf{N}^{\mathbf{w}}\mathbf{N}^{\mathbf{w}^\mathrm{T}}\,\,\hbox {d}\varOmega \nonumber \\&+\,\frac{\rho h^{3}}{252}\int _\varOmega \left( {\mathbf{N}_{,x}^\mathbf{w} {\mathbf{N}_{,x}^\mathbf{w}}^\mathrm{T}+\mathbf{N}_{,y}^\mathbf{w} {\mathbf{N}_{,y}^\mathbf{w}}^\mathrm{T}} \right) \,\,\hbox {d}\varOmega ,\nonumber \\ \mathbf{M}^{34}= & {} -\frac{4\rho h^{3}}{315}\int _\varOmega \mathbf{N}_{,x}^\mathbf{w} {\mathbf{N}^{\varvec{\upphi }_\mathbf{x}}}^\mathrm{T}\,\,\hbox {d}\varOmega ,\\ \mathbf{M}^{35}= & {} -\frac{4\rho h^{3}}{315}\int _\varOmega \mathbf{N}_{,y}^\mathbf{w} {\mathbf{N}^{\varvec{\upphi }_\mathbf{y}}}^\mathrm{T}\,\,\hbox {d}\varOmega ,\nonumber \\ \mathbf{M}^{44}= & {} \frac{17\rho h^{3}}{315}\int _\varOmega \mathbf{N}^{\varvec{\upphi }_\mathbf{x} }{N^{\varvec{\upphi }_{{\mathbf {x}}}}}^\mathrm{T}\,\,\hbox {d}\varOmega ,\nonumber \\ \mathbf{M}^{55}= & {} \frac{17\rho h^{3}}{315}\int _\varOmega \mathbf{N}^{\varvec{\upphi }_\mathbf{y} }\mathbf{N}^{\varvec{\upphi }_\mathbf{y} ^\mathrm{T}}\,\,\hbox {d}\varOmega .\nonumber \end{aligned}$$
(17)
where ‘\(x\)’ in subscript means derivation with respect to \(x\) and so on. \(\varOmega \) denotes surface. The linear stiffness matrix is introduced as
$$\begin{aligned}&\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {\mathbf{K}_\mathbf{L}^{11} }&{} {\mathbf{K}_\mathbf{L}^{12} }&{} {\mathbf{K}_\mathbf{L}^{13} }&{} \mathbf{0}&{} \mathbf{0} \\ &{} {\mathbf{K}_\mathbf{L}^{22} }&{} {\mathbf{K}_\mathbf{L}^{23} }&{} \mathbf{0}&{} \mathbf{0} \\ &{} &{} {\mathbf{K}_\mathbf{L}^{33} }&{} {\mathbf{K}_\mathbf{L}^{34} }&{} {\mathbf{K}_\mathbf{L}^{35} } \\ &{} &{} &{} {\mathbf{K}_\mathbf{L}^{44} }&{} {\mathbf{K}_\mathbf{L}^{45} } \\ \hbox {sym}&{} &{} &{} &{} {\mathbf{K}_\mathbf{L}^{55} } \\ \end{array} }} \right] \!=\!\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {\mathbf{K}_{\mathbf{L1}}^{11} }&{} {\mathbf{K}_{\mathbf{L1}}^{12} }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0} \\ &{} {\mathbf{K}_{\mathbf{L1}}^{22} }&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0} \\ &{} &{} {\mathbf{K}_{\mathbf{L1}}^{33} }&{} {\mathbf{K}_{\mathbf{L1}}^{34} }&{} {\mathbf{K}_{\mathbf{L1}}^{35} } \\ &{} &{} &{} {\mathbf{K}_{\mathbf{L1}}^{44} }&{} {\mathbf{K}_{\mathbf{L1}}^{45} } \\ \hbox {sym}&{} &{} &{} &{} {\mathbf{K}_{\mathbf{L1}}^{55} } \\ \end{array} }} \right] \nonumber \\&\quad +\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0} \\ &{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0} \\ &{} &{} {\mathbf{K}_{\mathbf{L2}}^{33} }&{} {\mathbf{K}_{\mathbf{L2}}^{34} }&{} {\mathbf{K}_{\mathbf{L2}}^{35} } \\ &{} &{} &{} {\mathbf{K}_{\mathbf{L2}}^{44} }&{} {\mathbf{K}_{\mathbf{L2}}^{45} } \\ \hbox {sym}&{} &{} &{} &{} {\mathbf{K}_{\mathbf{L2}}^{55} } \\ \end{array} }} \right] \!+\!\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0&{} 0&{} {\mathbf{K}_{\mathbf{L3}}^{13} }&{} 0&{} 0 \\ &{} 0&{} {\mathbf{K}_{\mathbf{L3}}^{23} }&{} 0&{} 0 \\ &{} &{} {\mathbf{K}_{\mathbf{L3}}^{33} }&{} 0&{} 0 \\ &{} &{} &{} 0&{} 0 \\ \hbox {sym}&{} &{} &{} &{} 0 \\ \end{array} }} \right] \end{aligned}$$
(18)
with in-plane linear stiffness terms as
$$\begin{aligned}&\left[ {{\begin{array}{c@{\quad }c} {\mathbf{K}_{\mathbf{L1}}^{\mathbf{11}} }&{} {\mathbf{K}_{\mathbf{L1}}^{\mathbf{12}} } \\ {\mathbf{K}_{\mathbf{L1}}^{\mathbf{21}} }&{} {\mathbf{K}_{\mathbf{L1}}^{\mathbf{22}} } \\ \end{array} }} \right] \nonumber \\&\quad =\int _\varOmega \left[ {{\begin{array}{c@{\quad }c@{\quad }c} {\mathbf{N}_{,x}^\mathbf{u} }&{} \mathbf{0}&{} {\mathbf{N}_{,y}^\mathbf{u} } \\ \mathbf{0}&{} {\mathbf{N}_{,y}^\mathbf{u} }&{} {\mathbf{N}_{,x}^\mathbf{u} } \\ \end{array} }} \right] \mathbf{A}\left( x \right) \left[ {{\begin{array}{c@{\quad }c} {\mathbf{N}_{,x}^{\mathbf{u} ^\mathrm{T}}}&{} \mathbf{0} \\ \mathbf{0}&{} {\mathbf{N}_{,y}^{\mathbf{u} ^\mathrm{T}}} \\ {\mathbf{N}_{,y}^{\mathbf{u} ^\mathrm{T}}}&{} {\mathbf{N}_{,x}^{\mathbf{u} ^\mathrm{T}}} \\ \end{array} }} \right] \,\,\hbox {d}\varOmega \nonumber \\ \end{aligned}$$
(19)
with
$$\begin{aligned} \mathbf{A}\left( x \right)= & {} \sum \limits _{k=1}^n h_k \left[ \left[ {\begin{array}{c@{\quad }c@{\quad }c} {U_1 } &{} {U_4 }&{} 0\\ {U_4 } &{} {U_1 }&{} 0\\ 0 &{} 0&{} {U_5 }\\ \end{array} }\right] +\,U_2 \left[ {\begin{array}{c@{\quad }c@{\quad }c} 1 &{} 0&{} 0\\ 0 &{} -1&{} 0\\ 0 &{}0&{}0\\ \end{array} }\right] \cos 2\theta _k \left( x \right) \right. \nonumber \\&+\,U_3 \left[ {\begin{array}{c@{\quad }c@{\quad }c} 1 &{} -1&{} 0\\ {-1} &{} 1&{}0\\ 0 &{}0&{}-1\\ \end{array} }\right] \cos 4\theta _k \left( x \right) \nonumber \\&+\,\frac{U_2 }{2} \left[ {\begin{array}{c@{\quad }c@{\quad }c} 0 &{}0&{}1\\ 0 &{}0&{}1\\ 1 &{}1&{}0\\ \end{array} } \right] sin2\theta _k \left( x \right) \nonumber \\&\left. +\,U_3 \left[ {\begin{array}{c@{\quad }c@{\quad }c} 0 &{}0&{}1\\ 0 &{}0&{}-1\\ 1 &{}-1&{}0\\ \end{array} }\right] \sin 4\theta _k \left( x \right) \right] , \end{aligned}$$
(20)
and with \(n\) representing the number of layers. Coefficients \(U_i , i=1-7,\) are given in Ref. [8]. The remaining components of the linear stiffness matrix are:
$$\begin{aligned} \mathbf{K}_{\mathbf{L1}}^{\mathbf{33}} \!=\!\int _\varOmega \left\lfloor {\begin{array}{ccc} {\mathbf{N}_{,xx}^\mathbf{w} }&{}\quad {\mathbf{N}_{,yy}^\mathbf{w} }&{}\quad {2\mathbf{N}_{,xy}^\mathbf{w} } \\ \end{array} }\right\rfloor \mathbf{F}\left( x \right) \left\{ {{\begin{array}{c} {\mathbf{N}_{,xx}^\mathbf{w}}^\mathrm{T} \\ {\mathbf{N}_{,yy}^{\mathbf{w}}}^\mathrm{T} \\ {2{\mathbf{N}_{,xy}^{\mathbf{w}}}^\mathrm{T}} \\ \end{array} }} \right\} \hbox {d}\varOmega \nonumber \\ \end{aligned}$$
(21)
where
$$\begin{aligned}&\mathbf{F}\left( x \right) =\sum \limits _{k=1}^n \left( {c^{2}\frac{z_k^7 -z_{k-1}^7 }{7}} \right) \left[ \left[ {\begin{array}{ccc} {U_1 } &{}{U_4 }&{}0 \\ {U_4 } &{}{U_1 }&{}0 \\ 0 &{}0 &{}{U_5 } \\ \end{array} } \right] \right. \nonumber \\&\quad +\,U_2 \left[ {\begin{array}{ccc} 1 &{}0 &{}0\\ 0 &{}{-1}&{}0\\ 0 &{}0 &{}0\\ \end{array} }\right] \cos 2\theta _k \left( x \right) +U_3 \left[ {\begin{array}{ccc} 1 &{}{-1}&{}0 \\ {-1}&{}1 &{}0 \\ 0 &{}0 &{}{-1} \\ \end{array} }\right] \cos 4\theta _k \left( x \right) \nonumber \\&\quad \left. +\,\frac{U_2 }{2}\left[ {\begin{array}{ccc} 0 &{}0 &{}1\\ 0 &{}0 &{}1\\ 1 &{}1 &{}0\\ \end{array} }\right] \sin 2\theta _k \left( x \right) +U_3 \left[ {{\begin{array}{ccc} 0 &{}0 &{}1 \\ 0 &{}0 &{}{-1} \\ 1 &{}{-1}&{}0 \\ \end{array} }} \right] \sin 4\theta _k \left( x \right) \right] .\nonumber \\ \end{aligned}$$
(22)
$$\begin{aligned}&\left[ {{\begin{array}{ccc} {\mathbf{K}_{\mathbf{L2}}^{\mathbf{33}} }&{} {\mathbf{K}_{\mathbf{L2}}^{\mathbf{34}} }&{} {\mathbf{K}_{\mathbf{L2}}^{\mathbf{35}} } \\ {\mathbf{K}_{\mathbf{L2}}^{\mathbf{43}} }&{} {\mathbf{K}_{\mathbf{L2}}^{\mathbf{44}} }&{} {\mathbf{K}_{\mathbf{L2}}^{\mathbf{45}} } \\ {\mathbf{K}_{\mathbf{L2}}^{\mathbf{53}} }&{} {\mathbf{K}_{\mathbf{L2}}^{\mathbf{54}} }&{} {\mathbf{K}_{\mathbf{L2}}^{\mathbf{55}} } \\ \end{array} }} \right] \nonumber \\&\quad =\int _\varOmega \left[ {{\begin{array}{cc} {\mathbf{N}_{,y}^\mathbf{w} }&{} {\mathbf{N}_{,x}^\mathbf{w} } \\ \mathbf{0}&{} {\mathbf{N}^{\varvec{\upphi }_\mathbf{x} }} \\ {\mathbf{N}^{\varvec{\upphi }_\mathbf{y} }}&{} \mathbf{0} \\ \end{array} }} \right] \mathbf{O}\left( x \right) \left[ {{\begin{array}{ccc} {\mathbf{N}_{,y}^{\mathbf{w}}}^\mathrm{T}&{} \mathbf{0}&{} {\mathbf{N}^{\varvec{\upphi }_\mathbf{y}}}^\mathrm{T} \\ {\mathbf{N}_{,x}^{\mathbf{w} }}^\mathrm{T}&{} {\mathbf{N}^{\varvec{\upphi }_\mathbf{x}}} ^\mathrm{T}&{} \mathbf{0} \\ \end{array} }} \right] \hbox {d}\varOmega \end{aligned}$$
(23)
where
$$\begin{aligned}&\mathbf{O}\left( x \right) =\sum \limits _{k=1}^n \left( {h_k -2c\left( {z_k^3 -z_{k-1}^3 } \right) +\frac{9}{5}c^{2}\left( {z_k^5 -z_{k-1}^5 } \right) } \right) \nonumber \\&\qquad \times \left[ U_6 \left[ {{\begin{array}{cc} 1 &{}0 \\ 0 &{}1 \\ \end{array} }} \right] +U_7 \left[ {{\begin{array}{cc} 1 &{}0 \\ 0 &{}{-1} \\ \end{array} }} \right] \cos 2\theta _k \left( x \right) \right. \nonumber \\&\qquad \left. -\,U_7 \left[ {{\begin{array}{cc} 0 &{}1\\ 1 &{}0\\ \end{array} }} \right] \sin 2\theta _k \left( x \right) \right] .\end{aligned}$$
(24)
$$\begin{aligned}&\left[ {{\begin{array}{cc} {\mathbf{K}_{\mathbf{L1}}^{\mathbf{34}} }&{} {\mathbf{K}_{\mathbf{L1}}^{\mathbf{35}} } \\ \end{array} }} \right] \nonumber \\&= \int _\varOmega \left[ {{\begin{array}{c@{\quad }c@{\quad }c} {\mathbf{N}_{,xx}^\mathbf{w} }&{} {\mathbf{N}_{,yy}^\mathbf{w} }&{} {2\mathbf{N}_{,xy}^\mathbf{w} } \\ \end{array} }} \right] \mathbf{L}\left( x \right) \left[ {{\begin{array}{ll} {\mathbf{N}_{,x}^{\varvec{\upphi }_\mathbf{x} }}^\mathrm{T}&{} \mathbf{0} \\ \mathbf{0}&{} {\mathbf{N}_{,y}^{\varvec{\upphi }_\mathbf{y}}} ^\mathrm{T} \\ {\mathbf{N}_{,y}^{\varvec{\upphi }_\mathbf{x} }}^\mathrm{T}&{} {\mathbf{N}_{,x}^{\varvec{\upphi }_\mathbf{y} }}^\mathrm{T} \\ \end{array} }} \right] \hbox {d}\varOmega \nonumber \\ \end{aligned}$$
(25)
where
$$\begin{aligned}&\mathbf{L}\left( x \right) =\sum \limits _{k=1}^n \left( {-c\frac{z_k^5 -z_{k-1}^5 }{5}+ c^{2}\frac{z_k^7 -z_{k-1}^7 }{7}} \right) \nonumber \\&\quad \times \left[ \left[ {{\begin{array}{c@{\quad }c@{\quad }c} {U_1 }&{}{U_4 } &{}0 \\ {U_4 }&{}{U_1 } &{}0 \\ 0 &{}0 &{}{U_5 } \\ \end{array} }} \right] \!+\!U_2 \left[ {{\begin{array}{c@{\quad }c@{\quad }c} 1 &{}0 &{}0\\ 0 &{}{-1}&{}0\\ 0 &{}0 &{}0\\ \end{array} }} \right] \cos 2\theta _k \left( x \right) \right. \nonumber \\&\quad +\,U_3 \left[ {{\begin{array}{c@{\quad }c@{\quad }c} 1 &{}{-1}&{}0 \\ {-1}&{}1 &{}0 \\ 0 &{}0 &{}{-1} \\ \end{array} }} \right] \cos 4\theta _k \left( x \right) \nonumber \\&\quad +\,\frac{U_2 }{2}\left[ {{\begin{array}{c@{\quad }c@{\quad }c} 0&{}0&{}1 \\ 0&{}0&{}1 \\ 1&{}1&{}0 \\ \end{array} }} \right] \sin 2\theta _k \left( x \right) \nonumber \\&\quad +\left. U_3 \left[ {{\begin{array}{c@{\quad }c@{\quad }c} 0&{}0 &{}1 \\ 0&{}0 &{}{-1} \\ 1&{}{-1}&{}0 \\ \end{array} }} \right] \sin 4\theta _k \left( x \right) \right] .\end{aligned}$$
(26)
$$\begin{aligned}&\left[ {{\begin{array}{c@{\quad }c} {\mathbf{K}_{\mathbf{L1}}^{\mathbf{44}} }&{} {\mathbf{K}_{\mathbf{L1}}^{\mathbf{45}} } \\ {\mathbf{K}_{\mathbf{L1}}^{\mathbf{54}} }&{} {\mathbf{K}_{\mathbf{L1}}^{\mathbf{55}} } \\ \end{array} }} \right] \nonumber \\&\quad \! =\!\int _\varOmega \left[ {{\begin{array}{l@{\quad }l@{\quad }l} {\mathbf{N}_{,x}^{\varvec{\upphi }_\mathbf{x} } }&{} \mathbf{0}&{} {\mathbf{N}_{,y}^{\varvec{\upphi }_\mathbf{x} } } \\ \mathbf{0}&{} {\mathbf{N}_{,y}^{\varvec{\upphi }_\mathbf{y} } }&{} {\mathbf{N}_{,x}^{\varvec{\upphi }_\mathbf{y} } } \\ \end{array} }} \right] \mathbf{J}\left( x \right) \left[ {{\begin{array}{l@{\quad }l} {\mathbf{N}_{,x}^{\varvec{\upphi }_\mathbf{x}}}^\mathrm{T}&{} \mathbf{0} \\ \mathbf{0}&{} {\mathbf{N}_{,y}^{\varvec{\upphi }_\mathbf{y}}}^\mathrm{T} \\ {\mathbf{N}_{,y}^{\varvec{\upphi }_\mathbf{x}}}^\mathrm{T}&{} {\mathbf{N}_{,x}^{\varvec{\upphi }_\mathbf{y}}}^\mathrm{T} \\ \end{array} }} \right] \hbox {d}\varOmega \nonumber \\ \end{aligned}$$
(27)
where
$$\begin{aligned} \mathbf{J}\left( x \right)= & {} \sum \limits _{k=1}^n \left( {\frac{z_k^3 -z_{k-1}^3 }{3}-2c\frac{z_k^5 -z_{k-1}^5 }{5}+c^{2}\frac{z_k^7 -z_{k-1}^7 }{7}} \right) \nonumber \\&\quad \times \left[ \left[ {{\begin{array}{ccc} {U_1 }&{}{U_4 }&{}0 \\ {U_4 }&{}{U_1 }&{}0 \\ 0 &{}0 &{}{U_5 } \\ \end{array} }} \right] +U_2 \left[ {{\begin{array}{ccc} 1&{}0 &{}0 \\ 0&{}{-1}&{}0 \\ 0&{}0 &{}0 \\ \end{array} }} \right] \cos 2\theta _k \left( x \right) \right. \nonumber \\&\quad +\,\,U_3 \left[ {{\begin{array}{ccc} 1 &{}{-1}&{}0 \\ {-1}&{}1 &{}0 \\ 0 &{}0 &{}{-1} \\ \end{array} }} \right] \cos 4\theta _k \left( x \right) \nonumber \\&\quad +\,\frac{U_2 }{2} \left[ {{\begin{array}{ccc} 0&{}0&{}1 \\ 0&{}0&{}1 \\ 1&{}1&{}0 \\ \end{array} }} \right] \sin 2\theta _k \left( x \right) \nonumber \\&\quad \left. +\,\,U_3 \left[ {{\begin{array}{ccc} 0&{}0 &{}1 \\ 0&{}0 &{}{-1} \\ 1&{}{-1}&{}0 \\ \end{array} }} \right] \sin 4\theta _k \left( x \right) \right] . \end{aligned}$$
(28)
Constant terms affected by imperfection are \(\mathbf{K}_{\mathbf{L3}}^{13} , \mathbf{K}_{\mathbf{L3}}^{23} \) and \(\mathbf{K}_{\mathbf{L3}}^{33} \) as
$$\begin{aligned}&\left[ {{\begin{array}{c} {\mathbf{K}_{\mathbf{L3}}^{\mathbf{13}} } \\ {\mathbf{K}_{\mathbf{L3}}^{\mathbf{23}} } \\ \end{array} }} \right] =\int _\varOmega \left[ {{\begin{array}{lll} {\mathbf{N}_{,x}^\mathbf{u} }&{} \mathbf{0}&{} {\mathbf{N}_{,y}^\mathbf{u} } \\ \mathbf{0}&{} {\mathbf{N}_{,y}^\mathbf{u} }&{} {\mathbf{N}_{,x}^\mathbf{u} } \\ \end{array} }} \right] \nonumber \\&\quad \times \,\mathbf{A}\left( x \right) \left[ {{\begin{array}{c} {w_{i,x} \mathbf{N}_{,x}^{\mathbf{w}}}^\mathrm{T} \\ {w_{i,y} \mathbf{N}_{,y}^{\mathbf{w}}}^\mathrm{T} \\ w_{i,x} {\mathbf{N}_{,y}^\mathbf{w}}^\mathrm{T}+w_{i,y} {\mathbf{N}_{,x}^\mathbf{w}}^\mathrm{T} \\ \end{array} }} \right] \hbox {d}\varOmega ,\end{aligned}$$
(29)
$$\begin{aligned}&\mathbf{K}_{\mathbf{L3}}^{\mathbf{33}} =\int _\varOmega \left[ {{\begin{array}{ccc} {w_{i,x} \mathbf{N}_{,x}^\mathbf{w} }&{} {w_{i,y} \mathbf{N}_{,y}^\mathbf{w} }&{} {w_{i,x} \mathbf{N}_{,y}^\mathbf{w} +w_{i,y} \mathbf{N}_{,x}^\mathbf{w} } \\ \end{array} }} \right] \nonumber \\&\quad \times \,\mathbf{A}\left( x \right) \left[ {{\begin{array}{c} {w_{i,x} \mathbf{N}_{,x}^{\mathbf{w}}}^\mathrm{T} \\ {w_{i,y} \mathbf{N}_{,y}^{\mathbf{w}}}^\mathrm{T} \\ w_{i,x} {\mathbf{N}_{,y}^\mathbf{w}} ^\mathrm{T}+w_{i,y} {\mathbf{N}_{,x}^\mathbf{w}} ^\mathrm{T} \\ \end{array} }} \right] \hbox {d}\varOmega , \end{aligned}$$
(30)
The elements of the nonlinear stiffness matrix are:
$$\begin{aligned}&\left[ {{{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \mathbf{0}&{} \mathbf{0}&{} {\mathbf{K}_{\mathbf{NL}}^{13} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} \mathbf{0}&{} \mathbf{0} \\ \mathbf{0}&{} \mathbf{0}&{} {\mathbf{K}_{\mathbf{NL}}^{23} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} \mathbf{0}&{} \mathbf{0} \\ {\mathbf{K}_{\mathbf{NL}}^{31} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} {\mathbf{K}_{\mathbf{NL}}^{32} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} {\mathbf{K}_{\mathbf{NL}}^{33} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} \mathbf{0}&{} \mathbf{0} \\ \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0} \\ \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0} \\ \end{array} }} }\right] \nonumber \\&\quad =\left[ {{{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \mathbf{0}&{} \mathbf{0}&{} {\mathbf{K}_{\mathbf{NL1}}^{13} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} \mathbf{0}&{} \mathbf{0} \\ \mathbf{0}&{} \mathbf{0}&{} {\mathbf{K}_{\mathbf{NL1}}^{23} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} \mathbf{0}&{} \mathbf{0} \\ {\mathbf{K}_{\mathbf{NL1}}^{31} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} {\mathbf{K}_{\mathbf{NL1}}^{32} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} {\mathbf{K}_{\mathbf{NL1}}^{33} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} \mathbf{0}&{} \mathbf{0} \\ \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0} \\ \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0} \\ \end{array} }} }\right] \nonumber \\&\qquad +\left[ {{{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0} \\ &{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0}&{} \mathbf{0} \\ &{} &{} {\mathbf{K}_{\mathbf{NL2}}^{33} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) }&{} \mathbf{0}&{} \mathbf{0} \\ &{} &{} &{} \mathbf{0}&{} \mathbf{0} \\ \hbox {sym}&{} &{} &{} &{} \mathbf{0} \\ \end{array} }} }\right] . \end{aligned}$$
(31)
with
$$\begin{aligned}&\left[ {{\begin{array}{c@{\quad }c} {\mathbf{K}_{\mathbf{NL1}}^{\mathbf{31}} }&{} {\mathbf{K}_{\mathbf{NL1}}^{\mathbf{32}} } \\ \end{array} }} \right] \nonumber \\&\quad =\int _\varOmega \left[ {{\begin{array}{c@{\quad }c@{\quad }c} {w_{, x}^0 \mathbf{N}_{,x}^\mathbf{w} }&{} {w_{, y}^0 \mathbf{N}_{,y}^\mathbf{w} }&{} {w_{, y}^0 \mathbf{N}_{,x}^\mathbf{w} +w_{, x}^0 \mathbf{N}_{,y}^\mathbf{w} } \\ \end{array} }} \right] \nonumber \\&\qquad \times \,\mathbf{A}\left( x \right) \left[ {{\begin{array}{c@{\quad }c} {{\mathbf{N}_{,x}^\mathbf{u}}^\mathrm{T}}&{} \mathbf{0} \\ \mathbf{0}&{} {{\mathbf{N}_{,y}^\mathbf{u}}^\mathrm{T}} \\ {{\mathbf{N}_{,y}^\mathbf{u}}^\mathrm{T}}&{} {{\mathbf{N}_{,x}^\mathbf{u}}^\mathrm{T}} \\ \end{array} }} \right] \hbox {d}\varOmega ,\end{aligned}$$
(32)
$$\begin{aligned}&\left[ {{\begin{array}{c} {\mathbf{K}_{\mathbf{NL1}}^{\mathbf{13}} } \\ {\mathbf{K}_{\mathbf{NL1}}^{\mathbf{23}} } \\ \end{array} }} \right] =0.5\left[ {{\begin{array}{c@{\quad }c} {\mathbf{K}_{\mathbf{NL1}}^{\mathbf{31}} }&{} {\mathbf{K}_{\mathbf{NL1}}^{\mathbf{32}} } \\ \end{array} }} \right] ^\mathrm{T}, \end{aligned}$$
(33)
These nonlinear terms are linearly dependent on the transverse deflection where the following nonlinear term is a quadratic function of generalized transverse coordinates.
$$\begin{aligned} \mathbf{K}_{\mathbf{NL1}}^{\mathbf{33}}= & {} \frac{1}{2}\int _\varOmega \left[ {{\begin{array}{ccc} {w_{, x}^0 \mathbf{N}_{,x}^\mathbf{w} }&{} {w_{, y}^0 \mathbf{N}_{,y}^\mathbf{w} }&{} {w_{, y}^0 \mathbf{N}_{,x}^\mathbf{w} +w_{, x}^0 \mathbf{N}_{,y}^\mathbf{w} } \\ \end{array} }} \right] \nonumber \\&\times \,\mathbf{A}\left( x \right) \left[ {{\begin{array}{c} {w_{, x}^0 \mathbf{N}_{,x}^{\mathbf{w}}}^\mathrm{T} \\ {w_{, y}^0 \mathbf{N}_{,y}^{\mathbf{w}}}^\mathrm{T} \\ {w_{, x}^0 {\mathbf{N}_{,y}^\mathbf{w}} ^\mathrm{T}+w_{, y}^0 {\mathbf{N}_{,x}^\mathbf{w}}^\mathrm{T}} \\ \end{array} }} \right] \hbox {d}\varOmega . \end{aligned}$$
(34)
The nonlinear term \(\mathbf{K}_{\mathbf{NL2}}^{33} \) is affected by imperfection and is linearly dependent on the transverse deflection,
$$\begin{aligned} \mathbf{K}_{\mathbf{NL2}}^{\mathbf{33}}= & {} \int _\varOmega \left( \frac{1}{2}\left[ {{\begin{array}{c@{\quad }c@{\quad }c} {w_{i,x} \mathbf{N}_{,x}^\mathbf{w} }&{} {w_{i,y} \mathbf{N}_{,y}^\mathbf{w} }&{} {w_{i,x} \mathbf{N}_{,y}^\mathbf{w} +w_{i,y} \mathbf{N}_{,x}^\mathbf{w} } \\ \end{array} }} \right] \right. \nonumber \\&\times \,\mathbf{A}\left( x \right) \left[ {{\begin{array}{c} {w_{, x}^0 {\mathbf{N}_{,x}^\mathbf{w}} ^\mathrm{T}} \\ {w_{, y}^0 {\mathbf{N}_{,y}^\mathbf{w}} ^\mathrm{T}} \\ {w_{, x}^0 {\mathbf{N}_{,y}^\mathbf{w}} ^\mathrm{T}+ w_{, y}^0 {\mathbf{N}_{,x}^\mathbf{w} }^\mathrm{T}} \\ \end{array} }} \right] \nonumber \\&+\,\left[ {{\begin{array}{c@{\quad }c@{\quad }c} {w_{, x}^0 \mathbf{N}_{,x}^\mathbf{w} }&{} {w_{, y}^0 \mathbf{N}_{,y}^\mathbf{w} }&{} {w_{, x}^0 \mathbf{N}_{,y}^\mathbf{w} +w_{, y}^0 \mathbf{N}_{,x}^\mathbf{w} } \\ \end{array} }} \right] \nonumber \\&\times \,\left. \mathbf{A}\left( x \right) \left[ {{\begin{array}{c} w_{i,x} {\mathbf{N}_{,x}^\mathbf{w}}^\mathrm{T} \\ w_{i,y} {\mathbf{N}_{,y}^\mathbf{w}} ^\mathrm{T} \\ w_{i,x} {\mathbf{N}_{,y}^\mathbf{w}} ^\mathrm{T}+ w_{i,y} {\mathbf{N}_{,x}^\mathbf{w}} ^\mathrm{T} \\ \end{array} }} \right] \right) \hbox {d}\varOmega . \end{aligned}$$
(35)
Appendix 2: Coefficients of statically condensed equation
Two terms \(\mathbf{K}_{\mathbf{LS}}^{33} \) and \(\mathbf{K}_{\mathbf{NLS}}^{33} \) are introduced here. \(\mathbf{K}_{\mathbf{LS}}^{33} \) is constant terms as
$$\begin{aligned} \mathbf{K}_{\mathbf{LS}}^{33} =\mathbf{K}_{\mathbf{L1}}^{33} +\mathbf{K}_{\mathbf{L2}}^{33} +\mathbf{K}_{\mathbf{L3}}^{33} +\mathbf{K}_{\mathbf{L4}}^{33} , \end{aligned}$$
(36)
where
$$\begin{aligned} \mathbf{K}_{\mathbf{L4}}^{33}= & {} -\left[ {{\begin{array}{c@{\quad }c} {\mathbf{K}_{\mathbf{L3}}^{\mathbf{13}}}^\mathrm{T}&{} {\mathbf{K}_{\mathbf{L3}}^{\mathbf{23}}}^\mathrm{T} \\ \end{array} }} \right] \left[ {{{\begin{array}{c@{\quad }c} {\mathbf{K}_{\mathbf{L1}}^{11} }&{} {\mathbf{K}_{\mathbf{L1}}^{12} } \\ {\mathbf{K}_{\mathbf{L1}}^{21} }&{} {\mathbf{K}_{\mathbf{L1}}^{22} } \\ \end{array} }} }\right] ^{-1}\nonumber \\&\times \left[ {{\begin{array}{c} {\mathbf{K}_{\mathbf{L3}}^{13} +\mathbf{K}_{\mathbf{NL1}}^{13} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) } \\ {\mathbf{K}_{\mathbf{L3}}^{23} +\mathbf{K}_{\mathbf{NL1}}^{23} \left( {\mathbf{q}_\mathbf{w} \left( t \right) } \right) } \\ \end{array} }} \right] . \end{aligned}$$
(37)
\(\mathbf{K}_{\mathbf{NLS}}^{33} \) is dependent linearly and quadratically on transverse deflection,
$$\begin{aligned} \mathbf{K}_{\mathbf{NLS}}^{33} =\mathbf{K}_{\mathbf{NL1}}^{33} +\mathbf{K}_{\mathbf{NL2}}^{33} +\mathbf{K}_{\mathbf{NL3}}^{33} , \end{aligned}$$
(38)
where
$$\begin{aligned} \mathbf{K}_{\mathbf{NL3}}^{33}= & {} -2\left[ {{\begin{array}{c@{\quad }c} {\mathbf{K}_{\mathbf{NL1}}^{13}}^\mathrm{T}&{} {\mathbf{K}_{\mathbf{NL1}}^{23}}^\mathrm{T} \\ \end{array} }} \right] \left[ {{\begin{array}{c@{\quad }c} {\mathbf{K}_{\mathbf{L1}}^{11} }&{} {\mathbf{K}_{\mathbf{L1}}^{12} } \\ {\mathbf{K}_{\mathbf{L1}}^{21} }&{} {\mathbf{K}_{\mathbf{L1}}^{22} } \\ \end{array} }} \right] ^{-1}\nonumber \\&\times \,\left[ {{\begin{array}{c} {\mathbf{K}_{\mathbf{NL1}}^{13} } \\ {\mathbf{K}_{\mathbf{NL1}}^{23} } \\ \end{array} }} \right] . \end{aligned}$$
(39)
