Appendix 1: 1-D inertial terms
The inertial terms are defined as:
$$\begin{aligned} I_{1}= & {} \oint _C m_{0} (\ddot{u}_{0} + z{\ddot{\phi }}-x{\dot{\phi }}^2-x\phi {\ddot{\phi }})\hbox {d}s=b_1{\ddot{u}}_0, \end{aligned}$$
(66)
$$\begin{aligned} I_{2}= & {} \oint _C m_{0} [\ddot{v}_{0} + x{\ddot{\theta }} _{z} + z\ddot{\theta }_{x} - F_{w}\ddot{\phi }']\hbox {d}s=b_1{\ddot{v}}_0, \end{aligned}$$
(67)
$$\begin{aligned} I_{3}= & {} \oint _C m_{0} (\ddot{w}_{0} - x{\ddot{\phi }}-z{\dot{\phi }}^2-z\phi {\ddot{\phi }} )\hbox {d}s=b_1{\ddot{w}}_0, \end{aligned}$$
(68)
$$\begin{aligned} I_{4}= & {} \oint _C m_{0} [z{\ddot{u}}_{0} + (x^2+z^2){\ddot{\phi }}-x{\ddot{w}}_0\nonumber \\&+\,x\phi {\ddot{u}}_0-z\phi {\ddot{w}}_0]\hbox {d}s\nonumber \\= & {} (b_4+b_5){\ddot{\phi }}, \end{aligned}$$
(69)
$$\begin{aligned} I_{5}= & {} \oint _C m_{0} [z{\ddot{v}}_{0} + zx{\ddot{\theta }}_z +z^{2}{\ddot{\theta }}_x - zF_{w}{\ddot{\phi }}']\hbox {d}s\nonumber \\= & {} b_4{\ddot{\theta }}_x, \end{aligned}$$
(70)
$$\begin{aligned} I_{6}= & {} \oint _C m_{0} \left[ x{\ddot{v}}_0 + x^2{\ddot{\theta }}_z + xz{\ddot{\theta }}_x - xF_{w}{\ddot{\phi }}'\right] \hbox {d}s\nonumber \\= & {} b_5{\ddot{\theta }}_z, \end{aligned}$$
(71)
$$\begin{aligned} I_{9}= & {} \oint _C m_{0} [-F_{w} {\ddot{v}}_{0}- xF_{w}{\ddot{\theta }}_z - zF_{w} {\ddot{\theta }}_x + F_{w}^{2}{\ddot{\phi }}' ]\hbox {d}s\nonumber \\= & {} b_{10}{\ddot{\phi }}', \end{aligned}$$
(72)
in which
$$\begin{aligned} m_{0}=\sum \limits _{k=1}^{m_l} \int _{h_{(k^-)}} ^{h_{(k^+)}}\rho _{(k)}\hbox {d}n. \end{aligned}$$
(73)
The inertial coefficients in Eqs. (27a–f) are defind as:
$$\begin{aligned} b_1= & {} \oint _{C} {m_{0} \hbox {d}s}, \quad (b_{4},\;b_{5} )={\oint _{C} {(z^{2}}} ,x^{2})m_{0} \hbox {d}s, \end{aligned}$$
(74)
$$\begin{aligned} b_{10}= & {} \oint _{C} m_0 F_w^2(s)\hbox {d}s. \end{aligned}$$
(75)
Appendix 2: Global stiffness quantities
The reduced stiffness coefficient\(K_{ij}\) are defined as:
$$\begin{aligned} K_{11}= & {} A_{22} - \dfrac{A_{12}^2}{A_{11}},\nonumber \\ K_{12}= & {} A_{26} - \dfrac{A_{12} A_{16}}{A_{11}}=K_{21}, \end{aligned}$$
(76)
$$\begin{aligned} K_{13}= & {} \Bigl (A_{26} - \dfrac{A_{12} A_{16}}{A_{11}}\Bigr )\psi (s) \nonumber \\&+\,2 \Big ( B_{26}-\dfrac{A_{12}B_{16}}{A_{11}} \Big ),\end{aligned}$$
(77)
$$\begin{aligned} K_{14}= & {} B_{22} - \dfrac{A_{12} B_{12}}{A_{11}}=K_{41},\nonumber \\ K_{22}= & {} A_{66} - \dfrac{A_{16}^{2}}{A_{11}}, \end{aligned}$$
(78)
$$\begin{aligned} K_{23}= & {} \Bigl (A_{66} - \dfrac{A_{16}^{2}}{A_{11}}\Bigr )\psi (s)\nonumber \\&+\,2 \Big ( B_{66} -\dfrac{A_{16}B_{16}}{A_{11}} \Big ),\end{aligned}$$
(79)
$$\begin{aligned} K_{24}= & {} B_{26} - \dfrac{A_{16} B_{12}}{A_{11}}=K_{42}, \end{aligned}$$
(80)
$$\begin{aligned} K_{43}= & {} \Bigl (B_{26} - \dfrac{B_{12} A_{16}}{A_{11}}\Bigr )\psi (s) \nonumber \\&+\,2 \Big ( D_{26}-\dfrac{B_{12}B_{16}}{A_{11}} \Big ), \end{aligned}$$
(81)
$$\begin{aligned} K_{44}= & {} D_{22} - \dfrac{B_{12}^{2}}{A_{11}},\nonumber \\ K_{51}= & {} B_{26} - \dfrac{B_{16} A_{12}}{A_{11}}, \end{aligned}$$
(82)
$$\begin{aligned} K_{52}= & {} B_{66}- \dfrac{B_{16} A_{16}}{A_{11}}, \end{aligned}$$
(83)
$$\begin{aligned} K_{53}= & {} \Bigl (B_{66}- \dfrac{B_{16} A_{16}}{A_{11}}\Bigr )\psi (s)\nonumber \\&+\,2 \Big ( D_{66}-\dfrac{{B_{16}}^2}{A_{11}} \Big ),\end{aligned}$$
(84)
$$\begin{aligned} K_{54}= & {} D_{26} - \dfrac{B_{12} B_{16}}{A_{11}}. \end{aligned}$$
(85)
The global stiffness quantities \(a_{ij}\)
\((=a_{ji})\):
$$\begin{aligned} a_{11}= & {} \oint _C K_{11} \hbox {d}s,\nonumber \\ a_{12}= & {} \oint _C \left( K_{11} x-K_{14}\dfrac{\hbox {d}z}{\hbox {d}s}\right) \hbox {d}s,\end{aligned}$$
(86)
$$\begin{aligned} a_{13}= & {} \oint _C \left( zK_{11}+K_{14}\dfrac{\hbox {d}x}{\hbox {d}s}\right) \hbox {d}s,\nonumber \\ a_{14}= & {} \oint _C K_{12} \dfrac{\hbox {d}x}{\hbox {d}s} \hbox {d}s, \end{aligned}$$
(87)
$$\begin{aligned} a_{15}= & {} \oint _C K_{12} \dfrac{\hbox {d}z}{\hbox {d}s}\hbox {d}s,\nonumber \\ a_{16}= & {} -\oint _C \left[ F_w K_{11}+K_{14} a(s)\right] \hbox {d}s, \end{aligned}$$
(88)
$$\begin{aligned} a_{17}= & {} \oint _C K_{13} \hbox {d}s,\nonumber \\ a_{18}= & {} \oint _C [K_{11}(x^2+z^2)+2r_n K_{41}]\hbox {d}s,\end{aligned}$$
(89)
$$\begin{aligned} a_{22}= & {} \oint _C \left[ x^{2}K_{11}-2x\dfrac{\hbox {d}z}{\hbox {d}s}K_{14} + \Bigl (\dfrac{\hbox {d}z}{\hbox {d}s}\Bigr )^{2}K_{44}\right] \hbox {d}s,\nonumber \\ \end{aligned}$$
(90)
$$\begin{aligned} a_{23}= & {} \oint _C \left[ xz K_{11}+x\dfrac{\hbox {d}x}{\hbox {d}s}K_{14}\right] \hbox {d}s\nonumber \\&-\oint _C \left[ z\dfrac{\hbox {d}z}{\hbox {d}s} K_{14}+\dfrac{\hbox {d}z}{\hbox {d}s}\dfrac{\hbox {d}x}{\hbox {d}s}K_{44}\right] \hbox {d}s,\end{aligned}$$
(91)
$$\begin{aligned} a_{24}= & {} \oint _C \left[ x\dfrac{\hbox {d}x}{\hbox {d}s}K_{12}-\dfrac{\hbox {d}z}{\hbox {d}s}\dfrac{\hbox {d}x}{\hbox {d}s}K_{42}\right] \hbox {d}s, \end{aligned}$$
(92)
$$\begin{aligned} a_{25}= & {} \oint _C \left[ x\dfrac{\hbox {d}z}{\hbox {d}s}K_{12}-\Bigl (\dfrac{\hbox {d}z}{\hbox {d}s}\Bigr )^{2}K_{42}\right] \hbox {d}s, \end{aligned}$$
(93)
$$\begin{aligned} a_{26}= & {} \oint _C \left[ -xF_wK_{11}-x a(s)K_{14}\right] \hbox {d}s \nonumber \\&+\,\oint _C \left[ K_{41}\dfrac{\hbox {d}z}{\hbox {d}s}F_w+a(s)\dfrac{\hbox {d}z}{\hbox {d}s}K_{44}\right] \hbox {d}s,\end{aligned}$$
(94)
$$\begin{aligned} a_{27}= & {} \oint _C \left[ xK_{13} - \dfrac{\hbox {d}z}{\hbox {d}s}K_{43}\right] \hbox {d}s, \end{aligned}$$
(95)
$$\begin{aligned} a_{28}= & {} \oint _C \left\{ x[K_{11}(x^2+z^2)+2r_n K_{41}]\right\} \hbox {d}s\nonumber \\&-\oint _C \left\{ \dfrac{\hbox {d}z}{\hbox {d}s}[K_{14}(x^2+z^2)+2r_n K_{44}]\right\} \hbox {d}s,\nonumber \\ \end{aligned}$$
(96)
$$\begin{aligned} a_{33}= & {} \oint _C \left[ z^{2}K_{11} + 2z\dfrac{\hbox {d}x}{\hbox {d}s}K_{14} + \Bigl (\dfrac{\hbox {d}x}{\hbox {d}s}\Bigr )^{2}K_{44}\right] \hbox {d}s,\nonumber \\ \end{aligned}$$
(97)
$$\begin{aligned} a_{34}= & {} \oint _C \left[ z\dfrac{\hbox {d}x}{\hbox {d}s}K_{12}+\Bigl (\dfrac{\hbox {d}x}{\hbox {d}s}\Bigr )^2 K_{42}\right] \hbox {d}s, \end{aligned}$$
(98)
$$\begin{aligned} a_{35}= & {} \oint _C \left[ z\dfrac{\hbox {d}z}{\hbox {d}s}K_{12}+\dfrac{\hbox {d}x}{\hbox {d}s}\dfrac{\hbox {d}z}{\hbox {d}s} K_{42}\right] \hbox {d}s, \end{aligned}$$
(99)
$$\begin{aligned} a_{36}= & {} \oint _C \left[ -zF_wK_{11}-a(s)zK_{14}\right] \hbox {d}s\nonumber \\&-\oint _C \left[ \dfrac{\hbox {d}x}{\hbox {d}s}F_wK_{41}+a(s)\dfrac{\hbox {d}x}{\hbox {d}s}K_{44}\right] \hbox {d}s, \end{aligned}$$
(100)
$$\begin{aligned} a_{37}= & {} \oint _C\left[ zK_{13} + \dfrac{\hbox {d}x}{\hbox {d}s}K_{43}\right] \hbox {d}s, \end{aligned}$$
(101)
$$\begin{aligned} a_{38}= & {} \oint _C \left\{ z[K_{11}(x^2+z^2)+2r_n K_{41}]\right\} \hbox {d}s\nonumber \\&+\,\oint _C \left\{ \dfrac{\hbox {d}x}{\hbox {d}s}[K_{14}(x^2+z^2)+2r_n K_{44}]\right\} \hbox {d}s,\nonumber \\ \end{aligned}$$
(102)
$$\begin{aligned} a_{44}= & {} \oint _C\left[ \Bigl (\dfrac{\hbox {d}x}{\hbox {d}s}\Bigr )^{2}K_{22} + \Bigl (\dfrac{\hbox {d}z}{\hbox {d}s}\Bigr )^{2}\bar{A}_{44} \right] \hbox {d}s, \end{aligned}$$
(103)
$$\begin{aligned} a_{45}= & {} \oint _C\left[ \dfrac{\hbox {d}x}{\hbox {d}s}\dfrac{\hbox {d}z}{\hbox {d}s}K_{22}-\dfrac{\hbox {d}x}{\hbox {d}s}\dfrac{\hbox {d}z}{\hbox {d}s}A_{44}\right] \hbox {d}s, \end{aligned}$$
(104)
$$\begin{aligned} a_{46}= & {} \oint _C\left[ -F_w\dfrac{\hbox {d}x}{\hbox {d}s}K_{21}-a(s)\dfrac{\hbox {d}x}{\hbox {d}s}K_{24}\right] \hbox {d}s, \end{aligned}$$
(105)
$$\begin{aligned} a_{47}= & {} \oint _C K_{23}\dfrac{\hbox {d}x}{\hbox {d}s} \hbox {d}s, \end{aligned}$$
(106)
$$\begin{aligned} a_{48}= & {} \oint _C \dfrac{\hbox {d}x}{\hbox {d}s}[K_{12}(x^2+z^2)+2r_n K_{42}]\hbox {d}s,\end{aligned}$$
(107)
$$\begin{aligned} a_{55}= & {} \oint _C\left[ \Bigl (\dfrac{\hbox {d}z}{\hbox {d}s}\Bigr )^{2}K_{22} + \Bigl (\dfrac{\hbox {d}x}{\hbox {d}s}\Bigr )^{2}\bar{A}_{44} \right] \hbox {d}s, \end{aligned}$$
(108)
$$\begin{aligned} a_{56}= & {} \oint _C\left[ -F_w\dfrac{\hbox {d}z}{\hbox {d}s}K_{21}-a(s)\dfrac{\hbox {d}z}{\hbox {d}s}K_{24}\right] \hbox {d}s, \end{aligned}$$
(109)
$$\begin{aligned} a_{57}= & {} \oint _C K_{23}\dfrac{\hbox {d}z}{\hbox {d}s},\end{aligned}$$
(110)
$$\begin{aligned} a_{58}= & {} \oint _C \dfrac{\hbox {d}z}{\hbox {d}s}[K_{12}(x^2+z^2)+2r_n K_{42}]\hbox {d}s,\end{aligned}$$
(111)
$$\begin{aligned} a_{66}= & {} \oint _C\left[ F_{w} ^{2}K_{11} + 2F_{w} a(s)K_{14} + a(s)^{2}K_{44}\right] \hbox {d}s,\nonumber \\ \end{aligned}$$
(112)
$$\begin{aligned} a_{67}= & {} -\oint _C\left[ F_wK_{13}+a(s)K_{43}\right] \hbox {d}s, \end{aligned}$$
(113)
$$\begin{aligned} a_{68}= & {} -\oint _C \left\{ F_w[K_{11}(x^2+z^2)+2r_n K_{41}]\right\} \hbox {d}s\nonumber \\&-\oint _C \left\{ a[K_{14}(x^2+z^2)+2r_n K_{44}]\right\} \hbox {d}s,\end{aligned}$$
(114)
$$\begin{aligned} a_{77}= & {} \oint _C \left( \psi (s)K_{23}+2 K_{53} \right) \hbox {d}s, \end{aligned}$$
(115)
$$\begin{aligned} a_{78}= & {} \oint _C [K_{13}(x^2+z^2)+2r_n K_{43}]\hbox {d}s, \end{aligned}$$
(116)
$$\begin{aligned} a_{88}= & {} \oint _C \left\{ (x^2+z^2)[K_{11}(x^2+z^2)+2r_n K_{41}]\right\} \hbox {d}s\nonumber \\&+\,\oint _C \left\{ 2r_n[K_{14}(x^2+z^2)+2r_n K_{44}]\right\} \hbox {d}s,\nonumber \\ \end{aligned}$$
(117)
where
$$\begin{aligned} {\bar{A}}_{44}=A_{44}-\dfrac{A_{45}^2}{A_{55}}. \end{aligned}$$
(118)
Appendix 3: Quadratic and cubic nonlinear terms
Quadratic nonlinear terms:
$$\begin{aligned}&N_u^2 \,:\; \left\{ a_{11} v'_0 u'_0 + a_{14} \theta _z u'_0 + a_{55} \theta _x \phi +(a_{55}-a_{44}) w'_0 \phi \right\} '\nonumber \\&\quad +\, \Big \{ a_{14} \Big ( \dfrac{3}{2} (u'_0)^2 + \dfrac{1}{2} (w'_0)^2 \Big ) \nonumber \\&\quad +\,a_{33} \theta '_x \phi ' +a_{37}(\phi ')^2 \Big \}'\nonumber \\&\quad +\, \left\{ a_{48}\dfrac{1}{2}(\phi ')^2 \right\} ', \end{aligned}$$
(119a)
$$\begin{aligned}&N_v^2 \,:\; \Big \{ a_{11} \left[ \dfrac{1}{2}(u'_0)^2 +\dfrac{1}{2}(w'_0)^2 \right] \nonumber \\&\quad -\, a_{14} w'_0 \phi + a_{18} \dfrac{1}{2}(\phi ')^2 \Big \}',\end{aligned}$$
(119b)
$$\begin{aligned}&N_w^2 \,:\; \left\{ a_{11} v'_0w'_0 +a_{14}\theta _z w'_0 -a_{14} v'_0 \phi - a_{44}\theta _z \phi \right\} '\nonumber \\&\quad +\, \left\{ a_{14} u'_0w'_0 -a_{22}\theta '_z\phi ' +(a_{55}-a_{44})u'_0 \phi \right\} ' , \end{aligned}$$
(119c)
$$\begin{aligned}&N_\phi ^2 \,:\; \left\{ a_{18} v'_0 \phi ' + a_{48} \phi ' \theta _z + 2 a_{37} u'_0 \phi ' +a_{33}u'_0 \theta '_x \right\} ' \nonumber \\&\quad +\, \left\{ -a_{22}w'_0 \theta '_z +a_{48}u'_0 \phi ' \right\} ' + a_{14} v'_0w'_0 + a_{44}w'_0\theta _z \nonumber \\&\quad -\,a_{55}u'_0\theta _x + (a_{44}-a_{55})u'_0w'_0 , \end{aligned}$$
(119d)
$$\begin{aligned}&N_x^2 \, :\; \left\{ a_{33} u'_0\phi ' \right\} ' - a_{55} u'_0 \phi ,\end{aligned}$$
(119e)
$$\begin{aligned}&N_z^2 \,: \; \left\{ -a_{22} w'_0\phi ' \right\} ' - a_{14} \left[ \dfrac{1}{2} (u'_0)^2 + \dfrac{1}{2} (w'_0)^2 \right] \nonumber \\&\quad +\, a_{44}w'_0 \phi - a_{48}\dfrac{1}{2}(\phi ')^2. \end{aligned}$$
(119f)
Cubic nonlinear terms:
$$\begin{aligned}&N_u^3 \,:\; \left\{ a_{11} \left[ \dfrac{1}{2} (u'_0)^3 + \dfrac{1}{2} u'_0 (w'_0)^2 \right] + a_{18} \dfrac{1}{2} u'_0 (\phi ')^2 \right\} ' \nonumber \\&\quad \left\{ -a_{44} u'_0 \phi ^2 -a_{14} \dfrac{1}{2}v'_0\phi ^2 - a_{44}\dfrac{1}{2} \theta _z \phi ^2 \right\} ' \nonumber \\&\quad + \,\Big \{ a_{33} u'_0 (\phi ')^2 -a_{14} u'_0 w'_0 \phi \nonumber \\&\quad -\,a_{22} \theta '_z \phi ' \phi +a_{55} u'_0 \phi ^2 \Big \}' , \end{aligned}$$
(120a)
$$\begin{aligned}&N_v^3 \,:\; \left\{ - a_{14} \dfrac{1}{2} u'_0 \phi ^2 \right\} ',\end{aligned}$$
(120b)
$$\begin{aligned}&N_w^3 \,:\; \left\{ a_{11} \left[ \dfrac{1}{2}w'_0(u'_0)^2 + \dfrac{1}{2}(w'_0)^3 \right] +a_{18}\dfrac{1}{2}w'_0(\phi ')^2 \right\} '\nonumber \\&\quad +\, \left\{ -a_{55}\dfrac{1}{2}\theta _x \phi ^2 +a_{22}w'_0(\phi ')^2 - a_{55}w'_0\phi ^2 \right\} ' \nonumber \\&\quad +\, \left\{ -a_{14} \left( \dfrac{1}{2}(u'_0)^2 \phi + \dfrac{3}{2}(w'_0)^2\phi \right) -a_{33}\theta '_x \phi '\phi \right\} ' \nonumber \\&\quad +\, \left\{ - a_{37}(\phi ')^2 \phi -a_{48}\dfrac{1}{2}(\phi ')^2\phi +a_{44} w'_0 \phi ^2 \right\} ' ,\nonumber \\ \end{aligned}$$
(120c)
$$\begin{aligned}&N_\phi ^3 \,:\; \left\{ a_{18} \left[ \dfrac{1}{2}\phi '(u'_0)^2 + \dfrac{1}{2}\phi '(w'_0)^2 \right] +a_{88}\dfrac{1}{2}(\phi ')^3 \right\} ' \nonumber \\&\quad +\, \left\{ a_{33} \phi '(u'_0)^2 +a_{22}\phi '(w'_0)^2 \right\} ' \nonumber \\&\quad -\, \Big \{ 2 a_{37} w'_0 \phi '\phi + a_{33}w'_0\theta '_x \phi +a_{22}u'_0\theta '_z \phi \nonumber \\&\qquad \quad +\, a_{48}w'_0\phi '\phi \Big \}'\nonumber \\&\quad +\, a_{33}w'_0\phi '\theta '_x +a_{37}w'_0(\phi ')^2 \nonumber \\&\quad +\, a_{22}u'_0\phi '\theta '_z+ a_{14} \left[ \dfrac{1}{2}(u'_0)^2 w'_0 + \dfrac{1}{2}(w'_0)^3\right] \nonumber \\&\quad +\, a_{48}\dfrac{1}{2}w'_0(\phi ')^2 + a_{14} v'_0u'_0 \phi \nonumber \\&\quad + \,a_{44}u'_0\theta _z \phi + a_{55}w'_0\theta _x \phi \nonumber \\&\quad +\, (a_{44}- a_{55}) \left[ (u'_0)^2\phi - (w'_0)^2\phi \right] ,\end{aligned}$$
(120d)
$$\begin{aligned}&N_x^3 \, :\; \left\{ - a_{33} w'_0\phi '\phi \right\} ' +\, a_{55} \dfrac{1}{2} w'_0 \phi ^2 , \end{aligned}$$
(120e)
$$\begin{aligned}&N_z^3 \,: \; \left\{ -a_{22} u'_0 \phi ' \phi \right\} ' + a_{44} \dfrac{1}{2} u'_0 \phi ^2 . \end{aligned}$$
(120f)
Appendix 4: Matrixes in Eqs. 29 and 39
Mass matrix
$$\begin{aligned} {\mathbf {M}}=\int _0^L \begin{bmatrix} b_1{\varvec{\Psi }}_u{\varvec{\Psi }}_u^T&0&0&0&0&0 \\ {}&b_1{\varvec{\Psi }}_v{\varvec{\Psi }}_v^T&0&0&0&0 \\ {}{} & {} b_1{\varvec{\Psi }}_w{\varvec{\Psi }}_w^T&0&0&0 \\ {}{} & {} {}&{\mathbf {M}}_{44}&0&0 \\ {}&{\text {Symm}}{} & {} {}&{\mathbf {M}}_{55}&0 \\ {}{} & {} {}{} & {} {}&{\mathbf {M}}_{66} \\ \end{bmatrix} \hbox {d}y.\nonumber \\ \end{aligned}$$
(121)
with
$$\begin{aligned} \left\{ \begin{array}{l} {\mathbf {M}}_{44}=(b_4+b_5){\varvec{\Psi }}_\phi {\varvec{\Psi }}_\phi ^T +(b_{10}+b_{18}){\varvec{\Psi '}}_\phi {\varvec{\Psi '}}_\phi ^T, \\ {\mathbf {M}}_{55}=(b_4+b_{14}){\varvec{\Psi }}_x{\varvec{\Psi }}_x^T,\\ {\mathbf {M}}_{66}=(b_5+b_{15}){\varvec{\Psi }}_z{\varvec{\Psi }}_z^T, \\ \end{array} \right. \end{aligned}$$
Stiffness matrix
$$\begin{aligned} {\mathbf {K}}=\int _0^L \begin{bmatrix} a_{44}{\varvec{\Psi }}'_u {{\varvec{\Psi }}'_u}^T&a_{14}{\varvec{\Psi }}'_u {{\varvec{\Psi }}'_v}^T&0&0&0&{\mathbf {K}}_{16} \\ {}&a_{11}{\varvec{\Psi }}'_v {{\varvec{\Psi }}'_v}^T&{\mathbf {K}}_{23}&0&{\mathbf {K}}_{25}&{\mathbf {K}}_{26} \\ {}{} & {} {\mathbf {K}}_{33}&0&{\mathbf {K}}_{35}&0 \\ {}{} & {} {}&{\mathbf {K}}_{44}&{\mathbf {K}}_{45}&{\mathbf {K}}_{46} \\ {}&{\text {Symm}}{} & {} {}&{\mathbf {K}}_{55}&0 \\ {}{} & {} {}{} & {} {}&{\mathbf {K}}_{66} \end{bmatrix} \hbox {d}y. \end{aligned}$$
(122)
with
$$\begin{aligned} \left\{ \begin{array}{l} {\mathbf {K}}_{16}=a_{44}{\varvec{\Psi }}'_u {{\varvec{\Psi }}_z}^T ,\\ {\mathbf {K}}_{23}=a_{15}{\varvec{\Psi }}'_v {{\varvec{\Psi }}'_w}^T ,\\ {\mathbf {K}}_{25}=a_{15}{\varvec{\Psi }}'_v {{\varvec{\Psi }}'_x}^T ,\\ {\mathbf {K}}_{26}=a_{14}{\varvec{\Psi }}'_v {{\varvec{\Psi }}_z}^T ,\\ {\mathbf {K}}_{33}=a_{55}{\varvec{\Psi }}'_w {{\varvec{\Psi }}'_w}^T ,\\ {\mathbf {K}}_{35}=a_{55}{\varvec{\Psi }}'_w {{\varvec{\Psi }}'_x}^T ,\\ {\mathbf {K}}_{44}=a_{77}{\varvec{\Psi }}'_\phi {{\varvec{\Psi }}'_\phi }^T+a_{66}{\varvec{\Psi }}''_\phi {{\varvec{\Psi }}''_\phi }^T, \\ {\mathbf {K}}_{45}=a_{37}{\varvec{\Psi }}'_\phi {{\varvec{\Psi }}'_x}^T ,\\ {\mathbf {K}}_{46}=a_{27}{\varvec{\Psi }}'_\phi {{\varvec{\Psi }}'_z}^T ,\\ {\mathbf {K}}_{55}=a_{33}{\varvec{\Psi }}'_x {{\varvec{\Psi }}'_x}^T+a_{55}{\varvec{\Psi }}_x {{\varvec{\Psi }}_x}^T, \\ {\mathbf {K}}_{66}=a_{22}{\varvec{\Psi }}'_z {{\varvec{\Psi }}'_z}^T+a_{44}{\varvec{\Psi }}_z {{\varvec{\Psi }}_z}^T. \\ \end{array} \right. \end{aligned}$$
External forces vector
$$\begin{aligned} {\mathbf {Q}}=\begin{Bmatrix} \int _{0}^{L} p_x {\varvec{\Psi }}_u \hbox {d}y + {\bar{Q}}_x {\varvec{\Psi }}_u (L) \\ \int _{0}^{L} p_y {\varvec{\Psi }}_v \hbox {d}y + {\bar{T}}_y {\varvec{\Psi }}_v (L) \\ \int _{0}^{L} p_z {\varvec{\Psi }}_w \hbox {d}y + {\bar{Q}}_z {\varvec{\Psi }}_w (L) \\ \int _{0}^{L} (m_y +b'_w) {\varvec{\Psi }}_\phi \hbox {d}y + [ {\bar{M}}_y {\varvec{\Psi }}_\phi (L) +{\bar{B}}_w {\varvec{\Psi }}'_\phi (L) ]\\ \int _{0}^{L} m_x {\varvec{\Psi }}_x \hbox {d}y + {\bar{M}}_x {\varvec{\Psi }}_x (L) \\ \int _{0}^{L} m_z {\varvec{\Psi }}_z \hbox {d}y + {\bar{M}}_z {\varvec{\Psi }}_z (L) \\ \end{Bmatrix}. \end{aligned}$$
(123)
Nonlinear terms
$$\begin{aligned} {\mathbf {N}}_{mn}=\left\{ {\begin{array}{*{20}{c}} {{\mathbf {N}}_{mn}^u}&{{\mathbf {N}}_{mn}^v}&{{\mathbf {N}}_{mn}^w}&{{\mathbf {N}}_{mn}^\phi }&{{\mathbf {N}}_{mn}^x}&{{\mathbf {N}}_{mn}^z} \end{array}} \right\} ^T, \end{aligned}$$
(124)
with
$$\begin{aligned} {\mathbf {N}}_{mn}^u= & {} \int _0^L {\varvec{\Psi }}'_u \Big \{ \dfrac{3}{2} a_{14} {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}'_u}^T {\mathbf {V}}_m^u} \nonumber \\&+\,a_{14} {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}_z}^T {\mathbf {V}}_m^z} \nonumber \\&+\,\dfrac{1}{2} a_{14} {{{\varvec{\Psi }}'_w}^T {\mathbf {V}}_n^w {{\varvec{\Psi }}'_w}^T {\mathbf {V}}_m^w} \nonumber \\&+\,(a_{55} - a_{44}){{{\varvec{\Psi }}'_w}^T {\mathbf {V}}_n^w {{\varvec{\Psi }}_\phi }^T {\mathbf {V}}_m^\phi } \nonumber \\&+\,\left( \dfrac{1}{2} a_{48} + a_{37}\right) {{{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_n^\phi {{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_m^\phi } \nonumber \\&+\, a_{33} {{{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_n^\phi {{\varvec{\Psi }}'_x}^T {\mathbf {V}}_m^x}\nonumber \\&+\,a_{55} {{{\varvec{\Psi }}_\phi }^T {\mathbf {V}}_n^\phi {{\varvec{\Psi }}_x}^T {\mathbf {V}}_m^x} \nonumber \\&+\,a_{11} {{{\varvec{\Psi }}'_v}^T {\mathbf {V}}_n^v {{\varvec{\Psi }}'_u}^T {\mathbf {V}}_m^u} \Big \} \hbox {d}y, \end{aligned}$$
(125a)
$$\begin{aligned} {\mathbf {N}}_{mn}^v= & {} \int _0^L {\varvec{\Psi }}'_v \Big \{ \dfrac{1}{2} a_{11} {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}'_u}^T {\mathbf {V}}_m^u} \nonumber \\&+\,\dfrac{1}{2} a_{11} {{{\varvec{\Psi }}'_w}^T {\mathbf {V}}_n^w {{\varvec{\Psi }}'_w}^T {\mathbf {V}}_m^w} \nonumber \\&-\, a_{14} {{{\varvec{\Psi }}'_w}^T {\mathbf {V}}_n^w {{\varvec{\Psi }}_\phi }^T {\mathbf {V}}_m^\phi } \nonumber \\&+\, \dfrac{1}{2} a_{18} {{{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_n^\phi {{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_m^\phi } \Big \} \hbox {d}y, \end{aligned}$$
(125b)
$$\begin{aligned} {\mathbf {N}}_{mn}^w= & {} \int _0^L {\varvec{\Psi }}'_w \Big \{ a_{14} {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}'_w}^T {\mathbf {V}}_m^w} \nonumber \\&+\,(a_{55} - a_{44}) {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}_\phi }^T {\mathbf {V}}_m^\phi }\nonumber \\&+\,a_{14} {{{\varvec{\Psi }}'_w}^T {\mathbf {V}}_n^w {{\varvec{\Psi }}_z}^T {\mathbf {V}}_m^z} \nonumber \\&-\,a_{22} {{{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_n^\phi {{\varvec{\Psi }}'_z}^T {\mathbf {V}}_m^z} -a_{44} {{{\varvec{\Psi }}_\phi }^T {\mathbf {V}}_n^\phi {{\varvec{\Psi }}_z}^T {\mathbf {V}}_m^z}\nonumber \\&+\, a_{11} {{{\varvec{\Psi }}'_v}^T {\mathbf {V}}_n^v {{\varvec{\Psi }}'_w}^T {\mathbf {V}}_m^w}\nonumber \\&-\, a_{14} {{{\varvec{\Psi }}'_v}^T {\mathbf {V}}_n^v {{\varvec{\Psi }}_\phi }^T {\mathbf {V}}_m^\phi } \Big \} \hbox {d}y,\end{aligned}$$
(125c)
$$\begin{aligned} {\mathbf {N}}_{mn}^\phi= & {} \int _0^L \Big \{ {\varvec{\Psi }}_\phi \big [ a_{44}-a_{55} \big ] {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}'_w}^T {\mathbf {V}}_m^w} \nonumber \\&+ \,{\varvec{\Psi }}'_\phi a_{33} {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}'_x}^T {\mathbf {V}}_m^x}\nonumber \\&+\, {\varvec{\Psi }}_\phi a_{55} {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}_x}^T {\mathbf {V}}_m^x} \nonumber \\&-\, {\varvec{\Psi }}'_\phi a_{55} {{{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_n^\phi {{\varvec{\Psi }}_x}^T {\mathbf {V}}_m^x} \nonumber \\&+\, {\varvec{\Psi }}_\phi a_{48} {{{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_n^\phi {{\varvec{\Psi }}_z}^T {\mathbf {V}}_m^z}\nonumber \\&-\, {\varvec{\Psi }}'_\phi a_{22} {{{\varvec{\Psi }}'_w}^T {\mathbf {V}}_n^w {{\varvec{\Psi }}'_z}^T {\mathbf {V}}_m^z} \nonumber \\&+\, {\varvec{\Psi }}_\phi a_{44} {{{\varvec{\Psi }}'_w}^T {\mathbf {V}}_n^w {{\varvec{\Psi }}_z}^T {\mathbf {V}}_m^z}\nonumber \\&+\, {\varvec{\Psi }}'_\phi \big [ 2 a_{37} +a_{48} \big ] {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_m^\phi } \nonumber \\&+\, {\varvec{\Psi }}'_\phi a_{18} {{{\varvec{\Psi }}'_v}^T {\mathbf {V}}_n^v {{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_m^\phi } \nonumber \\&+\, {\varvec{\Psi }}_\phi a_{14} {{{\varvec{\Psi }}'_v}^T {\mathbf {V}}_n^v {{\varvec{\Psi }}'_w}^T {\mathbf {V}}_m^w} \Big \} \hbox {d}y, \end{aligned}$$
(125d)
$$\begin{aligned} {\mathbf {N}}_{mn}^x= & {} \int _0^L \Big \{ {\varvec{\Psi }}'_x a_{33} {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_m^\phi } \nonumber \\&+\,{\varvec{\Psi }}_x a_{55} {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}_\phi }^T {\mathbf {V}}_m^\phi } \Big \} \hbox {d}y,\end{aligned}$$
(125e)
$$\begin{aligned} {\mathbf {N}}_{mn}^z= & {} \int _0^L \Big \{ {\varvec{\Psi }}_z \dfrac{1}{2} a_{14} \big [ {{{\varvec{\Psi }}'_u}^T {\mathbf {V}}_n^u {{\varvec{\Psi }}'_u}^T {\mathbf {V}}_m^u}\nonumber \\&+\, {{{\varvec{\Psi }}'_w}^T {\mathbf {V}}_n^w {{\varvec{\Psi }}'_w}^T {\mathbf {V}}_m^w} \big ] - {\varvec{\Psi }}'_z a_{22} {{{\varvec{\Psi }}'_w}^T {\mathbf {V}}_n^w {{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_m^\phi }\nonumber \\&- \,{\varvec{\Psi }}_z a_{44} {{{\varvec{\Psi }}'_w}^T {\mathbf {V}}_n^w {{\varvec{\Psi }}_\phi }^T {\mathbf {V}}_m^\phi } \nonumber \\&+ \,{\varvec{\Psi }}_z \dfrac{1}{2} a_{48} {{{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_n^\phi {{\varvec{\Psi }}'_\phi }^T {\mathbf {V}}_m^\phi } \Big \} \hbox {d}y. \end{aligned}$$
(125f)
