By assuming only transverse external load on the system (
\(F_{{s}{1}} =F_{S2} = 0\) and
\(F_{S3} = F)\) and defining nondimensional terms as
$$\begin{aligned} {x_{1}}^{*}= & {} \frac{x_{1} }{L},\,\,\,\,\,{u_{1}}^{*} =\frac{u_{1} }{h},\,\,\,\,{u_{3}}^{*} =\frac{u_{3} }{h},\,\,\,\,{u_{30}}^{*} =\frac{u_{30} }{h},\,\,\,\,\,{I_{00}}^{*} =\frac{I_{00} h^{2}}{I_{44} },\,\,\,\,\nonumber \\ {I_{55}}^{*}= & {} \frac{I_{55} h^{2}}{I_{44} },\,\,\,\,\,{I_{66}}^{*} =\frac{I_{66} }{I_{44} },\,\,\,\,\,{I_{77}}^{*} =\frac{I_{77} h^{2}}{I_{44} }, \nonumber \\ {I_{88}}^{*}= & {} \frac{I_{88} h^{2}}{I_{44} },\,\,\,\,\,{I_{99}}^{*} =\frac{I_{99} }{I_{44} },\,\,\,\,\,\eta =\frac{h}{L},\,\,\,\,\,F^{*} =\frac{FL^{4}}{C_{T} I_{44} h}, \end{aligned}$$
(22)
the nondimensional equilibrium equations are written as
$$\begin{aligned}{} & {} -8I_{00} \frac{1}{\eta ^{2}}u_{1x_{1} x_{1} } -8I_{00} \frac{1}{\eta }\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } u_{30x_{1} } } \right) \nonumber \\{} & {} \quad +12I_{00} \frac{1}{\eta }\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} }^{2}} \right) +24I_{00} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} } u_{3x_{1} } u_{30x_{1} } } \right) -4I_{77} \frac{1}{\eta }\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} }^{2}} \right) +12I_{00} \eta \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} }^{2}u_{30x_{1} }^{2}} \right) \nonumber \\{} & {} \quad +12\eta \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} x_{1} }^{2}} \right) \,\,+24I_{66} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} x_{1} } \phi _{x_{1} } \,} \right) +8I_{55} \frac{1}{\eta ^{2}}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } \phi } \right) +12I_{99} \frac{1}{\eta }\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {\phi _{x_{1} }^{2}} \right) +4I_{55} \frac{1}{\eta ^{3}}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {\phi ^{2}} \right) \, \nonumber \\{} & {} \quad +12I_{00} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} } u_{3x_{1} }^{2}} \right) +12I_{00} \eta \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{30x_{1} } u_{3x_{1} }^{3}} \right) +3I_{00} \eta \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} }^{4}} \right) =0, \end{aligned}$$
(23)
$$\begin{aligned}{} & {} -8I_{00} \frac{1}{\eta }\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} } u_{30x_{1} } } \right) -8I_{00} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } u_{30x_{1} }^{2}} \right) +8u_{3x_{1} x_{1} x_{1} x_{1} } -8I_{55} \,\frac{1}{\eta ^{2}}u_{3x_{1} x_{1} } +8I_{66} \frac{1}{\eta }\phi _{x_{1} x_{1} x_{1} } -8I_{55} \frac{1}{\eta ^{3}}\phi _{x_{1} } \nonumber \\{} & {} \quad +12I_{00} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} }^{2}u_{30x_{1} } } \right) -8I_{77} \frac{1}{\eta }\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} } u_{3x_{1} } } \right) +24I_{00} \eta \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} } u_{3x_{1} } u_{30x_{1} }^{2}} \right) \,-24\eta \frac{\textrm{d}^{2}}{\textrm{d}x_{1} ^{2}}\left( {u_{1x_{1} } u_{3x_{1} x_{1} } } \right) \nonumber \\{} & {} \quad -24I_{66} \frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {u_{1x_{1} } \phi _{x_{1} } } \right) +8I_{55} \frac{1}{\eta ^{2}}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} } \phi } \right) -12I_{77} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } ^{2}u_{30x_{1} } } \right) +12I_{00} \eta ^{2}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} }^{2}u_{30x_{1} }^{3}} \right) \nonumber \\{} & {} \quad +12\eta ^{2}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} x_{1} }^{2}u_{30x_{1} } } \right) -24\eta ^{2}\frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {u_{3x_{1} } u_{3x_{1} x_{1} } u_{30x_{1} } } \right) +24I_{66} \eta \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} x_{1} } \phi _{x_{1} } u_{30x_{1} } \,} \right) \nonumber \\{} & {} \quad +16I_{55} \,\frac{1}{\eta }\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } \phi u_{30x_{1} } } \right) -24I_{66} \eta \frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {u_{3x_{1} } \phi _{x_{1} } u_{30x_{1} } } \right) +12I_{99} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {\phi _{x_{1} }^{2}u_{30x_{1} } } \right) \nonumber \\{} & {} \quad +4I_{55} \,\frac{1}{\eta ^{2}}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {\phi ^{2}u_{30x_{1} } } \right) +12I_{00} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} }^{2}u_{3x_{1} } } \right) +36I_{00} \eta \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} } u_{3x_{1} }^{2}u_{30x_{1} } } \right) -4I_{88} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} }^{3}} \right) \nonumber \\{} & {} \quad +24I_{00} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} }^{3}u_{30x_{1} } ^{2}} \right) +12\eta ^{2}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } u_{3x_{1} x_{1} }^{2}} \right) -12\eta ^{2}\frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {u_{3x_{1} }^{2}u_{3x_{1} x_{1} } } \right) \nonumber \\{} & {} \quad +24I_{66} \eta \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } u_{3x_{1} x_{1} } \phi _{x_{1} } \,} \right) +12I_{55} \,\frac{1}{\eta }\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} }^{2}\phi } \right) -12I_{66} \eta \frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {u_{3x_{1} }^{2}\phi _{x_{1} } } \right) \, \nonumber \\{} & {} \quad +12I_{99} \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } \phi _{x_{1} }^{2}} \right) +4I_{55} \,\frac{1}{\eta ^{2}}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } \phi ^{2}} \right) +12I_{00} \eta \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} } u_{3x_{1} }^{3}} \right) +15I_{00} \eta ^{2}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{30x_{1} } u_{3x_{1} }^{4}} \right) \nonumber \\{} & {} \quad +3I_{00} \eta ^{2}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} }^{5}} \right) =F, \end{aligned}$$
(24)
$$\begin{aligned}{} & {} -8I_{66} \eta u_{3x_{1} x_{1} x_{1} } +8I_{55} \frac{1}{\eta }u_{3x_{1} } -8I_{99} \phi _{x_{1} x_{1} } +8I_{55} \frac{1}{\eta ^{2}}\phi \,+24I_{66} \eta ^{2}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} } u_{3x_{1} x_{1} } } \right) -8I_{55} u_{1x_{1} } u_{3x_{1} } \nonumber \\{} & {} \quad +24I_{99} \eta \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{1x_{1} } \phi _{x_{1} } } \right) -8I_{55} \frac{1}{\eta }u_{1x_{1} } \phi +24I_{66} \eta ^{3}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } u_{3x_{1} x_{1} } u_{30x_{1} } } \right) -8I_{55} \eta u_{3x_{1} }^{2}u_{30x_{1} } \nonumber \\{} & {} \quad +24I_{99} \eta ^{2}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } \phi _{x_{1} } u_{30x_{1} } } \right) \,-8I_{55} u_{3x_{1} } \phi u_{30x_{1} } +12I_{66} \eta ^{3}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} } ^{2}u_{3x_{1} x_{1} } } \right) -4I_{55} \eta u_{3x_{1} }^{3} \nonumber \\{} & {} \quad +12I_{99} \eta ^{2}\frac{\textrm{d}}{\textrm{d}x_{1} }\left( {u_{3x_{1} }^{2}\phi _{x_{1} } } \right) \,\,-4I_{55} u_{3x_{1} }^{2}\phi =0. \end{aligned}$$
(25)
where * is neglected from the parameters for the sake of brevity. By employing a series expansion, the degrees of freedom are written as
$$\begin{aligned} u_{1} \left( {x_{1} } \right)= & {} \sum \limits _{j=1}^M {\Re _{j} U_{j} } \left( {x_{1} } \right) , \end{aligned}$$
(26)
$$\begin{aligned} u_{3} \left( {x_{1} } \right)= & {} \sum \limits _{i=1}^N {\aleph _{i} W_{i} } \left( {x_{1} } \right) , \end{aligned}$$
(27)
$$\begin{aligned} \phi \left( {x_{1} } \right)= & {} \sum \limits _{i=1}^N {\kappa _{i} \psi _{i} } \left( {x_{1} } \right) , \end{aligned}$$
(28)
$$\begin{aligned} u_{30} \left( {x_{1} } \right)= & {} W_{0} \left( {x_{1} } \right) , \end{aligned}$$
(29)
and the equations of motion are discretised as
$$\begin{aligned}{} & {} K_{11}^{L} \Re +K_{12}^{L} \aleph +K_{11}^{NL} \Re ^{2}+K_{12}^{NL} \Re \aleph +K_{13}^{NL} \aleph ^{2}+K_{14}^{NL} \aleph \kappa \nonumber \\{} & {} \quad +K_{15}^{NL} \kappa ^{2}\,+K_{16}^{NL} \Re \aleph ^{2}+K_{17}^{NL} \aleph ^{3}+K_{18}^{NL} \aleph ^{4}=0, \end{aligned}$$
(30)
$$\begin{aligned}{} & {} K_{21}^{L} \Re +K_{22}^{L} \aleph +K_{22}^{L} \kappa +K_{21}^{NL} \Re ^{2}+K_{22}^{NL} \Re \aleph +K_{23}^{NL} \Re \kappa \,+K_{24}^{NL} \aleph ^{2}+K_{25}^{NL} \aleph \kappa +K_{26}^{NL} \kappa ^{2}+K_{27}^{NL} \Re ^{2}\aleph \nonumber \\{} & {} \quad +K_{28}^{NL} \Re \aleph ^{2}+K_{29}^{NL} \aleph ^{3}+K_{210}^{NL} \aleph ^{2}\kappa \,+K_{211}^{NL} \aleph \kappa ^{2}+K_{212}^{NL} r\aleph ^{3}+K_{213}^{NL} \aleph ^{4}+K_{214}^{NL} \aleph ^{5}=F, \end{aligned}$$
(31)
$$\begin{aligned}{} & {} K_{32}^{L} \aleph \,+K_{33}^{L} \kappa +K_{31}^{NL} \Re \aleph +K_{32}^{NL} \Re \kappa +K_{33}^{NL} \aleph ^{2}+K_{34}^{NL} \aleph \kappa +K_{35}^{NL} \aleph ^{3}+K_{36}^{NL} \aleph ^{2}\kappa =0. \end{aligned}$$
(32)
The linear stiffness coefficients (
\(K^{L}_{ij})\) of the bending equilibrium equations are defined as
$$\begin{aligned} K_{11}^{L}= & {} -8\frac{1}{\eta ^{2}}I_{00} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } {U}''_{i} \left( {x_{1} } \right) \textrm{d}x_{1} , \end{aligned}$$
(33)
$$\begin{aligned} K_{12}^{L}= & {} -8\frac{1}{\eta }I_{00} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) W_{0} ^{\prime }\left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(34)
$$\begin{aligned} K_{21}^{L}= & {} -8I_{00} \frac{1}{\eta }\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(35)
$$\begin{aligned} K_{22}^{L}= & {} -8I_{00} \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'^{2}_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} +8\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } W_{i}^{(4)} \left( {x_{1} } \right) \textrm{d}x_{1} \nonumber \\{} & {} -8I_{55} \,\frac{1}{\eta ^{2}}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } {W}''_{i} \left( {x_{1} } \right) \textrm{d}x_{1} , \end{aligned}$$
(36)
$$\begin{aligned} K_{23}^{L}= & {} +8I_{66} \frac{1}{\eta }\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } {\psi }'''_{i} \left( {x_{1} } \right) \textrm{d}x_{1} -8I_{55} \frac{1}{\eta ^{3}}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } {\psi }'_{i} \left( {x_{1} } \right) \textrm{d}x_{1} , \end{aligned}$$
(37)
$$\begin{aligned} K_{32}^{L}= & {} -8I_{66} \eta \int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } {W}'''_{i} \,\,\,\left( {x_{1} } \right) \textrm{d}x_{1} +8I_{55} \frac{1}{\eta }\int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } {W}'_{i} \left( {x_{1} } \right) \textrm{d}x_{1} , \end{aligned}$$
(38)
$$\begin{aligned} K_{33}^{L}= & {} +8I_{55} \frac{1}{\eta ^{2}}\int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } \psi _{i} \left( {x_{1} } \right) \textrm{d}x_{1} -8I_{99} \int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } {\psi }''_{i} \left( {x_{1} } \right) \textrm{d}x_{1} , \end{aligned}$$
(39)
and the nonlinear stiffness coefficients (
\(K^{NL}_{ij})\) are defined as
$$\begin{aligned} K_{11}^{NL}= & {} +12\frac{1}{\eta }I_{00} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {U}'_{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(40)
$$\begin{aligned} K_{12}^{NL}= & {} +24I_{00} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) W_{0}^{'}\left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(41)
$$\begin{aligned} K_{13}^{NL}= & {} -4\frac{1}{\eta }I_{77} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) W_{j} ^{\prime }\left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +12\eta I_{00} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) W_{j}^{'}\left( {x_{1} } \right) {W_{0}^{'2}}\left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +12\eta \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}''_{i} \left( {x_{1} } \right) {W}''_{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(42)
$$\begin{aligned} K_{14}^{NL}= & {} +24I_{66} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}''_{i} \left( {x_{1} } \right) {\psi }'_{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} +8\frac{1}{\eta ^{2}}I_{55} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( x \right) \psi _{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1},\nonumber \\ \end{aligned}$$
(43)
$$\begin{aligned} K_{15}^{NL}= & {} +12\frac{1}{\eta }I_{99} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{\psi }'_{i} \left( {x_{1} } \right) {\psi }'_{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} +4\frac{1}{\eta ^{3}}I_{55} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {\psi _{i} \left( {x_{1} } \right) \psi _{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} ,\nonumber \\ \end{aligned}$$
(44)
$$\begin{aligned} K_{16}^{NL}= & {} +12I_{00} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) W_{k}^{\prime }\left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(45)
$$\begin{aligned} K_{17}^{NL}= & {} +12\eta I_{00} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) W_{k}^{\prime }\left( {x_{1} } \right) W_{0}^{'}\left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(46)
$$\begin{aligned} K_{18}^{NL}= & {} +3\eta I_{00} \int \limits _0^1 {U_{p} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) W_{k}^{\prime }\left( {x_{1} } \right) {W}'_{m} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(47)
$$\begin{aligned} K_{21}^{NL}= & {} +12I_{00} \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {U}'_{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(48)
$$\begin{aligned} K_{22}^{NL}= & {} -8I_{77} \frac{1}{\eta }\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} -24\eta \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {{U}'_{i} \left( {x_{1} } \right) {W}''_{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +24I_{00} \eta \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'^{2}_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(49)
$$\begin{aligned} \,K_{23}^{NL}&{=}&{-}24I_{66} \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {{U}'_{i} \left( {x_{1} } \right) {\psi }'_{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} {+}8I_{55} \frac{1}{\eta ^{2}}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) \psi _{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(50)
$$\begin{aligned} K_{24}^{NL}= & {} -12I_{77} \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +12I_{00} \eta ^{2}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'^{3}_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +12\eta ^{2}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}''_{i} \left( {x_{1} } \right) {W}''_{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} -24\eta ^{2}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {{W}'_{i} \left( {x_{1} } \right) {W}''_{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(51)
$$\begin{aligned} K_{25}^{NL}= & {} +24I_{66} \eta \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}''_{i} \left( {x_{1} } \right) {\psi }'_{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +16I_{55} \,\frac{1}{\eta }\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) \psi _{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} -24I_{66} \eta \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {{W}'_{i} \left( {x_{1} } \right) {\psi }'_{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(52)
$$\begin{aligned} K_{26}^{NL}= & {} +12I_{99} \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{\psi }'_{i} \left( {x_{1} } \right) {\psi }'_{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +4I_{55} \,\frac{1}{\eta ^{2}}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {\psi _{i} \left( {x_{1} } \right) \psi _{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(53)
$$\begin{aligned} K_{27}^{NL}= & {} +12I_{00} \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {U}'_{j} \left( {x_{1} } \right) {W}'_{k} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(54)
$$\begin{aligned} K_{28}^{NL}= & {} +36I_{00} \eta \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'_{k} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(55)
$$\begin{aligned} K_{29}^{NL}= & {} -4I_{88} \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'_{k} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +24I_{00} \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'_{k} \left( {x_{1} } \right) {W}'^{2}_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +12\eta ^{2}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}''_{j} \left( {x_{1} } \right) {W}''_{k} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} -12\eta ^{2}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}''_{k} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(56)
$$\begin{aligned} K_{210}^{NL}= & {} +24I_{66} \eta \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}''_{j} \left( {x_{1} } \right) {\psi }'_{k} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +12I_{55} \,\frac{1}{\eta }\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) \psi _{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} -12I_{66} \eta \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}^{2}}{\textrm{d}x_{1}^{2}}\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {\psi }'_{k} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(57)
$$\begin{aligned} K_{211}^{NL}= & {} +12I_{99} \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {\psi }'_{j} \left( {x_{1} } \right) {\psi }'_{k} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} +4I_{55} \,\frac{1}{\eta ^{2}}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) \psi _{j} \left( {x_{1} } \right) \psi _{k} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(58)
$$\begin{aligned} K_{212}^{NL}= & {} +12I_{00} \eta \int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'_{k} \left( {x_{1} } \right) {W}'_{m} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(59)
$$\begin{aligned} K_{213}^{NL}= & {} +15I_{00} \eta ^{2}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'_{k} \left( {x_{1} } \right) {W}'_{m} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(60)
$$\begin{aligned} K_{214}^{NL}= & {} +3I_{00} \eta ^{2}\int \limits _0^1 {W_{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'_{k} \left( {x_{1} } \right) {W}'_{m} \left( {x_{1} } \right) {W}'_{n} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(61)
$$\begin{aligned} K_{31}^{NL}= & {} -8I_{55} \int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } {U}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) \textrm{d}x_{1} +24I_{66} \eta ^{2}\int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{j} \left( {x_{1} } \right) {W}''_{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} , \end{aligned}$$
(62)
$$\begin{aligned} K_{32}^{NL}= & {} +24I_{99} \eta \int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{U}'_{i} \left( {x_{1} } \right) {\psi }'_{j} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} -8I_{55} \frac{1}{\eta }\int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } {U}'_{i} \left( {x_{1} } \right) \psi _{j} \left( {x_{1} } \right) \textrm{d}x_{1} , \end{aligned}$$
(63)
$$\begin{aligned} K_{33}^{NL}= & {} +24I_{66} \eta ^{3}\int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}''_{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} -8I_{55} \eta \int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } {W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) \textrm{d}x_{1} , \end{aligned}$$
(64)
$$\begin{aligned} K_{34}^{NL}= & {} +24I_{99} \eta ^{2}\int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {\psi }'_{i} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \, \nonumber \\{} & {} -8I_{55} \int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } {W}'_{i} \left( {x_{1} } \right) \psi _{i} \left( {x_{1} } \right) {W}'_{0} \left( {x_{1} } \right) \textrm{d}x_{1} , \end{aligned}$$
(65)
$$\begin{aligned} K_{35}^{NL}= & {} +12I_{66} \eta ^{3}\int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } \frac{\textrm{d}}{\textrm{d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}''_{k} \left( {x_{1} } \right) } \right) \textrm{d}x_{1} \nonumber \\{} & {} -4I_{55} \eta \int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } {W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {W}'_{k} \left( {x_{1} } \right) \textrm{d}x_{1} , \end{aligned}$$
(66)
$$\begin{aligned} K_{36}^{NL}= & {} +12I_{99} \eta ^{2}\int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } \frac{\text {d}}{\text {d}x_{1} }\left( {{W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) {\psi }'_{k} \left( {x_{1} } \right) } \right) \text {d}x_{1} \nonumber \\ {}{} & {} -4I_{55} \int \limits _0^1 {\psi _{l} \left( {x_{1} } \right) } {W}'_{i} \left( {x_{1} } \right) {W}'_{j} \left( {x_{1} } \right) \psi _{k} \left( {x_{1} } \right) \text {d}x_{1} . \end{aligned}$$
(67)
which by solving the nonlinear polynomial equations of motion using the Newton–Raphson method, the static bending due to the external static force can be obtained.