The closed-form solutions for buckling and postbuckling behaviour of anisotropic shear deformable laminated doubly-curved shells by matching method with the boundary layer of shell buckling

Abstract

Based on a boundary layer theory of shell buckling, the semi-analytical solutions for nonlinear stability analysis of anisotropic laminated composite doubly-curved shells with rectangular planform subjected to lateral pressure are derived. A new shell model of arbitrary constant curvature and fibre stacking sequences but constant thickness is developed. The governing equations are based on an extended higher-order shear deformation shell theory with von Kármán-type of kinematic nonlinearity and including the effect on stiffness couplings. The nonlinear deformation and initial deflection of shells are both taken into account. The boundary layer equations of buckling for doubly-curved shells are introduced to match the asymptotic solutions satisfying the clamped or simply-supported boundary condition. The closed-form solutions for buckling and postbuckling analysis of an anisotropic shear deformable laminated doubly-curved panel are obtained by the two-step perturbation methods and the boundary layer theory for shell buckling, which is employed to determine interactive buckling loads and postbuckling equilibrium paths. At the same time, the internal quantitative relationship in the asymptotic sense between deflection and rotations of the normal to the middle surface is for the first time obtained. The influences of anisotropic lay-up, change in the stacking sequence, temperature variation, different types of elastic foundation and boundary condition on nonlinear stability behaviour are analysed and discussed. The study provides a good theoretical method for the load-carrying capacity design of composite shell structures.

Appendices

Appendix 1

In Eqs. (53)–(54)

$$\begin{aligned} C_{1} &= \tfrac{{\gamma _{{24}}^{2} + 2\gamma _{0} \gamma _{5} + \gamma _{0}^{2} }}{{C_{q} }},\;C_{2} = \tfrac{{2m^{2} n^{2} \beta ^{2} }}{{(g_{{210}}^{2} - g_{{220}}^{2} )}}\left[ {(m^{2} + \gamma _{0} n^{2} \beta ^{2} )g_{{210}} \pi _{3} + (g_{{210}}^{2} + g_{{220}}^{2} )\pi _{4} } \right],\;C_{3} = 1 - \tfrac{{g_{{210}} g_{{310}} - g_{{220}} g_{{320}} }}{{(m^{2} + \gamma _{0} n^{2} \beta ^{2} )g_{{210}} }}\varepsilon , \hfill \\ \Theta _{3} & = \tfrac{{\gamma _{{24}}^{2} - \gamma _{5}^{2} }}{{\gamma _{{24}}^{2} + 2\gamma _{0} \gamma _{5} + \gamma _{0}^{2} }}C_{1} \left[ {\varepsilon (\gamma _{{T2}} - \gamma _{5} \gamma _{{T1}} + \gamma _{{211}} \gamma _{{T12}} )\Delta T - \lambda _{q}^{{(0)}} } \right] + \tfrac{{\gamma _{{211}} }}{{\gamma _{{24}} }}\lambda _{s}^{{(0)}} , \hfill \\ \Theta _{4} & = \tfrac{1}{{C_{3} }}\left[ {C_{2} + \tfrac{{\gamma _{{24}}^{2} - \gamma _{5}^{2} }}{{\gamma _{{24}}^{2} + 2\gamma _{0} \gamma _{5} + \gamma _{0}^{2} }}C_{1} \lambda _{q}^{{(2)}} - \tfrac{{\gamma _{{211}} }}{{\gamma _{{24}} }}\lambda _{s}^{{(2)}} } \right], \hfill \\ \lambda _{q}^{{(0)}} &= \left\{ {\tfrac{{\gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )^{2} g_{{210}} }}{{(1 + \mu )(g_{{210}}^{2} - g_{{220}}^{2} )}}} \right. + \tfrac{{\gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )}}{{(1 + \mu )(g_{{210}}^{2} - g_{{220}}^{2} )}}\left[ {(g_{{210}} g_{{31}} + g_{{220}} g_{{32}} ) + \tfrac{{(m^{2} + \gamma _{0} n^{2} \beta ^{2} )(g_{{210}} g_{{310}} + g_{{220}} g_{{320}} )}}{{m^{2} (1 + \mu )}}} \right]\varepsilon \hfill \\ &\quad+ \tfrac{1}{{(1 + \mu )}}\left[ {g_{{110}} + F_{{11}} + \tfrac{{\gamma _{{14}} \gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )}}{{(1 + \mu )m^{2} }}} \right.\tfrac{{g_{{31}} (g_{{210}} g_{{310}} + g_{{220}} g_{{320}} ) + g_{{32}} (g_{{210}} g_{{320}} + g_{{220}} g_{{310}} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }} \hfill \\ &\quad \left. { - \tfrac{{\gamma _{{14}} \gamma _{{24}} \mu (m^{2} + \gamma _{0} n^{2} \beta ^{2} )^{2} }}{{(1 + \mu )m^{4} }}\tfrac{{2g_{{220}} g_{{310}} g_{{320}} + g_{{210}} (g_{{310}}^{2} + g_{{320}}^{2} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }}} \right]\varepsilon ^{2} - \tfrac{\mu }{{m^{2} (1 + \mu )^{2} }}\left[ {(g_{{110}} g_{{310}} + g_{{120}} g_{{320}} } \right.) \hfill \\ &\quad + \tfrac{{\gamma _{{14}} \gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )g_{{31}} }}{{(1 + \mu )m^{2} }}\tfrac{{2g_{{220}} g_{{310}} g_{{320}} + g_{{210}} (g_{{310}}^{2} + g_{{320}}^{2} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }} + \tfrac{{\gamma _{{14}} \gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )g_{{32}} }}{{(1 + \mu )m^{2} }}\tfrac{{g_{{220}} (g_{{310}}^{2} + g_{{320}}^{2} ) + 2g_{{210}} g_{{310}} g_{{320}} }}{{g_{{210}}^{2} - g_{{220}}^{2} }} \hfill \\ &\quad \left. { - \tfrac{{\gamma _{{14}} \gamma _{{24}} \mu (m^{2} + \gamma _{0} n^{2} \beta ^{2} )^{2} }}{{(1 + \mu )^{2} m^{6} }}\tfrac{{g_{{210}} g_{{310}} (g_{{310}}^{2} + 3g_{{320}}^{2} ) + g_{{220}} g_{{320}} (3g_{{310}}^{2} + g_{{320}}^{2} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }}} \right]\varepsilon ^{3} + \tfrac{{\mu ^{2} }}{{m^{4} (1 + \mu )^{3} }}\left[ {g_{{110}} (g_{{310}}^{2} + g_{{320}}^{2} ) + 2g_{{120}} g_{{310}} g_{{320}} } \right.\hfill \\&\quad + \tfrac{{\gamma _{{14}} \gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )g_{{31}} }}{{(1 + \mu )m^{2} }}\tfrac{{g_{{220}} g_{{320}} (3g_{{310}}^{2} + g_{{320}}^{2} ) + g_{{210}} g_{{310}} (g_{{310}}^{2} + 3g_{{320}}^{2} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }} \hfill \\ &\quad + \tfrac{{\gamma _{{14}} \gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )g_{{32}} }}{{(1 + \mu )}}\tfrac{{g_{{220}} g_{{310}} (g_{{310}}^{2} + 3g_{{320}}^{2} ) + g_{{210}} g_{{320}} (3g_{{310}}^{2} + g_{{320}}^{2} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }}\left. {\left. {\tfrac{{4g_{{220}} g_{{310}} g_{{320}} (g_{{310}}^{2} + g_{{320}}^{2} ) + g_{{210}} (g_{{310}}^{4} + 6g_{{310}}^{2} g_{{320}}^{2} + g_{{320}}^{4} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }}} \right]\varepsilon ^{4} } \right\}, \hfill \\ \end{aligned}$$

$$\begin{aligned} \lambda _{s}^{{(0)}} & = - \tfrac{1}{{2mn\beta }}\left\{ {\tfrac{{\gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )^{2} g_{{220}} }}{{(1 + \mu )(g_{{210}}^{2} - g_{{220}}^{2} )}}} \right. + \tfrac{{\gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )}}{{(1 + \mu )(g_{{210}}^{2} - g_{{220}}^{2} )}}\left[ {(g_{{220}} g_{{31}} + g_{{210}} g_{{32}} ) + \tfrac{{(m^{2} + \gamma _{0} n^{2} \beta ^{2} )(g_{{210}} g_{{320}} + g_{{220}} g_{{310}} )}}{{m^{2} (1 + \mu )}}} \right]\varepsilon \hfill \\ &\quad + \tfrac{1}{{(1 + \mu )}}\left[ {g_{{120}} + \tfrac{{\gamma _{{14}} \gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )}}{{(1 + \mu )m^{2} }}\tfrac{{g_{{31}} (g_{{220}} g_{{310}} + g_{{210}} g_{{320}} ) + g_{{32}} (g_{{210}} g_{{310}} + g_{{220}} g_{{320}} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }}} \right.\left. { - \tfrac{{\gamma _{{14}} \gamma _{{24}} \mu (m^{2} + \gamma _{0} n^{2} \beta ^{2} )^{2} }}{{(1 + \mu )m^{4} }}\tfrac{{2g_{{210}} g_{{310}} g_{{320}} + g_{{220}} (g_{{310}}^{2} + g_{{320}}^{2} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }}} \right]\varepsilon ^{2} \hfill \\ &\quad - \tfrac{\mu }{{m^{2} (1 + \mu )^{2} }}\left[ {(g_{{110}} g_{{320}} + g_{{120}} g_{{310}} } \right.) + \tfrac{{\gamma _{{14}} \gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )g_{{31}} }}{{(1 + \mu )m^{2} }}\tfrac{{2g_{{210}} g_{{310}} g_{{320}} + g_{{220}} (g_{{310}}^{2} + g_{{320}}^{2} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }} \hfill \\ &\quad+ \tfrac{{\gamma _{{14}} \gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )g_{{32}} }}{{(1 + \mu )m^{2} }}\tfrac{{g_{{210}} (g_{{310}}^{2} + g_{{320}}^{2} ) + 2g_{{220}} g_{{310}} g_{{320}} }}{{g_{{210}}^{2} - g_{{220}}^{2} }}\left. { - \tfrac{{\gamma _{{14}} \gamma _{{24}} \mu (m^{2} + \gamma _{0} n^{2} \beta ^{2} )^{2} }}{{(1 + \mu )^{2} m^{6} }}\tfrac{{g_{{220}} g_{{310}} (g_{{310}}^{2} + 3g_{{320}}^{2} ) + g_{{210}} g_{{320}} (3g_{{310}}^{2} + g_{{320}}^{2} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }}} \right]\varepsilon ^{3} \hfill \\ &\quad + \tfrac{{\mu ^{2} }}{{m^{4} (1 + \mu )^{3} }}\left[ {g_{{120}} (g_{{310}}^{2} + g_{{320}}^{2} ) + 2g_{{110}} g_{{310}} g_{{320}} } \right. + \tfrac{{\gamma _{{14}} \gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )g_{{31}} }}{{(1 + \mu )m^{2} }}\tfrac{{g_{{220}} g_{{310}} (g_{{310}}^{2} + 3g_{{320}}^{2} ) + g_{{210}} g_{{320}} (3g_{{310}}^{2} + g_{{320}}^{2} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }} \hfill \\ &\quad+ \tfrac{{\gamma _{{14}} \gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )g_{{32}} }}{{(1 + \mu )}}\tfrac{{g_{{220}} g_{{320}} (3g_{{310}}^{2} + g_{{320}}^{2} ) + g_{{210}} g_{{310}} (g_{{310}}^{2} + 3g_{{320}}^{2} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }} - \tfrac{{\gamma _{{14}} \gamma _{{24}} \mu (m^{2} + \gamma _{0} n^{2} \beta ^{2} )^{2} }}{{(1 + \mu )^{2} m^{6} }}\left. {\left. {\tfrac{{4g_{{210}} g_{{310}} g_{{320}} (g_{{310}}^{2} + g_{{320}}^{2} ) + g_{{220}} (g_{{310}}^{4} + 6g_{{310}}^{2} g_{{320}}^{2} + g_{{320}}^{4} )}}{{g_{{210}}^{2} - g_{{220}}^{2} }}} \right]\varepsilon ^{4} } \right\}, \hfill \\ \lambda _{q}^{{(2)}} &= \tfrac{{2n^{2} \beta ^{2} g_{{210}}^{2} }}{{(1 + \mu )(g_{{210}}^{2} - g_{{220}}^{2} )}}\left\{ {(\pi _{1} + \pi _{3} )\left[ {\tfrac{{\gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )^{2} (g_{{210}}^{2} + 3g_{{220}}^{2} )}}{{(g_{{210}}^{2} - g_{{220}}^{2} )g_{{210}} }}} \right.} \right.\left. { + \gamma _{{14}} \gamma _{{24}} \tfrac{{m^{4} }}{{g_{{210}} }}} \right]\left. { + (\pi _{2} + \pi _{4} )\tfrac{{\gamma _{{14}} m^{2} (1 + \mu )(g_{{210}}^{2} + g_{{220}}^{2} )}}{{g_{{210}}^{2} }}} \right\}, \hfill \\ \lambda _{s}^{{(2)}} &= - \tfrac{{2n^{2} \beta ^{2} g_{{210}}^{2} }}{{(1 + \mu )(g_{{210}}^{2} - g_{{220}}^{2} )}}\left\{ {(\pi _{1} + \pi _{3} )\left[ {\tfrac{{\gamma _{{24}} (m^{2} + \gamma _{0} n^{2} \beta ^{2} )^{2} g_{{220}} (3g_{{210}}^{2} + g_{{220}}^{2} )}}{{(g_{{210}}^{2} - g_{{220}}^{2} )g_{{210}}^{2} }}} \right.} \right.\left. { + \gamma _{{14}} \gamma _{{24}} \tfrac{{m^{4} g_{{220}} }}{{g_{{210}}^{2} }}} \right]\left. { + (\pi _{2} + \pi _{4} )\tfrac{{2\gamma _{{14}} m^{2} (1 + \mu )g_{{220}} }}{{g_{{210}} }}} \right\}, \hfill \\ \lambda _{q}^{{(T)}} &= \left( {\gamma _{{T1}} m^{2} + 6\gamma _{{T3}} mn\beta + \gamma _{{T2}} n^{2} \beta ^{2} } \right)\Delta T. \hfill \\ \end{aligned}$$

(55)

In the above equations (with $g_{ij}$ and $g_{ijk}$ defined as in Li and Lin [29]),

$$\begin{gathered} C_{b} = \gamma_{24} \left[ {(\gamma_{0} + \gamma_{5} )m^{2} + (\gamma_{24}^{2} + \gamma_{0} \gamma_{5} )n^{2} \beta^{2} } \right] \hfill \\ \quad\quad\quad+ \tfrac{2}{\pi }(\gamma_{24}^{2} - \gamma_{5}^{2} )\left[ {m^{2} (\vartheta_{1} \overline{b}_{01}^{(2)} - \varphi_{1} \overline{b}_{10}^{(2)} ) + n^{2} \beta^{2} (\vartheta_{2} \overline{b}_{01}^{(2)} - \varphi_{2} \overline{b}_{10}^{(2)} )} \right]\varepsilon^{1/2} , \hfill \\ F_{11} = K_{1} + K_{2} (m^{2} + n^{2} \beta^{2} ), \hfill \\ e_{11} = 4m^{2} ,\;e_{12} = - \tfrac{{4\gamma_{24} m^{2} (m^{2} + \gamma_{0} n^{2} \beta^{2} )g_{210} }}{{(1 + \mu )(m^{2} + \zeta_{1} n^{2} \beta^{2} )(g_{210}^{2} - g_{220}^{2} )}}, \hfill \\ e_{21} = 16m^{4} (1 + \gamma_{14} \gamma_{24} \gamma_{220} \varpi_{420} + \gamma_{14} \gamma_{24} \gamma_{230} \theta_{420} ),\;e_{22} = - 4\gamma_{24} m^{2} , \hfill \\ \chi_{11} = 4\gamma_{0} n^{2} \beta^{2} ,\;\chi_{12} = - \tfrac{{4\zeta_{1} \gamma_{24} n^{2} \beta^{2} (m^{2} + \gamma_{0} n^{2} \beta^{2} )g_{210} }}{{(1 + \mu )(m^{2} + \zeta_{1} n^{2} \beta^{2} )(g_{210}^{2} - g_{220}^{2} )}}, \hfill \\ \chi_{21} = 16n^{4} \beta^{4} (\gamma_{214} + \gamma_{14} \gamma_{24} \gamma_{223} \varpi_{302} + \gamma_{14} \gamma_{24} \gamma_{233} \theta_{302} ),\;\chi_{22} = - 4\gamma_{0} \gamma_{24} n^{2} \beta^{2} , \hfill \\ \xi_{1} = - \tfrac{{\gamma_{24} (1 + \mu )m^{2} n^{2} \beta^{2} (m^{2} + \gamma_{0} n^{2} \beta^{2} )}}{{g_{210} }},\;\xi_{2} = \tfrac{{\gamma_{24} (1 + 2\mu )m^{2} n^{2} \beta^{2} (g_{210}^{2} - g_{220}^{2} )}}{{2g_{210}^{2} }}, \hfill \\ \pi_{1} = \tfrac{{e_{11} \xi_{2} - e_{21} \xi_{1} }}{{e_{11} e_{22} - e_{12} e_{21} }},\;\pi_{2} = \tfrac{{e_{22} \xi_{1} - e_{12} \xi_{2} }}{{e_{11} e_{22} - e_{12} e_{21} }},\;\pi_{3} = \tfrac{{\chi_{11} \xi_{2} - \chi_{21} \xi_{1} }}{{\chi_{11} \chi_{22} - \chi_{12} \chi_{21} }},\;\pi_{4} = \tfrac{{\chi_{22} \xi_{1} - \chi_{12} \xi_{2} }}{{\chi_{11} \chi_{22} - \chi_{12} \chi_{21} }},\;\zeta_{1} = \tfrac{{\gamma_{24}^{2} + \gamma_{0} \gamma_{5} }}{{\gamma_{0} + \gamma_{5} }}, \hfill \\ \varpi_{302} = 4n^{2} \beta^{2} \tfrac{{\gamma_{223} (\gamma_{42} + \gamma_{432} 4n^{2} \beta^{2} ) - \gamma_{233} (\gamma_{32} + \gamma_{332} 4n^{2} \beta^{2} )}}{{(\gamma_{32} + \gamma_{322} 4n^{2} \beta^{2} )(\gamma_{42} + \gamma_{432} 4n^{2} \beta^{2} ) - (\gamma_{32} + \gamma_{332} 4n^{2} \beta^{2} )^{2} }}, \hfill \\ \theta_{302} = 4n^{2} \beta^{2} \tfrac{{\gamma_{223} (\gamma_{31} + \gamma_{322} 4n^{2} \beta^{2} ) - \gamma_{223} (\gamma_{32} + \gamma_{332} 4n^{2} \beta^{2} )}}{{(\gamma_{31} + \gamma_{322} 4n^{2} \beta^{2} )(\gamma_{42} + \gamma_{432} 4n^{2} \beta^{2} ) - (\gamma_{32} + \gamma_{332} 4n^{2} \beta^{2} )^{2} }}, \hfill \\ \varpi_{420} = 4m^{2} \tfrac{{\gamma_{220} (\gamma_{42} + \gamma_{430} 4m^{2} ) - \gamma_{230} (\gamma_{32} + \gamma_{330} 4m^{2} )}}{{(\gamma_{31} + \gamma_{320} 4m^{2} )(\gamma_{42} + \gamma_{430} 4m^{2} ) - (\gamma_{32} + \gamma_{330} 4m^{2} )^{2} }},\;\theta_{420} = 4m^{2} \tfrac{{\gamma_{230} (\gamma_{31} + \gamma_{320} 4m^{2} ) - \gamma_{220} (\gamma_{32} + \gamma_{330} 4m^{2} )}}{{(\gamma_{31} + \gamma_{320} 4m^{2} )(\gamma_{42} + \gamma_{430} 4m^{2} ) - (\gamma_{32} + \gamma_{330} 4m^{2} )^{2} }} ,\hfill \\ \end{gathered}$$

(56)

$$\begin{aligned} b_{1} &= (\gamma _{{320}} \gamma _{{430}} - \gamma _{{330}}^{2} )\left[ {\tfrac{{\gamma _{{14}} \gamma _{{24}} }}{{g_{{16}} }}} \right]^{{1/2}} ,\;c_{1} = \tfrac{1}{2}\gamma _{{14}} \gamma _{{24}} (\gamma _{{320}} \gamma _{{430}} - \gamma _{{330}}^{2} )\tfrac{{g_{{15}} }}{{g_{{16}} }}, \hfill \\ \vartheta _{1} & = [(b_{1} - c_{1} )/2]^{{1/2}} ,\;\varphi _{1} = [(b_{1} + c_{1} )/2]^{{1/2}} , \hfill \\ b_{2} &= \gamma _{0} (\gamma _{{322}} \gamma _{{432}} - \gamma _{{332}}^{2} )\left[ {\tfrac{{\gamma _{{14}} \gamma _{{24}} }}{{\bar{g}_{{16}} }}} \right]^{{1/2}} ,\;c_{2} = \tfrac{1}{2}\gamma _{0} \gamma _{{14}} \gamma _{{24}} (\gamma _{{322}} \gamma _{{432}} - \gamma _{{332}}^{2} )\tfrac{{\bar{g}_{{15}} }}{{\bar{g}_{{16}} \beta ^{2} }}, \hfill \\ \vartheta _{2} & = [(b_{2} - c_{2} )/2]^{{1/2}} ,\;\varphi _{2} = [(b_{2} + c_{2} )/2]^{{1/2}} , \hfill \\ C_{{16}} & = (\gamma _{{320}} \gamma _{{430}} - \gamma _{{330}}^{2} ) + \gamma _{{14}} \gamma _{{24}} \gamma _{{220}} (\gamma _{{220}} \gamma _{{430}} - \gamma _{{230}} \gamma _{{330}} ) + \gamma _{{14}} \gamma _{{24}} \gamma _{{230}} (\gamma _{{230}} \gamma _{{320}} - \gamma _{{220}} \gamma _{{330}} ), \hfill \\ C_{{17}} &= \gamma _{{240}} (\gamma _{{320}} \gamma _{{430}} - \gamma _{{330}}^{2} ) - \gamma _{{220}} (\gamma _{{310}} \gamma _{{430}} - \gamma _{{330}} \gamma _{{410}} ) - \gamma _{{230}} (\gamma _{{320}} \gamma _{{410}} - \gamma _{{310}} \gamma _{{330}} ), \hfill \\ C_{{18}} &= (\gamma _{{320}} \gamma _{{430}} - \gamma _{{330}}^{2} ), \hfill \\ \bar{C}_{{16}} &= (\gamma _{{322}} \gamma _{{432}} - \gamma _{{332}}^{2} ) + \gamma _{{14}} \gamma _{{24}} \gamma _{{223}} (\gamma _{{223}} \gamma _{{432}} - \gamma _{{233}} \gamma _{{332}} ) + \gamma _{{14}} \gamma _{{24}} \gamma _{{233}} (\gamma _{{233}} \gamma _{{322}} - \gamma _{{223}} \gamma _{{332}} ), \hfill \\ \bar{C}_{{17}} &= \gamma _{{244}} (\gamma _{{322}} \gamma _{{432}} - \gamma _{{332}}^{2} ) - \gamma _{{223}} (\gamma _{{313}} \gamma _{{432}} - \gamma _{{332}} \gamma _{{413}} ) - \gamma _{{233}} (\gamma _{{322}} \gamma _{{413}} - \gamma _{{313}} \gamma _{{332}} ), \hfill \\ \bar{C}_{{18}} &= (\gamma _{{322}} \gamma _{{432}} - \gamma _{{332}}^{2} ), \hfill \\ \end{aligned}$$

(57)

$$\begin{aligned} g_{{15}} &= \gamma _{{120}} (\gamma _{{220}} \gamma _{{430}} - \gamma _{{230}} \gamma _{{330}} ) + \gamma _{{130}} (\gamma _{{230}} \gamma _{{320}} - \gamma _{{220}} \gamma _{{330}} ) + \gamma _{{220}} (\gamma _{{310}} \gamma _{{430}} - \gamma _{{330}} \gamma _{{410}} ) \hfill \\ &\quad + \gamma _{{230}} (\gamma _{{320}} \gamma _{{410}} - \gamma _{{310}} \gamma _{{330}} ) - (\gamma _{{320}} \gamma _{{430}} - \gamma _{{330}}^{2} )(\gamma _{{140}} + \gamma _{{240}} ), \hfill \\ g_{{16}} &= \gamma _{{110}} (\gamma _{{320}} \gamma _{{430}} - \gamma _{{330}}^{2} ) - \gamma _{{120}} (\gamma _{{310}} \gamma _{{430}} - \gamma _{{330}} \gamma _{{410}} ) - \gamma _{{130}} (\gamma _{{320}} \gamma _{{410}} - \gamma _{{310}} \gamma _{{330}} ) \hfill \\ &\quad \times \left[ {(\gamma _{{320}} \gamma _{{430}} - \gamma _{{330}}^{2} ) + \gamma _{{14}} \gamma _{{24}} \gamma _{{220}} (\gamma _{{220}} \gamma _{{430}} - \gamma _{{230}} \gamma _{{330}} ) + \gamma _{{14}} \gamma _{{24}} \gamma _{{230}} (\gamma _{{230}} \gamma _{{320}} - \gamma _{{220}} \gamma _{{330}} )} \right] \hfill \\ &\quad + \gamma _{{14}} \gamma _{{24}} \left[ {\gamma _{{140}} (\gamma _{{320}} \gamma _{{430}} - \gamma _{{330}}^{2} ) - \gamma _{{120}} (\gamma _{{220}} \gamma _{{430}} - \gamma _{{230}} \gamma _{{330}} ) - \gamma _{{130}} (\gamma _{{230}} \gamma _{{320}} - \gamma _{{220}} \gamma _{{330}} )} \right] \hfill \\ &\quad \times \left[ {\gamma _{{240}} (\gamma _{{320}} \gamma _{{430}} - \gamma _{{330}}^{2} ) - \gamma _{{220}} (\gamma _{{310}} \gamma _{{430}} - \gamma _{{330}} \gamma _{{410}} ) - \gamma _{{230}} (\gamma _{{320}} \gamma _{{410}} - \gamma _{{310}} \gamma _{{330}} )} \right], \hfill \\ g_{{17}} & = \tfrac{{C_{{17}} }}{{C_{{16}} }},\;g_{{18}} = \tfrac{{C_{{18}} }}{{C_{{16}} }},\;g_{{19}} = - g_{{17}} - \tfrac{{\varphi _{1}^{2} - 3\vartheta _{1}^{2} }}{{b_{1}^{2} }}g_{{18}} ,\;g_{{20}} = g_{{17}} + \tfrac{{3\varphi _{1}^{2} - \vartheta _{1}^{2} }}{{b_{1}^{2} }}g_{{18}}, \hfill \\ \bar{g}_{{15}} &= \gamma _{{123}} (\gamma _{{223}} \gamma _{{432}} - \gamma _{{233}} \gamma _{{332}} ) + \gamma _{{133}} (\gamma _{{233}} \gamma _{{322}} - \gamma _{{223}} \gamma _{{332}} ) + \gamma _{{223}} (\gamma _{{313}} \gamma _{{432}} - \gamma _{{332}} \gamma _{{413}} ) \hfill \\ &\quad + \gamma _{{233}} (\gamma _{{322}} \gamma _{{413}} - \gamma _{{313}} \gamma _{{332}} ) - (\gamma _{{322}} \gamma _{{432}} - \gamma _{{332}}^{2} )(\gamma _{{144}} + \gamma _{{244}} ), \hfill \\ \bar{g}_{{16}} & = [\gamma _{{114}} (\gamma _{{322}} \gamma _{{432}} - \gamma _{{332}}^{2} ) - \gamma _{{123}} (\gamma _{{313}} \gamma _{{432}} - \gamma _{{332}} \gamma _{{413}} ) - \gamma _{{133}} (\gamma _{{322}} \gamma _{{413}} - \gamma _{{313}} \gamma _{{332}} )] \hfill \\ &\quad \times [(\gamma _{{322}} \gamma _{{432}} - \gamma _{{332}}^{2} ) + \gamma _{{14}} \gamma _{{24}} \gamma _{{223}} (\gamma _{{223}} \gamma _{{432}} - \gamma _{{233}} \gamma _{{332}} ) + \gamma _{{14}} \gamma _{{24}} \gamma _{{233}} (\gamma _{{233}} \gamma _{{322}} - \gamma _{{223}} \gamma _{{332}} )] \hfill \\ &\quad + \gamma _{{14}} \gamma _{{24}} [\gamma _{{144}} (\gamma _{{322}} \gamma _{{432}} - \gamma _{{332}}^{2} ) - \gamma _{{123}} (\gamma _{{223}} \gamma _{{432}} - \gamma _{{233}} \gamma _{{332}} ) - \gamma _{{133}} (\gamma _{{233}} \gamma _{{322}} - \gamma _{{223}} \gamma _{{332}} )] \hfill \\ &\quad \times [\gamma _{{244}} (\gamma _{{322}} \gamma _{{432}} - \gamma _{{332}}^{2} ) - \gamma _{{223}} (\gamma _{{313}} \gamma _{{432}} - \gamma _{{332}} \gamma _{{413}} ) - \gamma _{{233}} (\gamma _{{322}} \gamma _{{413}} - \gamma _{{313}} \gamma _{{332}} )], \hfill \\ \bar{g}_{{17}} & = \tfrac{{\bar{C}_{{17}} }}{{\bar{C}_{{16}} }},\;\bar{g}_{{18}} = \tfrac{{\bar{C}_{{18}} }}{{\bar{C}_{{16}} }},\;\bar{g}_{{19}} = - \bar{g}_{{17}} - \tfrac{{\varphi _{2}^{2} - 3\vartheta _{2}^{2} }}{{b_{2}^{2} }}\bar{g}_{{18}} ,\;\bar{g}_{{20}} = \bar{g}_{{17}} + \tfrac{{3\varphi _{2}^{2} - \vartheta _{2}^{2} }}{{b_{2}^{2} }}\bar{g}_{{18}}, \hfill \\ \end{aligned}$$

(57)

for the case of curved edges clamped:

$$\begin{aligned} \bar{a}_{{01}}^{{(1)}} &= a_{{01}}^{{(1)}} = 1,\;b_{{01}}^{{(2)}} = \gamma _{{24}} g_{{19}} ,\;b_{{10}}^{{(2)}} = \gamma _{{24}} \tfrac{{\vartheta _{1} }}{{\varphi _{1} }}g_{{20}} ,\;\bar{a}_{{10}}^{{(1)}} = \tfrac{{\vartheta _{2} }}{{\varphi _{2} }},\;\bar{b}_{{01}}^{{(2)}} = \gamma _{{24}} \bar{g}_{{19}} ,\;\bar{b}_{{10}}^{{(2)}} = \gamma _{{24}} \tfrac{{\vartheta _{2} }}{{\varphi _{2} }}\bar{g}_{{20}}. \hfill \\ \end{aligned} $$

for the case of curved edges simply supported

$$\begin{aligned} \bar{a}_{{01}}^{{(1)}} &= a_{{01}}^{{(1)}} = 1,\;a_{{10}}^{{(1)}} = \tfrac{1}{{2\vartheta _{1} \varphi _{1} }}\left[ {\tfrac{{g_{{17}} }}{{g_{{18}} }}b_{1}^{2} + c_{1} } \right],\;b_{{01}}^{{(2)}} = 0, \hfill \\ b_{{10}}^{{(2)}} &= - \tfrac{{\gamma _{{24}} }}{{2\vartheta _{1} \varphi _{1} }}\left[ {g_{{18}} + 2g_{{17}} c_{1} + \tfrac{{g_{{17}}^{2} }}{{g_{{18}}^{2} }}b_{1}^{2} } \right],\;\bar{a}_{{10}}^{{(1)}} = \tfrac{1}{{2\vartheta _{2} \varphi _{2} }}\left[ {\tfrac{{\bar{g}_{{17}} }}{{\bar{g}_{{18}} }}b_{2}^{2} + c_{2} } \right],\;\bar{b}_{{01}}^{{(2)}} = 0, \hfill \\ \bar{b}_{{10}}^{{(2)}} &= - \tfrac{{\gamma _{{24}} }}{{2\vartheta _{2} \varphi _{2} }}\left[ {\bar{g}_{{18}} + 2\bar{g}_{{17}} c_{2} + \tfrac{{\bar{g}_{{17}}^{2} }}{{\bar{g}_{{18}}^{2} }}b_{2}^{2} } \right]. \hfill \\ \end{aligned} $$

Appendix 2

In Eq. (51)

$$\begin{gathered} \Delta_{00} = \left| {\begin{array}{*{20}c} {r_{11} } & {r_{12} } & {r_{13} } & {r_{14} } \\ {r_{21} } & {r_{22} } & {r_{23} } & {r_{24} } \\ {r_{31} } & {r_{32} } & {r_{33} } & {r_{34} } \\ {r_{41} } & {a_{42} } & {r_{43} } & {r_{44} } \\ \end{array} } \right|,\;\Delta_{01} = \left| {\begin{array}{*{20}c} {e_{1} } & {r_{12} } & {r_{13} } & {r_{14} } \\ {e_{2} } & {r_{22} } & {r_{23} } & {r_{24} } \\ {e_{3} } & {r_{32} } & {r_{33} } & {r_{34} } \\ {e_{4} } & {r_{42} } & {r_{43} } & {r_{44} } \\ \end{array} } \right|,\;\Delta_{02} = \left| {\begin{array}{*{20}c} {r_{11} } & {r_{12} } & {e_{1} } & {r_{14} } \\ {r_{21} } & {r_{22} } & {e_{2} } & {r_{24} } \\ {r_{31} } & {r_{32} } & {e_{3} } & {r_{34} } \\ {r_{41} } & {r_{42} } & {e_{4} } & {r_{44} } \\ \end{array} } \right|, \hfill \\ \Delta_{03} = \left| {\begin{array}{*{20}c} {r_{11} } & {e_{1} } & {r_{13} } & {r_{14} } \\ {r_{21} } & {e_{2} } & {r_{23} } & {r_{24} } \\ {r_{31} } & {e_{3} } & {r_{33} } & {r_{34} } \\ {r_{41} } & {e_{4} } & {r_{43} } & {r_{44} } \\ \end{array} } \right|,\;\Delta_{04} = \left| {\begin{array}{*{20}c} {r_{11} } & {r_{12} } & {r_{13} } & {e_{1} } \\ {r_{21} } & {r_{22} } & {r_{23} } & {e_{2} } \\ {r_{31} } & {r_{32} } & {r_{33} } & {e_{3} } \\ {r_{41} } & {r_{42} } & {r_{43} } & {e_{4} } \\ \end{array} } \right|,\;\Delta_{05} = \left| {\begin{array}{*{20}c} {f_{1} } & {r_{12} } & {r_{13} } & {r_{14} } \\ {f_{2} } & {r_{22} } & {r_{23} } & {r_{24} } \\ {f_{3} } & {r_{32} } & {r_{33} } & {r_{34} } \\ {f_{4} } & {r_{42} } & {r_{43} } & {r_{44} } \\ \end{array} } \right|, \hfill \\ \Delta_{06} = \left| {\begin{array}{*{20}c} {r_{11} } & {r_{12} } & {f_{1} } & {r_{14} } \\ {r_{21} } & {r_{22} } & {f_{2} } & {r_{24} } \\ {r_{31} } & {r_{32} } & {f_{3} } & {r_{34} } \\ {r_{41} } & {r_{42} } & {f_{4} } & {r_{44} } \\ \end{array} } \right|,\;\Delta_{07} = \left| {\begin{array}{*{20}c} {f_{2} } & {r_{12} } & {r_{13} } & {r_{14} } \\ {f_{1} } & {r_{22} } & {r_{23} } & {r_{24} } \\ {f_{4} } & {r_{32} } & {r_{33} } & {r_{34} } \\ {f_{3} } & {r_{42} } & {r_{43} } & {r_{44} } \\ \end{array} } \right|,\;\Delta_{08} = \left| {\begin{array}{*{20}c} {r_{11} } & {r_{12} } & {f_{2} } & {r_{14} } \\ {r_{21} } & {r_{22} } & {f_{1} } & {r_{24} } \\ {r_{31} } & {r_{32} } & {f_{4} } & {r_{34} } \\ {r_{41} } & {r_{42} } & {f_{3} } & {r_{44} }, \\ \end{array} } \right|, \hfill \\ r_{11} = r_{22} = r_{34} = r_{43} = \gamma_{331} mn\beta ,\;r_{12} = r_{21} = r_{33} = r_{44} = \gamma_{32} + \gamma_{330} m^{2} + \gamma_{332} n^{2} \beta^{2} , \hfill \\ r_{13} = r_{24} = \gamma_{41} + \gamma_{430} m^{2} + \gamma_{432} n^{2} \beta^{2} ,\;r_{14} = r_{23} = \gamma_{431} mn\beta ,\;r_{31} = r_{42} = \gamma_{321} mn\beta , \hfill \\ r_{32} = r_{41} = \gamma_{31} + \gamma_{320} m^{2} + \gamma_{322} n^{2} \beta^{2} ,\;e_{1} = (\gamma_{231} m^{2} + \gamma_{233} n^{2} \beta^{2} )n\beta , \hfill \\ e_{1} = (\gamma_{231} m^{2} + \gamma_{233} n^{2} \beta^{2} )n\beta ,\;e_{3} = (\gamma_{221} m^{2} + \gamma_{223} n^{2} \beta^{2} )n\beta ,\;e_{4} = (\gamma_{220} m^{2} + \gamma_{222} n^{2} \beta^{2} )m, \hfill \\ f_{1} = - (\gamma_{41} - \gamma_{411} m^{2} - \gamma_{413} n^{2} \beta^{2} )n\beta ,\;f_{2} = - (\gamma_{32} - \gamma_{410} m^{2} - \gamma_{412} n^{2} \beta^{2} )m, \hfill \\ f_{3} = - (\gamma_{32} - \gamma_{311} m^{2} - \gamma_{313} n^{2} \beta^{2} )n\beta ,\;f_{4} = - (\gamma_{31} - \gamma_{310} m^{2} - \gamma_{312} n^{2} \beta^{2} )m. \hfill \\ \end{gathered} $$

(58)