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\begin{center}
\vspace*{-0.5cm}	
EXTENDED ABSTRACTS OF MINISYMPOZIUM

\vspace*{0.4cm}
ON ONE VARIANT OF THE IMPROVED, MORE ACCURATE THEORY
OF LAMINATED ELASTIC ORTHOTROPIC SHELLS OF ROTATION
IN UNSYMMETRIC DEFORMATIONS

\vspace*{0.3cm}
{\it E. Abramidze}

\vspace*{0.3cm}
{\it N.I. Musckhelishvili Institute of Computational Mathematics\\
of Georgian Academy of Sciences}
\end{center}

\vspace*{0.3cm}
\par For the multilayered shells, whose physical characters differ strongly from each 
other, the consideration of the deformation of transverse shear is essential. The application 
of the hypothesis of  Kirchhof-Love and of rectilinearity of an element for the whole 
package of shell does not allow us to consider the inhomogeneity of deformation 
of transverse shear, caused by difference of elastic properties of layers.
\par In this note to overcome the indicated deficiency,  the following approach is offered. 
It is based on the assumption of the presence of local corners in the rotation layers of the 
shell, caused by transverse shear, and on the satisfaction of the conditions of continuity of 
displacements and strains on the surface of contact of adjacent layers [1,2].
\par We shall write the basic equations of the theory of laminated orthotropic shells in 
geometrically nonlinear statement for quadratic approximation [3,4] taking into account 
the non-homogeneity of deformations of transverse shear in layers.
\par Denoting by $\gamma_\alpha^{(i)},$ $\gamma^{(i)}_\beta$ the transverse shears in the i-th layer, we obtain the 
expressions for the tangential displacements
%(1)
\begin{equation}
\begin{array}{c}
\displaystyle
u_\alpha^{(i)}=\frac{1+k_1\gamma}{1+k_1\gamma_i}u^{(i)}-
\frac{\gamma-\gamma_i}{A(1+k_1\gamma_i)}\frac{\partial \omega}{\partial \alpha}+
(\gamma-\gamma_i)\gamma_\alpha^{(i)},\\
\displaystyle
u_\beta^{(i)}=\frac{1+k_2\gamma}{1+k_2\gamma_i}v^{(i)}-
\frac{\gamma-\gamma_i}{B(1+k_2\gamma_i)}\frac{\partial \omega}{\partial \beta}+
(\gamma-\gamma_i)\gamma_\beta^{(i)},
\end{array}
\end{equation}
where $\omega$ is a  normal displacement, assumed unchanged in thickness; $u^{(i)}$, $v^{(i)}$ are 
tangential displacements of the surface  $\gamma=\gamma_i;$ $A,\;B$                  are the parameters of 
Lam$\'{e}$;           
$k_1\;\;k_2$ are the main curvatures of the coordinate surface.
\par Assuming the continuity of the tangent strains $\tau_{\alpha\gamma},$
$\tau_{\beta\gamma},$ determined by the 
transverse shears $\varepsilon_{\alpha\gamma}=\gamma_\alpha$,
$\varepsilon_{\beta\gamma}=\gamma_\beta$, we express $\gamma_\alpha^{(i)}$, 
$\gamma_\beta^{(i)}$, by $\gamma_\alpha^{(0)}$, $\gamma_\beta^{(0)}$,
using Hooke's law in the form
%(2)
\begin{equation}
\gamma_\alpha^{(i)}=\frac{G^{(0)}_{\alpha\gamma}}{G^{(i)}_{\alpha\gamma}}\gamma_\alpha^{(0)},\;\;
\gamma_\beta^{(i)}=\frac{G^{(0)}_{\beta\gamma}}{G^{(i)}_{\beta\gamma}}\gamma_\beta^{(0)},
\end{equation}
where $G^{(i)}_{\alpha\gamma}$, $G^{(i)}_{\beta\gamma}$ are the modulus of the shear of the material in the i-th layer.
\par Satisfying the conditions of continuity of displacements, we express $u^{(i)}$,
$v^{(i)}$ by the displacements of coordinate surface $u,$ $v$ and the total corners of rotation
$\Psi_\alpha$, $\Psi_\beta$ in the layer with index  $< <0> >$
%(3)
\begin{equation}
\Psi_\alpha=-\frac{1}{A}\frac{\partial\omega}{\partial\alpha}+k_1u+\gamma_\alpha^{(0)},\;\;
\Psi_\beta=-\frac{1}{B}\frac{\partial\omega}{\partial\beta}+k_2v+\gamma_\beta^{(0)}.
\end{equation}
\par The indicated approach allows to represent the expressions of the tangential 
displacements (1) in the form
%(4)
\begin{equation}
\begin{array}{c}
u_\alpha^{(i)}=u+a_1^{(i)}\gamma_\alpha^{(0)}+\gamma(\Psi_\alpha+a_2^{(i)}\gamma_\alpha^{(0)}),\;\;
u_\beta^{(i)}=v+b_1^{(i)}\gamma_\beta^{(0)}+\gamma(\Psi_\beta+b_2^{(i)}\gamma_\beta^{(0)}),
\end{array}
\end{equation}
where
$a_1^{(i)}\;\; a_2^{(i)}\;\;
b_1^{(i)}\;\; b_2^{(i)}$ are expressed by the values 
$G^{(i)}_{\beta\gamma}\;\;G^{(i)}_{\beta\gamma}$.
\par Taking into account the expressions (4), we represent the components of the 
deformations of the shell in the form
%(6)
\begin{equation}
\begin{array}{c}
\displaystyle
\varepsilon_{\alpha\alpha}^{(\gamma)}=\varepsilon_{\alpha\alpha}^{(i)}+
\gamma\varkappa_{\alpha\alpha}^{(i)},\;
\varepsilon_{\beta\beta}^{(\gamma)}=\varepsilon_{\beta\beta}^{(i)}+
\gamma\varkappa_{\beta\beta}^{(i)},\;
\varepsilon_{\alpha\beta}^{(\gamma)}=\varepsilon_{\alpha\beta}^{(i)}+
\gamma 2\varkappa_{\alpha\beta}^{(i)},\\\displaystyle
\varepsilon_{\alpha\gamma}^{(\gamma)}=\gamma_\alpha^{(i)},\;
\varepsilon_{\beta\gamma}^{(\gamma)}=\gamma_\beta^{(i)},\;
\varepsilon_{\gamma\gamma}^{(\gamma)}=0.
\end{array}
\end{equation}
\par Here
%(7)
$$
\begin{array}{c}
\displaystyle
\varepsilon_{\alpha\alpha}^{(i)}=\varepsilon_{\alpha\alpha}+\frac{1}{A}
\frac{\partial a_1^{(i)}\gamma_\alpha^{(0)}}{\partial\alpha}+\frac{1}{AB}
\frac{\partial A}{\partial \beta}b_1^{(i)}\gamma_\beta^{(0)},\;
\varepsilon_{\beta\beta}^{(i)}=\varepsilon_{\beta\beta}+\frac{1}{B}
\frac{\partial b_1^{(i)}\gamma_\beta^{(0)}}{\partial\beta}+\\
\displaystyle
+\frac{1}{AB}
\frac{\partial B}{\partial \alpha}a_1^{(i)}\gamma_\alpha^{(0)},\;\;
\varepsilon_{\alpha\beta}^{(i)}=\varepsilon^*_{\alpha\beta}+\frac{A}{B}
\frac{\partial }{\partial\beta}\left(\frac{a_1^{(i)}\gamma_\alpha^{(0)}}{A}\right)
+\frac{B}{A}\frac{\partial }{\partial \alpha}
\left(\frac{b_1^{(i)}\gamma_\beta^{(0)}}{B}\right),\\
\end{array}
$$
\begin{equation}
\begin{array}{c}
\displaystyle
\varkappa_{\alpha\alpha}^{(i)}=\varkappa_{\alpha}+\frac{1}{A}
\frac{\partial a_2^{(i)}\gamma_\alpha^{(0)}}{\partial\alpha}+\frac{1}{AB}
\frac{\partial A}{\partial \beta}b_2^{(i)}\gamma_\beta^{(0)},\;
\varkappa_{\beta\beta}^{(i)}=\varkappa_{\beta}+\frac{1}{B}
\frac{\partial b_2^{(i)}\gamma_\beta^{(0)}}{\partial\beta}+\\
\end{array}
\end{equation}
$$
\begin{array}{c}
\displaystyle
+\frac{1}{AB}
\frac{\partial B}{\partial \alpha}a_2^{(i)}\gamma_\alpha^{(0)},\;\;
2\varkappa_{\alpha\beta}^{(i)}=2\varkappa_{\alpha\beta}+\frac{A}{B}
\frac{\partial }{\partial\beta}\left(\frac{a_2^{(i)}\gamma_\alpha^{(0)}}{A}\right)
+\frac{B}{A}\frac{\partial }{\partial \alpha}\left(\frac{b_2^{(i)}\gamma_\beta^{(0)}}{B}\right)-\\
\\
\displaystyle
-\frac{k_1}{A}\left[\frac{\partial }{\partial \alpha}\left(b_1^{(i)}\gamma_\beta^{(0)}\right)
-\frac{1}{B}\frac{\partial A}{\partial\beta}\left(a_1^{(i)}\gamma_\alpha^{(0)}\right)
\right]
-\frac{k_2}{B}\left[\frac{\partial }{\partial \beta}\left(a_1^{(i)}\gamma_\alpha^{(0)}\right)
-\frac{1}{A}\frac{\partial B}{\partial\alpha}b_1^{(i)}\gamma_\beta^{(0)}
\right],
\end{array}
$$
where
%(8)
\begin{equation}
\begin{array}{c}
\displaystyle
\varepsilon_{\alpha\alpha}=\varepsilon_{\alpha}+\frac{1}{2}\theta_\alpha^2,\;
\varepsilon_{\beta\beta}=\varepsilon_{\beta}+\frac{1}{2}\theta_\beta^2,\;
\varepsilon_{\alpha\beta}^*=\varepsilon_{\alpha\beta}+\theta_\alpha\theta_\beta,\;
\gamma_\alpha^{(0)}=\Psi_\alpha-\theta_\alpha,\\
\\
\displaystyle
\gamma_\beta^{(0)}=\Psi_\beta-\theta_\beta,\;
\theta_\alpha=-\frac{1}{A}\frac{\partial \omega}{\partial \alpha}+k_1u,\;
\theta_\beta=-\frac{1}{B}\frac{\partial \omega}{\partial \beta}+k_2v.
\end{array}
\end{equation}
\par The values $\varepsilon_{\alpha},\;\varepsilon_{\beta},\;\varepsilon_{\alpha\beta},$
$\varkappa_{\alpha},\;\varkappa_{\beta},\;\varkappa_{\alpha\beta}$
characterize the deformation of  coordinate 
surface of the shell. Their expressions are given in [2].
\par Having the expressions (5), (6), on the bases of Hooke's law we obtain the relations 
of elasticity
$$
\begin{array}{l}
\displaystyle
N_\alpha=C_{11}\varepsilon_{\alpha\alpha}+C_{12}\varepsilon_{\beta\beta}+
K_{11}\varkappa_{\alpha}+K_{12}\varkappa_{\beta}+A_{11}\frac{\partial \gamma_\alpha^{(0)}}{\partial \alpha}
+A_{12}\gamma_{\alpha}^{(0)}+\\
\displaystyle
+B_{11}\frac{\partial \gamma_\beta^{(0)}}{\partial \beta}+B_{12}\gamma_{\beta}^{(0)},\;\;\;\;
N_\beta=C_{12}\varepsilon_{\alpha\alpha}+C_{22}\varepsilon_{\beta\beta}+
K_{12}\varkappa_{\alpha}+K_{22}\varkappa_{\beta}+\\
\displaystyle
+A_{21}\frac{\partial \gamma_\alpha^{(0)}}{\partial \alpha}+
A_{22}\gamma_{\alpha}^{(0)}
+B_{22}\frac{\partial \gamma_\beta^{(0)}}{\partial \beta}
+B_{22}\gamma_{\beta}^{(0)},
\\\displaystyle
N_{\alpha\beta}=C_{66}\varepsilon_{\alpha\beta}^*+2K_{66}\varepsilon_{\alpha\beta}+
k_{2}(K_{66}\varepsilon_{\alpha\beta}^*+2D_{66}\varkappa_{\alpha\beta})+
(A_{16}+k_2E_{16})\frac{\partial \gamma_\alpha^{(0)}}{\partial \beta}+\\
\displaystyle
+(A_{26}+k_2E_{26})\gamma_{\alpha}^{(0)}+(B_{16}+k_2F_{16})\frac{\partial \gamma_\beta^{(0)}}{\partial \alpha}+
(B_{26}+k_2F_{26})\gamma_{\beta}^{(0)},\\\displaystyle
N_{\beta\alpha}=C_{66}\varepsilon_{\alpha\beta}^*+2K_{66}\varepsilon_{\alpha\beta}+
k_{1}(K_{66}\varepsilon_{\alpha\beta}^*+2D_{66}\varkappa_{\alpha\beta})+
(A_{16}+k_1E_{16})\frac{\partial \gamma_\alpha^{(0)}}{\partial \beta}+
\end{array}
$$
\begin{equation}
\begin{array}{l}
\displaystyle
\hspace{-2.5cm}+(A_{26}+k_1E_{26})\gamma_{\alpha}^{(0)}+(B_{16}+k_1F_{16})\frac{\partial \gamma_\beta^{(0)}}{\partial \alpha}
+(B_{26}+k_1F_{26})\gamma_{\beta}^{(0)},
\end{array}
\end{equation}
$$
\begin{array}{l}
\displaystyle
M_\alpha=K_{11}\varepsilon_{\alpha\alpha}+K_{12}\varepsilon_{\beta\beta}+
D_{11}\varkappa_\alpha+D_{12}\varkappa_\beta+E_{11}\frac{\partial \gamma_\alpha^{(0)}}{\partial \alpha}+
E_{12}\gamma_\alpha^{(0)}+\\\displaystyle
+F_{11}\frac{\partial \gamma_\beta^{(0)}}{\partial \beta}+
F_{12}\gamma_\beta^{(0)},\;\;\;\;
M_\beta=K_{12}\varepsilon_{\alpha\alpha}+K_{22}\varepsilon_{\beta\beta}+
D_{12}\varkappa_\alpha+D_{22}\varkappa_\beta+\\\displaystyle
+E_{21}\frac{\partial \gamma_\alpha^{(0)}}{\partial \alpha}+
E_{22}\gamma_\alpha^{(0)}+F_{21}\frac{\partial \gamma_\beta^{(0)}}{\partial \beta}+
F_{22}\gamma_\beta^{(0)},\;\;\;
Q_\alpha=K_1\gamma_\alpha^{(0)},\;\;Q_\beta=K_2\gamma_\beta^{(0)},
\\\displaystyle
M_{\alpha\beta}=M_{\beta\alpha}=K_{66}\varepsilon_{\alpha\beta}^*+
2D_{66}\varkappa_{\alpha\beta}
+E_{16}\frac{\partial\gamma_{\alpha}^{(0)}}{\partial\beta}+
E_{26}\gamma_{\alpha}^{(0)}+F_{16}\frac{\partial\gamma_{\beta}^{(0)}}{\partial\alpha}+
F_{26}\gamma_{\beta}^{(0)},
\end{array}
$$
where $N_\alpha,\;N_\beta,\;N_{\alpha\beta},\;N_{\beta\alpha}$ are the tangential forces;
$Q_{\alpha},\;Q_\beta$ are intersecting forces;                  
$M_\alpha\;M_\beta$ are bending  moments; $M_{\alpha\beta},\;M_{\beta\alpha}$ are twisting moments;
$C_{ij},\;K_{ij},\;D_{ij},\;K_{1},\;K_{2}$ are 
rigidity characters, which are determined by elastic parameters of the layers and their 
thickness; and $A_{11},\;A_{12},\;...\;F_{16},\;F_{26}$ are values, depending on the geometrical 
and mechanical parameters of the layers of the shell [2].
\par The equations of the equilibrium of the element of shell have the following form:
%(10)
\begin{equation}
\begin{array}{l}
\displaystyle
\frac{\partial BN_\alpha}{\partial\alpha}+\frac{\partial AN_{\beta\alpha}}{\partial\beta}
+\frac{\partial A}{\partial\beta}N_{\alpha\beta}-\frac{\partial B}{\partial\alpha}N_{\beta}
+ABk_1Q_\alpha^*+ABq_1=0,\\
\displaystyle
\frac{\partial AN_\beta}{\partial\beta}+\frac{\partial BN_{\alpha\beta}}{\partial\alpha}
+\frac{\partial B}{\partial\alpha}N_{\beta\alpha}-\frac{\partial A}{\partial\beta}N_{\alpha}
+ABk_2Q_\beta^*+ABq_2=0,\\
\displaystyle
\frac{\partial BQ_\alpha^*}{\partial\alpha}+\frac{\partial AQ_\beta^*}{\partial\beta}
-ABk_1N_\alpha-ABk_2N_\beta+ABq_3=0,
\\\displaystyle
\frac{\partial BM_\alpha}{\partial\alpha}+\frac{\partial AM_{\beta\alpha}}{\partial\beta}
+\frac{\partial A}{\partial\beta}M_{\alpha\beta}-\frac{\partial B}{\partial\alpha}M_{\beta}
-ABQ_\alpha=0,
\end{array}
\end{equation}
$$
\begin{array}{l}
\displaystyle
\frac{\partial AM_\beta}{\partial\beta}+\frac{\partial BM_{\alpha\beta}}{\partial\alpha}
+\frac{\partial B}{\partial\alpha}M_{\beta\alpha}-\frac{\partial A}{\partial\beta}M_{\alpha}
-ABQ_\beta=0,
\end{array}
$$
where         
%
\begin{equation}
\begin{array}{l}
\displaystyle
Q_\alpha^*=Q_\alpha-(N_\alpha+k_1M_\alpha)\theta_\alpha-(N_{\alpha\beta}+k_1M_{\alpha\beta})\theta_\beta,
\\\displaystyle
Q_\beta^*=Q_\beta-(N_{\beta\alpha}+k_2M_{\beta\alpha})\theta_\alpha-(N_{\beta}+k_2M_{\beta})\theta_\beta.
\end{array}
\end{equation}
\par In the equations (9) the values $q_1,\;q_2\;q_3$ are the projections of the surface 
loaded in he direction of the coordinate lines $\alpha,\;\beta,\;\gamma$ respectively.
\par Let us consider the non -axially symmetric deformation of the laminated shells of 
rotation. We take the coordinate surface of the laminated shell of rotation before the 
deformation in the curvilinear orthogonal system $\alpha=s,$ $\beta=\theta$ where $s$ is the length 
of the arc of generator; $\theta$ is the central corner in the parallel disk. Starting from the fact 
that the functions $N_s,\;N_{s\theta},\;Q_s^*,\;M_s,\;M_{s\theta},$ $u,\;v,$
$\omega,\;\Psi_s,\;\Psi_\theta$
are taken as solving, from the 
general equations we obtain the solving system of differential equations in the form
%(12)
$$
\begin{array}{l}
\hspace{-1.7cm}
\displaystyle
\frac{\partial N_s}{\partial s}=-\frac{1}{r}\frac{\partial N_{s\theta}}{\partial \theta}-
\frac{1}{r}(k_1-k_2)\frac{\partial M_{s\theta}}{\partial \theta}-
\frac{cos\varphi}{r}(N_\theta-N_s)-k_1Q_s^*-q_1,\\
\hspace{-1.7cm}
\displaystyle
\frac{\partial N_{s\theta}}{\partial s}=-\frac{1}{r}\frac{\partial N_{\theta}}{\partial \theta}-
-\frac{2cos\varphi}{r}N_{s\theta}-\frac{cos\varphi}{r}(k_1-k_2)M_{s\theta}
-k_2Q_\theta^*-q_2,
\end{array}
$$
\begin{equation}
\begin{array}{l}
\hspace{-5.1cm}
\displaystyle
\frac{\partial Q_{s}^*}{\partial s}=-\frac{cos\varphi}{r}Q_{s}^*-
\frac{1}{r}\frac{\partial Q_{\theta}^*}{\partial\theta}+k_1N_s+k_2N_\theta-q_3,\\
\end{array}
\end{equation}
$$
\begin{array}{l}
\displaystyle
\frac{\partial M_{s}}{\partial s}=-\frac{1}{r}\frac{\partial_{s\theta}}{\partial\theta}
+\frac{cos\varphi}{r}(M_\theta-M_s)+Q_{s}^*+(N_s+k_1M_s)\Psi_s+(N_{s\theta}+k_1M_{s\theta})\Psi_\theta,\\
\displaystyle
\frac{\partial M_{s\theta}}{\partial s}=-2\frac{cos\varphi}{r}M_{s\theta}
-\frac{1}{r}\frac{\partial M_\theta}{\partial\theta}-K_1\left(\Psi_\theta+
\frac{1}{r}\frac{\partial\omega}{\partial\theta}-k_2v\right),
\frac{\partial u}{\partial s}=\varepsilon_s-k_1\omega,\\\displaystyle
\frac{\partial v}{\partial s}=\varepsilon_{s\theta}-\frac{1}{r}\frac{\partial u}{\partial\theta}-
\frac{cos\varphi}{r}v,\;\;
\frac{\partial \omega}{\partial s}=k_1u+\frac{1}{r}\frac{\partial \omega}{\partial\theta}-
k_2v,\;\;
\frac{\partial \Psi_s}{\partial s}=k_1\varepsilon_s+\varkappa_s,\\
\displaystyle
\frac{\partial \Psi_\theta}{\partial s}=2\varkappa_{s\theta}-\frac{1}{r}
\frac{\partial \Psi_s}{\partial \theta}+\frac{cos\varphi}{r}\Psi_\theta+
k_1\varepsilon_{s\theta}+(k_2-k_1)\left(\frac{1}{r}\frac{\partial u}{\partial\theta}-
\frac{cos\varphi}{r}\right)v.
\end{array}
$$
\par The values
$N_\theta,\;Q_\theta^*,\;M_\theta,\;\varepsilon_s,\;\varepsilon_{s\theta},\;
\varkappa_s,\;\varkappa_{s\theta}\;$                                              
involved in the equations (11)  
have the following form
%(13)
$$
\begin{array}{l}
\displaystyle
\varepsilon_s=b_{11}N_s+b_{12}M_s+b_{13}\varepsilon_{\theta\theta}+
b_{14}\varepsilon_{\theta}+b_{15}\frac{\partial \gamma_s^{(0)}}{\partial s}
+b_{16}\gamma_s^{(0)}+b_{17}\frac{\partial \gamma_\theta^{(0)}}{\partial \theta}
+
\\
\displaystyle
+b_{18}\gamma_\theta^{(0)}-\frac{1}{2}(\Psi_s-\gamma_s^{(0)})^2,\\\displaystyle
Q_\theta^*=K_2\left(\Psi_\theta+\frac{1}{r}
\frac{\partial\omega}{\partial\theta}-
k_2\omega\right)-(N_{s\theta}+k_1M_{s\theta})\Psi_s-
(N_\theta
+k_2M_\theta)\Psi_\theta,\\\displaystyle
\varkappa_s=b_{21}N_s+b_{22}M_s+b_{23}\varepsilon_{\theta\theta}+
b_{24}\varkappa_{\theta}+b_{25}\frac{\partial \gamma_s^{(0)}}{\partial s}
+b_{26}\gamma_s^{(0)}+b_{27}\frac{\partial \gamma_\theta^{(0)}}{\partial \theta}
+b_{28}\gamma_\theta^{(0)},
\end{array}
$$
$$
\begin{array}{l}
\displaystyle
N_\theta=b_{31}N_s+b_{32}M_s+b_{33}\varepsilon_{\theta\theta}+
b_{34}\varkappa_{\theta}+b_{35}\frac{\partial \gamma_s^{(0)}}{\partial s}
+b_{36}\gamma_s^{(0)}+b_{37}\frac{\partial \gamma_\theta^{(0)}}{\partial \theta}
+b_{38}\gamma_\theta^{(0)},
\end{array}
$$
$$
\begin{array}{l}
\displaystyle
M_\theta=b_{41}N_s+b_{42}M_s+b_{43}\varepsilon_{\theta\theta}+
b_{44}\varkappa_{\theta}+b_{45}\frac{\partial \gamma_s^{(0)}}{\partial s}+b_{46}\gamma_s^{(0)}+b_{47}\frac{\partial \gamma_\theta^{(0)}}{\partial \theta}
+b_{48}\gamma_\theta^{(0)},\\
\end{array}
$$
\begin{equation}
\begin{array}{l}
\hspace{-1.5cm}
\varepsilon_{s\theta}=b_{51}N_{s\theta}+b_{52}M_{s\theta}+b_{53}
\frac{\partial\gamma_s^{(0)}}{\partial\theta}+
+b_{54}\gamma_s^{(0)}+b_{55}\frac{\partial \gamma_\theta^{(0)}}{\partial s}
+b_{56}\gamma_\theta^{(0)}-\\\hspace{-1.5cm}
\displaystyle
-(\Psi_s-\gamma_s^{(0)})(\Psi_\theta-\gamma_\theta^{(0)}),\\\hspace{-1.5cm}
\displaystyle
2\varkappa_{s\theta}=b_{61}N_{s\theta}+b_{62}M_{s\theta}+b_{63}
\frac{\partial \gamma_s^{(0)}}{\partial\theta}+
b_{64}\gamma_s^{(0)}+b_{65}\frac{\partial \gamma_s^{(0)}}{\partial s}
+b_{66}\gamma_s^{(0)}.
\end{array}
\end{equation}
\par In the expressions (12) the values 
$\varepsilon_{\theta\theta},\;\varkappa_{\theta},\;\gamma_s^{(0)},\;
\gamma_\theta^{(0)},\;\frac{\partial\gamma_s^{(0)}}{\partial s},\;
\frac{\partial\gamma_s^{(0)}}{\partial \theta},\;
\frac{\partial\gamma_\theta^{(0)}}{\partial s},\;
\frac{\partial\gamma_\theta^{(0)}}{\partial \theta}$                                                                         
are expressed by the solving functions in the following form
%(14)
$$
\begin{array}{l}
\hspace{-4cm}\displaystyle
\varepsilon_{\theta\theta}=\frac{1}{r}\frac{\partial v}{\partial\theta}+
\frac{cos\varphi}{r}u+k_2w+\frac{1}{2}\left(k_2v-\frac{1}{r}\frac{\partial w}{\partial \theta}\right)^2,\\
\\\hspace{-4cm}
\displaystyle
\varkappa_\theta=\frac{1}{r}\frac{\partial \Psi_\theta}{\partial\theta}+
\frac{cos\varphi}{r}\Psi_s-k_2\left(\frac{1}{r}\frac{\partial v}{\partial\theta}+
\frac{cos\varphi}{r}u+k_2w\right),
\\
\\\hspace{-4cm}\displaystyle
\gamma_s^{(0)}=K_1^{-1}\left[Q_s^*+(N_s+k_1M_s)\Psi_s+(N_{s\theta}
+k_1M_{s\theta})\Psi_\theta\right],\\
\\\hspace{-4cm}\displaystyle
\gamma_\theta^{(0)}=\Psi_\theta+\frac{1}{r}\frac{\partial w}{\partial\theta}
-k_2w,
\end{array}
$$
\begin{equation}
\begin{array}{l}
\hspace{-3cm}
\displaystyle
\frac{\partial \gamma_s^{(0)}}{\partial\theta}=K_1^{-1}\left[
\frac{\partial Q_s^*}{\partial\theta}+\left(\frac{\partial N_s}{\partial \theta}
+k_1\frac{\partial M_s}{\partial \theta}\right)\Psi_s+
\left(\frac{\partial N_{s\theta}}{\partial \theta}+\right.\right.
\\\hspace{-3cm}\displaystyle
\left.\left.
+k_1\frac{\partial M_{s\theta}}{\partial \theta}\right)\Psi_\theta+(N_s+k_1M_s)
\frac{\partial\Psi_s}{\partial\theta}+
(N_{s\theta}+k_1M_{s\theta})\frac{\partial\Psi_\theta}{\partial\theta}\right],\;\;
\end{array}
\end{equation}
$$
\begin{array}{l}
\displaystyle
\frac{\partial \gamma_\theta^{(0)}}{\partial\theta}=\frac{\partial\Psi_\theta}
{\partial\theta}+\frac{1}{r}\frac{\partial^2 w}{\partial\theta^2}+k_2
\frac{\partial v}{\partial\theta},\\
\\\displaystyle
\frac{\partial \gamma_\theta^{(0)}}{\partial s}=
b_1N_{s\theta}+b_2M_{s\theta}+b_3\frac{\partial \gamma_s^{(0)}}{\partial \theta}+b_4\gamma_s^{(0)}+
b_5\gamma_\theta^{(0)}+\\
\\\displaystyle
+b_6(\Psi_s-\gamma_s^{(0)})(\Psi_\theta-\gamma_\theta^{(0)})+
b_7\frac{\partial\Psi_s}{\partial\theta}+b_8\frac{\partial u}{\partial\theta}+b_9\frac{\partial w}{\partial\theta}+
b_{10}\Psi_\theta+b_{11}^*u+b_{12}^*v,\\
\\
\displaystyle
\frac{\partial \gamma_s^{(0)}}{\partial s}=\frac{1}{c_0-c_1N_s-c_2M_s-c_3\Psi_s
-c_4\Psi_\theta-\frac{\partial \Psi_\theta}{\partial\theta}}(a_1N_s+a_2Q_s^*+
a_3M_s+a_4u+\\
\\\displaystyle+
a_5v+a_6w+a_7\Psi_s+a_8\Psi_\theta+a_9N_{s\theta}\Psi_s
+a_{10}M_{s\theta}\Psi_s+a_{11}M_{s}\Psi_s+a_{12}N_{s}\Psi_\theta)
\end{array}
$$
\par The coefficients $b_{11},\;b_{12},\;...,\;a_{11},\;a_{12}$ involved in the expressions (12), (13) are 
determined by geometrical and mechanical parameters of the layers of the shells.
\par Adding to the equations (11) the corresponding boundary conditions, we obtain 
the nonlinear boundary problem. At this the boundary conditions are given in the following 
way: the static ones - by forces and the moment - in the integral form, the kinematic ones 
in discrete number of points of the end-wall of the shell.
	

\vspace*{0.5cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.3cm}
\end{center}
\par
[1] Grigorenko Y. M., Vasilenko A.T., On Considering the Nonhomogeneity of Deformations 
of  the Transverse Shear in the Thickness in the Laminated Shells. //Appl. Mechanics, 1977,
13, N 10, pp. 36-42.
\par
[2] Grigorenko Y.M., Vasilenko A.T.,  The Theory of Shells of Variable Rigidity. Kiev: 
Naukova Dumka, 1981, 544 p.
\par
[3] Novojilov V.V., The Basis of Nonlinear Theory of Elasticity. M.: Gostexizdat, 1948, 212 p.
\par
[4] Shapovalov L.A., On One Simple Variant of Equations of Geometric Nonlinear Theory of 
Thin Shells.// Ing. jur. MTT. 1968, N 1, pp. 56-62.
\end{document}