These results show predicting ability of SSFE finite element based on N-T's model.
[C.sup.0] finite element discretization of the first-order N-T model applied the PPR method to solve problems of gradient discontinuities across edges of triangular elements during interpolation.
Equation (18) shows that the total deformation energy due to N-T model contains additional terms: coupled membrane and Gaussian bending ([E.sub.eQ], [E.sub.Qe]) and Gaussian deformation energy [E.sub.QQ].
As we mentioned above, when the thickness ratio 2[chi] is greater than 0.098, this additional energy ([E.sub.eQ], [E.sub.Qe]) influences global deformation energy and shows the difference between N-T and R-M models applied to the spherical thick shells.
The investigation of the variation of the thickness ratio h/R for certain values 0 < 2[chi] < 0.099 proves that the Kirchhoff-Love, Reissner-Mindlin, and N-T classical models have the same contribution of total deformation energy.
The N-T model handles spherical thin and thick shell properly because it clearly shows how the change of the third fundamental form enhances the total deformation energy when the ratio [chi] becomes greater.
We have investigated in this framework the influence of the third fundamental form proportional to [chi square] (with [chi square] = [(h/2R).sup.2]) included in N-T's model of elastic thick shells.
In the contrary, our finite element (CSFE3) is based on N-T's theoretical model of thick shells which is deduced from [[epsilon].sub.[alpha]3](u) = 0 obtained at limits analysis of 3-dimensional shell equations .
The successive scaling over the benchmark structure has presented divergence of displacements results as h/R increases for K-L kinematic model of shells and that of N-T. This is due to the influence of the additional terms found in N-T's shell theory in the deformation energy of the structure.
For Reissner-Mindlin and K-L classical model of shells, the global stiffness matrix [K.sup.R-M.sub.G] contains only [K.sub.m] and [K.sub.b] while in N-T's model, the global stiffness matrix [K.sup.N-T.sub.G] contains additional terms ([K.sub.mg] + [K.sub.gm] + [K.sub.g]) to those found in [K.sup.R-M.sub.G].
N-T's theoretical approach for the modelling of the displacement and stains in cylindrical structures have been examined and compared with that of KL for the case of self-weight loading cylindrical roof.
The comparison has revealed that N-T's shell theory is suitable for the analysis of cylindrical thin and thick structures from the perspectives of both accuracy and simplicity.