Mathematical Formulation of Laminated Composite Thick Conical Shells

The mathematical formulation of thick conical shells using third order shear deformation of thick shell theory are presented. The equations of motion are obtained using Hamilton’s principle. For present analysis, we consider shell's system transverse normal stress, rotary inertia and shear deformation.

Towards this end, in this paper, we propose our contribution towards the mathematical theory of third order shear deformation thick conical shell theory (see Zannon et al. (TSDTZ); Qatu et al., 2013) and its stress-strain deformation at the mid thick conical shell surface (Duc & Cong, 2015;Akbari et al., 2015;Jam & Kiani, 2015, Viola et al., 2016).

Mathematical Formulation of Conical Shell
The displacement components using the third-order shear deformation shell theory are given in (Asadi & Qatu, 2012;Leissa & Qatu, 2011).Conical shells are one form of engineering solids that are formed by revolving two non-paralleled lines, mostly a line and axis of revolution.We are interested mainly in a particular type of shells which have a circular cross-section (Qatu, 1994).
A closed conical shell with circular sides (Figure 2.1) has a closed shape and the open conical shell can be obtained by cutting the sides of the solid between 12 and  (Roh et al, 2008).An open conical shell with sides less than the half of the radius of the curvature then the solid is shallow (Qatu et al., 2010;Dung et al., 2014).
Figure 2.2 A a side view of the closed coincal shell (Qatu, 1994)

Mathematical Analysis
One cannot find an exact solution for a general lamination structure shell with general boundary conditions and/or lamination having series of sequence and layers (Qatu et al., 2013;Zannon et al., 2015;Qatu, 1994).Many researchers talked about the vibration of shells as in Leissa & Qatu, (2011), she considered a thin plate in her paper "vibration of shells".One can be permitted to obtain a fundamental frequency with good accuracy as in Qatu et al., (2013) by using the classical thin plate (CPT), now using the shear deformation plate theories (SDPTs) can largely eliminate the inaccuracies.Later Qatu et al., (2010) and Reddy (1994) developed this subject, Leissa & Qatu (2011) studied the exact solutions ''solutions which satisfy both the equations of motion, and boundary conditions'' for simply supported cross-ply thick shell.
The structural mass parameters { } ij N and the external applied load vector, as a function of time {} ij F are given (see Qatu et al., 2013;Zannon et al., 2015).

Conclusions
The mathematical analysis of the third order shear deformation theory (see Zannon et al., 2015;Qatu et al.,2013) are presented for simply supported with circular cross section of a thick conical shell.This solution will be used in further investigations to assess the results for free vibration analysis of the circular cross section thick shells.