Coordinates and displacements for a cylindrical thin shell

Accordingly, the strain measures as well as rotations have to be renamed. Then the introduction of Eqs. (6.11) into Eqs. (6.8) and (6.9) yields  

with Eqs. (6.12) and (6.15), the complete set of strain measures in terms of displacements is established for thin cylindrical shells. By neglecting the underlined terms of Eqs. (6.15), this formulation based on the theory of Sanders [158] and Koiter [114], may be reduced to the also well-known formulation of Donnell [64] or Girkmann [84]. Theories similar to the latter likewise have been developed by Mushtari and Vlasov, see Novozhilov [134]. The inspection of the constituents of terms involving the radius shows that 

The principle of virtual displacements, given by Eq. (3.45), may be utilized to determine the equations of equilibrium. We will refrain from considering external loads. For the two-dimensional shell structure still with the transverse shear strains 7° and 7 and associated internal transverse forces Qx and Qs, the principle of virtual displacements may then be reformulated as follows  

The virtual strain measures are related to the virtual displacements, just as it is given for the actual case in Eqs. (6.12), (6.13), and (6.14a). When these kinematic relations are substituted into the principle, different derivatives of the virtual displacements appear. These may be eliminated with the aid of integration by parts to summarize the contributions connected to every virtual displacement. In order to satisfy the principle, each of the resulting integrands needs to vanish. Not to be pursued here, the natural boundary conditions therewith can be determined. The sought-after equilibrium conditions in directions of the coordinates x, s, and n take the following form 

Since the transverse shear strains 7° and 7° will be neglected by virtue of Remark 6.2, the associated internal transverse forces and may be eliminated by substitution of Eqs. (6.18) into Eqs. (6.17b) and (6.17c). Thus, the set of equilibrium conditions then consists of Eq. (6.17a) and, considering that the radius i of a cylindrical shell is not a function of the coordinate x, of the following  

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Fig. 6.2. Orientation of coordinates and displacements for a cylindrical thin shell.

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