2. Benchmark solution

2.1. Calculation method used for the reference solution

The calculation method used for the reference solution was determined by F. Voldoire (EDF R&D/AMA) and is presented in the appendix.

The analytical reference results are:

• movements and rotations,

• axial stress, generalized forces (in shell theory),

in internal and external skins on sections \(\mathrm{AB}\) and \(\mathrm{CD}\).

In axisymmetric 2D, the complete solutions given in the appendix are such as \({\varepsilon }_{\mathrm{rz}}=0\) where \(r\) and \(z\) are the radial and axial directions of the cylinder, respectively. For the loads of uniform rotation and thermal expansion, the boundary conditions are chosen in such a way that the solutions do not depend on \(z\) (we have in particular \({\varepsilon }_{\mathrm{zz}}=0\)).

For shells, with equivalent boundary conditions, rotation \({\theta }_{\theta }\) around the orthoradial axis is zero for uniform rotation and thermal expansion loads, which is not the case for gravity loading where the rotation is constant (the cylinder then turns into a conical shape). On the other hand, in all cases, the transverse distortion is zero; thus the Love‑Kirchhoff theories and those of Hencky-Mindlin provide the same reference solution.

2.2. Reference quantity

\(\mathrm{DX}\): movement along the \(\mathrm{OX}\) axis,

\(\mathrm{DY}\): movement along the \(\mathrm{OY}\) axis,

\(\mathrm{DZ}\): movement along the \(\mathrm{OZ}\) axis,

\(\mathrm{NXX}\): normal force along the \(\mathrm{OX}\) axis,

\(\mathrm{NYY}\): normal force along the \(\mathrm{OY}\) axis,

\(\mathrm{MXX}\): moment around the \(\mathrm{OX}\) axis,

\(\mathrm{MYY}\): moment around the \(\mathrm{OY}\) axis,

\(\mathrm{SIXX}\): constraint along the \(\mathrm{OX}\) axis,

\(\mathrm{SIYY}\): constraint along the \(\mathrm{OY}\) axis.