2. Benchmark solution#

2.1. Calculation method used for the reference solution#

Isotropic material: Analytical solution [bib1], obtained with the thin shell hypothesis:

\({\sigma }_{\mathrm{zz}}=0\)

\({\sigma }_{\theta \theta }=\mathrm{P0}R\frac{(L-z)}{Le}\)

\({u}_{r}=\frac{\mathrm{P0}{R}^{2}}{Ee}\left[1-\frac{z}{L}\right]\)

\({u}_{z}=\frac{-\mathrm{P0}Rvz}{Ee}\left[1-\frac{z}{\mathrm{2L}}\right]\)

Radial displacement at the base of the cylinder: \({u}_{r}(z=0)=\frac{\mathrm{P0}{R}^{2}}{Ee}\)

Vertical movement at the top of the cylinder: \({u}_{z}(z=L)=\frac{-\mathrm{P0}RLv}{2Ee}\)

Circumferential stress at the bottom of the cylinder \({\sigma }_{\theta \theta }(z=0)=\frac{\mathrm{P0}R}{e}\)

Orthotropic material: The solution can be deduced from the previous one: the constraints being statically determined, it is sufficient to modify the law of behavior, and to integrate the deformations. In order for the solution to be independent of the various notations (the value \({E}_{T}\) does not have the same meaning depending on the orthotropy coordinate system), we use the \((r,\theta ,z)\) cylindrical coordinate system.

Radial displacement at the base of the cylinder: \({u}_{r}(z=0)=\frac{\mathrm{P0}{R}^{2}}{{E}_{\theta}e}\)

Vertical movement at the top of the cylinder: \({u}_{z}(z=L)=\frac{-\mathrm{P0}RL{v}_{\theta z}}{2{E}_{\theta }e}\)

Circumferential stress at the bottom of the cylinder \({\sigma }_{\theta \theta }(z=0)=\frac{\mathrm{P0}R}{e}\)

Isotropic material:
Radial displacement at the base of the cylinder:	\(\mathrm{Ur}(\mathrm{A1})=5.8017857E–05m\)
Vertical movement at the top of the cylinder:	\(\mathrm{Uz}(\mathrm{A3})=–2.442857E–05m\)
Circumferential stress at the bottom of the cylinder:	\(\mathrm{Stt}(\mathrm{A1})=2.1375E+06\mathrm{Pa}\)


Orthotropic material:
Radial displacement at the base of the cylinder:	\(\mathrm{Ur}(\mathrm{A1})=5.8017857E–05m\)
Vertical movement at the top of the cylinder:	\(\mathrm{Uz}(\mathrm{A3})=–6.107143E–06m\)
Circumferential stress at the bottom of the cylinder:	\(\mathrm{Stt}(\mathrm{A1})=2.1375E+06\mathrm{Pa}\)

Analytical solution.

PILKEY W.D.: « Formulas for Stress, Strain, and Structural Matrixes. » Wiley & Cons, New York, 1994.