2. Benchmark solution

2.1. Calculation method used for the reference solution

The deformation due to pressure alone is given by:

\({u}_{r}=\frac{{p}_{0}}{\mathrm{\nu }EL}\times \frac{1}{1-2\mathrm{\nu }}\times \frac{{z}^{2}}{4}\mathrm{sin}(\mathrm{\theta })\); \({u}_{z}=\frac{-{p}_{0}}{\text{}2\mathit{nu}EL}\mathit{rz}\mathrm{sin}(\mathrm{\theta })\); \({u}_{\mathrm{\theta }}=\frac{{p}_{0}}{\mathrm{\nu }EL}\times \frac{1}{1-2\mathrm{\nu }}\times \frac{{z}^{2}}{4}\mathrm{cos}(\mathrm{\theta })\).

The constraint field is equal to:

\({\sigma }_{\mathit{rr}}=-\frac{{p}_{0}r\mathrm{sin}(\theta )}{L}\); \({\sigma }_{\mathit{zz}}=-\frac{1-\nu }{\nu L}{p}_{0}r\mathrm{sin}(\theta )\); \({\sigma }_{\theta \theta }=-\frac{{p}_{0}r\mathrm{sin}(\theta )}{L}\);

\({\sigma }_{\mathit{rz}}=\frac{{p}_{0}z\mathrm{sin}(\theta )}{L}\); \({\sigma }_{r\theta }=0\); \({\sigma }_{\theta z}=\frac{{p}_{0}z\mathrm{cos}(\theta )}{L}\).

2.2. Benchmark results

Radial (\(\mathit{DX}\)), vertical (\(\mathit{DY}\)), and ortho-radial (\(\mathit{DZ}\)) displacements at points \(C\) and \(D\) for an angle \(\theta =45°\).

Constraints at points \(A\) and \(B\).

2.3. Uncertainty about the solution

• Analytical solution.