1. Reference problem

1.1. Geometry

Straight pipe of length \(80m\), modelled in shells between \(0\) and \(\mathrm{40m}\), and in beams between \(40\) and \(\mathrm{80m}\).

Outer radius: \(\mathrm{2m}\), thickness: \(\mathrm{0.1m}\).

The pipe axis is in plane \(\mathrm{Oxy}\), angled 30 degrees from \(\mathrm{Ox}\).

1.2. Material properties

\(E={2.10}^{11}\mathrm{Pa}\)

\(\nu =0.3\)

1.3. Boundary conditions and loads

« Beam-type » embedding in \(x=y=0\), made by a shell-beam connection between the \(\mathrm{C1}\) edge of the shell and a point \(\mathrm{P1}\) (located in \(O\)). It is this point that is blocked.

4 unit load cases applied to point \(\mathit{P2}(80\mathrm{\times }\mathit{cos30},80\mathrm{\times }\mathit{sin30},0)\)

Traction \(\mathrm{Fx1}=\mathrm{1N}\) along the \(\mathrm{Ox1}\) axis, i.e. \(\mathrm{Fx}=\mathrm{cos30.Fx}\) and \(\mathrm{Fy}=\mathrm{sin30.Fx1}\)

Torsional moment \(\mathrm{Mx1}=\mathrm{1Nm}\) around \(\mathrm{Ox1}\), i.e. \(\mathrm{Mx}=\mathrm{cos30.Mx1}\) and \(\mathrm{My}=\mathrm{sin30.Mx1}\)

Bending moment \(\mathrm{My1}=\mathrm{1Nm}\) around \(\mathrm{Oy1}\) i.e. \(\mathrm{Mx}=-\mathrm{sin30.My1}\) and \(\mathrm{My}=\mathrm{cos30.My1}\)

Bending moment around \(\mathrm{Oz}\), which is \(\mathrm{Mz}=\mathrm{1Nm}\)