1. Reference problem

1.1. Geometry

A system of 3 bars in \(U\), drawn here in a local \((x,y)\) coordinate system:

The area of the cross sections is \(A\mathrm{=}{\mathrm{1m}}^{2}\). The length of each of the 3 bars is \(L\mathrm{=}\mathrm{10m}\).

The coordinate system in which is drawn here is rotated by \(60°\) in relation to the laboratory coordinate system \((X,Y)\):

1.2. Material properties

\(E\mathrm{=}2.{10}^{11}\mathit{Pa}\) for the 3 bars.

\(\rho \mathrm{=}8000\mathit{kg}\mathrm{/}{m}^{3}\) only for the \(\mathit{CD}\) bar. For the other 2 bars, \(\rho \mathrm{=}0\).

1.3. Boundary conditions and loads

Embedding in \(A\) and \(B\).

To avoid rigid body movements, \(\mathit{DZ}\mathrm{=}0\) for all knots, and \(\mathit{DX}\mathrm{=}0\) in \(C\) and \(D\).

A single load is applied: gravity. Gravity is obviously linked to the laboratory’s coordinate system, so in the \(\mathrm{-}Y\) direction; we take a virtual acceleration of \(g\mathrm{=}\mathrm{20m}\mathrm{/}{s}^{2}\). In the frame of reference of the structure, gravity is therefore expressed \((\mathrm{sin}(60)g,\mathrm{-}\mathrm{cos}(60)g)\mathrm{=}(0.866\mathrm{\times }g,\mathrm{-}0.5\mathrm{\times }g\mathrm{,0})\), which is equivalent to \(g\mathrm{=}\mathrm{10m}\mathrm{/}{s}^{2}\), in the direction \(–y\).