2. Benchmark solution

2.1. Calculation method used for the reference solution

In each case, a preliminary calculation is carried out with the keyword LIAISON_DDL to introduce linear relationships between degrees of freedom. This calculation is used as a reference to the calculation with the LIAISON_MAIL keyword that generates these linear relationships.

To get the desired linear relationships with LIAISON_MAIL, we write:

Case 1:

LIAISON_MAIL :( NOEUD_2: E MAILLE_1 :Q1

CENTRE: B ANGL_NAUT: 90. TRAN :( -5. 0.))

This means that we eliminate the 2 degrees of freedom of the node \(E\) according to the degrees of freedom of the point \(E\text{'}\) obtained when we make \(E\) undergo a rotation of 90 degrees around \(B\) and then a vector translation (—5.0). So \(E\text{'}\) is in the middle of \(\mathit{CD}\). The displacement vector of \(E\) is identified (after rotation of 90 degrees) with that of \(E\text{'}\). So we get the 2 equations:

DX (E) = DY (E') = 0.5 DY (C) + 0.5 DY (D)

DY (E) = - XX (E') = -0.5 XX (C) - 0.5 XX (D)

Case 2:

LIAISON_MAIL :( MAILLE_2: S1 MAILLE_1 :( Q1, Q2)

DDL_2: “DNOR” DDL_1: “DNOR” CENTRE: B ANGL_NAUT: 180. TRAN :( +5. +10.))

This means that we eliminate the normal displacement of the nodes \(B\) and \(E\) (nodes of the segment \(\mathit{S1}\)) according to the degrees of freedom of the points \(B\text{'}\) and \(E\text{'}\) obtained when we make \(B\) and \(E\) undergo a rotation of 180 degrees around \(B\) and then a vector translation \((+\mathrm{5,}+10)\). So \(B\text{'}\) is in the middle of \(\mathit{CF}\) and \(E\text{'}\) is in the middle of \(\mathit{DC}\). The normal movement of B is identified (after rotating 180 degrees) to that of \(B\text{'}\). We’re doing the same for \(B\text{'}\). We then obtain the 2 equations:

:math:`\mathit{DY}(E)\mathrm{=}\mathrm{-}\mathit{DY}(E\text{'})\mathrm{=}\mathrm{-}0.5\mathit{DY}(C)\mathrm{-}0.5\mathit{DY}(D)`

:math:`\mathit{DY}(B)\mathrm{=}\mathrm{-}\mathit{DY}(B\text{'})\mathrm{=}\mathrm{-}0.5\mathit{DY}(C)\mathrm{-}0.5\mathit{DY}(F)`

2.2. Benchmark results

We observe the \(\mathit{DY}\) movement of the point \(F\):

case 1: \(\mathit{DY}(F)\mathrm{=}\mathrm{1.4153582447720D}+00\)

case 2: \(\mathit{DY}(F)\mathrm{=}\mathrm{1.0561898652983D}+00\)

These displacements are obtained with linear relationships between degrees of freedom introduced by the keyword LIAISON_DDL.

2.3. Uncertainties about the solution

No uncertainty.