3. Modeling A#
3.1. Characteristics of modeling#
The concrete beam is represented by 60 MECA_HEXA20 elements, supported by as many hexahedral meshes at 20 knots. The figure below gives a simplified representation of the mesh of the beam.
A concrete material is assigned to the elements, for which the behaviors ELAS (Young’s module \({E}_{b}=\mathrm{4,5}{.10}^{10}\mathrm{Pa}\)) and BPEL_BETON are defined: the characteristic parameters of this relationship are set to 0 because the tension losses along the prestress cable are neglected.
The degrees of freedom \(\mathrm{DX}\), \(\mathrm{DY}\), and \(\mathrm{DZ}\) of the nodes on face \(x=0\) are blocked.
The cable is represented by 30 MECA_BARRE elements, supported by as many 2-node segment meshes. The left and right endpoints are nodes \(\mathrm{NC000001}\) and \(\mathrm{NC000031}\) respectively.
An area of right cross section \({S}_{a}=\mathrm{2,5}{.10}^{-3}{m}^{2}\) is assigned to the elements, as well as a steel material for which the behaviors ELAS (Young’s modulus \({E}_{a}\mathrm{=}\mathrm{1,85}{.10}^{11}\mathit{Pa}\)) and BPEL_ACIER are defined: the characteristic parameters of this relationship are set to 0 (tension losses neglected), with the exception of the elastic limit stress for which a zero value is illegal (\({f}_{\mathit{prg}}\mathrm{=}\mathrm{1,77}{.10}^{9}\mathit{Pa}\)).
To avoid any redundancy with kinematic relationships, no blocking is imposed on node \(\mathit{NC000001}\) (see note in paragraph [§2.3]).
Tension \({F}_{0}\mathrm{=}{10}^{6}N\) is applied to node \(\mathit{NC000031}\). This tension value is consistent with the cross-section and elastic limit values, for a strand-type pretension cable.
The calculation of the equilibrium state of the girder and cable assembly is carried out in a single step, the behavior being elastic. An additional calculation is then carried out to determine the stresses at the nodes of the elements of the beam.
3.2. Calculation steps and functionalities tested#
The main calculation steps correspond to the functionalities that we want to validate:
operator DEFI_MATERIAU: definition of the behavioral relationships BPEL_BETON and BPEL_ACIER, in the particular case where the tension losses along the prestress cable are neglected (values by default parameters);
operator DEFI_CABLE_BP: determination of a constant tension profile along the prestress cable, the losses being neglected; calculation of the coefficients of the kinematic relationships between the degrees of freedom of the cable nodes and the degrees of freedom of the « neighboring » nodes of the concrete beam, in the case of a beam modelled by 3D elements;
operator AFFE_CHAR_MECA: definition of a RELA_CINE_BP load;
operator STAT_NON_LINE, option COMPORTEMENT: calculation of the equilibrium state taking into account the load of type RELA_CINE_BP, in the case of a beam modelled by 3D elements.
3.3. Model A results#
3.3.1. Displacements of the girder nodes#
The values extracted from field DEPL from STAT_NON_LINE are compared to the theoretical reference values. The tolerance for relative deviation from the reference is equal to:
3% for node \(\mathrm{NB010527}\);
1% for the nodes \(\mathrm{NB030127}\), \(\mathrm{NB050127}\), and \(\mathrm{NB050527}\);
0.1% for the other nodes.
Node |
Component |
Reference Value |
Tolerance (%) |
NB010105 |
DX |
—2,298342.10—4m |
\(\mathrm{1,0}\) |
NB010305 |
DX |
—1,237569.10—4m |
\(\mathrm{1,0}\) |
NB010505 |
DX |
—1,767956.10—5m |
\(\mathrm{1,0}\) |
NB030105 |
DX |
—1,502762.10—4m |
\(\mathrm{1,0}\) |
NB030305 |
DX |
—4,419890.10—5m |
\(\mathrm{1,0}\) |
NB030305 |
DY |
—7,955801.10—5m |
\(\mathrm{1,0}\) |
NB030305 |
DZ |
—1,060773.10—4m |
\(\mathrm{1,0}\) |
NB030505 |
DX |
+6,187845.10—5m |
\(\mathrm{1,0}\) |
NB050105 |
DX |
—7,071823.10—5m |
\(\mathrm{1,0}\) |
NB050305 |
DX |
+3,535912.10—5m |
\(\mathrm{1,0}\) |
NB050505 |
DX |
+1.414365.10—4m |
\(\mathrm{1,0}\) |
NB010116 |
DX |
—8,618785.10—4m |
\(\mathrm{1,0}\) |
NB010316 |
DX |
—4,640884.10—4m |
\(\mathrm{1,0}\) |
NB010516 |
DX |
—6,629834.10—5m |
\(\mathrm{1,0}\) |
NB030116 |
DX |
—5,635359.10—4m |
\(\mathrm{1,0}\) |
NB030316 |
DX |
—1,657459.10—4m |
\(\mathrm{1,0}\) |
NB030316 |
DY |
—1,118785.10—3m |
\(\mathrm{1,0}\) |
NB030316 |
DZ |
—1,491713.10—3m |
\(\mathrm{1,0}\) |
NB030516 |
DX |
+2,320442.10—4m |
\(\mathrm{1,0}\) |
NB050116 |
DX |
—2,651934.10—4m |
\(\mathrm{1,0}\) |
NB050316 |
DX |
+1,325967.10—4m |
\(\mathrm{1,0}\) |
NB050516 |
DX |
+5,303867.10—4m |
\(\mathrm{1,0}\) |
NB010127 |
DX |
—1,493923.10—3m |
\(\mathrm{1,0}\) |
NB010327 |
DX |
—8,044199.10—4m |
\(\mathrm{1,0}\) |
NB010527 |
DX |
—1,149171.10—4m |
\(\mathrm{3,0}\) |
NB030127 |
DX |
—9,767956.10—4m |
\(\mathrm{1,0}\) |
NB030327 |
DX |
—2,872928.10—4m |
\(\mathrm{1,0}\) |
NB030327 |
DY |
—3,361326.10—3m |
\(\mathrm{1,0}\) |
NB030327 |
DZ |
—4,481768.10—3m |
\(\mathrm{1,0}\) |
NB030527 |
DX |
+4.022099.10—4m |
\(\mathrm{1,0}\) |
NB050127 |
DX |
—4,596685.10—4m |
\(\mathrm{1,0}\) |
NB050327 |
DX |
+2,298343.10—4m |
\(\mathrm{1,0}\) |
NB050527 |
DX |
+9,193370.10—4m |
\(\mathrm{1,0}\) |
3.3.2. Displacements of the nodes of the prestress cable#
The values extracted from field DEPL from STAT_NON_LINE are compared to the theoretical reference values. The tolerance for relative deviation from the reference is equal to:
1% for node \(\mathrm{NC000031}\), component \(\mathrm{DZ}\);
0.1% for the other nodes.
Node |
Component |
Reference Value |
Tolerance (%) |
NC000006 |
DY |
—1,243094.10—4m |
\(\mathrm{1,0}\) |
NC000006 |
DZ |
—1,657459.10—4m |
\(\mathrm{1,0}\) |
NC000011 |
DY |
—4,972376.10—4m |
\(\mathrm{1,0}\) |
NC000011 |
DZ |
—6,629834.10—4m |
\(\mathrm{1,0}\) |
NC000016 |
DY |
—1,118785.10—3m |
\(\mathrm{1,0}\) |
NC000016 |
DZ |
—1,491713.10—3m |
\(\mathrm{1,0}\) |
NC000021 |
DY |
—1,988950.10—3m |
\(\mathrm{1,0}\) |
NC000021 |
DZ |
—2,651934.10—3m |
\(\mathrm{1,0}\) |
NC000026 |
DY |
—3,107735.10—3m |
\(\mathrm{1,0}\) |
NC000026 |
DZ |
—4,143646.10—3m |
\(\mathrm{1,0}\) |
NC000031 |
DY |
—4,475138.10—3m |
\(\mathrm{1,0}\) |
NC000031 |
DZ |
—5,966851.10—3m |
\(\mathrm{10,0}\) |
3.3.3. Normal force in the pretension cable#
The value extracted from field SIEF_ELNO from STAT_NON_LINE is compared to the theoretical reference value.
The component that the test focuses on is \(N\).
Node |
Mesh |
Reference Value |
Tolerance (%) |
\(\mathrm{NC000016}\) |
|
|
|