1. Reference problem#
1.1. Geometry#
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Figure 1.1-a: geometry of the reinforced concrete square plate
Length: \(l=1.0m\);
Thickness of the plate: \(e=0.1m\).
Steel diameter: \(0.01m\).
Coating the lower and upper reinforcing plies of the steels along the \(x\) axis: \(0.01m\).
For model DHRC, where the cross section is exactly represented, the steels along the \(y\) axis have a coating of \(0.021m\).
1.2. Material properties#
All the parameters of model GLRC_DM, elastic and non-linear, are identified from corresponding tests in the A and B models, except for the Young’s modulus modified in the D test (shear) in order to reduce the error in the linear domain in order to reduce the error in the linear domain and thus better validate the damage part. That is to say, we identify:
Effective Young’s modulus of membrane \({E}_{\mathrm{éq}}^{m}\) |
Effective Poisson’s Bending Ratio \({\nu }_{m}\) |
Effective Young’s modulus of flexure \({E}_{\mathrm{éq}}^{f}\) |
Effective membrane Poisson’s ratio \({\nu }_{f}\) |
Membrane force of the crack threshold under tension \({N}_{D}\) (noted \({\mathit{NYT}}_{\mathit{GLRC}}\)) |
Traction damage coefficient \({\mathrm{\gamma }}_{\mathit{mt}}\) |
Traction damage coefficient \({\mathrm{\gamma }}_{\mathit{mc}}\) |
Compression damage initiation coefficient \({\mathrm{\alpha }}_{c}\) |
Bending moment of the crack threshold during bending (\({M}_{D}\) noted \({\mathit{MYF}}_{\mathit{GLRC}}\)) |
Flexural damage coefficient \({\gamma }_{f}\) |
These parameters are calculated from the characteristics of steel materials (elastoplastic model \({E}_{a}\), \({\sigma }_{e}^{\mathrm{acier}}\), \({E}_{\mathrm{ecr}}^{\mathrm{acier}}\)) and concrete (via the model ENDO_ISOT_BETON \({E}_{b}\), \({\nu }_{b}\), \({\gamma }_{\mathrm{EIB}}\), \({\mathrm{SYT}}_{\mathrm{EIB}}\), see [R7.01.09]), and verified by calibration using the A modeling and thanks to the B. modeling
For the H, I, J and L, M, N models, the parameters of model GLRC_DM are calibrated from the parameters obtained for the model DHRC, by an automated parameter identification procedure described in document [R7.01.37], see [SU.10.01].
In summary, here are the values of the characteristics of the materials and the parameters GLRC_DM and EIB:
modeling |
A and B |
C |
D and E |
F |
G |
H, I, J, L, L, M and N |
N |
K |
|
\({E}_{a}\), \(\mathrm{MPa}\) |
200000 |
200000 |
200000 |
200000 |
200000 |
200000 |
200000 |
200000 |
|
\({\sigma }_{e}^{\mathrm{acier}}\), \(\mathrm{MPa}\) |
570 |
570 |
570 |
570 |
570 |
570 |
570 |
570 |
|
\({E}_{\mathrm{ecr}}^{\mathrm{acier}}\), \(\mathrm{MPa}\) |
300 |
300 |
300 |
300 |
300 |
300 |
300 |
300 |
300 |
\({E}_{b}\), \(\mathrm{MPa}\) |
32308 |
32308 |
32308 |
32308 |
32308 |
32308 |
32308 |
32308 |
|
\({\nu }_{b}\) |
0.2 |
0.2 |
0.2 |
0.2 |
0.2 |
0.2 |
0.2 |
0.2 |
|
D_ SIGM_EPSI |
|
|
|
|
|
|
|
||
\({\gamma }_{\mathrm{EIB}}\) |
|
|
|
|
|
|
|
||
\({\mathrm{SYT}}_{\mathrm{EIB}}\), \(\mathrm{MPa}\) |
3.4 |
3.4 |
3.4 |
3.4 |
3.4 |
3.4 |
3.4 |
3.4 |
|
\({E}_{\mathrm{éq}}^{m}\), \(\mathrm{MPa}\) |
35620 |
35700 |
35700 |
35700 |
32308 |
35700 |
42510 |
35625 |
35620 |
\({E}_{\mathrm{éq}}^{f}\), \(\mathrm{MPa}\) |
38700 |
38700 |
38700 |
38700 |
35700 |
73200 |
38660 |
38660 |
38700 |
\({\nu }_{m}\) |
0.18 |
0.18 |
0.18 |
0.18 |
0.18 |
0.16 |
0.18 |
0.18 |
|
\({\nu }_{f}\) |
0.17 |
0.17 |
0.17 |
0.17 |
0.18 |
0.12 |
0.17 |
0.17 |
|
\({\gamma }_{\mathrm{mt}}\) |
0.02 |
0.02 |
0.02 |
0.02 |
0.1 |
0.225 |
0.06 |
0.02 |
|
\({\gamma }_{c}\) |
1 |
1 |
1 |
1 |
1 |
1 |
0.8 |
0.02 |
|
\({\gamma }_{f}\) |
0.32 |
0.32 |
0.32 |
0.32 |
0.1 |
0.6 |
0.45 |
0.32 |
|
\({\mathit{NYT}}_{\mathit{GLRC}}\), \(N/m\) |
370000 |
370000 |
370000 |
370000 |
370000 |
360000 |
370000 |
370000 |
370000 |
\({\mathit{MYF}}_{\mathit{GLRC}}\), \(N\) |
9000 |
9000 |
9000 |
9000 |
5000 |
1600 |
9000 |
9000 |
|
\({\alpha }_{c}\) |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
100 |
In the C modeling, parameters whose presence allows the calculation of the internal variables V8 to V18 are added:
NYC |
|
EPSI_C |
0.00232424991421 |
EPSI_ELS |
2E-03 |
EPSI_LIM |
1E-02 |
RX |
0.8 |
OMX |
8E-04 |
EA |
2.E11 |
SY |
5.70000000E8 |
FTJ |
3.4E6 |
FCJ |
48E6 |
Note: we note that the value of \({\gamma }_{\mathrm{EIB}}\), cf. [R7.01.04], is comparable to the inverse of that used in the GLRC_DM law, cf. [R7.01.32].
The identification of the parameters of model DHRC is done automatically via an external procedure carried out beforehand, considering that the reinforced concrete plate is \(e=0.1m\) thick and that the steels have a diameter \(Ø=10\mathrm{mm}\), spaced by \(10\mathit{cm}\) apart. The elastic material characteristics for concrete and steel are those of model EIB summarized in the table above. The identification also requires damage parameters for concrete \({\alpha }_{t}\), \({\gamma }_{t}\),, \({\alpha }_{c}\) and \({\gamma }_{c}\) whose values are estimated from the usual results of concrete tension-compression tests, respectively: \(1\), \(-\mathrm{0,04}\),, \(\mathrm{1,9}\) and \(\mathrm{0,8}\). The threshold parameters for model DHRC are chosen from the thresholds for damage to concrete alone under tension and for steel-concrete sliding at \(\mathrm{1,7}\mathrm{MPa}\) and \(\mathrm{1,6}\mathrm{MPa}\), respectively. All of these values, which are inputs to the identification tool [SU1.10.01], are summarized in the table below:
\({E}_{a}\), \(\mathrm{MPa}\) |
|
|
, |
|
|
|
|
|
, |
, |
, |
, |
, \({\sigma }_{d}\) \(\mathrm{MPa}\) \({\tau }_{\mathrm{crit}}\) \(\mathrm{MPa}\) |
|||
200,000 |
0.2 |
32 308 |
32 308 |
0.2 |
0.2 |
0.04 |
1.9 |
0.8 |
1.7 |
1.7 |
1.6 |
1.3. Boundary conditions and loads#
Different models from A to P are considered for different types of characteristic loads and different plate behaviors. In all cases, the loads are movements (or rotations) imposed at the edges of the plate differently for each model.
The models considered are:
modeling A: traction - compression - pure traction and test of DEFI_GLRC; GLRC_DM
B modeling: pure alternating flexure; GLRC_DM
C modeling: tension coupling - compression and flexure; GLRC_DM
D modeling: pure shear and in-plane distortion; GLRC_DM
E modeling: coupling of flexure and shear in the plane; GLRC_DM
F modeling: traction — pure compression — with « kit_dll » of damaging elastoplastic behavior (GLRC_DM + Von Mises);
G modeling: pure shear — with « kit_ddl » of damaging elastoplastic behavior (GLRC_DM + Von Mises);
H modeling: traction — pure compression, high stresses; GLRC_DM and DHRC
modeling I: pure alternating flexure, high stresses; GLRC_DMet DHRC
J modeling: traction/compression and flexure coupling, high stresses; GLRC_DMet DHRC
K modeling: compression - traction with ALPHA_C =100; GLRC_DM
L modeling: pure shear and in-plane distortion, high stresses; GLRC_DM and DHRC
M modeling: coupling flexure and shear in the plane, high stresses; GLRC_DMet DHRC
N modeling: anticlastic flexure, high stresses; GLRC_DMet DHRC
O modeling: thermal load stresses; GLRC_DM, GLRC_DAMAGE, DHRC and ELAS.
1.4. Initial conditions#
Initially, travel and constraints were zero everywhere.