1. Reference problem#
1.1. Plate geometry#
1.2. Characteristics of the models#
This test case is composed of 8 models. The table below summarizes their characteristics:
Orientation of the frames: longitudinal direction \((L)\): \(\mathrm{OX}\); transverse direction \((T)\): \(\mathrm{OY}\)
Eccentricity = \(0m\)
Section per linear meter = \(\mathrm{0,01}{m}^{2}/\mathrm{ml}\) (same thickness in the transverse and longitudinal directions in case of presence of transverse reinforcements)
Modeling |
Law of behavior |
Presence of frame transversal |
Temperature application mode |
A |
linear isotropic |
Yes |
at the nodes |
B |
linear kinematics |
Yes |
at the nodes |
C |
Pinto Menegotto |
Yes |
At the knots |
D |
Pinto Menegotto |
Yes |
to the elements |
E |
Pinto Menegotto |
No |
to the elements |
F |
linear isotropic |
No |
at the nodes |
G |
linear kinematics |
No |
at the nodes |
H |
Pinto Menegotto |
No |
*** For test case H, temperature loading is replaced by mechanical loading (displacement imposed on the nodes).
1.3. Material properties#
1.3.1. Properties common to all models#
Young’s module: |
\(E={2.10}^{11}\mathrm{MPa}\) |
Poisson’s ratio: |
\(\nu =0\) |
Elastic limit: |
\({\sigma }_{y}={2.10}^{8}\mathrm{MPa}\) |
Coefficient of thermal expansion: \(\alpha ={10}^{-5}(°{C}^{-1})\)
1.3.2. Isotropic and kinematic plastic behavior#
For isotropic (GRILLE_ISOT_LINE) and kinematic (GRILLE_CINE_LINE) behaviors
Work hardening slope: \({E}_{T}={2.10}^{10}\mathrm{MPa}\)
1.3.3. Pinto Menegotto’s behavior#
For the PINTO MENEGOTTO (GRILLE_PINTO_MEN) behavior
EPSI_ULTM |
: |
3.0. 10—2 |
SIGM_ULTM |
: |
2.58. 108 |
EPSP_HARD |
: |
0.0023 |
R_PM |
: |
20,0 |
EP_SUR_E |
: |
0.01 |
A1_PM |
: |
18.5 |
A2_PM |
: |
0.15 |
ELAN |
: |
4.9 |
A6_PM |
: |
620.0 |
C_PM |
: |
0.5 |
A_PM |
: |
0.008 |
1.4. Boundary conditions and loading#
The plate is fully embedded for models A to G (thermo-mechanical). For H modeling (mechanical), we block all the movements and all the rotations at the nodes except \(\mathrm{UX}\) for the nodes \(\mathrm{NO2}\) and \(\mathrm{NO3}\).
The loading is of thermal origin for models A to G. The evolution of the temperature as a function of time is given for each model in the following table. The temperature is applied to the nodes or to the elements, depending on the modeling.
Instant |
Evolution \(A\) \(T°\) |
Evolution \(B\) \(T°\) |
Evolution \(C\) \(T°\) |
|
0 |
50 |
50 |
50 |
|
1 |
—50 |
—50 |
—300 |
|
2 |
—250 |
—250 |
—100 |
|
3 |
—150 |
—150 |
50 |
|
4 |
—250 |
—250 |
—150 |
|
5 |
—50 |
—50 |
—350 |
|
6 |
350 |
350 |
350 |
—200 |
7 |
50 |
150 |
||
8 |
—450 |
—450 |
||
9 |
—110 |
—250 |
||
10 |
550 |
650 |
||
11 |
50 |
450 |
For all tests, a reference temperature of \(50°\) was taken.
For the H modeling, a nodal force \(\mathrm{FX}\) is applied to the nodes \(\mathrm{NO2}\) and \(\mathrm{NO3}\) (directed according to the vector \(\mathrm{UX}\)) by driving the calculation by the displacement \(\mathrm{UX}\) of \(\mathrm{NO3}\) so that it follows the following evolution:
Instant |
0.001 |
0.0023 |
0.0023 |
0.03 |
0.2 |
0.4 |
6.4 |
7.92 |
|
\(\mathit{Ux}\) (\(m\)) |
0.001 |
0.0023 |
0.0023 |
0.03 |
0.0296667 |
0.02 |
-0.04 |
-0.033 |
|
Instant |
17 |
19 |
19 |
20 |
20 |
21 |
22 |
25 |
50 |
\(\mathit{Ux}\) (\(m\)) |
0.01 |
0.03 |
0.03 |
0.0298 |
0.026 |
0.022 |
0.025 |
0.05 |