9. G modeling#
9.1. Characteristics of modeling#
Loading is a combination of traction — compression and flexure.
Figure 9.1-a : mesh and boundary conditions
Modeling: DKTG
Boundary conditions: Traction — Compression and Flexion coupling:
\(\mathrm{DX}=0.0\) and \(\mathrm{DRY}=0.0\) on the \({A}_{1}\mathrm{-}{A}_{3}\) edge
\(\mathrm{DX}={U}_{0}\times f(t)\) and \(\mathrm{DRY}={R}_{0}\times f(t)\) on the \({A}_{2}-{A}_{4}\) edge,
where \({U}_{0}\mathrm{=}1.\times {10}^{\mathrm{-}3}m\), \({R}_{0}\mathrm{=}3.\times {10}^{\mathrm{-}2}\mathit{rad}\), and \(f(t)\) is the magnitude of the cyclic loading as a function of the (pseudo-time) parameter \(t\).
The following load is considered:
The same \(f\) loading function for membrane and flexure:
Figure 9.1-b : loading function
9.2. Characteristics of the mesh#
Number of knots: 9.
Number of stitches: 8 TRIA3; 8 SEG2.
9.3. Tested sizes and results#
We compare the forces along the axis \(\mathrm{Ox}\) in \(\mathit{A1}\mathrm{-}\mathit{A3}\), the displacements along the axis \(\mathrm{Oy}\) in \(\mathit{A4}\), the moments along the axis \(\mathrm{Oy}\) in \(\mathit{A1}\mathrm{-}\mathit{A3}\) and the rotations along the axis \(\mathrm{Ox}\) en \(\mathit{A4}\) obtained by the multilayer modeling (reference) and by those based on the model ENDO_ISOT_BETON, in in terms of relative differences; the tolerance is taken as an absolute value based on these relative differences:
Identification |
Reference type |
Reference value |
Tolerance |
|
FLEXION POSITIVE - ELASTIQUE \(t=\mathrm{0,25}\) |
||||
Relative difference \(\mathit{MY}\) |
|
1 10-6 |
||
Relative difference \(\mathit{DRX}\) |
|
1 10-6 |
||
Relative difference \(\mathit{FX}\) |
|
1 10-6 |
||
Relative difference \(\mathit{DY}\) |
|
1 10-6 |
||
FLEXION POSITIVE - ENDOMMAGEMENT \(t=\mathrm{1,0}\) |
||||
Relative difference \(\mathit{MY}\) |
|
1 10-6 |
||
Relative difference \(\mathit{FX}\) |
|
1 10-6 |
||
FLEXION POSITIVE - DECHARGEMENT \(t=\mathrm{1,5}\) |
||||
Relative difference \(\mathit{MY}\) |
|
1 10-6 |
||
Relative difference \(\mathit{FX}\) |
|
1 10-6 |
||
FLEXION NEGATIVE — ELASTIQUE \(t=\mathrm{2,25}\) |
||||
Relative difference \(\mathit{MY}\) |
|
1 10-6 |
||
Relative difference \(\mathit{FX}\) |
|
1 10-6 |
||
FLEXION NEGATIVE - ENDOMMAGEMENT \(t=\mathrm{3,0}\) |
||||
Relative difference \(\mathit{MY}\) |
|
1 10-6 |
||
Relative difference \(\mathit{FX}\) |
|
1 10-6 |
||
FLEXION NEGATIVE - **** DECHARGEMENT ** \(t=\mathrm{3,5}\) |
||||
Relative difference \(\mathit{MY}\) |
|
1 10-6 |
||
Relative difference \(\mathit{FX}\) |
|
1 10-6 |
Force comparative graphs \(\mathrm{FX}\) — displacement \(\mathrm{DX}\) for load \(f\) :
Comparative moment graphs \(\mathrm{MY}\) **— rotation:math:`mathrm{DRY}`**for loading**:math:`f`**: **
Comparative graphs displacement \(\mathrm{DY}\) (due to the Poisson effect) as a function of time:
Comparative rotation graphs \(\mathit{DRX}\) (due to the Poisson effect) as a function of time:
9.4. notes#
The test case carried out here aims to test model BETON_REGLE_PRsous with stresses that are significant enough for steels to effectively recover their stiffness.
The behavior is similar in bending and traction for the laws BETON_REGLE_PRet ENDO_ISOT_BETON under load: the differences appear for large loads due to the difference in behavior under compression.
The landfill response is not taken into account by law BETON_REGLE_PR (elastic response).
Rotation \(\mathrm{DRX}\) and displacement \(\mathit{DY}\) are zero with law BETON_REGLE_PR because the Poisson effect is not taken into account.