Modeling A ============== Characteristics of modeling ----------------------------------- The load is of the traction — compression — pure traction type. :math:`\mathrm{A6}` .. image:: images/1000000000000320000001E18AA1E5624797F771.jpg :width: 4.0665in :height: 2.4382in .. _RefImage_1000000000000320000001E18AA1E5624797F771.jpg: **Figure** 3.1-a **: mesh and boundary conditions.** Modeling: DKT Boundary conditions: * Embedding in :math:`{A}_{1}`; * :math:`\mathrm{DX}=0.0` on the :math:`{A}_{1}-{A}_{3}` edge; * * :math:`\mathrm{DX}={U}_{0}\times f(t)` on the :math:`{A}_{2}-{A}_{4}` edge; where :math:`{U}_{0}=2.0\times {10}^{-4}m` and :math:`f(t)` represent the amplitude of cyclic loading as a function of the (pseudo-time) parameter :math:`t`. To properly verify the model, two loading functions are considered as follows: +-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ | | | + .. image:: images/10000000000001F8000001201AA96929C8D1ABF2.png + .. image:: images/10000000000001F800000120F7A19CE7F7EAE366.png + | :width: 3.4437in | :width: 3.4453in | + :height: 1.9673in + :height: 1.9689in + | | | + + + | | | +-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ **Figure** 3.1-b **:** f **loading functions f1 (left) and f2 (right) .** *Note: the extreme deformation is:* :math:`2.0\times {10}^{\mathrm{-}4}` *, which is well below the transition to plasticity of steels.* *No* *integration time:* :math:`0.05` *.* Characteristics of the mesh ---------------------------- Number of knots: 9. Number of stitches: 8 TRIA3; 8 SEG2. Tested quantities and results for the f1 loading function ---------------------------------------------------------------- We compare the sum of the reaction forces along the axis :math:`\mathrm{Ox}` in :math:`\mathit{A1}\mathrm{-}\mathit{A3}` and the displacements along the axis :math:`\mathrm{Oy}` in :math:`\mathit{A4}` (Poisson effect) obtained by the multilayer modeling with the ENDO_ISOT_BETON law and by that based on the BETON_REGLE_PR law, in terms of relative differences, in terms of relative differences; the tolerance is taken as an absolute value on these relative differences: .. csv-table:: "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**" "**TRAC. - PHASE CHAR. ELAS .** :math:`t=\mathrm{0,25}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1 10-2" "*Relative difference in displacement* :math:`\mathrm{DY}` "," NON - REGRESSION ", "-", "1 10-6" "**TRAC. - PHASE CHAR. ENDO .** :math:`t=\mathrm{1,0}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1 10-2" "**TRAC. - PHASE DECHAR. ELAS .** :math:`t=\mathrm{1,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1.1" "**COMPR. - PHASE CHAR. ELAS .** :math:`t=\mathrm{3,0}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1 10-2" "**COMPR. - PHASE DECHAR. ELAS** :math:`t=\mathrm{3,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "3 10-2" **Comparative graphs** **efforts** :math:`{N}_{\mathrm{xx}}` **— displacement** :math:`\mathrm{DX}` **in traction/compression for load** :math:`\mathrm{f1}` **:**: ** .. image:: images/10000201000002D700000236D02AD3A549A5DBF1.png :width: 4.6457in :height: 3.9173in .. _RefImage_10000201000002D700000236D02AD3A549A5DBF1.png: **Comparative graphs** displacement :math:`\mathrm{DY}` (due to the Poisson effect) as a function of time: .. image:: images/10000201000002D8000002358CA1054777031475.png :width: 4.6457in :height: 3.9335in .. _RefImage_10000201000002D8000002358CA1054777031475.png: Tested quantities and results for the f2 loading function ---------------------------------------------------------------- We compare the reaction forces along the :math:`\mathrm{Ox}` axis and the displacements along the :math:`\mathrm{Oy}` axis in :math:`\mathit{A4}` obtained by the multilayer modeling with the ENDO_ISOT_BETON law and by the one based on the BETON_REGLE_PR law, in terms of relative differences; the tolerance is taken in absolute value on these relative differences: .. csv-table:: "**Identification**", "**Reference Type**", "**Reference Value**", "**Tolerance**" "**COMPR. - PHASE CHAR. ELAS .** :math:`t=\mathrm{0,25}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathit{xx}}` "," AUTRE_ASTER "," ", "0", "3 10-2" "**COMPR. - PHASE CHAR. ENDO .** :math:`t=\mathrm{1,0}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "5 10-2" "**COMPR. - PHASE DECHAR. ELAS .** :math:`t=\mathrm{1,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathit{xx}}` "," AUTRE_ASTER "," ", "0", "3 10-2" "**COMPR .** - PHASE DECHAR. **ELAS .** :math:`t=\mathrm{2,25}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1 10-2" "**TRAC. - PHASE CHAR. ELAS .** :math:`t=\mathrm{3,0}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1 10-1" "**TRAC. - PHASE DECHAR. ELAS .** :math:`t=\mathrm{3,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1.1" **Comparative graphs** :math:`{N}_{\mathrm{xx}}` **— displacement** :math:`\mathrm{DX}` **in traction/compression for load** :math:`\mathrm{f2}` **:**: ** .. image:: images/10000201000002D700000236BBD113F6CA9C3AD2.png :width: 4.6457in :height: 3.4925in .. _RefImage_10000201000002D700000236BBD113F6CA9C3AD2.png: notes --------- According to the previous curves, it can be seen that the multilayer model with law BETON_REGLE_PR represents the overall behavior of reinforced concrete under tension — pure compression in a satisfactory manner under load. However, in discharge, law BETON_REGLE_PR follows the same curve as the charge, unlike law ENDO_ISOT_BETON. The Poisson effect is not modelled by law BETON_REGLE_PR, so we obtain a zero displacement in the transverse direction DY. A symmetry of the response is observed depending on the chosen direction of compression-traction load or inverse, depending on the load :math:`\mathit{f1}` or :math:`\mathit{f2}` for law BETON_REGLE_PR. .. _refnumpara__30672705: