Modeling A ============== Characteristics of modeling ----------------------------------- Traction - compression - pure traction. .. image:: images/1000000000000320000001E18AA1E5624797F771.jpg :width: 5.2043in :height: 3.1201in .. _RefImage_1000000000000320000001E18AA1E5624797F771.jpg: Figure 3.1-a: mesh and boundary conditions. Modeling: DKTG 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: Load functions f1 (left) and f2 (right). *Note: the extreme deformation is:* :math:`2.0\times {10}^{-4}` *, which is well below the transition to plasticity of steels.* *No* *integration time:* :math:`0.05s` *.* Characteristics of the mesh ---------------------------- Number of knots: 9. Number of stitches: 8 TRIA3; 8 SEG2. Tested values and results for the f1 loading function -------------------------------------------------------------- The average reaction forces along the axis :math:`\mathrm{Ox}` and the average displacements along the axis :math:`\mathrm{Oy}` and :math:`\mathrm{A2}-\mathrm{A4}` obtained by multi-layer modeling (reference) and by that based on the model GLRC_DM are compared, in terms of relative differences; the tolerance is taken as an absolute value on these 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", "5 10-2" "*Relative difference in displacement* :math:`\mathrm{DY}` "," AUTRE_ASTER "," ", "0", "5 10-2" "TRAC. - PHASE CHAR. ENDO. :math:`t=\mathrm{1,0}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1.2 10-1" "*Relative difference in displacement* :math:`\mathrm{DY}` "," AUTRE_ASTER "," ", "0", "1.7 10-1" "TRAC. - PHASE DECHAR. ELAS. :math:`t=\mathrm{1,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1.2 10-1" "*Relative difference in displacement* :math:`\mathrm{DY}` "," AUTRE_ASTER "," ", "0", "1.7 10-1" "COMPR. - PHASE CHAR. ELAS. :math:`t=\mathrm{3,0}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "5 10-2" "*Relative difference in displacement* :math:`\mathrm{DY}` "," AUTRE_ASTER "," ", "0", "1.7 10-1" "COMPR. - PHASE DECHAR. ELAS :math:`t=\mathrm{3,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "5 10-2" "*Relative difference in displacement* :math:`\mathrm{DY}` "," AUTRE_ASTER "," ", "0", "1.7 10-1" **Comparative graphs** **efforts** :math:`{N}_{\mathrm{xx}}` **— displacement** :math:`\mathrm{DX}` **in traction/compression for load** :math:`\mathrm{f1}` **:**: ** .. image:: images/10000000000001F8000001203F343E78BB7FAF27.png :width: 5.2508in :height: 3.0008in .. _RefImage_10000000000001F8000001203F343E78BB7FAF27.png: **Comparative graphs** displacement :math:`\mathrm{DY}` (due to the Poisson effect) as a function of time: .. image:: images/10000000000001F8000001206D9EFDD91669AD32.png :width: 5.2508in :height: 3.0008in .. _RefImage_10000000000001F8000001206D9EFDD91669AD32.png: **Diagrams** of the evolution of the damage of model GLRC_DM (:math:`{d}_{1}` for the upper side and :math:`{d}_{2}` for the lower side) as a function of time: .. image:: images/10000000000001F800000120938A13AE01738FDE.png :width: 5.2508in :height: 3.0008in .. _RefImage_10000000000001F800000120938A13AE01738FDE.png: Using the damage variables, we also test the energy dissipated, which is written as: [:external:ref:`R7.01.32 §2.7 `]: :math:`E\mathrm{=}{k}_{0}\mathrm{\times }({d}_{1}+{d}_{2})` with here :math:`{k}_{0}\mathrm{=}8.89910J\mathrm{/}{m}^{2}` The energy dissipated therefore has the same profile as the curve above. Tested values and results for the f2 loading function -------------------------------------------------------------- The average reaction forces along the axis :math:`\mathrm{Ox}` and the average displacements along the axis :math:`\mathrm{Oy}` and :math:`\mathrm{A2}-\mathrm{A4}` obtained by multi-layer modeling (reference) and by that based on the model GLRC_DM are compared, in terms of relative differences; the tolerance is taken as an absolute value for these relative differences; the tolerance is taken as an absolute value for 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", "5 10-2" "*Relative difference in displacement* :math:`\mathrm{DY}` "," AUTRE_ASTER "," ", "0", "5 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" "*Relative difference in displacement* :math:`\mathrm{DY}` "," AUTRE_ASTER "," ", "0", "5 10-2" "COMPR. - PHASE DECHAR. ELAS. :math:`t=\mathrm{1,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "5 10-2" "*Relative difference in displacement* :math:`\mathrm{DY}` "," AUTRE_ASTER "," ", "0", "5 10-2" "TRAC. - PHASE CHAR. ELAS. :math:`t=\mathrm{3,0}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1.2 10-1" "*Relative difference in displacement* :math:`\mathrm{DY}` "," AUTRE_ASTER "," ", "0", "1.7 10-1" "TRAC. - PHASE DECHAR. ELAS. :math:`t=\mathrm{3,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," AUTRE_ASTER "," ", "0", "1.2 10-1" "*Relative difference in displacement* :math:`\mathrm{DY}` "," AUTRE_ASTER "," ", "0", "1.7 10-1" **Comparative graphs** :math:`{N}_{\mathrm{xx}}` **— displacement** :math:`\mathrm{DX}` **in traction/compression for load** :math:`\mathrm{f2}` **:**: ** .. image:: images/10000000000001FD0000012026EC3080587C9C49.png :width: 5.3028in :height: 3.0008in .. _RefImage_10000000000001FD0000012026EC3080587C9C49.png: **Comparative graphs** displacement :math:`\mathrm{DY}` (due to the Poisson effect) as a function of time: .. image:: images/10000000000001FD000001204EE458CF71760E5C.png :width: 5.3028in :height: 3.0008in .. _RefImage_10000000000001FD000001204EE458CF71760E5C.png: **Diagrams** of the evolution of the damage of model GLRC_DM (:math:`{d}_{1}` for the upper side and :math:`{d}_{2}` for the lower side) as a function of time: .. image:: images/10000000000001FD00000120C5A64D270C2CE8A3.png :width: 5.3028in :height: 3.0008in .. _RefImage_10000000000001FD00000120C5A64D270C2CE8A3.png: notes --------- According to the preceding curves, it can be seen that model GLRC_DM represents the overall behavior of reinforced concrete under tensile — pure compression in a satisfactory manner. The relative error of model GLRC_DM with respect to the reference solution is admissible. It should be noted that the difference between model GLRC_DM and ENDO_ISOT_BETON is the most important during the damage phase: the behavior of concrete under tension is then softening and we find a negative slope in the multi-layer reference model, despite the reinforcement (steel layers and layers ENDO_ISOT_BETON) while one of the hypotheses of the model GLRC_DM is not to model the softening of reinforced concrete. Being based on the isotropic equivalent material hypothesis (see [:external:ref:`R7.01.32 `]), the GLRC_DM model slightly overestimates the Poisson effect. We also check the symmetry of the response according to the chosen direction of compression-traction load or the opposite, depending on the load :math:`\mathit{f1}` or :math:`\mathit{f2}`. .. _refnumpara__30672705: