C modeling ============== Characteristics of modeling ----------------------------------- Loading is a combination of traction — compression and flexure. .. image:: images/1000020000000189000000EB5E0D602A193B4B87.png :width: 3.422in :height: 1.7992in .. _RefImage_1000020000000189000000EB5E0D602A193B4B87.png: **Figure** 5.1-a **:** m**mesh and boundary conditions** Modeling: DKT Boundary conditions: Traction — Compression and Flexion coupling: * :math:`\mathrm{DX}=0.0` and :math:`\mathrm{DRY}=0.0` on the :math:`{A}_{1}-{A}_{3}` edge * :math:`\mathrm{DX}={U}_{0}\times f(t)` and :math:`\mathrm{DRY}={R}_{0}\times f(t)` on the :math:`{A}_{2}-{A}_{4}` edge, where :math:`{U}_{0}\mathrm{=}1.5\times {10}^{\mathrm{-}4}m`, :math:`{R}_{0}\mathrm{=}5.\times {10}^{\mathrm{-}3}\mathit{rad}`, and :math:`f(t)` is the magnitude of the cyclic loading as a function of the (pseudo-time) parameter :math:`t`. Two types of loading are considered: * The same loading function :math:`\mathrm{f1}` for membrane and flexure (synchronous case): .. image:: images/10000000000001F8000001201AA96929C8D1ABF2.png :width: 3.2882in :height: 1.8783in .. _RefImage_10000000000001F8000001201AA96929C8D1ABF2.png: **Figure** 5.1-b **: f1 loading function** * The :math:`\mathrm{f2}` function of membrane loading twice as fast as that of flexure (in practice, the membrane frequencies of a slab are higher than those of flexure): .. image:: images/10000000000001f8000001205 BAC63A07247F1DC .png :width: 3.1535 in :height: 1.802 in .. _refImage_10000000000001F8000001205 BAC63A07247F1DC .png: **Figure** 5.1-c **: f2 loading function** Characteristics of the mesh ---------------------------- Number of knots: 9 Number of stitches: 8 TRIA3; 8 SEG2. Quantities tested and results: first loading (same loading function for membrane and flexure) ------------------ We compare the sum of the forces along the axis :math:`\mathit{Ox}` in :math:`\mathit{A1}\mathrm{-}\mathit{A3}`, the displacements along the axis :math:`\mathit{Oy}` in :math:`\mathit{A4}`, the moments along the axis :math:`\mathit{Oy}` in :math:`\mathit{A1}\mathrm{-}\mathit{A3}` and the rotations along the axis :math:`\mathit{Ox}` *en* :math:`\mathit{A4}` obtained by the multilayer modeling with the ENDO_ISOT_BETON law and by the one based on the law BETON_REGLE_PR, in terms of relative differences; the tolerances are taken in absolute values: .. csv-table:: "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**" "**PHASE ELASTIQUE** :math:`t=\mathrm{0,25}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in displacement* :math:`\mathrm{DY}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE ENDOMMAGEMENT** :math:`t=\mathrm{1,0}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in displacement* :math:`\mathrm{DY}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE DECHARGEMENT** :math:`t=\mathrm{1,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in displacement* :math:`\mathrm{DY}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE ELASTIQUE** :math:`t=\mathrm{2,25}` ", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE RECHARGEMENT** :math:`t=\mathrm{3,0}` ", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in displacement* :math:`\mathrm{DY}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE DECHARGEMENT** :math:`t=\mathrm{3,5}` ", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in displacement* :math:`\mathrm{DY}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" **Diagrams** **compared efforts** :math:`{N}_{\mathrm{xx}}` **—** **based on displacement** :math:`\mathrm{DX}` **imposed for load** :math:`\mathrm{f1}` **:**: ** .. image:: images/10000201000002D700000236F8397DAA0BC1D6B0.png :width: 4.6457in :height: 3.8508in .. _RefImage_10000201000002D700000236F8397DAA0BC1D6B0.png: **Comparative graphs** **moment** :math:`{M}_{\mathrm{yy}}` **based on the rotation** :math:`\mathrm{DRY}` **imposed for loading** :math:`\mathrm{f1}` **:**: ** .. image:: images/10000201000002D200000235ABEA33B747C9FDE8.png :width: 4.6457in :height: 4.0783in .. _RefImage_10000201000002D200000235ABEA33B747C9FDE8.png: **Comparative displacement graphs** :math:`\mathrm{DY}` **(due to the Poisson effect) for loading** :math:`\mathrm{f1}` **:** .. image:: images/10000201000002D800000235CEF8BC5072DBA5CA.png :width: 4.6457in :height: 3.7681in .. _RefImage_10000201000002D800000235CEF8BC5072DBA5CA.png: **Comparative rotation graphs** :math:`\mathrm{DRX}` **(due to the Poisson effect) for loading** :math:`\mathrm{f1}` **:** .. image:: images/10000201000002ED000002355D8F40C167936A3B.png :width: 4.6457in :height: 3.4217in .. _RefImage_10000201000002ED000002355D8F40C167936A3B.png: Quantities tested and results: second loading (membrane twice as fast as flexure) ------------------------------------------------------------------------------------------------- We compare the forces along the axis :math:`\mathrm{Ox}` in :math:`\mathit{A1}\mathrm{-}\mathit{A3}`, the displacements along the axis :math:`\mathrm{Oy}` in :math:`\mathit{A4}`, the moments along the axis :math:`\mathrm{Oy}` in :math:`\mathit{A1}\mathrm{-}\mathit{A3}` and the rotations along the axis :math:`\mathrm{Ox}` *en* :math:`\mathit{A4}` obtained by the multilayer modeling with the ENDO_ISOT_BETON law and by the one based on the law BETON_REGLE_PR, in terms of relative differences; the tolerances are taken in absolute values: .. csv-table:: "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**" "**PHASE ELASTIQUE** :math:`t=\mathrm{0,2}` ", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in displacement* :math:`\mathrm{DY}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE ELASTIQUE** :math:`t=\mathrm{0,25}` ", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE ENDOMMAGEMENT** :math:`t=\mathrm{0,5}` ", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in displacement* :math:`\mathrm{DY}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE ENDOMMAGEMENT** :math:`t=\mathrm{1,0}` ", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE DECHARGEMENT** :math:`t=\mathrm{1,5}` ", "", "" "*Relative difference in efforts* :math:`{N}_{\mathrm{xx}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in displacement* :math:`\mathrm{DY}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE ELASTIQUE** :math:`t=\mathrm{2,25}` ", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE ELASTIQUE** :math:`t=\mathrm{2,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathit{xx}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in displacement* :math:`\mathit{DY}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE RECHARGEMENT** :math:`t=\mathrm{3,0}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathit{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathit{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**PHASE DECHARGEMENT** :math:`t=\mathrm{3,5}` ", "", "", "" "*Relative difference in efforts* :math:`{N}_{\mathit{xx}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in displacement* :math:`\mathit{DY}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative moment difference* :math:`{M}_{\mathit{yy}}` "," NON_REGRESSION "," ", "-", "1 10-6" "*Relative difference in rotations* :math:`\mathit{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" **Comparative force graphs** :math:`\mathrm{FX}` **(** efforts :math:`{N}_{\mathrm{xx}}`) **— (** efforts) **— as a function of displacement** :math:`\mathrm{DX}` **imposed for load** :math:`\mathrm{f2}` **:** .. image:: images/10000201000002D700000236B23E287185A71AD1.png :width: 4.6457in :height: 3.7756in .. _RefImage_10000201000002D700000236B23E287185A71AD1.png: **Comparative graphs** **moment** :math:`{M}_{\mathrm{yy}}` **depending on the rotation** :math:`\mathrm{DRY}` **imposed for loading** :math:`\mathrm{f2}` **:**: ** .. image:: images/10000201000002D20000023562D7413FCF27C14E.png :width: 4.6457in :height: 3.7362in .. _RefImage_10000201000002D20000023562D7413FCF27C14E.png: **Comparative displacement graphs** :math:`\mathrm{DY}` **(due to the Poisson effect) for loading** :math:`\mathrm{f2}` **:** .. image:: images/10000201000002D800000235AE987DD76773BB5F.png :width: 4.6457in :height: 3.4484in .. _RefImage_10000201000002D800000235AE987DD76773BB5F.png: **Comparative rotation graphs** :math:`\mathrm{DRX}` **(due to the Poisson effect) for loading** :math:`\mathrm{f2}` **:** .. image:: images/10000201000002ED00000235C921A78F1376694F.png :width: 4.6457in :height: 3.0429in .. _RefImage_10000201000002ED00000235C921A78F1376694F.png: notes --------- According to the preceding curves, it can be seen that the multilayer model with law BETON_REGLE_PR represents the overall behavior of reinforced concrete under bending and traction in a satisfactory manner under load. The same damage thresholds are identified. 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 zero rotation in the DRX direction and zero DY displacement. For the second load, a similar behavior is observed for the membrane-flexure coupling between BETON_REGLE_PRet ENDO_ISOT_BETON under load (elastic response, damage and compression). The same damage thresholds are identified. .. _refnumpara__6723_1772789992: