B modeling ============== Characteristics of modeling ----------------------------------- The loading is of the pure alternating bending type. .. image:: images/10000000000001A9000001269D6B8497D226D3CE.png :width: 3.0992in :height: 1.9035in .. _RefImage_10000000000001A9000001269D6B8497D226D3CE.png: **Figure** 4.1-a **: mesh and boundary conditions** Modeling: DKT Boundary conditions: * :math:`\mathrm{DRY}=0.0` on the :math:`{A}_{1}-{A}_{3}` ridge * :math:`\mathrm{DRY}={R}_{0}\times f(t)` on the :math:`{A}_{2}-{A}_{4}` edge, where :math:`{R}_{0}=6\times {10}^{-3}` and :math:`f(t)` is the amplitude of the cyclic loading as a function of the (pseudo-time) parameter :math:`t`, to properly verify the model, we consider three loading functions as: +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | | | + .. image:: images/10000000000001F8000001201AA96929C8D1ABF2.png + .. image:: images/10000000000001F800000120F7A19CE7F7EAE366.png + | :width: 3.3681in | :width: 3.3681in | + :height: 1.9244in + :height: 1.9244in + | | | + **Figure** 4.1-b **:** f **negative flexion, then positive flexing** + **Figure** 4.1-c **:** f **positive flexion, then negative flexing** + | | | +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ .. image:: images/10000000000001FB00000120D2C7311DCD9F963A.png :width: 3.3437in :height: 1.8992in .. _RefImage_10000000000001FB00000120D2C7311DCD9F963A.png: **Figure** 4.1-d **:** d **two cycles of alternating flexure** Note: the extreme deformation of steels is: :math:`2.4\times {10}^{-3}`, i.e. below the transition to plasticity of the steels. *Integration increment:* :math:`0.05s` *.* Characteristics of the mesh ---------------------------- Number of knots: 9. Number of stitches: 8 TRIA3. Tested quantities and results for the f1 loading function ---------------------------------------------------------------- We compare the sum of the moments along the axis :math:`\mathrm{Oy}` in :math:`\mathit{A1}\mathrm{-}\mathit{A3}` and the rotations along the axis :math:`\mathrm{Ox}` 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 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**" "**FLEXION NEG - PHASE CHAR. ELAS .** :math:`t=\mathrm{0,25}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "3 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION NEG - PHASE CHAR. ENDO .** :math:`t=\mathrm{1,0}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "1 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION NEG - PHASE DECHAR. ELAS .** :math:`t=\mathrm{1,5}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "6 10-1" "*Relative difference in rotations DRX*", "NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION POS. - PHASE CHAR. ELAS .** :math:`t=\mathrm{2,25}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "3 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION POS. - PHASE CHAR. ENDO .** :math:`t=\mathrm{3,0}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "2 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "FLEXION POS. - PHASE DECHAR. ELAS. :math:`t=\mathrm{3,5}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "6 10-1" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" **Comparative moment/rotation diagrams in cyclic bending for load** :math:`\mathrm{f1}` **:** .. image:: images/10000201000002D2000002351644D27374A242D7.png :width: 4.6457in :height: 3.972in .. _RefImage_10000201000002D2000002351644D27374A242D7.png: refill (opposite bending) **Comparative graphs** **rotation** :math:`\mathrm{DRX}` **(due to the Poisson effect) as a function of time for loading** :math:`\mathrm{f1}` **:** .. image:: images/10000201000002ED000002351345ED56ACBE9458.png :width: 4.6457in :height: 3.9571in .. _RefImage_10000201000002ED000002351345ED56ACBE9458.png: Tested quantities and results for the f2 loading function ---------------------------------------------------------------- We compare the moments along the :math:`\mathrm{Oy}` axis in :math:`\mathit{A1}\mathrm{-}\mathit{A3}` and the rotations along the :math:`\mathrm{Ox}` *en* :math:`\mathit{A4}` axis obtained by multilayer modeling with the ENDO_ISOT_BETON law and by that based on the BETON_REGLE_PR law, in terms of relative differences; some tolerances are taken in absolute value: .. csv-table:: "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**" "**FLEXION NEG. - PHASE CHAR. ELAS .** :math:`t=\mathrm{0,25}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "3 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION NEG. - PHASE CHAR. ENDO .** :math:`t=\mathrm{1,0}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "1 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION NEG. - PHASE DECHAR. ELAS .** :math:`t=\mathrm{1,5}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "6 10-1" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION POS. - PHASE CHAR. ELAS .** :math:`t=\mathrm{2,25}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "3 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION POS. - PHASE CHAR. ENDO .** :math:`t=\mathrm{3,0}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "2 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "FLEXION POS. - PHASE **DECHAR. ELAS .** :math:`t=\mathrm{3,5}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "6 10-1" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" It is verified that these results are identical to those obtained with load :math:`\mathrm{f1}` (in the opposite direction). Tested quantities and results for the f3 loading function ---------------------------------------------------------------- We compare the moments along the :math:`\mathrm{Oy}` axis in :math:`\mathit{A1}\mathrm{-}\mathit{A3}` and the rotations along the :math:`\mathrm{Ox}` axis in :math:`\mathit{A4}` obtained by multilayer modeling with the ENDO_ISOT_BETON law and by the one based on the BETON_REGLE_PR law, in terms of relative differences: .. csv-table:: "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**" "**FLEXION NEG. - PHASE CHAR. ELAS .** :math:`t=\mathrm{4,25}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "7 10-1" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION NEG. - PHASE CHAR. ENDO .** :math:`t=\mathrm{5,0}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "3 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION NEG. - PHASE DECHAR. ELAS .** :math:`t=\mathrm{1,5}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "6 10-1" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION POS. - PHASE CHAR. ELAS .** :math:`t=\mathrm{2,25}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "3 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "**FLEXION POS. - PHASE CHAR. ENDO .** :math:`t=\mathrm{3,0}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathrm{yy}}` "," AUTRE_ASTER "," ", "0", "2 10-2" "*Relative difference in rotations* :math:`\mathrm{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" "FLEXION POS. - PHASE **DECHAR. ELAS .** :math:`t=\mathrm{3,5}` ", "", "", "" "*Relative moment difference* :math:`{M}_{\mathit{yy}}` "," AUTRE_ASTER "," ", "0", "6 10-1" "*Relative difference in rotations* :math:`\mathit{DRX}` "," NON_REGRESSION "," ", "-", "1 10-6" **Comparative moment/rotation diagrams in cyclic bending for load** :math:`\mathit{f3}` **:** .. image:: images/10000201000002D20000023508EEDDE4A5885D6F.png :width: 4.6457in :height: 3.7681in .. _RefImage_10000201000002D20000023508EEDDE4A5885D6F.png: **Comparative graphs** **rotation** :math:`\mathit{DRX}` **(due to the Poisson effect) as a function of time for loading** :math:`\mathit{f3}` **:** .. image:: images/10000201000002E500000235D594B585489C5472.png :width: 4.6457in :height: 3.2445in .. _RefImage_10000201000002E500000235D594B585489C5472.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 bending —+ 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 get zero rotation in the DRX direction. We observe a symmetry of the response according to the chosen direction of load (flexure — and +) or the opposite case, depending on the load :math:`\mathit{f1}` or :math:`\mathit{f2}` for law BETON_REGLE_PR. .. _refnumpara__6721_1772789992: