Benchmark solution ===================== Calculation method used for the reference solution -------------------------------------------------------- Modeling A ~~~~~~~~~~~~~~~ The equations governing behavior are nonlinear differential equations. To validate the response obtained with *Code_Aster* in nonlinear statics, the values obtained are compared with the analytical values. .. _RefNumPara__3869_1409840568: Benchmark results ---------------------- The comparison is carried out on the moments ('MZ') obtained as a function of the rotations' DRZ 'imposed for the various elements and the loads applied, with a calculation time step of :math:`1.0s`. The moments are in :math:`s` and the moments in :math:`N\mathrm{.}m`. The curves below make it possible to verify that threshold crossings and slope changes are detected and taken into account even with a very rough calculation step. .. image:: images/10000201000005F7000003E6629FC035A4B46810.png :width: 6.3374in :height: 4.1417in .. _RefImage_10000201000005F7000003E6629FC035A4B46810.png: **Figure** 2.2-a **: Curve** **Moment-rotation,** **element 1.** The answer given by*Code_Aster* is tested for the following values: **Table** 2.2-a **:** **rotations and** **moments,** **element 1.** .. csv-table:: "**Moment**", "*rotation* :math:`{\theta }_{z}` ", "*Moment* :math:`{M}_{z}`" "1.0000000000E+00", "1.0000000000E+03", "1.0000000000E-03", "1.0000000000E-03" "3.0000000000E+00", "-1.0000000000E+03", "-1.0000000000E-03", "-1.0000000000E-03" "5.0000000000E+00", "1.4000156777E+03", "3.0000000000E-03", "3.0000000000E-03" "7.0000000000E+00", "-1.6500032505E+03", "-3.0000000000E-03", "-3.0000000000E-03" "9.0000000000E+00", "1.8000256320E+03", "5.0000000000E-03", "5.0000000000E-03" "1.1000000000E+01", "-1.8500108811E+03", "-5.0000000000E-03", "-5.0000000000E-03" "1.3000000000E+01", "2.2000192257E+03", "1.1000000000E-02", "1.1000000000E-02" "1.5000000000E+01", "-2.4833453804E+03", "-1.1000000000E-02", "-1.1000000000E-02" "1.7000000000E+01", "2.3601391920E+03", "1.5000000000E-02", "2.3601391920E+03" "1.9000000000E+01", "-2.6166696872E+03", "-1.5000000000E-02", "-1.5000000000E-02" "2.1000000000E+01", "2.5600393177E+03", "2.0000000000E-02", "2.0000000000E-02" "2.3000000000E+01", "-2.7833327927E+03", "-2.0000000000E-02", "-2.0000000000E-02" .. image:: images/10000201000005F7000003E61295576838939B58.png :width: 6.3374in :height: 4.1417in .. _RefImage_10000201000005F7000003E61295576838939B58.png: **Figure** 2.2-b **: Curve** **Moment-rotation,** **element 2.** The answer given by*Code_Aster* is tested for the following values: **Table** 2.2-b **:** **rotations and** **moments,** **element 2.** .. csv-table:: "**Moment**", "*rotation* :math:`{\theta }_{z}` ", "*Moment* :math:`{M}_{z}`" "1.0000000000E+00", "-1.0000000000E+03", "-1.0000000000E-03", "-1.0000000000E-03" "3.0000000000E+00", "1.0000000000E+03", "1.0000000000E-03", "1.0000000000E-03" "5.0000000000E+00", "-1.6500000701E+03", "-3.0000000000E-03", "-3.0000000000E-03" "7.0000000000E+00", "1.4000143913E+03", "3.0000000000E-03", "3.0000000000E-03" "9.0000000000E+00", "-1.8500245881E+03", "-5.0000000000E-03", "-5.0000000000E-03" "1.1000000000E+01", "1.8000013063E+03", "5.0000000000E-03", "1.10000000E-03" "1.3000000000E+01", "-2.4500169230E+03", "-1.1000000000E-02", "-1.1000000000E-02" "1.5000000000E+01", "2.200015606060E+03", "1.1000000000E-02" "1.7000000000E+01", "-2.6166668190E+03", "-1.5000000000E-02", "-1.5000000000E-02" "1.9000000000E+01", "2.3601185361E+03", "1.5000000000E-02", "2.3601185361E+03" "2.1000000000E+01", "-2.7833347518E+03", "-2.0000000000E-02", "-2.0000000000E-02" "2.3000000000E+01", "2.560038303030E+03", "2.0000000000E-02" .. image:: images/10000201000005F7000003E6E477314F4FEE71E9.png :width: 6.3374in :height: 4.1417in .. _RefImage_10000201000005F7000003E6E477314F4FEE71E9.png: **Figure** 2.2-c **: Curve** **Moment-rotation,** **elements 5 and 6.** **Table** 2.2-c **:** **rotations and** **moments,** **elements 5 and 6.** .. csv-table:: "**Moment**", "*rotation* :math:`{\theta }_{z}` ", "*Moment* :math:`{M}_{z}`" "3.0000000000E+00", "1.8000004556E+03", "5.0000000000E-03", "1.8000004556E+03" "6.0000000000E+00", "2.1600162134E+03", "1.0000000000E-02", "2.1600162134E+03" "9.0000000000E+00", "2.3600225066E+03", "1.5000000000E-02", "2.3600225066E+03" "1.2000000000E+01", "2.5599993245E+03", "2.0000000000E-02" "1.5000000000E+01", "1.1054534813E+03", "1.5000000000E-02", "1.5000000000E-02" "1.8000000000E+01", "-1.2000046711E+03", "1.0000000000E-02", "1.2000046711E+03" "2.1000000000E+01", "-1.9499993974E+03", "5.0000000000E-03", "-1.9499993974E+03" **Table** 2.2-d **:** **rotations and** **moments,** **element 3.** .. csv-table:: "**Moment**", "*rotation* :math:`{\theta }_{z}` ", "*Moment* :math:`{M}_{z}`" "1.0000000000E+00", "1.1904121201E+03", "1.1545287124E-03", "1.1904121201E+03" "3.0000000000E+00", "-1.3331111483E+03", "-1.1545081204E-03", "-1.3331111483E+03" "5.0000000000E+00", "1.7237265601E+03", "3.4638528074E-03", "3.4638528074E-03" "7.0000000000E+00", "-1.9588403006E+03", "-3.4638188809E-03", "-3.4638188809E-03" "9.0000000000E+00", "2.2570514348E+03", "5.7731769015E-03", "2.2570514348E+03" "1.1000000000E+01", "-2.2255575302E+03", "-5.7731814447E-03" "1.3000000000E+01", "2.6190094112E+03", "1.2701327870E-02", "2.6190094112E+03" "1.5000000000E+01", "-2.9330146633E+03", "-1.2701282573E-02", "-2.9330146633E+03" "1.7000000000E+01", "2.8324211293E+03", "1.7320099228E-02", "2.8324211293E+03" "1.9000000000E+01", "-3.1107779579E+03", "-1.7320059070E-02", "-3.1107779579E+03" "2.1000000000E+01", "3.0989885500E+03", "2.3093563448E-02" "2.3000000000E+01", "-3.3330038571E+03", "-2.3093529688E-02", "-3.3330038571E+03" **Table** 2.2-e **:** **rotations and** **moments,** **element 4.** .. csv-table:: "**Moment**", "*rotation* :math:`{\theta }_{z}` ", "*Moment* :math:`{M}_{z}`" "1.0000000000E+00", "-1.3331111481E+03", "-1.1545081204E-03", "-1.3331111481E+03" "3.0000000000E+00", "1.1904121201E+03", "1.1545287124E-03", "1.1904121201E+03" "5.0000000000E+00", "-1.9588485809E+03", "-3.4638188656E-03", "-3.4638188656E-03" "7.0000000000E+00", "1.7237108264E+03", "3.4638528189E-03", "3.4638528189E-03" "9.0000000000E+00", "-2.2255289211E+03", "-5.7731814457E-03", "-2.2255289211E+03" "1.1000000000E+01", "2.2570794615E+03", "5.7731768999E-03", "2.2570794615E+03" "1.3000000000E+01", "-2.9330774561E+03", "-1.2701282533E-02" "1.5000000000E+01", "2.6190179987E+03", "1.2701327871E-02", "2.6190179987871E-02" "1.7000000000E+01", "-3.1108083764E+03", "-1.7320059051E-02", "-3.1108083764E+03" "1.9000000000E+01", "2.8325478530E+03", "1.7320099228E-02" "2.1000000000E+01", "-3.3330092804E+03", "-2.3093529684E-02" "2.3000000000E+01", "3.0989396258E+03", "2.3093563476E-02" **Table** 2.2-f **:** **rotations and** **moments,** **element 7.** .. csv-table:: "**Moment**", "*rotation* :math:`{\theta }_{z}` ", "*Moment* :math:`{M}_{z}`" "3.0000000000E+00", "2.2570241105E+03", "5.7731769185E-03", "2.2570241105E+03" "6.0000000000E+00", "2.5656146751E+03", "1.1546635072E-02" "9.0000000000E+00", "2.8326866571E+03", "1.7320099213E-02", "2.8326866571E+03" "1.2000000000E+01", "3.0989736939E+03", "2.3093563449E-02", "3.0989736939E+03" "1.5000000000E+01", "1.1994330950E+03", "1.7320334952E-02" "1.8000000000E+01", "-1.8045529425E+03", "1.1547265855E-02", "-1.8045529425E+03" "2.1000000000E+01", "-2.2218840472E+03", "5.7738233936E-03", "-2.2218840472E+03" **Table** 2.2-g **:** **rotations and** **moments,** **element 8.** .. csv-table:: "**Moment**", "*rotation* :math:`{\theta }_{z}` ", "*Moment* :math:`{M}_{z}`" "3.0000000000E+00", "4.6662898355E+06", "3.0801270189E-03", "4.6662898355E+06" "6.0000000000E+00", "9.3318646247E+06", "6.1602540378E-03" "9.0000000000E+00", "1.3997034885E+07", "9.2403810568E-03" "1.2000000000E+01", "1.8662205044E+07", "1.2320508076E-02" "1.5000000000E+01", "1.3996314269E+07", "9.2403810568E-03" "1.8000000000E+01", "9.3304234944E+06", "6.1602540378E-03" "2.1000000000E+01", "4.6637232079E+06", "3.0801270189E-03", "4.6637232079E+06" Uncertainty about the solution --------------------------- Modeling A ~~~~~~~~~~~~~~~ The reference solution is obtained by numerical integration of a nonlinear differential system.