2. Benchmark solution#
2.1. Calculation method used for the reference solution#
2.1.1. 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.
2.2. 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 \(1.0s\). The moments are in \(s\) and the moments in \(N\mathrm{.}m\).
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.
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.
Moment |
rotation \({\theta }_{z}\) |
Moment \({M}_{z}\) |
|
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 |
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.
Moment |
rotation \({\theta }_{z}\) |
Moment \({M}_{z}\) |
|
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 |
Figure 2.2-c : Curve Moment-rotation, elements 5 and 6.
Table 2.2-c : rotations and moments, elements 5 and 6.
Moment |
rotation \({\theta }_{z}\) |
Moment \({M}_{z}\) |
|
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.
Moment |
rotation \({\theta }_{z}\) |
Moment \({M}_{z}\) |
|
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.
Moment |
rotation \({\theta }_{z}\) |
Moment \({M}_{z}\) |
|
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.
Moment |
rotation \({\theta }_{z}\) |
Moment \({M}_{z}\) |
|
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.
Moment |
rotation \({\theta }_{z}\) |
Moment \({M}_{z}\) |
|
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 |
2.3. Uncertainty about the solution#
2.3.1. Modeling A#
The reference solution is obtained by numerical integration of a nonlinear differential system.