Benchmark solution
=====================


Benchmark solution
---------------------

The reference solution is:


* "ANALYTIQUE "for the resulting efforts and moments,


* "NON_REGRESSION "for displacements and constraints (modeling A)


* "AUTRE_ASTER "for movements and constraints (B and C models). The results obtained with modeling A serve as a reference solution.

Calculations of the resulting efforts and moments


* Line load case :math:`\mathit{fx}`

:math:`{R}_{x}\mathrm{=}{\mathrm{\int }}_{s}{f}_{x}\mathit{ds}\mathrm{=}{f}_{x}\mathrm{.}h\mathrm{.}l\mathrm{=}2000N`

:math:`{M}_{y}\mathrm{=}{\mathrm{\int }}_{s}{f}_{x}z\mathit{ds}\mathrm{=}\frac{({f}_{x}\mathrm{.}{h}^{2})}{2}l\mathrm{=}20000\mathit{N.mm}`

:math:`{M}_{z}\mathrm{=}{\mathrm{\int }}_{s}{f}_{x}y\mathit{ds}\mathrm{=}\frac{({f}_{x}\mathrm{.}{l}^{2})}{2}h\mathrm{=}200000\mathit{N.mm}`


* Line load case :math:`\mathit{fy}`

:math:`{R}_{y}\mathrm{=}{\mathrm{\int }}_{s}{f}_{y}\mathit{ds}\mathrm{=}{f}_{y}\mathrm{.}h\mathrm{.}l\mathrm{=}2000N`

:math:`{M}_{x}\mathrm{=}{\mathrm{\int }}_{s}{f}_{y}z\mathit{ds}\mathrm{=}\frac{({f}_{y}\mathrm{.}{h}^{2})}{2}l\mathrm{=}20000\mathit{N.mm}`

:math:`{M}_{z}\mathrm{=}{\mathrm{\int }}_{s}{f}_{y}L\mathit{ds}\mathrm{=}{f}_{y}\mathrm{.}h\mathrm{.}l\mathrm{.}L\mathrm{=}1500000\mathit{N.mm}`


* Line load case :math:`\mathit{fz}`

:math:`{R}_{z}\mathrm{=}{\mathrm{\int }}_{s}{f}_{z}\mathit{ds}\mathrm{=}{f}_{z}\mathrm{.}h\mathrm{.}l\mathrm{=}2000N`

:math:`{M}_{x}\mathrm{=}{\mathrm{\int }}_{s}{f}_{z}y\mathit{ds}\mathrm{=}\frac{({f}_{z}\mathrm{.}{l}^{2})}{2}h\mathrm{=}200000\mathit{N.mm}`

:math:`{M}_{y}\mathrm{=}{\mathrm{\int }}_{s}{f}_{z}L\mathit{ds}\mathrm{=}{f}_{z}\mathrm{.}h\mathrm{.}l\mathrm{.}L\mathrm{=}1500000\mathit{N.mm}`


Reference quantities
----------------------


* Resulting efforts: :math:`\mathit{RESULT}\text{\_}X`, :math:`\mathit{RESULT}\text{\_}Y`, :math:`\mathit{RESULT}\text{\_}Z`


* Resulting moments: :math:`\mathit{MOMENT}\text{\_}X`, :math:`\mathit{MOMENT}\text{\_}Y`, :math:`\mathit{MOMENT}\text{\_}Z`


* Travel: :math:`\mathrm{DX}`, :math:`\mathit{DY}`, :math:`\mathit{DZ}`


* Constraints: :math:`\mathit{SIXX}`


Benchmark results
----------------------


* Line load case :math:`\mathit{fx}`


+------------------+--------+---------------------------------+------------------+-------------------+
|**Identification**                                           |**Reference type**|**Reference value**|
+------------------+--------+---------------------------------+                  +                   +
|Point             |GROUP_MA|Components                       |                  |                   |
+------------------+--------+---------------------------------+------------------+-------------------+
|:math:`A`         |GAUCHE  |:math:`\mathit{RESULT}\text{\_}X`|'ANALYTIQUE'      |-2,000. N          |
+                  +--------+---------------------------------+------------------+-------------------+
|                  |DROITE  |:math:`\mathit{RESULT}\text{\_}X`|'ANALYTIQUE'      |2,000. N           |
+                  +--------+---------------------------------+------------------+-------------------+
|                  |GAUCHE  |:math:`\mathit{MOMENT}\text{\_}Y`|'ANALYTIQUE'      |-20,000. N.mm      |
+                  +--------+---------------------------------+------------------+-------------------+
|                  |GAUCHE  |:math:`\mathit{MOMENT}\text{\_}Z`|'ANALYTIQUE'      |-200,000. N.mm     |
+------------------+--------+---------------------------------+------------------+-------------------+


+------------------+---------------------+------------------+-------------------+
|**Identification**                      |**Reference type**|**Reference value**|
+------------------+---------------------+------------------+-------------------+
|Point :math:`A`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |1.067538 MPa       |
+------------------+---------------------+------------------+-------------------+
|Point :math:`B`   |:math:`\mathrm{DX}`  |'NON_REGRESSION'  |5.676675E-3 mm     |
+------------------+---------------------+------------------+-------------------+
|Point :math:`C`   |:math:`\mathit{DX}`  |'NON_REGRESSION'  |5.837494E-3 mm     |
+------------------+---------------------+------------------+-------------------+
|Point :math:`D`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |1.030237 MPa       |
+------------------+---------------------+------------------+-------------------+
|Point :math:`F`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |1.435778 MPa       |
+------------------+---------------------+------------------+-------------------+
|Point :math:`I`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |1.472049 MPa       |
+------------------+---------------------+------------------+-------------------+


* Line load case :math:`\mathit{fy}`


+------------------+--------+---------------------------------+------------------+-------------------+
|**Identification**                                           |**Reference type**|**Reference value**|
+------------------+--------+---------------------------------+                  +                   +
|Point             |GROUP_MA|Components                       |                  |                   |
+------------------+--------+---------------------------------+------------------+-------------------+
|:math:`A`         |GAUCHE  |:math:`\mathit{RESULT}\text{\_}Y`|'ANALYTIQUE'      |-2,000. N          |
+                  +--------+---------------------------------+------------------+-------------------+
|                  |DROITE  |:math:`\mathit{RESULT}\text{\_}Y`|'ANALYTIQUE'      |2,000. N           |
+                  +--------+---------------------------------+------------------+-------------------+
|                  |GAUCHE  |:math:`\mathit{MOMENT}\text{\_}X`|'ANALYTIQUE'      |20,000. N.mm       |
+                  +--------+---------------------------------+------------------+-------------------+
|                  |GAUCHE  |:math:`\mathit{MOMENT}\text{\_}Z`|'ANALYTIQUE'      |-1.50 E6 N.mm      |
+------------------+--------+---------------------------------+------------------+-------------------+


+------------------+---------------------+------------------+-------------------+
|**Identification**                      |**Reference type**|**Reference value**|
+------------------+---------------------+------------------+-------------------+
|Point :math:`A`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |4.55542676 MPa     |
+------------------+---------------------+------------------+-------------------+
|Point :math:`B`   |:math:`\mathit{DY}`  |'NON_REGRESSION'  |0.15476384 mm      |
+------------------+---------------------+------------------+-------------------+
|Point :math:`C`   |:math:`\mathit{DY}`  |'NON_REGRESSION'  |0.15475096 mm      |
+------------------+---------------------+------------------+-------------------+
|Point :math:`D`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |-4.43905267 MPa    |
+------------------+---------------------+------------------+-------------------+
|Point :math:`F`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |26.8162393 MPa     |
+------------------+---------------------+------------------+-------------------+
|Point :math:`I`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |-26.9483395 MPa    |
+------------------+---------------------+------------------+-------------------+


* Line load case :math:`\mathit{fz}`


+------------------+--------+---------------------------------+------------------+-------------------+
|**Identification**                                           |**Reference type**|**Reference value**|
+------------------+--------+---------------------------------+                  +                   +
|Point             |GROUP_MA|Components                       |                  |                   |
+------------------+--------+---------------------------------+------------------+-------------------+
|:math:`A`         |GAUCHE  |:math:`\mathit{RESULT}\text{\_}Z`|'ANALYTIQUE'      |2,000. N           |
+                  +--------+---------------------------------+------------------+-------------------+
|                  |DROITE  |:math:`\mathit{RESULT}\text{\_}Z`|'ANALYTIQUE'      |-2,000. N          |
+                  +--------+---------------------------------+------------------+-------------------+
|                  |GAUCHE  |:math:`\mathit{MOMENT}\text{\_}X`|'ANALYTIQUE'      |200,000. N.mm      |
+                  +--------+---------------------------------+------------------+-------------------+
|                  |GAUCHE  |:math:`\mathit{MOMENT}\text{\_}Y`|'ANALYTIQUE'      |-1.50 E6 N.mm      |
+------------------+--------+---------------------------------+------------------+-------------------+


+------------------+---------------------+------------------+-------------------+
|**Identification**                      |**Reference type**|**Reference value**|
+------------------+---------------------+------------------+-------------------+
|Point :math:`A`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |-132.297834 MPa    |
+------------------+---------------------+------------------+-------------------+
|Point :math:`B`   |:math:`\mathit{DZ}`  |'NON_REGRESSION'  |-6.2097302 mm      |
+------------------+---------------------+------------------+-------------------+
|Point :math:`C`   |:math:`\mathit{DZ}`  |'NON_REGRESSION'  |-6.2082328 mm      |
+------------------+---------------------+------------------+-------------------+
|Point :math:`D`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |-131.564485 Mpa    |
+------------------+---------------------+------------------+-------------------+
|Point :math:`F`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |37.0401177 MPa     |
+------------------+---------------------+------------------+-------------------+
|Point :math:`I`   |:math:`\mathit{SIXX}`|'NON_REGRESSION'  |36.1921083 MPa     |
+------------------+---------------------+------------------+-------------------+


Uncertainty about the solution
---------------------------

Analytical and numerical solutions.