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

Calculation method used for the reference solution
--------------------------------------------------------

Comparison to other numerical results obtained with *Code_Aster* (version 14.2 [:ref:`bib1 <bib1>`]) with a :math:`3D` mesh of the elbow and straight parts, connected at the ends to straight beams (C modeling). This :math:`\mathrm{3D}` mesh contains 1500 HEXA20 meshes. A modeling of the elbow in elements COQUE_3D gave results comparable to calculation :math:`\mathrm{3D}` (see [:ref:`§2.2 <§2.2>`]).

B modeling uses **Plasticity** behavior MFront instead of code_aster VMIS_ISOT_LINE behavior (3D calculation reference).

Quantities useful for seismic resistance are also calculated using the following formulas:

.. math::
 :label: eq-1

    \ mathit {EN} = {\ mathrm {\ epsilon}} _ {\ mathit {xx}}


.. math::
 :label: eq-2

    \ mathit {ET} =\ frac {{r} _ {\ mathit {moy}}\ times {\ mathrm {\ gamma}}} _ {\ mathit {twist}}}} {2}


.. math::
 :label: eq-3

    \ mathit {EFY} = {r} _ {\ mathit {moy}}\ times {\ mathrm {\ kappa}}} _ {y}


.. math::
: label: eq-4

    \ mathit {ESTAR} =\ sqrt {{\ mathit {EN}}} ^ {2}} + {\ mathit {ET}} ^ {2} + {\ left (\ frac {\ mathrm {\ pi}\\ times\ mathit {\ pi}\ pi}\ times\ pi}\ times\ mathit {EFY}}} {4}\ right)} ^ {2}} + {\ left (\ frac {\ mathrm {\ pi}\ times\ pi}\ times\ mathit {EFZ}} {4}\ right)} ^ {2}}

Calculation of the second type of quantity for the earthquake:

.. math::
: label: eq-5

    \ mathrm {\ lambda} =\ frac {e\ times {R} _ {c}} {{r} _ {\ mathit {moy}}} ^ {2}} ^ {2}}

and

.. math::
 :label: eq-6

    k2=\ mathit {max}\ left (\ mathrm {1,}\ frac {\ mathrm {1.65}} {\ mathrm {\ lambda}}\ right)

With:

.. math::
 :label: eq-7

    {\ mathit {EFY}} _ {2} =\ frac {{r} _ {\ mathit {moy}}\ times {\ mathrm {\ kappa}}} _ {y}} {y}} {k2}

And:

.. math::
 :label: eq-8

    {\ mathit {ESTAR}} _ {2} =\ sqrt {{\ mathit {EN}}} ^ {2} + {\ mathit {ET}} ^ {2} + {\ left (\ frac {\ mathrm {\ pi}\ times\ mathrm {\ pi}\ pi}\ times\ mathrm {\ gamma}}\ times\ mathit {EFY}}} {4}\ right)} ^ {2}} + {\ mathrm {\ pi}\ pi}\ pi}\ pi}\ times\ mathrm {\ gamma}\ times\ mathrm {\ gamma}\ times\ mathit {}}} {4}\ right)} ^ {2}} + {\ left (\ frac {\ mathrm {\ pi}\ times\ mathrm {\ gamma}\ times\ mathit {EFZ}} {4}\ right)}} ^ {2}}}


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

For a moment applied :math:`\mathrm{Mz}` in :math:`D`, moving :math:`\mathrm{DY}` from the same point :math:`D` is equal to [:ref:`bib1 <bib1>`]:

.. csv-table::

    "**Moment**", ":math:`\mathrm{Dy}` **point** :math:`D` :math:`(m)` **** **(3D)**", ":math:`\mathrm{Dy}` **point** :math:`D` :math:`(m)` **(** COQUE_3D **)**"
    "0. ", "0. ", "0."
    "3.08670D+06", "1.09257D—02", "1.08875D—02", "1.09257D—02"
    "3.48715D+06", "1.23431D—02", ""
    "3.88759D+06", "1.37775D—02", "1.37381D—02",
    "4.28804D+06", "1.52557D—02", ""
    "4.68848D+06", "1.67908D—02", ""
    "5.08892D+06", "1.83836D—02", ""
    "5.48937D+06", "2.00903D—02", ""
    "5.88981D+06", "2.20209D—02", ""
    "6.29026D+06", "2.42545D—02", ""
    "6.69070D+06", "2.68829D—02", ""
    "7.09115D+06", "3.01030D—02", ""



Precision on the reference results
----------------------------------------

Because the reference solution is numerical, the precision can be evaluated from [:ref:`§2.2 <§2.2>`] to :math:`\text{2\%}` by comparing the 3D and COQUE_3D solutions.


Bibliographical references
--------------------------

[:ref:`1 <1>`] J.M. PROIX, A. BEN HAJ YEDDER: "Project CACIP: study of a pipe bent under bending". Rating EDF/DER HI-75/98/001/0

[:ref:`2 <2>`] C. BARATTE (SEPTEN), MN. BERTON, N. BLAY (CEA), F. LE BRETON (FRAMATOME - ANP): "Project for the new codification of seismic pipe design criteria". Note EDF/SEPTEN E-N-ES-MS/01-01004-A.