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

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

The reference solutions selected to verify *Code_Aster* models are as follows:


* for POUTRE models: comparison with the codes: :math:`\text{POUX}`, :math:`\text{ADL}` and :math:`\text{TITUS-T}` :ref:`[1] <RefHeading__8605_460707812>`, using beam-type modeling,


*for TUYAU models: comparison with the :math:`\text{ABAQUS}` code, using pipe-type modeling. The Fourier mode number (:math:`M`) used when calculating the reference is the same as that used when calculating with*Code_Aster*.


* for beam models, a moment equivalent to point :math:`T1` is also defined as :math:`{M}_{\mathit{eq}}=\sqrt{({\mathit{MT}}^{2}+{\mathit{MFY}}^{2}+{\mathit{MFZ}}^{2})}`. The reference value is that obtained by hand with the results of modeling A (reference AUTRE_ASTER).


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

.. csv-table::

    "**Load Case**", "**Move to point 3** (T3 node group)", "**Beam Modeling**", "**Beam Modeling**
    (:math:`\text{POUX}`, :math:`\text{ADL}`, :math:`\text{TITUS}`)", "**Pipe Modeling:** :math:`M=3` (:math:`\text{ABAQUS}`)", "**Pipe Modeling:** :math:`M=6`
    (:math:`\text{ABAQUS}`)"
    "", ":math:`\mathrm{DX}` ", "—0.1658E—3", "—0.16517E—3", "—0.16512E—3"
    "**Own weight**", ":math:`\mathrm{DY}` ", "—0.2040E—4", "—0.13870E—4", "—0.13946E—4"
    "", ":math:`\mathrm{DZ}` ", "—0.8010E—5", "—0.80376E—5", "—0.80369E—5"
    "", "", "", "", ""
    "", ":math:`\mathrm{DX}` ", "—0.1651E—3", "—0.16445E—3", "—0.16441E—3"
    "**Nodal force**", ":math:`\mathrm{DY}` ", "—0.2080E—4", "—0.14245E—4", "—0.14320E—4"
    "", ":math:`\mathrm{DZ}` ", "—0.9516E—5", "—0.10047E—4", "—0.10047E—4"
    "", "", "", "", ""
    "", ":math:`\mathrm{DX}` ", "—6.1418E—3", "—6.3277E—3", "—6.3236E—3"
    "**Expansion**", ":math:`\mathrm{DY}` ", "—13.090E—3", "—13.092E—3", "—13.093E—3"
    "", ":math:`\mathrm{DZ}` ", "16.799E—3", "16.798E—3", "16.798E—3"



.. csv-table::

    "**Load case**", "**Moment equivalent to point** :math:`T1` ", "**Reference** AUTRE_ASTER **(modeling A)**"
    "Own weight", ":math:`{M}_{\mathit{eq}}` ", "189.76886594440944"



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

The uncertainty on the reference solution is set to :math:`\text{2\%}`.


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



.. _RefHeading__8605_460707812:

1. M.W. KELLOG Co. Design of Piping Systems. New York, 1956 - Problem #5.9