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

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

Since the solutions are specific to each model, they are described in the corresponding paragraphs. They are mainly taken from [:ref:`bib1 <bib1>`] and [:ref:`bib2 <bib2>`];


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

Here we describe the characteristics calculated by MACR_CARA_POUTRE [:ref:`R3.08.03 <R3.08.03>`]:

• Geometric characteristics of sections

◦ In the :math:`\mathrm{2D}` mesh description frame OYZ for the mesh provided by the user

▪ area: A_M

▪ center of gravity position: CDG_Y_M, CDG_Z_M

▪ moments and product of inertia of area, at the center of gravity G in the coordinate system GYZ: IY_G_M, IZ_G_M, IYZ_G_M

◦ In the same global coordinate system, for the mesh obtained by symmetrization if SYME_Y or SYME_Z:

▪ Area: A

▪ center of gravity position: CDG_Y, CDG_Z

▪ moments and product of inertia of area, at the center of gravity G in the coordinate system GYZ: IY_G, IZ_G, IYZ_G

◦ In the main inertia coordinate system Gyz. of the right section, whose name corresponds to that used in the description of the GX neutral fiber beam elements [:ref:`U4.24.01 <U4.24.01>`].

▪ main moments of inertia of air in the Gyz coordinate system, usable for calculating the flexural stiffness of the beam: IY and IZ

▪ angle of transition from coordinate system GYZ to the main inertia coordinate system Gyz: ALPHA

▪ characteristic distances, in relation to the center of gravity G of the section for maximum stress calculations: Y_ MAX, Y_ MIN, Z_ MAX, Z_ MIN and R_ MAX.

◦ In the global coordinate system, at a point :math:`P` provided by the user:

▪ Y_P, Z_P: point for calculating moments of inertia

▪ IY_P, IZ_P, IYZ_P: moments of inertia in the PYZ coordinate system

▪ IY_P, IZ_P: moments of inertia in the Pyz coordinate system.

• Mechanical characteristics:

.. csv-table::

    "**Identification**", "**Significance**"
    ":math:`\mathit{JX}` ", "Torsion constant"
    ":math:`\mathit{EY}` ", "Torsion/shear center position"
    ":math:`\mathit{EZ}` ", "Torsion/shear center position"
    ":math:`\mathit{PCTY}` ", "Eccentricity of the center of torsion in the :math:`\mathrm{GYZ}` coordinate system along the :math:`Y` axis"
    ":math:`\mathit{PCTZ}` ", "Eccentricity of the center of torsion in the :math:`\mathrm{GYZ}` coordinate system along the :math:`Z` axis"
    ":math:`\mathit{AY}` ", "Shear coefficient"
    ":math:`\mathit{AZ}` ", "Shear coefficient"
    ":math:`\mathit{JG}` ", "Warping constant"



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

Analytical solution.


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


1. PILKEY W.D.: "Formulas for Stress, Strain, and Structural Matrixes." Wiley & Cons, New York, 1994.


1. D. BLEVINS: Formulas for natural frequency and mode shape.