2. Benchmark solution#

2.1. 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 [bib1] and [bib2];

2.2. Benchmark results#

Here we describe the characteristics calculated by MACR_CARA_POUTRE [R3.08.03]:

  • Geometric characteristics of sections

◦ In the \(\mathrm{2D}\) 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 [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 \(P\) 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:

Identification

Significance

\(\mathit{JX}\)

Torsion constant

\(\mathit{EY}\)

Torsion/shear center position

\(\mathit{EZ}\)

Torsion/shear center position

\(\mathit{PCTY}\)

Eccentricity of the center of torsion in the \(\mathrm{GYZ}\) coordinate system along the \(Y\) axis

\(\mathit{PCTZ}\)

Eccentricity of the center of torsion in the \(\mathrm{GYZ}\) coordinate system along the \(Z\) axis

\(\mathit{AY}\)

Shear coefficient

\(\mathit{AZ}\)

Shear coefficient

\(\mathit{JG}\)

Warping constant

2.3. Uncertainty about the solution#

Analytical solution.

2.4. Bibliographical references#

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

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