Geometric characteristics ================================= **Hypothesis:** Only the cross sections of homogeneous and isotropic beams are treated here (same material characteristics for all points and in all directions). The MACR_CARA_POUTRE command can also calculate the geometric characteristics of a set of disjoint sections. Any section ------------------ Principle ~~~~~~~ Consider a section :math:`(S)` of surface :math:`S` in the plane :math:`(\mathrm{0,}y,z)` whose origin :math:`O` is the center of gravity :math:`G` of the section, [Figure]. .. image:: images/100025D6000069BB00005D1FD7DE7D300241B652.svg :width: 294 :height: 259 .. _RefImage_100025D6000069BB00005D1FD7DE7D300241B652.svg: .. _Ref438032671: Figure 1: **section in the plan** :math:`(\mathrm{0,}y,z)` The geometric moment of inertia of :math:`(S)` with respect to the :math:`(\mathrm{Oy})` axis (which passes through the center of gravity) is expressed by: :math:`{I}_{y}={\int }_{s}\mathrm{z²}\mathrm{dS}` with :math:`[I]={\int }_{s}\mathrm{OM}\otimes \mathrm{OM}\mathrm{dS}` In a similar way, the geometric moment with respect to :math:`(\mathrm{Oz})` is defined by: :math:`{I}_{z}={\int }_{s}{y}^{2}\mathrm{dS}` When the centrifugal geometric moment (often called the product of inertia of area) defined by is zero, :math:`{I}_{\mathrm{yz}}={\int }_{s}yz\mathrm{dS}` the axes :math:`(\mathrm{Oy})` and :math:`(\mathrm{Oz})` are main axes of the section :math:`(S)`. We will use this hypothesis for the future; :math:`{I}_{y}` and :math:`{I}_{z}` are then called the main geometric moments. In general, we must place ourselves in the main axes of a beam section for everything concerning its characteristics since the beam elements of *Code_Aster* are formulated in this coordinate system. Starting from an origin located at the center of gravity, it suffices, to pass from any axis system :math:`(G,y\text{'},z\text{'})` to the main axis system :math:`(G,y,z)`, to perform an angle rotation :math:`\theta` such as [Figure]: :math:`\theta =\frac{1}{2}\mathrm{Arctg}(\frac{2{I}_{y\text{'}z\text{'}}}{{I}_{z\text{'}}-{I}_{y\text{'}}})` .. image:: images/100000000000022B000001B4CE92C5312DFCDE51.png :width: 5.5945in :height: 3.7846in .. _RefImage_100000000000022B000001B4CE92C5312DFCDE51.png: .. _Ref438031262: Figure 2: **Main and any axes.** The polar geometric moment with respect to the center of gravity is given by: :math:`{I}_{p}={\int }_{s}\mathrm{r²}\mathrm{dS}` where r is the distance from the element :math:`\mathrm{dS}` to the center of gravity [Figure]. We naturally deduce :math:`{I}_{p}={I}_{y}+{I}_{z}`. The polar geometric moment is used in the calculation of the torsional stiffness of girders with a circular cross section (Saint-Venant twist). For the other shapes of sections, a torsional constant of the same dimension will be defined. In addition, the geometric moments can be calculated in another coordinate system :math:`(P,y,z)`, of any origin :math:`P`, different from the center of gravity :math:`G` (Huygens formula): :math:`\begin{array}{}{I}_{y}^{P}={I}_{y}^{G}+{(\mathrm{GP}\mathrm{.}Z)}^{2}\mathrm{.}S={\int }_{s}{z}^{2}\mathrm{dS}+{(\mathrm{GP}\mathrm{.}Z)}^{2}\mathrm{.}S\\ {I}_{z}^{P}={I}_{z}^{G}+{(\mathrm{GP}\mathrm{.}Y)}^{2}\mathrm{.}S={\int }_{s}{y}^{2}\mathrm{dS}+{(\mathrm{GP}\mathrm{.}Y)}^{2}\mathrm{.}S\\ {I}_{\mathrm{yz}}^{P}={I}_{\mathrm{yz}}^{G}+(\mathrm{GP}\mathrm{.}Y)(\mathrm{GP}\mathrm{.}Z)\mathrm{.}S={\int }_{s}\mathrm{yz}\mathrm{dS}+(\mathrm{GP}\mathrm{.}Y)(\mathrm{GP}\mathrm{.}Z)\mathrm{.}S\end{array}` in general, the Huygens formula gives: :math:`\begin{array}{}\left[I\right]={\int }_{s}(\text{PG}+\text{GM})\otimes (\text{PG}+\text{GM})\\ \phantom{\left[I\right]}\text{=}{\int }_{s}\text{PG}\otimes \text{PG}+{\int }_{s}\text{GM}\otimes \text{GM}\\ \phantom{\left[I\right]}+2{\int }_{s}\text{PG}\otimes \text{GM}\\ \phantom{\left[I\right]}\text{=}S(\text{PG}\otimes \text{PG})+{\int }_{s}\text{GM}\otimes \text{GM}\end{array}` Calculating geometric characteristics using MACR_CARA_POUTRE ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This macro-command allows the determination of the characteristics of a beam cross section from a 2D mesh of the [:external:ref:`U4.42.02 `] section. It allows you to build a table of values, usable in the AFFE_CARA_ELEM command (SECTION: 'GENERALE' [:external:ref:`U4.42.01 `]). Geometric characteristics can be calculated on the full mesh, half mesh with symmetry with respect to :math:`Y` or :math:`Z`, quarter mesh with two symmetries with respect to :math:`Y` and to :math:`Z` [Figure]. These characteristics are calculated in the table for the entire mesh and for each group of elements in the list specified by the user (case of a network of beams). The data corresponds to a half or a quarter of the section if the keywords SYME_Y or SYME_Z are present. .. image:: images/100000000000019E000001614D3811A6B950B48D.png :width: 3.7429in :height: 3.1102in .. _RefImage_100000000000019E000001614D3811A6B950B48D.png: **Figure** 3 **: Definition of geometric characteristics.** The results are grouped into four groups: • In frame :math:`\mathit{OYZ}` describing the 2D mesh 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 air, at the center of gravity :math:`G` in the GYZ coordinate system: 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 air, at the center of gravity :math:`G` in the GYZ coordinate system: IY_G, IZ_G, IYZ_G • In the main inertia coordinate system :math:`\mathrm{Gyz}`. in the right section, whose name corresponds to that used in the description of neutral fiber beam elements :math:`\mathrm{Gx}` [:external:ref:`U4.24.01 `]. • main moments of inertia of air in frame :math:`\mathrm{Gyz}`, usable for calculating the flexural stiffness of the beam: IY and IZ • angle of transition from coordinate system :math:`\mathit{GYZ}` to the main inertia coordinate system :math:`\mathrm{Gyz}`: ALPHA • characteristic distances, in relation to the center of gravity :math:`G` of the section for maximum stress calculations: Y_ MAX, Y_ MIN, Z_ MAX, Z_ MIN and R_ MAX. • In the global frame of reference, 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. Calculations made ~~~~~~~~~~~~~~~~~~ The list of commands called by MACR_CARA_POUTRE is shown in document [:external:ref:`U4.42.02 `]. The previous quantities are obtained by calling POST_ELEM, for the option 'CARA_GEOM'. In addition, you can add the keywords SYME_Y, SYME_Z, and ORIG_INER which define the point :math:`P`. The calculations are performed in POST_ELEM, for the entire mesh, then possibly for each group of elements, as follows: • Loop over the 2D elements (D_ PLAN modeling), with the call of the elementary option 'MASS_INER'. We get a CHAM_ELEM with one value per element (1 Gauss point) containing the components: :math:`{\int }_{\mathrm{élément}}\mathrm{dS},{\int }_{\stackrel{x}{\mathrm{élément}}}\mathrm{dS},{\int }_{\mathrm{élément}}\mathrm{dS}` :math:`{\int }_{s}\mathrm{x²}\mathrm{dS},{\int }_{s}\mathrm{y²}\mathrm{dS},{\int }_{s}\mathrm{xy}\mathrm{dS}`, • Summation of the previous elementary quantities to obtain: A_M, CDG_Y_M,, CDG_Z_M, IY_G_M, IZ_G_M, IYZ_G_M • Calculation of A, CDG_Y, CDG_Z, IY_G, IZ_G, IYZ_G (taking into account SYME_Y, SYME_Z) • Calculating IY, IZ, ALPHA • Calculation of Y_ MAX, Z_ MAX, Y_, Y_ MIN, Z_ MIN, R_ MAX • If you specify a particular point :math:`P` (keyword ORIG_INER), you also calculate the characteristics in the original global coordinate system :math:`P`: :math:`\mathit{PYZ}` .. _Ref437393983: .. _Ref437394028: Examples of use: Full rectangle (processed by test SSLL107G) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. image:: images/100000000000017400000102D74D6DBA1785E6B0.png :width: 3.5693in :height: 2.4138in .. _RefImage_100000000000017400000102D74D6DBA1785E6B0.png: Geometric characteristics obtained .. code-block:: text LIEU A_M CDG_Y_M CDG_Z_M IY_G_M IZ_G_M IYZ_G_M 0.000003 1.00E-03 4.24E-18 -3.39E-18 -3.39E-18 2.08E-07 3.33E-08 2.65E-23 GR1 5.00E-04 2.20E-17 -1.25E-02 2.60E-08 1.67E-08 3.97E-23 GR2 5.00E-04 -8.47E-18 1.25E-02 2.60E-08 1.67E-08 5.62E-23 LIEU A CDG_Y CDG_Z IY_G IZ_G IYZ_G IY IZ ALPHA 0.000003 1.00E-03 4.24E-18 -3.39E-18 -3.39E-18 2.08E-07 3.33E-08 2.65E-23 3.33E-08 2.08E-07 9.00E+01 GR1 5.00E-04 2.20E-17 -1.25E-02 2.60E-02 2.60E-08 1.67E-08 3.97E-23 1.67E-08 2.60E-08 9.00E-08 9.00E+01 GR2 5.00E-04 -8.47E-18 1.25E-02 2.60E-08 1.67E-08 5.62E-23 1.67E-08 2.60E-08 9.00E-08 9.00E+01 LIEU Y_P Z_P IY_P IZ_P IYZ_P IY_PRIN_P IZ_PRIN_P 0.000003 0.00E+00 0.00E+00 2.08E-07 3.33E-08 2.65E-23 3.33E-08 2.08E-07 GR1 0.00E+00 0.00E+00 1.04E-07 1.67E-08 -9.79E-23 1.67E-08 1.04E-07 GR2 0.00E+00 0.00E+00 1.04E-07 1.67E-08 3.31E-24 1.67E-08 1.04E-07 LIEU Y_ MAX Z_ MAX Y_ MIN Z_ MIN R_ MAX 0.000003 2.50E-02 1.00E-02 -2.50E-02 -1.00E-02 2.69E-02 GR1 2.50E-02 2.25E-02 -2.50E-02 2.50E-02 2.50E-03 3.36E-02 GR2 2.50E-02 -2.50E-03 -2.50E-03 -2.50E-02 -2.25E-02 3.36E-02 LIEU MX AU AZ AND EZ PCTY PCTZ RT 0.000003 - - - - - - - - - 1.93871E-2 GR1 3.43E-08 1.20E+00 1.20E+00 9.00E-17 -3.97E-18 2.60E-17 -1.25E-02 1.56391E-2 GR2 3.43E-08 1.20E+00 1.20E+00 -4.03E-17 1.19E-16 -1.27E-16 1.25E-02 1.56391E-2 Special case of rectangular and circular sections -------------------------------------------------------- Geometric characteristics are directly calculated in AFFE_CARA_ELEM based on user data. .. image:: images/10000000000001F00000012ED3D084BBAC9D327C.png :width: 4.4236in :height: 2.7327in .. _RefImage_10000000000001F00000012ED3D084BBAC9D327C.png: **Figure** 4 **: rectangular section.** In the case of the rectangular beam (Operand SECTION: 'RECTANGLE'), the calculation gives: :math:`{I}_{y}=\frac{1}{12}\left[{h}_{y}{h}_{z}^{3}-({h}_{y}-2{\mathrm{ep}}_{y}){({h}_{z}-{\mathrm{2ep}}_{z})}^{3}\right]` :math:`{I}_{z}=\frac{1}{12}\left[{h}_{z}{h}_{y}^{3}-({h}_{z}-2{\mathrm{ep}}_{z}){({h}_{y}-{\mathrm{2ep}}_{y})}^{3}\right]` .. image:: images/10000000000000F6000000C6C148799A9499DF1D.png :width: 2.3563in :height: 1.8382in .. _RefImage_10000000000000F6000000C6C148799A9499DF1D.png: **Figure** 5 **: circular section** For the circular section (Operand SECTION: 'CERCLE'), we get: :math:`{I}_{y}={I}_{z}=\frac{p}{4}\left[{R}^{4}-{(R-\mathrm{ep})}^{4}\right]` :math:`{I}_{p}=\frac{p}{2}\left[{R}^{4}-{(R-\mathrm{ep})}^{4}\right]`