Calculation of the warping constant
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The warping constant is used by the warping beam model (modeling POU_D_TG and POU_D_TGM), which is important to take into account for beams with thin open sections (cf. [R3.08.04]).

This coefficient (noted :math:`{I}_{w}` in [:external:ref:`R3.08.04 <R3.08.04>`], in :math:`{m}^{6}`) is involved in the expression of the virtual work of internal forces on torsional terms:

:math:`{W}_{\text{int}}={\int }_{0}^{2}({\theta }_{x,x}^{\text{*}}\mu \text{.}C\text{.}{\theta }_{x,x}+{\theta }_{x,\text{xx}}^{\text{*}}\text{.}E\text{.}{I}_{\omega }\text{.}{\theta }_{x,\text{xx}})\text{dx}`

Taking the approach of [:ref:`§3.1 <§3.1>`], and by placing ourselves in a coordinate system linked to the center of torsion :math:`C`, the kinematics of the torsion of any section is:

:math:`u(M)\text{=}\left[\begin{array}{c}u\\ v\\ w\end{array}\right]\text{=}\left[\begin{array}{c}\frac{\partial {\theta }_{x}}{\partial x}x\\ 0\\ 0\end{array}\right]\wedge \left[\begin{array}{c}x\\ y\\ z\end{array}\right]\text{+}\left[\begin{array}{c}\frac{\partial {\theta }_{x}}{\partial x}\xi (y,z)\\ 0\\ 0\end{array}\right]` :math:`\text{=}(\begin{array}{c}\frac{\partial {\theta }_{x}}{\partial x}\text{}\xi (y,z)\\ -\frac{\partial {\theta }_{x}}{\partial x}xz\\ \frac{\partial {\theta }_{x}}{\partial x}xy\end{array})`

where :math:`\xi (y\text{,}z)` is the warping function (which is only cancelled in the case of a circular section).

The expression for the stress field is (in elasticity):

:math:`\begin{array}{}{\sigma }_{\text{xx}}=E{\varepsilon }_{\text{xx}}=E\xi (y,z)\frac{{\partial }^{2}{\theta }_{x}}{\partial {x}^{2}}\\ {\sigma }_{\text{xy}}=2\mu {\varepsilon }_{\text{xy}}=\mu \frac{\partial {\theta }_{x}}{\partial x}(\frac{\partial \xi (y,z)}{\partial y}-z)\\ {\sigma }_{\text{xz}}=2\mu {\varepsilon }_{\text{xz}}=\mu \frac{\partial {\theta }_{x}}{\partial x}(\frac{\partial \xi (y,z)}{\partial z}-y)\end{array}`

Unlike [:ref:`§4 <§4>`], second-order terms in :math:`\frac{{\partial }^{2}{\theta }_{x}}{\partial {x}^{2}}` are no longer overlooked.

The first equilibrium relationship :math:`{(\text{div}\sigma )}_{x}={\sigma }_{\text{xx},x}+{\sigma }_{\text{xy},y}+{\sigma }_{\text{xz},z}=0` then involves the following condition on the warping function: :math:`\Delta \xi =0`

On the other hand, without external loading on the outline of the section, we must have :math:`\sigma \otimes n`, which can be written: :math:`\frac{\partial \xi }{\partial y}{n}_{y}+\frac{\partial \xi }{\partial z}{n}_{z}=z\mathrm{.}{n}_{y}-y\mathrm{.}{n}_{z}`, where :math:`{n}_{y}` and :math:`{n}_{z}` are the two components of the normal, or even in vector form: :math:`\text{grad}\xi \text{.}n=\frac{\partial \xi }{\partial n}=(n\wedge \text{CM}).x`

This determines the warping function to the nearest constant. To overcome this indeterminacy, for example, we write the expression for normal force (for a section where twisting produces warping):

:math:`N=\underset{S}{\int }{\sigma }_{\text{xx}}\text{ds}=\underset{S}{\int }E\xi \frac{{\partial }^{2}{\theta }_{x}}{\partial {x}^{2}}\text{ds}=0`

so the additional condition on the warping function is :math:`\underset{S}{\int }\xi \text{ds}=0`. In practice, in MACR_CARA_POUTRE, we place ourselves first and foremost in a coordinate system linked to the center of torsion :math:`c`.

We then calculate :math:`\xi` which must check:

:math:`\begin{array}{}\Delta x\text{=0}\\ \text{grad}x\text{.}n=\frac{\partial \xi }{\partial n}=(n\wedge \text{CM})\text{.}x\\ \underset{S}{\int }x\text{ds}=0\end{array}`

Warping inertia :math:`{I}_{\omega }` is then obtained by: :math:`{I}_{\omega }=\underset{S}{\int }{\xi }^{2}\text{ds}`

MACR_CARA_POUTRE uses the following basic commands:

• Translation of the coordinates of the nodes into the coordinate system linked to the center of torsion (calculated previously in table TCARS):

 CREA_MAILLAGE (MAILLAGE = my,

     REPERE = _F (TABLE = TCARS, NOM_ORIG = 'TORSION'))

• Assignment of a model (thermal plane), of a material field:

 AFFE_MODELE (MAILLAGE = my,

     AFFE = _F (TOUT = 'OUI', PHENOMENE = 'THERMIQUE',

             MODELISATION =' PLAN '))

 AFFE_MATERIAU (MAILLAGE = my,

     AFFE = _F (TOUT = 'OUI' MATER: mat))

• Boundary conditions on the outer contour :math:`\mathrm{G0}`: :math:`\frac{\partial \xi }{\partial y}{n}_{y}+\frac{\partial \xi }{\partial z}{n}_{z}=z\mathrm{.}{n}_{y}-y\mathrm{.}{n}_{z}`

 F1= DEFI_FONCTION (NOM_PARA =, VALE = (0., 0., 10., -10.))

 F2= DEFI_FONCTION (NOM_PARA =, VALE = (0., 0., 10., 10.))

 CH1 = AFFE_CHAR_THER_F (MODELE = mod,

     FLUX_REP = _F (GROUP_MA = G0, FLUX_X = F1, FLUX_Y = F2))

• Condition on the solution field: :math:`\underset{S}{\int }\xi \text{ds}=0`: creation of a unit source term throughout the mesh, and of the associated second member vector. LIAISON_CHAMNO then makes it possible to impose the desired condition.

 CHS = AFFE_CHAR_THER (MODELE =

     SOURCE = _F (TOUT = 'OUI' SOUR = 1.))

 VS = CALC_VECT_ELEM (OPTION = 'CHAR_THER' CHARGE = CHS...)

 MS = CALC_MATR_ELEM (MODELE =... OPTION = 'RIGI_THER')

 NUM = NUME_DDL (MATR_RIGI = MS)

 GO = ASSE_VECTEUR (VECT_ELEM = VS NUME_DDL = NUM)

 CH2 = AFFE_CHAR_THER (

     LIAISON_CHAMNO = _F (CHAM_NO = VA COEF_IMPO = 0.))

• Calculation of the warping function :math:`x`:

 THER_LINEAIRE (MODELE =...

     EXCIT = (

         _F (CHARGE: CH1),

         _F (CHARGE: CH2),

     )

 )

• Calculation of the warpage constant :math:`{I}_{\omega }=\underset{S}{\int }{\xi }^{2}\text{ds}` and enrichment of the table:

 TCARS = POST_ELEM (MODELE =...

     CARA_POUTRE = _F (CARA_GEOM = TCARS,

                 LAPL_PHI = KSI OPTION: 'CARA_GAUCHI'));