4. Calculation of the warping constant#

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 \({I}_{w}\) in [R3.08.04], in \({m}^{6}\)) is involved in the expression of the virtual work of internal forces on torsional terms:

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

Taking the approach of [§3.1], and by placing ourselves in a coordinate system linked to the center of torsion \(C\), the kinematics of the torsion of any section is:

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

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

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

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

Unlike [§4], second-order terms in \(\frac{{\partial }^{2}{\theta }_{x}}{\partial {x}^{2}}\) are no longer overlooked.

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

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

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):

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

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

We then calculate \(\xi\) which must check:

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

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

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 \(\mathrm{G0}\): \(\frac{\partial \xi }{\partial y}{n}_{y}+\frac{\partial \xi }{\partial z}{n}_{z}=z\mathrm{.}{n}_{y}-y\mathrm{.}{n}_{z}\)

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: \(\underset{S}{\int }\xi \text{ds}=0\): 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 \(x\):

THER_LINEAIRE (MODELE =…

EXCIT = (

_F (CHARGE: CH1),

_F (CHARGE: CH2),

)

)

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

TCARS = POST_ELEM (MODELE =…

CARA_POUTRE = _F (CARA_GEOM = TCARS,

LAPL_PHI = KSI OPTION: “CARA_GAUCHI”));