8. Implementation of element TUYAU in Code_Aster#

8.1. Description#

This new element (named METUSEG3) is based on a curvilinear SEG3ouSEG4 mesh. It assumes that the pipe cross section is circular. Unlike elements POU_D_E, POU_D_T, [R3.08.01] this element is not « exact » at the nodes for loads or twists concentrated at the ends. It is therefore necessary to mesh with several elements to obtain correct results.

8.2. Modeling data#

The element is used as follows:

AFFE_MODELE (MODELISATION = “TUYAU_3M”…)

4-knot meshes are generated from 3-knot meshes using:

MAIL = CREA_MAILLAGE (MAILLAGE = MAIL, MODI_MAILLE =_F (OPTION =” SEG3_4 “, TOUT =” OUI”))

We use routine INI090 for the shape functions, their derivatives and their second derivatives (for the shell part) at the Gauss points, as well as the corresponding weights.

The characteristics of the section are defined in AFFE_CARA_ELEM

AFFE_CARA_ELEM (

POUTRE = _F (SECTION = “CERCLE”, CARA = (“R” “EP”), VALE = (…),),

ORIENTATION =_F (GROUP_NO =D, CARA =” GENE_TUYAU “, VALE =( X Y Z),),

TUYAU_NCOU =” NOMBRE OF COUCHES “, TUYAU_NSEC =” NOMBRE OF SECTEURS”,),

)

R and EP represent, as for conventional beam elements, respectively the external radius and the thickness of the section. At one of the end nodes of the pipe line, we also define the vector whose projection onto the cross section is the origin of the angles for the Fourier series decomposition. This vector must not be collinear to the mean line from the elbow to the end node in question. At this level, the number of layers and angular sectors to be used for digital integration is also defined.

AFFE_CHAR_MECA (DDL_IMPO = _F (

DX=.. , DY=.. , DZ=.. , DRX =.. , DRY =.. , DRZ =.. , DDL of girder

UI2 =.. , VI2 =.. , WI2 =.. , UO2 =.. , VO2 =.. , WO2 =.. , DDL related to mode 2

UI3 =.. , VI3 =.. , WI3 =.. , UO3 =.. , VO3 =.. , WO3 =.. , DDL related to mode 3

WO=.. , WI1 =.. , WO1 =..,, DDL of swelling and mode 1 on :math:`W`

The loads supported in AFFE_CHAR_MECA are:

  • nodal forces (FORCE_NODALE), which only work on beam movements.

  • internal pressure (FORCE_TUYAU = _F (PRES =..))

  • gravity, (PESANTEUR)

  • linear forces (FORCE_POUTRE)

Since the internal pressure works on the swelling degree of freedom \(\mathit{WO}\), we then calculate:

\({W}_{\text{pres}}={\int }_{0}^{l}{\int }_{0}^{2\pi }p{w}^{o}{r}_{{}_{\text{int}}}d\phi \text{dx}={\int }_{0}^{l}{\int }_{0}^{2\pi }p\sum _{k=1}^{N}{H}_{k}{w}_{k}^{o}{r}_{{}_{\text{int}}}d\phi \text{dx}=\sum _{k=1}^{N}{H}_{k}\left[{\int }_{0}^{l}{\int }_{0}^{2\pi }p{r}_{{}_{\text{int}}}d\phi \text{dx}\right]{w}_{k}^{o}\)

8.3. Linear elasticity calculation#

The stiffness matrix and the mass matrix (options RIGI_MECA and MASS_MECA respectively) are numerically integrated into the TE0582. The calculation takes into account the fact that the terms corresponding to the beam degrees of freedom are classically expressed in the global coordinate system, and that the Fourier degrees of freedom are in the coordinate system local to the element. In the case where the element does not belong to any elbow, this local coordinate system is defined by the generator and the direction vector carried by the mean fiber of the element as indicated on [Figure 8.3-a]. In the case where the element belongs to an elbow, the local coordinate system is defined from the elbow plane as mentioned in [§2.1].

_images/1000243E000069BB00005632F7061E5B58236BCF.svg

Figure 8.3-a: Local coordinate system for a straight pipe

8.4. Nonlinear calculations#

The tangent stiffness matrix (options RIGI_MECA_TANG and FULL_MECA) as well as the plastic projection (options FULL_MECA and RAPH_MECA) are digitally integrated into the TE0586. All the plane stress laws available in Code_Aster can be used: if they are not integrated directly, it is always possible to use a law of behavior formulated in plane deformation, and to treat the hypothesis of plane stresses using the De Borst method.

The pipe elements should only be used in small deformations and small displacements.

8.5. Post-treatment#

The basic calculations currently available correspond to the options:

  • SIEF_ELGA that provide the constraints to the integration points in the user coordinate system. These values are stored as follows:

  • for each Gauss point in the length, (\(n\mathrm{=}\mathrm{1,}3\))

for each integration point in the thickness, (:math:`nmathrm{=}1`, * \(2{N}_{\mathit{COU}}+1\mathrm{=}7\))

for each integration point on the circumference, (:math:`nmathrm{=}1`, * \(2{N}_{\mathit{SECT}}+1\mathrm{=}33\))

  • 6 constraint components: SIXXSIYYSIZZSIXYSIXZSIYZ

where \(X\) refers to the direction given by the element’s two vertex nodes, \(Y\) represents the angle \(\phi\) describing the circumference, and \(Z\) represents the radius. SIZZ and SIYZ corresponding to \({\sigma }_{\text{rr}},{\sigma }_{\mathrm{r\phi }}\) are taken to be equal to zero.

  • EFGE_ELNO which allows you to obtain the generalized forces per element at the nodes in the frame of reference of the beam.

  • VARI_ELNO which calculates the internal variables field per element at the nodes for all layers and sectors, in the local coordinate system of the element.

  • EPSI_ELGA which provides the total deformations at the integration points in the element’s local coordinate system. The calculation is done in TE0584, and currently gives the values to the 693 integration points (for an element with 3 Fourier modes). These fields are called integration « sub-point » fields. These values are stored as follows:

  • for each Gauss point in the length, (\(n\mathrm{=}\mathrm{1,}3\))

for each integration point in the thickness, (:math:`nmathrm{=}1`, * \(2{N}_{\mathit{COU}}+1\mathrm{=}7\))

for each integration point on the circumference, (:math:`nmathrm{=}1`, * \(2{N}_{\mathit{SECT}}+1\mathrm{=}33\))

  • 6 deformation components: EPXXEPYYEPZZEPXYEPXZEPYZ

where \(X\) designates the direction given by the two vertex nodes of the element, \(Y\) represents the angle \(\phi\) describing the circumference and \(Z\) represents the radius. EPZZ and EPYZ corresponding to \({\epsilon }_{\text{rr}},{\epsilon }_{\mathrm{r\phi }}\) are taken to be equal to zero.

  • The options SIEQ_ELGA and EPEQ_ELGA allow the calculation of the invariants, (Von Mises, Von Mises signed, trace) at each integration point (fields with « subpoints »).

  • EFGE_ELNO provides the generalized efforts of classical girders: N, VY, VY, VZ, MT, MFY, MFZ. These efforts are given in the local curvilinear coordinate system of the element.

  • The POST_CHAMP/MIN_MAX_SP command makes it possible to extract, at each of the linear Gauss points of an element, the maximum and minimum values of a component of a field. The min/max is taken on all the sub-points of a point.

8.6. Quiz: SSLL106A#

It is a straight pipe with direction vector \((\mathrm{4,}\mathrm{3,}0)\) fixed at its \(O\) end and meshed with 18 TUYAU elements.

_images/Object_521.svg

The pipe is subjected to various types of load:

  • a traction force,

  • 2 shear forces,

  • 2 moments of bending,

  • 1 torsional moment,

  • internal pressure.

The displacements at point \(B\), the deformations and the stresses at certain integration points of the section containing \(B\), as well as the first natural modes are calculated.

This makes it possible to test the beam degrees of freedom, the swelling degree of freedom, and mode 1 of Fourier series development.