2. Modeling capabilities#
2.1. Reminder of the formulation#
2.1.1. Geometry of pipe elements#
Here we recall the methods and models implemented for the pipe elements and which are presented in the reference document [R3.08.06].
For pipe elements, we define a mean, straight or curved fiber (\(x\) defines the curvilinear coordinate) and a circular hollow section. This section should be small compared to the length of the pipe. Figure [Figure 2.1.1-a] illustrates the two different configurations. A local coordinate system \(\mathit{oxyz}\) is associated with the average fiber.
Figure 2.1.1-a: Straight or curved pipe.
2.1.2. Formulation of pipe elements#
Pipe kinematics [Figure] consists of shell kinematics that describe ovalization, swelling, and warpage, and girder kinematics that describe the overall movement of the pipe line. The displacement \(U\) [Figure] of a material point in the pipe consists of a macroscopic beam part (\({U}^{P}\)) and an additional local shell part (\({U}^{S}\)): \(U={U}^{P}+{U}^{S}\)
Figure 2.1.2-a: Breakdown of displacement into beam and shell fields.
The formulation of the elements is based on:
The**beam theory**for the kinematics of the average fiber. If we make the complete hypothesis of beam theory: the straight sections associated with beam movements (:math:`{U}^{P}`), which are perpendicular to the**mean fiber* of reference [Figure 2.1.2-b] remain perpendicular to the mean fiber after deformation. The straight section does not warp. This will be true on average in element TUYAU. Beam theory is only used to describe the movement of the average fiber: the average fiber of the pipe is equivalent to the average fiber of a beam. This kinematics makes it possible to describe the overall movement of the pipe line.
Shell theory to describe the deformation of cross sections around the middle fiber. Kinematics of cross sections: straight sections that are perpendicular to the mean reference surface remain straight. The material points located on the normal to the undeformed mean surface remain on a line in the deformed configuration. The formulation used is a LOVE - KIRCHHOFF formulation without transverse shear for the description of the behavior of cross sections. The thickness of the shell remains constant. The average area \(\omega\) of the pipe, located halfway through, is equivalent to the average surface area of a shell. This shell kinematics provides a description of the swelling, ovalization, and warping of the cross section.
Figure 2.1.2-b :Fiber and average surface area in the case of a straight pipe.
The additional displacements (\({U}^{s}\)) of the pipe surface are approximated by a Fourier series up to the order \(M\) (\(M=3\) for TUYAU_3M modeling and \(M=6\) for TUYAU_6M modeling).
\({u}^{s}(x,\Phi )=\sum _{m=2}^{M}{u}_{m}^{i}(x)\mathrm{cos}m\Phi +\sum _{m=2}^{M}{u}_{m}^{0}(x)\mathrm{sin}m\Phi\)
\({v}^{s}(x,\Phi )={w}_{1}^{i}(x)\mathrm{sin}\Phi +\sum _{m=2}^{M}{v}_{m}^{i}(x)\mathrm{sin}m\Phi -{w}_{1}^{0}(x)\mathrm{cos}\Phi +\sum _{m=2}^{M}{v}_{m}^{0}(x)\mathrm{cos}m\Phi\)
\({w}^{s}(x,\Phi )={w}^{0}+\sum _{m=2}^{M}{w}_{m}^{i}(x)\mathrm{cos}m\Phi +\sum _{m=2}^{M}{w}_{m}^{0}(x)\mathrm{sin}m\Phi\)
Where |
\({u}^{s}\): represents the axial displacement of the mean surface in the local coordinate system \(x\Phi \zeta\) |
\({v}^{s}\): represents the ortho-radial displacement of the mean surface in the local coordinate system \(x\Phi \zeta\) |
|
\({w}^{s}\): represents the radial displacement of the mean surface in the local coordinate system \(x\Phi \zeta\) |
|
\({w}^{0}\): represents swelling |
These elements therefore involve locally:
6 kinematic variables for the beam formulation: the movements \({u}^{p}\), \({v}^{p}\) and \({w}^{p}\) according to the reference fiber and the rotations around the local axes,
3 kinematic variables for the shell formulation: the additional displacements \({u}^{s}\), \({v}^{s}\) and \({w}^{s}\) in the mean surface coordinate system,
4 constraints in the thickness of the pipe noted \(\mathrm{SIXX}\) (\(\mathrm{sxx}\)), (), \(\mathrm{SIYY}\) (\(\mathrm{sff}\)), and \(\mathrm{SIXZ}\) (\(\mathrm{sxz}\)). \(\mathrm{SIXY}\) \(\mathrm{sxf}\) Constraint \(\mathrm{SIZZ}\) (\(\mathrm{szz}\)) is zero (hypothesis of plane constraints). The transverse shear stresses are zero (Love Kirchoff hypothesis),
4 deformations in the thickness of the pipe noted \(\mathrm{EPXX}\) (\(\mathrm{exx}\)), (), \(\mathrm{EPYY}\) (\(\mathrm{eff}\)), and \(\mathrm{EPXZ}\) (\(\mathrm{sxz}\)). \(\mathrm{EPXY}\) \(\mathrm{exf}\) Deformation \(\mathrm{EPZZ}\) (\(\mathrm{ezz}\)) is zero for the beam part.
Important note:
Beam kinematics is based on the Timoshenko hypothesis [R3.08.03]. The pipe element is not « exact » at the nodes for loads or twists concentrated at the ends, you have to mesh with several elements to obtain correct results.
Depending on the average fiber, these elements are of the isoparametric type. As a result, the displacements vary like polynomials of order 2 next \(X\) for elements with 3 nodes and of order 3 for 4 nodes.
2.2. Comparison to other items#
2.2.1. The differences between pipe elements#
The pipe elements TUYAU_3M and TUYAU_6M are line elements:
TUYAU_3Mà three or four knots.
TUYAU_6Mà four knots.
These elements differ only in terms of the approximation of the additional displacement field COQUE, which is made by a Fourier series decomposition:
TUYAU_3Mjusqu “in order 3.
TUYAU_6Mjusqu “to order 6.
Therefore the number of degrees of freedom is different:
TUYAU_3M21 per node (6 degrees of freedom of beam and 15 degrees of freedom of shell)
TUYAU_6M39 per node (6 degrees of freedom of beam and 33 degrees of freedom of shell)
Compared to modeling TUYAU_3M, modeling TUYAU_6M allows a better approximation of the behavior of the cross section in the case where it deforms in a high mode, for example in the case of thin tubes where the ratio of thickness to radius of the cross section is \(<0.1\), and in some cases in plasticity. In areas with low demand, TUYAU_3M modeling is recommended.
Finite elements of straight or curved pipes
There appears to be a difference in behavior on the elbows present in the structure between the curved and straight quadratic meshes. The middle node of the elements does not end up on the element’s curvature line as a result, for example, of a CREA_MAILLAGE/LINE_QUAD operation.
This difference strongly modifies the stiffness characteristics of the element and also the consideration of internal pressures (shape effect); it is therefore advisable to ensure that the mesh respects the curvature of the elbow as much as possible.
2.2.2. The differences between pipe elements and beam elements#
Like finite elements TUYAU, finite elements POUTRE are also part of the class of linear finite elements. In this section, the formulations and loads applicable for these two classes of elements are compared.
In terms of formulation:
Item POUTRE :
The formulation is based on an exact resolution of the equations of the continuous model performed for each element of the mesh. Several types of beam elements are available:
POU_D_E: transverse shear is neglected, as well as rotational inertia. This hypothesis is verified for strong impulses (Euler hypothesis),
POU_D_T: transverse shear and all inertia terms are taken into account. This hypothesis is to be used for low tendencies (Timoshenko hypothesis).
These elements use SEG2 meshes with 6 degrees of freedom per node, 3 displacements and three rotations. The wording of these elements is presented in the reference document [R3.08.01]. The section is constant, the only possible behavior of transverse sections is translation and rotation for all the points in the section. The section can be of any shape that is constant or variable over the length.
Item TUYAU :
The formulation combines both a beam formulation based on the Timoshenko hypothesis and a shell formulation based on the Love_Kirchhoff hypothesis making it possible to model the phenomena of swelling, ovalization and warping. The hollow section, which is circular in shape, is constant over the entire length of the element. The element is not « exact » at the nodes for loads or twists concentrated at the ends, so it is necessary to mesh with several elements to obtain correct results, in particular to represent the curvature.
These elements use SEG3 or SEG4 meshes with, for the beam kinematics, 6 degrees of freedom per node, 3 displacements and three rotations, and for the shell kinematics, 15 or 33 degrees of freedom for the displacement type.
In terms of applicable loads:
Item POUTRE :
Possible loads are extensional, flexural, and torsional loads. Internal pressure loading for hollow sections does not exist (the section is undeformable).
Item TUYAU :
Element TUYAU accepts conventional beam loads as well as the application of internal pressure.