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Finite elements of right pipe and curve


Date:
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Author (S):
P. MASSIN, J.M. PROIX, A. Key BEN HAJ YEDDER
:
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: 1/54

Organization (S): EDF-R & D/AMA
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
R3.08.06 document

Finite elements of right pipe and curve
with ovalization, swelling and warping
in elastoplasticity

Summary:

This document presents the modeling of a finite element of pipe usable in calculations of pipings in
elasticity or in plasticity. The pipes, curves or rights, can be relatively thick (thickness report/ratio on
radius of the transverse section up to 0.2) and are subjected to various combined loadings - internal pressure,
cross-bendings and anti-plane, torsion, extension - and can have a nonlinear behavior.

This linear element combines at the same time properties of hulls and beams. The average fiber of the pipe
comprise like a beam and the surface of the pipe like a hull. The element carried out is an element of
right pipe or curve in small rotations and deformations, with an elastoplastic behavior in
plane constraints.

Three modelings, corresponding to three various types of elements, are available:

· TUYAU_3M, which takes into account 3 modes of Fourier to the maximum, and which can rest on
meshs with 3 nodes or 4 nodes.
· TUYAU_6M, which takes as a count up to 6 modes of Fourier, and is pressed on meshs with 3 nodes.

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Finite elements of right pipe and curve


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Count

matters

1 Introduction ............................................................................................................................................ 4
2 various theories of hulls and beams for the finite elements of right or bent pipes 4
2.1 The pipe in theory of beam .......................................................................................................... 4
2.1.1 Case of a pipe bent ............................................................................................................. 4
2.1.2 Case of the right pipe .................................................................................................................. 8
2.1.3 Remarks ............................................................................................................................. 9
2.2 The pipe in linearized theory of hull ....................................................................................... 10
2.2.1 General case .......................................................................................................................... 10
2.2.2 Case of the right pipe ................................................................................................................ 12
2.2.3 Notice ............................................................................................................................. 13
2.3 Analyze right and bent pipes ............................................................................................. 15
3 mixed Elements hull-beam for the right and curved pipes ....................................................... 17
3.1 Kinematics ................................................................................................................................... 17
3.2 Law of behavior ..................................................................................................................... 18
3.3 Work of deformation ................................................................................................................... 19
3.4 Energy interns elastic elbow ............................................................................................... 19
3.5 Work of the forces and couples external ........................................................................................ 19
3.6 Principle of virtual work ................................................................................................................ 20
3.6.1 External efforts part and couples for the part beam ..................................................... 20
3.6.2 External efforts part and couples for the part hull ...................................................... 21
3.7 Generalized efforts ......................................................................................................................... 22
4 numerical Discretization of the variational formulations ................................................................. 23
4.1 Discretization of the fields of displacement and deformation for the part beam ................... 23
4.1.1 Beam curves ....................................................................................................................... 23
4.1.2 Right beam ......................................................................................................................... 25
4.2 Discretization of the fields of displacement and deformation for the additional part….26
4.2.1 Bend 27
4.2.2 Right pipe ............................................................................................................................ 29
4.3 Discretization of the field of total deflection ............................................................................. 32
4.4 Stamp rigidity .......................................................................................................................... 32
4.5 Stamp of mass .......................................................................................................................... 33
4.6 Functions of form ........................................................................................................................ 35
4.7 Numerical integration .................................................................................................................... 35
4.8 Discretization of external work .................................................................................................. 36
5 geometrical Characteristics of the pipe section ........................................................................ 37
6 Connection pipe-pipe ............................................................................................................................ 38
6.1 Construction of a particular generator ................................................................................... 38
6.2 Connection from one element to another ...................................................................................................... 39
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6.3 Numerical establishment ................................................................................................................. 40
7 Connections hull-pipe and 3D-pipe ..................................................................................................... 40
7.1 Followed step ............................................................................................................................ 40
7.2 Kinematics of the pipe. .................................................................................................................. 42
7.3 Kinematics of hull .................................................................................................................. 43
7.4 Calculation of average displacement on the section S ............................................................................ 44
7.5 Calculation of the average rotation of the section S .............................................................................. 44
7.6 Calculation of the average swelling of the section S ................................................................................ 45
7.7 Calculation of the coefficients of Fourier on the section S ....................................................................... 45
7.8 Establishment of the method ........................................................................................................... 47
8 Establishment of element TUYAU in Code_Aster ...................................................................... 49
8.1 Description ..................................................................................................................................... 49
8.2 Introduced use and developments ........................................................................................ 49
8.3 Calculation in linear elasticity ............................................................................................................ 50
8.4 Plastic design ........................................................................................................................ 51
8.5 Test: SSLL106A ........................................................................................................................... 51
9 Conclusion ........................................................................................................................................... 52
10
Bibliography .................................................................................................................................. 53
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:
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1 Introduction

There is an important bibliography on the modeling of pipings and many elements
stop pipes right and bent are available in the great codes of finite elements.
syntheses were already realized [bib1], [bib5], [bib6], in the past that one supplemented in
incorporating the last developments known in the field [bib11]. Important effects with
to take into account are swelling due to the internal pressure and the ovalization of the transverse sections
by combined inflections plane and anti-plane. One places oneself on the assumption of small rotations and
deformations within the framework of this document.

It is about a linear element with 3 or 4 nodes, of curved or right beam type with local plasticity
taking into account ovalization, warping and swelling. The kinematics of beam is
enriched by a kinematics of hull for the description of the behavior of the transverse sections.
This kinematics is discretized in M modes of Fourier of which the number M must at the same time be sufficient
to obtain good results in plasticity and not too large to limit the calculating time.
literature encourages us to use M=6 [bib9], [bib13] in plasticity. In elasticity, for thick pipes, one
can be satisfied with M=2 or 3.

2
Various theories of hulls and beams for
finite elements of right or bent pipes

One presents in this chapter the elements of kinematics in three-dimensional curvilinear geometry,
as their restrictions within the framework of the models of beam and hull. Indeed, to build
the finite element of piping enriched which answers the schedule of conditions defined in introduction, one exploits
a technique of decomposition of three-dimensional kinematics. The kinematics of hull y
bring the description of ovalization, swelling and warping, while kinematics
of beam described there the general movement of the line of piping.

The various theories of hulls and beams used for each element translate them
assumptions chosen a priori on the type of deformations and behaviors.

2.1
The pipe in theory of beam

2.1.1 Case of a bent pipe

A first approaches relatively simple come down to consider the elbow represented below
like a beam digs circular section. The beam is obtained by rotation of angle of
circular section around OZ. A point of the beam is located by its distance R compared to the axis
beam and by the two angles, where is the longitudinal angle with OY indicated above and
the trigonometrical angle with OZ measured on the circular section.
Z
Z
Z
X
2
2
·
X
ux
·
()
X

E
uz
R
E
U ·
y
·
1

·
1
er
Z
·

Z
y
Z
Z
Y
R
y
Y
R
y


X


X
O
O

Appear 2.1.1-a: Géométrie and kinematics of the elbow in theory of beam
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:
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In the curvilinear frame of reference (R,), the relations between displacements U of
points of the elbow of position R = OM = - E
R (
0
y) + Re (
R,) and deformations of
Green-Lagrange are given by the following tensor in the natural base (R,):

(0
R + U) (0
R + U) 0
R 0
R
2 F =
.
-
.
, (,) {R,}.




The vectors units in the directions (R,) are:

0
R
1 0
R
1 0
R
0
R 0
R
0
R R
E
0
R =
E
,
=
, E =
where A =
.

and B =
.
.


R
With

B



If one expresses the position of a point of the elbow in the local toric orthonormée base (E E
R, E)
by (yr,
y,
y) one has the following relations:

0
R
0
R
0
R
er =
E
, =
, E =
.
yr
y
y

The form of the tensor of the deformations of Green-Lagrange in this base is then:

(0
R + U) (0
R + U) 0
R


0
R
2 =
.
-
.
.
y
y
y y

Relations of passage between the expression of the deformations of Green-Lagrange in the system of
co-ordinates curvilinear and in the local toric base previously definite are:

= F
rr
rr
F
= A2
F
= B2
F
= AB
F R

R = A
F R

R = B
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:
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The use of this base is particularly interesting because the relations of behavior in
base toric orthonormée are simple of use. For elbow Ci above, if it is considered that them
deformations remain small, one obtains then [bib4] after linearization of the deformations of
Green-Lagrange:

U
R
rr = R
1 U

U


With

ur
=
+
+
With
AB R
1 U
ur
=
+
B
R

1 U

1 U



U With


2 = =
+
-
B
With
AB
1 U

U
U
R




2 R
= R
=
-
+
With
R
R


1 U

U
U


R


2 R
= R
=
-
+
B
R
R



with:
R + R sin
With = R + R sin, B = R, R =
, R = R.
sin

The expressions of the deformations established above are written then:

U
R
rr = R
1
U



=
(
+ U

cos + U sin)
R + R
R
sin
1 U



= (
+ U

)
R
R


1
U

1 U


2 = =
(
- U

cos

) +
R + R sin
R
1
U

U
R


2 =
=
(
- U

R
R
sin) +
R + R sin
R

1 U

U

R

2 = = (
- U

R
R
) +
R

R

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:
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Ur displacement,
U,
U of a point of the elbow in the toric base associated the transverse section
of observation can easily express itself according to displacements and rotations associated with
center transverse section. Indeed, if 1 is noted
U, u2, u3 displacement in the curvilinear base
local (O (), X (), y (), Z () associated the transverse section as indicated on [Figure 2.1.1-a]
there are the following, valid relations within the framework of the kinematics of the beams of Timoshenko
[R3.08.01]:

U (R,
1
,) = ux
() + Z
() R sin - y
() R cos
U (R,
2
,) = U y
() + X
() R cos

U (R,
3
,) = uz
() - X
() R sin

where ux, U y, uz are the displacement of translation of the section and X, y, Z the rotation of its center
O. The expression of the components of displacement in the local toric orthonormée base
(E E
R, E) are obtained by change of reference mark:

U (R,

,) = U (R,) = U (
X) + (
Z) R sin - (
y) R cos
1

U (R,

,) = U (R,) sin - U (R,) cos = U (
Z) sin - U (
y) cos - (
X)
3
2
R
U (R,
R
,) = [
- U (R,) cos + U (R,) sin] = [
- U (
Z) cos + U (
y) sin]
3
2

The introduction of this field of displacement into the expression of the linearized deformations us
allows to obtain the expression of the three-dimensional deformations associated the kinematics of
beam:

rr = 0
1
=
(U X - U y - R cos
X
+ R sin
Z
- R cos
y
)
R + R sin
,
,
,

= 0
1
2 =
(- U cos
X
- U
cos
y
+ U
sin
Z
- R X + R cos2
y
- R sin
Z
cos)
R + R sin
,
,
,


+ (cos
Z
+ sin
y
)
1
2 R =
(- U sin
X
- U
sin
y
- U
cos
Z
+ R sin
y
cos - R sin 2
Z
)
R + R sin
,
,

+ (sin
Z
- cos
y
)
2 R = 0

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2.1.2 Case of the right pipe

The expressions of the deformations established above also apply to the case of the right pipe, where one
replace by S where S is the curvilinear X-coordinate along average fiber of the pipe, with:

With =,
1 B = R 1
,/R =, 0 R = R.

The expressions given for the elbow are written then for the right pipe:

U
R
rr = R
U
X
xx = X
1 U



= (
+ U

)
R
R

U



1 U
X

2 X = X =
+
X

R
U

U
R
X

2 X-ray = X-ray =
+
X

R

1 U

U

R

2 = = (
- U

R
R
) +
R

R


2
Z
Z
·
U
U R
X
With
Z
X
U
X
ux=u1
U 2
O
U
·
3
X
Z
U
1
U
y
y


y
y
Z
Z
Y
Z
X
Transverse section: sight of 1 towards 2
O

Appear 2.1.2-a: Géométrie and kinematics of a right pipe in theory of beam
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:
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As previously, ur displacement, U U
X, of a point of the pipe in the associated toric base
with the transverse section of observation can easily express itself according to displacements and
rotations associated with the center with the transverse section. Indeed, if 1 is noted
U, u2, u3 displacement
in the local curvilinear base (O, X, y, Z) associated the transverse section as indicated on the figure
below there are the following relations:

U (R, X,
1
) = U (X)
X
+ (X) R sin
Z
- (X) R cos
y

U (R, X,
2
) = U (X)
y
+ (X) R cos
X


U (R, X,
3
) = U (X)
Z
- (X) R sin
X


and:

U (R, X,
X
) = U (R, X,) = U (X)
X
+ (X) R sin
Z
- (X) R cos
1
y

U (R, X,

) = U (R, X,) sin - U (R, X,) cos = U (X) sin
Z
- U (X) cos
y
- (X)
3
2
R
X

U (R, X,
R
) = [
- U (R, X,) cos + U (R, X,) sin] = [
- U (X) cos
Z
+ U (X) sin
y
]
3
2

The introduction of this field of displacement into the expression of the deformations given below
us allows to obtain the expression of the deformations associated with kinematics with beam:

rr = 0
xx = ux X + R sin
Z X
- R cos
,
,
y, X

= 0

2
X
= - R X X + (y + U) sin
Z X
+ (Z - U) cos
,
,
y, X

2 X-ray = (Z - U) sin
y X
- (y + U) cos
,
Z, X

2 R = 0

2.1.3 Remarks

The fact that rr, and R are simultaneously null shows that the kinematics of beam cannot
to represent the deformations of the transverse sections to average fiber of the pipe. Indeed, them
transverse sections are actuated by a rigid movement of body, which prohibits to model it
warping, swelling and ovalization.

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2.2
The pipe in linearized theory of hull

2.2.1 Case
General

The bent pipe is regarded as a thin hull of revolution (portion of core). Surface
average is obtained by rotation of angle of a circle of radius has whose center is at a distance
R of the axis of revolution OZ. One indicates by H the thickness of the elbow. One imposes on this thickness
to remain constant like with the section of the elbow being perfectly circular. A point on surface
average is characterized by the two angles, and its position - H/2 +h/2 compared to
surface average, where is the longitudinal, variable angle between 0 and, and the angle measured on
transverse section.


v y
H
W
R

O
U: Axial displacement of average surface
U
y
v: Orthoradial displacement of average surface

E
R
Z

W: Radial displacement of average surface
er
: Rotation of average surface compared to E


E
: Rotation of average surface compared to E

O

Z
E
X
R: Radius of curvature
R: Radius of the cross section
H: Thickness of the elbow
: With
ngle longitudinal
:
Angle of cross section
X

Appear 2.2.1-a: Géométrie and kinematics of the elbow in theory of hull

One places oneself first of all within the framework of the linearized theory of the hulls with shearing
transverse such as it was described for example in Washizu [bib14]. This choice had already was made
for the linear elements of hulls [R3.07.01]. It limits our study to the framework of small
deformations. Moreover, great rotations of average surface are not taken into account.
Displacements and rotations are thus defined compared to the initial geometry of the elbow. If them
displacements of the points of average surface in the three directions axial,
orthoradiale and radial is noted U, v and W those of any point of the elbow
are written in the following way:

U = U (

,) + (
,)
U = v (

,) - (
,)
U = (

W,)

where and are rotations compared to the vectors
E and
E respectively.
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The deformations in any point are thus given by [bib14]:


E
+
= 1+/R
E +
= 1+/R
2
E
+
2

2 = = (

1+/
R) (1+/
R)
2
E

2 = = (1+/R)
2nd

2 = = (1+/R)
with:
R + has sin
With = R + has sin, B = has, R =
, R =.
has
sin

where
E, E and
E are the membrane deformations of average surface, them
deformations of inflection of average surface and
E, E transverse distortions.
deformations of average surface are connected to displacements of average surface in
replacing the field of displacement of the preceding paragraph by that given above. One finds
then:

1 U
v A
W

E
=
+
+
With
AB

R
1 v
W
E =
+
B

R
1 U
1 v
U With
2
E
=
+
-
B
With
AB
1


With
=
-
With
AB
1
= - B
1
To 1
1 1 U
1
1 v
U

With
2 =
-
-
+ [
+
(
-
)]
B
AB A

R B

R
With
AB
1 W
U
2
E
= +
-
With

R
1 W
v
2nd = - +
-
B


R
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:
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That is to say still:

1
U

E =
(
+ v cos + wsin)
R + has sin
1
v

E = (
+)
W
has
1
v

1 U

=
(
- U cos) +
R + has sin
has
1

=
(


- cos

)
R + has sin
1
= -


has
1
1

sin
1 U

1
1
v

=
-
(
+ cos

) + [
+
(
- U cos)]
has
R + has sin
R + has sin has R + has sin has
1
W

= +
(
- U sin)
R + has sin
1
W

= - + (- v)
has


In this theory there are thus five unknown factors; 3 displacements U, v and W like two
rotations
,
. If the assumption of Coils-Kirchhoff is applied (thin tube) shearings
transverses are null and there are nothing any more but 3 displacements U, v and W since:

1
W

= -
(
- U sin)
R + has sin

1
W

= (- v)
has

2.2.2 Case of the right pipe

If one applies these equations to the case of the right pipe with:

With =,
1 B = has 1
,/R =,
0 R =.
has
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One finds the more usual expression for this kind of geometry:

U

Exx = X
1
v

E = (
+)
W
has
v

1 U

2Ex =
+
X

has



xx =
X

1 X
= - has
1
X
1 v

2 =
-
+ [
]
X
has
X

X has

W

2Ex = + X
1
W

2nd = - + (
- v)
X
has


In this theory there are thus five unknown factors; 3 displacements U, v and W like two
rotations,
X

. If the assumption of Coils-Kirchhoff is applied (thin tube) shearings
transverses are null and there are nothing any more but 3 displacements U, v and W since:

W

= - X

1
W

= (
v)
X
-
has

2.2.3 Notice

One can directly introduce the kinematics of hull into the field of deformation 3D. In it
case one a:
=
E
+
= E +

2 = = 2nd +
2

2 = = 2nd

2 = = 2nd

where expressions
E, E and
E for the membrane deformations, for
deformations of inflection and
E, E for the transverse distortions are given by the expression
following in the general case:
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:
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1
U

E =
(
+ v cos + wsin)
R + R sin
1
v

E = (
+)
W
R
1 U

1
v

2nd =
+
(
- U cos)
R R + R sin
1

=
(


- cos

)
R + R sin
1
= -


R
1
1

2 =
-
(
+ cos

)
R
R + R sin
R + has sin
1
W

2nd =
+
(
- U sin)
R + R sin
R + R sin
1 W has

2nd = - + (
- v)
R
R


It is noticed that to command 1 in the two ways of proceeding give identical results. It is
the definition of the deformation of membrane or inflection which changes. In the first case it is
independent of the position in the thickness and is calculated for the average radius of the section
transverse of the pipe, whereas it depends on it in the case on the approach 3D. The term between hook
in the expression of [§2.2.1]. represent a coupling between the inflection and the membrane which appears
when one expresses R + R sin and R according to R + has sin and A. Dans the continuation of our analysis
we will use this expression 3D degenerated of the kinematics of hull.
If moreover we use the assumption of Coils-Kirchhoff for transverse shearings,

E
=
E
= 0 one find well the following expressions of rotations:

1
W

= -
(
- U sin)
R + has sin

1
W

= (- v)
has
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and:
1
1
2
W U
cos

W
=
[-
(
-
sin

) -
(
- v)]
R + R sin
R + has sin
2


has

1
2
W

v
= -
(
-
)

2
rear

W
cos
cos has
2
= (
- U sin

) [
+
]

(R + R sin) (R + has sin) R (R + sin has) 2
2
W
1
1
-
[
+
]
(R + R sin) R has (R + has sin)
v
1
U
1
+
+ (
sin + U cos)
has (R + R sin)

R (R + has sin)

expressions for
E,
E
and

E remaining unchanged.

One can easily extend this remark to the case of the right pipe.

2.3
Analyze right and bent pipes

In conclusion of the two preceding analyzes one can model the pipe like an element of
beam whose section is a thin hull. This interpretation is made in the majority of the codes
([bib2], [bib8], [bib9], [bib10], [bib12], etc…). In the absence of warping of the transverse sections
(i.e the transverse sections remain plane) the axial displacement of beam gives the new position
transverse section and displacements of ovalization (it is enough to take then u=0 in
mean equations of hulls) make it possible to know how this one becomes deformed. Total deflection
is obtained like superposition of the deformations of beam and the deformations of ovalization.
field of displacement which one represents on the figure below writes:

p
S
U = U +U.
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In the first field of displacement the image of the transverse section is a transverse section
identical obtained by translation and rotation of the first. In the second field of displacement,
transverse section is deformed.

M
M
inflection-torsion of a right beam
In theory of the beams-Euler
In theory of the hulls
U
v
W
Cross
Transverse section
Cross
Transverse section
warping
ovalization

Appear 2.3-a: Décomposition of displacement in fields of beam and hull

Modeling finite element must thus give an account of two different mechanical answers: that
beam and that of the hull for ovalization, swelling and warping. These three
last modelings utilize degrees of freedom which are not nodal (decomposition
in Fourier series for example).
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3
Mixed elements hull-beam for the right pipes and
curves

3.1 Kinematics

One breaks up the field of displacement into a macroscopic part of “beam” and a part
additional local of “hull”. V is the useful space of the fields of displacements
three-dimensional definite on an unspecified section of pipe.

For the beam part, as in [R3.03.03], one introduces space T of the fields associated with a torque
(defined by two vectors):

T = {vV/(T,)

that

such

v (M) = T + GM}
For the fields of displacement of T, T is the translation of the section (or the point G),
infinitesimal rotation and fields v are displacements preserving the section S plane and not
deformation there (One uses still the assumptions of NAVIER-BERNOULLI).
T is a vectorial subspace of finished size equalizes to 6. It has additional orthogonal
for the scalar product on V:





T = v V/v W
. =
0 W

T.



S

Any field U of V breaks up then in a single way all in all of an element of T and one

element of T:
p
S p
S

U = U + U U T U
,
T.
One postulates then for displacements of surface of the pipe defined in [§2.2] the decomposition in
following Fourier series who check the preceding principle of orthogonality with displacements of
beam until command 3 in the thickness of the pipe:

M
U (X,) = I
um (X) cos m
m=2
M
+ O
um (X) sin m
m=2
M
v (X,) = I
wn (X
1
) sin + I
vnm (X) sin m
m=2
M
............ - O
wn (X
1
) cos + O
vnm (X) cos m
m=2
(
W X,) = wo (X)
N
(uniform radial expansion)
M
......... + I
wnm (X) cos m
m=1
M
......... + O
wnm (X) sin m
m=1

where X is the curvilinear X-coordinate along the elbow or of the right pipe, indifferently, and M the number of
modes of Fourier. Rotations (X,
X
) and (,
X) result from U (X,), v (X,) and (
W X,) by
the relations of Coils-Kirchhoff [§2.2.1].
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:
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Note:

One can note that in the decomposition of v (X,) and (
W X,) terms in cos and sin
are not completely independent because of orthogonality with displacements of
beam. This makes it possible moreover to avoid the movements of rigid body, because if
I
O
I
O
vn, v
1
n1 and
N
W 1, N
W 1 are independent, one can find a solution nonnull giving
null deformations. In addition in the expression of U (X,) one notes the absence of the terms
in cos and sin already present in the beam part.
If one neglects the variation of metric with the thickness of the pipe the conditions of orthogonality
rigorous between displacements of beam and those of the surface of the pipe are satisfied.
In the contrary case, to satisfy this condition rigorously one would need one
development in Fourier series of rotations (X,
X
) and (,
X) starting with
command 2. This is incompatible with the assumptions of Love_Kirchhoff for these rotations.

3.2
Law of behavior

The behavior of the new element is a behavior 3D in plane constraints, because it
total behavior of the structure is that of a thin hull. It results from it that = 0 and the law from
behavior is written in a general way in the following way:


xx
pxx + S

xx xx


p
S




+








p
S


X =
C
+


X

X =



C
X
p
S







+

R






X
p
S

xr +

X


X

In our case one will neglect transverse shearings for the hull part of our field of
displacement. It thus results from it that S
X = S
= 0. As in addition [§2.1.2] it was shown that
p
= 0
=
R
it results from it that
0

. For an elastic behavior one has as follows:

1
0
0
1
0
0
xx


xx




1
0
0




1
0
0

E
1 - v



E
1 - v


=
0 0
0
and C =
0 0
0
.
2








X 1 - v
2

X
1
2
v
2

-


1 - v




1 - v

X
0 0
0


X
0 0
0


2

2
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3.3
Work of deformation

The general expression of the work of deformation 3D for the element of elbow with the type of
above mentioned behavior is worth:

L 2 H/2
W
=
(
+ + + FD
def
xx xx
X
X
X
X)


0 0 - H/2

where L is the curvilinear X-coordinate which is worth L = (R + R sin
) for an elbow where is the traversed angle
to describe the elbow. In the case of an elbow, one has thus FD = (R + R sin)
D

rd
D and for one
right pipe FD =

dxrd
D where. is the position in the thickness of the elbow which varies between - H/2 and
+h/2. In the continuation, in order to reduce the notations, one will employ the second expression.

3.4
Energy interns elastic elbow

In the case of an elastic behavior, energy interns elastic elbow is expressed way
following:

L 2 H
1
/2
E
int =
(
(2
2
+ + 2) + G (2
2
+))FD

xx

xx
X
X

2
2

-
0 0 - H
1
/2

This energy can be broken up into part of energy of beam, part of energy for
H/2 L 2
surface pipe and terms of coupling of the type p
. S
xx xx FD.
- H/2 0 0

3.5
Work of the forces and couples external

With the decomposition of displacements stated at the head of paragraph, the work of the forces
being exerted on the pipe expresses itself in the following way:

L +h/2
2
L
2
+h/2
2
P
S
P
S
P
S
W
=
F
ext.
v. (U + U FD
)
+ S
F.(U + U) (± H has/)
2D dx
+ C
F.(U + U) rd D
=
0 - H/2 0
0 0
- H/2 0
L +h/2
2
L
2
+h/2
2
p
p
p
F
v U
.
FD + S
F U
.
H has/)
2D dx
+ C
F U
.
rd D
+

0 - H/2 0
0 0
- H/2 0
L +h/2
2
L
2
+h/2
2
S
S
S
p
S
F
v U
.
FD + S
F U
.
H has/)
2D dx
+ C
F U
.
rd D
= Wext +Wext
0 - H/2 0
0 0
- H/2 0

by simple linear decomposition, where v
F, S
F, C
F are the voluminal, surface efforts and of contour
being exerted on the pipe, respectively. One determines as follows:

L +h/2
2
L
2
+h/2
2
W p
p
=
F U
.
FD
p
+
F U
.
H has
p
ext.
v
S
/)
2
D dx + C
F U
.

rd
D
0 - H/2 0
0 0
- H/2 0
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and:
L +h/2
2
L
2
+h/2
2
W S
S
=
F U
.
FD
S
+
F U
.
H has
S
ext.
v
S
/)
2
D dx + C
F U
.

rd
D
0 - H/2 0
0 0
- H/2 0

The work of the external forces can thus be separate in two contributions distinct from same
forces, on the kinematics of beam and its additional.

3.6
Principle of virtual work

It is written in the following way:
p
S
ext.
W
= ext.
W
+ ext.
W
= int
W with:

L 2 H/2
W

=
(+
+
+) FD
int
xx xx
X
X
X
X

0 0 - H/2

3.6.1 External efforts part and couples for the beam part

The discretization of the principle of virtual work for the external efforts gives:

L
W
pext = (F
X U
X + F y U
y + F Z U
Z + mxx + my y + mzz dx
) +
0
[X U
X +y U
y +z U
Z + µxx + µ there y + µzz] 0, L

F X, F y, F Z: linear forces acting according to X, y and Z passing by the center of gravity of
transverse sections:

+h/2
2

2
fi =
F.
O C I

rd
D + F .e (has
S
I
± H/)
2
D


where ex, ey, ez are the vectors of the base
- H/2 0
0
curvilinear local.
MX, my, mz: linear couples acting around axes X, y and Z:
+h/2
2

2
semi =
(rxF.)
v I.E.(internal excitation)

rd
D + (rxF.)E (has
v
I
± H/)
2
D


where ex, ey, ez are the vectors of the base
- H/2 0
0
curvilinear local.
X, y, Z: concentrated forces acting according to X, y and Z passing by the center of gravity of
transverse sections:
+h/2
2
I =
F.
E.C.I.

rd
D

where ex, ey, ez are the vectors of the local curvilinear base.
- H/2 0
µx, µy, µz: moments concentrated around axes X, y and Z:
+h/2 2
µi = (rxF.) rd D
C I.E.(internal excitation)
where ex, ey, ez are the vectors of the local curvilinear base.
- H/2 0
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3.6.2 External efforts part and couples for the hull part

It is supposed that the external efforts applied to the elbow are independent the thickness of
bend:

L
2
W S =
(F
ext.
xu + Fv + Frw+ MX + M
X
)
D
D
0 0


2
+ [xu + v + rw + +] 0 L,
D
0
where:

Fx, F,

R
F: surface forces acting according to X, and R:
+h/2
F =
F .e rd
v
I
+ F .e (H has/)
2
I
S
I
±

where ex, E, E
R are the vectors of the local toric base.
- H/2
MR. M
X,
: surface couples acting around X and:
+h/2
M =
(E xF) .e rd
R
v
I
+ (± H/2nd xF) .e (H has/)
2
I
R
S
I
±

where ex, E, E
R are the vectors of the base
- H/2
toric local.

X,
R: linear forces acting according to X, and R:
+h/2
I = F. rd
C I.E.(internal excitation)
where ex, E, E
R are the vectors of the local toric base.
- H/2
,
X: linear couples acting around X and:
+h/2
I = (E xF) .e rd
R
C
I
where ex, E, E
R are the vectors of the local toric base.
- H/2

Note:

When the external forces applied are independent of external work on
kinematics of hull is null except that of the compressive forces corresponding to the forces
according to er. It is also noticed that the expressions of the moments linear and concentrated by
report/ratio with R are null. One finds although exerted moment ago
perpendicular to the plan of the hull.
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3.7 Efforts
generalized

If S is the surface of the transverse section S of the pipe, one poses:

NR = dS
xx: normal effort in the center of gravity of the transverse section.
S
T = dS = - (sin
+ cos
) dS
y
xy

X
X
and
S
S
T = dS = (sin
- cos
) dS
Z
xz

X
X
sharp efforts following y and Z.
S
S
M = (y - Z) dS = - dS
X
xz
xy
X: torque around X.
S
S
M = Z dS = - R cos dS
y
xx

xx
: bending moment around Y.
S
S
M = - y dS = R sin dS
Z
xx

xx
: bending moment around Z.
S
S
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4
Numerical discretization of the variational formulations

4.1 Discretization of the fields of displacement and deformation for
the beam part

In a point of average fiber, the field of displacement of beam is in the curvilinear reference mark
U
X


U y


room defined in [§2.1]:
p
U
U = Z



X

y


Z

This field can be discretized in the following way:

NR
NR
U = H
K
K
K
K () K
U
[X xk + K
U there y K + K
uz zk] and = HK () [X xk + y yk + Z zk]
K =1
K =1

It should be noted that the nodal values are given in the local reference marks attached to the nodes and
that U and must be expressed in the local reference mark associated the current point.

4.1.1 Beam
curve

One obtains then:


U
U (X .x) U (Y.X)


(X .x) (Y.X)
X
kx K + K

X
K
X
K
+ K

NR

y
K



NR

y
K



U
y =
H K
K
X (xk .y) + K
(Y. y)
y = H K () K
ux (xk .y) + K
U (Y. y)
y
K
and
()
y
K





K =


U
1

K
U Z




K =1
K
Z


Z


Z K

Z

Z
K


According to the kinematics of beam presented higher to [§2.1]:

rr = 0
1
=
(U X - U y - R cos
X
+ R sin
Z
- R cos
y
)
R + R sin
,
,
,

= 0
1
2 =
(- U cos
X
- U
cos
y
+ U
sin
Z
- R X + R cos2
y
- R sin
Z
cos)
R + R sin
,
,
,


+ (cos
Z
+ sin
y
)
1
2 R =
(- U sin
X
- U
sin
y
- U
cos
Z
+ R sin
y
cos - R sin 2
Z
)
R + R sin
,
,

+ (sin
Z
- cos
y
)
2 R = 0

Knowing that X, = - yety, = X with moreover
.
X xk =.
y y K = cos (- K) = Ck and.
y xk = -.
X y K = sin (- K) = Sk.
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That implies for the field of deformation:

NR

1
K
K
K
K
=
[H (kux

cos (-)
K - U y

sin (-)) + H (
K
K - U X

sin (-)
K - U y

cos (-))
R + R sin
K
K 1
=
- H (K
K U X

sin (-)
K
K + U y

cos (-)) - Rh cos
K
K

(kx

cos (-)
K - ky

sin (-))
K
- Rh cos
K

(kx

sin (-)
K + ky

cos (-)) - Rh cos
K
K

(kx

cos (-)
K - ky

cos (-))
K
+ Rh sin
K
K]
Z
= 0
NR

1
K
K
=
[- H cos
K
(ux

cos (-)
K - U y

sin (-))
R + R sin
K
K 1
=
- H cos
K
(K
ux

sin (-)
K
K + U y

cos (-)) - H cos
K
K
(K
U X

cos (-)
K
K - U y

sin (-))
K
K
+ H U sin -

K Z
rHk
(kx

cos (-)
K - ky

sin (-)) - Rh (
K
K
K
- X

sin (-)
K - ky

cos (-))
K
+ Rh cos2
K

(kx

sin (-)
K + ky

cos (-))
K
- Rh K K sin
Z
cos]
+ H sin
K

(kx

sin (-)
K + ky

cos (-))
K
+ H K K cos
Z

NR

1
K
K
=
[- H sin
R
K
(ux

cos (-)
K - U y

sin (-))
R + R sin
K
K 1
=
- H sin
K
(K
ux

sin (-)
K
K + U y

cos (-)) - H sin
K
K
(K
ux

cos (-)
K
K - U y

sin (-))
K
K
- H U cos
K Z
+ Rh sin
K
cos
(kx

sin (-)
K + ky

cos (-))
K
- Rh K K sin 2
Z
]
- H cos
K

(kx

sin (-)
K + ky

cos (-))
K
+ H K K sin
Z



Maybe in matric form:
K
U
X
K
U y
P



NR
K
U
P

p
U = Z
= P P
Bk The U.K. where K
is the field of displacement to the node K
K
P
K =




1
X



K

y
K
Z
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and

H



K Ck
- HK Sk
- R HK Sk cos
- R HkCk cos


R + R sin
R + R sin
R + R sin
R + R sin
R H

K sin


-
0
2H

K Sk
- 2HkCk
- 2rH
cos
2
cos
R

+ R sin

K Ck
R HK Sk

R + R sin
R + R
+
sin
R + R sin
R + R sin


0
0
0
0
0
0

- H


+ 2


K Sk cos
- HkC cos
H S (R sin
R)
H C (R sin + 2r
K
K K
K
K
)


B P = R + R sin
R + R sin
H sin
R + R sin
R + R sin
RH cos
K
K
K
- 2H

2


K Ck cos
H S
K K cos
R + R sin
R H

C
R H S
R + R sin
K
K
K K
+

R + R sin
R + R sin
-
+
R + R sin
R + R sin

- HS sin
- HC


K K
K
K sin


R + R sin
R + R sin
- H cos
K

- RH S cos
K K

- RH C cos
K
K

RH sin
K

- 2H C sin
2H S


K
K
K K sin
R + R sin
R + R sin
R + R sin
R + R sin

+

R + R sin
R + R sin

The matrix of passage of the deformations to the field of displacement is written as follows: P
B = (P
P
B
B

1 L
NR)

4.1.2 Beam
straight line

U
K
U

K

X
X
X
X

NR
NR
U


K
y
= HK (X) K


U y and y = H K (X)


y
K =


1
K
K =


1
K
uz
uz
Z
Z

According to the kinematics of beam presented higher [§2.1]:

rr = 0
xx = ux X + R sin
Z X
- R cos
,
,
y, X

= 0

2
X
= - R X X + (y + U) sin
Z X
+ (Z - U) cos
,
,
y, X

2 X-ray = (Z - U) sin
y X
- (y + U) cos
,
Z, X

2 R = 0

that implies for the field of deformation:

NR
'
'
'
xx =
K
H kux - H K R cos () K
y + H Kr sin () K
Z



k=1
= 0
NR

'
'
'
X = - HK cos () K
U y + HK sin () K
uz -
K
H K R X + H K sin () K
y + H K cos () K
Z



k=1
NR
'
'
X-ray = (- HK sin () K
U y - HK cos () K
uz - H K cos () K

y + H K sin () kz)
k=1
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
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Titrate:
Finite elements of right pipe and curve


Date:
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Author (S):
P. MASSIN, J.M. PROIX, A. Key BEN HAJ YEDDER
:
R3.08.06-B Page
: 26/54

Maybe in matric form:
K
U
X

K
P

U

y
X


P
NR
K
U
=
B U
p
U = Z
P
P P
K
K where
is the field of displacement to the node K

K
K
X
K =1
X
P


X
K

y
K
Z
and:

H K
0
0
0
- rcos () H rsin () H

K
K
0
0
0
0
0
0

P
B

K =

0
- cos () H
sin () H
K
K
- Rh K
sin () H K
cos () H K


0
- sin () H K - cos () H K
0
- cos () H K
sin () H K

The matrix of passage of the deformations to the field of displacement is written as follows: P
B = (P
P
B
B

1 L
NR)

4.2 Discretization of the fields of displacement and deformation for
the additional part

NR
One discretizes the field of displacement for the surface of the pipe in the form:
S
U = HK (X) S
The U.K.
K =1
with:
I
U
I



U
m
K m


I
v
VI
m
K m


I
W
wi
m
K m
O
O
U m
ukm


S
U = O
v and S
U =

K
vo
m =,
2 Mr.
m
K m


O
W
O

m
wkm
I
I
w1
W
k1
O
W
O


1
wk1




O
W
O


W
K
One has as follows:

U (X,) cos (m)
0
0
sin (m)
0
0
0
0
0



v (X,) =
0
sin (m)
0
0
cos (m)
0
sin () - cos () 0 S
U
W (X,) 0
0
cos (m)
0
0
sin (m) cos ()
sin ()
1




1
4
4
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
4
4
4
3
m=,
2 M
Handbook of Référence
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Finite elements of right pipe and curve


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:
R3.08.06-B Page
: 27/54

if the indices m of S
The U.K. are ordered in the following way:

I
U

km=2
I
v

km=2
I
wkm=2
O

ukm=2
O
v

km=2
O
wkm=2



M

I
U

S
The U.K. = km=M
I

vkm=M
I

wkm=M
O
U

km=M
O

vkm=M
O

wkm=M

I
W


k1


O
wk1

O

wk

The kinematics of hull presented higher to [§2.2] is:

=
E + X
= E +
= 2
E
+ 2
= 2
E
= 0
= 2nd = 0

4.2.1 Bend

With:

1
U
E
=
(
+ v cos

+ wsin)
R + R sin
1 v
E
= (
+)

W

R
1 U
1
v
2nd
=
+
(
- U cos

)
R R + R sin
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
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Finite elements of right pipe and curve


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:
R3.08.06-B Page
: 28/54

and:

1
1
2
W U
cos

W
=
[-
(
-
sin

) -
(
- v)]
R + R sin
R + has sin
2


has

1
2
W

v
= -
(
-
)

2
rear

W
cos
cos has
2
= (
- U sin

) [
+
]

(R + R sin) (R + has sin) R (R + sin has) 2
2
W
1
1
-
[
+
]
(R + R sin) R has (R + has sin)
v
1
U
1
+
+ (
sin + U cos)
has (R + R sin)

R (R + has sin)

allows to break up the field of deformation of hull on the modes of Fourier in the way
following:

S

xx
S
NR
=
B U
S
S S
K
K with:
X k=1
S
X
S
Bk = (if
if
so
so
sg
Bkm
B
=2
L
K m=M
B km
B
=2
L
K m=M
Bk)

where


H




K cos m sin
HK cosm


-

H

cos
sin
sin cos

R + R sin
(R + R sin) (R + has sin)
K
m
HK
m
(1+
)
1+


R + R sin
(R + has sin)
R + R sin


has
H




K m sin m cos

+

has (R + R sin)



m


1

m2


0
H
1
cos
1



K
+
m
HK
cosm
R


has
+
R


has

Bsi




km =
m
H cosm cos
-
H

K
K sin m -

R
R + R sin



mH



K sin m
mHk sinm
H 1+


+


K
sin m

cos sin
H







+

+


K cos m
aHk cosm
has
R (R has sin)
has (R R sin)
-
[
+
]

(R + has sin) (R + R sin) R (R + sin has)
R + R sin
cos


cosm H
cosm H has
+

[
K
K
+
]

(cosm cos - msin m sin)
R + has sin (R + R sin) R (R + has sin)
+ H
K

R (R + has sin)



0
0
0

Handbook of Référence
R3.08 booklet: Machine elements with average fiber
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Code_Aster ®
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Finite elements of right pipe and curve


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:
R3.08.06-B Page
: 29/54


H




K sin m sin
HK sinm


-

H

sin
sin
cos

R + R sin
(R + R sin) (R + has sin)
K
m
HK
m
[1+
]
1+
cos


R + R sin
(R + has sin)
R + R sin
has
H



K m cos m

-
cos

(R + R sin has

)


m


1

m2


0
-
H 1
sin

1


K
+
m
HK
sin m

R


has
+
R


has

Bso

km = m
H sin m cos


H

K
K cosm -

R
R + R sin



mH



K cosm
mHk cosm
H 1+

-
+


K
cosm

cos sin
H







+

+


K sin m
aHk sinm
has
R (R has sin)
has (R R sin)
-

[
+
]

R + has sin R + R sin
R (R + has sin)
R + R sin
cos
H


aHk sin
K sin m
+

m
[
+


]
(mcosm sin + sin m cos)
R + has sin
sin
(
sin)
+




R + R
R R + has

HK

R (R + has sin)



0
0
0

and
2H
2
2










K cos sin
Hkcos
HK (sin
- cos)
Hksin
HK sin

-
-

R + R sin
(R + R sin) (R + has sin)
R + R sin
(R + R sin) (R + has sin) (R + R sin)

2 H
2





K cos sin
2 H

cos
H



K
K
+
-


(R + R has
-
sin)
(R + R has
(R
sin)
+ sin has) (R + R sin)

2

2

H


1+
H

1+
H
K

K

K





cos
sin




Bsg
R
has
R
has
R

K =


2


2


1+
H cos
K



+

has

1


has

H sin
(R + R sin) (R + has sin)
K
[
]
- H cos
K
[
]

R +
+
R sin R (R + has sin)
R +
+
R sin R (R + has sin)


Ha cos
K



H cos

K

cos
cos has
H sin
K

cos
cos has
+
+
[
+
]
+
[
+
]
R (R + has sin) 2

R + has sin R + R sin R (R + has sin)
R + has sin R + R sin R (R + has sin)



0
0
0



4.2.2 Pipe
right

With:

U

Exx = X
1
v

E
= (
+ W

)
R
v

1 U

2Ex =
+
X

R




xx =
X

1 X
= - has
1
X

2 X =
-
R
X

W

= - X

1
W

= (
v)
X
-
has
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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Titrate:
Finite elements of right pipe and curve


Date:
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:
R3.08.06-B Page
: 30/54

the field of deformation of hull breaks up on the modes of Fourier in the following way:

NR M
xx =
H (X)
N
(cos I
m unm + sin O
m unm)
X N 1=m=2
2

NR

M

-
H (X)
N
O
wn +
I
m wnm +
O
m wnm
2
(cos
sin
)
X N 1
=


m 1
=


1 NR

M


I
O
I
O
=
H (X) sin
N

wn - cos
1
W 1n + (sin
m vnm + cos
m vnm)
R N 1
=


m=2


1 NR

M

+ H (X)
N
O
wn + (cos I
m wnm + sin O
m wnm)
R N 1
=


m 1
=


NR

M

+
H (X) sin
N

I
wn - cos O
I
O
1
W 1n + (sin
m vnm + cos
m vnm)
rear N 1
=


m=2


2

NR

M

-
H (X)
N
O
wn +
I
m wnm +
O
m wnm
2
(cos
sin
)
rear N 1
=


m 1
=


NR

M


I
O
I
O

X =
H (X)
N
sinwn - cos
1
W 1n + (sin
m vnm + cos
m vnm)
X N 1=


m=2


1 NR M
+
H (X)
N
(cos I
m unm + sin O
m unm)
R N 1
= m=2
NR

M

+
H (X) sin
N

I
wn - cos O
I
O
1
W 1n + (sin
m vnm + cos
m vnm)
X has

N 1
=


m=2


2


NR

M

- +
H (X)
N
O
wn + (cos I
m wnm + sin O
m wnm)
R
R
X N 1
=


m 1
=


X = 0
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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Titrate:
Finite elements of right pipe and curve


Date:
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P. MASSIN, J.M. PROIX, A. Key BEN HAJ YEDDER
:
R3.08.06-B Page
: 31/54

That is to say still:

NR M
xx = H (X)
N
(cos I
m unm + sin O
m unm)
N 1
= m=2
NR

M

- H (X)
N
O
wn + (cos I
m wnm + sin O
m wnm)
N 1
=


m 1
=


1 NR

M


I
O
I
O
= H (X)
N
coswn + sin
1
W 1n + (mcos
m vnm - msin
m vnm)
R N 1
=


m=2


1 NR

M

+ H (X)
N
O
wn + (cos I
m wnm + sin O
m wnm)
R N 1
=


m 1
=


NR

M

+
H (X)
N
cos I
wn + sin
O
I
O
1
W 1n + (mcos
m vnm - sin
m

m vnm)
rear N 1
=


m=2


NR
M

+
H (X)
N
(2
m cos
I
2

m wnm + m sin O
m wnm)
rear N 1
=

m 1
=


NR

M


I
O
I
O

X = H (X) sin
N

wn - cos
1
W 1n (sin
m vnm + cos
m vnm)
N 1
=


m=2


1 NR M
+ H (X)
N
(- sin
m
I
m unm + mcos O
m unm)
R N 1
= m=2
NR

M

+ H (X) sin
N

I
wn - cos
O
I
O
1
W 1n (sin
m vnm + cos
m vnm)
has

N 1
=


m=2



NR
M

- + H (X)
N
(- sin
m
I
m wnm + mcos O
m wnm)
R
N 1 has
=

m 1
=


X = 0

This gives in matric form:

S

xx
S
NR
=
B U
S
S S
K
K with:
X k=1
S
X

S
Bk = (if
if
so
so
sg
Bkm
B
=2
L
K m=M
B km
B
=2
L
K m=M
Bk)
Handbook of Référence
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Code_Aster ®
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Titrate:
Finite elements of right pipe and curve


Date:
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:
R3.08.06-B Page
: 32/54

where:

H cos (
K

m)
0
- H cos (
K

m)




m

1

2
m




0
H cos (
K

m) 1
H cos (m
K
) 1


+
+

if
Bkm
R

has
=
R

has,
m




- H sin (
K

m)
H sin (
K

m
) 1 +
+ H
m sin (
K

m)
R

has
R has


0
0
0


H sin (
K

m)
0
- H sin (
K

m)




m

1

2
m



0
H sin (
K

m) 1
H sin (m
K
) 1


-
+
+

so
Bkm
R

has
=
R

has
m




H cos (
K

m)
H cos (
K

m
) 1+
- + H
m cos (
K

m)
R

has
R
has


0
0
0

and

- H cos (
K
)
- H sin (
K
)
- H K


2

2

H K
1 + H cos (
K
)
1 + H sin (
K
)
sg
R
has
R
has
R

Bk
=





2


2

1 +
+ H sin (
K
) - 1 +
+ H cos (
K
)
0

has
R
has
R








0
0
0


4.3
Discretization of the field of total deflection

P
S

xx xx xx
P
S
NR
NR
NR
= + =
B U
B U
B U
B U
P
S
P P
K
K + S
S
K
K =
=
with



X

X X
K
K
K =1
K =1
K =1
P S




X X X
P
U
B = (P S
B
K
U =

K B K) K,
1
= NR and


S
The U.K.K, 1
= NR

4.4
Stamp rigidity

The variational formulation of the work of deformation is:



xx



L
2 H/2





W
=


def
(xx




X

X) rd

dxd


0 0 - H/2

X







X
Handbook of Référence
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Code_Aster ®
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Titrate:
Finite elements of right pipe and curve


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:
R3.08.06-B Page
: 33/54

that is to say still:


xx



L
2
H/2





W
=

C

def
(xx




X

X)
rd
dxd


0 0 - H/2

X







X
L
T

2
H/2 NR

NR

W
=
B U
C
B U

rd dxd


def




K K
K K




0 0 - H/2 k=1

k=1



L
2
H/2 NR
T
T
NR

= U B C B U



rd dxd


K
K
K K



0 0 - H/2 k=1
k=1






2
/2

L H
U



1


T
T T


= U
U
B
C B

1
L
M


NR

rd dxd
0 0 - H/2




U




NR




U
1

T
T
= U
U
K

1
L
NR



M

U




NR
The principle of virtual work is written UT then
KU = F U
where K is the matrix of rigidity which
is worth:

L
2
H/2
K = {BT}
CB rd

dxd
0 0 - H/2

Note:

One makes no assumption on the law of behavior. This expression is thus in
private individual validates the nonlinear behaviors in the case of (plasticity).

4.5
Stamp of mass

The terms of the matrix of mass are obtained after discretization of the variational formulation
following of the noncentrifugal terms of inertia:

u1 (X, R)

u2 (X, R)
L
2 +h/2
u3 (X, R)
W ac
U =

farmhouse =
u&.v

rdxd
D

with
.
U (X, R)
0 0 - H/2
v (X, R)



(
W X, R)

The notations used are those of [§2.1]: 1
U, u2 and u3sont displacements of beam in a point
section and U, v and W are displacements of average fiber of this section in this same
not.
Handbook of Référence
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Code_Aster ®
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Titrate:
Finite elements of right pipe and curve


Date:
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Author (S):
P. MASSIN, J.M. PROIX, A. Key BEN HAJ YEDDER
:
R3.08.06-B Page
: 34/54

The discretization gives then:
the U.K.
X


the U.K.
y


the U.K.
Z
K
X


K
y
K
Z


ui
NR
km
U = H


K NR K v I
km
K 1
=
wi
km

m =,
2 M
U okm
vo
km


wo

km
wi
k1


wo
k1


wo
K

where the Nk matrices have as an expression:

X .x Y.X 0 - R cos (X .y) - R cos (Y. y

K
K
K
K
) R sin
0
0
0
0
0
L



X .y Y. y 0
R cos

(X .x)
R cos (Y.X
K
K
K
K
)
0
0
0
0
0
0
L

0
0
1 - R sin (X .x) - R sin (Y.X)
0
0
0
0
0
0
L

NR
K
K


K = 0
0
cos (m
L
)
0
0
sin (m)
0
0
0
0
0
0
0
0
sin (m
L
)
0
0
cos (m)
0
sin () - cos () 0


0
0
0
0
L
cos (m)
0
0
sin (m) cos ()
sin ()
1
144444444444 2
4 4
3
44444444444
m=2, M
The matrix of mass has then as an expression:

L
2 +h/2
M =
T
NR NR

rdxd
D

.
0 0 - H/2
with
NR = (H K Nk) K, 1
= NR.

Note:

In the case of the right pipe, one has X .x
K
= Y. y
K
=1 and X .y
K
= Y.X
K
= 0.
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4.6
Functions of form

One chooses at least quadratic functions of form for the part beam (displacements and
rotations) in order to avoid the phenomena of numerical blocking [bib3]. This choice implies the use of one
finite element with three or four nodes. In the case of an element with 3 nodes, the functions of form are
quadratic, and for an element with 4 nodes, the functions of form will be cubic. For the part
additional, one chooses to take the same functions of form as for the beam part.

The quadratic functions of form (element with 3 nodes) are as follows:

X
X
H1 (X) 2



=
- 1 - 1


L



L

X
X
H2 (X) 2

=
- 1

L L

X X
H3 (X)


= - 4 -
1
L L


The cubic functions of form (element with 4 nodes) are related to Lagrange of command 3:

- - -
H1 (X) (
2) (
3) (
4)
= (1 - 2) (1 - 3) (1 - 4)
- - -
H2 (X)
(
1) (
3) (
4)
= (2 - 1) (2 - 3) (2 - 4)
- - -
H3 (X) (
1) (
2) (
4)
= (

3 - 1) (3 - 2) (3 - 4)
- - -
H 4 (X)
(
1) (
2) (
3)
= (4 - 1) (4 - 2) (4 - 3)
- 1 1

4.7 Integration
numerical

Integration is done by the method of Gauss along average fiber, the method of Simpson
in the thickness and on the circumference. For the integration of Gauss, one uses 3 points of integration
for the elements with 3 nodes, as for the elements with 4 nodes (those under-are thus integrated).
Integration in the thickness is an integration by layers of which the number could be fixed
later on by the user. For each layer one takes 3 points of Simpson, the 2 points
ends being common with the close layers. Thus for N layers one uses 2n+1 points.
a many sectors for integration on the circumference, could also be fixed later on by
the user. Currently the numbers of layers and sectors are fixed at their maximum value: 3
layers (7 points) and 16 sectors (33 points), which gives 693 points of integration on the whole.
The integration of Simpson amounts calculating the sum of the values of the function at the points
integrations (ends and medium of each layer or sector) affected of the weights given
by the table below.

Cordonnées of the points
Weight
- 3/5 = - 0,77459 66692 41483
5/9=0,55555 55555 55556
0
8/9=0,88888 88888 88889
3/5 =0,77459 66692 41483
5/9=0,55555 55555 55556
Integration of Gauss on average fiber
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1
4
2
4
2
2
4
2
4
1










3
3
3
3
3
..... 3 3 3 3 3

Weight of the points of integration for the method of Simpson
Thus for a function F (X,) on a right geometry one a:

L 2 H/2
1 2 H/2

~
F (X,) rdxd = L
D
F x~
(, R X
d~
)
dd
2
0 0 - H/2
- 1 0 - H/2

NPG 2NCOU +1 2NSECT +1

= L
H
2

[
~
W
~
K wn wm RN
F (xNPG, NR
, NR
)
COU
SECT
COU
]
2 2NCOU 2N SECT k=1 n=1
m=1


K
W
N
W
m
W being respectively weights of integration over the length, the circumference and
in the thickness, ordered as the two tables show it above.

4.8
Discretization of external work

The variational formulation of external energy for the beam part is written
:
L
W
pext = (F
X U
X + F y U
y + F Z U
Z + mxx + my y + mzz dx
) +
0
[X U
X +y U
y +z U
Z + µxx + µ there y + µzz] 0, L
and for the additional part she is written by taking into account only the loading of
L
pressure: W
sext = R
F dx
W + [R W
] 0, L
0
By taking account of the discretization of displacements, one can write:
NR L
Wp = (F H (X
K
K
K
K
K
ext.
X K) (xk.xux +yk.xu) + F H (X
y
y
K
) (xk.
y ux + yk .yu) + F H (X
y
Z
K

) uz +
K =1 0
m H (X
K
K
K
K
K
X
K
) (xk.
X X + yk.x) + m H (X
y
y
K
) (xk .yx + yk.
y
) + m H (X
y
Z
K

)
) dx
Z
+

[H (X
K

K
K
K
K
K
y HK (X
) uy + Z HK (X
) uz + µx HK (X
) X + µy HK (X
) y + µz HK (X
)
X
K

) ux +
Z] 0, L

NR
L
L
L

= H
K (X) (F X (X .x)
K
+ F y (X .y)
K
) dx+ [xHk (X)]0, L H
K (X) (fx (Y.X)
K
+ F y (Y. y)
K
) dx+ [yHk (X)]0, L fzHk (X) dx+ [µxHk (X)]0, L
K 1
= 0
0
0
L
L
L

H (X)
p


µ
µ
µ

K
(m (X .x)
K
+ m (X .y)
X
y
K
) dx+ [H (X)] H (X)
X
K
0, L
K (m (Y.X)
K
+ m (Y. y)
X
y
K
) dx+ [H (X)] m H (X) dx+ [H (X)]
y
K
0, L
Z K
Z
K
0, L
The U.K.

0
0
0

NR
P
p
P
p
=


K
F The U.K. = F U
K 1
=
and
NR
L


W S =
F H (X O
O


ext.
R
K

) W dx + [H (X
K
R
K

) wk] 0, L


K =1 0

NR
L


=
0 0 0 0 0 0
F H (X D
) X

+ [H (X

S
R
K
R
K
)]0, L the U.K.




K =1
0

NR
S
S
S
S
=
F
kUk = F U
K =1
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Note:

For the extreme nodes of the elbow there are X .x
K
= Y. y
K
=1 and X .y
K
= Y.X
K
= 0. In the case of it
right pipe, one has X .x
K
= Y. y
K
=1 and X .y
K
= Y.X
K
= 0 for all the element.

5
Geometrical characteristics of the pipe section

One has in this chapter some useful results to characterize the element pipe and which are
calculated by the option of calculation MASS_INER of Code_Aster. In the continuation the index D indicates them
results for the right pipe and the index C for the curved pipe.

L
2 H/2
V =
rdxdd = 2 lah
D


·
0 0 - H/2
Volume:
.
L
2 H/2

2 H/2
V =
rdxdd =
[R + (has +) sin (
] +) D dd has = 2 R
ah
C



0 0 - H/2
0 0 - H/2
· Center of gravity: The position of this last is calculated starting from the point medium to both
extreme nodes of the pipe section, in the reference mark associated with the node interns element
(cf [§2.1.1]). In this reference mark, the co-ordinates of the center of gravity are:
X

0

Gd
0
xGc



2



2

1
2
H

y
yGc = - R sin
1
(+
[has +
]) - cos
Gd = 0 and




2
2 2
R
4
2


zGd 0
Z


Gc

0

· Stamp inertia: It is relatively easy to calculate the matrix of inertia in the center of
curvature of the elbow O in the reference mark defined above. To have his expression one uses
then the fact that:
2
I (G) = I (O) - mb
xx
xx
I (G) = I (O)
yy
yy

2
I (G) = I (O) - mb
zz
zz
where B is the distance between the center of gravity and the center of curvature which is worth:
2

1
2
H
B = R
sin
1
(+
[2
has +
]).

2
2 2
R
4
In the case of a right pipe, the concept of center of curvature does not have a direction. O and G are confused
with the node interns element and the medium of the segment joining the two nodes node.
If one notes A the surface of the transverse section and I his inertia compared to the center of the section one can
to write:
C
I
2
I 1 sin
I (O) = R
(+ [AR + 3] [+
])
xx
I D (O) = Li
2
2 2
4
xx
C
I
2
I 1 sin
I D (O) = L
(I/2
2
+ Al/12)
yy
and I
(O) = R
(+ [AR + 3] [-
])
yy
.
2
2 2
4
I D (O) = L (I/2
2
+ Al/)
12
zz
C
I
I (O) = R
(
2
AR + 3)
zz
2
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6 Connection
pipe-pipe

In order to be able to represent a line of piping correctly where the elbows are not
coplanar, it is necessary to choose an origin common Des. Ainsi for two elbows belonging to two
perpendicular plans between them, it is necessary to be able to take into account the fact that displacements in
the plan of the first elbow are equal to displacements except plan of the second in the cross-section of
connection.
· ·
Generating line
2
·
·
· ·
1

Appear 6-a: Représentation of two noncoplanar elbows connected by a right pipe

In [bib12], this common origin is defined by a generating line continues along
piping as indicated above. This generator intersects each transverse section in one
not. The angle between Z defined on [Figure 2.1.1-a] and the line passing by the center of the section
transverse and this point are worth.

6.1
Construction of a particular generator

For a transverse section end of the line of piping, one defines a vector origin z1
unit in the plan of this section. The intersection enters the direction of this vector and surface
average of the elbow determines the trace of the generator on this section. One calls x1, y1, z1 it
direct trihedron associated this section where x1 is the unit vector perpendicular to the section
transverse built with [Figure 2.1.1-a]. For the whole of the other transverse sections, the trihedron
xk, yk, zk are obtained either by rotation of the trihedron xk 1, y
-
K 1, Z
-
K 1
- in the case of bent parts,
maybe by translation of the trihedron xk 1, y
-
K 1, Z
-
K 1
- for the right parts of piping. The intersection
between the transverse section and the line resulting from the center of this section directed by zk is the trace
of a generator represented below.
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Z
Z
B

z2
x2
generator
y2
O
Y
Z
1
Z
x1
With
y1
X

Appear 6.1-a: Représentation of the generator of reference

The origin of the commune to all the elements is defined compared to the trace of this generator
on the transverse section. The angle enters the trace of the generator and the current position on the section
transverse is then called.

6.2
Connection from one element to another

The kinematics of [§3.1] is given in the plan of the elbow. This one is determined by the arc of circle
generated by the axis of the elbow. The origin of the angles is the normal in the plan chosen as with [§2.1].
To define the origin starting from a generator makes it possible to raise the problems of continuity of
displacements of an element to another. Indeed if one postulates that relative displacements of
M
transverse sections are of type I
U p cos p + O
U p sin p where is the angle with the trace of
p=1
generator on the transverse section, the continuity of displacements is automatically assured
from one element to another.
One notes Z the vector perpendicular to the plan of the elbow corresponding at the selected origin of the angles
up to now. It is noticed that vectors Z and zk are in the plan of the section K. is the angle
defined compared to Z. If one introduces the angle counted starting from the trace of the generator on
transverse section (thus compared to zk) one with the following relation: = - K where = (Z,
)
K
Z K
angle between Z and zk in the plan of the transverse section. Thus displacements are from now on
M
type I
U p cos p (- K) + O
U p sin p (- K). It should be noted that for an elbow given the angle K
p=1
whatever the selected transverse section is identical. It is at the time of the passage from one elbow to another
that the value of K changes.

Note:

When piping consists of right elements colinéaires, one chooses arbitrarily
= 0.
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6.3
Numerical establishment

The line of piping is with a grid by right or curved elements to order. The first element
indicate the beginning of the line of piping. One determines for this element the associated trihedron
x1, y1, z1. If this element is right, one chooses
0
1 =
, if not one calculates 1 as indicated in
preceding paragraph. If the first element is right the trihedron associated with the first section
transverse of the second element x2, y2, z2 is obtained by translation of x1, y1, z1. If the first
element is curved, the associated trihedron x2, y2, z2 is obtained by rotation of x1, y1, z1 in the plan
elbow. In this case
0
2
2 =
if the second element is right and = (Z,
)
2
z2 if the second
element is curved where 2
Z is built like Z of [Figure 2.1.1-a]. The continuation of construction
results easily by recurrence from the preceding diagram.

7
Connections hull-pipe and 3D-pipe

7.1 Step
followed

One adopts here a step similar to the cases 3D-beam [R3.03.03], and hull-beam [R3.03.06]: it
acts to characterize the connection between a node end of an element pipe and a group of meshs of
edge of elements of hulls or 3D. This makes it possible to net part of piping (for example one
bend) in hulls or elements 3D, and the remainder in right pipes.


Appear 7.1-a: Liaison between a right grid COQUE_3D and pipes [HI75-98/001]

Thanks to the kinematics introduced into the element pipe, the connections hull - pipe and 3D - pipe must
to allow to net in elements of hulls or 3D only the elbow, without right parts,
since the damping of ovalization (and warping) is taken into account in the element
pipe.

The connection results in relations kinematics between the degrees of freedom of the nodes of S (which
represent the section of connection, modelled by elements of edge of hull or 3D), and it
node NR of pipe.
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So that the connection is effective, it is necessary [R3.03.03] that it checks the following properties:

1) to be able to transmit efforts of beam to the grid hulls or 3D, and to be able to transmit
all degrees of freedom of the element pipe (or duaux efforts of those),
2) not to generate in the elements of hulls or 3D of secondary stresses,
3) not to support the static relations kinematics or conditions ones by report/ratio
with the others,
4) to admit unspecified behaviors and to function in dynamics.

The linear relations will have the same form as in the case hull-beam, with equations
additional specific to the degrees of freedom of the pipe.

One already introduced with [§3.1] space T of the fields associated with a torque (defined by two vectors):

T = {vV/(T,)

that

such

v (M) = T + GM}

where for the fields of displacement of T, T is the translation of the section (or the point G),
infinitesimal rotation and fields v are displacements preserving the section S plane and not
deformation there (One uses still the assumptions of NAVIER-BERNOULLI).
The displacement of the pipe is worth then:

T
p
S
p
S

U = U + U
U T U
,
T
where:





T = v V/v W
. =
0 W

T



S

The step consists in breaking up the field of displacement of hull C
U or the field of
three-dimensional displacement 3D
U in three fields:
C
p
S

U = U + U + U
· a field of displacement following a kinematics of beam p
U (torque),
· a field of local displacement of the section according to the kinematics of pipe S
U (series of
Fourier) defined in [§3.1],
· an additional field
U orthogonal with the two first within the meaning of the scalar product.

Note:

When the decomposition in Fourier series of [§3.1] is infinite one has
U = 0.

To translate the equation above into linear relations, it is shown that the integrals should be calculated
following, for the hull (or the 3D) and the pipe:

· average displacement: C
U dS
S
· average rotation: GM C
U dS
S
GM
· average swelling:
C
U
. dS
S GM
GM
· modes of Fourier
: C
U cos p dS
,
C
U sin p dS
, CPU
cos pdS,
S
S
S
GM
GM
GM
GM
CPU
sin p dS
,
. C
U cos pdS,
. C
U sin p dS
. For
S
GM
S GM
S GM
modes of Fourier one will choose the relations simplest to exploit since some are
redundant (see remark of [§6.7]).
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Note:

One passes easily from the analytical expressions of the connection 3D - pipe to those of the connection
hull - pipe. It is enough to substitute 3D
U with C
U in the whole of the integrals of
connection proposed above. One will thus speak again of this connection only to [§7.8] treating
numerical establishment.

7.2
Kinematics of the pipe.

In the curvilinear base (O, X, y, Z) associated the transverse section of [§2.1] one notes displacements
1
U, u2 and u3 where:

U (R, X,
1
) = U (X)
X
- (X) R cos
y
+ (X) R sin
Z
+ U (X,) + (X,)
U (R, X,
2
) = U (X)
y
+ (X) R cos
X
- v (X,) cos - (
W X,) sin + (X,) cos.
U (R, X,
3
) = U (X)
Z
- (X) R sin
X
+ v (X,) sin - (
W X,) cos - (X,) sin

Once discretized this expression becomes:

NR
U (R, X,) = H (X) (X .x) the U.K.
K
+ H (X) (Y.X) the U.K.
K
- H (X
K
) (X .y)
K
R cos - H (X
K
) (Y. y)
K
R cos + H (X K
) R sin + U (X,) + (X


1
K
X
K
y
K
y
K
y
K
Z
,)
K =1
NR

U (R, X,) = H (X) (X .y) the U.K.
K
K
X R
+ HK X
y R
- v X
- W X
+ X

K
+ H (X) (Y. y) the U.K.
K
+ H (X
2
K
X
K
y
K
) (xk .x)
cos
() (yk .x)
cos
(,) cos
(,) sin
(,) cos
K =1
NR
U (R, X,
3
) = H (X) K
U - H (X) (X
K
K
K
Z
K
X R - HK X
y R + v X
- W X
- X

K .x)
sin
() (yk .x)
sin
(,) sin
(,) cos
(,) sin
K =1

where U (X,), v (X,), (
W X,), (X,) and (X,) are discretized as with [§3.1].

Displacement in a node K of X-coordinate xk end of pipe is written then:
M
U (R, X,) = the U.K.
K
- R
K
cos + R sin + (ui cosm + uo sinm) + (X


1
K
X
y
Z
km
km
K,)
m=2
M
M
U (R, X,) = the U.K.
K
+ R cos - cos (VI sinm + vo cosm) - sin (wi cosm + wo sinm
2
K
y
X
km
km
km
km
)
m=2
m=2
- wi sin2 + wo cos2 - wo
+


k1
k1
K sin
(
xk,) cos

M
M
U (R, X,


3
) = the U.K.
K
- R sin + sin
(VI sin m + vo cosm) - cos
(wi cosm + wo sin m
K
Z
X
km
km
km
km
)
m=2
m=2
- wi cos
1

2 - wo sin
1

2 - wo cos - (X



K
K
K
K,) sin

GM
For the pipe the kinetic vector moment GM U (M) and swelling
.u (M) has for
GM
respective expressions:

- U (X) R sin
Z
+ U (X) R cos
y
+ 2
R (X)
X
- rv (X,)

GM U (M) =
- Ru (R, X,
1
) cos
,


Ru (R, X,
1
) sin
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and:
GM .u (M) = U
- (X) cos
Z
- U (X) sin
y
+ (
W X,).
GM

Note:

The first component of the field of displacement utilizes U (X,) in an isolated way. It
in goes in the same way for the first component of the vector rotation with respect to v (X,) and of
swelling with respect to (
W X,). This remark will be used with [§5.6] to bind the modes of
Fourier with the degrees of freedom of the edges of hull.

7.3
Kinematics of hull

The kinematics of hull of Coils-Kirchhoff or of Naghdi-Mindlin is written in the thickness:

CPU (M) = CPU (Q) + (C
(Q) N
) .y3

·
C
U (Q) constitutes the vector displacement of average surface in Q,
·
C
(Q) constitutes the vector rotation in Q of the normal according to directions' T1 and t2 of the plan
tangent with Q.

x3
y3
N
M
y3
H
Q
Q y
T =e
2
E
1
1
3
E
Section of hull:
2
G
E
S = L × I
1
x1 L: line of the points Q on the average layer
H
H H
I = -, interval describing the thickness.


2 2


This displacement and this rotation are calculated in the total reference mark. It is possible by change
of reference mark to have their expressions in the curvilinear base (O, X, y, Z) of [§2.1] associated the section
transverse of the junction enters the hull and the pipe.

For each node, the program calculates the coefficients of the 9+6 (M-1) linear relations which connect:

· 6 ddl of the node of beam P of the pipe,
· 2+3x2 (M-1) ddl of Fourier of the pipe,
· the ddl of swelling of the pipe,
· with the ddl of all the nodes of the list of the meshs of the edge of hull.

These linear relations will be dualisées, like all the linear relations resulting for example from
key word LIAISON_DDL of AFFE_CHAR_MECA. They are built as for the connection 3D-beam has
to leave the assembly of elementary terms.
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

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Titrate:
Finite elements of right pipe and curve


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:
R3.08.06-B Page
: 44/54

7.4
Calculation of average displacement on the section S

It is a question of calculating the integral C
U dS, where C
U is the displacement of hull (comprising 6 ddl by
S
node), S is the edge of hull.

Average displacement on the section S is written:

CPU (
/2
M) dS =
H
H CPU Q DLL
C
Q N
y Dy DLL
S
() +
L
(())

L

3
3

- H/2


that is to say C
U (M) dS = H C
U Q DLL.
S
()
L
In addition one also has for the pipe part:
K
U
K
U
X
X
CPU (M) dS =
p
S
[U M
U M dS
]
U dS
S U
.
S

() + () = K
y
= K
S

y
S K
U

K
U
Z
Z

One establishes as well as the average displacement of the section of the pipe to the node K is equal to
displacement of beam of the node K. One can thus linearly bind the degrees of freedom of beam of
translation with the node K with the average of the degrees of freedom of displacement of the edge of the hull.

One neglects in this expression the variation of metric in the thickness of the hull.

7.5
Calculation of the average rotation of the section S

GM CPU (
/2
M) dS =
H
GQ y N Q
u.a. Q
C
Q N Q y dldy
S
L (+ 3 () (() + () () 3)
- H/2
3
=
/2
hGQ CPU (Q) DLL +
H
GQ
C
Q N Q DLL
y Dy

L

(() ())
L
- H/2 3 3
H
+ N (
/2
Q) CPU (Q)

H
y Dy DLL
N Q
C
Q
N Q 2 y2 Dy DLL
.
L

3
3

+


- H/2
() (() ())
L


3
3
- H2

3
C
C
H
that is to say GMu (M) dS
.
= hGQu (Q) DLL + C
(Q) DLL.
S
L
L
12

In addition one also has for the pipe part:

r2 K


K


X

X
GM CPU

(M) dS = GM
2
2
K
K
[
]
cos
.
S

up (M) + custom (M) ds =
S
R
y dS = I y
S r2
2
K
K
sin

Z



Z

where I is the tensor of inertia of the beam. One establishes as well as the average rotation of the section of the pipe
with the node K is equal to the rotation of beam to the node K. One can thus bind the degrees linearly of
freedom of beam of rotation to the node K with the degrees of freedom of rotation of the edge of the hull.

One neglects in this expression the variation of metric in the thickness of the hull.
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
Version
6.4
Titrate:
Finite elements of right pipe and curve


Date:
12/12/03
Author (S):
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:
R3.08.06-B Page
: 45/54

7.6
Calculation of the average swelling of the section S

C
C GM
GM
GQ
It is a question of calculating the integral U N
. dS = U.
dS, where N =
=
is the normal with
S
S
GM
GM
GQ
surface average hull.

Average displacement on the section S is written:


/2
CPU (M) N
. dS =
H
H CPU Q N
. DLL
C
Q N N
.
y Dy DLL
H CPU Q N
. DLL.
S
() +
L
(())

L

3
3

=
- H/2
()


L

In addition one also has for the pipe part:

GM C
GM
.u (M) dS
p
=
.[U (M)
S
+ U (M) dS
]
= wodS
K
S
S
GM
GM
S

One establishes as well as the average swelling of the section of the pipe to the node K is equal to the degree of
freedom of swelling of the pipe to the node K. One can thus linearly bind the degree of freedom of
swelling of pipe to the node K with the degrees of freedom of displacement of the edge of the hull.

One neglects in this expression the variation of metric in the thickness of the hull.

7.7
Calculation of the coefficients of Fourier on the section S

GM
It is a question of calculating the six integrals C
U cos p dS
, C
U sin p dS
, CPU
cos pdS,
S
S
S
GM
GM
GM
GM
CPU
sin p dS
,
. C
U cos pdS and
. C
U sin p dS
, where C
U is it
S
GM
S GM
S GM
displacement of hull (comprising 6 ddl by node), S is the edge of hull.

One with the following relation for displacements on the section S:

CPU (
/2
M) cos p dS
=
H
H CPU Q cos p DLL

C Q N
y Dy cos p DLL
3
3

S
()
+
L
(())

L




- H/2


that is to say C
U (M) cos p dS
= H C
U Q cos p DLL
and C
U (M) sin p dS
= H C
U Q sin p DLL
.
S
()
S
()
L
L

In addition one also has for the pipe part:
U (R, X
1
K,)


C
U (M) cos p dS
= U (R, X,
2
) cos p dS
K
.
S
S U (R, X

3
K,)

The first component of this relation then enables us to connect the coefficient linearly of
Fourier I
ukp with the components of displacements of the edge of hull in the following way:
- R K
cos2 dS


if p 1
=
y
C

H U (Q) cos
S
1
p DLL
= U (R, X,) cos
1
p dS
=




S
K
L
ui cos2 p dS


if p
kp


1

S
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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Titrate:
Finite elements of right pipe and curve


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:
R3.08.06-B Page
: 46/54

U (R, X
1
K,)


Of the same C
U (M) sin p dS
= U (R, X,
2
) sin p dS
K
from where one deduces from it that:
S
S U (R, X

3
K,)
R K
sin2 dS

if p

1
=
Z
C

H U (Q) sin
S
1
p DLL
= U (R, X,) sin
1
p dS
=




S
K
L
uo sin 2 p dS

if p

kp


1

S

One with the following relation for rotations on the section S:

+h/2
CPU (
GM
C
GQ
M).
cos p dS
= H U (Q)
cos p DLL
+ (C (Q) N) N (y Dy) cos p
3
3
DLL

.
S
L
L
GM
GQ
- H/2

The first component of this relation then enables us to connect the coefficient linearly of
Fourier O
vkp with the components of displacements and rotations of the edge of hull in the manner
following:
- [K
Ru cos2
y
+

O
rw cos2] dS if p
1
k1
=
GQ
H [CPU (Q)


] cos
S
1
p DLL
=

L
GQ

O
rv cos2 (p) dS
if p

kp


1

S
In the same way one a:
[K
Ru sin 2
Z
+

I
rw sin 2] dS

if p 1
k1
=
GQ
H [CPU (Q)


] sin
S
1
p DLL
=

L
GQ

I
rv sin 2 (p) dS

if p
kp


1

S
One with the following relation for swelling on the section S:

GM
GQ
. C
U (M) .cos p dS
= H
. C
U Q cos p DLL
.
S

()
L
GM
GQ

This relation enables us to connect linearly the coefficient of Fourier ikp
W with the components of
displacements of the edge of hull in the following way:

[- K
U cos2
Z
+

I
W cos2] dS if p
1
k1
=
GQ
H
. C
U (Q)

cos p DLL
= S
.
L GQ

I
W cos2 (p) dS
if p

kp


1

S
In the same way, one a:
- K
U sin 2
y
+

O
W sin 2 dS


if p 1
k1
=
GQ
H
. C
U (Q)

sin p DLL
= S

L GQ

O
W sin 2 (p) dS

if p
kp


1

S
One uses for all these relations the fact that cos p cosq dS
= 0
if p Q.
S
One neglects in this expression the variation of metric in the thickness of the hull.
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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Finite elements of right pipe and curve


Date:
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Author (S):
P. MASSIN, J.M. PROIX, A. Key BEN HAJ YEDDER
:
R3.08.06-B Page
: 47/54

Note:

One can note that some of the relations established in this paragraph for p=1 are
redundant with those established in the paragraphs [§7.4] and [§7.5]. On the six relations
established starting from the calculation of the integral forms
CPU cosdS

,
CPU sindS

,
S
S
C
GM
C
GM
GM
GM
U

cos dS,
U

sindS,
CPU
.
cos dS

and
CPU
.
sindS

, only
S
GM
S
GM
S GM
S GM
two among the four last are linearly independent of the others. Thus both
first were already established with [the §7.4] and of the combinations of the four last give again
those of [§7.5].

7.8
Establishment of the method

The calculation of the coefficients of the linear relations is done in three times:

· calculation of elementary quantities on the elements of the list of the meshs of edges of hulls
(mesh of type SEG2):
- surface
=
1;
X;
y;
Z.
elt
elt
elt
elt
-
summation of these quantities on (S) from where the calculation of:
-
WITH = S
- position
of
G.
· knowing G, elementary calculation on the elements of the list of the meshs of edges of
hulls of:
GM = {X, y, Z}
Ni;
xNi;
yNi;
zNi
:

where
elt
elt
elt
elt
Nor =

of

form

of

functions
element

It should be noticed that in the case of the connection hull - pipe, the integrals on the elements of edge
are to be multiplied by the thickness of the hull: NR H NR where L represents average fiber of
I =
I
elt
L
h3
the element of edge of hull. Moreover, one adds the additional term:
NR.
I
12 L
· “assembly” of the terms calculated above to obtain of each node of the section
of connection, coefficients of the terms of the linear relations,
· connection between the modes of Fourier and displacements of hull as shown at the beginning
[§7].

More precisely:

· for the connection hull - pipe, one carries out elementary calculations on all the elements of
edge of the section of connection S of the type:

u.a.
C
N
U
2
2 X
Nb

elements S 2
X
U
U
P

cm = 1

C cos (
1
C
1
m) D =
U cos

C


(m) D =
cos (m) U D
y


C

N
0
0
=

U
1
N
C
1


U
R
Z
and
u.a.
C
N
U
2
2 X
Nb

elements S 2
X
U
U
P

Sm = 1

C sin (
1
C
1
m) D =
U sin

C


(m) D =
sin (m) U D
y


C

N
0
0
=

U
1
N

C
1

U
R
Z
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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Titrate:
Finite elements of right pipe and curve


Date:
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Author (S):
P. MASSIN, J.M. PROIX, A. Key BEN HAJ YEDDER
:
R3.08.06-B Page
: 48/54

if m > 1 where P is the matrix of passage of the local reference mark of the element to the total reference mark and
1
position orthoradiale of the element. By expressing displacement according to the degrees of freedom
nodal:

I
U
N

N
U
m
Nbél

éments S 2
Nb _ nodes
X
U

cm =
1

O
v
cos m P
NR U
D
m =
() N () N


I
y
n=1
N
n=
W
1
1



N
U
m
Z
and
O
U
N

N
U
m
Nb

elements S 2
Nb _ nodes
X
U

Sm =
1

I
v
sin m P
NR U
D
m =
() N () N


O
y
n=1
N
n=
W
1
1


N
U
m
Z

where the NR are related to form of the element, one obtains for each calculation twice
9 coefficients with the nodes of the element running of S:

I
U
has
has
has
U
m
11
12
13

N


Nb _ nodes
X
O

ucm = v
has
has
has
U
.
m =

21 22 23 N
I
y


elements S
n=1


W
has
has
has



N
U
m
31
32
33

Z


2
L
1
1
N
has
cos m P NR D
cos MX P X NR X dx
ij =
() ij () N () = () ij () N ()

R


0
1

and an equivalent expression for U
where I is the length of the element of edge of hull.
Sm

· for the connection 3D - pipe, one carries out elementary calculations on all the elements of
edge of the section of connection S of the type:

U 3D
X
U
= 2 U 3D
cm
(
2
cos m) dS = U 3D cos

(m) rd D
S
S
S
S U 3D
R

3
N N

D
U N
Nb

elements S H
2 2
Nb _
X
nodes

= 2
cos (m) P NR (,) U3 rdd
N

D
y
S
n=1
N N
n=
H
1
3D
1 1
U Z
and
U 3D
X
usm = 2 3D
2
u.a. sin (m) dS = U 3D sin

(m) rd D
S
S
S
S U 3D
R

3
N N

D
U N
Nb

elements S H
2 2
Nb _
X
nodes

= 2
sin (m) P NR (,) U3 rdd
N

D
y
S
n=1
N N
n=
H
1
3D
1 1
U Z
if m > 1 where P is the matrix of passage of the local reference mark of the element to the total reference mark, the position
1
orthoradiale of the element, H its position radial and the NR are related to form of the element.
1
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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Titrate:
Finite elements of right pipe and curve


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:
R3.08.06-B Page
: 49/54

8
Establishment of element TUYAU in Code_Aster

8.1 Description

This new element (of name METUSEG3) is pressed on a mesh SEG3 or curvilinear SEG4. It supposes
that the section of the pipe is circular. Contrary to elements POU_D_E, POU_D_T, [R3.08.01] this
element is not <<exact>> with the nodes for loadings or torques concentrated with
ends. It is thus necessary to net with several elements to obtain correct results.

8.2
Introduced use and developments

The element is used in the following way:

AFFE_MODELE (MODELING = “TUYAU_3M'…)

The meshs with 4 nodes are generated starting from the meshs with 3 nodes using:

MALL =CREA_MAILLAGE (MAILLAGE=MAIL,
MODI_MAILLE=_F (OPTION = “SEG3_4”, ALL = “YES”)
)

One calls upon routine INI090 for the functions of form, their derivative and their derivative
seconds (for the hull part) at the points of Gauss, as well as the corresponding weights.

The characteristics of the section are defined in AFFE_CARA_ELEM

AFFE_CARA_ELEM (BEAM = _F (SECTION = “CIRCLE”,
CARA = (“R” “EP”),
VALE = (.......),),
ORIENTATION=_F (GROUP_NO=D, CARA=' GENE_TUYAU', VALE= (X Y Z),),

TUYAU_NCOU = ' A NUMBER OF COUCHES', TUYAU_NSEC = ' A NUMBER OF SECTEURS',),
)

R and EP represent, as for the elements of beams traditional, respectively the radius
external and the thickness of the section. One also defines on one of the nodes end of the line of
piping the vector whose projection on the transverse section is the origin of the angles for
decomposition in Fourier series. This vector should not be colinéaire with the average line of the elbow
with the node end considered.
One also defines in this level the number of layers and angular sectors to use for integration
numerical.

AFFE_CHAR_MECA (DDL_IMPO = _F (
DX =., DY =., DZ =., DRX =., DRY =., DRZ =., DDL of beam
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 W

FORCE_NODALE = _F (FX =., FY =., FZ =., MX =., MY =., MZ =. )

They are the conventional forces of beam, which work only on displacements of beam.

Addition of a key word in AFFE_CHAR_MECA (FORCE_TUYAU = _F (PRES =. )) for the calculation of
work of the internal pressure.

The pressure works on the DDL of swelling WO, one calculates then:

NR
L
2
L
NR
2
L

O
O

2
O
W
=
pw R ddx =
p
H W R ddx =



0 0
int
0 0
int
H
Pr D dx W
near
K
K
K
0
K

0
int


K =1
K =1
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

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Finite elements of right pipe and curve


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:
R3.08.06-B Page
: 50/54

8.3
Calculation in linear elasticity

The matrix of rigidity and the matrix of mass (respectively options RIGI_MECA and MASS_MECA)
are integrated numerically in the TE0582. Calculation holds account owing to the fact that the terms
corresponding to the DDL of beam are expressed classically in total reference mark, and that DDL of
Fourier are in the local reference mark with the element. In the case or the element does not belong to any elbow, it
locate local is defined by the generator and the directing vector carried by average fiber of the element
as indicated on [Figure 8.3-a]. If the element belongs to an elbow, the local reference mark
is defined starting from the plan of the elbow as mentioned with [§2.1].
Z
y
Z
X
Z
O
Y
generator
X

Appear 8.3-a: local Repère for a right pipe

Elementary calculations (CALC_ELEM) currently available correspond to the options:

· EPSI_ELGA_DEPL and SIEF_ELGA_DEPL which provide the strains and the stresses
at the points of integration in the local reference mark of the element. Calculation is carried out in the TE0584,
and currently gives the values to the 693 points of integration (for an element to 3 modes of
Fourier). These fields are called fields at “under-points” of integration. One stores these
values in the following way:
-
for each point of Gauss in the length, (n=1, 3)
-
for each point of integration in the thickness, (n=1, 2NCOU+1=7)
-
for each point of integration on the circumference, (n=1, 2NSECT+1=33)
-
6 components of strain or stresses:
EPXX EPYY EPZZ EPXY EPXZ EPYZ or SIXX SIYY SIZZ SIXY SIXZ SIYZ
where X indicates the direction given by the two nodes nodes of
the element, Y represents the angle describing the circumference and Z represents
the radius. EPZZ and EPYZ corresponding to,
in the case of them
rr
R
deformations and SIZZ and SIYZ corresponding to,
in the case of them
rr R
constraints are taken equal to zero.

· EFGE_ELNO_DEPL: who gives the efforts generalize of beam traditional: NR, VY, VZ,
MT, MFY, MFZ. These efforts are given in the local curvilinear reference mark of the element. This
option is calculated in the TE0585.
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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Finite elements of right pipe and curve


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:
R3.08.06-B Page
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· SIGM_ELNO_TUYO, and VARI_ELNO_TUYO allow, starting from fields SIEF_ELGA or
VARI_ELGA, and of the data of an angular position and a position in the thickness, of
to calculate the constraints or variables intern with the nodes of the elements, with this position. One
then obtains a field without under-points, i.e. a traditional field, usable for
operations of postprocessing (for example the statement of a component of constraints length
one line of piping).

· Options EQUI_ELGA_SIGM and EQUI_ELGA_EPSI allow the calculation of the invariants, in
each point of integration (fields at “under-points”)

· Option VALE_NCOU_MAXI makes it possible to extract, of each of the 3 linear points of Gauss
of an element, the values maximum and minimum of a component of a field.

· In version 7, one can extract from other fields in a point of the section: EPSI_ELNO_TUYO
for the deformations, SIEQ_ELNO_TUYO for the equivalent constraints,
EPEQ_ELNO_TUYO for the equivalent deformations.

Finally the TE0585 calculates also option FORC_NODA for operator CALC_NO.

8.4
Plastic design

The matrix of tangent rigidity (options RIGI_MECA_TANG and FULL_MECA) as well as projection
plastic (options FULL_MECA and RAPH_MECA) are integrated numerically in the TE0586. One makes
call to the option of calculation STAT_NON_LINE. All laws of plane constraints 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 assumption of
plane constraints using the method of Borst:

STAT_NON_LINE (…

COMP_INCR = _F (RELATION = ' “, ALGO_C_PLAN=' DEBORST”),
.....)

Elementary calculations (CALC_ELEM) currently available correspond to the options:

· SIEF_ELNO_ELGA which makes it possible to obtain the efforts generalized by element with the nodes in
locate beam. This option is calculated in the TE0587.
· VARI_ELNO_ELGA which calculates the field of internal variables by element with the nodes for
all layers and all sectors, in the local reference mark of the element. This option is
calculated in the TE0587.

8.5
Test: SSLL106A

It is about a right pipe of directing vector (4, 3, 0) fixed in its end O and which is with a grid with 18
elements TUYAU.

y
B
X
Y
3
L
Z
O
X
O
4
L = 5
Z

Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
Version
6.4
Titrate:
Finite elements of right pipe and curve


Date:
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Author (S):
P. MASSIN, J.M. PROIX, A. Key BEN HAJ YEDDER
:
R3.08.06-B Page
: 52/54

The pipe is subjected to different types of load:

· a tractive effort,
· 2 sharp efforts,
· 2 bending moments,
· 1 torque,
· an internal pressure.

Displacements at the point B, the strain and the stresses in certain points are calculated
of integration of the section containing B, as well as the first clean modes.
This makes it possible to test the DDL of beam, the DDL of swelling and modes 1 of the development in
Fourier series.

9 Conclusion

The finite elements of elbow which we describe here are usable for calculations of piping in
elasticity or in plasticity. Pipings can be subjected to various combined loadings -
internal pressure, cross-bendings and anti-plane, torsion, extension.
For the moment, the element carried out is a linear element of beam type, right or curve, to three
nodes, in small rotations and deformations, with a local elastoplastic behavior in
plane constraints. It makes it possible to take into account ovalization, warping and swelling. It
combine the properties of hulls and beams. The kinematics of beam for the axis of the elbow is
increased by a kinematics of hull, of type Coils-Kirchhoff without transverse shearing, for
description of the behavior of the transverse sections. This last kinematics is discretized in
M modes of Fourier, of which the number M, which the literature encourages us to choose equal to 6 [bib8], [bib13],
must at the same time be sufficient to obtain good results in plasticity and not too large to limit it
calculating time. In elasticity, for relatively thick pipings (the thickness report/ratio on
radius of the transverse section higher than 0.1), one can be satisfied with M=2 or 3.

Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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Titrate:
Finite elements of right pipe and curve


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:
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10 Bibliography

[1]
ABAQUS: “Theory manual” Hibbit, Karlsson and Sorensen Inc. (1984) chapter 4.4.1
[2]
K.J. BATHE and C.A. ALMEIDA: “A simple and effective pipe elbow element-linear analysis”,
Newspaper off Applied Mechanics, Vol.47 (1980) pp93-100.
[3]
J.L. BATOZ, G. DHATT: “Modeling of the structures by finite elements - Coques”, Vol.3,
Hermès, Paris, 1992.
[4]
W.B. BICKFORD, B.T. STROM: “Vibration off planes curved beams”, Journal off Sound and
Vibration (1975) 39 (2), pp.135-146.
[5]
J.T. BOYLE and J. SPENCE: “Inelastic analysis methods for piping systems”, Nuclear
Engineering and Design 57 (1980) 369-390.
[6]
J.T. BOYLE and J. SPENCE: “A state off the art review off inelastic (static & dynamic) piping
analysis methods, with particular application to LMFBR ", Glasgow, Scotland, the U.K. (1983).
[7]
P. GUIHOT: “Mechanical Behavior Accidentel of Circuit Primaire”, Note EDF
HP-52/96/012/A.
[8]
H.D. HIBBIT: “Special structural elements for piping analysis”, Proc. Conf., Presses Vessels
and Piping: Analysis and Computers, Miami, ASME, 1974.
[9]
H.D. HIBBIT & E.K. LEUNG: “Year approach to detailed inelastic analysis off thin-walled
pipelines ", ASME Special Pub., “Nonlinear Finite Element Analysis off Shells” (1981).
[10]
A. KANARACHOS, R.N. KOUTSIDES: “A new Approach off Shell displacements in A
Beam-Type pipe element “, Proc. 8th Int. Conf. SMIRT, paper B9/2, Brussels (1985).
[11]
P. MASSIN, A. BEN HAJ YEDDER: “Bibliography for the development of an element
bend with internal pressure and local plasticity ", Note HI-74/97/026, EDF-DER 1998.
[12]
A. MILLARD: “Year enriched beam finite element for accurate piping analysis”, Proc. 14th Int.
Conf. SMIRT, Lyon (1997).
[13]
H. OHTSUBO, O. WATANABÉ: “Stress analysis off pipe bends by boxing ring elements”, Trans.
ASME, Journal off Pressure Vessel technology, Vol.100 (1978) 112-122.
[14]
K. WASHIZU: “Variational Methods in Elasticity & Plasticity”, 3rd ED. Pergamon (1981).
[15]
J. PELLET: “Connection 3D-beam”, Documentation de Référence of Code_Aster [R3.03.03].
[16]
J.M. PROIX
: “Connection hull-beam”, Documentation de Référence of Code_Aster
[R3.03.06].
[17]
F. VOLDOIRE, C. SEVIN: “Axisymmetric thermoelastic Hulls and 1D”, Documentation
of Référence of Code_Aster [R3.07.01].
[18]
J.M. PROIX, P. MIALON, m.t. BOURDEIX: “Exact Elements of right and curved beams”,
Reference material of Code_Aster [R3.08.01].
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
Version
6.4
Titrate:
Finite elements of right pipe and curve


Date:
12/12/03
Author (S):
P. MASSIN, J.M. PROIX, A. Key BEN HAJ YEDDER
:
R3.08.06-B Page
: 54/54

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Handbook of Référence
R3.08 booklet: Machine elements with average fiber
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