Code_Aster
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Titrate:
Conditions of solid connection of body
Date:
12/02/01
Author (S):
J. PELLET
Key:
R3.03.02-A
Page:
1/12
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/01/001/A
Organization (S):
EDF/MTI/MN
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
Document: R3.03.02
Conditions of solid connection of body
Summary
One presents in this documentation a manner of modelizing indeformable parts of structure, in
small displacements and rotations, thanks to the key word
LIAISON_SOLIDE
of
AFFE_CHAR_MECA
.
Code_Aster
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Titrate:
Conditions of solid connection of body
Date:
12/02/01
Author (S):
J. PELLET
Key:
R3.03.02-A
Page:
2/12
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/01/001/A
Contents
1 Introduction ............................................................................................................................................ 3
2 Principle of the use of the key word .......................................................................................................... 3
3 Which are the treated cases of figure?..................................................................................................... 4
4 Processing of the cases 2DA and 3DA ............................................................................................................ 5
4.1 Case 2DA .......................................................................................................................................... 5
4.2 Case 3DA .......................................................................................................................................... 5
5 Processing of the case 2DB .......................................................................................................................... 6
5.1 General case ..................................................................................................................................... 6
5.2 Particular cases ................................................................................................................................ 6
6 Processing of the case 3DB .......................................................................................................................... 7
6.1 General case ..................................................................................................................................... 7
6.1.1 Processing of the points A, B, C and determination of the vector rotation
.................................... 7
6.1.2 Relations concerning a point M
(A, B, C) .......................................................................... 8
6.1.3 Summary of the equation to write ............................................................................................... 9
6.2 Particular cases ................................................................................................................................ 9
7 How to detect the particular cases?.............................................................................................. 11
Code_Aster
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Version
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Titrate:
Conditions of solid connection of body
Date:
12/02/01
Author (S):
J. PELLET
Key:
R3.03.02-A
Page:
3/12
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/01/001/A
1 Introduction
The key word
LIAISON_SOLIDE
controls
AFFE_CHAR_MECA
and (
AFFE_CHAR_MECA_F
) allows
to modelize an indeformable part of a structure.
The principle selected is to write linear relations between the ddls “solid” part; these relations
expressing the fact that the distances between the nodes are invariable.
Important remark:
The relations expressing the indeformability of a solid are in general not linear.
linearization of the problem supposes that the problem can be solved in theory of “small
displacements ". To be convinced some, let us take the example of a segment AB in 2D:
With
B
The indeformability of AB is written:
(
) (
)
[
]
(
) (
)
[
]
(
) (
)
(
) (
) (
) (
) (
) (
)
X
dx
X
dx
y
Dy
y
Dy
X
X
y
y
dx
dx
X
X
dx
dx
Dy
Dy
y
y
Dy
Dy
X there X y
dx Dy dx Dy
With
With
B
B
With
With
B
B
With
B
With
B
B
With
B
With
B
With
B
With
B
With
B
With
With
With
B
B
With
With
B
B
+
-
+
+
+
-
+
=
-
+
-
-
+
-
-
+
-
+
-
-
=
2
2
2
2
2
2
2
2
0
while noting
co-ordinates of A and B
displacements of A and B
,
,
,
,
,
,
it is seen that the expression is quadratic in
dx dx Dy
Dy
With
B
With
B
,
,
and
. To be able to linearize it,
it is necessary to eliminate the quadratic terms and for this reason, one is obliged to suppose that them
elements
dx dx Dy
Dy
With
B
With
B
,
,
and
are small compared to the length of AB.
This remark wants to say that one cannot use this key word when the structure becomes deformed (or
turn) too much. In such situations, “to rigidify” a solid part, one is obliged to use one
“hard” material (compared to the remainder of the structure).
2
Principle of the use of the key word
The key word
LIAISON_SOLIDE
is a key word factor répétable at will. With each occurrence of the word
key, the user defines a “piece of model” which it wishes to rigidify.
This “piece of model” defined by the key words
GROUP_MA
,
GROUP_NO
,
NET
and
NODE
, one
deduced the list from the nodes to be rigidified.
Once this drawn up list, one writes the linear relations necessary to express that the “piece
rigid " has nothing any more but the degrees of freedom of a solid (3 in 2D or 6 in 3D).
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Conditions of solid connection of body
Date:
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Key:
R3.03.02-A
Page:
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R3.03 booklet: Boundary conditions and loadings
HI-75/01/001/A
3
Which are the treated cases of figure?
According to ddls carried by the nodes of the list of the nodes to rigidify, one places oneself in one of the four
following cases of figure. If one does not find oneself in one of these cases of figure, the code stops in
fatal error
The cases 2DA and 2DB correspond to plane” or axisymmetric problems “.
The cases 3DA and 3DB correspond to 3D problems.
Case 2DA:
All the nodes of the list of the nodes to be rigidified carry the ddls DX, DY (and possibly
DRZ) but they do not carry DRX, DRY and DZ and there is at least a node of the list of
nodes to be rigidified which carries DRZ.
Case 2DB:
All the nodes of the list of the nodes to be rigidified carry DX, DY but they do not carry DRX,
DRY and DZ.
Case 3DA:
All the nodes of the list of the nodes to be rigidified carry DX, DY, DZ (and possibly DRX,
DRY, DRZ) and there exist a node of the list of the nodes to be rigidified which carries DRX, DRY, DRZ.
Case 3DB:
All the nodes of the list of the nodes to be rigidified carry DX, DY, DZ and there is not node
list of the nodes to be rigidified carrying at the same time DRX, DRY, DRZ.
Code_Aster
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Titrate:
Conditions of solid connection of body
Date:
12/02/01
Author (S):
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Key:
R3.03.02-A
Page:
5/12
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/01/001/A
4
Processing of the cases 2DA and 3DA
In these 2 cases of figure, one could find a node of the list of the nodes to be rigidified which carried all
degrees of freedom of the solid. That is to say this node.
in 2D: DX, DY, DRZ
in 3D: DX, DY, DZ, DRX, DRY, DRZ
That is to say a node M of the list of the nodes to be rigidified unspecified.
In theory of small displacements, the movement of a solid body is expressed by:
U
U
AM
U
M
With
With
=
+
where
is of displacement of A
the vector rotation of the solid
4.1 Case
2DA
The linear relations are written:
()
()
()
()
()
()
()
()
-
+
=
=
-
-
=
+
-
=
M
WITH DX M
DX A
y DRZ A
X
y
DY M
DY WITH
X DRZ A
M
DRZ
DRZ M
DRZ A
:
:
0
0
0
with
if carries
AM
4.2 Case
3DA
()
()
()
()
()
()
()
()
()
()
()
()
()
()
()
()
()
()
-
-
+
=
-
-
+
=
-
-
+
=
+
-
=
-
=
-
=
M
WITH DX M
DX A
DRY A Z DRZ A y
DY M
DY WITH
DRZ A X DRX A Z
DZ M
DZ A
DRX A y DRY A X
M
DRX DRY DRZ
DRX M
DRX A
DRY M
DRY A
DRZ M
DRZ A
:
.
.
.
.
.
.
,
,
:
0
0
0
0
0
0
if carries
Code_Aster
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Titrate:
Conditions of solid connection of body
Date:
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Key:
R3.03.02-A
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R3.03 booklet: Boundary conditions and loadings
HI-75/01/001/A
5
Processing of the case 2DB
5.1 Case
General
With
B
and
in the list of the nodes to be rigidified/
(
)
D WITH B
,
0
·
determination
of
:
That is to say
N AB K
=
=
N
N
X
y
(
K
unit vector according to
O
Z
).
(
)
(
)
(
)
U
U
K AB
U
U
AB
U
U
N
K AB N
B
With
B
With
B
With
-
-
-
=
-
-
=
.
.
.
0
0
·
since
AB
0
, one can determine
:
(
)
()
()
()
()
(
)
(
)
=
-
+
-
=
=
=
1
1
K AB N
K AB N
N
N
.
.
.
.
.
.
;
'
'
'
DX B N
DX A N
DY B N
DY B N
N
N
X
X
y
y
X
y
That is to say
·
equations to be written:
-
(
)
U
U
AB
B
With
With
B
-
=
.
(
)
0
1 equation for the 2 points and
-
(
)
=
M
WITH B
X
y
,
:
AM
()
()
()
()
()
()
(
)
()
()
()
()
()
()
(
)
DX M
DX A
y DX B N
DX A N
DY B N
DY With N
DY M
DY WITH
X DX B N
DX A N
DY B N
DY With N
X
X
y
y
X
X
y
y
-
+
-
+
-
=
-
-
-
+
-
=
.
.
.
.
.
.
'
'
'
'
'
'
'
'
0
0
5.2 Case
private individuals
·
list nodes to be rigidified =
{}
With
there is nothing to write
With
tear,
·
list nodes to be rigidified =
{}
With
I
where all them
With
I
the same co-ordinates have.
That is to say
With
O
the first node of the list of the nodes to be rigidified
With
With
I
O
it is necessary to write
()
()
()
()
DX A
DX A
DY WITH
DY WITH
I
I
-
=
-
=
0
0
0
0
Note:
is unspecified
Code_Aster
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Conditions of solid connection of body
Date:
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R3.03 booklet: Boundary conditions and loadings
HI-75/01/001/A
6
Processing of the case 3DB
6.1 Case
General
WITH B C
,
in the list of the nodes to be rigidified such as
ABC
that is to say a triangle of nonnull surface
6.1.1 Processing of the points A, B, C and determination of the vector rotation
With
B
C
B
C
B AB
C AC m AM
N B C
B B N C C N
=
=
=
=
=
=
;
;
'
; '
That is to say
-
=
M
M
With
, U
U
M
éq 6.1-1
·
for the points B and C:
U
U
B
U
U
C
B
With
C
With
-
=
-
=
U
U N
B
B
With
-
=
.
. '
éq 6.1-2
U
U.a.
N
B
With
-
=
.
.
éq 6.1-3
U
U B
B
With
-
=
.
0
éq 6.1-4
U
U N
C
C
With
-
=
.
. '
éq 6.1-5
U
U B
N
C
With
-
= -
.
.
éq 6.1-6
U
U.a.
C
With
-
=
.
0
éq 6.1-7
U
U.a. U
U B
B
With
C
With
-
+
-
=
.
.
0
éq 6.1-8
Of the 6 equations concerning the points B and C,
-
3 are to be written: [éq 6.1-4], [éq 6.1-7] and [éq 6.1-8] (they do not utilize
)
-
3 are used to determine
:
(
)
(
)
(
)
. '
.
. '
.
. '
.
B
U
U
N
C
U
U
N
N
U
U
C
=
-
=
-
=
-
B
With
C
With
B
With
éq
6.1-9
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6.1.2 Relations concerning a point M
(A, B, C)
(
)
(
)
That is to say
the vector
the vector
U
ABCM
With
With
With
B
B
C
M
M
M
U V W U V
W U
V
W
X
y Z
:
,
,
,
,
, .......,
,
,
,
,
,
R
R
12
3
The equation [éq 6.1-9] can be written:
M
MR. U
1
2
.
=
ABCM
with
is invertible bus
is of nonnull surface
and
M
M
1
1
2
0
0
0
0 0 0
0
0
0
0 0 0
0
0
0
0 0 0
=
=
-
-
-
-
-
-
-
-
-
B
B
B
C
C
C
N
N
N
M
ABC
N
N
N
N
N
N
N
N
N
N
N
N
C
C
C
C
C
C
X
y
Z
X
y
Z
X
y
Z
X
y
Z
X
y
Z
X
y
Z
X
y
Z
X
y
Z
X
y
Z
'
'
'
'
'
'
=
-
MR. MR. U
1 1
2
.
ABCM
éq 6.1-10
The equation [éq 6.1-1]
U
U
M
M
With
-
-
=
0
can be written:
MR. U
M
4
3
0
.
.
ABCM
+
=
éq 6.1-11
(
)
with
where
and
M
M
M
3
4
0
0
0
1
1
1
0
0
1
1
0
1
=
-
-
-
=
=
-
-
-
M
M
M
M
M
M
MR. M
M
Z
y
Z
X
y
X
X
y
Z
,
,
MR. U
MR. MR. MR. U
MR. U
4
3
1 1
2
5
0
0
.
.
.
ABCM
ABCM
ABCM
+
=
=
-
(
)
M
WITH B C
,
, the 3 equations should be written corresponding to the 3 lines of the matrix
M
M
M
MR. M
5
4
3
1 1
2
=
+
-
-
Code_Aster
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Conditions of solid connection of body
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R3.03 booklet: Boundary conditions and loadings
HI-75/01/001/A
6.1.3 Summary of the equation to be written
·
calculation
of
B C N B C
, “,”
·
calculation
of
MR. M
1 1
2
-
·
for the points B and C:
U
U B
U
U.a.
U
U.a. U
U B
B
With
C
With
B
With
C
With
-
=
-
=
-
+
-
=
.
.
.
.
0
0
0
·
(
)
M
WITH B C
,
:
- calculation
of
M
M
MR. MR. M
5
4
3
1 1
2
=
+
-
-
writing of the 3 equations corresponding to
M
6.2 Case
private individuals
·
list nodes to be rigidified =
{}
With
there is nothing to write
With
tear,
·
list nodes to be rigidified =
{}
With
I
where all them
With
I
the same co-ordinates have.
That is to say
With
O
the first point of the list of the nodes to be rigidified
With
With
I
O
()
()
()
()
()
()
DX A
DX A
DY WITH
DY WITH
DZ A
DZ A
I
I
I
-
=
-
=
-
=
0
0
0
0
0
0
is unspecified, which does not pose a problem.
·
list nodes to be rigidified =
{}
With
I
where all them
With
I
are aligned (right
).
The solid
{}
With
I
does not have any more whereas 5 movements of possible rigid bodies.
It misses rotation around
.
That is to say:
- two
points
With
B
and
/
AB
0
-
B AB
=
-
N
1
a vector not no orthogonal with
B
(thus with
)
-
N
B N
2
1
=
With
B
N
2
N
1
B
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·
not b:
U
U
B
B
With
-
=
(
)
U
U
B
B
With
-
=
.
0
éq 6.2-1
(
)
(
)
U
U
N
N
B
B
With
-
=
.
.
1
1
éq 6.2-2
(
)
(
)
U
U
N
N
B
B
With
-
=
.
.
2
2
éq 6.2-3
-
the equation [éq 6.2-1] is to be written
-
the equations [éq 6.2-2] and [éq 6.2-3] are used to calculate
The component of
on
B
is unspecified, one does not hold account of it:
(
)
(
)
(
)
(
)
====
1 1
2 2
1
2 1
2
1
2
2
1 2
1
1
2
2
1
1 1
2 2
1
2
0
N
N
N
N
N
N
N B
N
N
N
U
U
N
U
U
N
N
N
N
N
U
+
=
=
=
= -
-
= -
-
=
+
=
=
=
that is to say
That is to say
K
K
K
K
K
K
K
has
B
C
U V W U V
B
With
B
With
ABM
With
With
With
B
.
.
.
;
;
,
,
,
,
(
)
[
]
B
B
M
M
M
ABM
W U
V
W
K
B has
C has
B
has
B C
C
has
C
B
,
,
,
,
:
,
,
=
=
-
=
-
-
-
-
-
-
R
9
1
1
2
2
2
0
0
0
0
MR. U
M
M
M
M
with
·
(
)
(
)
-
-
=
=
=
+
=
=
-
-
-
=
-
-
-
=
=
+
M
WITH B
MR. M
M
M
M
M
M
M
M
K
M
With
X
y
Z
ABM
Z
y
Z
X
y
X
ABM
,
,
,
.
U
U
m
m
AM
MR. U
M
M
M
MR. U
M
M
MR. M
0
0
0
0
0
1
0
0
1
0
1
0
0
1
0
0
1
1
0
1
4
3
3
4
5
5
4
3
1
with
with
for each point
M
, the 3 equations should be written corresponding to the 3 lines of the matrix
M
5
.
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Conditions of solid connection of body
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HI-75/01/001/A
·
summary of the equations to be written:
- calculation
of
B N N
,
,
1
2
K
- calculation
of
M
1
-
for point b:
U
U B
B
With
-
=
.
0
-
(
)
M
WITH B
,
- calculation
of
M
M
MR. M
5
4
3
1
1
=
+
K
.
-
writing of the 3 equations corresponding to
M
5
7
How to detect the particular cases?
In the paragraphs [§6] and [§7], we saw that it could arrive of the particular cases when
certain nodes geometrically were confused or aligned on the same line.
The numerical criteria selected to detect these particular cases are:
·
2 points A and B are confused if:
AB
-
10
6
. DMIN
·
3 points A, B, C are aligned if:
(
)
AB AC
1 2
6
10
/
.
-
DMIN
where:
DMIN
note the length of smallest stops meshs of the mesh.
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Conditions of solid connection of body
Date:
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R3.03.02-A
Page:
12/12
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/01/001/A
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