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Titrate:
Connection 3D - Beam
Date:
19/12/95
Author (S):
J. PELLET
Key:
R3.03.03-A
Page:
1/10
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/95/042/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
Document: R3.03.03
Connection 3D - Beam
Summary:
This document explains the principle retained in Aster to connect a modeling continuous medium 3D and
a modeling beam.
This connection results in 6 linear relations connecting displacements of the whole of nodes “3D”
(3 degrees of freedom per node) dependant with the node of beam with the 6 degrees of freedom of this node.
Code_Aster
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Version
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Titrate:
Connection 3D - Beam
Date:
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Author (S):
J. PELLET
Key:
R3.03.03-A
Page:
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R3.03 booklet: Boundary conditions and loadings
HI-75/95/042/A
Contents
1 Presentation of the document ..................................................................................................................... 3
2 the connection 3D-beam ............................................................................................................................. 3
2.1 Objectives and excluded solutions .......................................................................................................... 3
2.2 Orientation ....................................................................................................................................... 4
2.3 Decomposition of displacement 3D on the interface .......................................................................... 5
2.4 Expression of the static condition of connection ............................................................................... 8
3 Establishment of the method of connection .................................................................................................. 9
4 Which uses can one make this modeling?.................................................................... 10
5 Bibliography ........................................................................................................................................ 10
Code_Aster
®
Version
3
Titrate:
Connection 3D - Beam
Date:
19/12/95
Author (S):
J. PELLET
Key:
R3.03.03-A
Page:
3/10
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/95/042/A
1
Presentation of the document
This document is shown of a note of S. ANDRIEUX [bib1] in the course of publication. We added there
some precise details concerning the establishment of modeling “connection 3D/Beam”. Order
AFFE_CHAR_MECA
[U4.25.01] key words
LIAISON_ELEM
and
OPTION
:
“3d_POU”
.
2
The connection 3D-beam
2.1
Objectives and excluded solutions
When one wishes to finely analyze part of a slim structure complexes [Figure 2.1-a], one
can, to minimize the size of the mesh to be handled, to want to represent the structure by a beam
“far” from the part being analyzed. The goal of schematization by a beam is to bring conditions
with the realistic limits at the edges of the part modelized and with a grid in continuous medium 3D. The connection
3D-beam must thus meet the following requirements:
P1
To be able to transmit the efforts of beam (torque) to the mesh 3D
P2
Not to generate “parasitic” stresses (even of stress concentration), because it
would then be necessary to place the connection far from the area to be sufficiently analyzed so that these
disturbances are attenuated in the area of study.
P3
Not to support the conditions kinematics or the static conditions of connection one by
report/ratio with the other. It must be equivalent to bring back a torque of effort or displacement to
limits of the field 3D.
P4
To admit unspecified behaviors on both sides of the connection (elasticity,
plasticity…) and to also allow a dynamic analysis.
modeling 3D
fissure
modeling beam
modeling beam
Appear 2.1-a
If these objectives are achieved one will be able to also use the rules of connection to deal with the problem
embedding of a beam in a solid mass 3D. However the distribution of the stresses in
solid mass around embedding will remain rather coarse and will have to be used with precaution. It is
preferable to net the connection in 3D then to prolong the starter of the mesh 3D of the section of
beam by one of the elements of beam with connection 3D/Beam [Figure 2.1-b].
Code_Aster
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Titrate:
Connection 3D - Beam
Date:
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Key:
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modeling beam
modeling 3D
Appear 2.1-b
Within sight of objectives 1 to 4, one can eliminate two current techniques of connection right now:
1) the first which brings back all the connection to the processing of conditions of connections between the points
in opposite with the intersection of the neutral axis of the beam and the solid 3D. Except the difficulty of
to correctly define the “specific” rotation of the material point pertaining to the solid 3D, one
concentrate the efforts (concentrated reaction, couple) in this point and one breaks symmetry
static kinematics/by privileging a particular kinematics.
2) the second solution which imposes completely a displacement of beam (NAVIER-BERNOULLI)
at the points of the solid mass 3D being with the intersection of the solid 3D and the section of the beam.
In elasticity, one knows that the assumption of indeformability of the sections in their plan is only one
approximation. Correct from the energy point of view for the beam, it leads to
stress concentrations in the vicinity of the limits of the section of junction for the solid
3D.
Note:
It goes without saying that all that is presented here is valid only on the assumption of
small disturbances (small displacements).
2.2 Orientation
We will leave the machine elements of the connection:
·
the field of vector forced
.n
defined on the trace of the section
S
beam on
solid mass 3D,
N
being the normal in the plan of
S
,
·
and the field of displacement
U
3D
defined on this same field,
for the three-dimensional solid, like:
·
torque
T
efforts of beam in the geometrical center of inertia
G
of
S
,
·
and the torque
D
displacements of beam in this same point,
for the beam.
These mechanical magnitudes are connected by:
·
conditions of kinematic continuity,
·
equilibrium conditions of the connection.
Code_Aster
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Titrate:
Connection 3D - Beam
Date:
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Author (S):
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Key:
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The first conditions are the conditions of connections to impose in an approach “displacement”,
the seconds result from the weak formulation of balance via the virtual work of the actions from
contact between beam and solid mass (which is not other than the expression of the “principle” of the action and the reaction
writing for the interface
S
). On surface
S
, one has indeed for any virtual displacement
(
)
v,
T
licit:
.
.
N v dS
S
=
+
F T M
.
éq 2.2-1
where:
·
T
and
are respectively the translation and the infinitesimal rotation of the beam:
(
)
D
=
T
,
·
F
M
and
are respectively the resultant and the moment in the beam at the point of connection:
(
)
T
=
F M
,
The first member of this equality will provide the scalar product thanks to which one will define
“component beam” of a field of displacement 3D defined on
S
. By using this scalar product, one
the symmetry of the approach between conditions kinematics and statics of connection (P3) will ensure like
the possibility of treating unspecified behaviors on both sides of the connection (P4) since none
aspect of behavior does not appear in the equality of balance used.
The step:
One will break up the field of displacement 3D into a part “beam” and a part
“complementary”. This will lead us to rather naturally define the conditions of connection
kinematics between beam and solid 3D like the equality of the displacement (torque) of beam and of
beam part of the field of displacement 3D [(§ 2.3)]. Once this made, the equality [éq 2.2-1] us
will allow to interpret in static term the conditions of connection and to thus reach the conditions of
static connection [(§2.4)].
2.3
Decomposition of displacement 3D on the interface
The junction enters the three-dimensional solid B and the beam of section
S
is supposed to be plane and of
normal
N
parallel with the tangent
with the beam at the point of contact
G
, geometrical center of inertia
section
S
[Figure 2.3-a].
G
S
N
=
(A) Normal with the solid = tangent with the beam
G
N
(b) Normal with the solid ° tangent with the beam
Appear 2.3-a
One thus excludes the case (b) where the beam “does not leave” by perpendicular to surface the solid. It is necessary
to understand well that this restriction is necessary to the coherence of the connection such as it is considered here
since the theory of the beams knows only cuts normal with average fiber: the condition
of balance [éq 2.2-1] no direction has if
S
is not the cross-section of the beam. If this
condition is violated, one will be able to modify the mesh to carry out it as the diagram indicates it
below.
Code_Aster
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Titrate:
Connection 3D - Beam
Date:
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Author (S):
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N
=
S
G
Appear 2.3-b
One notes:
G
S
X
2
X
1
·
(
)
G, E, E
1
2
a main reference mark of geometrical inertia of
S
having for origin the center
of inertia
G
, and
(
)
X X
1
2
,
associated co-ordinates,
·
N
or
E
3
the normal in the plan
S
, outgoing with the solid mass 3D,
·
(
)
3
=
E E E
3
,
,
the alternate shape of the mixed product of the basic vectors,
finally
I
the geometrical tensor of inertia of
S
(diagonal in the reference mark
(
)
E, E
1
2
) and
With
S
=
the surface
section
S
.
Let us recall that the tensor of inertia
I
can be defined in an equivalent way by an application
linear (mixed representative):
()
()
()
(
)
I U
GM
U GM
=
X
X dx
S
or a symmetrical bilinear application (covariant representative):
(
)
()
(
)
()
(
)
I U V
U GM
V GM
,
.
=
X
X dx
S
Code_Aster
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Version
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Titrate:
Connection 3D - Beam
Date:
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Author (S):
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Key:
R3.03.03-A
Page:
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R3.03 booklet: Boundary conditions and loadings
HI-75/95/042/A
These two expressions will be useful, in the reference mark
(
)
G, E, E, E
1
2
3
the matrix representative of
I
is:
[]
I
=
+
I
I
I
I
1
2
1
2
0
0
0
0
0
0
with
I
geometrical moment of inertia of
S
compared to the axis
(
)
G, E
. By convention indices
Greeks take values 1 or 2.
Useful space for the fields of displacements and vectors forced definite on
()
S
V
L S
is
=
2
3
. Space is introduced
T
fields associated with a torque (defined by two
vectors):
(
)
()
{
}
T
v
v
GM
=
=
+
V
M
/
,
T
T
such as
éq 2.3-1
For the fields of displacement of
S,
T
is the translation of the section (or point
G
),
infinitesimal rotation and fields
v
are displacements preserving the section
S
plane and not
deformation (Assumptions of NAVIER-BERNOULLI).
For the fields of vectors forced,
S
T
is the resultant
F
actions applied to
S
, and
()
I
is the resulting moment
M
in
G
. Fields
v
correspond then to distributions of
stresses closely connected in the section. Indeed, one a:
()
()
()
() (
)
()
F
T
T
M
=
=
=
=
.
.
N dS
dS
S
X
N dS
X
dS
S
S
S
S
GM
GM
GM
I
One used the fact here that
G
is geometrical center of inertia thus:
X dS
S
=
0
. The subspace
vectorial
T
being of finished size (equal to 6) has additional orthogonal for the product
scalar defined on
V
:
{
}
T
v
v. W
W T
=
=
V
dS
S
/
0
éq 2.3-2
Maybe, in a more explicit way:
{
}
T
v
v
GM v
=
=
=
V
dS
dS
S
S
/
0
0
and
éq 2.3-3
Any field of
V
all in all breaks up in a single way of an element of
T
and of an element of
T
.
U U
U
U
T
U
T
p
S
p
S
=
+
,
éq 2.3-4
Code_Aster
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Titrate:
Connection 3D - Beam
Date:
19/12/95
Author (S):
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Key:
R3.03.03-A
Page:
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R3.03 booklet: Boundary conditions and loadings
HI-75/95/042/A
One has moreover the following property:
For any couple of field 3D
()
U, v
defined on
S
,
U U
U
v. W
v. W
v. W
v v
v
p
S
p
p
S
S
p
S
=
+
=
+
=
+
dS
dS
dS
S
S
S
éq 2.3-5
The following definition is thus natural:
Definition:
One calls component of displacement of beam of a field
U
defined on the section
component
U
U
p
of
on the subspace.
The calculation of the beam part of a field 3D
U
take place by using the property of projection
orthogonal since
T
T
and
are orthogonal by definition.
If one notes
U
GM
p
U
U
=
+
T
, then:
(
) () (
)
T
T
T
U
U
U
GM
,
,
=
-
-
Argmin
2
S
éq 2.3-6
One will note in the passing the interpretation of the component beam of
U
: it is the field of displacement
of beam nearest to
U
within the meaning of least squares. The calculation of the led minimum
immediately with the characterization:
T
U
U
I
GM U
=
=
-
1
1
S
dS
dS
S
U
S
,
éq 2.3-7
The kinematic condition of connection sought is thus the following linear constraint between the field 3D
on
S
and elements of the torque of displacement of the beam in
G
:
()
S
dS
dS
S
S
T
U
I
GM U
-
-
=
,
0
éq 2.3-8
2.4
Expression of the static condition of connection
While returning to the weak formulation of the balance of the interface [éq 2.2-1], one can deduce them
conditions necessary and sufficient of static connection. Indeed, one a:
.
.
N dS
V
S
v
R.T
M
v
v
v
=
+
éq 2.4-1
Code_Aster
®
Version
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Titrate:
Connection 3D - Beam
Date:
19/12/95
Author (S):
J. PELLET
Key:
R3.03.03-A
Page:
9/10
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R3.03 booklet: Boundary conditions and loadings
HI-75/95/042/A
Thanks to the expressions [éq 2.3-7] and the decomposition of space
V
, and with the property [éq 2.3-5], one has
immediately three equations:
()
()
F
M
=
=
=
=
.
.
.
.
N dS
X
N dS
N
N dS
T
S
S
S
GM
v
v
S
0
0
or in an equivalent way
éq 2.4-2
The conditions of static connection are thus:
·
transmission of the torque of the efforts of beam, (satisfied the P1 property),
·
nullity of the complementary part (“not beam”) of the field of vector forced 3D on
section of the solid 3D (satisfied the P2 property).
One will also notice static and kinematic symmetry (P3 property) since conditions of
connection are also interpreted like:
·
equality within the meaning of least squares between displacement 3D and the displacement of the beam,
·
equality within the meaning of least squares between the field of vector forced and the elements of
reduction of the torque of the efforts of beam.
3
Establishment of the method of connection
For each connection, the user must define:
S:
the trace of the section of the beam on the solid mass 3D: it does it by the key words
MAILLE_1
and/or
GROUP_MA_1
; i.e. it gives the list of the meshs (
lma
) surface (affected
elements “edge” of modeling 3D) which represent this section geometrically.
P:
a node (key word
NOEUD_1
or
GROUP_NO_1
) carrying the 6 ddl conventional of beam:
DX,
DY, DZ, DRX, DRY, DRZ
Note:
·
the node P can be a node of element of beam or discrete element,
·
the list of the meshs
lma
must represent the section of the beam exactly. It is one
important stress for the mesh.
For each node, the program calculates the coefficients of the 6 linear relations [éq 2.3-8] which
connect:
·
6 ddl of the node P,
·
with the ddl of all the nodes of
lma
.
These linear relations will be dualisées, like all the linear relations resulting for example from
key word
LIAISON_DDL
of
AFFE_CHAR_MECA
.
Code_Aster
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Version
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Titrate:
Connection 3D - Beam
Date:
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Author (S):
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Key:
R3.03.03-A
Page:
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R3.03 booklet: Boundary conditions and loadings
HI-75/95/042/A
The calculation of the coefficients of the linear relations is done in several stages:
·
calculation of elementary quantities on the elements of
lma
: (
OPTION: CARA_SECT_POUT3
)
- surface
=
1
2
elt
elt
elt
elt
X
y
X
;
;
;
; !
·
summation of these quantities on
()
S
from where the calculation of:
-
With
S
=
- position
of
G
- tensor
of inertia
·
knowing
G
, elementary calculation on the elements of
lma
of:
(
OPTION:CARA_SECT_POUT4
)
{
}
Ni
xNi
yNi
zNi
X y Z
Ni
elt
elt
elt
elt
=
=
;
;
;
,
where:
functions of form of the element
GM
·
“assembly” of the terms calculated above to obtain of each node of
lma
, them
coefficients of the terms of the linear relations.
4
Which uses can one make this modeling?
In addition to the two aimed uses to [§2] [Figure 2.1-a] and [Figure 2.1-b], this connection can also be
used for:
·
to apply a torque of efforts to a known surface of a modeling 3D:
For that, the user defines the surface of load application (
lma
), it “connects it” with one
node (P) of discrete element (
DIS_TR_N
) without rigidity then it applies the torque wanted to it
node (
FORCE_NODALE
).
In this way, the torque is applied in “softness”, without generating secondary stresses
on surface.
·
“to retain” a structure without too the encaster:
For example, if there is with a grid in 3D a pipe and that one wants to prevent his movements of
solid body
S
()
P
one connects
()
S
with P then one locks the 6 ddls P.
The structure being then retained, without
()
S
that is to say embedded. In particular, the section
()
S
can ovalize itself.
5 Bibliography
[1]
S. ANDRIEUX: “Connections 3D/Beam, 3D/Hulls and other imaginations” (note to be appeared).