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Manual of Reference
R3.03 booklet: Boundary conditions and loadings
R3.03.06 document
Connection hull-beam
Summary:
Here the connection hull-beam is described, which makes it possible to connect two parts of mesh, one made up
elements of beams (or a discrete element), and the other with a grid one in elements of hulls (to represent
phenomena except kinematics of beam). This development thus functions under assumptions translating
that it is the same kinematics of beam which is transmitted between the two mesh, on both sides of
connection. It results in 6 linear relations connecting displacements of the whole of the nodes of the edge of
hull with the 6 degrees of freedom of the node end of the beam.
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Contents
1 Assumptions and applications .................................................................................................................... 3
1.1 Assumptions and limitations ................................................................................................................. 3
1.2 Applications concerned: ........................................................................................................................ 3
1.2.1 Modeling of pipings .................................................................................................. 3
1.2.2 Connection plates beam ................................................................................................. 3
1.2.3 Beam with symmetrical profile ...................................................................................................... 4
1.2.4 Application of a loading or boundary conditions of the type “beam” ........................ 4
1.2.5 Application not considered: ................................................................................................... 4
2 Application of the method of the connection 3D-beam. Equations of connection ................................................ 4
3 Integrals to be calculated. Kinematics of hull ......................................................................................... 6
3.1 Calculation of average displacement on the section S .............................................................................. 7
3.2 Calculation of the average rotation of the section S ................................................................................ 7
3.3 Calculation of the tensor of inertia ............................................................................................................... 7
3.4 Establishment of the method ............................................................................................................. 8
4 ................................................................................................................................................ Use 9
4.1 Modeling .................................................................................................................................... 9
4.2 Examples and tests ............................................................................................................................ 9
4.2.1 Test SSLX101 ........................................................................................................................ 9
4.2.2 Bending of a plate ............................................................................................................ 10
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1
Assumptions and applications
1.1
Assumptions and limitations
Here the connection hull-beam is described, which is used to connect two mesh, one comprising of the elements
hulls (or plates), the other comprising of the elements of beams. This functionality allows
to modelize a slim structure in two parts: a part with a grid with conventional elements of
beams, representing a kinematics and a behavior of beams, and the other part with a grid in
elements of hulls, to reveal other phenomena (ovalization, swelling, plasticity
located).
The following assumptions however are made:
1) the transverse sectional surface of the end of the mesh of hulls is identical to
right sectional surface of the element of beam which corresponds to him,
the 2) centers of gravity are identical,
3) the sections are plane and coplanar,
4) the normal with the section of hulls is confused with the axis of the beam.
Limitations:
1) one does not hold account in the connection of the ovalization of the cross-sections,
2) account of the roll is not taken.
1.2
Applications concerned:
1.2.1 Modeling of pipings
One of the major applications relates to pipings. The bent parts or prickings are
then with a grid in hulls, which makes it possible to reveal an ovalization, a behavior
elastoplastic room or a swelling in the event of internal pressure. This connection does not transmit
the ovalization of the pipes since the aforementioned is not modelized in the elements of beams. For it
to make, it is necessary to use the connection hull-pipe or to net a sufficient length of right piping in
elements of hulls so that ovalization on the level of the connection is negligible.
Circular piping of section (or rectangular…) with a grid in hull then in beam.
1.2.2 Connection plates beam
Connection plate-beam (mean rectangular section).
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1.2.3 Beam with symmetrical profile
Beam with symmetrical profile with a grid partly in hulls.
1.2.4 Application of a loading or boundary conditions of the type “beam”
At the end of a slim structure with a grid in hulls, it is often useful to impose is one
loading of the type “beam” i.e. a torque of efforts, is boundary conditions
(embedding) compatible with the kinematics of beam. One can then connect the section
transverse of end of the mesh hulls to a discrete element to which one will apply this torque
or this embedding.
1.2.5 Application not considered:
This functionality does not make it possible to modelize the `'prickings transverse or orthogonal '' of one
beam on a plate or a hull:
2
Application of the method of the connection 3D-beam. Equations
of connection
The step is identical to that of the connection beam-3D [R3.03.03]: the connection results in 6
linear relations connecting displacements of the whole of the nodes hull of the section of
connection (6 degrees of freedom per node, compared to 3 ddl by node in 3D) with the 6 degrees
of freedom of the node of beam. The section of connection of hull is made up of elements of edge
hulls (segments). On the section crosses connection, one breaks up the field of
displacement “hull” in a part “beam” and a “complementary” part. This brings us to
to define the conditions of kinematic connection between beam and hull like the equality of displacement
(torque distributer or kinematic torque) of beam and the beam part of the field of displacement
hull
As in [R3.03.03], one introduces space
T
fields associated with a kinematic torque (definite
by two vectors):
(
)
()
{
}
T
v
v
T
GM
=
= +
V
T
M
/
,
such as
éq 2-1
Here,
G
represent the center of gravity of the section of connection (having to be identical to that of
the beam). For the fields of displacement of
T
,
T
is the translation of the section (or point
G
),
infinitesimal rotation and fields
v
are displacements of the space of displacements
acceptable V preserving the section
S
there plane and not deformed (One uses still the Assumptions of
NAVIER-BERNOULLI).
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The vectorial subspace
T
being of finished size (equalizes to 6) has additional
orthogonal for the definite scalar product on
V
:
() ()
{
}
T
v
v
W
W T
=
=
V
M
M dS
S
/
.
0
éq 2-2
Maybe, in a more explicit way:
()
{
}
T
v
v
GM v
=
=
=
V
M dS
dS
S
S
/
0
0
and
éq 2-3
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 2-4
One has moreover the following property:
For any couple of field hull
()
W v
,
defined on
S
,
W
W
W
v
v
v
v W
v W
v W
p
S
p
S
p
p
S
S
=
+
=
+
=
+
.
.
.
dS
dS
dS
S
S
S
éq 2-5
Definition:
One calls component of displacement of beam of a field of hull
U
defined on the section
component
U
U
p
of
on the subspace
T
.
The characterization immediately is obtained:
T
U
U
U
I
GM U
=
=




-
1
1
S
dS
dS
S
S
,
éq 2-7
where
S
represent the surface of the section
S
and
I
the geometrical tensor of inertia of surface
S
, expressed
in
G
.
In other words, one can as say as the calculation of the beam part of a field hull
U
take place in
using the property of orthogonal projection since
T
T
and
are orthogonal by definition.
If one notes
U
GM
p
U
U
=
+
T
, then:
(
)
(
)
(
)
T
T
U
U
U
GM
,
,
=
-
-
Argmin
T
2
S
éq 2-6
The component beam of
U
can thus be interpreted like the field of displacement of beam it
nearer to
U
within the meaning of least squares.
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The kinematic condition of connection sought between the field hull on
S
and elements of the torque
of displacement of the beam in
G
is given by:
()
S
dS
dS
S
S
T
-
=
-
=
U
I
GM U
0
0
éq 2-8
The equation [éq 2-8] shows that the situation is identical to the case 3D-beam. Linear relations
will have the same form. The only difference comes from the integrals on S (which represents a curve here
corresponding to the section of the hull, modelized by elements of edge of hull). Moreover,
the field of displacement of hull utilizes DDL of rotation.
To translate the equation [éq 2-8] into linear relations, the two integrals should be calculated:
·
average displacement:
udS
S
·
average rotation:
GM U
dS
S
3
Integrals to be calculated. Kinematics of hull.
For each node, the program calculates the coefficients of the 6 linear relations [éq 2-8] which connect:
·
6 ddl of the node of beam P (geometrically confused with the center of gravity G of
transverse section of the mesh hulls)
·
with the ddl of all the nodes of the list of the meshs of the edge of hull.
These linear relations are dualisées, like all the linear relations resulting, for example, of the word
key
LIAISON_DDL
of
AFFE_CHAR_MECA
. They are built as for the connection 3D-beam with
to leave the assembly of elementary terms.
E
1
G
X
3
X
1
H
y
2
Q
H
Q
M
E
2
E
3
y
3
T
1
=e
1
y
3
N
Edge of the transverse section (of hull) of
connection
S L I
= ×
L
: line of the points
Q
on the average layer
I
H H
= -




2 2
,
.
interval describing the thickness
Kinematics of hull or linear plate in the thickness:
()
() ()
(
)
U
U
N
M
Q
Q
y
=
+



.
3
·
U
is the vector displacement of average surface in
Q
,
·
N
is the normal vector on the average surface of the hull in
Q
,
·



is the vector rotation in
Q
normal according to directions' T
1
and T
2
tangent plan
·
y
3
is the co-ordinate in the thickness (
y
H H
3
2 2
-




,
).
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3.1
Calculation of average displacement on the section S
It is a question of calculating the integral
udS
S
, where
U
is the displacement of hull (comprising 6 ddl by node),
S
is the edge of hull of the transverse section of connection.
Average displacement on the section
S
is written:
()
()
()
(
)
U M ds H U Q ds
Q
N
y Dy ds
H
H
L
L
S
=
+


-
3
3
2
2
/
/
that is to say
()
()
U M ds H U Q ds
S
L
=
One neglects in this expression the variations of metric in the thickness of the hull.
3.2
Calculation of the average rotation of the section S
()
()
(
)
() () ()
(
)
()
() ()
(
)
() ()
() () ()
(
)
GM U M ds
GQ y N Q
U Q
Q
N Q y dsdy
H GQ U Q ds
GQ
Q
N Q ds
y Dy
N Q
U Q
y Dy ds
N Q
Q
N Q
y Dy ds
H
H
L
S
H
H
L
L
H
H
H
H
L
L
=
+
+
=
+
+




+
-
-
-
-
3
3
3
2
2
3
3
2
2
3
3
2
2
32
2
2
3
.
.
/
/
/
/
/
/
that is to say
()
()
() () ()
(
)
GM U M ds H GQ U Q ds H
N Q
Q
N Q ds
L
S
L
=
+
3
12
.
3.3
Calculation of the tensor of inertia
The tensor of inertia is defined by [R3.03.03]:
()
(
)
I
GM
GM ds
S
=
while posing:
()
GM
GQ N Q y
=
+
.
3
One obtains:
()
(
)
()
()
(
)
I
H GQ
GQ ds H
N Q
N Q ds
L
L
=
+
3
12
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3.4
Establishment of the method
The calculation of the coefficients of the linear relations is done in two times:
·
calculation of elementary quantities on the elements of the list of the meshs of edges of hulls
(mesh of type SEG2 or SEG3):
-
the 9 terms are calculated:
-
ds xds
yds
X ds
y ds
Z ds
xyds
xzds
yzds
elt
elt
elt
elt
elt
elt
elt
elt
elt
;
;
;
;
;
;
;
;
2
2
2

as well as terms resulting from
()
(
)
I
H
N
N ds
L
:
3
12

what makes it possible to calculate:
(
)
H
N
N ds H
N N ds
y
Z
L
X y
L
3
2
2
3
12
12
+
,
,
etc
-
summation of these quantities on
()
S
from where the calculation of:
-
With
S
=
- position
of
G
- tensor
of inertia
I
·
knowing
G
, elementary calculation on the elements of the list of the meshs of edges of
hulls of:
{
}
NR ds
xN ds
yN ds
Zn ds
X y Z
NR
I
I
I
elt
elt
elt
I
elt
I
;
;
;
:
,
=
=
where
functions of form of the element
GM

(It should simply be noticed that in this case, the integrals on the elements of edge
are to be multiplied by the thickness of the hull:
NR ds H NR DLL
I
elt
I
L
=
where
L
represent the X-coordinate
curvilinear of average fiber of the element of edge of hull).

Moreover, one adds the terms additional coming from:
()
()
(
)
H
N Q
N Q ds
L
3
12

While noting
N
=
N
N
N
X
y
Z
and



=

X
y
Z
in the total reference mark one obtains:
()
()
(
)
(
)
(
)
(
)
N Q
N Q
N
N
N N
N N
N N
N
N
N N
N N
N N
N
N
y
Z
X
X there y
X Z Z
X there X
X
Z
y
y Z Z
X Z X
y Z y
X
y
Z
=
+
-
-
-
+
+
-
=
-
-
+
+
2
2
2
2
2
2
With




then:
()
()
(
)
() ()
(
)
H
N Q
N Q ds
H
S NR S ds
L
J
el
J
el
3
3
12
12
=
With



·
“assembly” of the terms calculated above to obtain of each node of the meshs
of edge, coefficients of the terms of the linear relations.
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4 Use
4.1 Modeling
For each connection, the user must define under the key word factor
LIAISON_ELEM
of
AFFE_CHAR_MECA:
S:
the trace of the cross-section of the beam on the hull: it does it by the key words
MAILLE_1
and/or
GROUP_MA_1
i.e. it gives the list of the linear meshs (affected of elements
“edge” of modeling hull) which represents 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
V:
the vector defining the axis of the beam, directed hull towards the beam, and defined by its
co-ordinates using the key word
AXE_POUTRE: (v1, v2, v3)
Note:
·
the node P can be a node of element of beam or discrete element,
·
the list of the meshs of edge of hull, defined by
NET
or
GROUP_MA
must represent
exactly the cross-section of the beam. It is an important stress for
mesh.
4.2
Examples and tests
4.2.1 Test
SSLX101
It is about a subjected right beam has unit efforts out of B (traction, moments bending and of
torsion). One takes a mean section of tube thickness H << R.
R
X
With
O
y
H
F
B
M
X
M
y
M
Z
Embedding out of O is carried out using a connection between the edge of the hull and a specific element
located out of O. This element is embedded (null translations and rotations).
This makes it possible to obtain in the hull a state of stresses very close to a solution “beam”: there is not
no disturbance of the stress field. The solution differs from the analytical solution (solution
RDM) of 3%, this being only due to the smoothness of the mesh in elements of hulls.
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4.2.2 Bending of a plate
Let us consider a sufficiently long thin section, length 2L, of width B, thickness H,
modelized by an element of hull OA and an element of beam on AB:
y
B
B
Z
D
With
C
H
X
L
L
O
·
1
era
condition of connection is written:
()
()
B H
With
H
y Dy
CD
U
U
=

the displacement of point A (pertaining to the beam) is the average of displacements of the edge CD
plate.
·
2
ème
condition of connection is written:
()
()
()
I
H
AQ U Q ds H
Q ds
CD
CD
=
+
3
12
In the case of a bending around
y
, the only term not no one is:
()
H
y Dy
B
B
3
2
2
12
-
/
Indeed,
()
H
AQ U Q ds H
U ydy
Z
B
B
CD
=

=
-
2
2
0
.x
For a bending around
y
, the connection is thus written:
()
I
With
bh
y y
y
=
3
12
because
y
is constant on CD.
This application is implemented in test SSLX100B: mixes 3d_coque_poutre.