Code_Aster ®
Version
5.0
Titrate:
Connection hull-beam
Date:
06/12/00
Author (S):
J.M. PROIX
Key:
R3.03.06-B
Page:
1/10
Organization (S): EDF/MTI/MN
Handbook of Référence
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 grid, 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 grids, 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.
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
HI-75/00/006/A

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Version
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Titrate:
Connection hull-beam
Date:
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Author (S):
J.M. PROIX
Key:
R3.03.06-B
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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 Inflection of a plate ............................................................................................................ 10
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
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Version
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Titrate:
Connection hull-beam
Date:
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Author (S):
J.M. PROIX
Key:
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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 grids, one comprising of the elements
hulls (or plates), the other comprising of the elements of beams. This functionality allows
to model a slim structure in two parts: a part with a grid with traditional 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 grid 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 warping 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 this one is not modelled 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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Connection hull-beam
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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 grid 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 model 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 of the fields associated with a kinematic torque (definite
by two vectors):
T = {v V/(
T,)
v
such as (M) = T + G}
M
éq 2-1
Here, G represents 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 of the point G),
infinitesimal rotation and fields v are displacements of the space of displacements
there acceptable V preserving the section S plane and not deformed (One uses still Hypothèses of
NAVIER-BERNOULLI).
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
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Titrate:
Connection hull-beam
Date:
06/12/00
Author (S):
J.M. PROIX
Key:
R3.03.06-B
Page:
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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/v (M) .w (M)
{
dS = 0 W


éq 2-2
S
} T
Maybe, in a more explicit way:
T = {v V/v (M) dS = 0 and GM vdS =

0
éq 2-3
S
S
}
Any field of V all in all breaks up in a single way of an element of T and an element of
T.
U up custom up
T
custom
=
+

T
,
éq 2-4
One has moreover the following property:
For any couple of field hull (W, v) definite on S,
W = wp + ws v wdS = vp.wpdS + vs.ws
.
éq 2-5
S
S

dS
v = vp + vs
S
Definition:
One calls component of displacement of beam of a field of hull U defined on the section
component up of U on the subspace T.
The characterization immediately is obtained:
1
T
- 1
U =
udS,

dS
éq 2-7
S
U = I
GM U
S



S


where S represents 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.S. 'operates in
using the property of orthogonal projection since T and T are orthogonal by definition.
If one notes up = T + GM
U
U
, then:
(T
2
U, U) = Argmin
éq 2-6
S (U - T - GM)
(T,)
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.
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
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Titrate:
Connection hull-beam
Date:
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Author (S):
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Key:
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The kinematic condition of connection sought between the field hull on S and the elements of the torque
of displacement of the beam in G is given by:
ST -
dS = 0
() - GM D
U.S. =
U
I

0
éq 2-8
S
S
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, modelled 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 grid 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.
x3
y3
N
M
y3
H
Q
Q y2
T
E
1=e1
3rd
Edge of the transverse section (of hull) of
2
connection
G
e1
x1 S = L × I
H
L: line of the points Q on the average layer
H H
I = -, interval describing the épaisseu .r




2 2
Kinematics of hull or linear plate in the thickness:
(
U M) = (
U Q) + ((
Q) N). y3
·
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 of the normal according to directions' T1 and t2 of the tangent plan
H H
·
y3 is the co-ordinate in the thickness (y3 -,
).

2 2
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Titrate:
Connection hull-beam
Date:
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3.1
Calculation of average displacement on the section S
It is a question of calculating the integral udS

, where U is the displacement of hull (comprising 6 ddl by node),
S
S is the edge of hull of the transverse section of connection.
Average displacement on the section S is written:
(
H/2
U M) ds = H

(uQ) ds + ((Q) N)
y Dy ds

S
L
L



H
3
3
/2


-
that is to say
(
U M) ds = H

(uQ) 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
H/2
GM (
U M) ds =

(GQ + y3 (nQ) ((uQ) + (Q) (nQ) .y3) dsdy
S
L - H
3
/2
H
=
/2
H GQ U

(Q) ds+ GQ

((Q) (nQ) ds y Dy

L
L
- H
3
3
/2
H
+ (
H/2
N Q) (
U Q)
y Dy ds
N
2




2
3
3

.
/2


-

+ (Q) ((Q) (nQ)
y Dy ds
L
H
H
L
3
3
- 2
h3
that is to say GM (
U M) ds = H GQ

(uQ) ds +

.
12 (
N Q) ((Q)

(nQ) ds
S
L
L
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
h3
One obtains: I () = H GQ

(GQ) ds+ 12 (nQ) (N (Q) ds
L
L
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
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Titrate:
Connection hull-beam
Date:
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Key:
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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;
x2ds;
y2ds;
z2ds;
xyds;
xzds;
yzds








elt
elt
elt
elt
elt
elt
elt
elt
elt
h3

as well as terms resulting from I ():
N (N) ds

12 L
h3
h3

what makes it possible to calculate:
+

,

, etc…
12 (N2 N2) ds
N N ds
y
Z
L
X y
12 L
-
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:
GM {X, y,}
Z
NR ds;
xN ds;
yN ds;
Zn ds
I
I

I
where:

=
elt
elt
elt
I
elt
Nor = functions of form of the element

(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
I where L represents the X-coordinate
elt
L
curvilinear of average fiber of the element of edge of hull).
h3

Moreover, one adds the terms additional coming from:
(nQ) (N (Q) ds
12 L
nx
X

While noting N = ny and = y in the total reference mark one obtains:
nz
Z
(N2 +n2
y
Z) - N N - N N
X
X there y
X Z Z

(
N Q) ((
N Q) = - N N
2
2
X there X
+ (N + N
X
Z) - N N
y
y Z Z
= A
- N N - N N
2
2
X Z X
y zy + (N + N
X
y) Z


then:
h3
h3

(nQ) ((nQ) ds =

With

12
12
((S) NR (S) ds
L
J
el
) J
el

· “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 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 traditional 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 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 MAILLE or GROUP_MA must represent
exactly the cross-section of the beam. It is an important constraint for
grid.
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.
y
H
M
R
Z
O
X
With
MX B
F
My
Embedding out of O is carried out using a connection between the edge of the hull and a specific element
located out of O. Cet 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 grid in elements of hulls.
Handbook of Référence
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Titrate:
Connection hull-beam
Date:
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Author (S):
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4.2.2 Inflection of a plate
Let us consider a sufficiently long thin section, length 2L, of width B, thickness H,
modelled by an element of hull OA and an element of beam on AB:
Z
y
O
L
B
D
With
L
B
X
H
C
·
1ère condition of connection is written:
B H U ()
With = H
U
(y) Dy
CD

the displacement of point A (pertaining to the beam) is the average of displacements of the edge CD
plate.

·
2nd condition of connection is written:
h3
I () = H
AQ U (Q) ds +


12
(Q) ds
CD
CD
h3 B 2
In the case of an inflection around y, the only term not no one is:
B
/(y) Dy
12 - 2
B

Indeed, H
AQ U

(Q) ds = H2 U ydy
B Z
.x 0
CD

2

=
-
For an inflection around y, the connection is thus written:
bh3
I ()
With
y y
=
y bus
12
y is constant on CD.
This application is implemented in test SSLX100B: mixes 3d_coque_poutre.
Handbook of Référence
R3.03 booklet: Boundary conditions and loadings
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