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Code_Aster
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Titrate:
Multifibre elements of beams (right)
Date
:
18/11/03
Author (S):
S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
1/18
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
Organization (S):
EDF-R & D/AMA, LMT Cachan















Manual of Reference
R3.08 booklet: Machine elements with average fiber
Document: R3.08.08



Multifibre element of beam (right)



Summary:

This document presents the elements of multifibre beam of Code_Aster based on a resolution of a problem
of beam for which each section of a beam is divided into several fibers. Each fiber behaves
then like a beam of Euler.
The beams are right (Element
POU_D_EM
). The section can be of an unspecified form.
The assumptions selected are as follows:
·
assumption of Euler: transverse shearing is neglected (this assumption is checked for forts
twinges),
·
the elements of beam introduces here do not make it possible to make correct calculation in torsion.
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Code_Aster
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Titrate:
Multifibre elements of beams (right)
Date
:
18/11/03
Author (S):
S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
2/18
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
Contents
1
Introduction ............................................................................................................................................ 4
2
Element of theory of the beams (recalls) ............................................................................................. 5
3
Equations of the movement of the beams ........................................................................................... 6
4
Element of right beam multifibre ........................................................................................................ 6
4.1
Element beam of reference ........................................................................................................... 6
4.2
Determination of the matrix of rigidity of the multifibre element ........................................................ 8
4.2.1
General case (beam of Euler) .................................................................................................. 8
4.2.2
Case of the multifibre beam ..................................................................................................... 9
4.2.3
Discretization of the section out of fibers ­ Calcul of K
S
............................................................ 12
4.2.4
Integration in the linear elastic case (
RIGI_MECA
) ...................................................... 12
4.2.5
Integration in the non-linear case (
RIGI_MECA_TANG
) ................................................... 13
4.3
Determination of the matrix of mass of the multifibre element ...................................................... 14
4.3.1
Determination of M
elem
......................................................................................................... 14
4.3.2
Discretization of the fiber section - Calculation of M
S
............................................................ 16
4.4
Calculation of the forces intern ............................................................................................................. 16
4.5
Nonlinear models of behavior usable ..................................................................... 18
5
Bibliography ........................................................................................................................................ 18
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Code_Aster
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Titrate:
Multifibre elements of beams (right)
Date
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S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
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R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
Notations
One gives the correspondence between this notation and that of the documentation of use.
DX, DY, DZ and DRX, DRY, DRZ are in fact the names of the degrees of freedom associated with the components
displacement
Z
y
X
W
v
U
,
,
,
,
,
.

E
Young modulus
E
Poisson's ratio
NAKED
G
modulate of Coulomb =
(
)
+
1
2
E
G
Z
y
I
I
,
geometrical moments of bending compared to the axes
Z
y
,
IZ
IY,
X
J
constant of torsion
JX
K
stamp rigidity
M
stamp of mass
Z
y
X
M
M
M
,
,
moments around the axes
Z
y
X,
,
MFZ
MFY
MT
,
,
NR
normal effort with the section
NR
S
surface of the section
With
W
v
U,
,
translations on the axes
Z
y
X,
,
DZ
DY
DX
Z
y
V
V,
sharp efforts along the axes
Z
y,
VZ
VY,
density
RHO
Z
y
X
,
,
rotations around the axes
Z
y
X,
,
DRZ
DRY
DRX
Z
y
X
Q
Q
Q
,
,
External linear efforts
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Titrate:
Multifibre elements of beams (right)
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S. MILL, L. DAVENNE, F. GATUINGT
Key
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R3.08.08-A
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R3.08 booklet: Machine elements with average fiber
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1 Introduction
The analysis of the structures subjected to a dynamic loading requires models of
behavior able to represent non-linearities of material.
Many analytical models were proposed. They can be classified according to two groups: has)
detailed models founded on the mechanics of the solid and their description of the local behavior of
material (microscopic approach) and b) of the models based on a total modeling of
behavior (macroscopic approach). In the first type of models, we can find them
conventional models E.F as well as the models of the type “fiber” (having an element of the beam type
how support).
While “conventional” models E.F are powerful tools for the simulation of
nonlinear behavior of the complex parts of the structures (joined, assemblies,…), them
application to the totality of a structure can prove not very practical because of a calculating time
prohibitory or of the size memory necessary to the realization of this calculation. On the other hand, a modeling
of multifibre beam type (see [Figure 1-a]), has the advantages of the simplifying assumptions
of a kinematics of the beam type of Euler - Bernoulli while offering a practical and effective solution
for a nonlinear analysis complexes composite elements of structures such as those which one
can meet for example out of reinforced concrete.
Moreover, this “intermediate” modeling is relatively robust and inexpensive in time calculation
because of use of nonlinear models of behavior 1D.


Appear 1-a: Description of a modeling of the multifibre beam type
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Titrate:
Multifibre elements of beams (right)
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S. MILL, L. DAVENNE, F. GATUINGT
Key
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R3.08.08-A
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R3.08 booklet: Machine elements with average fiber
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2
Element of theory of the beams (recalls)
One takes again here the elements developed within the framework of the elements of beam of Euler ([bib4]).
A beam is a solid generated by a surface of surface
S
of which the geometrical center of inertia
G
followed a curve
C
called the average fiber or neutral fiber. The surface
S
(section is the cross-section
transversal) or profile, and it is supposed that if it is evolutionary, its evolutions (size, form) are
continuous and progressive when
G
described the average line.
For the study of the beams in general, one makes the following assumptions:
·
the cross-section of the beam is indeformable,
·
transverse displacement is uniform on the cross-section.
These assumptions make it possible to express displacements of an unspecified point of the section, in
function of displacements of the point corresponding located on the average line, and according to one
increase in displacement due to the rotation of the section around the transverse axes.
The discretization in “exact” elements of beam is carried out on a linear element with two nodes and
six degrees of freedom by nodes. These degrees of freedom are the three translations
W
v
U,
,
and three
rotations
Z
y
X
,
,
[Figure 2-a]).

Z
y
X
1
2
U
X
v
y
W
Z
U
X
v
y
W
Z
Appear 2-a: Element beam

Waited until the deformations are local, it is built in each node of the mesh a base
local depending on the element on which one works. The continuity of the fields of displacements is
ensured by a basic change, bringing back the data in the total base.
In the case of the right beams, one traditionally places the average line on axis X of the base
local, transverse displacements being thus carried out in the plan
()
Z
y,
.
Finally when we arrange sizes related to the degrees of freedom of an element in a vector
or an elementary matrix (thus of dimension 12 or 12
2
), one arranges initially the variables for
node 1 then those of node 2. For each node, one stores initially the sizes related to
three translations, then those related to three rotations. For example, a vector displacement will be
structured in the following way:
4
4
4
4
3
4
4
4
4
2
1
4
4
4
3
4
4
4
2
1
2
node
1
node
2
2
2
1
1
1
,
,
,
,
,
,
,
,
,
,
,
2
2
2
1
1
1
Z
y
X
Z
y
X
W
v
U
W
v
U
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Code_Aster
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Titrate:
Multifibre elements of beams (right)
Date
:
18/11/03
Author (S):
S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
6/18
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
3
Equations of the movement of the beams
We will not include in this document all the equations of the movement of the beams. For
more complements concerning this part one can refer to documentation concerning
elements
POU_D_E
and
POU_D_T
([bib4]).


4
Element of right beam multifibre
One describes in this chapter obtaining the elementary matrices of rigidity and mass for the element
of multifibre right beam, according to the model of Euler. The matrices of rigidity are calculated with
options
“RIGI_MECA”
or
“RIGI_MECA_TANG”
, and matrices of mass with the option
“MASS_MECA”
for the coherent matrix, and the option '
MASS_MECA_DIAG'
for the matrix of diagonalized mass.
We present here a generalization [bib3] where the reference axis chosen for the beam is
independent of any geometrical, inertial or mechanical consideration. The element functions for
an unspecified section (heterogeneous is without symmetry) and is thus adapted to an evolution not
linear of the behavior of fibers.
One also describes the calculation of the nodal forces for the nonlinear algorithms:
“FORC_NODA”
and
“RAPH_MECA”
.
4.1
Element beam of reference
[Figure 4.1-a] the change of variable shows us realized to pass from the real finite element
[Figure 2-a] with the finite element of reference.
Z
y
X
1
2
0
L
Z
y
X
1
2
U
1
1
X
v
1
1
y
W
1
1
Z
U
2
2
X
v
2
2
y
W
2
2
Z
Appear 4.1-a: Element of reference vs real Elément
One will then consider the continuous field of displacements in any point of the average line by report/ratio
with the field of displacements discretized in the following way:
{}
[]
{}
.
U
NR
U
S
=
The index
S
indicate the quantities attached to average fiber.
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Titrate:
Multifibre elements of beams (right)
Date
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18/11/03
Author (S):
S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
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R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
By using the functions of forms of the element of reference, the discretization of the variables
() () () () () ()
X
X
X
X
W
X
v
X
U
sz
sy
sx
S
S
S
,
,
,
,
,
becomes:
()
()
()










































-
-
-
-
=












2
2
2
2
2
2
1
1
1
1
1
1
,
6
,
5
,
4
,
3
,
6
,
5
,
4
,
3
2
1
6
5
4
3
6
5
4
3
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Z
y
X
Z
y
X
X
X
X
X
X
X
X
X
sz
sy
sx
S
S
S
W
v
U
W
v
U
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
X
W
X
v
X
U
éq 4.1-1
With the following functions of interpolation:
2
,
6
2
3
2
6
3
2
,
5
3
3
2
2
5
2
,
4
2
3
2
4
3
2
,
3
3
3
2
2
3
,
2
2
,
1
1
6
2
;
12
6
;
2
3
6
4
;
2
12
6
;
2
3
1
1
;
1
;
1
L
X
L
NR
L
X
L
X
NR
L
X
L
NR
L
X
L
X
NR
L
X
L
NR
L
X
L
X
X
NR
L
X
L
NR
L
X
L
X
NR
L
NR
L
X
NR
L
NR
L
X
NR
xx
xx
xx
xx
X
X
+
-
=
+
-
=
-
=
-
=
+
-
=
+
-
=
+
-
=
+
-
=
=
=
-
=
-
=
éq
4.1-2
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Multifibre elements of beams (right)
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Key
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R3.08.08-A
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R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
4.2
Determination of the matrix of rigidity of the multifibre element
4.2.1 General case (beam of Euler)
Let us consider a beam of Euler, straight line directed in the direction
X
, subjected to efforts distributed
Z
y
X
Q
Q
Q
,
,
[Figure 4.2.1-a].



Y,
v
Z,
W
X,
U
Appear 4.2.1-a: Beam of Euler 3D
The fields of displacements and deformations take the following form then when it is written
displacement of an unspecified point of the section according to displacement
()
S
U
line of
average:
(
)
()
()
()
X
Z
X
y
X
U
Z
y
X
U
sy
sz
S
+
-
=
,
,

éq
4.2.1-1
(
)
()
X
v
Z
y
X
v
S
=
,
,
éq
4.2.1-2
(
)
()
X
W
Z
y
X
W
S
=
,
,
éq
4.2.1-3
()
()
()
X
Z
X
y
X
U
sy
sz
X
xx
'
'
'
+
-
=
éq
4.2.1-4
0
=
=
xz
xy
éq
4.2.1-5

Note:
·
Torsion is treated overall separately, one does not calculate
yz
here.
·
()
X
f'
indicate the derivative of
()
X
F
compared to
X
.
By introducing the equations [éq 4.2.1-4] and [éq 4.2.1-5] into the principle of virtual work one
obtains:
()
()
()
(
)
+
+
=
0
0
0
V
L
Z
S
y
S
X
S
xx
xx
dx
Q
X
W
Q
X
v
Q
X
U
FD
éq
4.2.1-6
Z
y
X
Q
Q
Q
,
,
indicating the linear efforts applied.
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Multifibre elements of beams (right)
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Key
:
R3.08.08-A
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R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
What gives by using the equation [éq 4.2.1-1]:
()
()
()
()
(
)
()
()
()
(
)
+
+
=
+
+
+
L
L
Z
S
y
S
X
S
sz
Z
sy
y
sx
X
S
dx
Q
X
W
Q
X
v
Q
X
U
dx
X
M
X
M
X
M
X
U
NR
0
0
'
'
'
'
éq 4.2.1-7
With:
-
=
=
=
S
S
S
xx
Z
xx
y
xx
dS
y
M
dS
Z
M
dS
NR
;
;
éq
4.2.1-8
Note:
·
Torque
X
M
is not calculated by integration but is not calculated directly with
to leave the stiffness in torsion (see [éq 4.2.2-4]).
·
The theory of the beam associated with an elastic material gives:
xx
xx
E
=
4.2.2 Case of the multifibre beam
We suppose now that the section
S
is not homogeneous [Figure 4.2.2-a].
Without adopting particular assumption on the intersection of the axis
X
with the section
S
or on
orientation of the axes
Y
,
Z
, the relation between the “generalized” stresses and deformation
“generalized”
S
D
becomes [bib2]:
S
S
S
D
K
F
=
éq 4.2.2-1
with:
(
)
() () () ()
(
)
T
sx
sz
sy
S
S
T
X
Z
y
S
X
X
X
X
U
D
M
M
M
NR
F
'
'
'
'
,
,
,
,
,
,
=
=
éq
4.2.2-2
Appear 4.2.2-a: Section S unspecified - multifibre beam
Material
1
Material 2
Material 3
Center
Cross-section
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The matrix
S
K
can then put itself in the following form:






44
33
23
22
13
12
11
0
0
0
S
S
S
S
S
S
S
S
K
sym
K
K
K
K
K
K
K
éq
4.2.2-3
with:
=
-
=
=
-
=
=
=
S
S
S
S
S
S
S
S
S
S
S
S
ds
Ey
K
Eyzds
K
dS
Ez
K
Eyds
K
Ezds
K
EdS
K
2
33
23
2
22
13
12
11
;
;
;
;
éq
4.2.2-4
where
E
can vary according to
y
and
Z
. Indeed, it may be that in modeling planes
section [Figure 4.2.2-a]), several materials cohabit. For example, in a section concrete reinforced,
there are at the same time concrete and reinforcements.
The discretization of the fiber section makes it possible to calculate the integrals of the equations [éq 4.2.2-4].
The calculation of the coefficients of the matrix
S
K
is detailed in the paragraph [§4.2.3] according to.
Note:
The term of torsion
X
S
GJ
K
=
44
is given by the user using the data of
X
J
.
The introduction of the equations [éq 4.2.2-1] to [éq 4.2.2-4] in the principle of virtual work leads to:
()
()
()
(
)
=
+
+
-
L
L
Z
S
y
S
X
S
S
S
T
S
dx
Q
X
W
Q
X
v
Q
X
U
dx
D
K
D
0
0
0
éq
4.2.2-5
The generalized deformations are calculated by (
S
D
is given to the equation [éq 4.2.2-2]):
{}
U
B
D
S
=
éq
4.2.2-6
With the matrix
B
following:








-
-
=
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,
2
,
1
,
6
,
5
,
4
,
3
,
6
,
5
,
4
,
3
,
2
,
1
X
X
xx
xx
xx
xx
xx
xx
xx
xx
X
X
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
NR
B
éq 4.2.2-7
Discretization of space
[]
L
,
0
with elements and the use of the equations [éq 4.2.2-5] returns
the equation [éq 4.2.1-6] equivalent to the resolution of a conventional linear system:
F
KU
=
éq
4.2.2-8
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Key
:
R3.08.08-A
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:
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R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
The matrix of rigidity of the element [Figure 4.2.2-b] and the vector of the efforts results are finally
given by:
dx
Q
NR
F
dx
B
K
B
K
L
T
L
S
T
elem
=
=
0
0
éq
4.2.2-9
Appear multifibre 4.2.2-b: Beam ­ Calcul of K
elem
With the vector
Q
who depends on the external loading:
(
)
T
Z
y
X
Q
Q
Q
Q
0
0
0
=
If we consider that the distributed efforts
Z
y
X
Q
Q
Q
,
,
are constant, we obtain the vector
nodal forces according to:
T
y
Z
Z
y
X
y
Z
Z
y
X
Q
L
Q
L
Lq
Lq
Lq
Q
L
Q
L
Lq
Lq
Lq
F




-
=
12
12
0
2
2
2
12
12
0
2
2
2
2
2
2
2
éq
4.2.2-10
Center
Cross-section
Points of integration
Under-points of integration




=
S
2
S
S
S
S
ds
y
E
ds
y
E
ds
y
E
ds
E
K
=
L
0
S
T
elem
dx
B
K
B
K
background image
Code_Aster
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Titrate:
Multifibre elements of beams (right)
Date
:
18/11/03
Author (S):
S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
12/18
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
4.2.3 Discretization of the section out of fibers ­ Calcul of K
S
The discretization of the fiber section makes it possible to calculate the various integrals which intervene
in the matrix of rigidity. Thus, if we have a section which comprises
N
fibers we will have them
approximation following of the integrals:
=
=
=
=
=
=
=
-
=
=
=
=
=
N
I
I
I
I
S
N
I
I
I
I
I
S
N
I
I
I
I
S
N
I
I
I
I
S
N
I
I
I
I
S
N
I
I
I
S
S
y
E
K
S
Z
y
E
K
S
Z
E
K
S
y
E
K
S
Z
E
K
S
E
K
1
2
33
1
23
1
2
22
1
13
1
12
1
11
;
;
;
;
éq 4.2.3-1
with
I
E
and
I
S
the initial or tangent module and the section of each fiber. The state of stress is
constant by fiber.
Each fiber is also identified using
I
y
and
I
Z
co-ordinates of the center of gravity of
fiber compared to the axis of the section defined by the control '
COO_AXE_POUTRE'
(document
“AFFE_SECT_MULTI”
).

4.2.4 Integration in the linear elastic case (
RIGI_MECA
)
When the behavior of material is linear, the element beam is homogeneous in its length,
the integration of the equation [éq 4.2.2-9] can be made analytically.
One obtains the matrix of following rigidity then:
























































-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
=
L
KL
K
L
K
L
K
L
K
L
K
L
K
SYM
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
L
K
K
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
elem
33
23
22
44
223
222
3 22
2 33
2 23
3 23
3 33
13
12
11
33
23
223
2 33
13
33
23
22
222
2 23
12
23
22
44
44
2 23
2 22
3 22
3 23
2 23
2 22
3 22
233
223
3 23
3 33
233
223
3 23
3 33
13
12
11
13
12
11
4
4
4
0
0
6
6
0
12
6
6
0
12
12
0
0
0
2
2
0
6
6
4
2
2
0
6
6
4
4
0
0
0
0
0
0
0
6
6
0
12
12
0
6
6
0
12
6
6
0
12
12
0
6
6
0
12
12
0
0
0
0
0
0
éq 4.2.4-1
with the following terms:
44
23
33
22
13
12
11
,
,
,
,
,
,
S
S
S
S
S
S
S
K
K
K
K
K
K
K
are given to the equation
[éq 4.2.2-4].
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Code_Aster
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Titrate:
Multifibre elements of beams (right)
Date
:
18/11/03
Author (S):
S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
13/18
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
4.2.5 Integration in the non-linear case (
RIGI_MECA_TANG
)
When the behavior of material is nonlinear, to allow a correct integration of
efforts intern (see paragraph [§4.4]), it is necessary to have at least two points of integration it
length of the beam. We chose to use two points of Gauss.
The integral of
elem
K
[éq 4.2.2-9] is calculated in numerical form:
()
() ()
I
L
I
I
S
T
I
I
S
T
elem
X
B
X
K
X
B
W
J
dx
B
K
B
K
=
=
=
0
2
1
éq
4.2.5-1
where
I
X
is the position of the point of Gauss
I
in an element of reference length 1,
1 ± 0,57735026918963)/2
I
W
is the weight of the point of Gauss
I
. One takes here
I
W
= 0,5 for each of the 2 points
J
one is Jacobien takes here
J
=
L
, the real element having a length
L
and the function of form
to pass to the element of reference being
L
X
.
()
I
S
X
K
is calculated using the equations [éq 4.2.2-3], [éq 4.2.2-4] (see paragraph [§4.2.3] for
the numerical integration of these equations)
The analytical calculation of
()
() ()
I
I
S
T
I
X
B
X
K
X
B
give:
































-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
33
2
4
23
2
4
22
2
4
44
2
1
23
4
2
22
4
2
22
2
2
33
4
2
23
4
2
23
2
2
33
2
2
13
4
1
12
4
1
12
2
1
13
2
1
11
2
1
33
4
3
23
4
3
23
3
2
33
3
2
13
3
1
33
2
3
23
4
3
22
4
3
22
3
2
23
3
2
12
3
1
23
2
3
22
2
3
44
2
1
44
2
1
23
4
2
22
4
2
22
2
2
23
2
2
12
2
1
23
3
2
22
3
2
22
2
2
33
4
2
23
4
2
23
2
2
33
2
2
13
2
1
33
3
2
23
3
2
23
2
2
33
2
2
13
4
1
12
4
1
12
2
1
13
2
1
11
2
1
13
3
1
12
3
1
12
2
1
13
2
1
11
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
K
B
K
B
K
B
K
B
K
B
B
K
B
B
K
B
K
B
B
K
B
B
K
B
K
B
K
B
B
K
B
B
K
B
B
K
B
B
K
B
K
B
B
K
B
B
K
B
B
K
B
B
K
B
B
K
B
K
B
B
K
B
B
K
B
B
K
B
B
K
B
B
K
B
K
B
K
B
K
B
K
B
B
K
B
B
K
B
K
B
K
B
B
K
B
B
K
B
B
K
B
K
B
B
K
B
B
K
B
K
B
K
B
B
K
B
B
K
B
B
K
B
K
B
K
B
B
K
B
B
K
B
B
K
B
B
K
B
K
B
B
K
B
B
K
B
B
K
B
B
K
B
éq 4.2.5-2
where them
I
B
are calculated with the X-coordinate
I
X
element of reference with:
L
X
L
NR
B
L
X
L
NR
B
L
X
L
NR
NR
B
L
NR
NR
B
I
xx
I
xx
I
xx
xx
X
X
6
2
6
4
12
6
1
,
6
4
,
4
3
2
2
,
5
,
3
2
,
2
,
1
1
+
-
=
=
+
-
=
=
+
-
=
-
=
=
=
=
-
=

éq
4.2.5-3
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Titrate:
Multifibre elements of beams (right)
Date
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18/11/03
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S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
14/18
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
4.3
Determination of the matrix of mass of the multifibre element
4.3.1 Determination of M
elem
In the same way, the virtual work of the efforts of inertia becomes [bib2]:
()
()
() ()
()
()
=




+
+
=
L
S
S
S
L
S
inert
dx
dt
U
D
M
U
dx
dS
dt
y
X
W
D
y
X
W
dt
y
X
v
D
y
X
v
dt
y
X
U
D
y
X
U
W
0
2
2
0
2
2
2
2
2
2
,
,
,
,
,
,
éq 4.3.1-1
with
S
U
the vector of “generalized” displacements.

What gives for the matrix of mass:














+
-
-
=
33
22
33
23
22
13
11
12
11
13
12
11
0
0
0
0
0
0
0
0
0
0
S
S
S
S
S
S
S
S
S
S
S
S
S
M
M
sym
M
M
M
M
M
M
M
M
M
M
M
éq
4.3.1-2
with:
=
-
=
=
-
=
=
=
S
S
S
S
S
S
S
S
S
S
S
S
ds
y
M
yzds
M
ds
Z
M
yds
M
zds
M
ds
M
2
33
23
2
22
13
12
11
;
;
;
;
éq
4.3.1-3
with
who can vary according to
y
and
Z
.
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Code_Aster
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Titrate:
Multifibre elements of beams (right)
Date
:
18/11/03
Author (S):
S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
15/18
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
As for the matrix of rigidity, we take into account the generalized deformations and
discretization of space [0, L]. What gives finally for the elementary matrix of mass:






































=
12
11
10
9
8
7
6
5
4
3
2
1
elem
elem
elem
elem
elem
elem
elem
elem
elem
elem
elem
elem
elem
M
M
M
M
M
M
M
M
M
M
M
M
M
with:


+
=


+
=


-
+
=


+
-
+
=


-
-
-
-
-
+
=


-
=


-
-
-
-
-
-
+
=


-
-
-
-
+
-
-
-
+
=


-
+
-
-
-
+
=


-
-
-
-
-
-
-
-
+
=


+
-
-
-
-
+
-
-
+
=


-
-
-
-
=
15
2
105
15
2
15
2
105
20
20
3
10
10
210
11
20
7
5
6
35
13
10
210
11
10
20
7
5
6
5
6
35
13
12
12
0
2
2
3
30
140
30
30
10
10
420
13
12
15
2
105
30
30
140
30
10
420
13
10
12
15
2
15
2
105
30
30
6
20
3
20
3
0
20
20
3
10
10
420
13
20
3
5
6
70
9
5
6
2
10
10
210
11
20
7
5
6
35
13
10
420
13
10
20
3
5
6
5
6
70
9
2
10
210
11
10
20
7
5
6
5
6
35
13
12
12
0
2
2
6
12
2
0
2
2
3
33
11
3
12
23
22
11
3
11
12
2
13
2
33
22
10
23
22
11
2
13
22
11
9
33
11
2
23
12
23
33
11
8
13
12
12
13
11
7
33
11
3
23
12
2
23
33
11
2
13
33
11
3
6
23
22
11
3
13
2
22
11
2
23
12
23
22
11
3
5
12
2
13
2
33
22
13
12
12
2
13
2
33
22
4
23
22
11
2
13
22
11
23
12
23
22
11
2
13
22
11
3
33
11
2
23
12
23
33
11
13
33
11
2
23
12
23
33
11
2
13
12
12
13
11
13
12
12
13
11
1
S
S
elem
S
S
S
elem
S
S
S
S
elem
S
S
S
S
S
S
elem
S
S
S
S
S
S
S
elem
S
S
S
S
S
elem
S
S
S
S
S
S
S
S
S
S
elem
S
S
S
S
S
S
S
S
S
S
S
elem
S
S
S
S
S
S
S
S
S
S
elem
S
S
S
S
S
S
S
S
S
S
S
S
S
S
elem
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
elem
S
S
S
S
S
S
S
S
S
S
elem
LM
M
L
sym
sym
sym
sym
sym
sym
sym
sym
sym
sym
sym
M
LM
LM
M
L
sym
sym
sym
sym
sym
sym
sym
sym
sym
sym
M
M
L
M
L
LM
LM
sym
sym
sym
sym
sym
sym
sym
sym
sym
M
M
M
M
L
LM
L
M
LM
sym
sym
sym
sym
sym
sym
sym
sym
M
M
M
L
M
LM
L
M
L
M
LM
sym
sym
sym
sym
sym
sym
sym
M
LM
LM
M
M
LM
sym
sym
sym
sym
sym
sym
M
LM
M
L
LM
M
L
M
M
M
L
LM
LM
M
L
sym
sym
sym
sym
sym
M
LM
LM
M
L
M
L
M
M
L
M
LM
LM
LM
M
L
sym
sym
sym
sym
M
M
L
M
L
LM
LM
LM
LM
M
L
M
L
LM
LM
sym
sym
sym
M
M
M
M
L
LM
L
M
LM
L
M
M
M
M
M
L
LM
L
M
LM
sym
sym
M
M
M
L
M
LM
L
M
L
M
LM
M
M
M
L
M
LM
L
M
L
M
LM
sym
M
LM
LM
M
M
LM
LM
LM
M
M
LM
M
éq 4.3.1-4
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Titrate:
Multifibre elements of beams (right)
Date
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18/11/03
Author (S):
S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
16/18
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
with the following terms:
23
33
22
13
12
11
,
,
,
,
,
S
S
S
S
S
S
M
M
M
M
M
M
who are given to the equation
[éq 4.3.1-3].
Note:
The matrix of mass is reduced by the technique of the concentrated masses ([bib4]). This
stamp of diagonal mass is obtained by the option
“MASS_MECA_DIAG”
of the operator
CALC_MATR_ELEM
.

4.3.2 Discretization of the fiber section - Calculation of M
S
The discretization of the fiber section makes it possible to calculate the various integrals which intervene
in the matrix of mass. Thus, if we have a section which comprises N fibers we will have them
approximation following of the integrals:
=
=
=
=
=
=
=
-
=
=
-
=
=
=
N
I
I
I
I
S
N
I
I
I
I
I
S
N
I
I
I
I
S
N
I
I
I
I
S
N
I
I
I
I
S
N
I
I
I
S
S
y
M
S
Z
y
M
S
Z
M
S
y
M
S
Z
M
S
M
1
2
33
1
23
1
2
22
1
13
1
12
1
11
;
;
;
;
éq
4.3.2-1
with
I
and
I
S
density and the section of each fiber.
I
y
and
I
Z
are the co-ordinates of
center of gravity of fiber defined as previously.

4.4
Calculation of the internal forces
The calculation of the nodal forces
int
F
had in a state of internal stresses given is done by
the integral:
dx
F
B
F
L
S
T
=
0
int
éq
4.4-1
where
B
is the matrix giving the generalized deformations according to nodal displacements
[éq 4.2.2-6] and where
S
F
is the vector of the generalized stresses given to the equation [éq 4.2.2-2],

background image
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Titrate:
Multifibre elements of beams (right)
Date
:
18/11/03
Author (S):
S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
17/18
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
Appear multifibre 4.4-a: Beam ­ Calcul of F
int
[
]
X
Z
y
T
S
M
M
M
NR
F
=
éq
4.4-2
Normal effort
NR
and bending moments
y
M
and
Z
M
are calculated by integration of the stresses
on the section [éq 4.2.1-8].
Behavior in linear torsion remaining, the torque is calculated with displacements
nodal:
L
GJ
M
X
X
X
X
1
2
-
=
éq.
4.4-3
The equation [éq 4.4-1] is integrated numerically:
() ()
=
=
=
L
I
I
S
T
I
I
S
T
I
X
F
X
B
W
J
dx
F
B
F
0
2
1
éq
4.4-4
The positions and weights of the points of Gauss as well as Jacobien are given in the paragraph
[§4.2.5].
The analytical calculation of
() ()
I
S
T
I
X
F
X
B
give:
() ()
[
]
[
]
Z
y
y
Z
Z
y
y
Z
T
I
S
T
I
M
B
M
B
M
B
M
B
NR
B
M
B
M
B
M
B
M
B
NR
B
X
F
X
B
4
4
2
2
1
3
3
2
2
1
0
0
-
-
-
=
éq 4.4-5
where it
I
B
are given to the equation [éq 4.2.4-1].
Center Section
Points of integration
Under-points of integration






=
S
S
int
S
ds
y
ds
F
=
L
0
int
S
T
int
dx
F
B
F
background image
Code_Aster
®
Version
6.4
Titrate:
Multifibre elements of beams (right)
Date
:
18/11/03
Author (S):
S. MILL, L. DAVENNE, F. GATUINGT
Key
:
R3.08.08-A
Page
:
18/18
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A
4.5
Nonlinear models of behavior usable
The supported models are on the one hand the relations of behavior 1D of the type
ECRO_LINE
and
PINTO_MENEGOTTO
[R5.03.09], in addition the model Labord_1D [R7.01.07] dedicated to the behavior of
concrete into cyclic. In addition all the laws 3D are usable thanks to
a routine “shunting” which
puts all the deformations other than the axial deformation (along fiber) at zero.
Note:
The internal, constant variables by fiber, are stored in the attached under-points
at the point of integration considered.


5 Bibliography
[1]
J.L. BATOZ, G. DHATT: Modeling of the structures by finite elements - HERMES.
[2]
J. GUEDES, P. PEGON & A. PINTO: With fiber Timoshenko bean element in CASTEM 2000 ­
Ispra, 1994
[3]
P. KOTRONIS: Dynamic shearing of reinforced concrete walls. Model simplified 2D and 3D
­ Thèse of Doctorate of the ENS Cachan ­ 2000
[4]
J.M. PROIX, P. MIALON, m.t. BOURDEIX: “Exact” elements of beams (right and
curves), Reference material of Code_Aster [R3.08.01]