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Multifibre elements of beams (right)

Date
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:
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Organization (S): EDF-R & D/AMA, LMT Cachan
Handbook of Référence
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 (Elément 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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Multifibre elements of beams (right)

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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 multifibre right beam ........................................................................................................ 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 fiber section ­ Calcul de Ks ............................................................ 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 Melem ......................................................................................................... 14
4.3.2 Discretization of the fiber section - Calcul of ms ............................................................ 16
4.4 Calculation of the forces intern ............................................................................................................. 16
4.5 Nonlinear models of behavior usable ..................................................................... 18
5 Bibliography ........................................................................................................................................ 18

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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 U, v, W, X, y, Z.

E
Young modulus
E


Poisson's ratio
NAKED
G
E
G
modulate of Coulomb = (

2 1+)
I
y,
y, I Z
geometrical moments of inflection compared to the axes
Z
IY, IZ
J X
constant of torsion
JX
K
stamp rigidity


M
stamp of mass

M
X, y,
X, M y, M Z
moments around the axes
Z
MT, MFY, MFZ
NR
normal effort with the section
NR

S
surface of the section
With
U, v, W
translations on axes X, y, Z
DX DY DZ
V
sharp efforts along axes y, Z
y, Vz
VY, VZ

density
RHO

rotations around axes X, y, Z
X, y, Z
DRX DRY DRZ



qx, qy, qz
External linear efforts


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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
traditional models E.F as well as the models of the type “fiber” (having an element of the beam type
how support).

While “traditional” 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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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 is the cross-section (section
transversal) or profile, and it is supposed that if it is evolutionary, its evolutions (size, form) are
continuous and progressive when G describes 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 U, v, W and the three
rotations X, y, Z [Figure 2-a]).


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

Waited until the deformations are local, it is built in each node of the grid 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 plan (y, Z).

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 122), 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:

U, v, W, U, v, W,
11 1 4
4
4 1 21x y 4
4
4 1 31z 12 2 4
4
4
4
2 22x y2 4
4
4
4
3
z2
node 1

node 2

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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 option “MASS_MECA”
for the coherent matrix, and 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.

y
y
1
2
1
2
X
0
L
X
u1 1 X
u2 2 X
Z
Z
v1 1 y
v2 2 y
w1
1 Z
w2 2 Z
Appear 4.1-a: Elément 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:

{Custom} = [NR] {U}.

The index S indicates the quantities attached to average fiber.
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By using the functions of forms of the element of reference, the discretization of the variables
custom (X), vs (X), ws (X), sx (X), sy (X), sz (X) becomes:
1
U


1
v
W
1
U

S (X)


N1
0
0
0
0
0
N2
0
0
0
0
0 1
X
v

S (X)




0
N3
0
0
0
N4
0
N5
0
0
0
N6 1y

ws (X)





0
0
N3
0
- N4
0
0
0
N5
0
- NR
0
6


=
1
Z


sx 0
0
0
N1
0
0
0
0
0
N2
0
0 u2
0
0
sy
- NR, 3
0
X
N4,
0
0
0
X
- NR
0
NR
0 v


,
5 X
6, X
2


NR
NR
NR
NR
W
sz
0
,
3
0
0
0
X
4,
0
X
,
5
0
0
0
X
6, X 2


x2
y2


z2
éq 4.1-1
With the following functions of interpolation:

X
1
N1 = 1
; NR, 1 = -
L
X
L
X
1
N2 =
; N2, =
L
X
L
2
3
X
X
6
X
N3 = 1 - 3
+ 2
; NR, 3 = -
+12
2
3
xx
2
3
L
L
L
L
2
3
X
X
4
X


éq
4.1-2
N4 = X - 2
+
; N4, = - + 6
2
xx
2
L
L
L
L
2
3
X
X
6
X
N5 = 3
- 2
; NR, 5 =
- 12
2
3
xx
2
3
L
L
L
L
2
3
X
X
2
X
N6 = -
+
; N6, = - + 6
2
xx
2
L
L
L
L

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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 direction X, subjected to efforts distributed
qx, qy, qz [Figure 4.2.1-a].


Y, v

X, U
Z, W

Appear 4.2.1-a: Poutre d' 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 the displacement (custom) of the line of
average:

U (X, y, Z) = custom (X) - ysz (X) + zsy (X) éq
4.2.1-1
v (X, y, Z) = vs (X)
éq
4.2.1-2
(
W X, y, Z) = ws (X)
éq
4.2.1-3
= u'
'
'
xx
X (X) - y sz (X) + Z sy (X)
éq
4.2.1-4
xy = xz = 0
éq
4.2.1-5

Note:

· Torsion is treated overall separately, one does not calculate yz here.
· f' (X) indicates the derivative of F (X) 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:


L
xx xx 0
FD =
U
X Q
v
X Q
W

X Q dx
éq
4.2.1-6
V
(S () X + S () y + S () Z)
0
0

qx, qy, qz indicating the linear efforts applied.
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What gives by using the equation [éq 4.2.1-1]:

L (NR u'
'
'
'
S (X) + M xsx (X) + M ysy (X) + M zsz (X))dx =



0
L (custom (X) qx + vs (X) qy + ws (X) qz) dx
0
éq 4.2.1-7

With:

NR = xxdS; M y = Z dS; M
y dS éq
4.2.1-8
S
xx
Z =
S
-
S
xx

Note:

· The torque M X 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 = E xx

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 axis X with the section S or on
orientation of the axes Y, Z, the relation between the “generalized” constraints and deformation
“generalized” Ds becomes [bib2]:

S
F = KS Ds




éq 4.2.2-1
with:
S
F = (NR, M y, M Z, M) T
X

éq
4.2.2-2
D
'
'
'
'
S = (custom (X), sy (X), sz (X), (X))T
sx
Center
Cross-section
Material 1
Material 3
Material 2

Appear 4.2.2-a: Section S unspecified - multifibre beam
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The matrix K S can then be put in the following form:

K 11
S
K 12
S
K 13
0
S




Ks22 Ks23
0
Ks

éq
4.2.2-3
K
0

s33

sym
Ks44

with:
Ks11 = EdS; Ks12 = Ezds; K
Eyds
S

s13 = -
S
S

éq
4.2.2-4
K
2
2
s22 = Ez dS; Ks23 = - Eyzds; K
Ey ds
S

s33 =
S
S

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 K S is detailed in the paragraph [§4.2.3] according to.

Note:

The term of torsion K s44 = GJ X is given by the user using the data of J X.

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 T
D
S KsDsdx -


0 éq
4.2.2-5
0
L (custom (X) qx + vs (X) qy + ws (X) qz) dx =
0

The generalized deformations are calculated by (Ds is given to the equation [éq 4.2.2-2]):

Ds = {
B U}
éq
4.2.2-6
With the following matrix B:

NR
0
0
0
0
0
NR
0
0
0
0
0
,
1 X
2, X



0
0
- NR
0
NR
0
0
0
,
3 xx
4, xx
- NR
0
NR
0
,
5 xx
6, xx

B =
0
NR
0
0
0
NR
0
NR
0
0
0
NR


,
3 xx
4, xx
,
5 xx
6, xx

0
0
0
NR
0
0
0
0
0
NR
0
0
,
1 X
2, X


éq 4.2.2-7

Discretization of space [,
0 L] 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 traditional linear system:

KU = F éq
4.2.2-8
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The matrix of rigidity of the element [Figure 4.2.2-b] and the vector of the efforts results are finally
given by:
L
K
=
BT K B dx
elem

S
0
éq
4.2.2-9
L
F =
NR T Q dx
0

Center
Cross-section
Points of integration
Under-points of integration
L

E ds
E y ds
T
S
S

K
B K B dx
elem =
K S =

S

2

0
E y ds
E y ds
S
S


Appear multifibre 4.2.2-b: Poutre ­ Calcul de Kelem

With the vector Q which depends on the external loading: Q = (qx qy Q 0 0 0) T
Z


If we consider that the efforts distributed qx, qy, qz are constant, we obtain the vector
nodal forces according to:

T
Lq Lq Lq
L2q L2q
Lq
Lq
Lq
L2q L2q

X
y
Z
Z
y
X
y
Z
Z
y
F

=
0 -
0
éq
4.2.2-10
2
2
2
12
12
2
2
2
12
12



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4.2.3 Discretization of the fiber section ­ Calcul de Ks

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
N
N
Ks11 = EiSi; Ks12 = EiziSi; Ks13 = I.E.(internal excitation) yiSi
i=1
i=1
i=1


éq 4.2.3-1
N
N
N
K
2
2
s22 = I.E.(internal excitation) zi If
; Ks23 = - I.E.(internal excitation) yi ziSi; Ks33 = I.E.(internal excitation) yi If
i=1
i=1
i=1

with I.E.(internal excitation) and If the initial or tangent module and the section of each fiber. The state of stress is
constant by fiber.
Each fiber is also located using yi and zi the co-ordinates of the center of gravity of
fiber compared to the axis of the section defined by command “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:

K
K
K
s11
s12
s13
- Ks11
- Ks12
- Ks13

0
0
0
0
0
0

L
L
L
L
L
L


12Ks33 - 12K
6K
6K
s23
s23
s33
- 12K
12K
6K
6K
s33
s23
s23
s33
0
0
0


L3
L3
L2
L2
L3
L3
L2
L2


12Ks22
- 6Ks22 - 6K
12K
s23
s23
- 12Ks22
- 6Ks22 - 6Ks23

0
0
0
3
2
2
3
3
2
2


L
L
L
L
L
L
L


Ks44
0
0
0
0
0
- Ks44
0
0


L
L


4K
4K
s22
s23
- Ks12 - 6K
6K
2K
2K
s23
s22
s22
s23

0
2
2


L
L
L
L
L
L
L


4Ks33
- Ks13 - 6K
6K
2K
2K
s33
s23
s23
s33
0


L
L
L2
L2
L
L

Kelem =
K
K
K


s11
s12
s13
0
0
0


L
L
L


12Ks33
- 12Ks23
- 6Ks23 - 6Ks33
0


L3
L3
L2
L2


12K
6K
6K


SYM
s22
s22
s23
0
3
2
2


L
L
L


Ks44

0
0

L


4K
4K


s22
s23

L
L


4Ks33


L



éq 4.2.4-1
with the following terms: K 11,
S
K 12,
S
K 13,
S
Ks22, Ks33, Ks23, Ks44 are given to the equation
[éq 4.2.2-4].
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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 Kelem [éq 4.2.2-9] is calculated in numerical form:
L
2
T
Kelem =
B K

S B dx = J I
W B (X) T
I
Ks (xi) B (xi) éq
4.2.5-1
0
i=1
where xi are 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. I here is taken
W = 0,5 for each of the 2 points
J is Jacobien One takes J here = L, the real element having a length L and the function of form
X
to pass to the element of reference being
.
L
Ks (xi) 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 B (X) T
I
Ks (xi) B (xi) gives:

21
B K 11
S
- 1
B 2
B K 13
S
1
B 2
B K 12
0
S
- 1
B 3
B K 12
S
- 1
B 3
B K 13
S
- 21
B K 11
S
1
B 2
B K 13
S
- 1
B 2
B K 12
0
S
- 1
B 4
B K 12
S
- B B K

1 4
13
S


2
2
2
B Ks33
2
B Ks23
0
2
B 3
B Ks23
2
B 3
B Ks33
1
B 2
B K 13
S
- 2
2
2
B Ks33
2
B Ks23
0
2
B 4
B Ks23
2
B 4
B Ks33

2
2
2

2
B Ks22
0
- 2
B 3
B Ks22 - 2
B 3
B Ks23 - 1
B 2
B K 12
S
2
B Ks23
- 2
B Ks22
0
- 2
B 4
B Ks22 -

2
B 4
B Ks23

2
1
B Ks44
0
0
0
0
0
- 21
B Ks44
0
0


2
2


3
B Ks22
3
B Ks23
1
B 3
B K 12
S
- 2
B 3
B Ks23
2
B 3
B Ks22
0
3
B 4
B Ks22
3
B 4
B Ks23

2
3
B Ks33
1
B 3
B K 13
S
- B B K
B B K
0
B B K
B B K


2 3 s33
2 3 s23
3 4 s23
3 4 s33
2


1
B K 11
S
- 1
B 2
B K 13
S
1
B 2
B K 12
0
S
1
B 4
B K 12
S
1
B 4
B K 13
S


2
2
B Ks33
- 22
B Ks23
0
- 2
B 4
B Ks23 - B B K

2 4 s33

2
2
B Ks22
0
2
B 4
B Ks22
2
B 4
B Ks23

2

B K
0
0

1
s44


2
2
4
B Ks22
4
B Ks23

2


4
B Ks33

éq 4.2.5-2

where I
B are calculated with X-coordinate xi of the element of reference with:
1
B = - NR
= NR
1
,
1 X
2, X = L
6
X
B = NR
= - NR
I
2
,
3 xx
,
5 xx = -
+12
L2
L2 éq
4.2.5-3
4
X
B = NR
I
3
4, xx = -
+ 6
L
L
2
X
B = NR
I
4
6, xx = -
+ 6
L
L
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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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
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4.3
Determination of the matrix of mass of the multifibre element

4.3.1 Determination of Melem

In the same way, the virtual work of the efforts of inertia becomes [bib2]:

L

D 2u X, y
D 2v X, y
D 2w X, y
inert
W
=,
,

,
0
U (X y)
() + v (X y) () + (wx y) () dSdx
S

2
2
2


dt
dt
dt



2
= L
D U.S.
U
Sm S
dx
0
dt 2
éq 4.3.1-1

with U.S. the vector of “generalized” displacements.

What gives for the matrix of mass:

M 11
0
0
S
M 12
S
M 13
0
S




M 11
0
0
0
S
- M 12
S


M 11
0
0
S
- M

M S =
13
S
éq
4.3.1-2

M S22 M S23
0


M
0


s33


sym
M S22 + M S33

with:

M s11 = ds; Ms12 =
zds

; M
yds

S

s13 = -
S
S

éq
4.3.1-3
M
2
2
s22 = Z
ds; M s23 = -
yzds

; M
y
ds
S

s33 =
S
S

with which can vary according to y and Z.
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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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
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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:

1
M

elem
2
M

elem
3
M elem


4
M elem
5
M elem
6
M elem
M elem =

7

M elem
8
M

elem
9
M

elem
10
M elem


11
M elem
12
M elem
with:

LM
M
M
LM
LM
LM
M
M
LM
LM
1

11
S
-
13
S
12
S
12
S
13
S
11
S
13
S
-
12
S
-
12
S
-
13
S

M elem =
0
0

3
2
2
2
12
6
2
2
12
12


LM
M
M
LM
M
L M
M
M
LM
M
M
LM
M
L M
M
2
13
6
11
S
s33 - 6
s23 - 7
11 2
12
S
s23
11
S
s33 -
9
6
6
13
S
11
S
s33
s23 - 3
12
S
s23 - 13 2

M
sym
elem =
+
+
-
11
S
+ s33

35
5L
5L
20
10
210
10
2
70
5L
5L
20
10
420
10

LM
M
LM
L M
M
M
M
M
LM
M
LM
L M
M
M
3
13
6
11
S
s22 - 7
S
- 11 2
13
11
S
s22 -
6
9
6
s23
12
S
s23
11
S
S 22 - 3
13 2
13
S
11
S
s22 -

M
sym sym
elem =
+
-
-
-
s23

35
5L
20
210
10
10
2
5L
70
5L
20
420
10
10

LM
LM
L M
L M
LM
LM
LM
LM
L M
L M
4
s22 +
2
s33
S
- 2
13
12
S
- 3
12
S
- 3
13
S
s22 +
s33 -
2
2

M
sym sym sym
elem =
0
13
S
12
S


3
20
20
20
20
6
30
30

3
L M
LM
LM
LM
M
L M
M
L M
L M
LM
LM
5
2
2
11
S
S 22
s23 -
12
S
- s23 - 13 2
2
11
S
s22
S
- 3
13
11
S
S 22 -

M
sym sym sym sym
elem =
+
+
-
-
s23

105
15
15
12
10
420
10
30
140
30
30


3
L M
LM
LM
L M
M
M
L M
LM
L M
LM
6
2
11
S
s33 -
13 2
13
S
11
S
s33
S
- 2
23
12
S
-
s23 -
3

M
sym sym sym sym sym
elem =
+
-
11
S
-
s33

105
15
12
420
10
10
30
30
140
30
LM
M
M
LM
LM
7

11
S
13
S
-
12
S
12
S
13
S

M
sym sym sym sym sym sym
elem =
0


3
2
2
12
12

LM
M
M
LM
M
L M
M
8
13
6
11
S
s33 - 6
S 23 - 7
12
S
- s23 - 11 2

M
sym sym sym sym sym sym sym
elem =
+
11
S
- s33

35
5L
5L
20
10
210
10

LM
M
LM
L M
M
M
9
13
6
11
S
s22 - 7
11 2

M
sym sym sym sym sym sym sym sym
elem =
+
13
S
11
S
+
s22
s23

35
5L
20
210
10
10

LM
LM
L M
L M
10
s22 +
s33 -
2
2

M
sym sym sym sym sym sym sym sym sym
elem =
13
S
12
S


3
20
20

3
L M
LM
LM
11
2
2

M
=
11
S
sym sym sym sym sym sym sym sym sym sym
elem
+
S 22
s23

105
15
15


3
L M
LM
12
2

M
=
11
S
sym sym sym sym sym sym sym sym sym sym sym
elem
+
s33

105
15


éq 4.3.1-4
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

Code_Aster ®
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6.4
Titrate:
Multifibre elements of beams (right)

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:
R3.08.08-A Page
: 16/18

with the following terms: M
, M, M, M
, M, M
who are given to the equation
11
S
12
S
13
S
s22
s33
s23
[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 option “MASS_MECA_DIAG” of the operator
CALC_MATR_ELEM.

4.3.2 Discretization of the fiber section - Calcul of ms

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
N
N
M
S; M
Z S; M
y S
s11 = I I
s12 = I I I
s13 = - I I I
i=1
i=1
i=1
éq
4.3.2-1
N
N
N
M
Z 2S; M
y Z S; M
y 2S
s22 = I I
I
s23 = - I I I I
s33 = I I
I
i=1
i=1
i=1

with and S density and the section of each fiber. y and Z are the co-ordinates of
I
I
I
I
center of gravity of fiber defined as previously.

4.4
Calculation of the internal forces

The calculation of the nodal forces F due in a state of internal stresses given is done by
int
the integral:

L
F

éq
4.4-1
int =
BT F dx

S
0

where B is the matrix giving the generalized deformations according to nodal displacements
[éq 4.2.2-6] and where F is the vector of the generalized constraints given to the equation [éq 4.2.2-2],
S

Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

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
: 17/18

Center Section
Points of integration
Under-points of integration
L
T

ds
F
B F
dx
int =
S int
F

S
= S
int

0

y ds
S


Appear multifibre 4.4-a: Poutre ­ Calcul de Fint

T
F = NR M
M
M
éq
4.4-2
S
[
y
Z
X]

The normal effort NR and the bending moments M and M are calculated by integration of the constraints
y
Z
on the section [éq 4.2.1-8].

Behavior in linear torsion remaining, the torque is calculated with displacements
nodal:

-
M = GJ
x2
x1
éq.
4.4-3
X
X
L

The equation [éq 4.4-1] is integrated numerically:
2
F = L T
B F dx
J
W B X
F X éq
4.4-4
I
S
=
0
I () T
I
S (I)
i=1
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 B (X) T F X gives:
I
S (I)

[B (X) T F X = - B NR B M - B M 0 B MR. B MR. B N-B MR. B M 0 B MR. B M
I
S (
)]T
I
[1 2 Z 2 y
3
y
3
Z
1
2
Z
2
y
4
y
4
Z]
éq 4.4-5

where B are given to the equation [éq 4.2.4-1].
I
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

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

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 Doctorat of the ENS Cachan ­ 2000
[4]
J.M. PROIX, P. MIALON, m.t. BOURDEIX: “Exact” elements of beams (right and
curves), Documentation de Référence of Code_Aster [R3.08.01]
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/03/005/A

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