Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
1/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Organization (S):
EDF-R & D/AMA
Manual of Reference
R3.08 booklet: Machine elements with average fiber
R3.08.04 document
Element of beam with 7 ddl for the taking into account
roll
Summary:
This document presents the element
POU_D_TG
who is a finite element of right beam with taking into account of
roll of the sections. It allows the calculation of the beams mean transverse sections and opened profile, with
constrained or free torsion.
With regard to the bending, the normal and sharp efforts, this element is based on the element
POU_D_T
,
who is an element
of right beam with transverse shearing (model of Timoshenko).
For the element
POU_D_TG
, the section is supposed to be constant (of an unspecified form) and the material is
homogeneous and isotropic, of linear or elastoplastic elastic behavior (behaviors
VMIS_POU_LINE
and
VMIS_POU_FLEJOU
).
This reference material is based on the general reference material of the beams, in
linear elasticity [R3.08.01] and in élasto - plasticity [R5.03.30]. It describes specificities of the element of beam
straight line with roll
POU_D_TG
.
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
2/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Count
matters
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
3/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
1 Field
of use
The development of the elements of beam of Timoshenko with roll (modeling
POU_D_TG
) in Code_Aster was carried out initially with an aim of calculating the behavior
pylons. The first development was made within the framework of a thesis at the Department MN. It
mainly acted to calculate formed structures of beams with open mean profile
(angles), for which the roll is important. Plasticity was introduced into
the element
POU_D_TG
[R5.03.30], but the nonlinear behavior relates only to traction,
bending and torsion. Shearing due to the sharp effort, as well as the roll and the Bi-moment
(effort related to the roll) remain dependant by an elastic behavior, fault of being able to express one
nonlinear behavior on these sizes. This is why the description of torsion with
roll is valid for the use of the element
POU_D_TG
with the linear operators
(
MECA_STATIQUE
,
DYNA_LINE_TRAN
,…) or not linear (
STAT_NON_LINE
,
DYNA_NON_LINE
,…).
2 Notations
The notations used here correspond to those used in [R3.08.01] and [R3.08.03]. One gives here
correspondence between this notation and that of the documentation of use.
GRX
DRZ
DRY
DRX
DZ
DY
DX
and
,
,
,
,
,
are the names of the degrees of freedom associated with
components of displacement
X
X
Z
y
X
W
v
U
,
,
,
,
,
,
,
.Ils is expressed in total reference mark, except the degree
of freedom associated with the roll GRX, which is expressed in local reference mark.
Notation used
Significance
Notation
of
documentation
of use
S
surface of the section
With
Z
y
I
I,
geometrical moments of bending compared to the axes
X and Y.
IZ
IY,
C
constant of torsion
JX
I
constant of roll
JG
Z
y
K
K
,
coefficients of shearing
AZ
AY
1
1
Z
y
E
E
,
eccentricity of the center of torsion/shearing by
report/ratio in the center of gravity of the cross-section
EZ
EY,
NR
normal effort with the section
NR
Z
y
V
V
,
sharp efforts along axes y and Z
VZ
VY,
Z
y
X
M
M
M
,
,
moments around axes X, y and Z
MFZ
MFY
MT
,
,
M
Bi-moment
BX
W
v
U
,
,
translations on axes X, y and Z
DZ
DY
DX
Z
y
X
,
,
rotations around axes X, y and Z
DRZ
DRY
DRX
X
X,
rotary derivative of torsion according to X
GRX
E
Young modulus
E
Poisson's ratio
NAKED
(
)
µ
=
+
= 1
2
E
G
modulate of Coulomb (identical to the coefficient of Lamé)
G
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
4/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
3
Kinematics specific to torsion with roll
Kinematics used to represent the displacement of the sections of beam is identical to that
right beams of Timoshenko [R3.08.01] with regard to the traction and compression, and
bending - shearing. Only torsion here is detailed.
Two possibilities are to be considered for the modeling of behavior in torsion of the sections not
circulars [bib1], which always produces a roll of the cross-section.
·
torsion is free (torsion of Saint-Coming): the roll of the cross-sections is not
no one (it can even be important for an open mean section), but it is independent of
the position on axis X of the beam, (constant according to X) and it does not have there a stress
axial due to torsion.
·
Torsion is constrained (Vlassov): the roll is nonnull, and moreover of the stresses
axial not uniforms (from which the effort resulting Bi-moment is called) exist in the beam.
The element
POU_D_TG
allows to treat these two configurations: torsion can be free or constrained.
The user will have access to the roll in both cases, on the other hand the Bi-moment will not be nonnull
that in the case of constrained torsion. It should be noted that at the place of the connection of the beams,
transmission of the roll depends on the type of connection. In general, torsion in an assembly
beams is constrained. The roll can then be locked at the points of connection.
Note:
With elements without modeling of the roll (
POU_D_T
,
POU_D_E
), one can
to treat the case of free torsion (displacements other than the roll will be
correct), but not the case of constrained torsion.
One can uncouple the effects of torsion and bending in a local reference mark (relocated main reference mark
of inertia) having for origin the center of torsion. The center of torsion is the point which remains fixed when
the section is subjected to the only torque. It is also called center of shearing because one
effort applied in this point does not produce rotation around X.
Displacements in the plan of the section will thus be expressed in this reference mark. Displacements
axial remain expressed in the main reference mark of inertia related to the center of gravity G, to keep one
decoupling of displacements of bending and traction and compression.
The displacement of an unspecified point of the cross-section is written then in general form (torsion
free or constrained):
(
)
(
)
(
)
()
()
()
() ()
(
) ()
(
) ()
ent
gauchissem
with
torsion
bending/Z
bending/y
membrane
T
déplacemen
+
+
+
=
-
-
-
+
-
+
+
=
X
y
y
X
Z
Z
X
Z
y
X
v
X
y
X
W
X
Z
X
U
Z
y
X
W
Z
y
X
v
Z
y
X
U
X
C
X
C
X
X
Z
y
G
,
,
0
)
(
)
(
0
0
0
,
,
,
,
,
,
The components are expressed in the main reference mark of inertia (centered in G): X is directed according to
the axis of the beam, y and Z are the two other main axes of inertia.
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
5/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
The term
() ()
X
Z
y
X
X,
,
represent axial displacement due to the roll of the cross-section.
()
Z
y,
is related to roll (expressed in m ², but which does not have physical interpretation
obvious).
The deformations of an unspecified point of the section are then:
(
)
(
)
(
)
()
()
()
()
()
(
)
(
)
()
(
)
(
)
()
y
X
xz
Z
X
xy
X
X
C
Z
X
X
C
y
xx
X
xy
X
Z
xz
X
y
X
G
xz
xy
xx
W
X
v
X
X
y
y
X
Z
Z
X
Z
y
X
X
y
X
X
Z
X
U
Z
y
X
Z
y
X
Z
y
X
+
=
-
=
-
+
-
-
+
-
+
+
=
,
,
,
,
,
,
,
,
,
,
)
(
)
(
,
0
)
(
)
(
0
0
0
,
,
2
,
,
2
,
,
Deformation = membrane + bending/y + bending/Z + torsion with roll
The term
()
()
X
Z
y
xx
X,
,
is null in the case of free torsion: one has indeed
()
0
,
=
X
xx
X
, since
the roll is independent of X. It is considerable in the case of constrained torsion.
The law of elastic behavior isotropic is written (by making the assumption of the plane stresses in
directions y and Z):
(
)
(
)
(
)
(
)
(
)
(
)
=
Z
y
X
G
Z
y
X
G
Z
y
X
E
Z
y
X
Z
y
X
Z
y
X
xz
xy
xx
xz
xy
xx
,
,
2
.
,
,
2
.
,
,
.
,
,
,
,
,
,
The efforts generalized in the section are expressed according to the stresses for a section
homogeneous by [bib1]:
(
)
(
)
(
)
(
)
(
)
(
) (
) (
) (
)
(
)
(
)
ent)
gauchissem
with
(associate
moment
-
Bi
torsion
of
moment
Z
of
around
bending
of
moment
y
of
around
bending
of
moment
Z
according to
edge
effort
y
according to
edge
effort
normal
effort
=
-
-
-
=
-
=
=
=
=
=
S
xx
S
xy
C
xz
C
X
S
xx
Z
S
xx
y
S
xz
Z
S
xy
y
S
xx
ds
Z
y
X
X
M
ds
Z
y
X
Z
Z
Z
y
X
y
y
X
M
ds
Z
y
X
y
X
M
ds
Z
y
X
Z
X
M
ds
Z
y
X
X
V
ds
Z
y
X
X
V
ds
Z
y
X
X
NR
,
,
.
)
(
,
,
.
,
,
.
)
(
,
,
.
)
(
,
,
.
)
(
,
,
)
(
,
,
)
(
,
,
)
(
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
6/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
)
(X
M
represent the generalized effort associated with the roll. It is expressed in N.m
2
. One can in
to give an illustration as in [bib1] for a beam to section in
I
(the Bi-moment acts here according to
Z only):
X
Y
Z
xx +
xx+
-
-
xx
xx
X
Y
Z
M
For an isotropic and homogeneous elastic behavior in the section, the efforts generalized
thus express themselves directly according to displacements by the following relations:
(
)
(
)
X, xx
X, X
X
X
Z
Z
Z
X
y
y
y
X
Z
Z
Z
X
y
y
X
I
E
X
M
J
G
X
M
I
E
X
M
I
E
X
M
W
S
Gk
X
V
v
S
Gk
X
V
U
S
E
X
NR
.
.
)
(
.
.
)
(
.
)
(
.
)
(
)
(
)
(
.
.
)
(
,
,
,
=
=
=
=
+
=
-
=
=
,
y,
where
Z
y
K
K,
are the coefficients of shearing. The roll does not intervene on the level of the efforts
edges, because those are expressed in the reference mark related to the center of shearing. Indeed, the function
of roll
is such as:
()
()
()
0
,
.
0
,
.
0
,
=
=
=
S
S
S
ds
Z
y
Z
ds
Z
y
y
ds
Z
y
And the constant of roll is expressed according to
by:
()
I
ds
Z
y
S
=
,
2
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
7/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
4
Element of right beam with roll: matrices of
rigidity and of mass
Elementary matrices of rigidity and mass for the element
POU_D_TG
are identical to those
element of right beam of Timoshenko (
POU_D_T
) with regard to the terms of traction -
compression and of bending - shearing [R3.08.01]. The step is identical, one recalls
simply the result.
This implies that, in the case of free torsion, one preserves the properties of exactitude of the solution
with the nodes for the degrees of freedom of bending and traction and compression.
On the other hand, we will see that with regard to obstructed torsion, one carries out an approximation which
does not allow to find this property in the general case.
The matrices of rigidity are always calculated with the option
'
RIGI_MECA'
, and matrices of mass
with the option
“MASS_MECA”
. But the option '
MASS_MECA_DIAG'
(matrix of diagonalized mass) does not have
not realized for this element (this option is especially useful for the problem of dynamics
rapid, which is not the preferential field of application of this element).
The degrees of freedom of the element are those of the beams of Timoshenko, plus a degree of freedom by
node allowing to calculate the terms relating to the roll:
In each of the two nodes of the element, the degrees of freedom are:
W
v
U,
,
translations on the axes
Z
y
X,
,
DZ
DY
DX
Z
y
X
,
,
rotations around the axes
Z
y
X,
,
DRZ
DRY
DRX
X, X
rotary derivative of torsion according to
X
GRX
The local co-ordinates are expressed in the main reference mark of inertia. The element
POU_D_TG
thus comprise 14 degrees of freedom. The element of reference is defined by: 0 < X < L
X
Y
Z
U
v
W
X
Y
Z
Z
X
y
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
8/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
4.1
Traction and compression the degree of freedom are U or DX
The matrix of rigidity of the element is:
-
-
=
1
1
1
1
L
ES
K
The matrix of mass (coherent) is written:
2
1
1
2
6
SSL
=
M
4.2 Bending in the plan (Gxz) the degrees of freedom concerned are W,
y or DZ, DRY
The matrix of rigidity is written for the movement of bending in the main plan of inertia (Gxz):
(
)
(
)
(
)
(
)
-
-
-
-
12
4
2
1
Sym
12
2
2
12
4
2
1
2
1
1
12
2
2
2
3
L
+
L
L
L
L
+
L
L
+
L
I.E.(internal excitation)
=
y
y
y
y
y
K
Transverse shearing is taken into account by the term:
2
12
SGL
K
I.E.(internal excitation)
Z
y
y
=
For the matrix of mass,
()
T
X,
W
and
()
T
X,
y
are discretized on the basis of function tests
introduced for the calculation of the matrix of rigidity, that is to say:
() ()
()
() ()
()
() ()
()
() ()
()
)
(
)
(
)
,
(
)
(
)
(
)
,
(
2
1
2
1
8
2
7
6
1
5
4
2
3
2
1
1
T
X
T
W
X
T
X
T
W
X
T
X
T
X
T
W
X
T
X
T
W
X
T
X
W
y
y
y
y
y
+
+
+
=
+
+
+
=
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
9/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
The function of interpolation used for the translations (
1
with
4
) are polynomials of Hermit of
degree 3, that which is used for rotations (
5
with
8
) are degree 2: for 0 < X < L, they are
defined by [R3.08.01]:
(
)
(
)
(
)
(
)
(
)
(
)
-
-
-
-
-
-
+
-
-
+
-
-
+
-
-
L
X
+
+
L
X
+
=
(X)
L
X
+
L
X
+
L
X
+
L
=
(X)
L
X
L
X
+
L
=
(X)
L
X
+
L
X
+
L
X
+
=
(X)
+
+
L
X
L
X
+
=
(X)
L
X
+
L
X
+
L
X
+
L
=
(X)
L
X
L
X
+
L
=
(X)
+
L
X
L
X
L
X
+
=
(X)
y
y
y
y
y
y
y
y
y
y
y
y
y
y
y
y
y
y
2
3
1
1
2
2
2
1
1
1
6
3
2
1
1
1
4
3
1
1
2
2
2
4
1
1
1
6
1
3
2
1
1
2
8
2
3
4
7
2
3
3
2
6
2
3
2
5
2
3
1
éq 4.2-1
The form of the matrix of mass is:
(
)
M =
S
+
y
y
y
y
y
L + L + L
11L
11L
L
L + L + L
L + L + L24
L + L + L
L
3L
40
L
24
L
L
L
120
L + L + L
L + L
y
y
y
y
y
y
y
y
2
y
y
y
y
1
2
13
35
7
10
3
210
120
24
9
70
3
10
6
13
420
3
40
105
60
120
13
420
140
60
13
35
7
10
3
11
210
11
2
2
2
2
2
2
2
2
2
2
3
3
3
2
2
2
2
3
3
3
2
2
2
2
-
-
-
-
-
-
-
-
-
-
(
)
y
y
y
y
+ L
L + L + L
sym
+
I
+
L
+
L
+
L + L + L
L
L + L
L
sym
L + L + L
y
y
y
y
y
y
y
y
y
y
y
y
120
24
105
60
120
2
2
3
3
3
2
1
6
5
1
10
2
6
5
1
10
2
2
15
6
3
1
10
2
30
6
6
6
5
1
10
2
2
15
6
3
2
2
2
2
-
-
-
-
-
-
-
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
10/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
4.3
Bending in the plan (Gxy) the degrees of freedom concerned are
, Z
or DY, DRZ
In the same way, for the movement of bending around the axis (Gz), in the main plan of inertia (Gxy),
stamp rigidity is written:
(
)
(
)
(
)
(
)
-
-
-
-
12
4
sym
2
1
12
2
2
12
4
2
1
2
1
1
12
2
2
2
3
L
+
L
L
L
L
+
L
L
+
L
I.E.(internal excitation)
=
Z
Z
Z
Z
Z
K
Transverse shearing is taken into account by the term:
2
12
SGL
K
I.E.(internal excitation)
y
Z
Z
=
To calculate the matrix of mass,
()
T
X,
v
and
()
T
X,
Z
are discretized by:
() () ()
() ()
() ()
() ()
()
() ()
() ()
() ()
() ()
T
X
T
v
X
T
X
T
v
X
T
X,
T
X
T
v
X
T
X
T
v
X
T
X,
v
Z
Z
Z
Z
Z
2
1
2
1
8
2
7
6
1
5
4
2
3
2
1
1
+
+
=
+
=
-
-
-
-
We obtain the matrix of following mass then:
(
)
(
)
+
-
-
-
+
-
-
-
-
-
-
-
-
-
+
+
-
-
-
+
+
3
2
6
15
2
2
10
1
5
6
6
2
6
30
2
10
1
3
2
6
15
2
2
10
1
5
6
2
10
1
5
6
2
120
3
60
105
24
120
11
210
11
3
10
7
35
13
120
3
60
140
24
40
3
420
13
120
60
105
24
40
3
420
13
6
10
3
70
9
24
120
11
210
11
3
10
7
35
13
2
1
1
2
3
3
2
2
2
2
2
2
3
3
2
2
2
2
2
3
3
3
2
2
2
2
2
2
2
2
2
2
Z
L
+
Z
L
+
L
sym
Z
L
Z
L
+
Z
L
L
Z
Z
L
+
Z
L
+
L
Z
L
Z
L
Z
Z
L
+
L
+
L
sym
L
L
L
L
+
L
+
L
L
L
L
L
Z
L
L
L
+
Z
L
+
L
L
L
L
L
+
L
+
L
L
Z
L
L
L
+
L
+
L
Z
+
I
+
+
S
=
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
M
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
11/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
4.4 Torsion and roll the degrees of freedom are
X, X, X or
DRX, GRX
With regard to torsion, the formulation is obviously different from that of the beams without
roll of the reference [R3.08.01]. The virtual work of the interior efforts is written for
torsion [bib1]:
dx
I
E
J
G
W
X, xx
X, xx
L
O
X, X
X, X
.
.
.
.
.
.
*
*
int
+
=
The functions of interpolation of the rotation of torsion must be of C2 class, since they must
to allow to interpolate the derivative second rotation.
By using the equilibrium equations, one shows in [bib1] that the analytical solution utilizes
function of interpolation hyperbolic in X. This then makes it possible to obtain exact results with
nodes. It is not the choice made for Code_Aster: one chose, by preoccupation with a simplicity for
numerical integration like avoiding the numerical problems of evaluation of the function
hyperbolic, of the polynomials of degree 3 of Hermit type, of the same kind as those used for
bending [éq 4.2-1]. One writes them here on the element of reference [- 1,1] according to [bib1] (instead of 0<x<L
previously):
1
1
1
2
-
-
=
L
X
()
(
) (
)
()
(
)
(
)
()
(
) (
)
()
(
)
(
)
2
4
2
3
2
2
2
1
1
1
8
2
1
4
1
1
1
8
2
1
4
1
+
-
+
=
-
+
=
-
-
=
+
-
=
L
NR
NR
L
NR
NR
The interpolation for the rotation of torsion and its derivative is:
()
()
()
()
()
2
4
2
3
1
2
1
1
,
,
X
X
X
X
X
X
NR
NR
NR
NR
X
+
+
+
=
()
()
()
()
()
2
,
4
2
,
3
1
,
2
1
,
1
,
,
,
X
X
X
X
X
X
NR
NR
NR
NR
X
X
X
X
X
X
+
+
+
=
The reference [bib1] note which this approximation corresponds to a borderline case of the interpolation
hyperbolic, obtained for
0
I.E.(internal excitation)
GJ
. However, this parameter not being without dimension, it is
difficult to define a priori the values for which the approximation is acceptable. Tests
numerical carried out show that one converges quickly towards the solution when the size of
elements decreases.
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
12/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
The matrix of rigidity corresponding to this approximation is written then:
-
-
-
+
-
-
-
-
=
+
2
2
2
3
2
2
2
4
sym
6
12
2
6
4
6
12
6
12
4
sym
3
36
3
4
6
36
3
36
30
L
L
L
L
L
L
L
L
I.E.(internal excitation)
L
L
L
L
L
L
L
L
GJ
=
K
K
K
T
The matrix of mass can be obtained in several ways [bib1]:
·
the most complete method would consist in calculating the terms of inertia with the functions
of interpolation above, by taking account of the additional term:
·
dx
I
W
L
O
iner
-
=
X
X,
X
X,
&&
.
.
.
*
·
in Code_Aster, the simplest method was selected: the matrix of mass is
identical to that of the element
POU_D_T
. One preserves the already definite terms for traction -
compression and the bending - shearing and one use a linear approximation for torsion.
The coefficients of the matrix of mass associated with the roll are null with this
approach.
4.5
Eccentricity of the axis of torsion compared to the neutral axis
In the center of torsion C, the effects of bending and torsion are uncoupled, one can thus use them
results established in the preceding chapter.
The co-ordinates of the point C are to be provided to
AFFE_CARA_ELEM
: one gives the components of
vector GC (G being the center of gravity of the cross-section) in the main reference mark of inertia:
Z
y
E
E
0
=
GC
One can numerically determine them starting from the plane mesh of the section using the operator
MACR_CARA_POUTRE
[R3.08.03].
Once the point C determined, one finds as in [R3.08.01] the components of displacement with
center of gravity G by considering the rigid relation of body:
rotation
vector
with
0
0
=
+
(C)
=
(G)
X
GC
U
U
-
.
E
W
=
W
E
+
v
=
v
U
=
U
X
y
C
G
X
Z
C
G
C
G
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
13/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
The change of variables is in the same way written that for
POU_D_T
, with 2 degrees of freedom
additional:
-
-
2
2
2
2
2
2
2
1
1
1
1
1
1
1
2
2
2
2
2
2
2
1
1
1
1
1
1
1
,
,
,
,
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
X
X
Z
y
X
Z
y
X
X
X
Z
y
X
Z
y
X
y
Z
y
Z
X
X
Z
y
X
Z
y
X
X
X
Z
y
X
Z
y
X
U
U
U
U
U
U
E
E
E
E
U
U
U
U
U
U
C
C
C
C
C
C
C
C
C
C
C
C
C
C
=
4
4
4
4
4
4
4
4
4
4
3
4
4
4
4
4
4
4
4
4
4
2
1
P
From the elementary matrices of mass and rigidity calculated previously in the reference mark
(C, X, y, Z) where the movements of bending and torsion are uncoupled, one obtains these matrices in
reference mark related to the neutral axis (G, X, y, Z) by the following transformations:
P
K
P
K
=
C
T
.
P
M
P
M
C
=
T
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
14/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
5
Geometrical rigidity - prestressed Structure
This matrix is calculated by the option: “
RIGI_GEOM
“. It is used to deal with problems of
buckling or of the vibrations of prestressed structures. In the case of a prestressed structure,
thus subjected to initial efforts (known and independent of time), one cannot neglect in
the equilibrium equation terms introduced by the change of geometry of the unconstrained state with
the prestressed state. This change of geometry modifies the equilibrium equation only by the addition of one
function term of displacements and prestressed with which the matrix associated is called matrix
of geometrical rigidity and which is expressed by:
FD
X
v
X
U
W
J
D
K
V
O
ij
I
D
K
G
O
3
3
=
where
O
ij
indicate the tensor of prestressing. This term appears naturally if the tensor is introduced
deformations of GREEN-LAGRANGE in the virtual work of the deformation:
+
+
+
+
=
+
=
+
+
+
+
=
+
=
+
+
+
=
+
=
Z
U
X
U
Z
U
X
U
Z
U
X
U
X
U
Z
U
E
y
U
X
U
y
U
X
U
y
U
X
U
X
U
y
U
E
X
U
X
U
X
U
X
U
E
D
Z
D
Z
D
y
D
y
D
X
D
X
D
Z
D
X
xz
xz
xz
D
Z
D
Z
D
y
D
y
D
X
D
X
D
y
D
X
xy
xy
xy
D
Z
D
y
D
X
D
X
xx
xx
xx
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
3
2
3
2
3
3
2
2
2
2
2
2
2
1
In the expression of these deformations, the terms quadratic
Z
U
X
U
y
U
X
U
X
U
D
X
D
X
D
X
D
X
D
X
3
3
3
3
2
3
,
and
are neglected here, according to the assumption usually carried out
by the majority of the authors [bib.3]. For a model of beam, the tensor of initial stresses
tiny room in the local axes of the beam to the components
xz
xy
xx
and
,
. Kinematics is used
introduced with [§2]:
(
)
()
()
() () ()
(
)
() (
) ()
(
)
() (
) ()
-
+
=
-
-
=
+
-
+
=
X
y
y
X
W
Z
y
X
U
X
Z
Z
X
v
Z
y
X
U
X
Z
y
X
y
X
Z
X
U
Z
y
X
U
X
C
C
D
Z
X
C
C
D
y
X
X
Z
y
G
D
X
,
,
,
,
,
,
,
3
3
,
3
and the expression of the efforts generalized according to the stresses:
-
=
=
=
=
=
S
S
S
S
oxz
S
oxy
oxx
ds
y
M
ds
Z
M
ds
V
ds
V
ds
NR
xx
Z
xx
y
Z
y
0
0
0
0
0
0
0
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
15/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
It is supposed, moreover, that
0
NR
,
0
y
V
,
0
Z
V
are constant in the discretized element (what is inaccurate
for example for a vertical beam subjected to its actual weight). The moments are supposed
to vary linearly:
(
)
01
01
02
0
y
y
y
y
M
+
L
X
M
M
M
-
=
0
0
0
=
V
X
M
Z
y
-
(
)
01
01
02
0
Z
Z
Z
Z
M
+
L
X
M
M
M
-
=
0
0
0
=
V
X
M
y
Z
+
These assumptions make it possible to express
G
W
for a right beam with roll in the way
following:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
X
X
X
X
X
X
y
Zr
Z
yr
X
X
X
X
y
Zr
C
Z
yr
C
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Z
C
y
C
X
X
X
X
X
X
X
X
X
X
X
X
X
X
C
X
X
X
X
X
X
C
X
X
X
X
C
C
Z
y
X
X
X
X
L
O
G
dx
DM
I
I
dx
DM
I
I
M
I
I
Z
M
I
I
y
v
v
V
W
W
V
V
Z
V
y
v
v
M
W
W
M
W
W
NR
y
v
v
NR
Z
Z
y
With
I
I
NR
W
W
v
v
NR
W
y
Z
Z
y
Z
y
Z
y
,
,
0
2
0
2
,
,
0
2
0
2
,
,
0
,
,
0
,
,
0
0
,
,
,
,
0
,
,
,
,
0
,
,
,
,
0
,
,
,
,
0
,
,
2
2
0
,
,
,
,
0
2
2
+
+
-
+
+
-
+
-
+
+
-
+
+
+
-
-
+
-
+
-
+
-
+
+
+
+
+
+
+
=
°
°
with the terms
(
)
(
)
ds
Z
y
Z
I
ds
Z
y
y
I
S
Zr
S
yr
+
=
+
=
2
2
2
2
2
2
who represent not - symmetry of the section. If the section has two axes of
symmetry (thus C is confused with G), these terms are null.
Attention, these terms (which name IYR2 and IZR2 in control AFFE_CARA_ELEM) are not
currently not calculated by MACR_CARA_POUTRE. The user must thus inform them from
values tabulées for each type of section (corner, right-angled,…).
Moreover, to be able to deal with the problems of discharge of thin beams, requested
primarily by moments bending and efforts normal, it is necessary to add the assumption of
rotations moderated in torsion [bib2], [bib3].
This results in the following amendment of the field of displacements (only for the calculation of
geometrical rigidity):
(
)
()
()
() ()
(
)
()
() ()
(
)
() ()
X
Z
y
X
X
X
y
X
X
X
Z
X
U
Z
y
X
U
X
X
y
X
Z
Z
X
y
G
D
X
,
3
,
,
,
+
-
-
+
+
=
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
16/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
The origin of this expression cannot be here detailed. It is the subject of the thesis of CITY OF
GOYET [bib2] on the buckling of the beams with open mean sections. The assumption of rotations
of torsion moderate (and not infinitesimal) allows to modelize discharge correctly
of a thin beam of section in torsion (coupling torsion - bending).
The assumption of moderate rotations results in adding with
0
G
W
the term
1
G
W
(
)
(
)
(
)
(
)
Z
X
O
Z
y
X
O
y
X
X
Z
X
Z
X
O
y
X
X
y
X
y
X
L
O
OZ
G
V
V
M
M
W
+
+
+
+
+
-
=
,
,
,
,
1
2
1
Finally, one obtains the geometrical matrix of rigidity while discretizing
1
0
G
G
G
W
W
W
+
=
using
same functions of interpolation as the matrix of rigidity of [§4.4]. After having calculated these matrices, it
is necessary to carry out a change of reference mark as with [§4.5]. A matrix of rigidity then is obtained
geometrical of the form:
3
2
2
1
With
With
With
With
K
G
=
The blocks of the matrix are clarified hereafter. One uses to simplify the expressions:
Z
y
Zr
Z
y
Z
yr
y
Z
y
Z
y
O
Z
Z
Z
y
y
y
Z
Z
Z
y
y
y
Z
O
ez
y
O
ey
E
I
I
I
E
I
I
I
E
E
S
I
I
NR
K
M
M
M
M
M
M
L
M
M
M
L
M
M
M
L
E
NR
NR
L
E
NR
NR
2
~
2
~
~
2
2
2
.
1
2
.
1
2
2
2
2
02
01
0
02
01
0
02
01
0
02
01
0
0
0
-
=
+
-
=
+
+
+
=
-
=
-
=
+
=
+
=
=
=
A1
2
1
v
3
1
W
4
1
X
5
1
y
6
Z
1
7
X X
, 1
2
1
v
L
NR
O
2
.
1
L
M
M
NR
y
y
ez
0
0
0
2
.
1
2
+
+
NR
O
10
E NR
LM
M
Z
y
y
0
0
2
0
10
+
-
3
1
W
L
NR
O
2
.
1
L
M
M
NR
Z
Z
ey
0
0
0
2
.
1
2
+
+
-
10
O
NR
-
-
+
-
E NR
LM
M
y
Z
Z
0
0
2
0
10
4
1
X
(
)
0
2
0
2
0
0
2
2
~
~
~
2
.
1
y
y
Zr
Z
Z
Z
yr
y
Z
y
y
Z
M
I
I
E
M
I
I
E
I
M
I
M
K
L
+
+
+
-
-
-
2
10
01
02
0
0
Z
Z
Z
y
M
M
M
L
NR
E
-
+
+
2
10
01
02
0
0
y
y
y
Z
M
M
M
L
NR
E
+
-
-
~
~
~
K
MR. I
MR. I
Z
y
y
Z
+
+
2
0
2
0
10
5
1
y
2
15
L NR
O
(
)
2
15
3
30
0
1
0
2
0
E LN
L M
M
y
Z
Z
-
-
6
1
Z
sym
2
15
L NR
O
(
)
2
15
3
30
0
1
0
2
0
E LN
L M
M
Z
y
y
-
-
7
1
, X
X
(
)
(
)
4
3
3
30
1
0
2
0
1
0
2
0
~
~
~
K L
LI
M
M
LI
M
M
y
Z
Z
Z
y
y
-
-
-
-
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
17/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
A2 9:
v
2
10
:
W
2
11
:
x2
12:
y2
13:
z2
14
:
X X
, 2
2
v
1
- 12
. NL
O
-
+
-
NR
M
M
L
ez
y
y
0
0
0
2
12
.
NR
O
10
E NR
LM
M
Z
y
y
0
0
1
0
10
-
+
3
W
1
- 12
. NL
O
NR
M
M
L
ey
Z
Z
0
0
0
2
12
+
-.
- NR
O
10
-
-
+
E NR
LM
M
y
Z
Z
0
0
1
0
10
4
x1
-
-
-
NR
M
M
L
ez
y
y
0
0
0
2
12
.
NR
M
M
L
ey
Z
Z
0
0
0
2
12
-
-.
- A1 (4,4)
E NR
LM
M
y
Z
Z
0
0
1
0
10
-
-
E NR
LM
M
Z
y
y
0
0
1
0
10
+
+
~
~
~
K
MR. I
MR. I
y
Z
Z
y
-
-
1
0
1
0
10
5
y1
NR
O
10
-
-
-
E NR
LM
M
y
Z
Z
0
0
2
0
10
- LN
O
30
-
+
+
E LN
L M
L M
y
Z
Z
0
0
2
0
30
60
6
z1
- NR
O
10
-
+
+
E NR
LM
M
Z
y
y
0
0
2
0
10
- LN
O
30
-
-
-
E LN
L M
L M
Z
y
y
0
0
2
0
30
60
7
X X
,
-
-
+
E NR
LM
M
Z
y
y
0
0
2
0
10
E NR
LM
M
y
Z
Z
0
0
2
0
10
-
+
- -
-
~
~
~
K
MR. I
MR. I
y
Z
Z
y
2
0
2
0
10
-
+
-
E LN
L M
L M
y
Z
Z
0
0
2
0
30
60
-
-
+
E LN
L M
L M
Z
y
y
0
0
2
0
30
60
-
+
+
~
~
~
K L
L MR. I
L MR. I
y Z
Z
y
0
0
30
30
A3
9:
v
2
10:
W
2
11:
x2
12:
y2
13:
z2
14
:
X X
, 2
2
v
2
12
. NL
O
NR
M
M
L
ez
y
y
0
0
0
2
12
-
+.
- NR
O
10
-
+
-
E NR
LM
M
Z
y
y
0
0
1
0
10
3
W
2
12
. NL
O
-
-
+
NR
M
M
L
ey
Z
Z
0
0
0
2
12
.
NR
O
10
E NR
LM
M
y
Z
Z
0
0
1
0
10
+
-
4
x2
12
12
2
2
0
0
2
0
2
0
.
~
.
~
~
K
L
M
L I
M
L I
E
I
I
M
E
I
I
M
Z
y
y
Z
y
yr
Z
Z
Z
Zr
y
y
-
+
+
+
-
+
-
+
+
-
E NR
LM
M
M
y
Z
Z
Z
0
0
1
0
2
0
10
2
-
-
-
+
E NR
LM
M
M
Z
y
y
y
0
0
1
0
2
0
10
2
- +
+
~
~
~
K
MR. I
MR. I
y
Z
Z
y
1
0
1
0
10
5
y2
2
15
L NR
O
(
)
2
15
3
30
0
1
0
2
0
E LN
L M
M
y
Z
Z
-
-
6
z2
2
15
L NR
O
(
)
2
15
3
30
0
1
0
2
0
E LN
L M
M
Z
y
y
+
-
7
X X
, 2
sym
(
)
(
)
4
3
3
30
1
0
2
0
1
0
2
0
~
~
~
K L
LI
M
M
LI M
M
y
Z
Z
Z
y
y
-
-
-
-
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
18/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
6 Loadings
Various types of loading available for the element
POU_D_TG
are:
Types or options
CHAR_MECA_FR1D1D
loading broken down by actual values
CHAR_MECA_FF1D1D
loading broken down by function
CHAR_MECA_PESA_R
loading due to gravity
CHAR_MECA_TEMP_R
“thermal” loading
CHAR_MECA_EPSI_R
loading by imposition of a deformation (of stratification type
thermics)
The loadings are in the same way calculated that for the elements without roll
[R3.08.01]. There is thus nothing in particular to the element
POU_D_TG
. Other types of loading
described in [R3.08.01] are not available for this element.
With regard to the roll, it is possible to give boundary conditions making
to intervene the degree of freedom
GRX
(what makes it possible to modelize constrained torsion: GRX=0), but by
against, nothing is designed to affect a loading of the Bi-moment type, of which physical interpretation
is difficult to establish.
Concerning connection between elements, the transmission of the roll is an open question
as the reference [bib1] announces it: the continuity of variable GRX from one element to another (of which
the roll depends directly) depends in fact on technology on the connection between
various beams (welding in the axis, in which case the roll can be transmitted
completely, connection by bracket,…).
For an assembled structure such as a lattice, it seems more reasonable to suppose than torsion
is obstructed, therefore that the roll is null at the ends. To determine the influence of this
assumption, one will be able to refer to the test SSLL102 (beam of corner section) of which modelings
C and D use element POU_D_TG, with free torsion for modeling C, and torsion obstructed for
modeling D [V3.01.102B].
It is noted that for the loading of bending, the variation on displacement is weak (2.5%), but for
a loading in torsion, one obtains for this section a side displacement not no one (discharge)
from which the value differs notably according to the assumption taken:
5
10
2
.
2
-
=
Z
U
for free torsion and
5
10
62
.
2
-
=
Z
U
for constrained torsion.
In the same way, rotation strongly varies:
4
10
79
,
3
-
=
X
for free torsion and
4
10
39
,
6
-
=
X
for constrained torsion (
GRX
is null with
ends).
6.1 Loadings distributed
: options
CHAR_MECA_FR1D1D
and
CHAR_MECA_FF1D1D
The loadings are given under the key word
FORCE_POUTRE
, that is to say by actual values in
AFFE_CHAR_MECA
(option
CHAR_MECA_FR1D1D
), that is to say by functions in
AFFE_CHAR_MECA_F
(option
CHAR_MECA_FF1D1D
). The loading is given only by forces distributed, not by
moments distributed.
The second associate member with the loading distributed with traction and compression is:
()
()
dx
L
X
X
F
F
dx
L
X
X
F
F
F
F
ext.
ext.
=
-
=
1
0
2
1
0
1
2
1
1
with
Code_Aster
®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
19/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
For a loading constant or varying linearly, one obtains:
.
+
,
+
3
6
6
3
2
1
2
1
2
1
N
N
L
=
F
N
N
L
=
F
X
X
2
1
N
N
and
are the components of the axial loading as in points 1 and 2 coming from the data of
the user replaced in the local reference mark.
If
2
1
2
,
,
1
Z
Z
y
y
T
T
T
T
and
are those of the shearing action, one a:
.
+
,
+
,
+
,
+
,
+
+
+
+
-
-
20
30
20
7
20
3
30
20
20
3
20
7
20
30
20
7
20
3
30
20
20
3
20
7
2
1
2
2
1
2
2
1
1
2
1
1
2
1
2
2
1
2
2
1
1
2
1
1
2
2
2
2
Z
Z
y
Z
Z
Z
Z
Z
y
Z
Z
Z
y
y
Z
y
y
y
y
y
Z
y
y
y
T
T
L
=
M
T
T
L
=
F
T
T
L
=
M
T
T
L
=
F
T
T
L
=
M
T
T
L
=
F
T
T
L
=
M
T
T
L
=
F
6.2
Loading of gravity: option “
CHAR_MECA_PESA_R
“
The force of gravity is given by the module of acceleration G and a normalized vector
N indicating
direction of the loading.
Remarks (simplifying assumption):
The functions of form used for this calculation are those of the Euler-Bernoulli model.
The step is similar to that used for the forces distributed, with the proviso of transforming initially
the vector loading due to gravity in the local reference mark with the element. One obtains in the reference mark
room of the beam:
-
L
X
=
L
X
=
dx
S
=
F
L
O
I
X
I
2
,
1
1
X
G
from where:
2
not
with
1,
not
with
+
+
3
6
6
3
2
1
S
S
L
=
F
S
S
L
=
F
X
X
X
G
X
G
Code_Aster
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Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
20/24
Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Bending in the plan (Gxz):
=
=
-
=
=
20
30
20
7
20
3
30
20
20
3
20
7
2
2
2
2
1
1
S
S
L
M
S
S
L
F
S
S
L
M
S
S
L
F
y
Z
y
Z
+
+
+
+
Z
G
Z
G
Z
G
Z
G
Bending in the plan (Gxy):
-
=
=
=
=
20
30
20
7
20
3
30
20
20
3
20
7
2
2
2
2
1
1
S
S
L
M
S
S
L
F
S
S
L
M
S
S
L
F
Z
y
Z
y
+
+
+
+
y
G
y
G
y
G
y
G
6.3
Thermal loading: option:
“
CHAR_MECA_TEMP_R
“
To obtain this loading, it is necessary to calculate axial displacements induced by the difference of
temperature
reference
T
T
-
:
(
)
(
)
(
)
thermics
dilation
of
T
coeffician
reference
reference
:
2
1
T
T
L
=
U
T
T
L
=
U
-
-
-
Then, one calculates simply the forces induced by
U
K
F =
.
Like
K
is the matrix of local rigidity to the element, one must then carry out a change of
identify to obtain the values of the components of the loading in the total reference mark.
Code_Aster
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Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
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Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
6.4
Loading by imposed deformation option
“CHAR_MECA_EPSI_R
“
One calculates as for the elements
POU_D_T
the loading starting from a state of deformation (this
option was developed to take into account the thermal stratification in pipings).
model takes into account only one work in traction and compression and pure bending (not of effort
edge, not of torque).
The deformation is given by the user using the key word
EPSI_INIT
in
AFFE_CHAR_MECA
. In
being given
X
X
X
U
Z
y
and
,
on the beam, one obtains the second elementary member associated with it
loading:
X
I
E
M
X
I
E
M
X
U
S
E
F
X
I
E
M
X
I
E
M
X
U
S
E
F
Z
Z
Z
y
y
y
X
Z
Z
Z
y
y
y
X
2
2
2
2
2
1
1
1
1
1
2
1
,
,
,
,
,
:
2
node
with
:
1
node
with
=
=
=
=
=
=
7
Torque of the efforts - nodal Forces and reactions
7.1 Options
available
Various options of postprocessing available for the element
POU_D_TG
are:
Types or options
EFGE_ELNO_DEPL
torque of the efforts to the 2 nodes of each element
SIEF_ELGA_DEPL
field of efforts necessary to the calculation of the nodal forces (option
“FORC_NODA”)
and of the reactions (option
“REAC_NODA”
).
FORC_NODA
nodal forces expressed in the total reference mark
REAC_NODA
nodal reactions
Code_Aster
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Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
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J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
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Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
7.2
The torque of the efforts
7.2.1 Generalized efforts, option: “
EFGE_ELNO_DEPL
“
One seeks to calculate with the two nodes of each element “beam” constituting the mesh of
studied structure, efforts exerted on the element “beam” by the remainder of the structure. Values
are given in the local base of each element. By integrating the equilibrium equations, one obtains
efforts in the local reference mark of the element:
(
)
2
2
2
2
2
2
2
1
1
1
1
1
1
1
,
,
,
,
,
,
,
,
,
,
M
M
M
M
V
V
NR
M
M
M
M
V
V
NR
=
=
Z
Y
T
Z
Y
Z
Y
T
Y
LOC
LOC
LOC
LOC
LOC
LOC
LOC
Z
E
E
E
,
,
,
+
:
where
-
-
-
-
-
-
-
-
R
F
U
M
U
K
R
&&
E
LOC
K
elementary matrix of rigidity of the element beam,
E
LOC
M
elementary matrix of mass of the element beam,
E
LOC
F
vector of the efforts “distributed” on the element beam,
LOC
U
vector “degree of freedom” limited to the element beam,
LOC
u&
&
vector “acceleration” limited to the element beam.
One changes then the signs of the efforts to node 1.
Indeed, by taking for example the case of the traction and compression, one shows [R3.08.01] that them
efforts in the element (option
EFGE_ELNO_DEPL
) are obtained by:
()
()
[]
()
()
-
=
-
2
1
F
F
L
U
O
U
K
L
NR
O
NR
7.2.2 Generalized efforts, option: “SIEF_ELGA_DEPL”
The option
“SIEF_ELGA_DEPL”
is established for reasons of compatibility with other options.
It is used only for calculation of the nodal forces. It produces fields of efforts by elements.
It is calculated by:
LOC
LOC
LOC
E
=
U
K
R
7.3
Calculation of the nodal forces and the reactions
7.3.1 Nodal forces, option: “
FORC_NODA
“
This option calculates a vector of nodal forces on all the structure, expressed in total reference mark.
It produces a field with the nodes in the control
CALC_NO
by assembly of the terms
elementary.
For this calculation, one uses the principle of virtual work and one writes [R5.03.01]:
T
Q
F
=
where
T
Q
symbolically represent the matrix associated with the operator divergence. For an element, one
writing agricultural work of virtual deformations:
()
()
()
*
*
*
U
U
U
T
=
U
Q
kinematically acceptable
Code_Aster
®
Version
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Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
02/05/05
Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
Page:
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Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
For the elements of beam, one calculates simply the nodal forces by assembly of the forces
nodal elementary calculated by the option
SIEF_ELGA_DEPL
, which is expressed by:
[
] [
] [
]
LOC
LOC
LOC
U
K
F
=
7.3.2 Nodal reactions, option: “
REAC_NODA
“
This option, called by
CALC_NO
, allows to obtain the reactions
R
with the supports, expressed in
total reference mark, starting from the nodal forces
F
by:
iner
tank
F
F
F
R
+
-
=
iner
tank
F
F
and
being nodal forces respectively associated with the loadings given
(specific and distributed) and with the efforts of inertia.
8 Bibliography
[1]
J.L. BATOZ, G. DHATT. “Modeling of the structures by finite elements” - HERMES.
[2]
V. OF TOWN OF GOYET. “Nonlinear static analysis by the finite element method
formed space structures of beams with nonsymmetrical sections “Thesis of
the University of Liege. 1989.
[3]
J. SLIMI. “Simulation of ruin of pylon” Report/ratio SERAM N°14.033, ENSAM June 1993.
Code_Aster
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Titrate:
Element of beam with 7 ddl for the taking into account of the roll
Date:
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Author (S):
J.L. FLEJOU, J.M. PROIX
Key
:
R3.08.04-B
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Manual of Reference
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
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