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Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
1/24
Organization (S): EDF-R & D/AMA
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
R3.08.04 document
Element of beam with 7 ddl for the taking into account
warping
Summary:
This document presents the element POU_D_TG which is a finite element of right beam with taking into account of
warping 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 inflection, the normal and sharp efforts, this element is based on element POU_D_T,
who is an element of right beam with transverse shearing (model of Timoshenko).
For 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 warping POU_D_TG.
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
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Code_Aster ®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
2/24
Count
matters
1 Field of application .............................................................................................................................. 3
2 Notations ................................................................................................................................................ 3
3 Kinematics specific to torsion with warping ..................................................................... 4
4 Element of right beam with warping: matrices of rigidity and mass ................................ 7
4.1 Traction and compression the degree of freedom are U or DX .............................................................. 8
4.2 Inflection in the plan (Gxz) the degrees of freedom concerned are W, y or DZ, DRY ...................... 8
4.3 Inflection in the plan (Gxy) the degrees of freedom concerned are, Z or DY, DRZ .................... 10
4.4 Torsion and warping the degrees of freedom are X, X, X or DRX, GRX ............................... 11
4.5 Eccentricity of the axis of torsion compared to the neutral axis ............................................................. 12
5 geometrical Rigidity - Structure prestressed .................................................................................... 14
6 Loadings ........................................................................................................................................ 18
6.1 Loadings distributed: options CHAR_MECA_FR1D1D and CHAR_MECA_FF1D1D .......................... 18
6.2 Loading of gravity: option “CHAR_MECA_PESA_R” .......................................................... 19
6.3 Thermal loading: option: “CHAR_MECA_TEMP_R” ............................................................. 20
6.4 Loading by imposed deformation option “CHAR_MECA_EPSI_R” ......................................... 21
7 Torque of the efforts - Forces nodal and reactions ............................................................................... 21
7.1 Options available ....................................................................................................................... 21
7.2 The torque of the efforts ..................................................................................................................... 22
7.2.1 Generalized efforts, option: “EFGE_ELNO_DEPL” ................................................................ 22
7.2.2 Generalized efforts, option: “SIEF_ELGA_DEPL” ................................................................ 22
7.3 Calculation of the nodal forces and the reactions ................................................................................... 22
7.3.1 Nodal forces, option: “FORC_NODA” ................................................................................ 22
7.3.2 Nodal reactions, option: “REAC_NODA” ........................................................................... 23
8 Bibliography ......................................................................................................................................... 23
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
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Code_Aster ®
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Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
3/24
1 Field
of use
The development of the elements of beam of Timoshenko with warping (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 in Département MMN. It
mainly acted to calculate formed structures of beams with open mean profile
(angles), for which warping is important. Plasticity was introduced into
the element POU_D_TG [R5.03.30], but the nonlinear behavior relates only to traction,
inflection and torsion. Shearing due to the sharp effort, as well as warping and the Bi-moment
(effort related to warping) 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
warping is valid for the use of 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.
DX, DY, DZ, DRX, DRY, DRZ and GRX are the names of the degrees of freedom associated with
components of displacement U v
, W, X, y, Z, X, X .Ils are expressed in total reference mark, except the degree
of freedom associated with the warping GRX, which is expressed in local reference mark.
Notation used
Significance
Notation
of
documentation
of use
S
surface of the section
With
I
geometrical moments of inflection compared to axes IY, IZ
y, I Z
X and Y.
C
constant of torsion
JX
I
constant of warping
JG
K
coefficients of shearing
1 1
y K
, Z
AY AZ
E
EY, EZ
y E
, Z
eccentricity of the center of torsion/shearing by
report/ratio in the center of gravity of the cross-section
NR
normal effort with the section
NR
V
sharp efforts along axes y and Z
VY VZ
,
y V
, Z
M
moments around axes X, y and Z
MT, MFY, MFZ
X, M y, M Z
M
Bi-moment
BX
U v
, W
translations on axes X, y and Z
DX DY DZ
rotations around axes X, y and Z
DRX DRY DRZ
X
, y
, Z
rotary derivative of torsion according to X
GRX
X, X
E
Young modulus
E
Poisson's ratio
NAKED
modulate of Coulomb (identical to the coefficient of Lamé) G
=
E
G
(
2 1+) = µ
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Code_Aster ®
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Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
4/24
3
Kinematics specific to torsion with warping
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
inflection - 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 warping of the cross-section.
·
torsion is free (torsion of Saint-Venant): the warping 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 constraint
axial due to torsion.
·
Torsion is constrained (Vlassov): warping is nonnull, and moreover of the constraints
axial not uniforms (from which the effort resulting Bi-moment is called) exist in the beam.
Element POU_D_TG makes it possible to treat these two configurations: torsion can be free or constrained.
The user will have access to warping 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 warping depends on the type of connection. In general, torsion in an assembly
beams is constrained. Warping can then be blocked at the points of connection.
Note:
With elements without modeling of warping (POU_D_T, POU_D_E), one can
to treat the case of free torsion (displacements other than warping will be
correct), but not the case of constrained torsion.
One can uncouple the effects of torsion and inflection in a local reference mark (relocated principal 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 principal reference mark of inertia related to the center of gravity G, to keep one
decoupling of displacements of inflection and traction and compression.
The displacement of an unspecified point of the cross-section is written then in general form (torsion
free or constrained):
U (X, y, Z) uG (X) Z y (X) - yz (X) (y, Z) X, X (X)
v (X, y, Z)
= 0 + 0 + v (X) + - (Z - zc) X (X)
W (X, y, Z)
0 W (X)
0
(y - teststemyç) X (X)
displacement = membrane + inflection/y + inflection/Z +
with
torsion
warping
The components are expressed in the principal reference mark of inertia (centered in G): X is directed according to
the axis of the beam, y and Z are the two other principal axes of inertia.
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
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Code_Aster ®
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Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
5/24
Term (y, Z) X, X (X) represents axial displacement due to the warping of the cross-section.
(y, Z) is related to warping (expressed in m ², but which does not have physical interpretation
obvious).
The deformations of an unspecified point of the section are then:
,
,
,
,
-
,
xx (X y Z)
uG (X)
Z
X
y
X
y Z
X
X
y X ()
Z X ()
() X, xx ()
2 xy (X, y, Z) = 0 +
0
+ xy (X) + (, y - (Z - zc)) X, X (X)
2 xz (X, y, Z)
0
xz (X)
0
(
, Z + (y - teststemyç) X, X (X)
xy (X) = v, X - Z
xz (X) = W, X + y
Deformation = membrane + inflection/y + inflection/Z + torsion with warping
Term (y, Z) X, xx (X) is null in the case of free torsion: there are indeed X (X
xx
) 0
,
=, since
warping 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 constraints in
directions y and Z):
xx (X, y, Z) E. xx (X, y, Z)
xy (X, y, Z)
= G.2 xy (X, y, Z)
xz (X, y, Z)
G.2 xz (X, y, Z)
The efforts generalized in the section are expressed according to the constraints for a section
homogeneous by [bib1]:
NR (X) = xx (X, y, Z) ds
normal
effort
S
V (X) =
y
xy (X, y, Z) ds
y
according to
edge
effort
S
V (X) =
Z
xz (X, y, Z) ds
Z
according to
edge
effort
S
M (X) =
y
Z. xx (X, y, Z) ds
y
of
around
inflection
of
moment
S
M (X) = -
Z
y. xx (X, y, Z) ds
Z
of
around
inflection
of
moment
S
M (X) =
X
(
(y - teststemyç) .xz (X, y, Z) - (Z - zc) .xy (X, y, Z))ds
torsion
of
moment
S
M (X) =
.
xx (X, y, Z) ds
Bi -
ent)
gauchissem
with
(associate
moment
S
Handbook of Référence
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Code_Aster ®
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Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
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M (X)
represent the generalized effort associated with warping. It is expressed in N.m2. One can in
to give an illustration as in [bib1] for a beam to I-section (the Bi-moment acts here according to
Z only):
Z
Z
xx+
Y
xx -
Y
M
X
X
xx -
xx +
For an isotropic and homogeneous elastic behavior in the section, the efforts generalized
thus express themselves directly according to displacements by the following relations:
NR (X) = E S
. U
. , X
Vy (X) = GkyS (v, X - Z)
Vz (X) = GkzS (W, X + y)
M y (X) = E I. yy, X
M Z (X) = E I. zz, X
M X (X) = G.J .x, X
M (X) = E I
. .x, xx
where K y K
, Z are the coefficients of shearing. Warping 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 warping is such as:
(y, Z) ds = 0
S
.
y (y, Z) ds = 0
S
Z.(y, Z) ds = 0
S
And the constant of warping is expressed according to by: 2 (y Z) ds = I
,
S
Handbook of Référence
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Code_Aster ®
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Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
7/24
4
Element of right beam with warping: matrices of
rigidity and of mass
The elementary matrices of rigidity and mass for 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 inflection - 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 inflection 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 option “RIGI_MECA”, and the matrices of mass
with option “MASS_MECA”. But 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 warping:
In each of the two nodes of the element, the degrees of freedom are:
U v
, W
translations on axes X, y, Z
DX DY DZ
rotations around axes X, y, Z
DRX DRY DRZ
X
, y
, Z
rotary derivative of torsion according to X
GRX
X, X
The local co-ordinates are expressed in the principal reference mark of inertia. Element POU_D_TG
thus comprise 14 degrees of freedom. The element of reference is defined by: 0 < X < L
Z
Z
Z
W
y
v
Y
Y
X
X
X
U
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Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
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:
R3.08.04-B
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4.1
Traction and compression the degree of freedom are U or DX
ES 1
-
1
The matrix of rigidity of the element is: K =
L - 1 1
SSL 2 1
The matrix of mass (coherent) is written: M =
6 1 2
4.2 Inflection 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 inflection in the principal plan of inertia (Gxz):
L
L
1
-
- 1
-
2
2
(4+y) 2L L (2-y)
2
L
12 I.E.(internal excitation)
=
y
K
3
L (1+y)
12
2
12
L
Sym
1
2
(4+y)
2
L
12
12 I.E.(internal excitation) y
Transverse shearing is taken into account by the term: y =
2
K SGL
Z
For the matrix of mass,
(
W T
X,) and y (T
X,) are discretized on the basis of function tests
introduced for the calculation of the matrix of rigidity, that is to say:
(
W X, T)
= 1 (X) 1
W (T)
+ 2 (X)
(T)
y
+ 3 (X) 2
W (T) + 4 (X)
(T)
1
y2
(X, T)
y
= 5 (X) 1
W (T)
+ 6 (X)
(T)
y
+ 7 (X) 2
W (T) + 8 (X)
(T)
1
y2
Handbook of Référence
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Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
9/24
The function of interpolation used for the translations (1
to 4) are polynomials of Hermite of
degree 3, that which is used for rotations (5 to 8) are degree 2: for 0 < X < L, they are
defined by [R3.08.01]:
3
2
1
6
=
(X)
1
X
X
X
X
X
2 -
3 -
+
y
(1+y)
=
(X)
1
+
1
5
y L
L
L
(L +
1 y) -
L L
L
X 3 4 +
2
2
y
X
2+
1
=
(X)
y
-
X
X
X
2
+
-
=
(X)
6
3 - (4+y)
+ (+
1 y)
+
1 y L
2 L
2 L
+
1 y L
L
3
2
1
X
X
X
- 6
X
X
=
(X)
3
-
2 +
3 +y
=
(X)
1
+
1
7
y
L
L
L
(L +
1 y) -
L L
L
X 3 2 -
2
2
1
=
(X)
4
y X
y
-
X
X
X
+
+
=
(X)
8
3 + (2+y)
-
+
1 y L
2 L
2 L
+
1 y L
L
éq 4.2-1
The form of the matrix of mass is:
2
2
2
2
2
2
2
13L 7L
2
2
9
3
13 2
3
y
L
11L
11L
y
y
L
L
L
L
L
L
L
+
+
y
-
-
-
+
y +
y
+
y +
y
35
10
3
210
120
24
70
10
6
420
40
24
2
3
3
2
2
3
3
2
L3
L
13 2
2
3
y
L
L
3L
y
y
L
L
L
L
y
y
y
S
+
+
-
-
-
-
-
-
M = (
105
60
120
420
40
24
140
60
120
2
2
2
2
2
1+
13L
7L
L
11L2
11L
L
y
y
y
y
y)
+
+
-
+
+
35
10
3
210
120
24
3
3
2
L3
L
L
y
y
sym
+
+
105
60
120
6
1
6
1
-
+ y
-
-
+ y
5L
10
2
5L
10
2
2L
L
L 2
2
y
y
1
y
L
L
L
y
y
I
+
+
-
-
-
+
+
y
15
6
3
10
2
30
6
6
(
2
1+
6
1
y
y)
-
5L
10
2
2L
L
L 2
sym
+
y +
y
15
6
3
Handbook of Référence
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Version
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Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
10/24
4.3
Inflection in the plan (Gxy) the degrees of freedom concerned are, Z
or DY, DRZ
In the same way, for the movement of inflection around the axis (Gz), in the principal plan of inertia (Gxy),
stamp rigidity is written:
L
L
1
- 1
2
2
(4+z) 2
L
L (2 - Z)
2
L
12 I.E.(internal excitation)
-
Z
12
2
12
=
K
3
L (1+z)
L
1
-
2
(4+z) 2
L
sym
12
Transverse shearing is taken into account by the term:
12 I.E.(internal excitation) Z
Z =
2
K SGL
y
To calculate the matrix of mass, v (T
X,) and (T
X,
Z
) are discretized by:
v
(T
X,)
= 1 (X) v
1 (T) -
2 (X)
Z (T)
+ 3 (X) v
2 (T) - 4 (X)
Z (T)
1
2
Z (T
X,) = - 5 (X) v
1 (T) +6 (X)
Z (T) - 7 (X) v
2 (T) +8 (X)
Z (T)
1
2
We obtain the matrix of following mass then:
2
2
2
2
2
2
2
2
2
2
13L
7L
L
11L
11L
L
9L
3L
L
13L
3L
L
+
Z +
Z
+
Z +
Z
+
Z +
Z
-
-
Z -
Z
35
10
3
210
120
24
70
10
6
420
40
24
3
3
3
2
2
2
2
2
3
3
3 2
L
L
L
13L
3L
L
L
L
L
Z
+
+
Z
+
Z +
Z
-
-
Z -
Z
S
105
60
120
420
40
24
140
60
120
M = (
2
2
2
2
2
1+
Z)
2
13L
7L
L
11L
11L
L
Z
Z
Z
Z
+
+
-
-
35
10
3
210
120
24
3
3
3 2
L
L
L
sym
+
Z +
Z
105
60
120
6
1
6
1
Z
Z
-
-
-
5L
10
2
5L
10
2
2
2
2L L
L
L
L
L
Z
Z
+
+
- 1 + Z -
-
Z
Z
+
I
Z
15
6
3
10
2
30
6
6
+ (
1+
Z)
2
6
- 1 + Z
5L
10
2
2
2L L
L
Z
Z
sym
+
+
15
6
3
Handbook of Référence
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Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
11/24
4.4 Torsion and warping 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
warping of the reference [R3.08.01]. The virtual work of the interior efforts is written for
torsion [bib1]:
L
W
=
* G
. .J
.
* E
. I
.
int
+
.
dx
X, X
X, xx
X, xx
O
X, X
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 Hermite type, of the same kind as those used for
inflection [éq 4.2-1]. One writes them here on the element of reference [- 1,1] according to [bib1] (instead of 0<x<L
previously):
2
= X - 1
L
- 1 1
NR () 1
= (1 -) 2
1
(2 +)
4
NR () = L
(1 -) (
2
2
1 -)
8
NR () 1
= (1+) 2
3
(2 -)
4
NR () = L
(1+) (
2
4
- 1+)
8
The interpolation for the rotation of torsion and its derivative is:
() = NR
X
() 1
1
+ NR () 1
2
+ NR () 2
3
+ NR () 2
4
X
X, X
X
X, X
X, X () = NR X () 1
,
1
+ N2 X () 1
,
+ NR X () 2
,
3
+ N4 X () 2
,
X
X, X
X
X, X
The reference [bib1] note which this approximation corresponds to a borderline case of the interpolation
GJ
hyperbolic, obtained for
0. However, this parameter not being without dimension, it is
I.E.(internal excitation)
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.
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
12/24
The matrix of rigidity corresponding to this approximation is written then:
36
3L
- 36
6L
12
6L
- 12
6L
GJ
2
4L
- 3L - 2
L I.E.(INTERNAL EXCITATION)
2
4L
-
2
6L 2L
K = K + K =
+
T
3
30L
36
- 3L
L
12
- 6L
2
2
sym
4L
sym
4L
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:
L
·
W
= -
*. .I.& & dx
iner
O
X, X
X, X
·
in Code_Aster, the simplest method was selected: the matrix of mass is
identical to that of element POU_D_T. One preserves the already definite terms for traction -
compression and the inflection - shearing and one use a linear approximation for torsion.
The coefficients of the matrix of mass associated with warping 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 inflection 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 principal reference mark of inertia:
0
GC = ey
ez
One can numerically determine them starting from the plane grid 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:
U
=
(G)
U (C)+ GC
X
with
= 0 vector
rotation
0
U
U
=
G
C
v
v
=
+ E
G
C
Z X
W
W
=
- E.
G
C
y
X
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
13/24
The change of variables is in the same way written that for POU_D_T, with 2 degrees of freedom
additional:
U
ux
X
1
C
1
U
U y 1 0 0
0
0 0 0
y
1
C
1
U
E
U
Z
0 1 0 -
0 0 0
Z
C
Z
1
1
0 0 1
E
0 0 0
0
X
y
X
1
C
1
0 0 0
1
0 0 0
y
y
1
C
1
0 0 0
0
1 0 0
Z
Z
1
C
1
0 0 0
0
0 1 0
X, X
C
1
0 0 0
0
0 0 1
X, x1
ux
=
C
U
2
X
1 0 0
0
0 0 0
U
2
y
C
U
2
0 1 0 - E
0 0 0
Z
y2
uz
C
uz
2
0 0 1
E
0 0 0
y
2
X
c2
0
0 0 0
1
0 0 0 x2
y
c2
0 0 0
0
1 0 0 y2
Z
C
0 0 0
0
0 1 0
2
z2
X, X
C
0 0 0
0
0 0 1
2 1
4
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
4
4
3 X, x2
P
From the elementary matrices of mass and rigidity calculated previously in the reference mark
(C, X, y, Z) where the movements of inflection and torsion are uncoupled, one obtains these matrices in
reference mark related to the neutral axis (G, X, y, Z) by the following transformations:
K = PT K P
C
M = PT M
.
P
C
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
14/24
5
Geometrical rigidity - Structure prestressed
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:
U 3D
v 3D
W
K
O
K
=
FD
G
X
ij
X
I
J
Vo
where O
ij indicates the tensor of prestressing. This term appears naturally if the tensor is introduced
deformations of GREEN-LAGRANGE in the virtual work of the deformation:
2
2
3D
3D
3D
2
U
1
3
X
U
U
X
y
U D
E
= + =
+
+
+
Z
xx
xx
xx
2
X
X
X
X
U 3D
U 3D
3 3
3 3
3
3
X
y
U D U D
U D U D
D
D
2nd
=
2
+
2
=
+
+ X
X
+
y
y
+ U
U
Z
Z
xy
xy
xy
y
X
X
y
X
y
X
y
3D
3D
U
U
U 3D
U 3D
U 3D
U 3D
3D
3D
2nd
=
2
+
2
=
X
+
Z
+ X
X
+
y
y
+ U
U
Z
Z
xz
xz
xz
Z
X
X
Z
X
Z
X
Z
In the expression of these deformations, the terms quadratic
2
U 3D
U 3D
U 3D
U 3D
U 3D
X
X
X
X
X
,
and
are neglected here, according to the assumption usually carried out
X
X
y
X
Z
by the majority of the authors [bib.3]. For a model of beam, the tensor of initial constraints
tiny room in the local axes of the beam to components xx
, xy
and
xz. Kinematics is used
introduced with [§2]:
U 3D
X (X, y, Z) = uG (X) + Z y (X) - y Z (X) + (y, Z) X, X (X)
U 3D
y (X, y, Z) = vC (X) - (Z - zc) X (X)
U 3D
Z (X, y, Z) = WC (X) + (y - teststemyç) X (X)
and the expression of the efforts generalized according to the constraints:
NR 0 = O
ds
V 0 =
0
0
0
0
0
xx
y
O ds V =
xy
Z
O ds
M =
xz
y
Z ds M =
xx
Z
- y ds
xx
S
S
S
S
S
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
15/24
It is supposed, moreover, that
0
NR, 0
V, 0
V are constant in the discretized element (what is inaccurate
y
Z
for example for a vertical beam subjected to its actual weight). The moments are supposed
to vary linearly:
0
M
0
X
M
y
0
y = (
0
0
M Y2 - M 1
y)
0
+ M 1
y
- V = 0
L
X
Z
0
0
X
M
M
Z
0
Z = (
0
0
M Z2 - M 1Z)
0
+ M 1z
+V = 0
L
X
y
These assumptions make it possible to express G
W for a right beam with warping in the way
following:
L
I + I
0
0
2
2
=
+
+
+
+
G
W
NR
(v, X v, X W, X W, X)
y
Z
NR
teststemyç zc X, X X, X
O
With
+ Z
0
0
C NR (v, X X, X + X, X v
, X) - ycN (W, xx, X +x, X W
, X)
- M 0 (W
0
, X, X +, X W
, X) - M (v, X, X +, X v
, X)
y
Z
- y 0
0
0
cV
- Z
°
cV° (X, X X + X X, X) + V (W, X X + X W
, X)
y
Z
y
- V 0 (
I 2
0
I
v
+
+
-
+
2
0
-
+
, X X
X v
, X)
yr
Zr
2yc
M
2zc
M
X, xx, X
Z
I
Z
y
I
y
Z
0
0
I
DM
DM
yr2
I
Z
zr2
y
+ -
+
(X, xx +xx, X)
I Z
dx
I y
dx
with the terms
I
= y 2
2
yr2
(y + Z) ds
S
I
= Z 2
2
zr2
(y + Z) ds
S
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 command 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 modification of the field of displacements (only for the calculation of
geometrical rigidity):
U 3D
X (X, y, Z) U
= G (X) + Z (y (X) +x (X) Z (X))- y (Z (X) - X (X) y (X))+ (y, Z) X, X (X)
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
16/24
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 model discharge correctly
of a thin beam of section in torsion (coupling torsion - inflection).
The assumption of moderate rotations results in adding with
0
W term 1
G
G
W
1 L
1
O
G
W =
- M
Z (X y, X + there X, X)
O
+ M y (xz, X +zx, X)
O
+ Vy (X y)
O
+ Vz (xz)
O
2
Finally, one obtains the geometrical matrix of rigidity while discretizing
0
1
W = W + W using
G
G
G
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:
A1 A2
K G =
A2 A3
The blocks of the matrix are clarified hereafter. One uses to simplify the expressions:
O
O
NR E
0
y
NR E
0
Z
Ney = 1 2
.
Nose = 2
.
1
L
L
M 0
0
0
0
+
0
y
M
1
y2
M
0
Z + M
1
Z
M
2
y =
M Z =
L
L
M 0
0
0
0
-
0
y
M
1
y2
M
0
Z - M
1
Z
M
2
y =
M
Z =
2
2
~
I + I
O
y
Z
K = NR
+ e2
2
y + ez
S
I
~
yr2
~
I Zr
I
2
y = -
+ E
2 there I Z =
- E
2 Z
I Z
I y
A1
2 1
v 3 1
W 4
1x 5
1y
6 z1 7
X, x1
0
0
0
2
0
0
+
-
1
v
NR O
NR O
E NR
LM
M
1.2
M
M
0
y
Z
y
y2
L
NR
y
ez +
+ 1.2
2
L
10
10
3
0
0
O
0
0
0
1
W
NR O
- E NR + LM - M
.
1 2
0
M
M
Z
-
+
+
NR
-
y
Z
z2
L
NR
Z
ey
1 2
.
2
L
10
10
0
0
0
0
0
0
~
0 ~
0 ~
4
.
1 2 (~
0~
0~
K - M
eyN + LM Z + M z2
E NR - LM - M
K + MR. I + MR. I
Z I y - M there I Z)
1
X
Z
y
y2
z2 y
y2 Z
L
10
10
10
I
0
yr2
0
I
M
0
-
+
+
zr2
0
E
+
1
Z
-
M
y
M
E
M
1
y
2
Z
Z
I
2
y
+
2
Z
I
y
2
5
2 L NR O
2
0
(3 0 0
E LN
L M -
1
M
y
Z
z2)
1
y
15
-
15
30
sym
6
2 L NR O
2
0
(3 0
0
E LN
L M -
1
M
Z
y
y2)
1
Z
-
15
15
30
7
~
~
0
0
~
0
0
X, 1
X
4K L - LI (3M -
-
-
1
M2) LI (3M 1 M
y
Z
Z
Z
y
y2)
30
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
17/24
A2 9: v2 10
:
w2 11
:
x2
12: y2
13: z2 14
:
X, x2
2
NR O
M 0
M 0
0
0
0
0
y
y
NR O
E NR - LM + M
Z
y
y1
v
- 12
.
- NR +
-
ez
12
.
1
L
2
L
10
10
3
NR O
M 0
M 0
- E NR 0 - LM 0 +
0
0
Z
Z
NR O
M
y
Z
z1
W
- 12
.
NR +
-
ey
12
.
-
1
L
2
L
10
10
4
M 0
M 0
0
0
0
0
0
0
0
0
+
+
~
0 ~
0 ~
0
M
M
-
-
E NR
LM
M
K - MR. I - MR. I
- N0 - y -
y
NR
Z
Z
-
- 12
.
- A1 (4,4)
E NR
LM
M
y
Z
z1
Z
y
y1
y1 Z
z1 y
ez
12
.
ey
x1
2
L
2
L
10
10
10
5
NR O
- E NR 0 - LM 0 - M 0
- E LN 0 + LM 0
2
0
y
Z
L M
y
Z
z2
- LN O
+
Z
y1
10
10
30
30
60
6
NR O
- E NR 0 + LM 0 + M 0
E LN 0
L M 0
L2 M 0
-
Z
y
y2
- LN O
-
-
Z
y -
y
z1
10
10
30
30
60
7
-
~
0 ~
0 ~
E NR 0 - LM 0 + M 0
0
0
0
-
+
- K - MR. I - MR. I
~
E LN 0
LM 0
- E LN 0 - LM 0
- kL +
0 ~
LM I
Z
y
y2
E NR
LM
M
y
Z
z2
y2 Z
z2 y
-
+
y
Z
Z
y
y Z
X, X
10
10
10
30
30
30
L2 M 0
L2 M 0
0 ~
-
Z
+
y
L MR. I
Z
y
60
60
+
30
A3
9: v2 10:
11: x2
12: y2
13: z2 14
:
X, x2
w2
2
NR O
0
0
- E NR 0 + LM 0 - M 0
12
.
M
M
NR O
Z
y
y1
v
0
y
y
L
NR -
+12
.
2
ez
-
2
L
10
10
3
NR O
0
0
0
0
0
12
.
0
M
M
Z
Z
NR O
E NR + LM - M
W
- NR -
+
y
Z
z1
L
ey
12
.
2
2
L
10
10
4
~
~
0 ~
0 ~
0
0
M
- E NR 0 + LM 0 + M 0
- E NR 0 - LM 0 - M 0
- K + MR. I + MR. I
K
Mz ~
y ~
y
Z
z1
Z
y
y1
12
.
- 12
.
I +
I
y1 Z
z1 y
x2
L
L y
L
Z
10
10
10
M 0
0
z2
M
I
-
+
y2
yr2
0
Izr2
0
+ E +
M - E +
M
2
2
y
2I
Z
Z
2I
y
Z
y
5
2 L NR O
2
0
(0 - 3 0
E LN
L M 1
M
y
Z
z2)
-
y2
15
15
30
6
2 L NR O
0
0
E LN
L M - 3 0
M
2 Z
(y1
y2)
+
z2
15
15
30
7
sym
~
~
~
4K L - LI (0
M - 3
0
0
0
-
-
1
M2) LI (M
3
1
M
y
Z
Z
Z
y
y2)
X, x2
30
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
18/24
6 Loadings
The various types of loading available for 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 warping
[R3.08.01]. There is thus nothing in particular to element POU_D_TG. Other types of loading
described in [R3.08.01] are not available for this element.
With regard to warping, it is possible to give boundary conditions making
to intervene the degree of freedom GRX (what makes it possible to model 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 warping is an open question
as the reference [bib1] announces it: the continuity of variable GRX from one element to another (of which
warping depends directly) depends in fact on technology on the connection between
various beams (welding in the axis, in which case warping 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 warping 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 inflection, 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
U
2.2 10
=
-
Z
for free torsion and
5
U = 2.62 10
Z
for constrained torsion.
In the same way, rotation strongly varies:
4
79
,
3
10
=
-
X
for free torsion and
4
= 39
,
6
10
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 key word FORCE_POUTRE, that is to say by actual values in
AFFE_CHAR_MECA (option CHAR_MECA_FR1D1D), is 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:
f1
1
with
F = F
1 ext. ()
X
X 1 - dx
F
2
0
L
1
F = F
2 ext. () X
X dx
0
L
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
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7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
19/24
For a loading constant or varying linearly, one obtains:
1
N
N2
F
=
L
X
+
,
1
3
6
1
N
N2
F
=
L
X
+
.
2
6
3
1
N
N
and
2 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 T
,
y T y2
, T1 Z
and
T 2 are those of the shearing action, one a:
1
Z
7 T
3
y
T y
T y
T y
F =
1
L
+
2
2
M =L
1 + 2
1
y
1
Z
20
20
20
30
3 T
7
y
T y
T y
T y
F =
1
L
+
2
M
- 2
= L
1 + 2,
y2
z2
20
20
30
20
7 T
3
Z
tz
tz
tz
F =
1
L
+
2
,
M
- 2
= L
1 + 2,
1
Z
1
y
20
20
20 30
3 T
7
Z
tz
tz
tz
F =
1
L
+
2
,
2
M =L
1 + 2.
z2
y2
20
20
30 20
6.2
Loading of gravity: option “CHAR_MECA_PESA_R”
The force of gravity is given by the module of acceleration G and a vector normalized 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:
S
S
L
F
=
S G
F
= G
X L
+
not
with
1,
X
X
X
I
dx
O
I
1
3
6
X
X
from where:
=1-
, =
S
S
1
2
L
L
F
= G
X L
+
not
with
2
2
X
6 3
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
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7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
20/24
Inflection in the plan (Gxz):
7 S
3 S
F
= G
Z L
+
1
Z
20
20
S
S
M
= - G
2
Z L
+
1
y
20 30
3 S
7 S
F
= G
Z L
Z
+
2
20
20
S
S
M
= G
2
Z L
y
+
2
30
20
Inflection in the plan (Gxy):
7 S
3 S
F
= G
y L
+
1
y
20
20
S
S
M
= G
2
y L
+
1
Z
20 30
3 S
7 S
F
= G
y L
y
+
2
20
20
S
S
M
= - G
2
y L
Z
+
2
30 20
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 T - reference
T
:
U
= -
1
L (T - reference
T
)
U
=
2
L (T - reference
T
)
(coeffician
:
thermics
dilation
of
T
)
Then, one calculates simply the forces induced by F
= K U
.
As K is the matrix of local rigidity to the element, one must then carry out a change of
locate to obtain the values of the components of the loading in the total reference mark.
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
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Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
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6.4
Loading by imposed deformation option “CHAR_MECA_EPSI_R”
One calculates as for 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 inflection (not of effort
edge, not of torque).
The deformation is given by the user using key word EPSI_INIT in AFFE_CHAR_MECA. In
U
y
being given
Z
,
and
on the beam, one obtains the second elementary member associated with it
X
X
X
loading:
U
:
1
node
with
F
= E S
X
,
1
1 X
M
= E I
y
y
y
,
1
1
X
M
= E I
Z
Z
Z
,
1
1
X
U
:
2
node
with
F
= E S
X
,
2
2 X
M
= E I
y
y
y
,
2
2
X
M
= E I
Z
Z
Z
2
2
X
7
Torque of the efforts - Forces nodal and reactions
7.1 Options
available
The various options of postprocessing available for 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
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
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7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
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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 grid 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:
R
=
E
E
E
K
U
+ M
u&
- F
LOC
LOC
LOC
LOC
LOC
LOC
where: R
=
-
, -
-
-
-
-
, -
,
LOC
(1 1 1 1 1 1 1 2 2 2 2 2 2 2
NR
V, V, M, M, M
M
NR, V, V, M, M, M,
Y
Z
T
Y
Z
M
Y
Z
T
Y
Z
)
E
K
elementary matrix of rigidity of the element beam,
LOC
E
M
elementary matrix of mass of the element beam,
LOC
E
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.
LOC
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:
- NR (O)
U O f1
K
NR (L)
=
[] ()
U (L) -
f2
7.2.2 Generalized efforts, option: “SIEF_ELGA_DEPL”
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:
E
R
= K
U
LOC
LOC
LOC
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 command 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
F = Q
where
T
Q symbolically represents the matrix associated with the operator divergence. For an element, one
writing agricultural work of virtual deformations:
(T
Q) *
U = (U)
(*
U)
*
U
kinematically acceptable
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
23/24
For the elements of beam, one calculates simply the nodal forces by assembly of the forces
nodal elementary calculated by option SIEF_ELGA_DEPL, which is expressed by:
[F
= K
U
LOC]
[LOC] [LOC]
7.3.2 Nodal reactions, option: “REAC_NODA”
This option, called by CALC_NO, makes it possible to obtain the reactions R with the supports, expressed in
total reference mark, starting from the nodal forces F by:
tank
iner
R = F - F
+ F
tank
iner
F
and F
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 “Thèse of
Université of Liege. 1989.
[3]
J. SLIMI. “Simulation of ruin of pylon” Rapport SERAM N°14.033, ENSAM Juin 1993.
Handbook of Référence
R3.08 booklet: Machine elements with average fiber
HT-66/05/002/A
Code_Aster ®
Version
7.4
Titrate:
Element of beam with 7 ddl for the taking into account of warping
Date:
02/05/05
Author (S):
J.L. Key FLEJOU, J.M. PROIX
:
R3.08.04-B
Page:
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Handbook of Référence
R3.08 booklet: Machine elements with average fiber
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