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Organization (S):
EDF/IMA/MN, EP/AMV
Manual of Reference
R3.08 booklet: Machine elements with average fiber
Document: R3.08.02
Modeling of the cables in Code_Aster
Summary:
The cables are flexible structures which can undergo great displacements. Their analysis is thus not
linear. From the mechanical point of view, a cable cannot support any moment and is not the seat that of an effort
normal called voltage. The expression of virtual work and its differentiation compared to displacements
lead to modeling in finite elements: stamp rigidity depending on the displacement of the nodes and
stamp of constant mass. One presents the iterative algorithms statics and dynamics. Two examples are
given: one, static, is the search of the figure of balance of a cable subjected to a horizontal voltage
data; the other, dynamic, is the comparison between calculations by finite elements and of the test results of
short-circuits. Finally four appendices treat: calculation of the forces of Laplace, change of the temperature
of a cable subjected to the Joule effect, force exerted by the wind and of the modeling of the operation of installation.
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Tables of content
1 Notations ................................................................................................................................................ 3
2 Introduction ............................................................................................................................................ 4
3 mechanical Assumptions ........................................................................................................................ 4
4 Application of the Principle of Virtual Work ......................................................................................... 5
5 Linearization ........................................................................................................................................... 6
6 numerical Realization by the finite elements ........................................................................................ 7
7 particular Cases of the elements of cable with two nodes ........................................................................... 8
8 Use in Code_Aster ............................................................................................................... 9
9 static Problem .................................................................................................................................. 9
9.1 Iterative algorithm ............................................................................................................................. 9
9.2 ......................................................................................................................................... Example 10
10 dynamic Problem ......................................................................................................................... 11
10.1 Iterative algorithm of temporal integration .................................................................................. 11
10.2 Comparison of calculations and tests of short-circuits ................................................................ 12
11 Conclusion ......................................................................................................................................... 14
12 Bibliography ...................................................................................................................................... 15
Appendix 1 Calculation of the forces of Laplace between conductors .................................................................... 16
Appendix 2 Calculation of the temperature of the cables ...................................................................................... 20
Appendix 3 Calculation of the force exerted by the wind .................................................................................... 23
Appendix 4 Modeling of the installation of the cables ........................................................................................ 25
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1 Notations
With
surface of the cross-section of the cable.
B
stamp deformation.
C
v
voluminal heat of conducting metal.
E
Young modulus to traction.
E
has
current module of the cable.
E
C
modulate with compression.
F
ext.
external forces data.
F
iner
inertias.
F
int
internal forces.
G
measure of Green of relative lengthening compared to the situation of
reference.
H
thermal convection coefficient of a cable with outside.
I
instantaneous intensity of current.
K
coefficient of variation of the resistivity with the temperature.
()
L
I
function of form relating to the node
I
.
[]
L
()
()
[
]
L
L
I
J
1
1
,
!
[]
L
'
dL
D
D
ds
dL
D
D
ds
I
O
J
O
1
1
,
!
NR
voltage of the cable.
S
O
curvilinear X-coordinate on the cable in situation of reference.
T, T
O
temperature in current situation and situation of reference.
()
U.S., T
O
vector displacement at the moment
T
compared to the situation of reference.
()
X S
O
vector position in situation of reference.
thermal expansion factor.
,
parameters of Newmark.
density.
resistivity.
1
stamp unit of command 3.
S
O
1
stamp diagonal
ds, ds, ds
O
O
O
.
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2 Introduction
The essential characteristic of the cables is that these are flexible structures, of which analysis
mechanics is nonlinear because they are likely to undergo great displacements. They in
undergo during the operation of installation, when one adjusts arrows to respect stresses
of environment.
Thereafter, the cables can be animated movements of great amplitude under the impulse
blast pressure, fall of sleeves of ice or, in the event of incident, the forces of
Laplace resulting from the currents of short-circuit. They exert then on their supports of the efforts
much higher than the static efforts. One must hold account in the design of it of
works.
For the old works, which can be subjected to intensities of short-circuit increased because
extension of the network, it should be checked that reliability is always assured.
3 Assumptions
mechanics
The cables are regarded as perfectly flexible wire, which cannot support any
moment, neither bending, nor of torsion, and are the seat only of one normal voltage. This voltage plays
the role of a generalized stress.
One wants to calculate the field of displacement
()
U.S., T
O
at the moment
T
compared to the situation of
reference. The aforementioned is a static configuration of the subjected cable, for example, with gravity and with
temperature
T
O
; it is defined by the field of vectors position
()
X S
O
.
X
+
U
+
X
S
O
ds
O
+
U
S
O
ds
O
X
+
U
ds
X
+
X
S
O
ds
O
ds
O
X
Appear 3-a: Length of cable in situations of reference and moved
As [bib1], one takes for deformation the measurement of Green of relative lengthening compared to
situation of reference [Figure 3-a]:
G
ds
ds
ds
O
O
=
-
2
2
2
2
.
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G
must remain small. The second member lends himself to calculation because it comprises only squares of
elementary lengths. One sees on [Figure 3-a] that:
G
S.S
S
O
O
O
=
+
X
U
U
1
2
2
.
éq 3-1
The relation of behavior is:
(
)
[
]
NR
E With G
T
T
has
O
=
-
-
éq 3-2
with:
E
E
NR 0
E
NR 0
has
C
=
>
if
if
.
4
Application of the Principle of Virtual Work
If one does not hold depreciation account, the virtual work of the whole of the forces applied to
a length of cable during virtual displacement
U
is:
()
()
()
()
W,
W
,
W
,
W
,
int
iner
ext.
U U
U U
U U
U U
=
-
-
""
,
éq 4-1
by distinguishing work from the interior forces, the inertias and the external forces.
According to [éq 3-1]:
()
(
)
(
)
(
)
W
,
NR. G ds
NR
S
S
ds
NR.
ds
int
S
S
O
O
S
S
S
S
1
2
1
2
1
2
U U
X U
U
B U
=
=
+
=
.
.
,
éq 4-2
where:
(
)
B
1 X U
1
=
+
S
S
O
T
O
,
éq 4-3
while indicating by the superscript
T
transposed of a matrix.
()
(
)
W
,
With
ds
iner
""
"".
S
S
1
2
U U
U U
= -
.
éq 4-4
In all the cases, we consider work
W
ext.
like independent of
U
during a pitch of
time, bus:
·
or it is it really, in the case of conservative forces like gravity;
·
or, in the case of the forces of Laplace, the force applied to an element of cable depends
not only of the displacement of this element (conventional following force), but still of
displacements of the whole of the cables. It is considered whereas, during a pitch of time,
the force is constant and equal to its value at the end of the pitch of previous time.
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5 Linearization
With balance:
()
W,
U U
=
0
.
éq 5-1
If, short of a tolerance, the preceding equation is not satisfied, a correction is sought
U
of
U
such as:
()
()
W
DW
U U
U U
U
,
,
.
+
=
0
,
()
DW U U
U
,
.
being the directional derivative of
()
W U U
,
in the direction
U
[bib2] and [bib3].
According to [éq 3-2], one has obviously:
DNN.
E In Dg
E WITH
has
has
U
U
B U
=
=
.
.
According to [éq 4-3]:
D.
D
ds
D
ds
O
T
O
B
U
1
U
1
=
.
Thus:
()
()
{
}
D W
,
E WITH
ds
D
ds
NR D
ds
ds
int
T
has
S
S
O
T
O
S
S
U U
U
B U
B U
1 U
1
U
.
.
=
+
1
2
1
2
éq 5-2
According to [éq 4-4]:
()
(
)
D W
With
ds
iner
T
S
S
"",
.
""
U U
U
U
U
= -
1
2
.
éq 5-3
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6
Numerical realization by the finite elements
One notes by the subscript
H
matrices discretized in finite elements.
If
v
is a vector defined on the cable (position, displacement, acceleration,…) one has, at the current point
of a finite element of nodes I, J,…:
[]
v
L v
=
E
,
v
E
being the vector made up of the components of
v
with the nodes.
In the same way:
[]
S
'
O
H
E
1
v
L v
=
.
According to [éq 4-3]:
(
)
B
X U
L L
H
E
T
T
'
'
=
+
.
Internal forces
F
int
E
of a finite element
E
of structure are the forces which it is necessary to exert in its
nodes to maintain it in its current deformed configuration. According to the theorem of work
virtual for the continuous mediums, the work of these specific forces is equal to the work of the stresses
in the element, i.e. with
W
int
, for any field of virtual displacement. One thus has, according to
[éq 4-2]:
(
)
F
B
L L
X U
int
E
=
=
+
NR
ds
NR
ds
H
T
S
S
T
E
S
S
1
2
1
2
'
'
.
In addition, one replaces the inertias distributed in the element by specific forces with
nodes
F
iner
E
such that their work is equal to that of the real inertias for any field of
virtual displacement. According to [éq 4-4], one thus has:
F
L
L
iner
E
T
E
S
S
With
ds U
= -
""
1
2
.
In the same way, the external forces distributed are replaced by concentrated nodal forces
F
ext.
E
equivalent within the meaning of virtual work.
The differential of the virtual work of the interior forces of a finite element of cable is written, according to
[éq 5-2]:
()
() (
)
D
H
E
T
M
G
E
W
int
U U
U
U
K
K
U
,
.
=
+
,
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with:
K
B
B
K
L
L
M
H
T
has
H
S
S
G
T
S
S
E WITH
ds
NR
ds
=
=
1
2
1
2
'
'.
K
M
and
K
G
matrices of material and geometrical rigidity of the element are called.
(
)
K
K
U
M
G
E
+
is the main part of the variation
F
int
interior forces with the nodes due
with the correction of displacements
U
E
.
The differential of the virtual work of the inertias results from [éq 5-3]:
()
()
D W
H
iner
E
T
E
"",
.
""
U U
U
U
M U
= -
with:
M
L
L
=
T
S
S
With
ds
1
2
.
M
is the matrix of mass of the element.
-
M U
""
E
is the variation
F
iner
F
iner
inertias
with the nodes due to the correction of acceleration
""
U
E
.
7
Particular case of the elements of cable with two nodes
These elements are of the 1
Er
degree: they are thus right in position of reference and remain right in
deformed position.
No moment is applied in their ends and they are the seat only of one uniaxial stress.
They are thus elements of bar.
In other words: to modelize a cable by elements with two nodes comes down comparing it to a channel
whose links (elements of cable) would be articulated perfectly between them.
On the other hand, the elements of cable having more than two nodes have a curvature in general
variable with the deformation. One cannot thus treat them as elements of bar.
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8
Use in Code_Aster
This paragraph indicates how one introduces the cables into the controls concerned of Aster.
Order
Key word factor
Key word
Argument
AFFE_MODELE
AFFE
GROUP_MA
Name of the group of meshs
supporting a cable.
PHENOMENON
“MECHANICAL”
.
MODELING
“CABLE”
.
DEFI_MATERIAU
CABLE
E
Value of the module of material.
EC_SUR_E
Report/ratio of the module of compression
(very weak and being able to be null) on
modulate.
AFFE_CARA_ELEM
CABLE
GROUP_MA
Name of the group of meshs
supporting a cable.
SECTION
Value of the section of the cable.
STAT_NON_LINE
COMP_ELAS
GROUP_MA
Name of the group of meshs
supporting a cable.
RELATION
“CABLE”
DEFORMATION
“GREEN”
9 Problem
statics
This problem is that of the search of the balance of a structure of cables in unspecified position and
subjected to a system of forces given.
9.1 Algorithm
iterative
The equilibrium equation, forms discretized [éq 5-1] and [éq 4-1] without the term of inertia, which must be
satisfied in each node, is:
F
F
int
=
ext.
éq 9.1-1
Let us suppose that one has just calculated the field of displacement of the cables,
()
U
N O
S
, with the iteration
N
:
·
if this field makes it possible to satisfy, except for a tolerance, with [éq 9.1-1], one considers that
line:
() ()
X
U
S
S
O
N O
+
is the figure of balance of the cables;
·
if not, corrections of displacement are calculated
U
N
+
1
by the linearized system:
[
]
K
K
U
F
F
M
N
G
N
N
ext.
N
+
=
-
+
1
int
.
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Displacement with the iteration
()
N
+
1
is:
U
U
U
N
N
N
+
+
=
+
1
1
.
It is looked at if [éq 9.1-1] is satisfied by the field
U
N
+
1
and so on.
9.2 Example
One wants to calculate the figure of balance of a heavy cable [9.2-a] of which an end
With
is fixed and of which
the other
B
, of level with
With
, is subjected to a given horizontal force.
This problem is dealt with in [bib4], where it is regarded as highly nonlinear.
B
With
B
O
C
- 17,7 m
0
46,4 m
61,0
F = 25,7 NR
Extensionnelle rigidity (E.A): 4,45 X 10
5
NR
Linear weight: 1,46 NR/m
Appear 9.2-a: Balances of a heavy cable subjected to a horizontal voltage
At the beginning, the cable, modelized by 10 elements of the 1
Er
degree, is supposed in weightlessness and has one
horizontal rectilinear position
WITH B
O
. One simultaneously subjects it to the action of gravity and to
horizontal force
F
applied in
B
O
. The static position of balance
WITH C B
is reached in
8 iterations only.
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10 Problem
dynamics
This problem is that of the calculation of the evolution of a structure of cables.
10.1 Iterative algorithm of temporal integration
The form discretized of [éq 5-1] and [éq 4-1] supplements, which must be satisfied in each node and with
each moment is:
()
()
()
F
F
F
int
T
T
T
iner
ext.
-
=
éq 10.1-1
The algorithm of temporal integration is of Newmark type [bib1] and [bib5]. Let us suppose that the state of
cable (fields
U U
U
, “""
and
with the nodes) is known at the moment
T
and that one has just calculated a value
approached these fields to N
ième
iteration of the moment
T
T
+
.
·
If these values satisfy [éq 10.1-1], except for a tolerance, one takes them for values of
fields at the moment
T
T
+
.
·
If not, the correction of displacement is sought
U
N
+
1
, to which correspond, according to
the algorithm of Newmark, corrections speed and acceleration:
“U
U
N
N
T
+
+
=
1
1
and
""
,
U
U
N
N
T
+
+
=
1
2
1
1
such as:
(
)
(
)
(
)
K
K
M
U
F
F
F
M
N
G
N
N
ext.
iner
N
T
T
T
T
T
T
T
+
+
=
+
-
+
+
+
+
1
2
1
int
N
.
In the analysis of the movement of the cables, the algorithm of Newmark can be unstable. This is why
we use the algorithm says HHT, defined in [bib7], in which two parameters of Newmark
are related to a third parameter
:
()
=
-
=
-
1
2
1
4
0
2
.
For
=
0
, the algorithm is that of Newmark, known as “regulates trapezoid”. But for
slightly
negative
(
)
-
0 3
,
, it appears numerical damping which stabilizes calculation.
Determination of initial acceleration and the initialization of the fields at the beginning of a new pitch of
times are indicated in [bib5].
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10.2 Comparison of calculations and tests of short-circuits
To validate this modeling of the cables, we compared dynamic calculations by
Code_Aster with test results of short-circuits [bib8]. Those were carried out at the Laboratory of
Electric engineering of EDF on an experimental structure representative of the configurations of post
[Figure 10.2-a]. Three cables tended between two gantries apart 102 m are shorted-circuit, with
foreground, by a shunt laid out on insulating columns.
On the level of the other gantry, they are fed by a three-phase current of 35 kA during 250 ms. One
recorded the evolution:
·
voltage of the cables to their anchoring on the gantries, using dynamometers;
·
displacement of the points mediums of the ranges, identified by spheres of indication, with
the aid of fast cameras. One sees the cage of glass of the one of these cameras installed on one
gantry, on the left of [Figure 10.2-a].
[Figure 10.2-b] the comparison for a voltage of anchoring and the displacement of the medium gives
of a cable.
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Appear 10.2-a: Seen general of the testing facility of short-circuits
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A-evolution of a voltage of anchoring
B-trajectory of a point medium of cable
Appear 10.2-b: Comparisons of calculations by Code_Aster
and of tests of short-circuits
11 Conclusion
The modeling of the cables presented above is powerful (a reasonable number of iterations by
no time or to reach a balance static) and precise: it is adapted to the analysis of
long cables. For the short cables, on the other hand, flexional rigidity is not negligible, especially
with anchorings, and modeling must be done by elements of beam in great displacements and
great rotations [R5.03.40].
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12 Bibliography
[1]
J.L. LILIEN: Stresses and electromechanical consequences related to the passage of one
intensity of current in the structures in cables. Thesis. University of Liege (1983).
[2]
J.C. SIMO, L. VU-QUOC: With three-dimensional finite-strain rod model. Leaves II: computational
aspects, Comput. Meth. appl. Mech. Engng, vol. 58, p. 79-116 (1986).
[3]
A. CARDONA, Mr. GERADIN: With beam finite element nonlinear theory with finite rotations, Int.
J. Numer. Meth. Engng. Vol. 26, p. 2403-2438 (1988).
[4]
R.L. WEBSTER: One the static analysis off structures with strong geometric non-linearity.
Computers & Structures 11, 137-145 (1980).
[5]
Mr. AUFAURE: Dynamic nonlinear algorithm. Document R5.05.05 (1995).
[6]
K.J. BATHE: Finite element procedures in engineering analysis. Prentice-hall (1982).
[7]
H. Mr. HILBER, T.J.R. HUGHES, R.L. TAYLOR: Improved numerical dissipation for time
integration algorithms in structural dynamics. Earthq. Engng Struct. Dyn. 5, 283-292 (1977).
[8]
F. DURAND: Range of three-phase line (102 m) for posts VHV. Report/ratio of tests. EDF
(1990).
[9]
Mr. AUFAURE: A finite element of cable-pulley. Document R3.08.05 (1996).
[10]
Mr. AUFAURE: Procedure Pose-cable. Document U4.66.01 (1994).
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Appendix 1 Calculation of the forces of Laplace between conductors
Any conductor traversed by a current creates a magnetic field in its vicinity. This field
magnetic, acting on the current conveyed by another conductor, induced on the aforementioned a force known as of
Laplace.
ds
2
I
2
(T)
I
1
(T)
R
P
E
2
E
1
Appear A1-a: Provision of two close conductors
Let us take a conductor
!
traversed by the current
I T
1
()
[A1-a Figure], located in the vicinity of the conductor
“
traversed by the current
I T
2
()
. At the point P of the conductor
!
, where the unit tangent directed in the direction of
current is
E
1
, the linear force of Laplace induced by the conductor
“
is:
F
E
R
()
() ()
P
I T I T
R ds
=
×
×
-
10
7 1
2
1
2
3
2
E
.
One is interested only in the forces due to the very intense currents of short-circuit, the forces of Laplace in
normal mode being negligible.
F ()
P
can obviously put itself in the shape of the product of a function of time by a function of
space.
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A1.1 Function of the time of the forces of Laplace
This function,
G T
()
, is, except for a factor, the product of the intensities in the conductors
!
and
“
:
G T
I T I T
()
() ()
=
-
2 10
7 1
2
éq A1.1-1
where:
(
)
[
]
I T
I
T
E
J
ej
J
T
J
()
cos
cos
/
=
+
-
-
2
éq A1.1-2
with:
I
ej
:
effective intensity of the current J;
:
pulsation of the current (
= 100
for a current of 50 Hz);
J
:
phase depending on the moment when the short-circuit occurs;
:
time-constant of the shorted-circuit line dependant on its characteristics
electric (coil, capacity and resistance).
Very often, the complete function is replaced
G T
()
[éq A1.1-1] and [éq A1.1-2] by its average -
that the part is called continues - by neglecting the cos terms (
T
+…) and cos (
2
T
+…). The catch in
count these terms would require a pitch of very small time and the corresponding forces, to 50 and
100 Hz, are almost without effect on the cables whose frequency of oscillation is about the hertz.
Thus:
(
)
G
T
I I
E
continuous
E
E
T
T
()
cos
cos
cos
=
-
+
-
+
2
1
2
1
2
1
2
1
2
1
2
A1.2 Function of space
This function is:
H ()
.
P
R ds
=
×
×
1
2
1
2
3
2
E
E
R
The integral is calculated analytically when the conductor is cut out
“
in rectilinear elements.
length of such an element
MR. M
1
2
[A1.2-a Figure], there is an effect:
(
)
R
y
R
ds
Dy
m
m
3
2
2 32
2
2
2
=
+
×
=
×
=
;
;
.
E
R
E
R
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R3.08 booklet: Machine elements with average fiber
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M
1
M
2
P
2
1
y
R
m
R
E
2
Appear A1.2-a: Forces of Laplace induced by a rectilinear element of conductor
Like:
(
)
(
)
Dy
y
R
R
y
y
R
m
y
y
m
m
y
y
2
2 32
2
2
2 12
1
2
1
2
1
+
=
+
,
one a:
(
)
1
2
1
2
1
2
3
2
1
2
2
2 12
1
2
1
2
E
E
R
E
E
R
×
×
=
×
×
+
R Dy
R
y
y
R
M
M
m
m
m
y
y
.
The hook of the second member is also equal to:
sin
sin
2
1
-
.
A1.3 Realization in Code_Aster
The function of space
()
H P
previously definite is calculated by an elementary routine which evaluates
for each element of the conductor
!
, the contribution of all the elements of the conductor
“
who
act on him.
This contribution is evaluated to the points of Gauss (1 only for the elements with 2 nodes) of the element
conductor
!
.
The elementary routine has 2 parameters of input:
·
the load card of the element of the conductor
!
including/understanding the list of the meshs of
conductor
“
acting on him;
·
the name of the geometry, variable in the course of the time, which at every moment makes it possible to evaluate them
quantities
R
m
, sin, sin
1
2
.
The function of time
()
G T
is calculated by a specific operator of Code_Aster which produces one
concept of the function type.
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A1.4 Use in Code_Aster
·
Definition of the function of time
()
G T
Order
Key word factor
Key word
Argument
“COMPLETE”
DEFI_FONC_ELEC
SIGNAL
or
“CONTINUOUS”
COURT
INTE_CC_1
I
E
1
TAU_CC_1
1
PHI_CC_1
1
INTE_CC_2
I
E
2
TAU_CC_2
2
PHI_CC_2
2
INST_CC_INIT
Moment of beginning of short-circuit.
INST_CC_FIN
Moment of end of short-circuit.
·
Definition of the function of space
()
H P
Order
Key word factor
Key word
Argument
AFFE_CHAR_MECA
MODEL
Name of the model.
INTE_ELEC
GROUP_MA
Name of the group of meshs of
conductor
!
.
GROUP_MA_2
Name of the group of meshs of
conductor
“
.
·
Taking into account of the forces of Laplace
Order
Key word factor
Key word
Argument
DYNA_NON_LINE
EXCIT
CHARGE
name of
()
H P
FONC_MULT
name of
()
G T
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Manual of Reference
R3.08 booklet: Machine elements with average fiber
HI-75/96/024/A
Appendix 2 Calculation of the temperature of the cables
When the cables are traversed by a current of short-circuit, they strongly become overheated by Joule effect.
When the current is tripped, their temperature drops because of the thermal losses towards outside.
A2.1 Formulation
One supposes:
·
that at a moment given the temperature is uniform in any length of cable traversed by
even running;
·
that any node common to several cables is at the average temperature of these cables.
The change of the temperature is given by the equation of heat [bib1]:
(
)
dT
dt
H p T
T
WITH C
J
C
ext.
v
v
+
-
=
2
éq A2.1-1
where
p
is the perimeter of the cable:
p
With
=
2
,
J
the density of current, is, by neglecting the effect of skin:
J
I
With
=
and:
(
)
[
]
=
+
-
O
ref.
K T T
1
.
The expression of the current is given by [éq A1.1-2].
Thus:
(
)
(
)
I
I
T
E
T
E
E
T
T
2
2
2
2
2
1
2
2
2
2
1
2
=
+
-
+
+ +
-
-
cos
cos cos
cos
.
If one neglects the last term of the second member, who decreases exponentially in the time and which
is even null if
=
2
, the integral of
I
2
over one duration
T
, much higher than the period of the current,
is practically equal to:
I
T
e2
.
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R3.08 booklet: Machine elements with average fiber
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In other words, under this simplification, the instantaneous voluminal heat source to the second member
of [éq A2.1-1] is:
J
I
With
E
2
2
2
=
.
The differential equation [éq A2.1-1] is then with constant coefficients and is integrated analytically. It
is written:
dT
dt
C B T
has
D
+
-
=
+
2
2
with:
(
)
has
C
K T
I
With
B
K C IA
C
H p
WITH C
D
C T
O
v
ref.
E
O
v
E
v
ext.
=
-
=
=
=
1
2
2
2
2
2
2
.
Its solution, summons general solution of the equation without second member and a solution
particular of the equation with second member is:
·
with the heating, during the existence of the current of short-circuit:
()
()
T
T
B
C T T
has
D
B
C
B
C T T
I
I
I
=
-
-
+ +
-
-
-
exp
exp
2
2
2
2
1
T
I
being the temperature at the moment
T
I
;
·
with cooling, as from the moment
T
FCC
of end of short-circuit
(
)
B has
= =
0
:
(
)
[
]
(
)
[
]
{
}
T
T
C T T
T
C T T
FCC
FCC
ext.
FCC
=
-
-
+
-
-
-
exp
exp
1
.
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A2.2 Use in Code_Aster
·
Calculation of the function
T
Order
Key word factor
Key word
Argument
DEFI_THER_JOULE
LIST_INST
List moments of calculation.
INST_CC_INIT
Moment of beginning of short-circuit.
INST_CC_FIN
Moment of end of short-circuit.
TEMP_EXT
T
ext.
TEMP_RESI_REF
T
ref.
PARA_COND_1D
INTE_CC
I
E
With
With
RESI_R
O
RESI_R1
K
CP
C
v
COEF_H
H
TEMP_INIT
T
I
·
Assignment of the function
T
with the nodes subjected to the Joule effect
Order
Key word factor
Key word
Argument
AFFE_CHAM_NO
MESH
Name of the mesh.
SIZE
“TEMP_F”
AFFE
GROUP_NO
Name of the group of nodes
subjected to the Joule effect.
NOM_CMP
“TEMP”
FUNCTION
Name of the function
T
.
·
Assignment
of one
EVOL_THER
with the preceding nodes
Order
Key word factor
Key word
Argument
CREA_RESU
TYPE_RESU
“EVOL_THER”
NOM_CHAM
“TEMP”
CHAM_GD
LIST_INST
List moments of calculation.
CHAM_NO
Name of the nodal field of function
T
.
·
Assignment of the corresponding thermal load
Order
Key word factor
Key word
Argument
AFFE_CHAR_MECA
MODEL
Name of the model.
TEMP_CALCULEE
Name of
EVOL_THER
.
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R3.08 booklet: Machine elements with average fiber
HI-75/96/024/A
Appendix 3 Calculation of the force exerted by the wind
A3.1 Formulation
It is admitted that a wind speed
V
exert in the vicinity of the point
P
of a cable [A3.1-a Figure] one
aerodynamic linear force
F
having the following characteristics:
·
F
with the direction and the direction of the component
V
N
speed of the wind in the normal plan of
cable;
·
F
a module proportional squared of that has of
V
N
.
cable
P
V
N
F
V
V
T
Appear A3.1-a: speed of the wind in the vicinity of a cable
The payments of calculation of the lines define the force of a wind by the pressure
p
that it exerts on
a plane surface normal with its direction. For a cable, placed normally at the direction of the wind, these
payments prescribe to take for linear force:
F
p
2
=
,
being the diameter of the cable. That amounts considering that the cable offers to the wind a plane surface
equal to its Master-couple. An increase of the force thus is obtained because the cable, cylindrical,
have a less resistance to the air than a plane surface.
If speed
V
wind forms an angle
with the cable, its component in the plan perpendicular to
cable has as a module:
V
V
N
=
sin
.
Therefore, the linear force is:
F
p
=
sin
2
.
Of course, the linear force exerted by the wind depends on the position of the cable: it is “following”.
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A3.2 Use in Code_Aster
Here how one introduces the force of the wind into Code_Aster. The unit vector having the direction and
the direction the speed of the wind has as components
v v v
X
y
Z
,
.
Order
Key word factor
Key word
Argument
AFFE_CHAR_MECA
FORCE_POUTRE
TYPE_CHARGE
“WIND”
FX
p
v
X
FY
p
v
y
FZ
p
v
Z
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Appendix 4 Modeling of the installation of the cables
A cable in the course of installation in a canton (several ranges between posts) [A4-a Figure] is fixed at the one of
supports of stop. It rests on pulleys placed at the bottom of the insulators suspension and it is retained by one
force on the level of the second support of stop.
Appear A4-a: Poses of a cable in a canton with two ranges
While exploiting this force - or by moving its point of application - one adjusts the arrow of the one of the ranges, that
who is subjected to stresses of environment. Then one removes the pulleys and one fixes the cable at
insulators: the length of the cable in the various ranges is then fixed. It is on this configuration that one
install possibly additional components: spacers, descents on equipment, them
specific masses,… to give to the canton its final form.
This scenario is carried out by Code_Aster in the following way.
One constitutes a first mesh of the line, supposed in weightlessness (rectilinear conductors), comprising
cable-pulleys [bib9] and also of the meshs representing the additional components. These last meshs
will not be taken into account in calculation intended to determine the lengths of cable, but they will be useful
in the procedure
POSE_CABLE
[U4.66.01] called later on. Meshs of cable-pulleys, meshs
cables related to cable-pulleys and the meshs of the additional components must belong to
groups respecting a specific nomenclature [bib10] to be able to be recognized and treated by
POSE_CABLE
.
The operator
STAT_NON_LINE
[U4.32.01] the preceding structure subjected to gravity calculates, with control
arrow, and constitutes a structure of data of the type
evol_noli
.
Procedure
POSE_CABLE
constitute then a new mesh, not comprising more cable-pulleys but
containing the additional components and where the length of the cables is deduced from displacements from
precedent
evol_noli
.
One analyzes finally this real structure, that is to say by
STAT_NON_LINE
if one is interested only in the effect of the wind and/or
ice, is by
DYNA_NON_LINE
[U4.32.02] if one wants to know the evolution caused by the forces of
Laplace due to currents of short-circuit, or by the fall of sleeves of ice.
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