The mechanical effects of short-circuit currents in - Montefiore

(2.71)
2 tube
tube
4 tube
U ∂U
∂ U
+
2 tube
tube 4 ∑ i
∂t
i=
1..
3
y
( ρS)
c + ( EI)
= F = 0
tube
∂
∂t
tube
Consequently : U = 0 = ∑u
∀t∀x
i
tube
i
∂x
if the sequence of the connecting pieces are the
same on each phase.
This relation of functions ui being true at any
moment, whatever the value of x, one could easily
deduce that the forces and moments deduced from
the function derivatived at connection level also
meet the relations below :
(2.72) ∑
i=
(2.73) ∑
i=
1..
3
1..
3
y
i,
k
f
M
z
i,
k
= 0
= 0
This is true for any tube k.
Consequently, as the efforts at top of the column
are the vectorial addition of the contributions of
tubes set on each side, they meet similar relations.
(2.74) ∑
i=
y
f i
1..
3
z
i
= 0
(2.75) ∑ M = 0
i=
1..
3
Dynamic response of insulator column :
With respect to the bending of the jth column of
phase i, one has the following equation available to
calculate the dynamic response of the system :
(2.76)
( ρS)
∂
2
u
column
column
j
∂u
j
column + c
column
+
∂t
2
∂t
∂
4
u
column
column
∂z
4
y
i,
j o
j
( EI)
= f δ ( z − z )
z0 being the point of application in the middle of
the connection.
At bottom of the jth column, it will also be possible
to check the following relations :
(2.77) ∑
i=
(2.78) ∑
i=
1..
3
1..
3
F
M
y
i,
j
x
i,
j
= 0
= 0
With respect to the torsion of the jth column of
phase i, one has the following equation to calculate
the dynamic response of the system :
29
( )
I
(2.79)
+
θ column
∂ θ
column
j
z
( GJ)
= M δ(
z −z
)
column
2 column
j
2
∂t
2
∂ θ
∂z
2
+ c′
column
∂θ
i,
j
column
j
∂t
z0 being the point of application in the middle of
the connection.
I θ : Mass inertia relative to the axis per unit of
length,
G : Coulomb or torsion modulus,
J : Polar inertia of straight section.
At bottom of the jth column of phase i, one could
also check the following relations :
(2.80) ∑
i=
Conclusions :
1..
3
M
z
i,
jbase
= 0
The wrenches (force and moment) at bottom of two
columns in which the phases differ determine the
third wrench. The other contributions of each phase
are consequently linked to stresses other than
electrodynamic (dead weight, wind, ...) The
supporting structures are subject to differential
loadings. Their sizing will most often depend on the
other design hypotheses (wind, ice, earthquake,
etc.)
Example :
Considering a structure as illustrated in Figure 2.18,
we have plotted in Figure 2.19 the time variations
vs. time, forces and moments in the three
directions.
The dimensional data for this structure are as
follows :
a = 5 m , Icc = 31.5 kA,
busbar:
bar length = 15 m, diameter = 120 mm, thickness =
8 mm, material : Aluminium,
connection : clamped, pinned on each post
insulator,
insulating glass column. C4
One can easily check on Figure 2.19 that, at bottom
of the column :
- the sum of the moments in the direction x is zero
(electrodynamic load),
- the moments in the direction y are practically
constant (they correspond to the dead weight),
- the sum of the moments to z is zero
(electrodynamic load),
- the efforts to x are zero (electrodynamic load),
- the sum of the forces to y is zero (electrodynamic
load),
- the forces to z are practically constant (they
correspond to the dead weight).
For the supporting structure given below, we have
calculated the loads under static conditions from the
maximum dynamic values and dead weight. The
results are as follows :
o