So far, we have only looked at level spans. It is necessary now to consider how sag is affected on non-level, i.e. hilly spans.
For calculation of a catenary over hilly ground, it can be shown from Figure 7.4 that CA and CB are each catenary curves of half spans X and (L− X).
D1= TH/w[cosh(wX/TH)− 1] (7.16)
D2= TH/w[cosh(w[L − X]/TH)− 1] (7.17)
and
total T (at B)= THcosh[w(L − X)/TH] (7.18) It should be noted that for tower lines, the calculation of a catenary over hilly ground is complicated by the displacement of the suspension insulator set and the reader is directed to the paper by Bradbury, Kuska and Tarr [3] for a full treatise of the required mathematics which addresses the displacement effect.
The equation of a parabola is of the type Y = KZ2. If D3= sag, and L/2 = half span length, then:
D3= K(L/2)2= KL2/4 or K = 4D3/L2 (7.19) Inserting values for K:
Y = 4D3/L2× Z2 (7.20) G L X (L – X) Y Z C D1 H D2 H/2 D3 A B
Weather loads, conductor sags and tensions 103
Slope of parabola dy/dx = 8D3/L2Zand the slope of an inclined parabola is
H /L. Therefore: H /L= 8D3/L2− Z (7.21) and, hence: Z= HL/(8D3) (7.22) Y = KZ2= 4D3/L2× {H L/(8D3)}2 = 4D3/L2× {H2L2/(64D32)} = H2/(16D3) (7.23) Referring to Figure 7.4, the total sag on the equivalent complete span is:
D2= H/2 + D3+ Y (7.24)
Inserting the value of Y from equation (7.23):
D2= H/2 + D3+ H2/(16D3)= H/2[1 + H /(8D3)] + D3 (7.25) If G = complete span length, then G/2 = L/2 + Z. Inserting the value of
Z= HL/(8D3)gives:
G/2= L/2 + H L/(8D3) (7.26)
From normal sag calculations D3= WL2/(8T ); inserting the value of D3into equation (7.26) gives:
G/2= L/2 + HL/8 × 8T /(WL2)= L/2 + H T /WL (7.27)
G= L + 2T H /(WL) (7.28)
and
X= L/2 − (T H /WL) (7.29)
If the value of X is negative, the lowest point of the sag is outside the actual span. Equation (7.26) is used to establish the position in the span for the lowest sag. This is of assistance in establishing ground clearance in a hilly area, and clearances to obstacles where a line profile is not readily to hand.
7.8
Creep
7.8.1
General
Elastic increase and decrease in conductor length will, of course, be apparent as the conductor metal expands and contracts due to temperature change. However, conductors will also permanently lengthen due to non-elastic stretch, or creep.
Creep is generally acknowledged as having three components: two types of geometrical settlement due to distortion and bedding-in and a metallurgical extension. When a conductor is stranded, the wires in alternate layers are given an opposing lay (see chapter 8) to ensure that the complete conductor retains its shape throughout
its life and avoid the possibility of bird-caging. This means that each layer has only point contact with the wire(s) beneath it. Tensioning the conductor tightens the wires that tend to crush against each other. Slight deformation occurs as the conductor is tensioned such that the area of each strand reduces and the length increases until balance is achieved.
This geometrical settlement is more marked as the number of layers increases and as the diameter of the individual wires decreases. Also, the wires in each layer will settle together more intimately than during the stranding process once a load is applied. Both of these geometrical settlements will exhaust their influence over a short period of time.
There is also a permanent metallurgical extension of the conductor caused by changes in its internal molecular structure as the relative positions of metal molecules slide to take up a new arrangement, different to that at the time of manufacture, when a permanent load is applied. It is known that creep is a function of the metal used: all aluminium conductor, steel reinforced (ACSR) creep the most, all aluminium alloy (AAA) and all aluminium (AA) less and copper significantly less.
Laboratory measurements are made on actual conductors and the results achieved over a period of one to two months are extrapolated for a creep life of 10–25 years.
By temperature shift we mean that the design tension is not calculated at ambient temperature, but is shifted by a few degrees that effectively dictates that a higher tension (erection tension) is used. This temperature shift is to allow for the conductor creep (section 7.8.2).
Table 7.8 (an extract from IEC 1597 Table 5) provides mean values of temperature shift based on a creep period of ten years derived from many creep tests under- taken on stranded conductors. This table indicates suggested creep and temperature shifts.
A creep period of ten years is assumed since creep is usually small from ten to fifty years, and this level of creep may have already occurred during the stringing process.
Since there are many conductor constructions in use with many differing basic spans (and hence differing erection tensions), only a few such laboratory tests have been undertaken. As a consequence, a number of predictor equations have been
Table 7.8 Extract from IEC 1597 Table 5
Conductor Creep Mean value of (mm/km) temperature
shift (◦C) All aluminium conductors 800 −35 All aluminium alloy conductors 500 −22 Aluminium conductor, alloy reinforced 700 −30 Aluminium conductor, steel reinforced 500 −25 Compacted covered conductor 250 −12
Weather loads, conductor sags and tensions 105
developed. These equations are typically of the following type for metallurgical creep:
metallurgical creep= KSaHbTc (7.30)
where
K= constant depending on the conductor material S= average conductor stress (kg/mm2)
H= time (hours) T = temperature (◦C)
a, b and c= experimentally determined coefficients
Calculations need to be made for a series of time intervals since the tension does not remain static. Computer programs have been developed which assume a time interval increment of, say, 15 per cent and the value of creep summated for the whole creep period.
Clearly, if creep is not considered during the line design, it is possible for the conductors to over-sag and infringe ground clearance resulting in an expensive opera- tion to re-regulate the conductors to a higher tension. In view of this, a number of different creep compensation routines are employed, used either in isolation or in part combination.