Mechanical Design of
Transmission Lines
1
Introduction
The energized conductors of transmission and
distribution lines must be placed to totally eliminate
the possibility of injury to people.
Overhead conductors, however, elongate with time,
temperature, and tension, thereby changing their
original positions after installation.
Despite the effects of weather and loading on a line,
the conductors must remain at safe distances from
buildings, objects, and people or vehicles passing
beneath the line at all times.
2
To ensure this safety, the shape of the terrain along the
right-of-way, the height and lateral position of the
conductor support points, and the position of the
conductor between support points under all wind, ice,
and temperature conditions must be known.
Bare overhead transmission or distribution conductors
are typically quite flexible and uniform in weight along
their length.
Because of these characteristics, they take the form of
a catenary between support points.
3
The shape of the catenary changes with conductor
temperature, ice and wind loading, and time.
To ensure adequate vertical and horizontal clearance
under all weather and electrical loadings, and to
ensure that the breaking strength of the conductor is
not exceeded, the behavior of the conductor catenary
under all conditions must be known before the line is
designed.
4
The future behavior of the conductor is determined
through calculations commonly referred to as sag-
tension calculations.
Sag-tension calculations predict the behavior of
conductors based on recommended tension limits
under varying loading conditions.
These tension limits specify certain percentages of the
conductor’s rated breaking strength that are not to be
exceeded upon installation or during the life of the
line.
5
These conditions, along with the elastic and
permanent elongation properties of the conductor,
provide the basis for determining the amount of
resulting sag during installation and long-term
operation of the line.
Accurately determined initial sag limits are essential in
the line design process.
Final sags and tensions depend on initial installed sags
and tensions and on proper handling during
installation.
6
The final sag shape of conductors is used to select
support point heights and span lengths so that the
minimum clearances will be maintained over the life
of the line.
If the conductor is damaged or the initial sags are
incorrect, the line clearances may be violated or the
conductor may break during heavy ice or wind
loadings.
7
Sag and Tension Calculation
A bare-stranded overhead conductor is normally held
clear of objects, people, and other conductors by
periodic attachment to insulators.
The elevation differences between the supporting
structures affect the shape of the conductor catenary.
The catenary’s shape has a distinct effect on the sag
and tension of the conductor, and therefore, must be
determined using well-defined mathematical
equations.
8
Level Spans
The shape of a catenary is a function of the conductor
weight per unit length, w, the horizontal component
of tension, H, span length, S, and the maximum sag of
the conductor, D.
Conductor sag and span length are illustrated in the
next slide for a level span.
9
The catenary curve for level spans
10
𝐻 𝑤 𝑤(𝑥)2
𝑦 𝑥 = cosh 𝑥 −1 =
𝑤 𝐻 2𝐻
Note that x is positive in either direction from the low
point of the catenary.
The expression to the right is an approximate parabolic
equation based upon a MacLaurin expansion of the
hyperbolic cosine.
𝐻 𝑤𝑆 𝑤(𝑆)2
𝐷 = cosh −1 =
𝑤 2𝐻 8𝐻
11
The ratio, H/w, which appears in all of the preceding
equations, is commonly referred to as the catenary
constant.
An increase in the catenary constant, having the units
of length, causes the catenary curve to become
shallower and the sag to decrease.
Although it varies with conductor temperature, ice and
wind loading, and time, the catenary constant
typically has a value in the range of several thousand
feet for most transmission-line catenaries.
12
The approximate or parabolic expression is sufficiently
accurate as long as the sag is less than 5% of the span
length.
The horizontal component of tension, H, is equal to
the conductor tension at the point in the catenary
where the conductor slope is horizontal.
For a level span, this is the midpoint of the span
length.
13
At the ends of the level span, the conductor tension, T,
is equal to the horizontal component plus the
conductor weight per unit length, w, multiplied by the
sag, D, as shown in the following:
𝑇 = 𝐻 + 𝑤𝐷
Application of calculus to the catenary equation allows
the calculation of the conductor length, L(x),
measured along the conductor from the low point of
the catenary in either direction.
14
The resulting equation becomes:
𝐻 𝑤𝑥 𝑥2 𝑤2
𝐿 𝑥 = sinh =𝑥 1+
𝑤 𝐻 6𝐻2
The conductor length corresponding to x=S/2 is half of
the total conductor length and the total length, L, is:
2𝐻 𝑆𝑤 𝑆2 𝑤2
𝐿= sinh =𝑆 1+
𝑤 2𝐻 24𝐻2
15
The parabolic equation for conductor length can also
be expressed as a function of sag, D, by substitution of
the sag parabolic equation, giving:
8𝐷2
𝐿=𝑆+
3𝑆
The difference between the conductor length, L, and
the span length, S, is called slack.
The parabolic equation for slack:
𝑤 2 8
3 2
𝐿−𝑆 =𝑆 2
=𝐷
24𝐻 3𝑆
16
While slack has units of length, it is often expressed as
the percentage of slack relative to the span length.
Note that slack is related to the cube of span length for
a given H/w ratio and to the square of sag for a given
span.
For a series of spans having the same H/w ratio, the
total slack is largely determined by the longest spans.
It is for this reason that the ruling span is nearly equal
to the longest span rather than the average span in a
series of suspension spans.
17
Inclined Spans
Inclined spans may be analyzed using essentially the
same equations that were used for level spans.
The catenary equation for the conductor height above
the low point in the span is the same.
However, the span is considered to consist of two
separate sections, one to the right of the low point and
the other to the left as shown in the next slide.
18
Inclined catenary span
19
𝐻 𝑤 𝑤(𝑥)2
𝑦 𝑥 = cosh 𝑥 −1 =
𝑤 𝐻 2𝐻
Note that x is considered positive in either direction
from the low point.
The horizontal distance, xL, from the left support point
to the low point in the catenary is:
𝑆 ℎ
𝑥𝐿 = (1 + )
2 4𝐷
20
The horizontal distance, xR, from the right support
point to the low point of the catenary is:
𝑆 ℎ
𝑥𝑅 = (1 − )
2 4𝐷
The midpoint sag, D, is approximately equal to the sag
in a horizontal span equal in length to the inclined
span, S1.
21
Knowing the horizontal distance from the low point to
the support point in each direction, the preceding
equations for y(x), L, D, and T can be applied to each
side of the inclined span.
The total conductor length, L, in the inclined span is
equal to the sum of the lengths in the xR and xL sub-
span sections:
2
𝑤
𝐿 = 𝑆 + 𝑥𝑅 3 + 𝑥𝐿 3
6𝐻2
22
In each sub-span, the sag is relative to the
corresponding support point elevation:
𝑤𝑥𝑅 2 𝑤𝑥𝐿 2
𝐷𝑅 = 𝐷𝐿 =
2𝐻 2𝐻
In terms of sag, D, and the vertical distance between
support points:
ℎ 2 ℎ 2
𝐷𝑅 = 𝐷(1 − ) 𝐷𝐿 = 𝐷(1 + )
4𝐷 4𝐷
23
The maximum tension is:
𝑇𝑅 = 𝐻 + 𝑤𝐷𝑅 𝑇𝐿 = 𝐻 + 𝑤𝐷𝐿
In terms of upper and lower support points:
𝑇𝑢 = 𝑇𝑙 + 𝑤ℎ
The horizontal conductor tension is equal at both
supports.
The vertical component of conductor tension is greater
at the upper support and the resultant tension, Tu, is
also greater.
24
Ice and Wind Conductor Loads
When a conductor is covered with ice and/or is
exposed to wind, the effective conductor weight per
unit length increases.
During occasions of heavy ice and/or wind load, the
conductor catenary tension increases dramatically
along with the loads on angle and dead end structures.
Both the conductor and its supports can fail unless
these high-tension conditions are considered in the
line design.
25
Certain utilities in very heavy ice areas use glaze ice
thicknesses of as much as two inches to calculate iced
conductor weight.
Similarly, utilities in regions where hurricane winds
occur may use wind loads as high as 34 lb/ft2.
The degree of ice and wind loads varies with the
region.
Some areas may have heavy icing, whereas some areas
may have extremely high winds.
26
Ice Loading
The formation of ice on overhead conductors may take
several physical forms (glaze ice, rime ice, or wet snow).
The formation of ice on overhead conductors has the
following influence on line design:
Ice loads determine the maximum vertical conductor loads
that structures and foundations must withstand.
In combination with simultaneous wind loads, ice loads also
determine the maximum transverse loads on structures.
In regions of heavy ice loads, the maximum sags and the
permanent increase in sag with time
(difference between initial and final sags) may be due to ice
loadings.
27
The calculation of ice loads on conductors is normally done
with an assumed glaze ice density of 57 lb/ft3.
The weight of ice per unit length is calculated with the
following equation:
𝑤𝑖𝑐𝑒 = 1.244𝑡(𝐷𝐶 + 𝑡)
where
t is thickness of ice, in.
Dc is conductor outside diameter, in.
wice resultant weight of ice, lb/ft
The ratio of iced weight to bare weight depends strongly
upon conductor diameter.
28
Wind Loading
Wind loadings on overhead conductors influence line
design in a number of ways:
The maximum span between structures may be
determined by the need for horizontal clearance to edge
of right-of-way during moderate winds.
The maximum transverse loads for tangent and small
angle suspension structures are often determined by
infrequent high wind-speed loadings.
Permanent increases in conductor sag may be
determined by wind loading in areas of light ice load.
29
Wind pressure load on conductors, Pw, is commonly
specified in lb/ft2.
The relationship between Pw and wind velocity is given
by the following equation:
𝑃𝑤 = 0.0025(𝑉𝑤 )2
where Vw is the wind speed in miles per hour.
30
The wind load per unit length of conductor is equal to
the wind pressure load, Pw , multiplied by the
conductor diameter (including radial ice of thickness
t, if any), is given by the following equation:
(𝐷𝐶 + 2𝑡)
𝑊𝑤 = 𝑃𝑤
12
31
Combined Ice and Wind Loading
If the conductor weight is to include both ice and wind
loading, the resultant magnitude of the loads must be
determined vectorially.
The weight of a conductor under both ice and wind
loading is given by the following equation:
𝑤𝑤+𝑖 = (𝑤𝑏 + 𝑤𝑖 )2 +(𝑊𝑤 )2
where wb, wi, Ww and ww+i are bare conductor, ice,
wind, and resultant of ice and wind loads per unit
length in lb/ft respectively.
32
The NESC prescribes a safety factor, K, in pounds per
foot, dependent upon loading district, to be added to
the resultant ice and wind loading when performing
sag and tension calculations.
Therefore, the total resultant conductor weight, w, is:
𝑤 = 𝑤𝑤+𝑖 + 𝐾
33
Conductor Tension Limits
The NESC recommends limits on the tension of bare
overhead conductors as a percentage of the
conductor’s rated breaking strength.
The tension limits are: 60% under maximum ice and
wind load, 33.3% initial unloaded (when installed) at
600F, and 25% final unloaded (after maximum loading
has occurred) at 600F.
It is common, however, for lower unloaded tension
limits to be used.
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Reading Assignment
Sag Template
Stringing Chart
Stringing and Sagging Procedures
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