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BLAW-KNOX TRANSMISSION POLE

(Patents Applied For)

FOMM NO 941

TRANSMISSION
TOWERS

Being a reprint of a paper read
by E. L. Gemmill, Chief Engi-
neer of the Transmission Tower
Department of Blaw-Knox
Company, Pittsburgh, before the
Engineering Society of the same
company.

To which have been added
many tables of properties of
wires, sags, loads and curves,
formulae, etc., to make it a most
complete reference book for all
interested in the subject.

BLAW-KNOX COMPANY

GENERAL OFFICES: PITTSBURGH, PA.

DISTRICT OFFICES

NEW YORK CHICAGO SAN FRANCISCO

165 Broadway Peoples Gas Bldg. Monadnock Bldg.

BOSTON DETROIT

Little Bldg. Lincoln Bldg.

Fig. A— Double Circuit Tower, for 110,000 Volt Line

TRANSMISSION LINE
TOWERS

It is only within the last twenty-five to thirty years that it has been
considered advisable to carry overhead electric power transmission
lines on anything else than wood poles. But with the ever increasing
tendency to concentrate power house units, and consequently to make
fewer and larger installations, spaced farther apart, it has become
necessary to transmit electrical energy over greater distances. This,
in turn, has made it advisable to set a higher limit for the voltage at
which the electrical energy will be conveyed from one point to another,
in order to reduce to the lowest possible minimum the loss in transmis-
sion. The using of these higher voltages has, of course, brought in its
train the necessity of making more careful provisions for supporting
the conductors by means of which the electrical energy is transmitted
from one point to another. Naturally, the first change made in the
general scheme in vogue was to place the conductors farther apart,
which necessitated the use of better cross arms for supporting them.
At the same time it was also imperative that, with increased voltage,
more clearance be allowed between the ground and the lowest conduc-
tor wires under the worst possible conditions of operation. This
could best be accomplished by making the supporting structures
higher.

So long as the wires were kept only a short distance above the
ground, the wood poles made an ideal support for them under ordinary
conditions; but when higher supports had to be considered, transmis-
sion line engineers began looking about for other supporting structures
which would lend themselves more readily to all the varying condi-
tions of service.

The steel structure was immediately suggested as the proper support
to take the place of the wood poles, and many arguments were ad-
vanced in its favor.

But these supports when built of steel were more expensive than the
wood poles had been, and in order to keep the total cost of the line
equipment down to a minimum, and to make such an installation com-
pare favorably with a similar line using the wood poles, it became
necessary to space the steel supports farther apart, so as to use fewer
of them to cover the same length of line.

4fl44<)9

4 Transmission Towers

The steel support, however, had come to stay, and the whole prob-
lem resolved itself into a matter of making a careful investigation and
study of each installation, in order that there might be used that sys-
tem which apparently worked out the best in each particular case.
From these several projects there have been evolved the different
types of structures in use today for transmission line work. They
may be roughly divided into three general types, namely:

Poles

Flexible Frames

Rigid Towers
POLES

All supports that are relatively small at the base or ground line are
generally classified as Poles. In plan at both ground line and near the
top they are made in several different shapes. They may be round,
square, rectangular, triangular, or of almost any other section. As a
rule, their general outline is continued below the ground line to the
extreme bottom of the anchorage. They are usually intended merely
to take care of the vertical loads combined with horizontal loads
across or at right angles to the direction of the line. They may have
greater strength transverse to the line than in the direction of the line,
but they are often made of the same strength in each direction. Poles
are very rarely designed to take care of any load in the direction of the
line when combined with the specified load across the line. They
must be spaced closer together than the heavier structures but can be
spaced much farther apart than wood poles. A very common spacing
for steel poles is about 300 feet apart.

FLEXIBLE FRAMES

Flexible Frames are heavier structures than the poles, and are
intended to take care of longer spans. Like the poles, their chief func-
tion is to take care primarily of transverse loads with a small margin of
safety so that under unusual conditions of service they could also pro-
vide a little resistance in the direction of the line; i. e., in a measure,
distribute a load coming in this direction over a number of supporting
structures, and transfer such a load to the still heavier structures
placed at regular intervals in the line. Or the flexible frames may
transfer all loads coming on them in the direction of the line to a point
where they will be resisted, by a frame of similar construction and

Transmission Towers 5

strength, but which is made secure against the action of such loads by
being anchored in this direction with guy lines.

These flexible frames are almost always rectangular in plan. Gener-
ally, the parallel faces in both directions will get smaller as the top is
approached, but often the two faces parallel to the direction of the line
will be of the same width from the bottom to the top. But the two
faces transverse to the line almost always taper from the ground line
up, and get smaller toward the top. The two faces parallel to the line
are generally extended below the ground line to form the anchorages.

RIGID TOWERS

Rigid Towers are the largest and heaviest structures made for
transmission line supports, and, as would be implied by the designa-
tion given them, they are intended to have strength to carry loads
coming upon them, either in the direction of the line or at right angles
to this direction. They are usually designed to take a combination of
loads in both directions. These towers are built in triangular, rec-
tangular, and square types, depending upon the particular conditions
under which the structure is to be used. When a plan of the tower at
the ground line is square in outline, each side of the square will be very
much larger than in the case of either poles or flexible frames. The
width of one side of a rigid tower, measured at the ground line, will
vary somewhere between about one-seventh and one-third of the total
height of the structure. This dimension is usually determined by the
construction which will give the most economical design, especially
when there are a large number of the towers required; but it often
happens that the outline of one or more of the structures will be deter-
mined by local conditions which are entirely foreign to the matter of
economy of design. Then, too, the conditions of loading may be such
as to make a special outline the most economical design.

LOADINGS

There are three kinds of loads which come upon transmission line
supports:

(1) The dead load of the wires together with any coating on

them; also the dead weight of the structure itself.

(2) Wind loads on the wires and the structure transverse to

the direction of the line.

6 Transmission Towers

(3) Pulls in the direction of the line caused by the dead load
and the wind load on the wires.

The dead load on the wires consists of the weight of the wire itself,
plus the weight of any insulating covering, plus the weight of any coat-
ing of snow or sleet. In most installations the conductors are not
covered with any insulating material, and hence at the higher tempera-
tures the dead load will be the weight of the wire only. At the lower
temperatures the wires may be coated with a layer of ice, varying up
to a thickness of 1 " or more, all around the wire. In some instances
ice has been known to accumulate on conductor wires until the thick-
ness of the layer would be as much as 1}^" all around the wire. But
such instances are very rare, especially on wires carrying high voltages
because there is generally enough heat in these wires to interfere with
the accumulation of much ice on them. But the heaviest coating of
ice alone does not often produce the worst conditions of loading for
the conductor and the supporting structure. The worst condition of
loading is that resulting from the strongest wind blowing against a
conductor covered with that coating which offers the greatest area of
exposed surface to the direction of the wind under all the several con-
ditions obtaining. This will almost always be true when the wind is
blowing horizontally and at right angles to the direction of the line.
In this case the total horizontal load on the supporting structure from
the wires is the combination of the wind load against the wires and the
unbalanced pull in the direction of the line, which is produced by the
resultant of the horizontal wind load and the weight of the wire itself
and any covering. But it does not follow that this condition will
always give the maximum load on the structure. In mountainous
districts it may happen that a transmission line will be subjected to a
gust of wind blowing almost vertically downward, in which case this
pressure, being added directly to the weight of the wire and the ice
load, may lead to much more serious results than a wind of equal or
even greater intensity blowing horizontally across the line. It may
happen in some districts where large sleet deposits are to be encoun-
tered, that the vertical load from the dead weight of the wire and its
coating of ice will be so great as to produce in the wire a tension large
enough to break the wire, even without any added load from the wind.
This is especially true if the wire is strung with a very small sag.

Since the design of the transmission line supports is determined very
largely by the loads which it is assumed will come upon them, and

Transmission Towers 7

since the load resulting from the pull in the direction of the line is very
often the dominating factor; and, further, since this load is a function
of the resultant load on the wire produced by the wind load and the
dead load, it naturally follows that the assumptions made regarding the
amount of this resultant loading are a matter of prime importance.
For this reason some very extensive experimenting has been done to
determine the amount of wind pressure against wires, either bare or
covered, under extreme conditions of velocity, density of air and tem-
perature. Careful observations have also been made to find out, as
near as possible, what is the maximum quantity of ice that will adhere
to a wire during and after a heavy storm. It not infrequently happens
that the temperature falls and the wind velocity increases immediately
after a sleet storm. The falling temperature, of course, tends to make
the ice adhere more closely to the wires. On the other hand, a rising
wind will tend to remove some of the ice from the wires.

In places where the lower temperatures prevail, the wind velocity
rarely gets to be as high as in the warmer districts where sleet cannot
form. On the other hand, a moderate wind acting on a wire covered
with a coating of ice, will oftentimes put much more stress into the
wire than a higher wind acting on the bare wire. This means that the
conditions of loading are altogether different for different sections of
the country. It is now generally assumed that in those districts where
sleet formation is to be met, the worst condition of loading on the wire
will be obtained when the wires are covered with a layer of ice Yl"
thick, the amount of the wind pressure on them, of course, depending
upon the wind velocity and the density of the air.

WIND PRESSURE ON PLANE SURFACES

The wind pressure per unit area on a surface may be obtained by the
following formula:

V2W

P = K — — in which
2 g

v = velocity of wind in feet per second;
W = weight of air per unit cube;
g = acceleration of gravity in corresponding units;
K = coefficient for the shape of the surface.

v2W
The factor — — is called the velocity head.

8 Transmission Towers

In considering the pressure on any flat surface normal to the direc-
tion of the wind, the pressure may be regarded as composed of two

parts :

(1) Front Pressure

(2) Back Pressure

The front pressure is greatest at the center of the figure, where its
highest value is equal to that due to the velocity head. It decreases
toward the edges. The following conclusions are generally regarded
as fair and reliable deductions from the results of many experiments
made by several investigators, to determine the amount and distribu-
tion of wind pressures on flat surfaces :

(1) The gross front pressure for a circle is about 75% of that

due to the velocity head, while for a square it is about
70%, and for a rectangle whose length is very long
compared with its width it is somewhere between 83%
and 86%.

(2) The back pressure is nearly uniform over the whole area

except at the edges.

(3) This back pressure is dependent on the perimeter of the

surface and will vary between negative values of 40%
and 100% of the velocity head.

(4) The maximum total pressure on an indefinitely long rec-

tangle of measurable width may be taken at 1.83 times
the velocity head pressure. For a very small square,
the coefficient may be as small as 1 . 1 .

Using the value for W corresponding to a temperature of freezing,
or 32° F., and a barometric height of 30 inches, which is 0.08071
pounds per cubic foot, and changing the wind velocity from feet per
second to miles per hour, the formula for normal pressure per square
foot on a flat surface of rectangular outline becomes :

P _ i o, x 0-08071 5280 5280

3 X 2 x 32.2 X 60760 X 60^60 X V
or P = 0.0049335 V2

WIND PRESSURE ON WIRES

In the case of cylindrical wires the pressure per square foot of pro-
jected area is less than on flat surfaces. The coefficient by which the
pressure on flat surfaces must be multiplied to obtain the pressure on

Transmission Towers 9

the projected surface of a smooth cylinder, varies, according to different
authorities, from 45% to 79%. Almost all Engineers in this country
assume this coefficient to be one-half, and, on this assumption our
formula becomes

P = 0.00246675 V2

for the pressure per square foot on the projected area of the wire, with
any coating it may have on it.

Mr. H. W. Buck has given the results of a series of wind pressure
experiments made at Niagara on a 950 ft. span of .58 inch stranded
cable, erected so as to be normal to the usual wind. From the data
obtained, the following formula was derived:

P = 0.0025 V2
in which

P = Pressure in pounds per sq. ft. of projected area
V = Wind velocity in miles per hour.

For solid wire previous experimenters had derived the formula

P = 0.002 V2

It is to be noted that Mr. Buck's formula gives values for pressures
25% in excess of the other formulas, which might be attributed to the
fact that for a given diameter, a cable made up of several strands, pre-t
sents for wind pressure a different kind of surface than a single wire.
If we could be sure that this difference exists, then it would be well
worth while to take this into consideration when determining the loads
for which a tower is to be designed, and to make a careful distinction
between towers which are to support solid wires and those which are to
carry stranded cables. Almost all Engineers are inclined to accept the
formula given by Mr. Buck, and to assume it to be correct for both
types of conductors. The fact that this formula agrees so closely with
the formula arrived at by assuming that the pressure on the projected
area of a cylindrical surface is 50% of the pressure on a rectangular
flat surface, would seem to warrant accepting it as being correct.

WIND VELOCITY

In assuming the loadings for which a line of towers are to be designed,
the first thing to be determined is the probable wind velocity which
will be encountered under the worst conditions. Our calculations, of
course, should be based on actual velocities. This is mentioned be-
cause it is necessary to distinguish between indicated and true wind

10 Transmission Towers

velocities. The indicated velocities are those determined by the
United States Weather Bureau. Their observations are made with
the cup anemometer and are taken over five minute intervals. The
wind velocities over these short periods of time are calculated on the
assumption that the velocity of the cups is one-third of the true velo-
city of the wind, for both great and small velocities alike. As the result
of considerable investigation, it has been found that this assumption
is not correct, but that the indicated velocity must be corrected by a
logarithmic factor, to convert it into the true velocity. The actual
wind velocities corresponding to definite indicated velocities, as given
by the United States Weather Reports, are as follows :

Indicated Actual Indicated Actual

10 9.6 60 48.0

20 17.8 70 55.2

30 25.7 . 80 62.2

40 33.3 90 69.2

50 40.8 100 76.2

It is generally conceded that the wind pressure increases with the
height above the ground, and that it is more severe in exposed posi-
tions, and where the line runs through wide stretches of open country,
than it is in places which are more or less protected by their sur-
roundings.

If we accept the theory advanced by some, to the effect that the
ground surface offers a resistance to the wind, which materially lessens
its force, then we must conclude that after a certain altitude has been
reached the effect of this resistance becomes negligible, and that
beyond that altitude the rate of increase in wind pressure must be
small. This is especially true, because the density of the air is less in
the higher altitudes, which tends to counteract some of the effect of
increases in velocity. But experimental data bearing on this matter
are very limited, so that the rate of increase in wind pressures for higher
elevations above the ground, must in each case be determined by the
judgment of the Engineer who is designing the installation.

The curve on Fig. 1 shows the relationship between Indicated
velocity and Actual velocity, and the curve on Fig., 2 shows the pres-
sure in pounds per square foot of projected area of wire, corresponding
to actual velocities in miles per hour. By placing above the curve
given on Fig. 2 a similar curve corresponding to the indicated velocities,
a direct comparison between the two different velocities may be made
in terms of pressure. This is shown in Fig. 3.

Transmission Towers

For the general run of transmission line work no special allowance is
made for the pressures on towers at different elevations; but pressures
are used which are considered to be fair average values for the par-
ticular location of the line and for towers of heights which usually
prevail. But, of course, there is a distinction made between require-
ments for a low pole line and for a line on high steel towers. This
applies both to the wind pressures, which it is assumed will be en-
countered, and also to the factor of safety expected in the construction
throughout.

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VELOCITY,

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:f .... ...

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ACTUAL

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WN OF

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WIND VELOCITIES

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m - 2 - - --

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INDICATED VELOCITY; miles per hour.

Fig. 1

Transmission Towers

§ §

Transmission Towers

STANDARD PRACTICE FOR WIND AND ICE LOADS

The Committee on Overhead Line Construction, appointed by the
National Electric Light Association of New York, assumes an ice
coating %* thick all around the wire, for all sizes of conductors, and
maximum wind velocities of 50 to 60 miles per hour, as being an aver-
age maximum condition of loading. This Committee states that 62
miles per hour is a velocity not likely to be exceeded during the cold
months.

Three classes of loading are considered by the Joint Committee on
Overhead Crossings, as follows:

Class of
Loading:

Vertical Component of
Load on Wire:

Horizontal Component of Load
on Wire, or Wind Load
Across Line:

A
B
C

Dead
Dead + Y2" Ice
Dead + M" Ice

15 Lbs. per Sq. Ft.
8 Lbs. per Sq. Ft.
11 Lbs. per Sq. Ft.

For the Class "B" Loading the ordinary range of temperature is
given as— 20° to 120° F.

For the calculation of pressures on supporting structures the require-
ments are 13 Ibs. per sq. ft. on the projected area of closed or solid
structures, or on V/% times the projected area of latticed structures
The same Joint Committee allows a maximum working stress on cop-
per of 50% of the ultimate breaking stress; in other words, the wires
may be stressed to a point very near to the elastic limit.

An analysis of these three classes of loadings would seem to suggest
that Class "A" be used for lines in the extreme Southern part of the
United States, and that Class "B" be used for all other lines in this
country, unless it be for a few lines which might be located in regions
where especially cold weather is to be encountered, along with very
severe wind storms. For such lines Class "C" would certainly be
ample to take care of the most extreme conditions.

Interpreting these loadings in terms of wind velocities, class "A"
would allow for an indicated wind velocity of 101.8 miles per hour, or
an actual velocity of 77.46 miles per hour, acting against the bare con-
ductor. Class "B" provides for an indicated wind velocity of 71.96
miles per hour, or an actual velocity of 56.57 miles per hour, applied

Transmission Towers

to the projected area of the wire covered with a layer of ice yy thick
all around. Class "C" assumes an indicated velocity of 85.9 miles per
hour, or an actual velocity of 66.33 miles per hour, against the wire
covered all around with a layer of ice %" thick.

It has been contended by some Engineers that sleet does not deposit
readily on aluminum, owing to the greasy character of the oxide which
forms on the surface of aluminum conductors, and that because of this
fact the wind loads acting on such lines should not be taken so high as
when copper wires are used. But the experience and observation of
many other Engineers does not confirm this assumption.

CURVES ASSUMED BY WIRES

When the wires are strung from one structure to another throughout
the line, they assume definite curves between each two of the struc-
tures, these several curves, of course, depending upon the different
conditions attending the stringing.

If a heavy uniform string which is considered to be perfectly flexible,
is suspended from two given points, A and B, and is in equilibrium in a
vertical plane, the curve in which it hangs will be found to be the
common catenary. This is shown in Fig. 4.

Tension, T,

CATENARY

i X

DlRKTIHX-1 I

ON-X N

Fig. 4

16 Transmission Towers

CATENARY

Let D be the lowest point of the catenary, i. e., the point at which the
tangent is horizontal. Take a horizontal straight line O X as the X
axis, whose distance from D we may afterwards choose at pleasure.
Draw D O perpendicular to this line, and let O be the origin of co-
ordinates. Let 0 be the angle the tangent at any point P makes with
O X. Let To and T be the tensions at D and P respectively, and let
the arc D P = Z. The length D P of the string is in equilibrium
under three forces, viz: the tensions T0 and T, acting at D and P in
the directions of the arrows, and its weight w Z acting at the center of
gravity G of the arc D P.

Resolving horizontally we have

T cos e = To (1)

Resolving vertically we have

T sin 0 = w Z (2)

Dividing equation (2) by equation (1)

dy w Z

dx " To (3)

If the string is uniform w is constant, and it is then convenient to
write : To = w C. To find the curve we must integrate the differential
equation (3).

We have,

z dz

/. dy = ±

/. y + A = db

We must take the upper sign, for it is clear from (3) that, when x
and Z increase, y must also increase. When Z = O, y + A = C.
Hence, if the axis of X is chosen to be at a distance C below the lowest
point D of the string, we shall have A = O. The equation now

takes the form,

y2 = Z2 + C2 (4)

Transmission Towers 17

Substituting this value of y in (3), we find,

Cdz

V Z2 -f C2 = dx'
where the radical is to have the positive sign. Integrating,

C log (z + VZ2 + C2) = x -f B
But x and Z vanish together, hence B = C log C.
From this equation we find,

v z2 + c2 + z = c e c

Inverting this and rationalizing the denominator in the usual manner,
we have

V Z2 + C2 — Z = CC~'
Adding and subtracting, we deduce by (4)

The first of these is the Cartesian equation of the common catenary.
The straight lines which have here been taken as the axes of X and Y
are called, respectively, the directrix and the axis of the catenary.
The point D is called the vertex.
Adding the squares of (1) and (2), we have by help of (4),

T2 = w2 (Z2 + C2) = w2y2;

/. T = w y (6)

The equations (1) and (2) give us two important properties of the
curve, viz: (1) the horizontal tension at every point of the curve is
the same and equal to w C; (2) the vertical tension at any point P is
equal to w Z, where Z is the arc measured from the lowest point. To
these we join a third result embodied in (6), viz: (3) the resultant
tension at any point is equal to w y, where y is the ordinate measured
from the directrix.

Referring to Fig. 4, let PN be the ordinate of P, then T = w PN.
Draw NL perpendicular to the tangent at P, then the angle P N L = 0
Hence,

PL = PN sin 0 = Z by (2)
N L = PN cos o = C by (1)

18 Transmission Towers

These two geometrical properties of the curve may also be deduced
from its cartesian equation (5).
By differentiating (3) we find,

1 do 1 dz C

1 " /"** • • I 1 /i

cos2# dz C do cos20 (7)

v2
f> is also = •=;

We easily deduce from the right-angled triangle P N H, that the
length of the normal, viz: PH, between the curve and the directrix, is
equal to the radius of curvature, viz., p, at P. At the lowest point of

C2
the curve D, the radius of curvature, /», = — = C. It will be noticed

that these equations contain only one undetermined constant, viz., C;
and when this is given, the form of the curve is absolutely determined.
Its position in space depends on the positions of the straight lines
called its directrix and axis. This constant C is called the parameter of
the catenary. Two arcs of catenaries which have their parameters
equal are said to be arcs of equal catenaries.

Since /> cos2 0 = C, it is clear that C is large or small according as
the curve is flat or much curved near its vertex. Thus, if the string is
suspended from two points A and B in the same horizontal line, then C
is very large or very small compared with the distance between A and
B, according as the string is tight or loose.

The relationship between the quantities y, Z, C, />, and 0 and T
in the common catenary may be easily remembered by referring to the
rectilineal figure P L N H. We have PN = y, PL = Z, NL = C,
PH = />, T = w- PN and the angles LNP, NPH are each equal to 0.
Thus the important relations (1), (2), (3), (4) and (7) follow from the
ordinary properties of a right-angled triangle.

The co-ordinates of the center of curvature for the catenary are:

a (abscissa) = x — Vy2 — C2
ft (ordinate) = 2y

When two or more unequal catenaries have similar outlines so that

*~^

the ratio ^ is the same for all of them, the curvature between the

points D and P will also be the same for all these catenaries. From

Transmission Towers 19

this it follows that, at similar points on the different catenaries, the
several radii of curvature will vary directly as the values of x for the
different curves. The radius of curvature at the lowest point D has
already been shown to be equal to C, the parameter of the catenary.
Since C and y — C both vary directly as the value of x for these
unequal but similar catenaries, it is evident that y must also vary in
the same manner. It will be seen from the triangle PLN, that when
C and y both vary in the same manner, LP or Z, which is the length of
the arc DP, must also vary in the same manner.

ELASTIC CATENARY

When a heavy elastic string is suspended from two fixed points and
is in equilibrium in a vertical plane, its equation may be found as
follows :

Using the same figure as for the inelastic string and denoting the
unstretched length of arc D P by Zi, let us consider the equilibrium of
the finite part D P;

Tcosfl = To (1) . dy _ _ wzi .. Zi , ,

(2) •' dx • To " C

From these equations we may deduce expressions for x and y in
terms of some subsidiary variable. Since Zi = C tan 0 by (3), it will
be convenient to choose either Zi or 0 as this new variable. Adding
the squares of (1) and (2), we have,

-P = W2 (C» + Z!2) (4)

Since — = cos 0 and -p = sin 0,
dz dz

we have by (1) and (2)

IT * - -T

where the constants of integration have been chosen to make

CV
x = O and y = C + -^-

at the lowest point of the elastic catenary. The axis of X is then the
statical directrix.

20 Transmission Towers

We have the following geometrical properties of the elastic catenary :

(2)
(3)

All of these reduce to known properties of the common catenary
when E is made infinite.

These equations have value only from an academic viewpoint.
They are too unwieldy to be of any practical value in determining the
properties of curves, assumed by transmission line wires under different
working conditions. These equations would be still further compli-
cated, if we attempted to make them take care of changes resulting
from conditions of loading due to different temperatures.

PARABOLA

If we consider the weight of the wire to be uniformly distributed
over its horizontal projection, instead of along its length, its equation
will be found to be that of a parabola.

PARABOLA

Fig. 5

Transmission Towers 21

By referring to Figure 5 ^nd considering the equilibrium of any
part OP of the wire, beginning at the lowest point O, the forces acting
on this part are seen to be the horizontal tension H at O, the tension T
along the tangent at P, and the total weight W of the wire, OP. As
this weight is assumed to be uniformly distributed over the horizontal
projection OP1 = x of OP,
the weight is W = w x, and bisects OP1.

Resolving the forces in the horizontal and vertical directions, we
find as conditions of equilibrium,

— H + T ^ = O, — wx + T ^ = O,
dz dz

whence, eliminating dz, -p- =77 x-

Integrating and considering that x = O when y = O, we get

w 2 H

y = 777 x2, which may be put in the form x2 = y. This is the

2 ri w '

equation to a parabola.

If we substitute - for x, and S for y, in the equation for the curve,

//y 2H

\2/ w

it becomes ( - ) = S or w /2 = 8 HS,

w/2
from which H = -r^-,

which is the well known equation for determining the horizontal ten-
sion in the wires, when the two points of support are in the same hori-
zontal plane. In that case - equals one-half of the span, and S

equals the sag or deflection of the wire below the plane of the supports.

The three forces H, T, and W, are in equilibrium; they must inter-
sect in a point R which bisects OP1, and the force polygon must be
similar to the triangle RPP1.

Drawing such a force diagram K L M, and making L M equal to W
or w x, and MK equal to H, KL will be the value of T and equal to
A/H2 + (w x)2.

Substituting for H and x their values in terms of w, / and S, this

//W/2Y,/ 1Y /w2/4

becomes ^(- ) + (wjj = ^-

+ law2/22 w/

^- = - /2 + 16S2.

Transmission Towers

A quantity \/A2 + a2, when a is very small relatively to A, may be

a2
approximated by using A -f -r-r ; hence, an approximation for the

above value of T is,

w/2

w / /7 . 16 S2\ w /2 c

8sV/ + ^rj'or'^s + wS-

In this form it is very similar to the expression for the tension in the
wire at the insulator supports, derived by assuming the curve to be a
catenary. It will readily be seen from the above that for very small
sags in short spans the maximum tension at the insulator supports is
very little more than the tension at the middle of the span.

But it must be noted that in order that the above assumption may
be warranted, it is essential that the span considered, be short, and that
the length of wire be little more than the span. This, of course, means
that the sag in the wire must be rather small.

? 14"

The equation for the parabola x2 = y has, for the coefficient

of y, a constant which is equal to four times the distance between the
directrix of the parabola and the vertex O, as shown in Fig. 6.

Fig. 6

Transmission Towers

The directrix is shown passing through the point A, and is parallel to
the X axis. The line OY is the axis of the curve. If a line is drawn tan-
gent to the curve at any point P, this tangent will intersect the Y axis at
a point B, such that the distance BO will equal the distance OC, where
C is the point of intersection of the Y axis with a line drawn through
the point P, parallel to the X axis. The length BC is the subtangent
and is equal to twice the ordinate of the point of contact. The line
PD drawn through the point P and perpendicular to the tangent BP,
will intersect the Y axis at point D. The length CD is the subnormal
of the curve, and is constant for all points on the curve. It is equal to
one-half the co-efficient of y in the original equation, and is therefore

equal to — . The angles TRX and ORB are each equal to the angle

PDC or e.

Tan# =

R P

PARABOLIC ARC WITH SUPPORTS AT DIFFERENT
ELEVATIONS

The curve in which a suspended wire hangs, may be considered to
extend indefinitely in both directions, and the suspended wire may be
secured to rigid supports at any two points, such as N and U, lying on
this curve (Fig. 7), without altering the tension in the wire. The law
of this parabola is

PARABOLA

:2 = Ky,

SPAN SUPPORTS AT
DIFFERENT LEVELS

Span measured horizontally =

^^ i i "^ 1 y

24 Transmission Towers

and in the case of a suspended wire the multiplier K is directly propor-
tional to the tension H, and inversely proportional to the density of
the conductor material. The value of K in terms of the horizontal
tension and the weight of the conductor has already been found to

u 2H

be - .

Let S = sag below level of lower support,

B = horizontal distance of lowest point of wire

from lower support,

h = difference in level of the two supports,
/ = length of span measured horizontally,

all as indicated on Fig. 7; then, by inserting the required values in
equation x2 = Ky, the following equations are derived therefrom:

B2 = K S,

(/ — B)2 = K (S + h), or, /2 — 2/B + B2 = KS + Kh,
from which B2 on one side and its equivalent KS on the other side can-
cel out, leaving I2 — 2/B = Kh.

Therefore,

— Kh , B2

2 ' K

From an inspection of the formula B = - —, — , it is seen that if

2> I

Kh = /2, the lowest point of the wire coincides with the lower support
N, while if Kh is greater than /2, the distance B is negative, and there
may be a resultant upward pull on the lower insulator N — a point to
bear in mind when considering an abrupt change in the grade of a
transmission line.

We may consider the curve of the wire between the two supports N
and U as being made up of two distinct parts, NO and OU. The part
NO will be equivalent to one-half of a curve whose half span measured
horizontally is B, and whose sag is S. Similarly, the part OU will be
equivalent to one-half of a curve whose half span measured hori-
zontally is / — B, and whose sag is S -)- h.

It is possible, and sometimes convenient, to express the formulas for
wires suspended from supports not at the same level, in terms of the
equivalent sag (Se) of the same wire, subjected to the same horizontal
tension when the horizontal span (/) is unaltered, but the supports are
on the same level.

Transmission Towers 25

For such a condition, the equation to the curve becomes

from which

^ = 45;

If we substitute for K in the above formulas, its equivalent value

K =— ,

then we get the following set of formulas, in which B, S, /, and h, are all
as indicated in Fig. 7.

COMPARISON OF CATENARY AND PARABOLA

If a straight line is drawn through the point P and any other point K
on the parabola, shown in Fig. 6, and this chord KP is bisected at the
point M, a line drawn through this point M and parallel to the Y axis
will bisect all other chords which are parallel to the chord KP. From
this it follows that a line SU drawn tangent to the parabola and
parallel to the chord KP, will be tangent to the curve at a point L
which lies on a line that is drawn parallel to the Y axis and through the
point M. Another property of the parabola is, that the tangents to
the curve at the points K and P will intersect at the point N, which
also lies on the line that passes through the points L and M. If the
horizontal projection of the chord KP be designated by /, then the
horizontal projection of KM, MP, KN, and NP will each be equal to

1AL

The total tension T in the wire at any point on the curve equals

+ (wx)2,
or,

T,.ygy+-

But it is seen from the figure that

V(!7

+ x2 = DP,
/. T = w DP .

26 Transmission Towers

In the case of the catenary the total tension in the wire is

T = w y

in which

y = VC2 + Z2
But, when we compare a catenary and a parabola having equivalent

horizontal tensions. C = — .it will be seen that the two formulas for

total tension in the wire differ only in that the value Z, which is the
true length of the arc, is used in the one case, -where x, which is the
horizontal projection of the length, is used in the other. But T will
always be greater for the catenary except at the lowest point of the
curve.

The radius of curvature of the parabola

2H ^

x2 = y is p = w

w ' c.

SmV

in which y is the angle which the tangent to the curve makes with the
Y axis.

/> sin3 <f = — = DP sin y, or, /> sin2 y = DP

from which DP ^

DZ.

sn <p

If we substitute C for—, the radius of curvature is,

C C

sin3 <f> cos3 9
In the case of the catenary,

cos2 o

A comparison of these two, shows that the radius of curvature for
the parabola, at a point where the tangent makes an angle 0 with the

horizontal, is times that for a similar point on the catenary.

Transmission Towers

At the lowest point of the curve the radius of curvature is the same for
both the parabola and the catenary, when the horizontal tension is the
same. It is also true that, when the horizontal tension at the lowest
point of the parabola is equivalent to that at the lowest point of the
catenary for the same span and loading conditions, the sag at this
point below the plane of the supports will be very nearly the same for
the two curves.

But the outlines of the two curves differ at all points between the
lowest point and the point of support. This difference between the
outlines of the two curves becomes greater as the spans and the sags
are made larger. It is because of this difference in the outlines of the
two curves that the sags will be nearly equal for only one loading con-
dition. Any change in the loading condition will produce different
changes in the lengths of the two curves, and hence, will make the sags
different.

REACTIONS FOR SPANS ON INCLINES

When wires are strung on towers that are located on steep grades, it
is very necessary that we determine carefully the reactions at the
points of support and also the deflection of the wire away from a
straight line joining the two points of support for any given span.
This case is shown in Fig. 8.

PARABOLA

Fig. 8

28 Transmission Towers

If we have given the horizontal distance, /, between the supports A
and B and also the vertical distance, h, that B is above A, together
with the maximum tension, T, in the wire at the point of support B,
we can determine the reactions at both of the supports and also the sag
in the wire.

The wire ADRB is in equilibrium under three forces; viz., the ten-
sions acting at A and B in the directions of the tangents, and its weight
w/1 acting at its center of gravity. These three forces intersect at the
point Z, and the vertical line through this point passes also through
the point C on the line AB and midway between A and B. On this
vertical line lay off the distance OU equal to W or w/1, and let it be
bisected at the point C so that OC equals CU or J/£ w/1. Complete the
force diagram by drawing UV and OV parallel respectively to ZB and
AZ. UV will then be the tension in the wire at the point B, and OV
will be the tension at the point A. The vertical component of the re-
action at B is the vertical component of UV and is equal to UM.
The vertical component of the reaction at A is the vertical component
of OV and is equal to MO or UO — UM.

Complete the parallelogram of forces by drawing OF parallel to UV
and FU parallel to OV. The points F and V must lie on the line AB.
Let the tension UV in the wire at the point B be denoted by T. Let
0 be the angle made by the line AB with the horizontal line AX, and
let </> be the angle between the lines FO and AB. Let ,3 be the angle
which the tangent at the point B makes with the horizontal plane.

Angle O C F = ft + 90°.

Sin <p = - Sin (90° + *) = cos
But

1 =

. -'/wA

an ^j

The horizontal component of the stress in the wire at either point of
support, is H = T Cos fi. This is also the total tension in the wire at
the point (if any) where the slope of the wire is zero. The total stress
in the wire is greatest at the highest point B, and the vertical com-
ponent of the reaction at this point is UM = NO = T Sin fi.

Transmission Towers 29

The weight supported by the lower tower is MO = w/1 -- T Sin /?.

In some cases this may be zero, or it may even be a negative quan-
tity, in which case the wire will exert an upward pull on the lower sup-
port. In that case the sag S below the point A will be zero. This is
not an unusual condition on a steep incline, or on a moderate incline if
the spans are short.

The position of the lowest point D in the span is determined by the
condition that the vertical component of the force acting at either point
of support is the weight of that part of the wire between the point D
and the point of support. This is true because the tension in the wire,
at the point D, has no vertical component. The horizontal distance
of the point D from the support B, is,

ON / - T Sin ,3 _ T Sin ,3 • cos "
* OU = w/1 w

It may happen that /B will be found to be equal to or greater than /,
in which case the support A will be the lowest point in the span. If
the vertical component of the force at the support B is greater than the
total weight of the span (w/1), it follows that the resultant force at the
support A will be in an upward direction.

The total sag is S + h. The value of this sag may be determined by
considering /„ to be one-half of a span having supports at the same
level, and having the sag,

Q m -J5L_ (2_W! w/,1

cos 0 ' 8 H 2 H cos ft '

STRINGING WIRES IN SPANS ON STEEP GRADES

When transmission lines are carried up steep grades, and are strung
on towers in such a manner that the lower support A is the lowest point
of the span, it is of considerable advantage in stringing the wires, to
know the maximum deflection of the wire from the straight line AB as
observed by sighting between the points A and B. Such a condition
is shown in Fig. 9.

A line drawn tangent to the curve and parallel to the line AB will be
tangent at the point R, which is on a vertical line drawn through the
point C at the middle of the line AB. The wire will, therefore, have
its maximum deflection from the line AB at this point R. The hori-
zontal projections of AR and RB are equal, and have the value Yd
when the horizontal projection of the span AB is /.

Transmission Towers

Fig. 9

Let the deflection of the wire, at the point R, from the straight line
AB be denoted by S1. By taking moments about the point A, and
putting their sum equal to zero, we may determine the value of this
deflection.

w/1

S1 = ^f

/
4°r

w/1

w/2
8H

Comparing this span with a span having the same horizontal pro-
jection but supports A and B1 at the same level, it will be seen that
when a wire is strung between supports on a slope, the maximum deflec-
tion S1 of that wire from the straight line joining the two points of sup-
port, is exactly the same as the maximum sag S of the same wire when
strung between points on the same level ; provided the span measured
horizontally and the horizontal component H of the tension are the
same in both cases.

The above analysis will also show that this relationship between

Transmission Towers 31

spans having the same horizontal projections will be true even though

the lowest point on the wire is not coincident with the lower support.

These formulas for lines carried up steep inclines are all based on the

assumption that the total weight of wire is the weight per foot of length

of wire multiplied by the length of the line AB (which is I1 = - — ).

cos "

This, of course, is an approximation which is the more nearly correct
as the sag is kept small in proportion to the span.

Having determined the value of S1, and knowing the value of 0,
we may determine the value of CR. This distance can then be
measured down vertically below the points of support A and B, as
shown at F and K, and the wire when strung between these two sup-
ports may be drawn up until it becomes tangent to the line FK parallel
to AB and at the distance CR vertically below AB. This may be
observed by sighting from F to K.

The length of the wire between fixed supports at the same horizontal
level is approximately,

8 S2

L = / + %

in which / is the distance between supports and S the sag at the center,
both expressed in feet. In the case of a wire between supports which
are not on the same level, the total length may be considered to be
made up of two distinct parts of the parabolic curve.

RELATION BETWEEN STRESS, TEMPERATURE AND SAG

All of the formulas so far deduced for determining working condi-
tions for the wires on the basis of using parabolic curves, have been
obtained by ignoring the fact that the wires are elastic and will there-
fore stretch under tension, and that the length will also be affected by
changes in temperature.

It is customary to assume that the material of the conductors is
perfectly elastic up to a certain critical stress, known as the elastic
limit, and that the process of elongation and contraction follows a
straight line law. Therefore, the length of the wire will be changed by
the amount of elongation or contraction which will be,

-L ^

L . £,

in which Le is the elongation or contraction, L is the unstressed length

32 Transmission Towers

of wire, Te is the stress per square inch in the wire, and E is the modulus
of elasticity.

The change in length due to differences of temperature will be,

Lt = Lo (1 + mt),

in which L0 is the original length, t is the number of degrees Fahrenheit
change in temperature, and m is the coefficient of expansion for the
material in the conductor. These two changes in length of wire are
not independent of each other. They act simultaneously and are
inter-related, and must be considered together. Any temperature
variation causes a change in the length of wire, which, in turn, changes
the sag condition, and hence changes the stress, which, in turn, will
affect the amount of change in length of wire due to the stress in it.

In the case of long spans it is always necessary to make proper allow-
ances for the changed outline of curve assumed by wires, due both to
their elasticity and to the elongation or contraction resulting from
changes in temperature. This is also advisable oftentimes in the case
of comparatively short spans, especially where a minimum clearance
is required under the lowest wires under the worst conditions of load-
ing. This matter has been the subject of quite extensive investiga-
tion on the part of several different men, and several different solutions
of the problem have been offered. No theoretically correct analytical
solution that is easy of application has yet been found, but every one is
based on assumptions which involve some approximations. The first
assumption generally made is that the parabola approximates near
enough to the true curve in which the wire hangs. For short spans
this is not much in error, but on the longer spans the difference in re-
sults obtained by figuring the curve first as a parabola and then as a
catenary, is very considerable, especially when changes due to the
elasticity of the wire and to differences in temperature are taken into
account.

Some mathematical expressions have been worked out for taking
care of these conditions approximately, but they all involve the first
and third powers of the unknown quantity. It is a tedious matter to
solve such equations, which is another reason why they are not satis-
factory, especially when it is known in advance that the result, when
obtained, will not be accurate.

Transmission Towers 33

THOMAS' SAG CALCULATIONS

After having studied several of these different solutions, the writer
is of the opinion that the best one is the semi-graphical method offered
by Percy H. Thomas in his article on "Sag Calculations for Suspended
Wires," which was presented at the 28th Annual Convention of the
American Institute of Electrical Engineers, and which was published
in their "Transactions for 1911." Thomas uses the true catenary, and,
at the same time, takes care of all changes in the loading conditions.
He attacks the given problem by first reducing the span in size, with-
out changing the shape of the curve, until the span is one foot. Having
determined all the conditions attending the problem for the similar
span of one foot length, it is then an easy matter to convert these into
corresponding quantities for the given span.

When the span is reduced in size without changing the shape of the
curve, the sag will be reduced in direct proportion to the reduction of
span; in other words, the percentage of sag will remain the same.
The stress in the wire and the length of wire, also, will be reduced in
the same ratio. Again, the stress in the wire for a given span for a
definite sag is directly proportional to the total load per foot on the
wire.

Taking advantage of this fact, two curves can be plotted which will
give the sag, stress and length relationships for a wire on which the
total load is one pound per foot when* strung over a one-foot span, as
shown in Fig. 10. These relationships will be directly proportional
to those obtaining for the longer spans and for the varied loadings
per foot of length on the wire used. Different sets of curves may be
plotted for different proportions of sags to spans. A careful study of
these curves will show how the stress changes with increase of sag. It
will be noted that after the sag has reached 15% to 20% there is little
reduction in stress by further increase of sag, and that an actual in-
crease of stress soon results. This is of extreme importance when
considering the use of very light wires on long spans.

The sine of the angle made by the wire with the horizontal, at the
point of support, is one-half the length of wire in the span divided by
the stress with one pound per foot weight of wire, and may be obtained
from the length curve.

Transmission Towers

THOMAS' CURVES FOR SAG AND

These curves are plotted from computations made on the assump-
tion that the sag curve is a catenary. They have the properties of a
catenary assumed by a wire weighing one pound per foot of length
when strung between supports at the same level and one foot apart.
The abscissas of these curves give the sag and length corresponding
to one ordinate which is the stress factor that is common to both
of them.

The following example will demonstrate the use of the curves.
Span = 500 feet; maximum stress allowed in the wire at the points
of support = 1800 pounds; weight per foot of wire, including any
ice or insulation coating combined with wind load =1.5 pounds
per foot. The stress factor for the curves is the equivalent stress
in a wire weighing one pound per foot and having a one-foot span .
This is obtained by dividing the allowable stress in the given wire by
the product of the span in feet and the weight per foot of the wire

In this case it is

1800

2.4. This is the

500 X 1.5

stress factor which is the ordinate to both of the curves.
The horizontal line through this value, 2.4, intersects the
sag curve at a point of which the abscissa is .0535, and the
length curve at a point of which the abscissa is 1.0076.
These values are for a one-foot span, and the required values
for the given span are obtained by multiplying these values
by the length of the given span. The required sag for the
500 foot span is therefore 500 x .0535 = 26.75 feet, and the
length is 500 x 1.0076 = 503.80 feet .In case the allowable
or length of wire had been given instead of the
stress, the operation would have been reversed.
The abscissa corresponding to the given value
would then locate a point on the curve, and a hori-
zontal line through this point would intersect the
stress factor line of ordinates at a point whose value
when multiplied by the span in feet and the weight
foot of the wire in pounds would equaj
the stress in the wire at each point of sup-
port. Had the sag of 26.75 feet been given,
the abscissa for the sag curve would be
obtained by dividing 26.75 by
500 giving a quotient of .0535.
A horizontal line through the sag
curve at a point
having abscissa
= .0535 would

WOO

1.00!

Transmission Towers

STRESS CALCULATIONS (FIG. 10)

intersect the stress factor line of ordinates at 2.4. This value when multiplied by the span in feet and
the weight of the wire per foot in pounds would give the desired stress, thus;

2.4 x 500 x 1.5 - 1800
Variations in Loading

Assume the load to decrease from 1.5 to 0.5 Ib. per ft., the temperature remaining the same. The
resulting conditions of Stress, Sag and Length are determined as follows. If all the load could be
removed from the wire it would contract to its "unstressed" Ungth, called Lo, from its full-load length,
L = 1.0076 ft. per foot of span. If the sectional area of the conductor = .03 sq. in.; the coefficient of
elasticity — E = 16,000,000. and the total stress = 1800 Ib.. then,

1800

L = 1.0076 = Lo + «* ,^/wwv X U,

= 1.003836 tt. for one-foot span.

1.0076
1.00375

T 16,000,000
from which,

T 1.0076

1800

1 . ~03"
T 16,000,000

Plot this on the zero Stress Factor line, at Lo. Then the line LoL is the "stress-stretch" line for this
particular span and loading. If the load of 1.5 Ib. per ft. stretches the wire for one-foot span from Lo to

L, a load of 0.5 Ib. per ft. will stretch it p| (L — Lo) = H (1.0076—1.003836) = .001255 ft., which,

added to Lo, gives the length of the wire for the lighter load, 1.005091 ft. for one-foot span. Plot this
value on the same Stress Factor line as for the preceding load, S. F. = 2.4, and through this point and
Lc draw a line to the Length curve. Its intersection, Li = 1.00533, is the length of the wire for one-
foot span for a load of 0.5 Ib. per ft. For this new condition the properties of the 500 ft. span will be:
Stress = 500 x 2.85 x 0.5 =713 Ib.; Length = 500 x 1.00533 = 502.665 ft.; and Sag = 500 x .0445 -
22.25 ft. This operation is reversed when working from a light to a heavier load, the principle being
the same in all cases.

Temperature Variation

The preceding methods assume a constant temperature, but every change of temperature causes a
readjustment of Stress, Sag and Length in any span. To determine the new conditions, first find Lo.
the "zero stress" length of wire for one-foot span, as above described. Then compute the change in this
length resulting from the change of temperature, and plot this variation from Lo along the zero
Stress Factor line. This gives the "unstressed" length for the new temperature, and through this point
draw a line parallel to the "stress-stretch" line for the load then existing. Its intersection with the
Length curve gives the new length of wire for one-foot span, and determines the other factors. For
example, if the above computation was for a temperature of 0° F., and the properties of the span are
required for 20° intervals to 100° F., and the coefficient of heat expanison is .0000096, the length at
20° F. will be.

LJOO = L» (1 + .0000096 t) - 1.003836 (1 + .0000096 X 20) = 1.004029 ft.

Through this new length draw a line parallel to line LoLi. It intersects the Length curve at 1.005505,
the coincident values being: Stress Factor = 2.8, Sag = .0453. Then for the 500 ft. span, Stress =
500 X 2.80 X .5 = 700 Ib., Length = 500 X 1.005505 = 502.7525 ft.. Sag = 500 X .0453 = 22.65 ft.
Similarly for successive 20° intervals, OF for any other temperature changes.

10."

Transmission Towers

VALUES USED FOR PLOTTING CURVES FOR WIRE WEIGHING ONE
POUND PER FOOT OF LENGTH WHEN SUSPENDED IN ONE-FOOT SPAN

Y =^iec

Stress

Sag = Y - C

Length = 2 X

Stress

Sag

Length

2c
c

Stress

Sag

Length

.0050
.0055
.0060
.0065

100.001 3
90.910 5
83.334 8
76.924 7

.001 250
.001 375
.001 500
.001 625

1.000 004 2
1.000 005 1
1.000 006 1
1.000 007 1

.080
.085
.090
.095

6.270 0
5.903 6
5.578 1
5.286 9

.020 01
.021 26
.022 52
.023 77

1.001 066
1.00 1 205
1.001 351
1. 001 503

.0070
.0075
.0080
.0085

71.430 3
66.668 5
62.502 0
58.825 7

.001 750
.001 875
.002 000
.002 125

1.000 008 2
1.000 009 4
1.000 010 7
1.000 012 0

.100
.105
.110
.115

5.025 0
4.788 2
4.573 0
4.376 6

.025 02
.026 27
.027 53
.028 78

1.001 668
1.001 839
1.002 017
1.002 205

.0090
.0095
.010
.011

55.557 8
52.633 9
50.002 5
45.457 3

.002 250
.002 375
.002 50
.002 75

1.000 013 5
1.000 015 0
1.000 017
l.COO 020

~120
.125
.130
.135

4.196 7
4.031 3
3.878 7
3.734 2

.030 04
.031 29
.032 55
.033 80

1.002 402
1.002 606
1.002 819
1.003 040

.012
.0125
.013
.014

41.669 7
40.003 1
38.464 8
35.717 8

.003 00
.003 13
.003 25
.003 50

1.000 025
1.000 026
1.000 028
1.000 033

.140
.145
.150
.170

3.606 5
3.484 6
3.370 9
2.983 8

.035 06
.036 31
.037 57
.042 60

1.003 270
1.003 508
1.003 754
1.004 825

.015
.016
.017
.0175

33.337 1
31.254 0
29.416 0
28.575 8

.003 75
.004 00
.004 25
.004 38

1.000 037
1.000 043
1.000 048
1.000 051

.200
.220
.25
.27

2.550 2
2.328 0
2.062 8
1.919 8

.050 17
.055 22
.062 83
.067 91

1.006 680
1.008 086
1.010 444
1.012 194

.018
.019
.020
.022

27.782 3
26.320 5
25.005 0
22.732 8

.004 50
.004 75
.005 00
.005 50

.000 054
1.000 060
1.000 067
1.000 081

.30
.32
.35
.37

1.742 2
1.643 2
1.517 0
1.444 9

.075 56
.080 68
.088 40
.093 56

1.015 068
1.017 154
1.020 542
1.022 973

.024
.025
.026
.028

20.839 3
20.006 3
19.237 3
17.864 1

.006 00
.006 25
.006 50
.007 00

1.000 096
.000 104
.000 113
.000 131

.40
.42
.45

.47

1.351 3
1.297 0
1.225 5
1.183 5

.101 34
.106 55
.114 41
.119 68

1.026 881
1.029 660
1.034 093
1.037 224

.030
.032
.034
.036

16.674 2
15.633 0
14.714 4
13.897 9

.007 50
.008 00
.008 50
.009 00

.000 150
.000 171
.000 193
1.000 216

.50
.55
.60
.70

1.127 6
1.050 1
.987 9
.896 5

.127 63
.141 00

.154 55
.182 26

1.042 19
1.051 19
1.061 09
1.083 69

.038
.040
.043
.047

13.167 4
12.510 0
11.638 7
10.650 1

.009 50
.010 00
.010 75
.011 75

1.000 241
1.000 267
1.000 308
1.000 368

.80
.90
1.00
1.10

.835 8
.796 2
.771 54
.758 42

.210 83
.240 61
.271 54
.303 87

.110 13
1.140 57
.175 20
.214 23

.050
.055
.060
.065

10.012 5
9.104 7
8.348 3
7.708 4

.012 50
.013 75
.015 00
.016 26

1.000 417
1.000 504
1.000 600
1.000 704

1.20

1.30
1.40
1.50

.754 44
.758 04
.768 18
.784 14

.337 77
.373 43
.411 04
.450 80

.257 88
.306 45
.360 21
.419 52

.070
.075

7.160 4
6.685 4

.017 51
.018 76

1.000 817
1.000 938

1.60

.805 46

.492 96

1.484 73

Transmission Towers 37

Considering the case of a s£>an having supports at unequal heights
above a given horizontal plane, if the horizontal distance from the
higher support to the lowest point of the wire is known, the stress and
sag in this part of the span can be determined by considering this part
as one-half of a span equal to twice this distance. The smaller stress
in the other part can be determined in the same manner.

The following formulas, based upon the catenary, give the horizontal
distance from the higher support to the lowest point of the wire,

. I , hT h_« (A)

1>U

-h + V~d (B)

where,

/ = the span in feet.
/u = the horizontal distance in feet from the higher

support to the lowest point of the wire,
h = the difference in height of the two supports in

feet.
T = the stress in pounds in the wire at the higher

support, with one pound per foot load on the

conductor,
d = the sag in feet measured from the higher point

of support.

Formula (A) is useful when the span and the stress to be allowed in
the wire are given, and formula (B) when the span and the sag are
given.

These formulas are approximate in that the horizontal projection of
the wire is substituted for the actual length of it. Formula (A) is
correct within from 2% to 4%, when neither sag nor difference in
heights of supports exceeds 15% of the span. Formula (B) has an
error of less than 1% under these conditions.

SPACING OF TOWERS

The problem of determining the type and the spacing of the towers
to be used, is one that requires considerable study of all the foregoing,
as the towers are only a part of the complete installation, and a saving
on one item may easily be more than offset by an increased cost of

38 Transmission Towers

some other items affected by the same conditions which made the
initial saving. In other words, it is a case of balancing one condition
against another, to determine what is the best possible combination.
The supporting structures must, of course, be placed as far apart as
possible; but an analysis of the various sag conditions for the wires
makes it evident that there are definite limits to be observed.

SPACING OF CONDUCTORS

After the spacing of towers has been determined, together with the
size of wires to be used and the voltage to be carried by them, the next
thing to consider is the spacing of the several wires and the minimum
clearance from the ground line to the lowest wire under the worst
loading condition. The maximum sag to be allowed must then be
determined, and this condition, along with the assumed loading across
the line, will determine the pull which may occur in the direction of the
line on the wire. The spacing of the wires in both horizontal and
vertical directions is dependent upon the voltage carried and upon the
length of spans. The minimum spacing, especially in the horizontal
direction, will obtain when the wires are supported on pin insulators,
or are attached to the cross arms by means of strain insulators. For
this condition, it is recommended that, for conductors carrying alter-
nating current, the minimum separation of these conductors, at the
points of support, shall be one inch for every twenty feet of span, and
one inch additional for each foot of normal sag, but in no case shall the
separation be less than :

Line Voltage Clearance

Not exceeding 6600 volts .. . 14J/2 inches

Exceeding 6600 volts but not exceeding 14000 volts, 24 inches
14000 " " " " 27000 " 30

27000 •" " " 35000 " 36

35000 " " " 47000 " 45

47000 " " " 70000 " 60

For voltages higher than 70000 the minimum separation should be
60 inches plus 0.6 inch for every additional 1000 volts.

Transmission Towers 39

When conductors are supported by suspension insulators, the separa-
tion of them horizontally must be made greater than when they are
supported on pin type insulators. The amount of this increase is
empirical, and is more or less a matter of judgment on the part of the
Engineer who designs the line. When the conductor wire is supported
from the cross arm by strain insulators, it is frequently assumed that
the jumper wire will swing to a position, making an angle of thirty
degrees with the vertical. It is usually assumed that the maximum
swing of a suspension insulator string will be to an angle of forty-five
degrees, but this depends upon the size and weight of the conductor,
and also upon the assumed maximum loading. It is possible that
under unusually severe conditions, two wires suspended from the same
cross arm may swing toward each other until each of them will make
an angle of about thirty degrees with the vertical. Or even though
they do not both swing the same amount, it is a safe assumption that
twice the horizontal projection of the length of one insulator string
when swung to thirty degrees from the vertical, will be equivalent to
the sum of the horizontal projections of the two wires when swinging
toward each other under the worst conditions of service. This means
that when wires are supported by suspension insulators instead of on
pin type insulators, the horizontal separation should be increased by
the length of one insulator string.

It is generally recommended that the minimum clearance in any
direction between the conductors and the tower, shall not be less than :

Line Voltage Clearance

Not exceeding 14000 volts 9 inches

Exceeding 14000 but not exceeding 27000 15 "

27000 " " 35000 18 "

35000 " " 47000 21 "

47000 " V 7000° ....24 "

Usually the suspension insulators are made sufficiently long so that
when swung out to the assumed position of maximum swing, the ver-
tical distance between the conductor and its supporting cross arm,
or any other part of the tower, will meet all the requirements for
clearance. The overhead ground wire, or wires, should be, in general,
not more than forty-five degrees from the vertical through the adjoin-
ing conductor.

40 Transmission Towers

The several wires must be spaced far enough apart vertically so that
under the worst conditions the wires will not come so close together as
to make trouble electrically. This must have careful consideration,
especially on the very long spans, because it is entirely possible during
storms for the lowest wires to be free from ice loading or to be suddenly
relieved of such loading, when they might swing up, close to the wires
directly above them, which might be heavily loaded with ice and
hence have considerable sag.

The arrangement and spacing of the wires almost always fixes, with-
in certain limits, the general type of the supporting structure to be
used. This is at least true of the upper part of it. The outline of the
structure below the lowest cross arm will be made that which is the
most economical, unless such an outline is prohibitive on account of
right-of-way or other limiting conditions.

Where the transmission line consists of three conductor wires, with
or without a ground wire, it very often works out to very good advan-
tage to put the three conductor wires in the same horizontal plane,
which means that the middle one will pass through the tower. When
suspension insulators are used with this arrangement of wires, the
tower must be made wide enough to allow ample clearance from the
conductor when swung to maximum position either way. But if
strain insulators are used, then a much narrower jtower may be used by
attaching the jumper wire to a pin on the center line of the tower.
The narrower tower makes a much more economical construction.
When six conductor wires, with or without ground wires, are used,
three of the conductors are placed on each side of the tower. These are
generally placed so that the three wires in each set are in the same
vertical plane, but sometimes the middle one will be put farther from
the center line of the tower than the other two wires.

The design of the supporting structures from this point on, consists in
determining just what loads are to be considered as coming on the
structures, what unit stresses shall be used throughout, and whether a
com para tirely large or small investment shall be put into them. In
other words, it is a matter of first importance whether these struc-
tures are to be regarded for a temporary proposition, and hence made
as cheaply as possible, or whether they are to be considered as part of
a permanent construction and therefore figured a little more liberally.

Transmission Towers 41

TEMPORARY STRUCTURES

For a temporary proposition the structures are, of course, made as
light as possible and are almost always painted. They are rarely gal-
vanized. In such cases the assumed loadings are kept very low, and
are intended to take care of only normal conditions, on the theory that
if some of the structures should be subjected to loadings of unusual
intensity resulting from specially severe storms, it will be more eco-
nomical to replace some of them that might be destroyed than to provide
additional strength in all the supports. For the same reason the unit
stresses are always run as high as possible.

PERMANENT STRUCTURES

On the other hand, where permanency of construction is wanted,
the design is made more liberal in every way. To start with, the
assumed loadings are such as will be expected to take care of more than
ordinary conditions of service. They will be made sufficiently high to
be in themselves an insurance against possible interruptions of service
due to breakdowns caused by storms. Also, the unit stresses will be
kept lower and heavier material will be specified. Generally, but not
always, such structures will be required to be galvanized instead of
painted, so that the structure will be in service for a longer time.

THICKNESS OF MATERIAL

When the material is required to be galvanized, many specifications
will allow web members to be made of material only y% "thick, but will
require a minimum thickness of •£$* or possibly %" for the main posts.
Almost all specifications require a minimum thickness of material of
TS* for all members when painted; but some specify that no material
less than % " thick shall be used when painted ; while others demand a
minimum thickness of y\r" for all material, regardless of whether it is
painted or galvanized.

GALVANIZE FOR PERMANENCY

The history of transmission line structures proves that where per-
manency of construction is desired, they should always be galvanized,
not painted. At least all parts of the structure in close proximity to
the conductor wires should be galvanized, irrespective of what kind of
a protective coating is given to the balance of the structure. This is
especially true in those cases where high voltages are used.

42 Transmission Towers

SPECIFICATIONS FOR DESIGNS

There is no such thing as a standard practice among Engineers
today regarding the method to be pursued in preparing specifications
for transmission line towers on which competitive bids are to be re-
ceived. Usually for a line requiring several towers, the Engineer in
charge of the installation will determine the arrangement of all the
wires, the limiting dimensions for the structures, and the loadings for
them; but the design of the structures will be left generally to the
Manufacturer, subject, however, to those provisions of the specifica-
tions which are intended to insure that the towers or poles will all be
designed to have the same strength. Different Engineers seek to
accomplish this result in as many different ways. Some will specify
the loadings under which they expect the towers to be used, and will
stipulate that the design shall provide sufficient strength to take care
of these loadings with a given factor of safety; others wi'.l state unit
stresses which shall be used in determining the sections in the design,
to take care of the stresses resulting from the above loadings. Other
Engineers will increase the desired working loads by some factor which
they will introduce as a margin of safety, and will then give these in-
creased loadings instead of the working loadings, and will require that
the structures be designed to withstand these loadings without failure.
Still other Engineers will specify that certain unit stresses shall be used
in determining a design, which shall support the specified loadings with
a given factor of safety; and, further, that the completed structure
must support loads that are twice as large as those specified, but with-
out any restriction regarding unit stresses to be employed.

FACTOR OF SAFETY

The term "Factor of Safety" is in reality a misnomer, and, because
of this, it is not always interpreted in the same way by different men.
Literally speaking, the structure which is properly designed with a
factor of safety of three, should sustain without failure loads three
times as great as those which are expected to be the working loads
under normal conditions. But the term "Factor of Safety," as it is
usually interpreted and applied, means that the unit stresses used
throughout shall be one-third the ultimate strength of the material
entering into the construction. In actual practice the results of such
an interpretation are very disappointing. In a composite structure

Transmission Towers 43

made up of a large number of different pieces, some of which are under-
going compression while others are in tension, the action of this body
as a whole against outside forces will differ radically from what would
be expected of any one of its component parts under a similar test.
This, of course, is accentuated in the case of transmission towers,
because they are always made as light as possible for the work required
of them, and, hence when under load, they deflect considerably from
their original outlines, and this in turn produces a rearrangement and
entirely different distribution of stresses. The net result of all this is,
that all such structures will fail when the loading on them reaches the
point where some, if not all, of the members making up the construc-
tion are stressed to the elastic limit for their material.

Since the elastic limit for steel in either tension or compression is
about one-half its ultimate strength, it follows that the structure whose
members are determined by using unit stresses equal to one-third of
the ultimate strength of the material, will have a total strength only
50% in excess of that required to take care of actual working condi-
tions; so that, instead of having the so-called "Factor of Safety of
Three" it has an actual factor of safety of one and one-half.

This fact is recognized by those who first multiply the required work-
ing loads by a factor which will provide a margin of safety, and then
specify that the towers shall support without failure these increased
stipulated loads. It is not often that, under these conditions, the
specifications will call for the employment of definite unit stresses in
determining the several sections of material to be used. But in all
such cases, when unit stresses are specified, it will almost always be
found that those recommended are close to the elastic limit for the
material.

UNIT STRESSES

The unit stress for a member in either tension or compression is the
quotient of the total load divided by the cross sectional area of the
member supporting the load. This is given in pounds per square inch.
The unit stress for a member in compression is less than that for a
member in tension by a quantity which is a function of the ratio
between the unsupported length of the member and its least radius
of gyration. Usually this unit stress is determined by a straight-line
formula, such as

44 Transmission Towers

Sc = S — C — in which

Sc = the desired unit stress in compression,

S = the unit stress allowed in tension,

L = the unsupported length of the member in inches,

R = the least radius of gyration for the member, in inches,

C = a constant determined by experimental investigation.

The elastic limit in tension is about 27000 pounds per square inch of

net section. The straight-line formula 27000 — 90 =r for unit stresses

in compression, gives values which have been proven by actual tests
to be approximately the elastic limit for the material.

Where the so-called ' 'Factor of Safety of Three" is wanted, the unit
stress generally specified for members in tension is 18000 pounds per
square inch of net section, while, for unit stress for members in com-
pression, the formula

18000 — 60 £ is specified.
K

It will be noted that these values are just two- thirds of those im-
mediately preceding, and, therefore, offer a margin of safety of 50%.
It is not often that unit stresses smaller than these are specified for
tower work, but occasionally we find specifications which are very
severe, considering the infrequency of maximum or even full loads on
this type of structure.

It is common practice among Engineers, when specifying that the
towers shall safely support certain loads, to refrain from putting any
limitations on the design, such as what relationship shall be allowed as
a maximum between the length of any compression member and its
least radius of gyration. On the other hand, when it is stipulated that
the structures shall be figured for carrying certain loads by using given
unit stresses, it is almost always also stipulated that the ratio of length
of compression members to their least radius of gyration shall be
limited to a certain maximum value.

BOLT VALUES

Bolts stressed to 24000 pounds per square inch in shear have value*
comparable with the strength of members which are figured on the
basis of 27000 pounds per square inch of net section in tension, or 27000

Transmission Towers 45

- 90 •=? pounds per square inch of gross section in compression. From

IN.

this it follows that bolts need not be stressed lower than 16000 pounds
per square inch in shear to get values corresponding to those resulting
from using 18000 pounds per square inch of net section in tension or

18000 — 60 TT pounds per square inch of gross section in compression,
K.

for members which are to be connected by means of these bolts. It is
evident that smaller values for bolts are unwarranted. Consistent
practice in designing requires that the values assumed for bolts shall
bear the same ratio to their elastic limit as the ratio obtaining between
the working value assumed and the elastic limit for the several mem-
bers which are connected by the bolts.

LOADINGS

In regard to the specific loadings for which the structures shall be
designed, considerable depends upon where they are to be used, as
there are several factors entering into this question.

The first thing that should be determined is the kind and maximum
value of the vertical load to be taken care of at the end of the cross arm.
If the line runs through a comparatively level country, there is no
reason why there should ever be any uplift at the end of the cross arm ;
but if the line runs along steep grades, then there may be times when
the vertical load will be upward rather than downward. This is of
considerable consequence in the designing of the tower. The vertical
load at the end of the cross arm is usually supported by members
which run from the end of the cross arm to the main post angles at
some point above the cross arm. If the vertical load is downward,
these supporting members will act in tension, but if the load can ever
be upward instead of downward, then, such members must be capable
of taking stress in compression. In cases where the cross arms are
long, which is almost always true when suspension insulators are used,
these members must be made much heavier to take the stress in com-
pression, rather than tension.

ANGLE TOWERS

The next thing to determine, if possible, is, how many towers will
have to take care of angles in the line, and what will be the maximum
angle encountered. If this angle should be very large, it will be neces-

46 Transmission Towers

sary to provide special structures for such points in the line; but if the
angle is very small, provision for it may be made by using one of the
straight line towers at this point and shortening the span on each side
of it. This shortening of the span reduces the wind load on the wires
transverse to the direction of the line, and at the same time reduces the
pull in the wires in the direction of the line, if the sag is made a greater
percentage of the shortened span than it is in the case of the adjoining
spans.

In Fig. 11 there is shown a graphical diagram of the components of
the tension in the wire, parallel to the faces of the tower, when its axis
parallel to the cross arm bisects the given angle in the line. It will be
seen that when the wires leading off in both directions from the end of
the cross arm have equal stresses, the component "Y" in one wire
balances the corresponding component from the other wire, but that
the component "X" is twice what it is when only one wire leads off
from the cross arm. This means that in the one case, marked condi-
tion "B," the load on the tower is twice the component "X" from one
wire, but that for condition "A," the load on the tower is the sum of the
components "Y" and "X" from one wire.

It will be noted that for condition "B" the total load on the tower
from the pull in the direction of the line will just equal this pull when
the tower bisects an angle of sixty degrees in the line, and that this load
increases to double the pull on one wire, as a maximum limit, when the
angle in the line reaches one hundred eighty degrees. For condition
"A" the total load will always be greater than the pull in the wire, no
matter how small the angle in the line, and the worst loading will occur
when the tower bisects an angle of ninety degrees in the line. When
the angle in the line is as large as ninety degrees, it will often be more
desirable to construct a special tower, and to set it normal to the direc-
tion of the line.

SPECIAL TOWERS

Having determined whether it will be necessary to provide special
towers to take care of angles in the line, the next step should be to
determine how many, if any, special towers should be provided to take
care of such special cases as railroad crossings, and what specifications
must govern in the design of these special structures. The Railroad
Companies have their own specifications for these structures, and they

Transmission Towers

Value of COMPONENT Y fir tension of 1000 Ib. in win
(tor Condition A. For Condition B the components balance and their sum 'a zero)

TRANSVEBX AHD Lff COMPONENTS

ANGLE TOWERS

B/sfcr THf AHGLC IH THE LINE

too too ioo4ooxo60oioo6oo9oo 1000 (Condition A]

tOO *W tOO 800 IOOO ItOO MOO I6OO 1800 tOOO /

Value of COMPONENT X for tension of IOOO Ik in wire.

Fi*. 11

48 Transmission Towers

insist that all wires carried over their crossings shall be supported by
structures complying with all their requirements as to loadings and unit
stresses to be employed. Their specifications are generally very severe
and, hence, special designs almost always are required for those
particular points in the line. Of course, one thing always to be kept
in mind, is to make as few different designs as circumstances will allow,
so that there will be as much duplication as possible in the structures.
This is an especial advantage for economical fabrication in the shop,
and is also a big advantage when it comes to erecting the towers in the
field.

Every line must be carefully studied and designed for its own
particular requirements. A line wrhich is taken through a city must be
built in a different way from one going through an open country. The
working loads might not need to be any heavier, but either the design
loads should be heavier or the unit stresses lower, and the towers
should be spaced closer together.

REGULAR LINE TOWERS

The average line of any length should have three different types of
towers. These may be designated as — Standard or Straight Line,
Anchor, and Dead End Towers.

All towers should be designed to take care of the dead weight of the
structures and also the vertical loads at the ends of all the cross arms,
in addition to and simultaneously with, the horizontal loadings speci-
fied below.

•

STANDARD TOWERS

The Standard, or Straight Line, Towers should predominate, and
should be designed to support without failure the required horizontal
loads transverse to the direction of the line, combined with a horizontal
pull in the direction of the line applied at any one insulator connec-
tion equivalent to the value of the wire when stressed to about one-half
its ultimate strength. These loads transverse to the line should be
large enough to include the wind load across the wires and that against
the tower itself, with a little margin of safety.

ANCHOR TOWERS

The Anchor Tower should be designed to support without failure
any one of the following horizontal loadings:

Transmission Towers 49

(1) The same horizontal loads as those specified for the Standard
Tower.

(2) An unbalanced horizontal pull in the direction of the line equiva-
lent to the working loads of all the conductor wires and the ground
wires, applied at the points of connection of the wires to the tower,
combined with the transverse horizontal loads on the wires and the
tower.

(3) An unbalanced horizontal pull parallel to the line equivalent to
the working loads of the wires, applied at one end of each cross arm,
all on the same side of the tower and all acting in the same direction,
combined with the horizontal transverse loads on the wires and the
tower.

DEAD END TOWERS

The Dead End Towers should be designed to support the same load-
ings as those specified for the Anchor Towers, but the sections should
be determined by using smaller unit stresses. Unit stresses of 18000

pounds per sq. in. in tension and 18000 — 60 -=r for compression, would

give these towers approximately 50% more strength than the anchor
towers would have when stressed just within the elastic limit.

It will be noted that under the above specification, the standard
tower will be required to take care of the torque resulting from an un-
balanced horizontal pull equivalent to the allowable tension (which
is one-half the ultimate strength) of one wire, applied at one end of any
cross arm and acting parallel to the. direction of the line; while, the
anchor and the dead end towers are both required to take care of either
the torque as given above for the standard tower or the torque result-
ing from an unbalanced horizontal pull equivalent to the working
loads (actual tension in the wire under the working conditions) of all
the wires on either side of the center line of the tower, applied at one
end of each of the cross arms, and all acting parallel to the line and in
the same direction. If one anchor tower is placed in the line for every
ten or twelve standard towers, all conditions resulting from broken
conductor wires should be localized to the territory between two an-
chor towers. The reason for using lower unit stresses in the dead end
towers than in the anchor towers for exactly the same loadings, is that
the dead end towers may have to support a large part of this total load-
ing at all times, and all of it very frequently, while the anchor tower

50 Transmission Towers

may have to support the same loading only once in a great while, and
then for only a very brief time.

One of the aims to be kept constantly in mind in designing a trans-
mission structure, is to get a finished tower in which all the stresses can
be determined definitely. We usually determine the stresses graphic-
ally. The stresses resulting from the horizontal loads applied as so
much shear must be determined separately from the stresses resulting
from torque. These stress diagrams cannot be combined except in
those cases where the slope of the post does not change between the
horizontal planes bounding that part of the tower for which the dia-
grams are wanted. This is true because the horizontal loads which act
as so much shear, may be assumed to be acting in a plane containing
both posts of the face of the tower, parallel to the direction of the load,
in which case the posts may or may not (depending upon the slope of
the posts) take up a part of this shear directly. On the other hand,
the torque is a moment acting in a horizontal plane and is constant
between any two parallel planes, unless it is either increased or de-
creased by an additional torque of the same or opposite kind.

ANCHORAGE DESIGNS

The members for anchoring the structure to the footings are gener-
ally the last part of the design to be considered.

The first question to be determined is whether concrete footings shall
be used. These are more simple, and involve much less steel work
than any other type of footing used for transmission line structures.
The weight of the concrete itself reacts against the tendency of the post
to pull away from the base because of the tension in the post on one
side of the tower. It also offers more bearing surface against the earth
around the footings and introduces the passive resistance of a larger
volume of earth against the uplifting tendency of the post on the ten-
sion side of the tower. Of course, the saving in the cost of steel in the
structure must be balanced against the expense involved in putting
the concrete in place, to determine whether or not it is advisable to
use concrete footings. This will depend upon many circumstances
which must be very carefully considered before reaching a conclusion.
It is impossible to overestimate the importance of good anchorages.
An otherwise excellent construction may be made inadequate by using
footings which are not substantial. If one of the footings should be

Transmission Towers

5plice ang/e

Concrete —-

Anchor bolts,
40to50diam.
in concrete

Concrete Anchors

Concrete pad-

Anchor

grouted in drilled holes.

Rock Anchor

Splice angk^& Ground line^

Earth Anchor

Fiji. 12

52 Transmission Towers

insufficient to take care of the loads for which the superstructure is
intended, it would be very apt to yield under full loading, and, in doing
so, would bring about a new distribution of stresses among the mem-
bers, and would put on some of the members stresses which were not in
keeping with those for which the members were designed. Such a re-
arrangement of stresses may very easily be so vital as to bring about
the failure of the superstructure. In view of this fact, it is recom-
mended that, where there is any doubt as to whether concrete footings
should be used, the benefit of any small doubt should always be given
in favor of such footings. But, it may be that the structures are to be
used where such footings would be practically impossible. Under such
circumstances, other provisions must, of course, be made.

In the case of poles, the regular outline is generally continued below
thie surface of the ground whether concrete footings are used or not;
but if concrete is not used, then additional steel must almost always
be used to get more bearing area against the earth.

In the case of towers there is provided a separate footing for each of
the posts. When concrete footings are used the posts are connected
to them in one of two ways: In the first method, extensions of the post
sections, which are called anchor stubs, may be built in these footings
with just sufficient length extending above the concrete so that the
lower post sections of the tower may be connected directly to them.
These anchor stubs may extend almost to the bottom of the footing, or
they may extend into the footings only far enough that the adhesion of
the concrete to them will develop their full strength, in which case it
will be necessary to add steel reinforcing bars from this point to the
bottom of the concrete. This is necessary because provision must be
made to bind the concrete together so that it will not break apart
under the uplifting force in the post, and thus defeat its purpose. The
other method used with the concrete footing is to have a base at the
lower end of the post section which will bear directly on the mass of
concrete in the footing, and which will at the same time be connected
directly to this concrete by means of long bolts or rods extending well
into the mass of concrete. These rods, in this case, would be brought
into action only when the post is under tension. If these rods are
straight for their full length, and fairly large, they should be imbedded
in the concrete for a length equal to fifty times their diameter, in order
to develop their full breaking strength. But if these rods are bent a
little near their lower ends, their breaking strength will be developed

Transmission Towers 53

by imbedding them in the concrete for a length equal to forty times
their diameter. Provision for binding together the concrete in the
footing must be made when anchor rods are employed, just the same
as when anchor stubs are used.

With any type of footing, there must be provided sufficient bearing
surface against the earth to resist the maximum compression in the
post, and also an arrangement to lift enough earth to resist the maxi-
mum uplifting tendency in the post under the worst condition of
loading.

The most positive and direct way to determine the size and outline
of a footing for any given loading, is to increase this loading by the
desired factor of safety, and then to determine a footing of which the
ultimate resisting value will be sufficient to meet the conditions to be
imposed. We recommend that the footing be so designed that its
ultimate resisting value will be not less than 25% in excess of what is
necessary to sustain the loading specified for the pole or tower.

For specially heavy towers which are required to dead-end heavy
wires on long spans, it sometimes becomes a troublesome matter to
provide adequate footings to take care of the uplift from the posts on
the tension side of the tower under the assumed condition of maximum
loading. This often happens in the case of ^River-Crossing Towers.
Footings for such cases, if built in the ordinary way, would have to be
made very deep and would require a large amount of concrete. It will
often be found to be economical to design these footings with special
outline and construction.

The following rather unique method has been successfully employed
for taking care of cases involving unusually large uplifts, when the
footings are built in clay or in mixed clay and sand that is compara-
tively free of gravel. A square pit is dug deep enough that its bottom
will be below the frost line and large enough to afford sufficient bearing
area against earth to sustain any possible downward pressure where
the tower post may be subjected to either tension or compression. In
the center of this pit a hole about twenty inches in diameter is bored
with an earth-auger to the depth desired (this depth has been made as
much as twenty feet below the bottom of the square pit). Dynamite
is then placed in the bottom of this hole and connected with a firing
magneto; then the hole is filled with concrete of 1 :2 :4 mixture, medium
wet, and the charge of dynamite is fired immediately. The charge of
dynamite that is generally used for this purpose consists of eight one-

54 Transmission Towers

half pound sticks of 60% dynamite. Reinforcing bars with their ends
bent are then pushed down through the concrete to the bottom of the
hole and then raised three inches and securely held in this position to
prevent them from sinking through the concrete and coming in con-
tact with the earth before the concrete has set. The hole is then re-
filled with concrete, and the footing in the square pit is also poured and
finished. From the moment the dynamite is placed and connected
with the firing magneto, it is essential that all the subsequent opera-
tions be conducted as rapidly as possible. Not more than five minutes
should be allowed between the time when the first pouring of concrete
is started and when the dynamite is fired.

The average displacement from such an explosion of dynamite is
about one and one-half cubic yards, this, of course, being dependent
upon the depth of the hole and the nature of the surrounding earth.
Experimental footings placed in this manner show that the enlarged
base takes an almost spherical form with its center above the bottom of
the excavated hole a distance equal to about one-fourth the horizontal
diameter of the enlarged base. This diameter is sometimes almost four
feet. It is evident that a footing of this kind can be made to resist a
very large uplifting pull.

In the case of light towers it is sometimes considered advisable to put
the tower in its erect position above the ground before the anchors are
set, and to then bolt these footing members to the lower end of the main
tower legs and put concrete or earth back fill around them while the
tower is being supported independent of them. But in the case of
heavy towers it is generally considered more economical to set the foot-
ing members exactly in their position first, and to then erect the towers
and connect them to their footings. This latter method of erection
requires that the anchor stubs be aligned as accurately as possible, as
any inaccuracy in the setting of these anchors will make the subsequent
assembling of the tower more difficult and less satisfactory. If the
anchor stubs are not set accurately to their true positions, there will be
introduced in the tower, additional stresses for which the tower mem-
bers were not designed. An accurate alignment of the anchors can be
accomplished only by using rigid templates that will hold the anchors
in their definite positions until they have been secured by either the
back fill or concrete.

Almost all towers are built smaller at the top than at the ground
line, and the tower leg inclines from the vertical as determined by this

Transmission Towers

outline of the structure. The anchor stub generally follows the. direc-
tion of the main tower leg, but when it is put in this position and sus-
pended from a template it has a
tendency to swing to the vertical
position. To obviate this condi-
tion the setting template should
be trussed as shown in Fig. 13.

TCMPLATT

fOK

ANCHORS

• aoft pests art ^•^•^L

Fig. 13

ERECTION

Transmission towers are
erected in one of two ways: they
may be erected by assembling
the members one at a time in
their proper positions in the com-
pleted structure, or by assemb-
ling the complete structure in a
prone position, and raising it to its vertical position by swinging it
about two hinge points on or near two anchor stubs.

If the first of these two methods is used, there will generally be re-
quired a crew of eight men, including one foreman. The following
equipment will generally suffice:

One light gin-pole, about 25 feet long.

One set of two-sheave and three-sheave blocks for % * diameter rope.
About 300 feet of %" diameter rope; four hand lines, each about
150 feet long; four small gate blocks for the hand lines.

The post members are erected with the gin-pole and tackle, but all
the other members are pulled up from the ground with the hand lines.
The time required will be about the same whether the tower is light or
heavy. The time required will, however, depend upon both the
accuracy of the fabrication of the material and the accuracy of the
alignment of the anchor stubs.

If the second method is used, the actual work erecting the tower does
not consume more than ten or fifteen minutes after all the prepara-
tions have been made. These preparations and the erection consist of
three distinct operations:

(1) Leveling the ground where required for the erection
equipment, and blocking up the tower on rough ground
and for side-hill extensions. A crew of seven or nine
men including a foreman is required.

56 Transmission Towers

(2) Rigging up erection equipment, and bolting erection

shoes and struts in place, etc. A crew of about twelve
men including a foreman is required.

(3) The actual raising of the tower. Sometimes horses are

used for this operation, but it is often found to be more
satisfactory to use a caterpillar tractor, especially for
raising the heavier towers. One team of horses will
generally suffice for this work, but it often requires four
and sometimes six horses especially in rough country
and for raising towers that are unusually heavy. The
Tractor gives a much steadier pull, and will permit of
holding the load at any desired point more satisfac-
torily than when horses are used. A substantial A--
frame usually built up of steel pipes is generally em-
ployed for raising the tower from the prone to the up-
right position. A steel cable should also be used in
preference to a manilla rope for this purpose in the case
of the heavier towers.

When concrete footings are used, and this method of erection is em-
ployed, there is an advantage in having the anchor stubs set and con-
creted in position in advance of the assembling of the tower. When
this is done, the tower can be assembled close to the anchor stub and
can be raised about hinges fastened to the tops of the anchor stubs;
but when the tower is assembled before the concrete is placed around
the anchor stubs, it is necessary to assemble the tower a few feet away
from the stubs, and then to skid the tower into the position from which
it is to be raised. This process of skidding the tower is costly, and is
also likely to injure the tower members.

SPACING OF TOWERS

The trend of American practice today in the designing of transmis-
sion line installations is to make the spans between supporting struc-
tures as great as possible. As the result of considerable study extend-
ing over several years of experience with lines having spans some of
which were very short while others were exceptionally long, it has been
determined that the best and most economical lines, all things con-
sidered, are those in which the supporting structures are spaced
far apart.

Transmission Towers 57

This is true even though the first investment for the original installa-
tion is somewhat larger in the case of long spans than where short
spans are used. It has been determined from comparative records
that the maintenance of lines having the long spans is much less than
was the maintenance of the same lines during previous periods when
shorter spans were used. This decreased cost of maintenance has
been proved to be sufficiently important to warrant making larger
initial investments on original projects. The maintenance is not only
less expensive with the long spans but it is also less troublesome,
because there is less interference with continuous service along the line.
This is a matter worthy of careful consideration, as the value of elec-
trical service in almost every case is dependent upon the assurance of
its continuity.

By using long spans the number of insulators required is reduced;
and, as there is always a chance that a flash-over will occur at the
insulator, it is obviously advisable to reduce the number of insulators
to a minimum in order to eliminate, as far as possible, this source of
trouble for the service.

Another advantage derived from the use of long spans is that the
variations of stress in the wires resulting from large changes in tem-
perature will be much less than under similar conditions of loading on
short spans. The constant changing of stress in the wires is produc-
tive of more trouble than higher stresses which are more uniformly
applied.

The long span is especially advantageous for a line carried along a
hillside, because it will generally permit of such an arrangement of
towers that there will not be any upward pull on any of them. The
upward pulls are always a source of trouble, and they should be elim-
inated wherever conditions will permit an alternative construction.
The upward pull causes not only mechanical but also electrical troubles,
because, during a rain storm, water will run down along the wire into
the insulator, which, of course, immediately produces electrical trouble.

The voltages used on present day high tension lines are such that
the suspension-type and strain-type insulators are rapidly displacing
the pin-type insulators. This, of course, means longer and heavier
cross arms and higher supporting structures. It is also true that the
cost of wood is steadily increasing, and will continually increase as the
wood becomes less plentiful. These conditions when combined with
the tendency for long span construction as described above, mean that

58 Transmission Towers

the wood pole construction is being rapidly superseded by the better
and more permanent steel tower construction.

When the Manufacturer is expected to design the structures for a
line of any considerable length, he is generally furnished very definite
and complete specifications regarding loadings and unit stresses; but
when he is asked for quotations on only a few structures, it is not often
that full and complete information regarding working conditions are
furnished him. Nor will this information always be forthcoming,
even when the customer is requested to give more definite data. As a
rule, a part of the necessary information will be furnished by the cus-
tomer, and it becomes the task of the Manufacturer to complete the
design by making his own assumptions regarding the missing data.

The customer will very often profit financially by making as com-
plete as possible the information he gives to the Manufacturer, and it is
always much more satisfactory to the designer to know positively what
working conditions are to determine the design.

Transmission Towers

« |

Ultimate
Strength Lbs. pe
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Transmission Towers

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Transmission Towers

ed— B

TABLE 3
re — Stran

MAX. LOAD PER
LIN. FOOT PLANE
OF RESULTANT

LIN. F
ONTAL

D PE
HOR

R LIN.
RTICAL

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OO^Ovoo r- t>. O 10 u?Tt'*f*? t*?c1r1c1

— odd dddd dddd dddd

O\l~— i/}1^ OJOOO'O ^f*5CN«^ *-^OO^O^

<N fN fN f>3 'S fS — — ! — r-.' —' — — — OO

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fOfjfotN (N — — — — qqq qqqq

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dddd o'ddd do" do" dddd

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Transmission Towers

•** oo io io ro CM" i~»"

O\ 00 CM ON ON «-< •*

ON l^ NO Tj« CO fO CM

8 8

222"33

°s°;?0*S

£ 2 S

ON 2- ?0 ^f

ri« io io >o

MAX. LOAD PER
LIN. FOOT PLANE
OF RESULTANT

CM 1O 1^- tf)

fN O f«: ro
10 fO CN —

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— ' ^ -,4 d do

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6600

CM CM t-^ IO 1-^ CO

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d d d d do

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ro" CM" ^<" -*" *+

sss ss

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do'dd dodo do

popo — M »o -*• »o\o

Transmission Towers

iO so •* PN •* 10 00 iO O'*
CN IO Os 00 lOCsOOfN

SO O O — Os Os
PN O r~~ -f 10 O

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Transmission Towers

•oc

i
£

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&

2
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LIN. FOOT PLA
OF RESU

D PER LIN.
HORIZONTA

IN.

CAL

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Transmission Towers

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66 Transmission Towers

SAGS

In the following tables are given sags at which conductors shall be
strung in order that, when loaded with the specified requirement of
one-half inch of ice and a wind load of 8.0 pounds per square foot of
projected area at 0 degrees Fahrenheit, the tension in the conductor
will not exceed the allowable value of one-half the ultimate strength of
the conductor as given in preceding tables. The sags given in the
tables for 120 degrees Fahrenheit are greater in every case than the
vertical component of the sags at 0 degrees Fahrenheit under the
maximum wind and ice load.

Transmission Towers

Minimum Sags for Stranded Hard -Drawn Bare Copper Wires

No. 4/0 B. & S.

SPAN IN FEET

100

or Less

125

150

200

250

300

400

500

600

Inches

Feet

—20

3.5

6.5

10.5

11.5

4.5

8.5

5.5

13.5

100

14.5

120

10.5

No. 3/0 B. & S.

SPAN IN FEET

100

or Less

125

150

200

250

300

400

500

600

Inches

Feet

—20

7.5

12.5

4.5

8.5

13.5

9.5

6.5

10.5

15.5

100

120

7.5

No. 2/0 B. & S.

SPAN IN FEET

100

or Less

125

150

200

250

300

400

500

600

Inches

Feet

—20

4.5

9.5

15.5

5.5

6.5

11.5

17.5

100

12.5

18.5

120

8.5

13.5

19.5

No. 0 B. & S.

SPAN IN FEET

100

or Less

125

150

200

250

300

400

500

600

Inches

Feet

Feet %

Feet

—20

2.5

5.5

11.5

18.5

2.5

6.5

12.5

19.5

3.5

7.5

20.5

4.5

8.5

14.5

21.5

100

' 37

120

9.5

15.5

22.5

Transmission Towers

Minimum Sags for Solid Hard -Drawn Bare Copper Wire

No. 1 B. & S.

SPAN IN FEET

Temp.

100

or Less

125

150

200

250

300

400

500

600

Inches

Feet

—20

14.5

3.5

8.5

23.5

4.5

9.5

24.5

5.5

10.5

25.5

100

120

11.5

26.5

No. 2 B. & S.

SPAN IN FEET

100

or Less

125

150

200

250

300

400

500

600

Inches

Feet

—20

10.5

18.5

4.5

29.5

11.5

19.5

5.5

30.5

12.5

20.5

6.5

100

21.5

31.5

120

7.5

13.5

No. 3 B. & S.

SPAN IN FEET

Temp.

100

or Less

125

150

200

250

300

400

500

600

Inches

Feet

—20

37.5

3.5

6.5

14.5

24.5

37.5

4.5

7.5

15.5

25.5

38.5

5.5

8.5

100

5.5

8.5

16.5

120

16.5

26.5

39.5

No. 4 B. & S.

SPAN IN FEET

Temp.

100

or Less

125

150

200

250

300

400

500

600

Inches

Feet

—20

5.5

18.5

31.5

9.5

31.5

46.5

6.5

46.5

6.5

19.5

32.5

10.5

19.5

32.5

47.5

100

• 7.5

32.5

47.5

120

7.5

20.5

Transmission Towers

Minimum Sags for Stranded Bare Aluminum Wires

No. 4/0 B. & S.

SPAN IN FEET

Temp.

or Less

100

125

150

200

250

300

400

500

600

Inches

Feet

—20

2.5

5.5

19.5

29.5

3.5

12.5

20.5

4.5

7.5

13.5

21.5

31.5

5.5

100

8.5

14.5

22.5

120

6.5

33.5

No. 3/0 B. & S.

SPAN IN FEET

Temp.

or Less

100

125

150

200

250

300

400

500

600

Inches

Feet

—20

5.5

33.5

3.5

6.5

13.5

22.5

4.5

34.5

7.5

14.5

23.5

5.5

35.5

8.5

15.5

24.5

100

6.5

36.5

120

9.5

16.5

25.5

No. 2/0 B. & S.

SPAN IN FEET

Temp.

or Less

100

125

150

200

250

300

400

500

600

Inches

Feet

—20

8.5

16.5

2.5

5.5

28.5

42.5

17.5

3.5

6.5

9.5

29.5

18.5

29.5

43.5

4.5

10.5

100

7.5

19.5

30.5

44.5

120

5.5

11.5

44.5

No. 0 B. & S.

SPAN IN FEET

Temp.

F.P

or Less

100

125

150

200

250

300

400

500

Inches

Feet

Feel

Feet

—20

3.5

10.5

36.5

21.5

36.5

4.5

7.5

11.5

8.5

22.5

37.5

5.5

8.5

12.5

100

120

9.5

13.5

23.5

-38.5

70 Transmission Towers

GALVANIZING IRON AND STEEL

We recommend the specifications adopted by the National Electric
Light Association, which are as follows:

These specifications give in detail the test to be applied to galva-
nized material. All specimens shall be capable of withstanding these
tests.
a — Coating

The galvanizing shall consist of a continuous coating of pure zinc
of uniform thickness, and so applied that it adheres firmly to the sur-
face of the iron or steel. The finished product shall be smooth.
b — Cleaning

The samples shall be cleaned before testing, first with carbona,
benzine or turpentine, and cotton waste (not with a brush), and then
thoroughly rinsed in clean water and wiped dry with clean cotton
waste.

The samples shall be clean and dry before each immersion in the
solution.
c — Solution

The standard solution of copper sulphate shall consist of commercial
copper sulphate crystals dissolved in cold water, about in the propor-
tion of 36 parts, by weight, of crystals to 100 parts, by weight, of water.
The solution shall be neutralized by the addition of an excess of
chemically pure cupric oxide (Cu O). The presence of an excess of
cupric oxide will be shown by the sediment of this reagent at the bot-
tom of the containing vessel.

The neutralized solution shall be filtered before using by passing
through filter paper. The filtered solution shall have a specific
gravity of 1.186 at 65 degrees Fahrenheit (reading the scale at the level
of the solution) at the beginning of each test. In case the filtered
solution is high in specific gravity, clean water shall be added to re-
duce the specific gravity to 1.186 at 65 degrees F. In case the filtered
solution is low in specific gravity, filtered solution of a higher specific
gravity shall be added to make the specific gravity 1.186 at 65 degrees
Fahrenheit.

As soon as the stronger solution is taken from the vessel containing
the unfiltered neutralized stock solution, additional crystals and
water must be added to the stock solution. An excess of cupric oxide
shall always be kept in the unfiltered stock solution.

Transmission Towers 71

d — Quantity of Solution

Wire samples shall be tested in a glass jar of at least two (2) inches
inside diameter. The jar without the wire samples shall be filled with
standard solution to a depth of at least four (4) inches. Hardware
samples shall be tested in a glass or earthenware jar containing at
least one-half (J/£) pint of standard solution for each hardware sample.

Solution shall not be used for more than one series of four immer-
sions.

e — Samples

Not more than seven wires shall be simultaneously immersed, and
not more than one sample of galvanized material, other than wire, shall
be immersed in the specified quantity of solution.

The samples shall not be grouped or twisted together, but shall be
well separated so as to permit the action of the solution to be uniform
upon all immersed portions of the samples.

f— Test

Clean and dry samples shall be immersed in the required quantity of
standard solution in accordance with the following cycle of immersions.

The temperature of the solution shall be maintained between 62
and 68 degrees Fahrenheit at all times during the following test.

First — Immerse for one minute, wash and wipe dry.

Second — Immerse for one minute, wash and wipe dry.

Third — Immerse for one minute, wash and wipe dry.

Fourth — Immerse for one minute, wash and wipe dry.

After each immersion the samples shall be immediately washed in
clean water having a temperature between 62 and 68 degrees Fahren-
heit, and wiped dry with cotton waste.

In the case of No. 14 galvanized iron or steel wire, the time of the
fourth immersion shall be reduced to one-half minute.

g— Rejection

If after the test described in Section "f" there should be a bright
metallic copper deposit upon the samples, the lot represented by the
samples shall be rejected.

Copper deposits on zinc or within one inch of the cut end shall not
be considered causes for rejection.

In the case of a failure of only one wire in a group of seven wires
immersed together, or if there is a reasonable doubt as to the copper
deposit, two check tests shall be made on these seven wires, and the
lot reported in accordance with the majority of the set of tests.

Transmission Towers

USEFUL DATA

Given, ax2 + bx -f c = 0; X

-b =fe V b2 — 4 ac
2a

e = Base of Napierian Logarithms = 2.7182818285

Log]0 C = 0.4342944819

(ev

One inch = 2.540005 centimeters
One centimeter = 0.3937 inches
One foot = 0.3048006 meter
One meter = 3.2808333 feet
One pound (avoirdupois) = 0.45359 kilograms
One pound per foot = 1.488161 kilograms per meter.
One pound per square inch = 0.0703067 kilograms per square centi-
meter

One inch-pound = 1.152127 kilogram-centimeters
One kilogram per meter = 0.67197 pounds per foot
One kilogram per square centimeter = 14.2234 pounds per square inch
One kilogram-centimeter = 0.86796 inch-pounds

Trigonometrical Formulae

Radius, 1 = sin2 A -f cos2 A

= sin A cosec A = cos A sec A = tan A cot A

cos A 1 . , * ^ /

-- cotan A— — *j sine

rt T

cot A

A sin A

cosec A

.-^^

ver.
•in A

/__ Cosine

Tangent
I

5 Cotangent
-l_.i Secant
Cnsprant

r cin A rot A \/ 1 cin2 A

tan A
. sin A

sec A

= =sin A sec A
cot A

f AqkleA j~~*

A -
cos A

» cos A

sin A
A_ tanA

~ tan A
1

* radius -i — J

sin A
A cotA

cos A

Transmission Towers
NATURAL TRIGONOMETRIC FUNCTIONS

Degrees

Sines

Cosines

Tangents

Cotangents

Secants

Cosecants

Degrees

0
1
2
3

0.00000
0.01745
0.03490
0.05234

1.00000
0.99985
0.99939
0.99863

0.00000
0.01746
0.03492
0.05241

57.28996
28.63625
19.08114

1.00000
1.00015
1.00061
1.00137

57.29869
28.65371
19.10732

90
89
88
87

4
5
6
7

0.06976
0.08716
0.10453
0.12187

0.99756
0.99619
0.99452
0.99255

0.06993
0.08749
0.10510
0.12278

14.30067
11.43005
9.51436
8.14435

1.00244
1.00382
1.00551
1.00751

14.33559
11.47371
9.56677
8.20551

86
85
84
83

8
9
10
11

0.13917
0.15643
0.17365
0.19081

0.99027
0.98769
0.98481
0.98163

0.14054
0.15838
0.17633
0.19438

7.11537
6.31375
5.67128
5.14455

1.00983
.01247
.01543
.01872

7.18530
6.39245
5.75877
5.24084

82
81
80
79

12
13
14
15

0.20791
0.22495
0.24192
0.25882

0.97815
0.97437
0.97030
0.96593

0.21256
0.23087
0.24933
0.26795

4.70463
4.33148
4.01078
3.73205

.02234
.02630
.03061
1.03528

4.80973
4.44541
4.13357
3.86370

78
77
76

16
17
18
19

0.27564
0.29237
0.30902
0.32557

0.96126
0.95630
0.95106
0.94552

0.28675
0.30573
0.32492
0.34433

3.48741
3.27085
3.07768
2.90421

1.04030
1.04569
1.05146
1.05762

3.62796
3.42030
3.23607
3.07155

74
73
72
71

20
21
22
23

0.34202
0.35837
0.37461
0.39073

0.93969
0.93358
0.92718
0.92050

0.36397
0.38386
0.40403
0.42447

2.74748
2.60509
2.47509
2.35585

1.06418
1.07115
1.07853
1.08636

2.92380
2.79043
2.66947
2.55930

70
69
68
67

24
25
26

0.40674
0.42262
0.43837
0.45399

0.91355
0.90631
0.89879
0.89101

0.44523
0.46631
0.48773
0.50953

2.24604
2.14451
2.05030
1.96261

1.09464
1.10338
1.11260
1.12233

2.45859
2.36620
2.28117
2.20269

66
65
64
63

28
29
30
31

0.46947
0.48481
0.50000
0.51504

0.88295
0.87462
0.86603
0.85717

0.53171
0.55431
0.57735
0.60086

1.88073
1.80405
1.73205
1.66428

1.13257
1.14335
1.15470
1.16663

2.13005
2.06267
2.00000
1.94160

62
61
60
59

32
33
34
35

0.52992
0.54464
0.55919
0.57358

0.84805
0.83867
0.82904
0.81915

0.62487
0.64941
0.67451
0.70021

1.60033
1.53987
1.48256
1.42815

1.17918
1.19236
1.20622
1.22077

1.88708
1.83608
1.78829
1.74345

58
57
56
55

36
37
38
39

0.58779
0.60182
0.61566
0.62932

0.80902
0.79864
0.78801
0.77715

0.72654
0.75355
0.78129
0.80978

1.37638
1.32704
1.27994
1.23490

1.23607
1.25214
1.26902
1.28676

.70130
.66164
1.62427
.58902

54
53
52
51

40
41
42
43

0.64279
0.65606
0.66913
0.68200

0.76604
0.75471
0.74314
0.73135

0.83910
0.86929
0.90040
0.93252*

1.19175
1.15037
1.11061
1.07237

1.30541
1.32501
1.34563
1.36733

.55572
.52425
.49448
.46628

50
49
48
47

0.69466
0.70711

0.71934
0.70711

0.96569
1.00000

1.03553
1.00000

1.39016
1.41421

1.43956
1.41421

Degrees

Cosines

Sines

Cotangents

Tangents

Cosecants

Secants

Degrees

74 Transmission Towers

Properties of the Circle

Circumference of Circle of Diameter 1 = - = 3.14159265
Circumference of Circle = 2 * r
Diameter of Circle = Circumference X 0.31831
Diameter of Circle of equal periphery as

square = side X 1.27324

Side of Square of equal periphery as circle = diameter X 0.78540
Diameter of Circle circumscribed about square = side X 1.41421

Side of Square inscribed in circle = diameter X 0.70711

Arc» a = TSTT = 0.017453 r A

loU

4b2 + c2 4 02

Radius r = — T-T — Diameter, d

,
80 40

Chord, c = 2V 2br — b2 = 2 r sin y

Rise, b = r—Y2 V 4 r2 — c2 = £ tan ~- = 2 r sin2 4

24 4

Rise, 6 = r '+ y — V r2 — ^c2. ^ = 0 — r + V^2 — ^2

^ = V r2 — (r + 3; — 0)2

TT = 3. 14159265, log = 0.4971499

^ = 0.3183099, log =7.5028501
7T2 = 9.8696044, log = 0.9942997
^ = 0.1013212, log =7.0057003
= 1.7724539, log = 0.2485749
= 0.5641896, log =7.7514251

— = 0.0174533, log = 2.2418774

loU

1 80

— = 57.2957795, log = 1.7581226

Transmission Towers 75

Pyramid and Cone

Volume of any Pyramid or Cone whether regular or irregular equals
product of area of base by one-third perpendicular height, or

V = iBh
in which

V = Volume

B = Area of Base

h = Perpendicular height

Volume of Frustrum of any Pyramid or Cone with parallel ends
equals sum of areas of base and top plus square root of their products,
all multiplied by one-third the perpendicular height or distance
between the two parallel ends, or

V = i h (B + \/Bb + b)
in which

V = volume

h = perpendicular distance between parallel ends

B = area of base

b == area of top

Transmission Towers

Ellipse

Area = * ab

Center of Gravity of part mnc is at point G

cGl = ! a • 3 = 0.4244 • a = abt. H a
cGn = G'G = ! • b • - = 0.4244 • b = abt.

Parabola

4 Area = | sh

\jh Center of Gravity at point G

Hft|bd

L-W—i— tt_.

Semi-Parabola — abd or cbd

Center of Gravity at Point G1

dG =|h

GG1 =|-W

For the area included between the semi-para-
bola abd and its enclosing rectangle aebd, or
between the semi-parabola cbd and its en-
closing rectangle cfbd, the center of gravity
is at the point m.

km= — w
4

Circular Quadrant

Center of Gravity at point G

V~2

CG = iRad. X

Rad. 0.6002

CX = XG = J Rad. X £ = Rad X 0.4244 or abt.
Rad. X

Transmission Towers

Fig. B— Towers for Double Circuit 130,000 Volt Line

Transmission Towers

Fig. C — Method of Erecting Towers from Prone Position

Transmission Towers

Fig. D — Method of Erecting Flexible A Frames from Prone Position

Transmission Towers

I A

Fig. E— Method of Erecting Towers in Position

Transmission Towers

Fiji. F— Double Circuit Towers, for 66,000 Volt Line

Transmission Towers

Fig. G— Special Strain Tower, for Double Circuit 110,000 Volt Line

Transmission Towers

Fig. H— Transposition Tower, for Double Circuit 130,000 Volt Line

Transmission Towers

Fig. I— Railroad Crossing Poles, for 6,600 Volt Line

Transmission Towers

Fig. J— Flexible A Frame, for Double Circuit 66,000 Volt Line

Transmission Towers

Fig. K— Flexible A Frame, for Single Circuit 66,000 Volt Line

Transmission Towers

L — Poles, for Double Circuit 6,600 Volt Line

INDEX

Anchor Towers

Anchorage Designs 50, 51

Angle Towers 45

Bolt Values.

. 44

Catenary 16

Comparison of Parabola and . .

Diagram 15

Elastic 19

Circle, Properties of

Conductors, Spacing of

Cone, Volume of 75

Dead End Towers.

. 49

Ellipse ' 76

Erection 55,78,79,80

Factor of Safety 42

Flexible A Frames, Illustration . 85, 86
Use of.. . 4

Galvanizing 41, 70-71

Ice and Wind Loads, Standard
Practice for 14

Loads, Kinds of 5-6

Specific 45

Standard Practice for Wind and
Ice 14

Parabola 76

Comparison of Catenary and .... 25

Diagram 20,22,23,27,30

Parabolic Arc 23

Semi 76

Poles, Illustration 84, 87

Railroad Crossing, Illustration ... 84
Use of. 4

Pressure and Wind Velocity, Rela-
tion between 12, 13

Pyramid, Volume of 75

Quadrant, Circular 76

Sag Calculations, Thomas' 33

Curves for 34-35

Relation Between Stress, Tem-
perature and 31

Tables.. . 66-69

Spacing 56

Spans, Reactions for, on Inclines. . . 27
Stringing Wires in, on Steep

Grades 29

Specifications for Designs 42

Stress Calculations, Thomas'

Curves for 34-35

Relation Between Temperature,

Sag and 31

Unit.. . 43

Temperature, Relation Between
Stress, Sag and 31

Towers, Anchor 48

Anchorage Designs 50

Angle 45

Dead End 49

Erection 55, 78, 79, 80

Factor of Safety 42

Installations 2, 77-87

Permanent 41

Regular Line 48

Rigid, Use of

Spacing of 37, 56

Special 46

Specifications for Designs 42

Standard 48

Strain, Illustration of 82

Temporary 41

Thickness of Materials for 41

Transposition, Illustration of. ... 83

Trigonometrical Formulae 72

Functions 73

Useful Data.

72-76

Wind and Ice Loads, Standard
Practice for 14

Pressure on Plane Surfaces 7

On Wires . 8

Velocities, Comparison of Indi-
cated and Actual 11

Velocity and Pressure, Relation

Between 12, 13

Wires, Curves Assumed by 15

Loadings Recommended for 59

Materials, Properties of 59-65

Stringing, in Spans on Steep
Grades 29

Tension in, Diagram of Com-
ponents of 47

Values Used for Plotting Curves
for 36

Wind Pressures on 8

Memoranda

PRODUCTS OF
THE BLAW-KNOX COMPANY

FABRICATED STEEL

Fabricated steel, one of the principal products of Blaw-Knox Company, includes
mill buildings, manufacturing plants, bridges, crane runways, trusses and other con-
struction of a highly fabricated nature.

A corps of highly trained engineers is maintained for consulting and designing
services.

TRANSMISSION TOWERS

Four legged straight line or suspension towers, anchor and dead end towers, latticed
and channel A-frames, river crossing towers, outdoor sub-stations, switching stations,
signal towers, steel poles, derrick towers.

We specialize in the design and fabrication of high tension transmission lines.

PLATE WORK

Riveted, pressed and welded steel plate products of every description, including:
accumulators, agitators, water boshes, annealing boxes, containers, digesters, filters,
flumes, gear guards, kettles, ladles, pans, penstocks, air receivers, stacks, standpipes,
miscellaneous tanks, miscellaneous blast furnace work, etc.

BLAW BUCKETS

Clamshell buckets and automatic cableway plants for digging and rehandling
earth, sand, gravel, coal, ore, limestone, tin scrap, slag, cinders, fertilizers, rock
products, etc.

For installation on derricks, overhead and locomotive cranes, monorails, dredges,
steam shovels, ditchers, cableways, ships for handling cargo and coal, etc.

BLAWFORMS

Steel forms for every type of concrete construction: aqueducts, bridges, cisterns,
columns, culverts, curbs and gutters, dams, factories, floors, foundations, houses,
locks, manholes, piers, pipe, reservoirs, roads, sewers, shafts, sidewalks, subways,
tanks, tunnels, viaducts, retaining walls, warehouses, etc.

FURNACE APPLIANCES

Knox patented water cooled doors, door frames, front and back wall coolers, ports,
bulkheads, reversing valves, etc., for Open Hearth. Glass and Copper Regenerative
Furnaces; water-cooled standings, boshes and shields for Sheet and Tin Mills.

THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW

AN INITIAL FINE OF 25 CENTS

WILL BE ASSESSED FOR FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO SO CENTS ON THE FOURTH
DAY AND TO $1.OO ON THE SEVENTH DAY
OVERDUE.

MAR 2 1946

LD 21-100/n-7,'40(6936s)

UNIVERSITY OF CALIFORNIA LIBRARY

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