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ment will soon find his eye measurements closely checking
his table measurements. When a quantity of rails are to be
curved for curves of different degrees, it is a good plan to

mark the degree of the curve of each rail in white paint on
the web of the rail on the concave side. There should be
ample force to handle the rails with dispatch, else much time
will be wasted. The use of sledges in curving rails should
under no circumstances be allowed. There is great danger
of fracture, and often a flaw is caused which at the time is
not perceptible, but which may, under the stresses caused
by frost and heavy trains at high speed, result in a broken
rail, with serious consequences.

In track work it is often necessary to ascertain the degree
of a curve, though no transit is available for measuring it.
The following table contains the middle ordinates of a one
degree curve for chords of various lengths:


Length of Chord
in Feet.

Middle Ordinate
of a 1 Curve.

20 ft.

4 in.




i in.



62 ft.

1 in.

100 ft.

2f in.

120 ft.

3f in.


The lengths of the chords are varied so that a longer or
shorter chord may be used, according as the curve is regular
or not.

The table is applied as follows: Suppose the middle ordi-
nate of a 44-foot chord is 3 inches. We find in the table
that the middle ordinate of a 44-foot chord of a one-degree
curve is $ inch. Hence, the degree of the given curve is
equal to the quotient of 3 -h ^ = 6 curve.

Additional examples are given as follows:

1. The middle ordinate of a 100-foot chord is 14f inches;
what is the degree of the curve ? Ans. 5.6, nearly.

The degree of the curve is probably 5 30'.

2. The middle ordinate of a 50-foot chord is 5 inches;
what is the degree of the curve ? Ans. 8.4.

The degree of the curve is probably 8 30'.

3. Calculate by rule 1 the difference in lengths between
the inner and the outer rails of a 7 curve 475 feet in
length. Ans. 34.29 in. = 2.857 ft.

4. Solve Example 3 by rule 2. Ans. 2.827ft.

1666. Springing Rails into Curve. Rails should
never be sprung and spiked to a curve; the elastic force of
the steel is constantly acting, and is sure to force the track
out of line. Each passing train, through its centrifugal
force, aids the rails to regain their original form. The re-
sult is that in a short time the curve, especially if a sharp
one, will show an angle at each joint. The effect at these
angles is to cause a sudden lurch of the car at each joint,
causing not only discomfort to passengers, but serious and
constant wear and strain upon the roiling stock.

1667. Widening Gauge of Curves. In passing
over curved track, the car wheels bind hard against the out-
side rail at the curve. The reason for this is that the differ-
ence between the gauge of the track and the gauge of the
wheels is taken up by the wheel base, which forms a chord
to the curve of the track, instead of being parallel to the
rails, as is the case on a straight line. To lessen this friction,


the gauge is usually widened on curves to the amount
of ^ inch per degree, but never to exceed 1 inch on
any curve. The increase in gauge is usually made in
quarter-inches, that being the amount allowed for 4 degrees.
The necessity for widening the gauge on sharp curves is
still more apparent when we consider that provision must
be made to accommodate cars of both standard gauge (4
feet 8 inches) and for those of 4 feet 9 inches gauge, com-
mon to Southern roads.

When the gauge is not widened, a wide-gauged car is
liable to mount the rail, especially if the flanges of the
wheels are badly worn and sharp. The effect of all curva-
ture is to increase the train resistance, and on sharp curves,
this resistance, due to friction, becomes so great as to
largely reduce the train load. All train loads are limited
by the maximum resistance which they must overcome.
This maximum resistance may be concentrated upon a
single curve, and it is at once apparent that a railroad com-
pany might well incur heavy expense in reducing this curv-
ature, if by so doing they could add one extra car to each
train load. Another charge against curvature is the loss of
time to passenger trains which can not run over sharp
curves, except at reduced speed. All curves exceeding eight
degrees, besides their resistance to trains, cause a direct
loss of time to all fast passenger trains.

1668. Guard Rails on Short Curves. On straight
track, laid to exact gauge, the guard rail is spaced 1| inches
from the gauge rail; but when the gauge is widened, as on
sharp curves, the amount of the increase in gauge must
be added to the space between the gauge and the guard

1 669. Lining Curves. A common habit of trackmen
when lining curves is to throw the curve outwards to line.
The effect of this, in time, is to reduce the degree of curva-
ture at the ends of the curve and sharpen it at the cen-
ter, besides crowding the roadway on the outside of the


A safe rule is to always throw the track inwards, i. e., tow-
ards the center of the curve. It is at once apparent that
the effect of the cen-
trifugal force of the
train in passing over
a curve is to throw
the track outwards,
and in lining curves,
the track should be
thrown inwards, if
for no other purpose
than to overcome this
effect of the trains.
The effect of throw-
ing the track out-
wards when lining a
curve is shown in
Fig. 518, in which FIG. sis.

ABC represents the true line of the curve and A E C the
position of the tracks due to improper lining.

When track is first laid, there should be a track center
stake driven at every 50 feet and carefully centered with a
tack. Before and after ballasting, the track should be care-
fully lined to the center stakes, and if the rails have been
properly curved the track will hold its line, with occasional
retouching, for years.

In the case of a badly lined curve, select a piece of track
60 feet in length, which appears to be in good line. There
are few curves, however badly out of line, but will show at
least 60 feet of good line. At each end of the 60 feet of
good track set an accurate center stake, and one in the cen-
ter of the track midway between them. In Fig. 519, A
and B represent the center stakes 60 feet apart, and C
the stake midway between them. Stretch a cord from A
to B, and measure the distance from Z, its middle point,
to C. The distance C L is the middle ordinate of a 60-foot
chord. Next, mark the middle point L of the chord, and
move the end A of the chord to C. Measure from B the


distance B M= C L, and carry the measuring cord forwards,
stretching it taut, and in the line C M, as determined by the
offset B M. The forward end D of the cord will mark the
spot for another track center. Then, move ahead as before,
measuring another offset and stretching the cord to locate
another center stake at E. In this way a perfect curve may
be run in without the use of an instrument. It is better
policy to set the track centers in line with the faces of the
stakes for line rather than the tack centers, as the cord is
sure to line properly to the faces of the stakes, but in order

FIG. 519.

to line their centers they must be practically of the same
height, which is sometimes difficult to obtain, especially if
the ballast contains stone.

Having set all the track centers, select a track gauge
which is square and true, and mark a point midway between
the gauge lines. Then, place the gauge on the track close
to the track center, and direct the men to move the track
until the middle point of the track gauge coincides with the
track center. Line up the track at each track center until
the entire curve has been moved to line; then, repeat the
operation, giving the final touches, as a second lining should
be sufficient.

1 67O. Elevation of Curves. To counteract the cen-
trifugal force which is developed when a car passes around
a curve, the outer rail is elevated. The amount of elevation
will depend upon the radius of the curve and the speed at


which trains are to be run. There is, however, a limit
in track elevation, as there is a limit in widening gauge,
beyond which it is not safe to pass.

When we consider that the centrifugal force of a car in-
creases as the degree of curvature, and as the square of the
speed, we readily see how a slight decrease in speed will
equalize a great increase in curvature.

To illustrate: A car passing around an 8-degree curve
will have double the centrifugal force of a car passing around
a 4 degree curve at the same speed. But to neutralize the
effect of sharpening the curve from 4 to 8 degrees, it is not
necessary to halve the speed, but only to reduce it in an inverse
proportion to the square root of the degrees of curvature.
Thus, if a speed of 60 miles per hour is admissible on a 4-de-
gree curve, the speed on an 8-degree curve is obtained by the
proportion 60 : x = |/8~ : |/4, or ;r = 42.43 miles per hour.
If we again double the degree of the curve to 16 degrees,
we only reduce the admissible speed of equal safety to 30
miles per hour. Hence, it will be seen that the centrifugal
force developed by an increase in speed is not proportional
to the centrifugal force developed by an increase in curva-
ture. In consequence of this varying relation between curva-
ture and speed, no fixed rule can be followed for elevating
the outer rail of curves.

It is a safe rule to elevate all curves to suit the highest
speed of trains passing over that part of the track. Ordi-
narily freight trains require the same track elevation as pas-
senger trains. All railroad men know that freight trains
repeatedly run at passenger train speed. The aim of every
freight train conductor is to "make time," and he makes
it whenever the grades and train loads permit.

On rolling grades it is often necessary to run down a grade
at top speed in order to acquire sufficient momentum to
carry the train to the summit of the following grade. Every
day fast running is necessary in order to make up for time
lost through unavoidable delays; hence, if a curved track is
elevated to meet the requirements of passenger trains, freight
trains will be equally well served. All curves, when possible,


should have an elevated approach on the straight main
track, of such length that trains may pass on and off the
curve without any sudden or disagreeable lurch. The
length of the approach should be in proportion to the
elevation of the curve and not to its degree.

A good rule for curve approaches is the following: For
each half-inch or fraction thereof of curve elevation, add
30 feet or 1 rail length to the approach ; that is, if a curve
has an elevation of 2 inches, the approach will have as many
rail lengths as is contained in 2, which is 4 times. The
approach will, therefore, have a length of 4 rails of 30 feet
each, or 120 feet.

The following formula by Searles, viz.,

c = 1.5S7V, (113.)

gives the length of the chord c, whose middle ordinate is
equal to the proper elevation of the outer rail of the curve
for any velocity V in miles per hour.

EXAMPLE. The curve is 8, and the velocity 40 miles pei hour ; what
is the proper elevation for the outer rail of the curve ?

SOLUTION. Substituting the given values in formula 113,

^ = 1.587 y,
we have c = 1.587 X 40 = 63.48 feet, the length of the required chord.

To find the middle ordinate of this chord, we apply formula 112.

We have just found c = 63.48 feet, and R = the radius of an 8
curve = 716.78 feet.

Substituting these values of c and R in the above formula, we have

This result is too great. The best authorities on this
subject place the maximum elevation at | the gauge, or
about 8 inches for standard gauge of 4 feet 8 inches. The
gauge on a 10 curve elevated for a speed of 40 miles an
hour should be widened to 4 feet 9^ inches.

The following table for elevation of curves is a com-
promise between the extremes recommended by different
engineers. It is a striking fact that experienced trackmen
never elevate track above 6 inches, and many of them
place the limit at 5 inches:



of Curve.

Length of


Width of

Speed of Trains.


GO ft.

1 in.

4 ft. 8 in.

GO mi. per hr.


1-20 ft.

2 in.

4 ft. 8 in.

60 mi. per hr.


150 ft.


4 ft. 8f in.

60 mi. per hr.


180 ft.

af in.

4 ft. 8| in.

55 mi. per hr.


180 ft.

3 in.

4 ft. 8f in.

50 mi. per hr.


210 ft.


4 ft. 8| in.

45 mi. per hr.


210 ft.

31 in.

4 ft. 9 in.

40 mi. per hr.


240 ft.

3f in.

4 ft. 9 in.

35 mi. per hr.


240 ft.

4 in.

4 ft. 9 in.

30 mi. per hr.


270 ft.

4 in.

4 ft. 9 in.

25 mi. per hr.


270 ft.

4 in-

4 ft. 9^ in.

20 mi. per hr.


270 ft.

4| in.

4 ft. 9i in.

15 mi. per hr.


240 ft.

4 in.

4 ft. 9i in.

10 mi. per hr.


240 ft.

4 in.

4 ft. 9i in.

10 mi. per hr.


240 ft.

4 in.

4 ft. 9 in.

10 mi. per hr.


240 ft.

4 in.

4 ft. 9 in.

10 mi. per hr.

Many persons overrate the objections to sharp curves,
especially where the grades are low. Their great objection
is not in their being an obstacle to high speed, but in their
great resistance to tractioa. Freight trains, which are
usually heavily loaded, are much more impeded by sharp
curves than passenger trains, which are generally lighter
and made up of cars which more readily adjust themselves
to irregularities in line and surface.

No curve exceeding 10 degrees should be placed in the
main line of any railroad. The additional cost of operating
and maintaining a sharper curve would pay for tne addi-
tional outlay necessary to bring the degree within the 10-
degree standard. Many roads place the maximum curve at
6 degrees, and though beyond the reach of many roads, it is
a safe standard.


Besides the loss of time necessitated by running slowly on
short curves, there is a much greater loss due to the wear
and tear on rolling stock and upon the rails themselves.
The friction of the wheel flanges against the rails rapidly
wears them out, and the continual lurching and rolling of
the cars detract greatly from the comfort of passengers.

Most of the trunk lines in the United States have been
greatly improved since their first construction, especially in
their alinement, some of them being practically rebuilt.
The Pennsylvania R. R. between Philadelphia and Harris-
burg is a striking instance of the great improvement, both
in alinement and grade, of a line originally cheaply and
poorly built. Many of the original curves have been re-
moved, and all of them lightened. In many places the
original line has been entirely abandoned, and a new and
better one adopted. This road is, however, an exceptional
case, as few lines in the world could afford to make slight
changes involving so great cost.

1671. The Elevation of Turnout Curves. The

speed of all trains in passing over turnout curves and cross-
overs is greatly reduced, so that an elevation of ^ inch
per degree is amply sufficient for all curves under 16 degrees.
On curves exceeding 16 degrees, the elevation may be held
at 4 inches until 20 degrees is reached, and on curves ex-
ceeding 20 degrees, -fa of an inch of elevation per degree
may be allowed until the total elevation amounts to 5 inches,
which is sufficient for the shortest curves.

1672. Curve Approaches Between Reverse
Curves. If possible, there should be a level piece of track,
at least 60 feet in length, between reverse curves, besides
the elevated approaches to the curves. When the whole of
the intermediate tangent is required in making the elevated
approaches to the curves, commence at the middle of the
intermediate tangent, if both curves are of the same degree.
If, however, they are of different degrees, make the ap-
proach to each curve in proportion to its degree. In ele-
vating the approaches to the curves, give to the first rail


length an elevation of |- inch, after which give inch addi-
tional elevation per rail length, or, if necessary, 1 inch
additional elevation, so as to make the total elevation of the
approach equal to the elevation of the outer rail of the

When a curve is compounded, commence to increase or
decrease the elevation far enough back from the point of
compound curvature to give to the second branch of the
compound curve the elevation which it requires. This in-
crease or decrease in elevation is made at the rate of inch
per rail length, precisely as in elevating the approach to a
regular curve. When the changes in a compound curve are
frequent and abrupt, it is best to elevate the outer rail for
the highest degree of the curve and carry this elevation
uniformly throughout the curve.

1673. Putting the Elevation in Curves. If the

track is in good surface, first catch up all the low joints on
the inner rail of the curve. The elevation of the outer rail
is determined by means of the track level shown in Fig.
520. For leveling track, the edge a b of the track level is

placed upon the rails, and when perfectly level the bubble c
of the spirif level will rest in the middle of the tube. The
steps d, e, etc., of the track level are made 1 inch in height,
so that when the step d is placed on the outer rail of a curve
and the rail raised until the bubble of the spirit level rests
in the middle of the tube, the outer rail has an elevation of
1 inch. Similarly, the step e, when brought to a level,
would indicate a track elevation of 2 inches, etc.

Having determined the amount of elevation required for
the curve, the outer rail is raised with the track jack and the
ballast thoroughly tamped under the ties. The elevation


should be about inch in excess of that required, in order
that provision may be made for settlement.

In dressing the track after the elevation has been made,
make the crown of the ballast at not more than one-third
of the width of the gauge from the outer rail, in order to
secure drainage. The raising of the outer rail reduces the
outer slope and increases the inner slope of the ballast. If
the curve is sharp, the ballast on the outer half of the track
is practically level and holds water, instead of shedding it.
By crowning the ballast as directed, thorough drainage is in-

1674. The Effects of Curved Track upon Loco-
motive and Car Wheels. The effect of all curved track,
however easy the curve, is to wear the flanges and treads of
car wheels. This effect is due to the centrifugal force which
forces the flanges of the wheels against the head of the out-
side rail of the curve.

The elevation of the outer rail, the widening of the gauge,
and the coning of the car wheels, all combine to reduce this
friction and consequent wear.

Where the elevation is insufficient, the friction increases,
and if the gauge is the same as on straight track, there is
great danger of the wheels mounting the rails, especially if
the flanges are badly worn. The conclusion from many
years of experiment and close observation is that the wear
of rails on curved track is largely due to the driving wheels
of the engine. When the tires become worn, the wear of
the rails rapidly increases, and hence the importance of
careful and repeated inspection of the driving wheels. As
soon as they show considerable wear, the tires should be
turned off to true lines. Besides preventing unnecessary
wear of rails, this greatly increases the tractive power of
the engine. When the treads of car wheels become badly
worn, especially at the flanges, there is bound to be more or
less slipping of the wheels. For the outer rail, being the
circumference of a greater circle, should require a wheel of
greater diameter than the inner wheel, if both are to make



the same number of revolutions. This increased diameter is
given by the coning of the wheels, shown in Fig. 521, in
which the rail a is on the outside of the curve. An inspec-
tion of the figure will show that the cone-shaped tread of the
wheel b gives a greater diameter to the wheel at c d than at
e f. In passing around the curve, the flange of the wheel b
is forced against the rail #, while the flange of the wheel h
recedes from the rail g. This increases the diameter of the
wheel <, while decreasing that of the wheel /i, and so the ex-

\d /

cess in length of the outer rail of the curve is at least par-
tially covered.

Careful experiment proves that under the most favoring
conditions some slipping of the wheels is bound to occur.
The friction between wheels and rails rapidly increases as the
rails become worn, and, as soon as the head of the outer rail
of a curve becomes badly worn, the outer rail should be
taken up and placed on the inside of the curve, and the
inner rail put in its place. This furnishes almost new wear-
ing surfaces to the wheel, and the life of the rails is greatly

1675. Care of Curved Track. As curved track
offers greater resistance and greater danger to passing
trains than straight track, special effort and pains should be
taken to maintain it in perfect order. All trackmen know
that a low spot on a curve will cause every car in a train to



lurch heavily towards the low side. By careful watching,
]"* and by prompt and thorough repairs,
curved track may be kept in perfect or-
der. It is highly important that the ele-
vation of the outer rail be kept uniform,
and no foreman, however experienced,
should place dependence upon his eye in
estimating curve elevation.

Both the civil engineer and the track
foreman will do well to cultivate each
other, the engineer imparting theoretical
knowledge in exchange for practical
knowledge. The result will certainly pro-
mote mutual respect and enhance the
efficiency of both.



1676. Turnouts. A turnout is a
device for enabling an engine and train to
pass from one track to another. It con-
sists of two lines of rails a b and c d (see
Fig. 522), so laid as to form a reversed
curve uniting the two tracks A B and
C D. The several parts of a turnout are
as follows: The switch rails e f and
gJi, the frog, and the two guard-rails
/ m and n o. The stationary ends e and
g of the switch rails are called the heels,
and the movable ends f and h are called
the toes. The distance f p, through
which the toes /"and h move, is called the
throw. The throw must equal the width
of the head of the rail, with sufficient
additional width to allow the flanges of the
wheels to pass freely between the main rails r s and / u and


the turnout rails a b and c d. The throw on tracks of stand-
ard gauge is 5 inches; that is, the toes /"and h are moved 5
inches from their original position in the main track in
forming the turnout curve on which the train is to pass
from the main track A B to the siding C D.

The movement of the switch rails is effected by means of
a lever.

1677. The Frog. The frog is a device by means of
which the rail at the turnout curve crosses the rail of the
main track. The frog shown in Fig. 523 is made of rails
having the same cross-section as those used in the track.
Its parts are as follows: The wedge shaped part A is the
tongue, of which the extreme end a is the point. The
space b, between the ends c and <^of the rails, is the mouth,

FIG. 523.

and the channel which they form at its narrowest point e is
the throat. The curved ends/ and g are the wings.

That part of the frog between A and A' is called the
heel. The width h of the frog is called its spread. Holes
are drilled in the ends of the rails c, d, k, and / to receive the
bolts used in fastening the rail splices, so that the rails of
which the frog is composed form a part of the continuous

1678. The Frog Point. The theoretical point of
frog a' (see Fig. 523) and the actual point a are quite dis-
similar. The reason for making a the point of frog is that
if the theoretical and actual point of frog were the same,
the point would be so small that the first blow inflicted by
a passing locomotive or car would completely destroy it.
The frog point is accordingly placed at a, where its width
is about of an inch.


1679. The Frog Number. The number of a frog is

Online LibraryInternational Correspondence SchoolsThe elements of railroad engineering (Volume 2) → online text (page 34 of 35)