The simplest way of designating a railroad curve is by giving the
length of the radius--_i. e._, the distance from the center to the
outside of the circle, or one-half the diameter. The shorter the radius
the sharper the curve. The length of the radius is usually stated in
feet; but English engineers often state the radius in chains (one chain
= 66 feet). The length of the radius of a railroad curve is measured to
the center of the track.
Civil engineers designate railway curves by degrees (using the sign °
for degrees and " for minutes, there being 60 minutes in one degree).
The sharpness of the curve is determined by the "degree of curve," or
the number of degrees of the central angle subtended by a chord of
100 feet. Or, in other words, let two lines start from the center of
a circle in the shape of a V, so that the angle at the point of the V
is one degree (equivalent to 1/360 of a complete circle), then, if the
two sides of the V are prolonged until they are 100 feet apart, any
part of a circle made by using one of these lines for its radius is a
"one-degree curve." The exact length of radius which with an angle of
one degree has a chord of 100 feet is found to be 5,729.65 feet. For
sake of convenience 5,730 feet is usually taken as the radius of a
one-degree curve. If the angle at the point of the V is two degrees and
the sides are prolonged until 100 feet apart, the length of each side
is (almost exactly) one-half as long as when the angle is one degree,
or one-half of 5,730==2,865 feet. For a three-degree curve the radius
is one-third of 5,730; for a four-degree curve one-fourth of 5,730; and
so on. For perfect exactness the length of 100 feet should be measured
not along a straight line connecting the ends of the V, but along the
line of the circle of which the sides of the V are radii--_i. e._,
the arc should be used and not the chord. The difference, however, is
so slight for any curves ordinarily used on main lines of standard
gauge railroad as to be ignored in practice. But for extremely sharp
curves, such as our locomotives both wide and narrow gauge are built
for, a considerable mathematical error would be involved by the use of
100-foot chords and calculating the length of the radius by dividing
5,730 by the degree of curve. The ratio of this error increases with
the degree of curve, since the error is caused by neglecting the
difference between the length of the chord and of the arc (_e. g._,
a 60-degree curve and 100-foot chord mathematically compels 100-feet
radius instead of 95½ feet; a 90-degree curve and 100-foot chord,
71+feet radius instead of 63.6 feet).
In practice, however, the formula of dividing 5,730 by the degree of
curve (R==5730/D) is almost universally used, and the mathematical
error is avoided by using two 50-foot chords for curves ranging from 10
to 16 degrees, and four 25-foot chords for curves ranging from 17 to 30
degrees, and further sub-dividing for sharper curves, since this almost
exactly balances the error, and it is also a practical necessity in
laying out sharp curves to use short chords.
For extremely sharp curves, or say 100 feet radius or less, it is usual
to express the curve by feet radius rather than by degrees. The table
following is computed by the formula R==5730/D, and fractions of feet
are not taken into account.
NOTE--The above engineers' method of designating the rate of
curvature of a railway curve must not be confounded with the number
of degrees of a circle occupied by the curved portion of the track;
thus a curved track making a quarter turn, equivalent to a right
angle, will always be 90 degrees of a circle (360 degrees--the
whole circle) no matter whether the curve is an easy one with a
long radius or a sharp one with a short radius.
Table Showing Lengths of Radius in Feet (Fractions Disregarded)
for Curves from One to Sixty Degrees