ABC's of Railroading
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Grades and curves

Railroading's weapons in the battle against gravity and geography
By Robert S. McGonigal
Published: May 1, 2006
Given a choice, railroads will always follow a straight, level path. Trains use less energy, speeds are higher, and there's less wear on equipment when railroads can build on an arrow-straight line. But the land rises and falls, obstacles must be avoided, and the ideal is more the exception than the rule. That requires grades to compensate for changes in elevation and curves to reorient the direction of the tracks.

Grades: uphill and down

In North America, gradient is expressed in terms of the number of feet of rise per 100 feet of horizontal distance. Two examples: if a track rises 1 foot over a distance of 100 feet, the gradient is said to be "1 percent;" a rise of 2 and-a-half feet would be a grade of "2.5 percent." In other parts of the world, particularly Britain and places with heavy British influence, gradients are expressed in terms of the horizontal distance required to achieve a 1-foot rise. This system would term the above examples "1 in 100" and "1 in 40," respectively.

On main lines, grades are generally 1 percent or less, and grades steeper than about 2.2 percent are rare.

The steepest grade on a major railroad's main track (as opposed to industrial spurs) was historically said to be on the Pennsylvania Railroad north of Madison, Ind. Now operated by short line Madison Railroad, the track rises 413 feet over a distance of 7012 feet - a 5.89-percent grade. The title for steepest main-line grade long rested with Norfolk Southern (and predecessor Southern Railway) for its 4.7-percent grade south of Saluda, N.C. With Saluda's closing in 2002, BNSF's 3.3-percent Raton Pass grade in New Mexico became the steepest main-line grade in North America.

The effect of grades on train operations is significant. For each percent of ascending grade, there is an additional resistance to constant-speed movement of 20 lbs. per ton of train. This compares with a resistance on level, straight track of about 5 lbs. per ton of train. A given locomotive, then, can haul only half the tonnage up a .25-percent grade that it can on the level. Descending grades carry their own penalties in the form of equipment wear and tear and increased fuel consumption.

The term "ruling grade" is used to describe the limiting grade between two terminals. It determines the maximum load that can be pulled over that portion of line by a given locomotive. The concept is analagous to that of the weakest link in a chain; no matter how many lesser grades a train can handle, if it can't make the ruling grade, it won't be able to complete the run.

A ruling grade is not necessarily the absolute steepest grade between two endpoints; it is assumed that trains will surmount certain steeper grades with momentum from descending grades or with the aid of helper locomotives.

For grades that are short relative to the total length of a train's run, helper engines - extra locomotives added to the front, rear, or even middle of a train - are employed. While the superior power of diesel locomotives has eliminated many helper districts, dieselization has brought helpers for use on trains going downhill, where dynamic braking is used to control speed on the descent.

If a train cannot make a grade, and no helpers are available, it may have to "double the hill," a practice in which the train is taken up the grade in two separate pieces. On some hills, "tripling" is necessary.

When a grade is steep enough to render the conventional "adhesion" system unworkable, a rack (or cog) or cable system may be used. Though there are some isolated examples, such alternative methods of negotiating hills are not found in the U.S. rail network.

Watch those curves

Railroad track is either "tangent" (straight) or curved.

Curves are best thought of as portions of circles. Curvature on railroads is not expressed in terms of radius, as it is on model layouts. (It would be impractical to strike such a large arc in the field.) Rather, it is given as the angle between two lines drawn from the center of the circle of which the curve is a part to two points on the circumference 100 feet apart. Since curve measurement is the description of an angle, the units used are the familiar ones from geometry class: degrees, minutes, and seconds. (Remember from geometry class that a circle contain 360 degrees.)

Curvature can be expressed in terms of the number of degrees traversed by 100 feet of track. For example, a relatively gentle 5-degree curve encompasses 5 degrees of a circle for each 100 feet of track; a sharper 15-degree curve covers 15 degrees in each 100 feet. The radius (distance from center point to edge) of a curve is obtained with the following conversion equation: radius in feet = 5729 divided by the degrees of curvature. This is known as the "arc" definition of curvature, which is normally used by highway designers.

Railroad designers use the "chord" definition of curvature, which is based on the degrees encompassed by a 100-foot line segment whose endpoints fall on the arc described by the curved track. An approximate method of determining curvature this way involves stretching a 62-foot-long string between two points on the inside face of the outer rail head. The number of inches between the center point of the string and the rail corresponds to the degrees of curvature: 1 inch equals 1 degree, 2 inches equals 2 degrees, and so on.

For the purposes of the casual observer, the difference between the arc and chord methods of measurement are small: the radius of a 15-degree arc-definition (highway) curve is approximately 382 feet, while the radius of a 15-degree chord-definition (railroad) curve is about 383 feet.

Curves of 1 or 2 degrees are the most common on mainline railroads; the sharpest curve a common four-axle diesel can take is about 20 degrees when coupled to other rolling stock, more than 40 degrees when by itself. Mountainous territory, however, generally dictates curves of 5 to 10 degrees, or even sharper. Branch lines and minor spurs may have an even greater number of sharper curves.

Just as grades impose additional resistance on trains, so do curves. However, wheel- and rail-wear are more significant (in terms of cost) than added fuel consumption. While it may seem that a long, gentle curve is preferable to a short, sharp one, the resistance is in fact the same as long as the central angle is the same, regardless of the radius.

In addition to reducing severe grades, many line relocations have reduction of total degrees of curvature as their goal.

Because of the resistance produced by curves, they pose an added difficulty when located on grades. To keep the combined resistance of grade and curve from overwhelming trains, grades are often "compensated" by being reduced on curves so resistance remains constant. A grade so treated would be termed, say, "1.7 percent, compensated."

Curves are often used to avoid undesirably heavy grades. By stretching out a given rise in elevation over a longer distance of track, loops and horseshoe curves (among other, less extreme, examples) keep grades manageable.

An important feature of a railroad curve is the extent to which it is "superelevated," or banked. To counteract centrifugal force as a train rounds a curve, the outer rail is raised to a higher level than the inner one. The difference in elevation between the two rails - called the "cross-level" - is how civil engineers measure superelevation. On main lines, the maximum difference in "cross-level" between the two rails can be as much as 6 inches, which is a superelevation good for 95 mph on a 1-degree curve, 45 mph on a 5-degree curve.

Since a train traversing a heavily superelevated curve at a relatively slow speed tends to cause excessive wear on the low rail, many railroads reduced curve superelevation when their passenger trains disappeared. This practice has worked against the reinstatement or speeding up of passenger service.

Curves aren't just portions of circles with tangents at each end; instead, a smooth transition in the form of a spiral is used. In a spiral, curvature and superelevation are gradually increased until the amounts needed for the curve itself are reached. Spirals may be more than 600 feet long in high-speed territory.
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