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Fundamentals of Railway Curve Superelevation
By Jeffrey G. Hook
Web Site Page 2 of 3  4-20-2010
(Scroll down for all text and illustrations.)

Equilibrium Elevation.

In order to counteract the effect of centrifugal force Fc the outside rail of the curve may be elevated above the inside rail effectively moving the center of gravity of the rolling stock laterally toward the inside rail. This procedure is generally referred to as superelevation or simply elevation when referring to a railway curve. If the combination of lateral displacement of the center of gravity provided by the superelevation, velocity of the rolling stock and radius of curve is such that resulting force Fr becomes centered between and perpendicular to a line across the running rails the downward pressure on the outside and inside rails of the curve will be the same. The superelevation that produces this condition for a given velocity and radius of curve is know as the balanced or equilibrium elevation Ee. Drawing DRTRK13, Figure B, illustrates the previously described.

drtrk13b

The following formula may be used to determine the centrifugal force developed when rolling stock rounds a curve:

Fc = ( W v 2 ) / ( g  R )   (Formula 1)

where:

Fc = Centrifugal force in pounds.
W = Weight of rolling stock in pounds.
v = Velocity in feet per second.
g = Acceleration due to gravity taken as 32.2 feet per second per second.
R = Radius of curve in feet.

Equilibrium Elevation, One-Eighth Scale Model Practice.

Using Deerfield & Roundabout Railway Engine No. 284 as an example with the following:

W = 1682 pounds.
v = 4.4 feet per second (3.0 miles per hour).
R = 60 foot radius curve.

results in:

Fc = 16.85 pounds = ( 1682 * 4.4 2 ) / ( 32.2 * 60 )

Knowing Fc the following formula may be used to determine the equilibrium elevation:

Ee = B / W Fc   (Formula 2)

where:

Ee = Equilibrium elevation in inches.
B = Distance between bearing points of wheels on track taken as 8 inches.

Using the previously determined Fc for DRRY Engine No. 284 results in:

Ee = 0.08 inches = 8 / 1682 * 16.85

Formulas 1 and 2 may be combined and simplified to produce the following:

Ee = 0.5344 V 2 / R   (Formula 3)

where:

Ee = Equilibrium elevation in inches for a railway having nominal 7.5 inch gage track.
V = Velocity in miles per hour.
R = Radius of curve in feet.

A velocity of 3.0 mph on a 60 foot radius curve results in the following:

Ee = 0.08 inches = 0.5344 * 3.0 2 / 60

Equilibrium Elevation, Full Scale Practice.

For a railway using 56.5 inch gage track (4 feet 8.5 inches) Formula 1 can be used without modification and Formula 2 can be used if the distance between bearing points of wheels on track is taken to be 58.8 inches.

Using a New York, Chicago & St. Louis Railroad, Class S1, 2-8-4 type locomotive with the following:

W = 416,000 pounds.
v = 35.2 feet per second (24.0 miles per hour).
R = 480 foot radius curve.

results in:

Fc = 33,349 pounds = ( 416,000 * 35.2 2 ) / ( 32.2 * 480 )

Formula 2 may then be used to determine the equilibrium elevation with the following:

B = Distance between bearing points of wheels on track taken as 58.8 inches.
W = 416,000 pounds.
Fc = 33,349 pounds.

results in:

Ee = 4.7 inches = 58.8 / 416,000 * 33,349

In full scale railway practice in the United States curves are generally designated by degree of curvature (100 foot chord basis) instead of by radius. The following formula, circa 1980, is generally accepted for use on 56.5 inch gage track to determine equilibrium elevation based on degree of curvature.

Ee = 0.0007 D V 2   (Formula 4)

where:

Ee = Equilibrium elevation in inches.
D = Degree of curvature (Decimal degrees 100 foot chord basis).
V = Velocity in miles per hour.

Using the following:

D = 11.94 degrees (480 foot radius).
V = 24.0 miles per hour.

results in:

Ee = 4.8 inches = 0.0007 * 11.94 * 24.0 2

Limitation of Superelevation, Full Scale Practice.

Typical railway operation results in rolling stock being operated at less than equilibrium velocity or coming to a complete stop on curves. Under such circumstances excess superelevation may lead to a downward force sufficient to damage the inside rail of the curve or cause derailment of rolling stock toward the center of the curve when draft force is applied to a train. Routine operation of loaded freight trains at low velocity on a curve superelevated to permit operation of higher velocity passenger trains will result in excess wear of the inside rail of the curve by the freight trains. For these reasons full scale practice superelevation is generally limited to not more than 6 inches.

Limitation of Curve Velocity, Full Scale Practice.

Maximum velocity on a curve may exceed equilibrium velocity, but must be limited to provide a margin of safety before overturning velocity is reached or a downward force sufficient to damage the outside rail of the curve is developed. This velocity is generally referred to as maximum safe velocity or safe speed. Although operation at maximum safe velocity will avoid overturning of rolling stock or rail damage, a passenger riding in a conventional passenger car will experience centrifugal force perceived as a tendency to slide laterally on their seat creating an uncomfortable sensation of instability. To avoid passenger discomfort maximum velocity on a curve is therefore limited to what is generally referred to as maximum comfortable velocity or comfortable speed. Operating experience with conventional passenger cars has lead to the generally excepted full scale practice, circa 1980, of designating the maximum velocity for a given curve to be equal to the result for the calculation of equilibrium velocity with three inches added to the actual superelevation that will be applied to the curve. Rephrasing, the actual superelevation applied to a curve is therefore three inches less than what would provide equilibrium elevation for the curve based on the radius and maximum velocity permitted. In the foregoing the difference between the actual superelevation and the equilibrium elevation is referred to as the unbalanced or cant deficiency.

The following formula, circa 1980, is generally used in full scale 56.5 inch gage practice to determine the maximum velocity permitted on curved track:

Vmax = Sqrt. of ( ( Ea + Cd ) / ( 0.0007 D ) )  (Formula 5)

where:

Vmax = Maximum comfortable velocity in miles per hour.
Ea = Actual superelevation in inches.
Cd = Cant deficiency taken as 3 inches.
D = Degree of curvature (Decimal degrees 100 foot chord basis).

When the maximum velocity permitted for a given curve is greater than equilibrium velocity, but limited to maximum comfortable velocity the actual superelevation applied to the curve is referred to as comfortable elevation Ec or unbalanced elevation Eu.

From Formula 5 therefore:

Ec = 0.0007 D Vmax 2 - Cd   (Formula 6)

where:

Ec = Comfortable elevation in inches.
D = Degree of curvature (Decimal degrees 100 foot chord basis).
Vmax = Maximum comfortable velocity in miles per hour.
Cd = Cant deficiency taken as 3 inches.

Using the following:

D = 11.94 degrees (480 foot radius).
Vmax = 24 miles per hour.

results in:

Ec = 1.81 inches = 0.0007 * 11.94 * 24 2 - 3

If resulting comfortable elevation Ec from Formula 6 is less than zero the curve is maintained at zero cross level.

Limitation of Velocity on Curved Track at Zero Cross Level.

The concept of maximum comfortable velocity may also be used to determine the maximum velocity at which rolling stock is permitted to round curved track without superelevation and maintained at zero cross level. The lead curve of a turnout located between the heel of the switch and the toe of the frog is an example of curved track that is generally not superelevated. Other similar locations would included yard tracks and industrial tracks where increased velocity made possible by superelevation is not required. In such circumstances the maximum comfortable velocity for a given curve may also be the maximum velocity permitted on tangent track adjoining the curve.

In full scale practice an American Railway Engineering Association No. 8 lateral turnout using a 16 foot 6 inch straight split switch has a lead curve radius of 487.3 feet or degree of curvature of 11 degrees 46 minutes 44 seconds. The straight split switch having an angle of 1 degree 46 minutes 22 seconds is calculated as having an effective curve radius of 533.3 feet using an excepted A.R.E.A. formula.

Using Formula 5 and an A.R.E.A. No. 8 lateral turnout maintained at zero cross level with the following:

Ea = 0 inches.
Cd = Cant deficiency taken as 3 inches.
D = 11.76 degrees (487.3 foot radius).

results in:

Vmax = 19.1 mph = Sqrt. of ( ( 0 + 3 ) / ( 0.0007 * 11.76 ) )

In the previous example the No. 8 lateral turnout lead curve radius governs the maximum velocity permitted because the lead curve radius is less than the effective curve radius of the straight split switch.

Height of Center of Gravity.

Operation on a curve at equilibrium velocity results in the center of gravity of the rolling stock coinciding with a point on a line that is perpendicular to a line across the running rails and the origin of which is midway between the rails. Under such condition the height of center of gravity is of no consequence as resulting force Fr coincides with the perpendicular line described. When rolling stock stops on a superelevated curve or rounds a curve under any condition of non equilibrium resulting force Fr will not coincide with the perpendicular line previously described and the height of the center of gravity then becomes consequential in determining the location of resulting force Fr relative to the center line of the track. The elasticity of the suspension system of rolling stock under conditions of non equilibrium will introduce a roll element that effects the horizontal displacement of the center of gravity that must also be considered when determining the location of resulting force Fr. Formulas 5 and 6 used in full scale practice, circa 1980, were developed by assuming 84 inches as the height of center of gravity and consideration given to the effects of suspension system displacement of conventional rolling stock when rounding a curve.

Detailed studies of the behavior of suspension systems of one-eighth scale model practice rolling stock is limited at this time. Although many years of construction and operating experience have proven that currently accepted design practice provides sufficient rolling stock stability at typical operating velocities. Unless the suspension system is unusually soft or an unusually high or laterally offset center of gravity exists, the location of the center of gravity of one-eighth scale model practice rolling stock is generally not considered.

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