Previous Page

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

Table No. 1
Comparison of Curve Equilibrium Elevations Between One-Eighth
 Scale Model Practice and Full Scale Practice.
From Formulas 3 and 4.

7.5 Inch Gage Track
Curve Radius 60 Feet

Dimensional
Ratio 1:8*

56.5 Inch Gage Track
Curve Radius 480 Feet

 
Actual MPH

Equilibrium
Elevation Inches

Curve Super
Elevation Ratio

 Equilibrium Elevation Inches

 
Actual MPH

1

0.009

1:59

0.535

8

2

0.036

1:59

2.139

16

2.37

0.050

1:60

3.000

18.95

3

0.080

1:60

4.813

24

4

0.143

1:60

8.557

32

5.17

0.238

1:60

      14.288**

41.35

* Dimensional ratio of 1:8 applies exactly to curve radius and nominally to gage of track.

** For comparison purposes only. Comfortable elevation greater than 6 inches generally not permitted in full scale practice.

Results from Table No. 1 reveal the relationship between equilibrium elevation for one-eighth scale model practice and full scale practice varies approximately by the ratio of 1:60. Taking full scale practice superelevation data and dividing by eight will result in excess superelevation being applied in one-eighth scale model practice.

Maximum Comfortable Curve Velocity, One-Eighth Scale Model Practice.

Experience in one-eighth scale model practice has proven that when rolling stock is operated at a velocity of 2.37 miles per hour on a 60 foot radius curve maintained at zero cross level, operating personnel or passengers seated on the rolling stock at typical heights above top of rail will generally not experience a sensation of discomfort resulting from the perceived effects of centrifugal force. Given the previous, a maximum comfortable velocity formula based on a cant deficiency of 0.05 inches may be developed for one-eighth scale model practice.

Vmax = Sqrt. of ( ( Ea + Cd ) R / 0.5344 )  (Formula 7)

where:

Vmax = Maximum comfortable velocity in miles per hour.
Ea. = Actual superelevation in inches.
Cd = Cant deficiency taken as 0.05 inches.
R = Radius of curve in feet.

Using the following:

Ea. = 0 inches.
R = 60 foot radius.

results in:

Vmax = 2.37 mph = Sqrt. of ( ( 0 + 0.05 ) * 60 / 0.5344 )

Practical Application of Superelevation, One-Eighth Scale Model Practice.

In full scale practice superelevation may be practically applied and maintained in increments of one quarter inch or less up to a typical maximum of 6 inches. In one-eighth scale model practice a similar multitude of increments of superelevation is generally not practical. The Deerfield and Roundabout Railway has adopted the practice of maintaining tangent track, turnouts and crossings at zero cross level and curves at a superelevation of 0.188 inches.

Table No. 2
Deerfield and Roundabout Railway
One-Eighth Scale Model Practice Maximum Curve Velocity.
0.05 Inch Cant Deficiency Basis.
From Formula 7.

 
Curve Radius in Feet

Zero Cross Level
Maximum Curve
Velocity in Actual MPH

0.188 Inch Super Elevation
Maximum Curve
Velocity in Actual MPH

40 to less than 50

1.9

4.2

50 to less than 60

2.2

4.7

60 to less than 70

2.4

5.2

70 to less than 80

2.6

5.6

80 to less than 90

2.7

6.0

90 to less than 100

2.9

6.3

100 and greater than 100

3.1

6.7

Note: Provisions of other Deerfield and Roundabout Railway documents shall supersede Fundamentals of Railway Curve Superelevation Table No. 2 maximum velocities inconsistent therewith.

Transition Between Zero Cross Level and Superelevation.

When superelevation is applied to a curve consideration must be given to the procedure that will be used to introduce the change from zero cross level condition of the tangent track preceding the curve to the full superelevation of the curve and then back to zero cross level condition of the tangent track following the curve. The previously described is referred to as run off, run out or run in, regardless of whether the change in rail elevation is considered to be increasing or decreasing. In current full scale practice superelevation run off is preferably accomplished in conjunction with the use of easement curves, also known as transition curves or spirals, that are located before and after a circular curve. The fundamental element of an easement curve is its varying radius that gradually introduces a change in alignment from tangent to circular curve or vice versa. This is taken advantage of by applying superelevation to an easement curve at a rate that corresponds to the rate of change of curvature of the easement curve. Rolling stock traversing at constant velocity an easement curve with run off applied will experience a change in the centrifugal force developed as the radius changes and a generally simultaneous counteracting effect produced by the corresponding change in superelevation. It should be noted that when the word curve is used individually it is generally understood to mean a circular curve or simple curve, that being a curve of constant radius.

During the early period of railway construction and operation in the United States, circa 1900, easement curves before and after a circular curve may not necessarily have been used. Superelevation run off would instead be applied to a given length of tangent track before and after a circular curve. The foregoing is not as satisfactory a method of accomplishing run off as compared to the use of an easement curve due to the fact that the maximum acceptable deviation from zero cross level condition on a tangent track run off may limit the superelevation on the circular curve and thus the velocity on the circular curve. In some cases the run off would be divided with a portion of the run off on the tangent and the remaining run off on the circular curve in an effort to increase the superelevation on the circular curve and thus the velocity. W. M. Camp addresses this issue in his book "Notes on Track," the first edition published by the author in 1903, which includes the following: "The value of easement or transition curves is greatest where sustained high speed is practicable. Elevation for simple circular curves can be run in quite satisfactorily for good speed, [The foregoing refers to run in applied to tangents before and after a circular curve] and it is only where extraordinary results are desired that the greater expense and care necessary to maintain the easement curve can be justified. The most practical or satisfactory application of the easement curve is then not so much to curves so sharp that in any event speed must be reduced in running around them, but to those curves of comparatively smaller degree where, with the aid of the transition curve, the slackening of speed may be avoided; and while by using the transition curve a slightly higher speed on curves of, say, about 6 or 8 deg., might be had with a feeling of greater comfort or security, perhaps, still its use on curves less than 6 or 8 deg. must no doubt be the more justifiable practice. Not necessarily, then, are transition curves best suited to roads of heaviest curvature. Furthermore, it will usually be found that the surroundings which determine the location of a sharp curve will allow of but little room for easements. In any case the easement should be no longer than to give sufficient distance in which to run out the elevation. Any available room beyond this had better be used in reducing the curvature of the central or circular portion of the curve."

The Deerfield and Roundabout Railway from 1977 to 2007 in an effort to simplify aspects of track design and construction did not generally install easement curves. This scheme might be thought of as following the 1903 "Camp philosophy" described for "roads of heaviest curvature." A typical main line curve on the Deerfield and Roundabout Railway has a radius of 75 feet and is equivalent to a 600 foot radius curve in full scale practice having a degree of curvature of 9.6 degrees. In early 2008 the engineering department of the Deerfield and Roundabout Railway concluded that in order to better demonstrate full scale railway engineering practices typically used after the early decades of the Twentieth Century, new main line curve construction on the Deerfield and Roundabout Railway will include easement curves and, where practical and when time permits, existing curves without easements will be realigned and provided with easement curves.

Supplemental Benefit of Superelevation, One-Eighth Scale Model Practice.

Typical ballasted railway track that has been surfaced to zero cross level or to an established superelevation may after it has been placed in service deviate from the original surfaced condition due to disturbing forces that act upon the track structure. The direction, magnitude and rate of deviation from the original surfaced condition depends on the stability of the track, ballast and roadbed to resist the disturbing forces.

In one-eighth scale model practice passengers may experience an uncomfortable sensation when rounding a curve when the elevation of the outside rail of the curve is lower than the elevation of the inside rail. This might be attributable to a passengers expectation of leaning toward the center of the curve and instead the passenger car tilts toward the outside of the curve. In a case where the outside rail of a curve is gradually lowering in elevation relative to the inside rail, the initial application of superelevation provides a period of time before the elevation of the outside rail falls below the elevation of the inside rail.

Additional Considerations When Applying Superelevation.

For design and survey purposes the reference for the horizontal alignment of railway track as it relates to tangents, horizontal curves, easement curves, turnouts, crossings, etc., is generally taken to be the center line of track. The reference for the vertical surface of railway track as it relates to longitudinal level condition, gradients or vertical curves is generally taken to be the top of the running rails assuming the rails to be at zero cross level. The application of superelevation by its nature disturbs the zero cross level condition of the track, therefore a scheme must be used that provides continuity of reference for the vertical surface of the track. This is generally accomplished by maintaining the inside rail of a curve at the established reference for vertical surface and elevating the outside rail the required superelevation relative to the inside rail. In the foregoing the inside rail of the curve is referred to as the reference rail.

In typical one-eighth scale model practice track construction the track will be brought to proper horizontal alignment, vertical surface and to a zero cross level condition relative to offset grade stakes that are set at a known distance from the center line of track and marked to indicate the elevation of top of rail without consideration for superelevation. Superelevation, if used, is then applied as a second step by raising the outside rail of the curve relative to the inside rail by the use of a spirit level to determine when then desired superelevation is achieved. Run off applied to an easement curve or tangent track before and after a circular curve is accomplished in a similar manner using the spirit level.

Installation of turnouts and crossings on curved track requires that special attention be given to the effect that superelevation of one track may have on another. In these circumstances potential superelevation of a given track, and thus maximum velocity, might be limited by the constraints imposed by a turnout or crossing track. Motor vehicle roadway crossings of superelevated railway track or tracks may cause similar disturbances to aspects of the vertical surface of the roadway.

Of consideration more in full scale practice than in one-eighth scale model practice is the effect that superelevation may have on the tilting of rolling stock with resulting effect on the clearance between rolling stock on adjacent tracks or between track side structures, including tunnel walls, overhead bridge supports, etc.

Definitions.

CIRCULAR CURVE - A curve having a radius of constant dimension. Also known as a simple curve. When the word curve is used alone it is generally understood to mean a circular curve.

EASEMENT CURVE - A curve having a radius of changing dimension. The rate of change of radius being in one direction and may be determined by a number of accepted formulas. Also known as a transition curve, spiral curve or spiral.

SUPERELEVATION - The banking of track by raising or superimposing the outside rail above the inside rail at a curve. The desired speed and curve degree or curve radius determine the amount of superelevation. Also known as elevation or raise when referring to railway track. More recently may also be known as cant. Should not be confused with canted rail.

BALANCED SUPERELEVATION - The superelevation applied to a curve of given radius on which rolling stock when operated at a given velocity results in an equal downward force on both rails. Also known as balanced elevation or equilibrium elevation. More recently may also be known as balanced cant.

EQUILIBRIUM VELOCITY - The velocity of rolling stock when operated on a curve of given radius and superelevation that results in an equal downward force on both rails. Also known as balanced velocity.

UNDER BALANCED SUPERELEVATION - The superelevation applied to a curve on which rolling stock is permitted to operate at greater than equilibrium velocity. Also known as unbalanced superelevation or unbalanced elevation. More recently may also be known as unbalanced cant.

CANT DEFICIENCY - The amount that the under balanced superelevation of a curve is less than what would be the balanced superelevation of the curve.

RUN OFF, SUPERELEVATION - The gradual and uniform transition from zero cross level track to superelevation or visa versa. Also known as run in or run out. The use of the terms run off, run in or run out may be used interchangeably regardless of whether the direction of movement is considered to be toward or away from a curve.

CENTRIFUGAL FORCE - 1. The apparent force that is felt by an object moving in a curved path that acts outwardly away from the center of rotation. 2. An outward force on a body rotating about an axis, assumed equal and opposite to the Centripetal Force and postulated to account for the phenomena seen by an observer in the rotating body. See other reference sources for detailed explanations of Centrifugal Force and Centripetal Force.

Note: Current Deerfield and Roundabout Railway Definitions of Terms Relating to Track Work, Document DRTRK1, shall supersede Fundamentals of Railway Curve Superelevation definitions inconsistent therewith.

References.

  1. William W. Hay, Mgt. E., MS, Ph.D., "Railroad Engineering," Second Edition, 1982.
  2. Ralph P. Johnson, ME, "The Steam Locomotive," 1942.
  3. W. M. Camp, "Notes on Track, Construction and Maintenance," 1903.
  4. Federal Railroad Administration, Track Safety Standards Compliance Manual, Chapter 5, Track Safety Standards Classes 1 through 5, July 27, 2006.
  5. J. G. Hook, "One-Eighth Scale Model Railway Circular Curve and Spiral Calculator
        Excel Workbook," 2009.
  6. Deerfield and Roundabout Railway, Standards for Level Condition of Tangent and
        Other Track and Elevation of Curved Track, Document DRTRK33, 2009
  7. Dictionary of Railway Track Terms, 2003;
  8. Dictionary . com;
  9. Merriam-Webster online dictionary.

Previous Page