Drawing Index

Deerfield & Roundabout Railway Track & Wheel Data

Comparison of Designations of Curvature
Degree of Curve vs. Radius

Railway curves in full scale practice in the United States are generally designated by degree of curvature or degree of curve instead of by radius. This is a natural result of the use of surveying methods and instruments that use deflection angles to locate the survey stations of a curve. In one-eighth scale model practice curves are generally designated by radius. This due to the lineal measuring techniques typically used to locate the survey stations of a curve. Due to the different designations of curvature the relationship between curves in full scale practice and one-eighth scale model practice is not as easily understood as the basic one-eighth scale dimensional relationship.

Figure 1 and Formula 1 illustrate the derivation of the designation of a curve by degree of curvature by chord basis. The actual length of a curve may be greater than or less than the standard chord C and is of no consequence in determining the degree of curvature.

D = 2 * arcsin ( C / 2 / R )  (Formula 1)

where:

D = Degree of curvature or degree of curve in decimal degrees based on the standard
       chord length C.
C = Standard chord length in feet.
R = Radius of curve in feet.

An alternate method of designating degree of curvature is by arc basis, where a standard distance, similar in purpose to C, is measured along the arc of the curve instead. Typical railway engineering practice uses chord basis, therefore further references to any degree of curvature shall be understood to mean chord basis and in the case of full scale practice it shall be understood to mean a standard chord of 100 feet.

Figure 2 illustrates the principle of a surveying method that employs deflection angles to stake out the survey stations on a curve. The variables C and D of Formula 1 also apply to Figure 2. The red dashed lines illustrate the standard chord length C that determines the distance between survey stations on the curve starting at point of curve (P.C.). Angles D / 2, 2 * ( D / 2 ) and 3 * ( D / 2 ) represent the deflection angles to the progressive survey stations on the curve. The vertex of all deflection angles being at P.C. and all measured from the dashed black line that is tangent to the curve and extends to the right of P.C. and is a prolongation of the solid black line to the left of P.C. If the degree of curvature D in Figure 2 is assumed to be 16.0 degrees, then the deflection angle to the first curve survey station beyond P.C. is 8.0 degrees, the deflection angle to the second curve survey station beyond P.C. is 16.0 degrees, the deflection angle to the point of tangent (P.T.)  is 24.0 degrees.

When a curve such as in Figure 2 is referred simply as a 16 degree curve, it is generally understood to mean the degree of curvature. This should not be confused with the total angle of a curve from P.C. to P.T. Unlike Figure 2, in actual practice the circumstances that dictate the required length of a curve from P.C. to P.T. generally does not result in a multiple number of exact standard chord lengths C. In such a case the first chord after P.C. or the last chord before P.T. or both will be sub chords of a length less than the standard chord C. An explanation of sub chords is beyond the scope of this article.

Table No. 1 was prepared using Formula 2 and the variables described in Formula 1.

R = ( C / 2 ) / sin ( D / 2 )  (Formula 2)

For full scale practice the standard chord length of 100 feet was used for C. For direct comparison purposes between full scale practice and one-eighth scale model practice a standard chord length of one-eighth that of full scale practice was used, therefor C being 12.5 feet. Although the use of a standard chord length of 12.5 feet provides for direct comparison, the use of a standard chord length of 10 feet for C in one-eight scale model practice provides for simplification when staking out curves using deflection angles.

Document DRTRK12-C   Table No. 1   Revision 10-03-2010 

The Deerfield and Roundabout Railway was originally constructed to provide a maximum main line degree of curvature of 9.56 degrees (12.5 foot chord basis) or minimum radius of 75.0 feet. Latter realignment projects have resulted in the construction of one main line curve having a degree of curvature of 11.05 degrees (12.5 foot chord basis) or 64.9 foot radius.

For comparison purposes Horseshoe Curve west of Altoona Pennsylvania on the former Pennsylvania Railroad main line provides an interesting example for study. Data published in a 1995 Conrail track chart (excerpt shown below), the owner of Horseshoe Curve at that time, illustrates the horizontal alignment (tangents and curves) of one of the tracks of the existing three tracks of Horseshoe Curve. The horizontal alignment line on the track chart indicates tangent track by a horizontal line. A curve to the left when proceeding in the direction of increasing mile posts (westward) is indicated by an upward arc in the horizontal alignment line. A curve to the right when proceeding in the direction of increasing mile posts (westward) is indicated by an downward arc in the horizontal alignment line. The degree of curvature of curves indicated on the horizontal alignment line are then noted above in degrees, minutes notation. Mile post locations are indicated by the numbers inside the diamond icons.

Horseshoe curve consists of two adjoining curves one having a degree of curvature of 9.00 degrees (9 degrees 0 minutes) and the other having a degree of curvature of 9.42 degrees (9 degrees 25 minutes).

< WESTWARD    Track Chart    EASTWARD >

The aerial image below of Horseshoe Curve is from the United States Geological Survey, date  unknown. The track on the upper right side of the image leads to Altoona, eastward. The point on this track at the edge of the image is approximately mile post 241.30. The track on the lower right side of the image leads to Gallitzin, westward. The point on this track at the edge of the image is approximately mile post 243.00. The white line that appears to run under Horseshoe Curve is the PUBLIC RD at mile post 242.04 that does in fact run through a highway tunnel under Horseshoe Curve.

PDF File of Track Chart Explanation

References

  1. C. Frank Allen, S.B., "Railroad, Curves and Earthwork," 1931.
  2. William W. Hay, Mgt. E., M.S., "Railroad Engineering," Volume One, 1953.
  3. John Clayton Tracy, Ph B., C.E., "Surveying Theory and Practice," 1948.

Credits

Thank you to Robert W. Kurth for proof reading and suggesting improvements to this article.