AREA 10-Chord Spiral

American Railway Engineering Association (AREA) was formed in 1899 as a forum for the development and study of recommended practices for the newly-integrated standard-gauge North American railway network. The AREA dealt with the many engineering challenges through standing technical committees. The AREA issued its first Manual of Recommended Practices in 1905.

The AREA was merged into the American Railway Engineering and Maintenance-of-Way Association (AREMA) in 1997 along with three other railway engineering associations.

About the AREA 10-Chord Spiral

The AREA 10-Chord Spiral was adopted in North America early in the 20th century. Spirals were calculated without computers from tables. Now we can plug the formulas into a computer program or spreadsheet and make the calculation instantly. We can use any convenient number of chords, or stations along the spiral.

Material on the Definitions, Notation, and Formulas tabs is quoted directly from an old C. & N. W. RY. Engineering Department booklet STANDARD 10 CHORD INSTRUMENT SPIRAL (A. R. E. A.) which itself quoted from a full AREA document. No year given.

Selected definitions related to the AREA 10-Chord Spiral:

  • 10-Chord Spiral: An approximate spiral measured in ten equal chords and whose change of degree of curve is directly proportional to the length measured along the spiral by such chords.
  • Easement Curve: A curve whose degree varies either uniformly or in some definitely determined manner so as to give a gradual transition between a tangent and a simple curve, which it connects, or between two simple curves. (AREA)
  • Spiral (When used with respect to track): A form of easement curve in which the change of degree of curve is uniform throughout its length. (AREA)
  • Elevation (of curves): The vertical distance the outer rail is raised above the inner rail. Sometimes called Superelevation. (AREA). AKA banking or cant.

AREA 10-Chord Spiral Notation

Quoted from STANDARD 10 CHORD INSTRUMENT SPIRAL, reformatted for better presentation.

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For curve points, the first initial represents the alignment on the side toward station zero, the second that away from station zero.

T.C. The point of change in alignment from tangent to circular curve.
C.T.The point of change from circular curve to tangent.
C.C.The point of change in degree of circular curve; the point of compound curve, the point of reverse curve.
T.S.The point of change from tangent to spiral.
S.C.The point of change from spiral to circular curve.
C.S.The point of change from circular curve to spiral.
S.T.The point of change from spiral to tangent.
S.S.The point of change from one spiral to another.

The symbols T.C. and C.T., T.S. and S.T., and S.C. and C.S. become transposed when the direction of stationing is changed.

aThe angle between the tangent at the T.S. and the chord from the T.S. to any point on the spiral.
AThe angle between the tangent at the T.S. And the chord from the T.S. and the S.C.
bThe angle at any point on the spiral, between the tangent at that point and the chord from the T.S.
BThe angle at the S.C. between the chord from the T.S. and the tangent at the S.C.
cThe chord from the T.C. to any point of the spiral.
CThe chord from the T.S. to the S.C.
dThe degree of curve at any point on the spiral.
DThe degree of central circular curve.
IThe angle between the initial and final tangents; the total central angle of circular curve and spirals.
kThe increase in degree of curve per station on the spiral.
lThe length of the spiral in feet from the T.S. to any given point.
LThe length of the spiral in feet from the T.S. to the S.C.
oThe ordinate of the offset T.C.; the distance between the tangent and a parallel tangent to the offset curve.
rThe radius of the osculation circle at any given point of the spiral.
RThe radius of the central circular curve.
sThe length of the spiral in stations from the T.S. to any given point.
SThe length of the spiral in stations from the T.S. to the S.C.
uThe distance of the tangent from the T.S. to the intersection with a tangent through any given point of the spiral.
UThe distance of the tangent from the T.S. to the intersection with a tangent through the S.C.; the longer spiral tangent.
vThe distance on the tangent through any given point from that point to the intersection with the tangent through the T.S.
VThe distance on the tangent through the S.C. from the S.C. to the intersection with the tangent through the T.S.; the shorter spiral tangent.
xThe abscissa or tangent distance of any given point, referred to the T.S.
XThe abscissa or tangent distance of the S.C., referred to the T.S.
yThe ordinate or tangent offset of any point on the spiral.
YThe ordinate or tangent offset of the S.C.
ZThe abscissa or tangent distance of the offset, T.C. referred to the T.S.
The central angle of the spiral from the T.S. to any given point.
The central angle of the whole spiral.
Ts The tangent distance of the spiraled curve; distance from T.S. to P.I. (point of intersection of tangents).
EsThe external distance of the offset curve.

AREA 10-Chord Spiral Formulas

Quoted from STANDARD 10 CHORD INSTRUMENT SPIRAL, reformatted for better presentation.

 

Begin quote:

Formulas for the Exact Determination of the Functions of the Ten-Chord Spiral When the Central Angle Does Not Exceed 45 Degrees

(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)R = 50 / sin ( 1/2 D )
(11)Z = X – R sin 
(12)o = Y – R vers 
(13)Ts = (R + o) tan( 1/2 I ) + Z
(14)Es = (R + o) exsec( 1/2 I ) + o

Formulas for Field Use

The formulas presented above are best adapted for the preparation of tables.  For use in the field, the following empirical formulas are sufficiently accurate and have the advantage that they do not require the computation of the long chord.  The formulas can all be applied for the functions of any parts of the spiral without serious error, thought they are derived for the completed spiral.

(15)a=1/3 
A=1/3 
(16) a=10 ks minutes
A=10 kS minutes

Formulas (15) and (16) are sufficiently accurate for turning deflection when  (or ) does not exceed 15 degrees.

A similar approximation may be used when the transit is set at an intermediate point on the spiral if the included central angle from the transit point to the point of sight, less the included angle from the T.S. to the transit point, does not exceed 15 degrees.

(17)X=L – L(1/3 vers 3/4  + 1/22 vers 1/2 )
(18)Y=(L/39) (20 sin 1/2  + 3 sin )
(19)U=L (2/3 + 10/39 exsec 1/2  + 1/10 vers 1/4  )
(20)V=L (1/3 + 10/39 exsec 1/2 )
(21)o=L/10 (sin 1/2  + sin 1/3 ) cos 1/2 D
(22)Z=L (0.5 – .12885 vers 1/2 ) – .073 D sin 
(23)L=(370.82 / (cos 21/60 D)) (1 + .000018 Do) sqrt o/D