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Spiral tables prepared for the Canadian Pacific Railroad Sullivan, J. 1908

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     SPIRAL TABLES
PREPARED FOR THE
CANADIAN PACIFIC RAILROAD
BY
J. G. SULLIVAN
C. E.  Cornell University,  1888
Ass't Chief Engineer, Canadian Pacific Railway
Member American Society of Civil Engineers
Member Canadian Society Civil Engineers
Member Amer. Railway and Maintenance of Way Ass'n
Reprinted from the Engineering Record
[                                               NEW YORK
McGRAW PUBLISHING COMPANY
239 WEST 39th STREET
i                                          1908
L   	 Copyright, 1908,
BY THE
McGRAW PUBLISHING COMPANY
New York EDITOR'S NOTE
It is generally admitted by railroad engineers
that most of the spirals which are now in use
give such little differences in alignment as to
make the latter a negligible factor in deciding
the relative values of different spirals. The
adoption of a particular easement curve depends,
therefore, largely on the ease of computing the
angles and distances required in the field and
in the platting. Many roads use more or less
elaborate tables, while others have adopted simple formulas or rules for figuring offsets and deflections. In the latter cases the formulas or
rules are generally chosen because of their simplicity, rather than from strict theoretical considerations.
Some roads have prepared blueprint tables
which are adaptations or amplifications of well-
known spirals, and in this class are the tables
based on Holbrook's spiral, which have been prepared by Mr. J. G. Sullivan, now assistant chief
engineer of the Canadian Pacific Ry., for the
engineering department of that road. They have
been in use since 1894 and have given entire
satisfaction. The data and tables given herewith are published in blue-print form for the
company's engineers. Mr. Sullivan states that he
is unable to give credit to the engineer who
originated the idea of these tables. In 1894 he
received copies of seven of the twenty tables
given herein, but without any instructions or explanations, from Mr. M. W. Ensign, then chief
engineer of the B., A. & P. Ry.  Spiral Tables of the  Canadian Pacific
Railway.
The formula, N2L/10, for figuring deflections for these tables from the beginning of spiral
in minutes, is easily remembered and so simple
that if the tables are not at hand, the deflections
can be rapidly figured. N in this formula
is the number of subchords from the transit to the point to be set, and L is the length of
subchord in feet. The fact that L is in feet
should be especially noted, since this symbol in
curve work is often taken in stations.
For staking out trestles on a spiral, the formula
dVioL is derived, as being easier to figure for
the fractional chord lengths used in giving alignment on trestle bents. In this expression d is
not a unit or subchord length, but the distance
in feet to the point to be located.
The formula N2L/io, shows that the deflections
vary as the length of the subchord, and this fact
is of special importance since the tables constructed with these values can be used for any
other subchord, by multiplying the tabulated
values by the ratio of the two subchords. This
rule applies also to the spiral angle, A, and approximately to C, the distance from the point of
spiral to the B. C. of the circular curve produced back to a parallel tangent.   R', the radius 6 SPIRAL TABLES
of the circular curve spiralled, will, of course, be
the same in both cases. The other quantities
would have to be figured independently of the
tabulated values. Examples of the use of the
tables for a subchord other than the one for
which they are calculated, are given hereafter.
By the use of this method the tables here given
can be made applicable to an indefinite number of
spirals, sufficient, indeed, to meet most cases.
Notation.—R    and   R' = radii of curves with
common center, O.    (Fig. I.)
D and D' 2= degrees of curves having radii R
and R', respectively.
0 = total central angle.
A — angle used in spiral.
S = R — R' = shift, or offset between original tangent and parallel tangent of
spiralled    circular    curve    produced
back.
C = distance   on   tangent   from  beginning
of spiral to B. C. of curve of radius R.
L == length of spiral subchord in feet.
N = number of subchords from transit to
point being set.
L N = length  of spiral    (N   may   be   fractional).
d = distance in feet to the point to be set.
(Used in staking out trestles.)
E X and X Y = co-ordinates.
S = XY— (R1 — tf'cosA).
C = EX — R's'mA
AB—R tan 1/2 0,  and can be taken from
tables of subtangents as for a simple
curve, using <P and D.
Deflection angle in minutes from beginning of SPIRAL TABLES
spiral = N2L/io,   or   for   staking  out  trestles =
dVioL.
Principles on Which the Tables are Based.—I.
The spiral is twice the length of the simple
curve which it replaces.
2. The angle of the spiral equals the angle of
the simple curve which it replaces.
3. The curvature of the spiral increases uniformly, and directly as the distance from the beginning of  same.
4. The offsets from the tangent and from the
circular curve produced vary as the cube of the
distances (approximately).
5. The deflection from the beginning of spiral
varies as the square of the distance measured on
the spiral, and the deflection from the beginning
to any point on the spiral equals one-third the
spiral angle up to that point. To turn tangent at
any point on the spiral, deflect from the beginning
of spiral two-thirds the spiral angle used up to
the given point.    (See Fig. 2.)
6. The distance required for one degree change
in curvature is chosen as the length of the sub-
chord and varies in these tables from 5 to 120 ft.
Length of Spiral.—On the main line of the
Canadian Pacific Ry., where the curves are usually not sharper than 4 deg., a spiral 100 ft. long
per degree of curvature is used and the track is
elevated at the rate of about 1 in. in 90 ft. On
mountain and branch lines the length of spiral
chosen should, where possible, be 60 ft. for each
inch of elevation to be given the circular curve
so that the beginning of elevation may start at
the beginning of spiral and proceed at the uni- SPIRAL TABLES
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II SPIRAL TABLES 9
form rate of increase of y2 in. per rail length.
If a different rate of increase is used choose spiral
length to suit.
Value of Deflection Angle.— (I) That the deflection from the beginning of spiral to any point
varies as the square of the distance measured on
the spiral can be shown as follows:
Let o, J, 2, 3, 4, 5, Fig. 3, represent a cubic
spiral having subchords of length a, and let the
offset from tangent O T at end of first chord
equal b. Then from the definition of this spiral,
the offests at 2, 3, 4, 5, etc., will be, respectively,
distances of 8b, 2jb, 64b, I2$b} etc., from tangent  OT.
From the figure:
Sin T O 1 = b/a
Sin T O 2 = 8b/2a = 4b/'a
Sin T O 3 = 2yb/sa == gb/a
Sin T O 4 = 64b/4a =±: i6b/a
Sin T 0 5 = i2Sb/sa = 256/a
Assuming that the length of spiral and length
of long chord are the same, and taking the sines
of small angles to vary as the angles, the deflections for points 2, 3, 4, 5, etc., equal 4, 9, 16, 25,
etc., times the deflection for the first point, or,
stated in words, vary as the square of the distances.
This is not absolutely correct; for instance,
the spiral for a 5-deg. curve, using 60-ft. sub-
chords, would be 300 ft., while the long chord
would be 299.76 ft. Deflecting for the end of this
spiral 2 deg. 30 min., according to the above,
there is an error in the offset of (0.24 ft. X
sin 20  30') =: 0.0105 ft-    This error is  certainly 10
SPIRAL TABLES
allowable, but if more accuracy is desired, the
transit can be moved to the third point, when
the total error in setting the last point would be
reduced to about one-tenth of the above. Some
tables have been compiled for this spiral which
give corrections to be made to the deflection
angles, but as this means considerable work, Mr.
Sullivan's practice is to move up the transit
when the spiral is long or the angle large due to
a sharp spiral.
(II) The second part of Principle 5 states that
the deflection angle is one-third the spiral angle
up to the point to be set. This is derived as follows, reference being to Fig. 3:
Since the length of spiral is twice the length
of the circular curve replaced.
0X5 = 2^5 (measured on curve)
0 X = 1/2 OX5 (approximately)
Therefore   D X ~i/8T 5 (From Principle 4.)
D A =:2DX = 1/4 T5 — TC=z shift.
A 5 = 1/2 0 5.     (Approximately.    In   example
discussed  above  for  5-chord  spiral  300-ft.  long
the error would be about 0.1 ft.)
Since     C A 5 = 1/2 A
C5 = 3/4^5 = ^5sini/2A
Since the sines of small angles vary directly as
the angles,
A 5 sin 1/2 A = 0 5 sin 1/4 A
.-. O 5 sin 1/4 A = 3/4 T5
or 0 5sin 1/3 A = 75
But T5-* 0 5 = sin TO5
.-.r05 = i/3A
It is evident that the same reasoning applies to
any point on the spiral as well as to point 5,
which has been here chosen for illustration.
Fig. 3 also shows graphically that the distances SPIRAL TABLES 11
from the circular curve to the spiral beginning at
5, equal the distances from the tangent to the
spiral.
Formula N2L/io.—The formula, N2L/io, used
in figuring the deflection angle is derived thus:
From   Principle  I,  NL = twice the length  of
the simple curve replaced, and from Principle 2,
A NLD'        A,_NLP'60
2 X 100 2X 100
Figure  4.
DO"
Since the degree of curvature D', at any point
on the spiral is equal to N  (each subchord representing a change of one degree in curvature),
and since the deflection angle is 1/3A,
Deflection  angle   (in minutes) =
N2L60      = N2L
3X2X100 10 12
SPIRAL TABLES SPIRAL TABLES
13
It will be noted that the deflection in minutes
for the first point of this spiral equals i/io of the
number of feet in the subchord (the distance for
a change of i deg. in curvature).
Deflections from a Point on the Spiral.—From
Principle 4 it follows that the spiral leaves the
circular curve at the same rate that it leaves the
tangent. This is true of the spiral and any curve
that may run through any point of the spiral and
tangent to it. Therefore, the formula N2L/io
gives the difference between the deflection angle
of the simple curve and that of the spiral when
the transit is at any point on the spiral. The deflection will then be equal to the deflection for an
equal length of simple curve of the degree of
curvature of the spiral at the transit point, plus
or minus the amount given by the formula N2L/io
as the points are respectively ahead or back of the
transit when approaching the sharp end of the
spiral.
This fact gives some trouble occasionally to instrument men and the following demonstration
is, therefore, given to show it clearly.
In Fig. 4 P P' is a spiral having a curvature of
a radius P' Of at P\ Let P" O" be the radius at
any point P", and PX = P'X' = P" X" (distances measured on spiral).
P" 0" :P'Of : :PP' :PP"
P" O" = (P' 0'XPPf)-t-P P"
Then from Principle 4 y = yf = yn.
Deflection for X from P =z
N2L/io = d2/io L=z(P Xy/10 L.
Deflection for X' from tangent at P' = deflection for curve of radius O'P' for distance P'X'
minus (P'X')2/ioL. 14
SPIRAL TABLES
= deflec-
distance
Deflection for X" from tangent at P"
tion for curve of radius P"0" for
P"X" plus  (P"X")7ioL.
Trestles.—In staking out trestles on spirals use
the following formula for deflections:
N2L
X-
(NLy
d2
10 L        10L 10 L
where d is the distance to the point in feet. That
is, the deflection from the beginning of the spiral
in minutes for any point equals the square of the
distance to the point in feet divided by io times
the length of subchord.
Use of the Tables.—The blank spaces in the
deflection tables denote the position of the transit
and deflections from that point to all other points
on the spiral will be found on the same horizontal
line.
It will be noted from the formula N2L/io that
the deflections depend directly on L. If an L be
chosen, which is a multiple of the L of a particular table, the deflection can be readily taken from
that table either by multiplying the deflections
under corresponding numbers by the ratio of the
two subchord lengths or dividing the deflections
under corresponding distances by the same ratio.
For example: In the table for 30-ft. subchords,
multiplying the deflections 03', 12', 27', 48', i° 15'
by 2 gives 06', 24', 54', 1 ° 36', 20 30', while dividing 12', 48', i° 48', 30 12', 50 00' by 2 gives 06', 24',
54', i° 36', 20 30', the same as will be found in the
first line of table for 60-ft. subchords.
It will be noticed that all deflections in the table
for 60-ft. subchords are twice those in corresponding columns of the 30-ft. subchord, and three
times those of the 20-ft. subchord; also that the
deflections in the 30-ft.  subchord table for dis- SPIRAL TABLES
15
tances corresponding to distances in the 6o-ft sub-
chord table are twice the latter, while corresponding deflections in the 20-ft. subchord table
are three times those in the 6o-ft. subchord table.
If it is desired to use an 80, 90, 100, 120 or
150-ft. subchord, use tables for 40, 45, 50, 60 and
75-ft. subchords, respectively, using ratio 2, as explained. If it is desired to use an 82^, 90, 97^,
105, 112^, 120 or 135-ft. subchord, use tables
VVz, 30, 32/4 35, 37V2, 40 and 45-ft. subchords,
respectively, using ratio 3.
The following examples will show how the
spiral is used:
1. To spiral an 11-deg. curve using 15-ft. subchord.
Given point of intersection at station 1895 + 64,
and 0 = 43 deg. 20 min. Table for 15-ft. subchord
gives A — 9 deg. 04.5 min.; R = 523.84 ft.; sub-
tangent = 523.84 X tan 210 40' = 208.11. [This
could be obtained from tables of subtangents,
using the arguments 430 20' and io° 57' (£>)].
208.11 + 82.31 (C) = 290.42. Station 1895 + 64 —
290.42 = Sta. 1892 -f 73.6 (P. S.). The transit
notes would be as shown in the accompanying
table, only every other point being set.
Transit Notes.
Station.
Point.
B. S.
Deflection.
P. T. (Hub)
3° 01.5'
6° 03'
1898+ 17.5
5° 45'
+ 87.5
5°oo'
+ 57-5
4° 03'
1897 + 27.5
20 54'
+ 97-5
i°33'
+ 67.5
P. S. (Hub)
3° 42.7'
12° 35.5'
1896
(Hub)
3° 22.6'
8° 52.6'
+ 50.0
6° 07.6'
1895
(Hub)
o°oo'
3° 22.6'
+ 50.0
o° 37-6'
+ 38.6
P. C. (Hub)
2° 15'
2° 01.5'
1894 + 23-6
i° 18'
+ 93-6
(Hub)
3°  12'
i°36'
+ 63.6
o° 54'
+ 33-6
o° 24'
1893 + 03.6
o° 06'
1892 + 73.6
P. S. (Hub)
o° 00' 16
SPIRAL TABLES
An examination of these transit notes will show
the method of turning tangent at the sharp end of
the spiral when there has been an intermediate set
up. For example, when the transit reaches the
P. C, it is then at the eleventh point on the spiral,
while the last hub is at the eighth point. Looking along the horizontal line for transit at n in
the table for a 15-ft. subchord, the back-sight
2° 15' is found under the column for 8. If the
P. S. were visible, the same could be accomplished
by turning 2/3 A from the P. S.
2. Example showing how to use the tables in
putting a spiral between the two simple curves of
a compound curve. To put in spiral with 25-ft.
subchords between a 3-deg. and an 8-deg. curve:
Length of spiral = 125 ft. S, the shift for a 5-deg.
curve, with 25-ft. subchords = 0.57 ft. The ends
of the spiral will be 62^ ft. from P. C. C.
A = 6° 52^'. Deflections from P. S. to beginning
of the 8-deg. curve for every 25 ft. are 25', 55',
i°3o', 20 10', 20 55', taken from table for 25-ft.
subchords, on line for 3 deg. columns 4, 5, 6, 7
and 8, respectively. The deflections running the
curve the other way are S?1/^, i° 5<>', 2° 37%',
30 20', 30 57^', on line for 8 deg., columns 7, 6,
5, 4 and 3, respectively.
3. If the circular curve is not in even degrees,
the Holbrook method is still quite easy of application, despite the fact that such cases are not
covered by the tables. For example, if it is necessary to spiral a 40 12' curve using 300-ft. of
spiral, the spiral angle would be A° = N L D' -*-
200, or (iy2 X40 12') 1=6° 18'. The length of
subchord, L, would be 300 ft. divided by 40 12'
or 71 3/7 ft., the deflection from tangent to the SPIRAL TABLES
17
end of the first chord in minutes is one-tenth of
this chord, or 71/7 min., for the second four
times, or 28 4/7 min.; for the third, nine times of
i° 04 2/7't and for the fourth, 16 times, or
i° 542/7'. For the end of the spiral the deflection
would be (4 1/5)2 times 7 1/7 min. = 20 06'. This
last deflection would not have to be computed
since A —6° 18' and one-third of this is 2° 06'.
By subtracting the above deflections from the deflections for a 40 12' curve for distances of 71 3/7,
142 6/7, etc., would give the deflections from tangent at the sharp end of curve.
Spirals of any Length for any Degree of Curve.
—The use of the deflection tables for fixed sub-
chords, such as the twenty given herewith, is restricted to a length of spiral which is a multiple
of the length of chord of the table selected, since
the number of subchords is made equal to the degree of curvature of the adjoining circular curve.
There is a considerable demand, however, for
spirals of specific lengths, with or without regard to the degree of the circular curve, and in
order to make this particular easement curve applicable to all cases, and give greater flexibility
the accompanying table of coefficients has been
devised by Mr. Sullivan. With its use deflections can be obtained from any point of a given
number on the spiral to all the other points, and
any number of subchords may be chosen up to 25.
The resulting curves will have the same properties
as those taken from the deflection tables, since the
coefficients and method are based on the same
principles.
The table is based on finding the deflection for
the first subchord point on the spiral, and when 18
SPIRAL TABLES
this has been found the deflections for all other
points are secured by multiplying it by the tabulated coefficients. Since the theory of this curve
is based on the fact that each subchord represents a change of i deg. of curvature, the deflection angle for the length of spiral representing
this change of i deg. must first be found, and then
from it, as a second step, the deflection for the
length of subchord it is desired to use. Since
the deflections from the beginning of the spiral
vary as the square of the distance, the deflection
for the first length of subchord chosen is obtained
by multiplying the deflection for a change of I
deg. by the square of the ratio that the adopted
subchord length bears to the length of spiral for
unit change of curvature. The value so obtained
for the deflection to the end of the first subchord
is used as a constant by which the tabulated coefficients are multiplied.
The following example will make the method
clear: Required to spiral a 6° 15' curve, using
400 ft. of spiral. From the P. S. to the P. C.
the degree of curvature changes from zero to
6° 15', or at the rate of 1 deg. in 64 ft. The formula N2L/10, therefore, gives the deflection for
the first 64 ft. as 6.4 min. The spiral can now be
divided into as many subchords as is desired
(the table presented herewith being limited, however, to 25 subchords). Six different chord
lengths are chosen for illustration.
Deflection, first 16 ft. = (16/64)2 X 6.4 min. =     2/5     min.
"    20 ft. = (20/64)2 X 6.4    "    =     5/8
"    25 ft. = (25/64)2 X 6.4    "    =125/128    "
"    40 ft. = (40/64)2 X 6.4    "    =21/2
"    50 ft. =s (50/64)2 X 6.4    "    =3 29/32      "
"   100 ft. =(ioo/64)2X 6.4    "    =153/8 '
SPIRAL TABLES
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SPIRAL TABLES
In this case 25 chords, each 16 ft. long, may be
used and the deflections in minutes from any one
of these 25 points on the spiral to any other may be
obtained by multiplying the corresponding number
in the table by 2/5 min. This applies no matter
which one of these 25 points the transit is made
to occupy along the curve. In like manner the
table can be used for the other subchord lengths,
the deflections being easily figured for all intermediate points as well as for the center and end
points, which are here worked out for purposes
of illustration, the values being given in the following table.
Deflec. Deflec.
Using.              Argument         at Center, at End,
in Min.              in Min. in Min.
20   20-ft. ch'ds     5/8            100X5/8=62.5 400X5/8=250
16   25-ft.     "     125/12864X125/128=62.5256X125/128=250
10   40-ft.     "     2  1/2          25X2 1/2=62.5 100X2 1/2=250
8   50-ft.     "   3 29/32    16X3 29/32=62.5 64X3 29/32=250
4 100-ft.     "   15 5/8          4X15 5/8=62.5 16X15 5/8=250
As stated above, this table of coefficients is
based on the same principles as the other tables
given herewith. The theoretical developments of
this curve show that the deflection from the P. S.
to any point on the spiral is equal to one-third
of the deflection for a simple curve of equal
length, having a degree of curve equal to that of
the spiral at the point being set. The deflection,
therefore, for any chord length of a simple curve
of the same degree as the spiral at the end of
any given chord, will be three times the spiral deflection from the P. S. for this first chord. Hence
three is the constant to be used in making a universal table of coefficients, such as the one here
given.
To Line in a Spiral by Middle Ordinates.—Let I,
2, 3, 4, etc., Fig. 5, be points on a spiral, 30 ft. SPIRAL TABLES
21
3
So
; * 22
SPIRAL TABLES
s|s   t-|oo
3
bJ3
ul     ul      ^ SPIRAL TABLES 23
apart (one rail length). Let the middle ordinate
of the first 60 ft. of curve be b. This distance, b,
varies in different spirals and is given at the bottom of each table of spirals in decimals of a foot
and in inches. Now the offset from tangent
produced to point 2, 60 ft. from beginning of
spiral is approximately 2 2/3 b. The offset for
point 1 (30 ft. from beginning of spiral) is one-
eighth of the offset for point 2. With these two
points established, stretch cord between 1 and 3
and move 3 until the middle ordinate equals 2b;
then stretch cord between 2 and 4 and move 4
until middle ordinate equals 3b. Proceed in this
manner up to end of spiral. If end of spiral
comes at a rail-joint, then the middle ordinate of
a 60-ft. chord (30 ft. on the spiral and 30 ft. on
the regular curve) would be 3/4 b greater than the
last middle ordinate on spiral, and the middle
ordinate of 60 ft. of the regular curve is b
greater than the last middle ordinate on spiral.
If the regular curve is lined up and it is desired
to run the spiral toward the tangent, stretch a
chord 60 ft. long between two points, one 30 ft.
on spiral and one 30 ft. on regular curve; then
move points on spiral until middle ordinate is 3/4 b
less than that of the regular curve. Then move
up 30 ft. and move leading point until the middle
ordinate equals that of the regular curve less b.
Proceed in this way diminishing the middle ordinate by b at each move until the tangent is
reached.
To Replace Simple Curve by Spiralled Curve
and not Change the Length of Track.—Let A =
spiral angle, 0 = total angle between tangents,
D = degree of original simple curve, and D' =
degree of curve to b used,   (See Fig. 6.) 24
SPIRAL TABLES
Then D'=[i + 1.896 (A/0)2 — 0.672 (A/0)3] D.
Values of D'  for Different Ratios of A/0.
A/0 =
D'    =
A/e =
D'    =
A/e =
D'     =
1/20
1.0048D
1/8
1.0283D
i/3
1.1858D
i/i5
I.0086D
1/6
1.0496D
2/5
1.2603D
1/12
1.0128D
i/5
1.0705P
1/2*
1.3900D
1/10
1.0183D
i/4
1.1086Z?
i/9
1.0225D
3/io
1.1524JD
*In  this case the  simple curve is entirely replaced by
two spirals.
(Taken from Paper No. 927, by Mr. C. C.
Wentworth, in the "Transactions" of the American Society of Civil Engineers.)
Elevation of Track on Curves.—The general
rules for gaining super-elevation are given above
in connection with the selection of the length of
spiral. The accompanying diagram, Fig. 7, is
also used where the schedule time of trains exceeds 40 miles per hour, the elevation being attained according to either line of values of E.
The dotted lines show the result of putting up
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