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of the ^
'Unlverdtt^ of Wiecondtn
Utbrarp
of the
xantverdUi^ of Wisconsin
FIELD-MANUAL
FOR
RAILROAD ENGINEERS.
BY
J. C. NAGLE, M.A., M.O.E.,
Frofeaaor of Civil Bnoinwring in the Agricultural
and MechamccU Colleye of lexers.
SECOND EDITION, REVISED,
TOTAL ISSUE, ELEVEN THOUSAND.
; NEW YORK
; JOHN WILEY & SONS, Inc.
London: CHAPMAN & HALL, Limited
1913
Copyright. 1897,
BY
J. C. NAGLE.
THE 8CICNTIFIC PRCSS
ROBERT DRUMMONO AND COMPANY
BROOKLYN, N. Y. '
196790
JUL 20 1915
<^^f^^^Q
PREFACE.
Ea9E of reference and uniformity of notation are essential in a
book that is to be consulted in the field. With this in mind an
effort has been made in the following pages to secure a systematic
arrangement of the subject-matter and uniformity of terms and
notation. Except for a few cases Greek letters have been avoided
and a single letter is used to designate an angle. In so far as
practicable each figure is intended to be self-explanatory, so that
the explanations necessary in connection with the problems have
been reduced to a minimum. Algebraic equations stand each in
a distinct line, thus rendering them more easily read.
A knowledge of the elements of geometry and trigonometry has
been assumed, and only in the derivation of a few formulas in
connection with the theory of transition-curves will any higher
mathematics be needed. But these formulas may be accepted by
the reader who is unfamiliar with the calculus without in any
way affecting his ability to understand their applications or to
follow subsequent reasoning.
One can most readily turn to what he wants In a book after hav-
ing become familiar with its contents in the classroom. Keeping
this in mind this book has been written so that it may be used as
a text as well as for reference in the field. Wherever practicable
solutions to problems have been given in a rigid, general form,
followed by illustrative examples, so that the student need not
lose sight of the principle involved while following the solution
for a particular case. Wherever approximate solutions seemed
preferable they have also been given and their limitations pointed
out.
Free use has been made of the Table of Functions of a One-
degree Curve, thus reducing the labor of field computations. By
defining the degree of curve with reference to short chords for
m
IV PREFACE,
sharp curves — and, with tables of Radii, Long Chords, Mid-
ordinates, etc., based on appropriate equations — the errors result-
ing from assuming the radius to vary inversely with the degree
of curve will generally be found to be quite small.
Chapter I gives briefly the general method of making Re-
connoissance; Chapter II treats of Preliminary Surveys; while
Chapter III relates to Location.
Chapter IV, on Transition-curves, follows the method adopted
by Professor Crandall, and enables one to locate the transition-
curve with rigid accuracy where such is necessary. Approximate
methods are also given by means of which the curve may be as
easily located as any of the more limited easement curves ordi-
narily met with.
Chapter V, on Frogs and Switches, contains all that is necessary
for their location. The formulas have been arranged to give the
desired quantities in terms of the frog number whenever the re-
sulting equations would be easier of application than the trigono-
metric ones usually given. The turnout tables are unusually full
and give not only the theoretical lead but the stub lead as well,
from which the practical lead can be at once found when the
length of switch-rail is known.
Chapter VI, on Construction, tells how to set slope-stakes, and
gives simple methods for computing areas and volumes either
directly or by the use of tables. A short table of prismoidal
corrections is given for end sections level, and also a formula for
three-level sections, by means of which a suitable table ma-y be
computed if desired.
The tables at the end of this book have been arranged with a
view to ease of reference, for, whatever the character of the text,
the chief value of a field-book must depend upon the ease with
which the tables may be consulted and upon their extent and
accuracy. Table IX— Functions of a One-degree Curve— sepa-
rates the logarithmic functions on the one side from the natural
functions on the other and will be of assistance in locating these
tables. Table XVI — Transition-curve Table — reading lengthwise
of the page, likewise serves to separate the trigonometric tables
from the miscellaneous tables that follow.
Some engineers object to the use of logarithmic tables in the
field, but for them the natural functions are at hand; while for
those who prefer logarithms the five-place tables of logarithmic
sines, cosines, etc., will be found .easy to consult and interpolate
between.
PREFACE. T
All trigonometric tables are five -place, and others were carried
to as many decimal places as their character demanded.
Tables I, III, IV, and V have been computed to agree with
the definition of the degree of curve requiring curves sharper
than 7" to be run with chords less than 100 feet in length, as
described in the text. Tables XVII and XVIII were also com-
puted expressly for this book.
Tables VI and XXVII are from electrotypes from Carhart's
Field Book for Civil Engineers and were furnished by Qinn & Co.
Electrotypes of Tables II, X, XII, XIII, XIX, XX, XXIV, XXV,
XXVI, and also XVI — this last being from Crandall's book,
T7i6 Transition Curve — were furnished by John Wiley & Sons.
Of the others, some were arranged from standard tables and
others adapted in part and extended to increase their usefulness.
It will be noticed that vertical lines have been omitted wher-
ever practicable, thus rendering it easier to refer to the tables.
Acknowledgments are due my associate. Professor D. W.
Jpence, for aid in making the tabular computations and in reading
proof.
J. C. Naqlb.
CoLLKOB Station, Texas, May, 1897.
PREFACE TO THE SECOND EDITION.
In this edition some of the typographical and other minor errors
that appeared in the first edition have been eliminated. Tables
XXVIII and XXIX have been added in order to increase the use-
fulness of the book, and are from electrotypes of tabl. s in Traut-
wine's Pocket Book, A suggestion has been made by one who
has had occasion to use the tables quite freely that Table XIX be
extended so as to give quantities for variations of one tenth of a
foot in center heights, but such extension would have increased
the size of the book unduly. When closer approximations are
wanted than are given by Table XIX the area for the given center
height can be taken from Table XVII and by entering Table XX
iirith this as argument the quantity can be at once read off. For
center heights greater than those given in Table XVII we may
refer to books devoted exclusively to earthwork computations.
J. C. N.
OoLLBGB Station, Texas, January, 1899.
CONTENTa
CHAPTER I.
BSCONKOISSANCB.
AiKTiOLai i. Objects of Rbconnoissancb— How Mads.
IKCnON PAOK
1. Relative Importance of the Work of ReconDoissance and Location.. 1
2. Object of Reconnoissance S
3. The Instruments 2
4. Use of Maps 4
b. Making the Reconnoissance 4
CHAPTER II.
PRELIMINARY SURVEYS.
Abtxglb 8. Objects; Thb Field Corps; Duties of the Chief.
(I. Objects of Preliminary Surveys
7. The Exploration-line
8. Data Sought in Making Preliminary Surveys
9. The Field Corps
10. The Chief of Party, Duties of
Article S. The Transit Party.
A. DUTIES OF the MEB1BER8.
11. Composition of the Transit Party 8-
12. The Transitman 8
1&-17. Other Members of the Party g
18. Instruments 9
B. TRANSIT ADJUSTMENTS-^THE YERNIER.
19. Kindof Transit 8
80. To Ad just the Plate Levels IC
21. Parallax .• IC
22. To Adjust the Line of Collimation IC
28. To Adjust the Standards 11
vii
Till CONTENTS,
8BCTION . PAOK
84. To Ad j U8t the Level on Telescope o 12
25. Direct and Retrograde Verniers 13
2(J. The Least Count of a Vernier o Id
27. To Read a Vernier 14
C. ACCESSORIES.
(1«) The Oradienter,
28. Description and Method of Using Qradienter 14
(2?) The Stadia^ or Telemetmr.
29. Principle of the Stadia 15
30. Formula for Line of Sight Horizontal 15
31. Formulas for Line of Sight Inclined 16
32. The Instrumental Constant, To Find 17
33. Reducing the Notes 4 17
D. FIELD-WORK.
34. station Numbers 18
85. Hubs or Plugs 18
36. Reference-points 18
37. Alignment ..-1 18
88. Form of Transit Notes 19
39. Stadia Methods for Preliminary Surveys 19
B. OBSTACLES IN TANGENT.
41. To Pass an Obstacle by Means of Parallel Lines. . • 30
42. To Pass an Obstacle by Angular Deflections 20
43. To Measure across a River 21
Article 4. The Level Party.
44. Make-up and Instruments 23
45. Work of theLeveler 23
46. Work of the Rodman 23
ADJUSTMENTS OF THE LEVEL.
47. To Adjust the Line of Collimation 23
48. To Adjust the Level-bubble 24
49. To Adjnn the Wyes 25
B. THEORT OF LEVELING.
jO. True and Aooarent Level 25
51. The Ersor Due to Curvature 25
62. The Difference of Elevation of Two Points 26
C. FIELD-WORK.
63. The Datum 27
54 Bench-marks 27
66. Work in the Field 28
CONTBIJTa. IX
8ICCTION * PAGB
^6. Tbe Level Notes 28
57. Precautions when UsiDg Level S9
58. TheRod 29
Article 5. The Topooraphic Party.
69. Instruments Used; Area to be Mapped 30
60. Methods of Recording Data 80
61. Topographers' Field-sheets 81
62. Use of the Slope-level 31
63. Cross section Rods 82
64. The Transit and Stodia in Topographical Surveying 32
Article 6. Pbbuminart Estimates.
66. Map of Preliminary Lines 83
67. The Profile . ." 88
68. Preliminary Estimates of Quantities 88
69. Reportof the Locating Engineer •• 84
CHAPTER ni.
LOCATION.
Article 7. Projecting Location.
70. Problems Involved in the Paper Location 85
71 . Hints Regarding Methods of Projecting the Line 85
72. The Curve-protractor 86
73. Work in the Field 87
Article 8. Simple Curves.
A. definitions and formulas.
74. Deflnftions 88
75. To Find the Radius /?, the Degree of Curve Being Known 40
76. To Find the Length of Curve 42
77. The Functions of a One-degree Curve 42
79. To Find D, ^ and C Being Known 48
80. To Find the Tangent Distance T, I and R Being Known 43
81. To Find «, Given i and r 44
82. Given i and A to Find the Long Chord L.C? 44
83. Ordinates from Chord 45
84-86. To Find the External .B 48
87. To Find i?, iff and 1 Given 49
88. To Find r.JJ audi Given 49
89. To Find the Deflection Offset from Chord Produced 49
90. To Find the Tangent Deflection Offset 60
91. The Sub-tangential Deflection Offset 51
92. To Find the Tangent Offset z 52
98. Difference in Length of Arc and Long Chord 53
Z COKTEKTS.
B. LOGATINO SIMPLE OURTRS.
BICTIOir PAGE
94. To Locate a Curve with the Chain by Offsets from Chords Produced 56
95. To Locate a Curve by Offsets from Tangent 51
96. To Locate a Curve by Offsets from a Long Chord 58
97. To Locate a Curve with Transit and Chain 59
96. The Index-angle 60
99. Subdeflection-angles 60
100-lOL Transit Notes 61
C. OBSTACLES.
102. To Pass an Obstacle on a Curve 63
108. To Locate a Curve when the P. C. is Inaccessible 64
104. To Pass to Tangent when the P.T. is Inaccessible 67
106-107. To Pass a Curve through a Given Point 69
108. To Locate a Tangent to a Curve from an Outside Point 71
109. To Run a Tangent to Two Curves of Contrary Flexure 72
D. CHANGE OP LOCATION. _. ^!
110. To Locate a Curve Parallel to a Qiven Curve 73
111. To Change P.O. in Order to Make P.T. Fall in a Parallel Tangent. . , 74
112. To Change R and P.O. to make P.T. Fall in Parallel Tangent, on
Same Radial Line 75
113. To Find Change in P.O. or E for a Qiven Change in / 76
114. Required the Change in P.O. and B for a Qiven Change in /, the
P. r. Unchanged 77
115. To Find New Radius for a Qiven Change in T 77
116. To Find New R to Connect P. C. with a Parallel Tangent 78
Article 9. Compound Curves.
A. LOCATION problems.
117. Qiven Both Tangents and One Radius, to Find the Other Radius ... 80
118. Qiven One Radius, the Long Chord and the Angles it Makes with
Tangents, to Find the Other Radius and Central Angles 82
119. Qiven the Radii and Central Angles, to Find the Tangents, the Long
Chord,and the Angles it Makes with Tangents. 82
120. Given the Long Chord and Angles Made with Tangents, to Find
Both Radii when Common Tangent is Parallel to Long Chord 83
B. OBSTACLES.
121. ToLocateSecondBranch when P. C7. is Inaccessible 84
C. CHANGE OP LOCATION.
122. To Compound a Simple Curve so P.T. shall Fall in a Parallel Tan-
gent 85
128. To Find Change in P.CC, Necessary to Make P.T. Fall in a Par-
allel Tangen t 86
124. To Change P.C.C. and Second Radius so P.T. shall Fall in a Par-
allel Tangent, on Sjsme Radial Line 89
.1
CONTENTS. n
CHMITION PAGK
125. To Change P.C.C. and Second Radius to Cause P.T. to Fall at a
New Point in Same Tangent »1
126. To Substitute a Three-centered Compound Curve for a Simple One. M
127. To Substitute a Cunre for a Tangent Uniting Two Curves 05
Article 10. Track Problems.
188. Reversed Curves, Where to Use 98
129. To Connect a Located Curve with au Intersecting Tangent 97
130. To Locate a Y 100.
131. A Reversed Cur^e between Parallel Tangents 102
133. A Crossover between Parallel Tracks when a Fixed Length of Tan-
gent is Inserted 106
133. A Reversed Curve with Unequal Angles 106
134. A Reversed Curve between Fixed Points 106
135. To Connect Two Divergent Tangents by a Reversed Curve 107
136. To Change P.R.C. so P. T. shall Fall in a Parallel Tangent 106
187. To Find the Radius of a Curved Track 100
CHAPTER IV.
TRANSITION-CURVES.
Abticlb 11. Theory of the Tbansitiok<;drte.
188. Elevation of Outer Rail on Curves 110
139 Requirements of the True Traaisition-curve Ill
140. Notation Employed ill
141. Equation of Transition-curve. 112
142. Transition-curve Angle, / 114
143. Coordinates of Points 114
144. Deflection-angles 115
145. Explanation of Transition-curve Tables 118
146. To Unite the Branches of a Compound Curve by a Transition-
curve 119
147. Length of Transition-curve to be Taken 121
Article 12. Field-work.
▲. field formitlas.
14d. When to Use the Simplified Formulas 122
149. Simplified Formulas for Transition^urves 122
150. Offsets l"^
151 . Compou nd Curves 1
B. SETTING OUT TRANSITION-CURVES.
158. Location by Offsets 1
154. Location by Deflection -angles 1
155. Form of Transit Notes for Transition-curves. 1
XU CONTENTS.
Abticls 13. Transition curve Problems,
section paob
156. Tangent Distances and External for Equal Offsets 139
157. Tangent Distances, Offsets Unequal 190
158. Transition-curves Inserted without Changing the Vertex of Cir-.
cular Curve 131
159. Transition-curves Inserted with Least Deviation from Old Track.... 133
160. Transition-curves Inserted at Ends of Long Circular Curve, Cen-
tral Portion Undisturbed 133
161. Transition-curve Inserted at P.C.C. by Changing Radius of Second
Branch 136
163. To Insert Transition-curves at the Ends of Two Circular Curves
United by a Common Tangent 188
163. To Unite a Tangent and Circular Curve when the Offset Cannot be
Directly Measured 139
164. Inserting Transition -curves in Old Track 140
165. Remarks on Tabular Interpolations 140
CHAPTER V.
FROGS AND SWITCHES.
Article 14. Turnouts.
▲. TURNOUTS from STRAIGHT LINES.?
166. Definitions 143
167. To Find the Lead, I, and Radius, /?, in Terms of the Frog Number,
J\r, and Gauge, g 144
168. Given R and g, to Find N, i, and Frog-angle, F 146
169. To Find Theoretic Length of Switch-rail 146
170. To Find Lead and Number of Crotch-frog for a Double Turnout to
Opposite Sides of Main Track 147
171. To Find Turnout Radius and Lead of Crotch-frog in Terms of
Crotch frog Number 148
172. To Find Radius of Curve from Point of Middle Frog to Point ot
Main Frog, Given i^,, JV, and N' 148
173. Double Turnout to Same Side of Main Track 150
174. To Find Radius of Curve between Frog-points for a Double Turn-
out to Same Side of Main Track 151
175. To Unite Main Track with Siding, Reversing Point Opposite Frog . . 152
176. To Lay Out a Ladder-track 153
B. TURNOUTS FROM CURVES.
177. To Find Lead and Radius for Turnout to Concave Side of Main
Line ." 154
178. To Find Lead and Radius, Turnout to Convex Side 157
1 79. To Find Theoretic I-«ngth of Switch-rail 158
180. To Unite Main Track with a Concentric Siding 160
CONTENTS. Xlll
O. THE rfTUB LEAD.
SECTION PAGE
181. Definitions 16*2
18J. Given iV, <, and p, to Find the Stub Lead 262
183. Turnout Table and Explanation 163
184. To Stalce Out a Turnout 165
185. Curving Ralls 166
Abticle 15. Crossovers.
186. Crossover between Parallel Straight Tracks, a Tangent between
Frog-Rotnts 166
187. A Cro.«sover in the Form of a Reversed Curve 168
188. A Crossover with Fixed Length of Intermediate Tangent 168
189. A Crossover between Curved Main Tracks 168
Article 16. CROssmo- frogs and Crossimchbufs.
A. CROS8INO-FROOS.
191. Length of Rail Intercepted between Two Intersecting Straight
Tracks 170
192. Angles of a Set of Crossing frogs, One Track Curved 170
193. Angles of a Set of Crossing-frogs, Both Tracks Curved 171
B. CROSSING-SLIPS.
195. Length and Radii of Slip-rails, Both Tracks Straight 172
196. Length and Radii of Slip-rails, One Track Curved 172
197. Length and Radii of Slip-rails, Both Tracks Curved' 173
CHAPTER VI.
CONSTRUCTION.
Article 17. Definitions ; General Considerations ; Vbrtigal
Curves ; Elevation of Outer Rail.
<99. The Division Engineer 176
800. The Resident Engineer 176
201-204. Definitions 177
205. To Find the Grade- point, Longitudinal Slope Uniform 178
206. Vertical Curves iTe
207 Elevation of Outer Rail on Curves 182
206. Easing Grade on Curves 183
Article 18. Earthwork.
A. setting SLOPE-STAKES.
209. The Distance Out for Level Sections 186
210. To Find Position of Slope-stakes for Surface Inclined 184
211. Cross-section Notes j86
212. Irregular Sections 187
U3. Staking Out Openings 187
XIV CONTENTS.
BBCnON PAOB
214. Manner of Marking: Stakes 187
215. Shrinkage— Growth 187
216. Borrow-pits, Drainage of , etc 188
B. ABBAS OF SBCTI0N8.
218. Area of Three-level Section. 186
219. Area of Five-level Section 18£
220. General Formula for Areas 19C
221. Explanation of Table of Areas of Level Sections and the Three-
level Correction 191
C. VOLUME OP BARTHWOBK.
228. Where Cross-sections should be Taken 192
228. Volume by Averaging End Areas 192
224. ThePrismoidal Formula 193
fbib. Form of Record 195
226. The Prismoidal Correction 195
227. Computation of Volumes when Passing from Cut to Fill 198
228. Use of Tables of Volumes in Making Preliminary Estimates 199
229 Side Ditches 199
230. Earthwork on Curves 199
231. Overhaul 201
Article 19. Grade and Ballast Stakes, Cxtlvbrts, Bridgss,
AND Tunnels.
232. Grade and Center Stakes 202
233. Ballast-stakes 202
285. Openings of , for Culverts. Trestles, etc 1 202
236. Bridge Piers and Abutments .203
237. Tunnels 204
Article 20. Monthly and Final Estimates.
238. Monthly Estimates 206
239. Measurements for Earthwork..^ • 206
240. Classiflcation of Earthwork 206
241. The Progress Profile 207
24*2. Masonry Estimates 207
213. Bridge Estimates 207
244. Track Material 207
245. Blank Estimate Sheets 208
246. Monthly Payments 208
247. Extras 208
248. Final Esthnate 206
249. Acceptance 809
TABLES.
Table Showing Length of _TransItioD-curve to be Taken 12l
Table of Values of ^- Vpt for Stub Lead 163
Turnout Table 164
Table of Corrections for Vertical Curves 181
CONTENTS. X7
PAGK
'Table of Elevation of Outer Rail on Curves 182
Table of Prlsmoidft* Oorrectioiis for Level SectionB 198
I. Radii of Curves 212
. II. Minutes in Decimals of a Degree 316
in. Tangential Offsets 216
IV. Long Chords and Actual Arcs 217
V. Mid-ordinates to Long Chords 218
VI, Logarithms of Numbers.. 220
VII. Logarithmic Sines and Cosines 238
VIII. Logarithmic Tangents and Cotangents 253
IX. Functions of a One-degree Curve 268
X. Natural Sines and Cosines 296
XI. Natural Secants and Cosecants 307
XII. Natural Tangents and Cotangents 320
xni. Natural Versines and Exsecants : 332
XIV. Coordinates for Transition-curves 355
XV. Deflection-angles for TransitionnSurves. ... 356
XVI. Transition-curve Table 358
XVIL Area* of Level Sections 371
XVIII. Corrections for Three-level Ground 375
XIX. Cubic Yards per 100 ft. in Terms of Center Height 376
XX. Cubic Yards per 100 ft. in Terms of Sectional Area 382
XXI. Rise per Mile of Various Grades 386
XXII. Slopes for Topography 387
XXIII. Material Required for One Mile of Track 387
XXIV. Mutual Conversion of Feet and Inches into Meters and Centi-
meters 388
XXV. Mutual Conversion of Miles and Kilometers 389
XXVI. Length of V Arc of LatitudA and Longitude 389
XXVn. Trigonometric and Miscellaneous Formulas 890
XXVIII. Square Roots and Cube Roots of Numbers from .1 to 28 895
XXIX. Squares, Cubes, Square Roots, and Cube Roots, of Numbers
from! to 1000 896
FIELD-MANUAL FOR RAILROAD
ENGINEERS.
CHAPTER I.
REC0NN01S8ANCE.
Article 1. Objects op Reconnoissance— How Made.
1. The question of the selection of the proper route for n line
of railway is essentially an economic one, involving not only tlie
cost of construction, but of maintenance and operation, and a
consideration of the immediate and future traffic likely to pass
over the completed road.
The engineer upon whom devolves the duty of making the
surveys for a railroad is not often called upon to determine
whether it should or should not be built, though his prelim iuaiy
estimate may decide those whose duty it is to do so : the problem
confronting him is 7iow to secure the best line, answering a gicen
purpose, for the least cost. Keeping in mind the proper working
of the completed road, the problem may be divided into two gen-
eral parts :
First, The selection of the general route between terminal
points, and in some cases the selection of the terminals them
selves.
Second, The fitting of the line to the ground in such a manner
as will render the cost of constructing and operating the road a
minimum.
The first is by far the more important and difficult operation,
requiring the highest giade of engineering skill— a fact too sel-
dom recognized by those selecting engineers for this work. The
acquirement of the necessary skill can result only from long
practice and close observation, coupled with the ability to fully
2 A FIELD-MANUAL FOR RAILROAD ENGINEERa
grasp and weigh all the complex features of the questioD. A
passing reference only can be made to it in this little volume,
which is intended to furnish hints and aids to the better execution
of the second part. For the beuefit of the beginner who has to
do with the location and construction a few definitious and hints
relating to reconuoissance will be given before going on to the
special problems arising in the work of the railroad engineer.
2. The Reconnoissance is a rapid, general survey of the area
through which the proposed railroad must pass, made only with
such instruments as can be easily carried, and which should ena-
ble the engineer to restrict the more accurate instrumental work
that follows to one or two general lines. The time required for
this part of the work will in general be only a small fraction of
the time consumed in location, involving the service of very few
men; yet there is no part of the work more rapidly and im-
properly done — not always because the engineer in charge under-
estimates its importance, but because he is not usually allowed
sufllcient time in which to study thoroughly the area under con
sideration.
Properly the reconnoissance includes the determination of the
terminal points of the road, but the locating engineer is usually
relieved from the necessity of selecting these points, and the
question reduces to that of finding the best available line which
admits of being built, maintained, and operated at the least cost
betioeen two given points.
The reconnoissance must be made over an area — not a line or
lines. Even what seems the most unpromising portion should
be carefully studied, for the engineer can never be satisfied he
has selected the best route until he has convinced himself by care-
ful study that all others are inferior. Too much haste on recon-
noissance means either a poor line or a much greater expenditure
of time and money on the preliminary. No amount of notes or
topography can take the place of an intimate personal knowledge
of the problems to be encountered, and hence the reconnoissance
and preliminary survey should be made by the engineer who is
to locate the road.
3. The Instruments needed will rarely be more than a pocket-
compass, hand-level, aneroid barometer, field-glasses, and some-
times a pedometer or an odometer.
(a) The Pocket-compass is used to obtain the magnetic bear-
ings of lines and the angles tlicy inake with each other.
RECOKNOISSANCB. 9
(b) The Hand-level enables one to obtain differences of ele-
vation between points not far apart.
(e) The Aneroid Barometer gives approximate heights of the
mercury column, and serves to roughly determine the difference
of elevation of given points. In addition to the scale giving
readings in inches, it should have also a scale graduated to give
readings in feet. If two aneroids, which have been previously
compared, are read simultaneously, one at each of the points
whose difference of elevation is desired, or if the same aneroid is
read at each successively at a short interval of time, during which
the atmospheric pressure has not sensibly altered, we may find
the difference of elevation by the formula*
d = eQm(}og H-iogK)(i + I^±^^), . . (1)
in which d is the difference of altitude in feet, H and h the
barometric readings in inches — the logarithms being of the com-
mon or Briggs kind, 7^ and t the temperatures of the two stations
in Fahrenheit degrees.
If the sum of the temperatures, T+t,m taken as 105', formula
(1 ) reduces to
ci = 63000(logir-logA) (1')
Example. — The reading of the barometer at the foot of a
mountain is 28.8 inches, and at the top 26.7 inches. Required
the height of the mountain.
By (1'). d = 63000 (log 28.8 - log 26.7) = 2071 feet.
The effect of. temperature on the metal of the instrument
should be considered in the barometric formula when very pre-
cise work is to be done ; but this correction, being small, may be
neglected in the rough work of reconnoissance, particularly since
the makers of the instrument construct it in such a way as to
compensate, as closely as possible, for such changes of tem-
perature.
(d) The Pedometer is an instrument which automatically
counts the number of steps made by a person when the instru-
ment is attached to his belt ; then, knowing the average length
of step, the distance passed over can be readily computed.
The Odoftieter registers the number of revolutions of a wheel
to which it is attached, and the number of revolutions multiplied
Uy the circumference of the wheel gives the space pdlssed over.
* See Plyinpton*8 Aneroid Barometer, p. 38, for formula (1).
4 A FIELD- jIAKUAL FOR RAILROAD ENGINEERS.
4. The Map. — Before beginning the reconnoissance the engi-
neer should provide himself with the best available map of the
region to be traversed ; if this is a topographic one, he can at
once determine from it the lines that are likely to justify an
examination ; and even if it is only a sketch-map, he can get
material assistance by observing the courses of the streams and
remembering that their positions indicate the relative elevations
of the portion of the region through which they flow. Thus the
large streams follow the lines of least elevation, and the manner
in which the lateral streams unite with the principal one indi-
cates the general trend of the terrain. Two streams flowing
nearly parallel approacli or recede from each other according as
the intervening land diminishes or increases in altitude. Two
streams flowing away from each other on opposite sides of a
divide, and having their source therein, approach each other
closest at the point of least elevation, and indicate the position of
a pass or the lowest point of the dividing ridge. The study of
any good contour map covering sufficient area will illustrate the
laws governing the courses followed by streams.
The elevations of a few correctly mapped points, when obtain-
able, from the map or otherwise, serve as a guide in tentatively
fixing on the maximum gradient to be employed and the amount
of development needed.
A skillful engineer will thus be enabled to project his lines
with sufficient accuracy to enable him to select on the ground the
most feasible route or routes for his preliminaries in the least
possible time. He should guard against the conviction, however,
that it is unnecessary for him to look elsewhere than along the
projected routes ; for the inaccuracies of the map, local peculiari-
ties, the nature of the excavation and embankment, the number
and cost of bridges and other mechanical structures, — all these
may conspire to make the most promising map-line inferior to
some other whose advantages * have to be sought for on the
ground.
6. Having tentatively decided on the limiting grades and cur-
vature to be employed, the engineer goes carefully over the
ground, examining the entire area that seems likely to afford
passage, in order to determine whether a suitable line may be
secured for the grades and curves previously assumed. With his
pocket-compass he takes the bearings of lines, and by means of
the hand-level and aneroid determines differences of elevation.
RECONNOISSANCE. 5
Distances are estimated by the eye, paced, and the count taken
irom the pedometer, or, if the country admits of the use of a
vehicle, taken from the odometer readings. If a well-gaited
saddle-horse is used, very good results may be gotten by timing
him, or by the use of the pedometer if bis stride is uniform.
But in all cases much dependence must be placed on the ability
to estimate with the eye differences of elevation and distances.
The ability to do this with even reasonable accuracy comes only
from long practice and careful observation, even to the most
gifted in this uespect. New and unexpected conditions some-
times deceive even the most practiced eye, but uuder ordinary
conditions almost any one can train his eye to estimate horizontal
distances fairly well. Vertical heights are more deceptive, pos-
aibly because we have less practice in this line, and the mind
seems naturally to exaggerate the vertical as compared with the
horizontal ; practice, however, will enable us to make allowance
for the natural tendency to overestimate heights and slopes.
The ground should be gone over in both directions, for the ap-
pearance may be quite different when approached from different
quarters. Ruling points, such as a pass in the mountains, the
crossing of a large stream, or a town or city through which the
road must be built, serve to reduce the problem to a number of
special ones, each having its own solution.
In a mountainous region offering a limited number of possible
routes, but heavy -construction work, it may often happen that
the location of a line is a much less diflBcult operation than in an
open, rolling country offering a score of possible lines, between
which the engineer making the reconnoissance must decide,
selecting only those that in his judgment seem to justify an
accurate instrumental survey.
The engineer must keep constantly in mind all the factors of
the general problem of economic location and maintenance, and
successful operation of trains. One line may cost more for con-
struction and maintenance than another, but less for operation,
or may invite less traflfic. In all cases, however, the question
of grades, curvature, length of line, earthwork, and mechanical
structures are the controlling elements to be considered.
Having decided upon the route or routes over which to run
preliminaries, these are marked on the map, and the engineering
party organized and put in the field, with all the ' necessary
instruments.
CHAPTER n.
PRELIMINARY SURVEYS.
A RTicLE 2. Objects; The Field Corps ; Duties op the Chief.
6. The Ol]jects of the preliminary surveys are to secure all the
data necessary to determine which one of the routes selected on
reconnoissance is the most feasible, all things considered, and the
approximate cost of construction. In rough country it will be
economical to make two, or even three, surveys over the route se-
lected for location before beginning to place the line in the position
it is finally to occupy. The first of these is often omitted, and is
called an " exploration-line " ; it will frequently save the making
of the more expensive "preliminary" over one or more of the
routes.
7. The Exploration-line may be made with either transit or
compass, and consists of a rapidly run line, made for the purpose
of determining the maximum curvature and gradients with which
to project the preliminary. It will not be necessary to make a
detailed study of the region at this time, the distances and eleva-
tions, with such sketch topography as may be easily taken, being
all that is needed. The magnetic bearing of lines is taken by
the compassman, and the chainmen align each other with the fiag
set by the flagman. As the progress of the level party will be
slower than that of the compass party, it will be economical to add
an extra rodman, and sometimes a recorder. The compassman
may sketch in the features adjacent to the line while waiting for
his chainmen, who may be either in front of or behind the com-
pass.
The stadia method of surveying — to be spoken of later — would
seem to offer exceptional advantages for this work — only three or
four men being needed in addition to the chief. With it, by set-
ting the transit over alternate stations, very rapid progress may be
made, and obstacles avoided with as much or greater ease than
with the compass.
The exploration -line will more than pay for itself in showing
6
PRELIMINARY SURVEYS. 7
what routes it will be unnecessary to make preliminaries over,
and in indicating the most feasible one. It should be run over all
the routes selected on reconnoissance.
8. The Preliminary Survey follows the exploration, or, when
this is omitted, comes next after the reconnoissance. It may, with
advantage, be made |in two parts — first and second preliminary>
It is made with such instrumental accuracy as the nature of the
case may demand, sufficient data being obtained to determine the
best line on which to locate and the approximate cost of construc-
tion. The rapidity with which this work can be done will depend
on the care with which the reconnoissance was made. The pre-
liminary line should approximate, as closely as the eye can deter-
mine, to the position the located line should occupy, and forms the
base on which the topographic work rests. In reasonably easy
country, where exploration-lines have been run, one preliminary
should suffice for each route, but in difficult regions it will be best
to run a second preliminary. If portions of the route are easy, fol-
lowed by difficult parts, it will often be sufficient to " back up '*
and re-run the difficult portion until a reasonably satisfactory line
has been obtained.
9. The Field Corps consists of a chief of party, transitman,
leveler, rodman, two chainmen, rear rodman or "back- flag,"
stakeman, and two or more axemen. If a topographic pai*ty is
added, as it should be in any but the easiest country, there will be
also a topographer with two or more assistants. A cook and
teamster will be needed with the camp outfit.
The corps is usually divided into the following parties :
(a) The Transit Party.
(6) The Level Party.
(c) The Topographic Party.
10. The Chief of Party receives his orders from the chief en-
gineer, or such other officer as may be in charge, directs the mo-
tions of the surveying corps, and is responsible for their conduct
and progress. He provides accommodations and supplies, pays all
expenses, taking receipts or vouchers for all outlays — ^in dupli-
cate when required. In the less thickly populated sections he
must provide tents, wagons, cook, and all necessary camping outfit
and supplies. He must direct the field operations in person, keep-
ing in advance of the transit, establish turning-points or hubs,
and direct the transitman in the proper course. He should keep
8 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
a record — or direct the transitinan and topographer to do so— of
the character of earthwork likely to be encountered, the places
where drains, culverts, bridges, cattle-guards, etc., are needed;
the nature of material for embankment, piling, etc., adjacent to
the line ; the probable amount of clearing and grubbing, and all
other features likely to affect the cost of construction. He should
see that the names of property owners and residents along the
line and the positions and bearings of property lines, when
possible, are noted.
He should have authority to discharge assistants — except transit-
man, leveler, and topographer — whose services are unsatisfactory,
and in many cases it will be best for him to have entire control,
engaging or discharging any member of the corps as circumstances
may require.
Article 3. The Transit Party.
A. Duties of the Members.
11. The Transit Party should consist of a transitman, head
chainman, rear chainman, rear flagman, stakeman, and as many
axemen as may be required — rarely less than two even for open
country.
12. The Transitman cares for his instrument, keeping it in aa-
justment; directs the chainmen into line; notes the angle between
successive tangents as read on plates; notes also the bearings of
tangents, of highways, streams, and property lines (on location),
with the plus at which the line crosses them. If there is no
topographic party he must make sketches, on the right-hand page
of note-book, of the surface features adjacent to the line; the
red line down the middle of page represents the transit line,
whether straight, broken, or curved, to which the sketches are
adjusted. He must see that the axemen keep in line, in order
that no unnecessary chopping may be done. Large trees need
rarely be felled on preliminary, even when a given general course
has to be followed, for small angles may be turned to avoid them,
the dellections to right being made to approximately balance those
to left.
When the chief of party is absent the transitman is ranking
man, and will take temporary charge.
13. The Head Chainman carries a range-pole or "flag," and
drags the chain, which he must see is straight and horizontal
PRELIMINARY SURVEYS. 9
when setting a point for a stake. He directs the stakeman where
to drive his stake, calling out the number after the rear cliainman
has read and called out the number on his stake; he keeps the
axemen in line by setting his flag and going ahead, directing them
cohere to cut by keeping them in line with the flag and transit.
The speed of the party is dependent on the rapidity and accuracy
with which he can set his flag in position, by ranging with stakes
already set between him and transit, and in seeing that the
axemen make all their work count.
14. The Rear Chainman must be careful to hold his end of
the chain in the proper place, and that it is kept straight and taut
when the head chainman is setting a stake. He must give all
pluses, note -the number on each stake as he comes up to it, and
see that the stakeman has marked it correctly; he must make a
note of pluses for roads, fences, streams, etc., to be given to the
transitman later on.
16. The Stakeman must keep himself supplied with stakes
about 1|" X 3" X 24", marking the number on them plainly, and
driving them as directed by the head chainman.
If sawed stakes are not provided, he must cut the stakes and
face them for the numbers. He must keep on hand a number of
plugs or **hubs^" to be driven flush with the ground and having
the point where flag rested marked with a tack. About ten oi
twelve inches to the left of and facing the hub a guard stake is
driven, on which is marked the station number, and which enables
one to And the hub at any time.
16. The Axemen do all necessary clearing and chopping in
order that the transit and level parties may have a clear sightway,
and yet restrict the work of clearing to a minimum. One of them
may be detailed to keep the stakeman supplied with stakes.
17. The Rear Flagman holds his flag on the last turning-
point for the transitman to use in back-sighting.
18. The Instruments used by the party are the transit (or
compass), one-hundred-foot chain or tape, range-poles, and the
necessary axes and hatchet for axemen and stakeman.
B. Transit Adjustments — The Vernier.
19. For railroad work the transit is usually plain, but it is
often convenient to have a clamp and tangent movement to tele-
10 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
scope, a vertical circle, a level on telescope, stadia wires, and a
gradienter; the solar attachment will rarely be needed.
20. To Adjust the Plate Levels.— The axis of the instrument
is set at right angles to the plates by the manufacturer, so that
when the axis is made vertical the plates will be horizontal.
In making adjustments remember that a complete reversal
always doiibles any existing error.
Place the bubble-tube parallel to a diagonal pair of leveling-
screws, and bring the bubble to the centre of its run. Revolve
the instrument 180" on the vertical axis, and the level- tube will
be parallel to the same pair of leveling-screws as before, but
reversed. If the bubble has moved from its central position
bring it half-wAj back by means of the capstan- headed screws at
the ends of the tube. Relevel and repeat until the bubble remains
at the centre after reversal. Do the same for the other bubble.
Both bubbles should remain at the centres of^their tubes during a
complete reversal.
21. Parallax is an apparent movement of the cross- wires with
respect to the object sighted when the eye is moved from side to
side of the eyepiece, and shows that the image does not fall in the
plane of the cross-wires. In precise measurements it should be
removed before making an observation with the telescope. To do
this, first bring the cross-wires clearly into view when the object,
glass is turned towards the sky, then, when sighting an object,
note if there is any relative movement of cross- wires and image
when the eye is moved from side to side at the eyepiece ; if there
is, refocus the object-glass until this movement disappears.
22. To Adjust the Line of Collimaiion is to make the line
joining the intersection of cross-wires and optical center of objec-
tive describe a plane perpendicular to the horizontal axis of instru-
ment.
First Method. — Level the instrument and clamp the move-
ments on vertical axis. Sight some well-defined object distant
about the length of an average sight, and in the same horizontal
plane as telescope. Reverse the telescope on its horizontal axis,
and fix a point about as far from instrument as first point, and in
the same horizontal plane. Revolve the instrument on its vertical
axis and sight the first point; then reverse the telescope and note
if line of sight cuts the second point. If not, loosen the capstan-
headed screws holding cross- wire ring and move the vertical wire
PRELIMINARY SURVEYS.
over ons fourth the apparent error — since there were two revei i
— remembering that the image of the cross-wires is inverted, w i
that of the object appears in its trne position. Test by repetit :
SscoND Method. — If the limb graduations can be relied
they may be nsed in adjusting the vertical wire. With the ins i
ment level sight a well-defined point, then revolve 180° by vern (
plate, reading both verniers; reverse telescope, and note if lint
sight cuts the point. If not, correct one half the apparent erroi
moving diaphragm ; then test by repetition.
The manufacturers adjust the object-glass slide so that the :
jective travels in the telescope axis, and this adjustment is ;:
liable to serious derangement. It is well, however, to sometic]
test by adjusting the line of collimation for both near and distii
objects. If not correct for both, move the ring which guides 1
rear end of object-glass slide until the adjustment is correct i[
both positions.
Next make the vertical wire vertical by noting if it coincid
throughout its length with a plumb-line, or by observing if it c
vlates from a point, on which the intersection has been fixed, wh
the telescope is elevated or depressed. Any error is corrected
turning the ring after slightly loosening the screws holding it.
The horizontal wire should also be adjusted so that the vdXk
section of the cross-wires will be in the axis of the telescope ;
the transit is to be used as a leveling instrument this adjustme
is essential.
Drive a stake close to the instrument, and with the telesco
clamped as nearly horizontal as can be conveniently done reac
rod held on top of the stake ; about 300 feet distant, and in li
with first stake and instrument, drive a second stake and read t
rod on it. Revolve 180** on vertical axis, reverse the telescope a
bring the horizontal wire to the former reading when the rod
held on first stake ; if the reading on the second stake is not t
same as before, correct oiie half the apparent error by moving t
cross- wire ring. Repeat as a test. The vertical wire should age
be tested lest the movement of the ring may have caused it
change.
23. To Adjust the Standards is to make the plane describ
by the line of collimation vertical. Set up the transit about as i
in front of some high building, or other tall object, as the high<
point that can be sighted is above the base. Level the instrumc
and fix the intersection of the cross- wires on the highest point tl
12 k FIELD-MANUAL FOR RAILROAD ENGINEERS.
can be easily sighted. Depress the telescope and fix a point near
the base of the building at about the height of the telescope. Un-
clamp and revolve on the vertical axis until the telescope reversed
cuts the lower point. Clamp the plates and raise the telescope
until the cross- wires are at the height of the upper point. If they
gut it the standards are in adjustment. If they do not, bring
them half-way back by means of the adjustable screws at the top
of one of the standards. Repeat ^.s a test.
24. To Adjust the Level on Telescope is to make the bubble
stand at the center of its run when the line of sight is horizontal.
Bring the telescope as nearly horizontal as may be convenient, and
take readings on the tops of two pegs in the same vertical plane
with, and equidistant from, the instrument — say 300 feet. The
difference of readings will equal the difference of elevation of the
pegs; this difference may be obtained with the wye-level if pre-
ferred.
Move the instrument to a point beyond one of the pegs and in
line with both. Set up as close to nearer peg as convenient, but
not so close that the rod cannot be easily read. Bring the tele-
scope as nearly horizontal as possible, and read on both pegs. If
the difference of readings equals their difference of elevation the
line of sight is horizontal, and the bubble may be brought to the
center by means of the adjustable screws attaching the level- tube
to the telescope. If this is not the case, we must set the telescope
so the reading on second peg equals the reading on first peg plus
the difference of elevation ; then read again on first peg and pro-
ceed as before until the condition is satisfied. Or we may proceed
as follows :
In Fig. 1 let the transit be at 0, and A and B be the pegs. AC
is a horizontal through A, so that CB is the difference of elevation
of A and B, Suppose line of sight to cut the rods at ^and i>,
we must find DG bo that the target may be set at the proper read-
PRELIMINARY SURVEYS. 13
ing to make the line of sight horizontal. Let 0F= a, FO = 6,
EA =r, DB = r\ GB = k. Draw DH parallel to GA and 00)
then JSH=r + k — r',
From similar triangles
a4-b
--m-
Set the target at a reading GB = QD + r*, sight to G, and the
line of sight will be horizontal. Bring the bubble to the center of
its run while the telescope is in this position, and the adjust-
ment is complete.
If desired, a correction for the curvature of the earth and re-
fraction may be introduced, but for short sights this is a useless
refinement.
26. The Vernier is an auxiliary scale for measuring smaller
divisions than those graduated on the limb. There are two
classes, the direct-reading and the retrograde, according as the
fractional parts of limb readings are taken on that side of the
zero of vernier scale towards which the vernier has moved with
respect to the limb, or the reverse. On the direct vernier a cer-
tain number of divisions on the vernier equals the same number
of divisions on the limb, less one ; on the retrograde there is one
more division on limb than on vernier when the same space is
covered by both.
26. The Least Count of a vernier is the smallest subdivision of
limb graduation that can be read by it, and equals the difference
of one space on limb and one on vernier.
Let I = value of one space on limb ;
V = value of one space on vernier ;
n = number of spaces on vernier.
Then for the direct vernier
nv = (71 — 1)1 ;
from which we get the least count,
n
For the retrograde vernier
nv = (n-\- ly.
14 A FIELD-MANUAL FOR RAILROAD ENGINEERS,
from which the least count is
the same result as found for the direct vernier.
So, to find the least count : Divide the value of one limb space h^
the number of spaces on the vernier.
For example : If the limb of a transit is divided to half-degrees
and the number of spaces on the vernier is 30, the least count
will be J divided by 30, or ^ of a degree — that is, 1 minute.
27. To Read a Vernier, take the number of the last division on
limb back of the vernier zero, then look along the vernier until a
line is found to coincide with a line on the limb ; add the number
of this vernier line, multiplied by the least count, to the scale
reading, and the result will be the required reading.
C. Accessories,
(V) The Qradienter.
28. The Gradienter consists of a tangent-screw having a
micrometer-head, attached to one of the standards of the transit
and capable of being clamped to the horizontal axis of the tele-
scope. It is used — as its name indicates — in running grades, and
it accurately measures a small vertical angle in terms of its tan-
gent. The screw is so cut that one revolution moves the tele-
scope through an angle whose tangent at one hundred feet from
the instrument has a certain value, usually one foot. The grad-
uated head is divided into 100 parts, so that one division corre-
sponds to 0.01 ft. at 100 feet from instrument.
To run a given gradient, bring the telescope level and read the
micrometer-head of screw; then turn the screw as many divisions
as there are hundredths of a foot rise or fall in 100 feet, and with
a target set at the height of the horizontal axis, points on the
surface corresponding to the given grade can be found.
For example : To run a 0.76 per cent grade, move the microm-
eter milled head 75 graduations from the horizontal.
When used as a Telemeter, we may either measure the space
on the rod moved over by the line of sight for a given number of
revolutions of the screw, or we may note the number of revolu-
tions required to move the line of sight over a certain space on
rod. The second method is the more accurate, particularly for
long sights.
PRELIMIKARY SURVEYS.
15
(2°) The Stadia, or Telemeter.
29. The Stadia is an instrument for determining the distance
of a point from the observer by noting the space intercepted on a
rod by a given visual angle, as determined by two auxiliary wires
parallel to, and equidistant from, the horizontal wire of the transit
telescope. When used with an ordinary leveling-rod the wires
should be adjustable ; if they are fixed (which for some reasons
is preferable), the rod must be graduated to correspond. In
addition to the distance of a point from the instrument, the differ-
ence of elevation is determined by observing the angle made by
line of sight with the horizontal when the middle horizontal wire
cuts a point on the rod as high above the ground as is the centre
of the telescope.
The horizontal position of the point is determined from its
magnetic bearing, or the azimuth of line of sight with reference
to some fixed line, usually the north-south line.
30. Line of Sight Horizontal.— In Fig. 2 let a and b be the
stadia wires, AB the intercept on the rod. The secondary axes
A
Fio. 2.
aA and bB pass through the optical center 0, Let h = ab,
r = AB, d = distance of cross- wires from objective, JD ~ distance
of rod from objective.
From similar triangles,
d
From optics,
d "^ d'
1_
'r
in which /is the focal length of objective.
Eliminating d from these two equations.
j>=f+
f.
16 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Let c be the mean distance of objective from center of instru-
ment. Adding this to D gives, for the distance of the rod from
the center of the instrument,
l = c+f+ir.
(2)
J- may be made constant, when (2) becomes
1 = a + kr.
(2)
31. Line of Sight Inclined. — When the line of sight is not
level it is difficult to hold the rod perpendicular thereto ; hence
the rod is held vertical, the angle of inclination measured, and a
correction applied. In Fig. 8
Q.
let r = CD be the reading on rod held vertical ;
r' = FE, the reading perpendicular to line of sight ;
// = AO, the horizontal distance from A to B ;
V= BO, the difference of elevation between A and B ;
n = BAG, the angle of inclination of line of sight.
Assume angles AFB and AEB = 90°, from which they rarely
differ more than 15' to 17'. Then, since FBC = n,
r' = r cos n.
PRELIMIKARY SURVEYS. 17 ^
Hence AB = a-\- kr cos n.
From triangle ABG
H = AB cos n
.•. H = a cos n-{-kT cos' n (8)
F=-4-Ssin«.;
. •, V =■ a^m n-\-kr sin w cos n.
But 2 sin n cos n = sin Sn. .
Hence F = a sin n + \kr sin 2» (4)
32. The Instrumental Constant a [r= c +/ of (3)] maj be
found by measuring the distance from center of instrument to
mean position of objective, which equals c ; then focusing on a
very distant object, preferably a star, and measuring from center
of objective to plane of cross- wires, which equals/. The sum of
these distances is a in formulas (3) and (4).
If the stadia wires are fixed, k may be found by measuring for-
ward on level ground the distance a from plumb-line, and from
this point a further distance b ; then note carefully the stadia
reading r when the telescope is level. Then, remembering (2)',
a-\'h =za-\-kT.
. '. A; = — , a constant ratio.
r
If the stadia wires are adjustable, we may so adjust k that any
desired reading may be had for a given length of base. A con-
venient value of k Is 100, which corresponds to an intercept of
1 foot on the rod at 100 feet from a point a feet in front of the
instrument, 2 feet at 200 feet in front, etc.
33. A Stadia Table based on formulas (8) and (4) is published
by the D. Van Nostrand Company in Winslow's Stadia Surveying ^
and can be used more rapidly than the formulas. Johnson's Re-
dtiction Diagram, by John Wiley & Sons, gives values of // and V
graphically. Colby's SHde-rule, manufactured by Mahn & Co.,
St. Louis, gives values of V for distances in feet, yards, or meters
to tenths of a foot, and can be used with great rapidity.
18 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
D. Field-work.
34. Station Numbers should begin with zero for the ioitial
stake, and are marked on rear side of stake, from the top down-
ward, the number of the preliminary, A, B, C, stc, being marked
on the forward side. The marking should be with kiel, or crayon
that will withstand the action of sun and rain. Stakes may be
set every hundred feet or only at even stations, as preferred.
36. Hubs, or Plugs, are transit turning-points, and are short,
flat- topped stakes driven into the ground flush witli the surface.
The flag is held on the top and carefully aligned, the position of
the point being marked by a tack. A special tack with coucavt
head offers a foothold for point of flag when used in backsight
ing.* About 10 inches to the left of and with numbered sid«
facing the hub is driven a guard-stake to mark its position.
36. Reference-points are two or more hubs, with guard -stakes
in each of two lines making a good intersection angle at thb
point whose position they serve to locate. They should be driver^
beyond reach of disturbance, and are used in replacing a dis
located hub.
These need rarely be used on preliminary.
37. Alignment. — It is not intended that the preliminary aniV
location lines occupy exactly the same position ; hence consider-
able latitude is allowable in the size and number of angles
turned, care being taken, however, that the maximum curvature
need not be exceeded on location. Large trees and other obstruc-
tions may be avoided by turning a small angle until the obstacle
has been passed, then m&king a deflection in the opposite sense.
Bearings of tangents are taken with the needle, to serve as a
check on the angle read on the plates.
In easy country not requiring a topographic party large angles
should not be turned, a succession of small ones with short inter-
vening tangents being substituted in order to make the prelimi-
nary profile approximate more closely to the location profile.
These short tangents may conveniently be the long chords of the
curve that is to follow.
* Such a tack is manufactured by the A. S. Aloe Co., St. Louis.
PRELIN^INARY SURVEYS.
19
38. The Transit Notes may be kept in the form below, which
shows both pages of the note -book. The notes run from the
bottom up, the right-hand page being reserved for sketches ; the
red line up the middle of the page represents the transit line,
whether straight or broken, to which the sketches must be
adjusted.
Sta.
Angle.
68
670
ao^cxL.
66
65
64
630
6«2'R.
62
61
Calculated
Course.
N. lo 48' W.
N. 180 12' E.
Magnetic
Course.
N. !• 4^ W.
N, 18« 15' E.
Bemarks and Sketches.
39. Stadia Methods for Preliminary Surveys. — Preliminary
lines are usually run with the transit, but the compass will
answer nearly as well in most cases, besides admitting of more
rapid work. The transit and stadia method might well be em-
ployed, and would effect considerable saving in the cost of pre-
liminary surveys. For some reason railroad engineers have not
regarded it with favor, though it is extensively employed in
topographic surveying where the map is to be used for work that
is often more precise than needed for railroad preliminaries.
Particularly is this method applicable to exploration lines.
With the transit and stadia the entire surveying corps need not
exceed five or six men, the instrument man acting as transitman,
leveler, and topographer all in one. The only objection would
seem to be in the amount of reduction the notes would need ;
however, with tables and slide-rule (see 33) this work may be
very rapidly done. For vertical angles of less than one degree
the horizontal reduction can be neglected, and with side readings
for topography the angle may be 5 or 10 degrees without necessi-
tating the correction. Vertical heights are found by the slide-
rule or by charts.
This method would really necessitate the making of a topo-
p^raphic map along a narrow strip of country, from which the
profile could readily be taken. With a skilled observer and two
to four rodmen the progress may be more rapid, and fully as
good for the purpose intended as the more expensive method
usually employed,
20 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
The transit need only be set at alternate stations (wliicb may be
any length within the reading limits of the wires), the bearings to
other stations and points off the line being taken with the needle.
The horizontal angle should also be read on the plates for points
on stadia line, as a check on the bearings.
£. Obstac/es in Tangent
40. Obstructions to vision and measurement in tangent may be
avoided in a number of ways, a few of which are given in the
following problems. Other methods of avoiding them will sug-
gest themselves in special cases.
The same devices may be used on location, but it is more im-
portant to maintain a clear sightway then ; so, when possible, we
should remove the obstruction.
41. To Pass an Obstacle by Means of Parallel JUnes.— In
Fig. 4, is the obstruction, AB the obstructed line. At B set
E F
Q H
r S
i
\ i
3 D
Fio. 4.
transit ; turn 90*" and measure BF long enough to clear obstruct
tion. Set transit at F, make BFO = 90*, and measure FG,
Move to Cf and backsight to F, making FOC = 90**. Measure
GC=FB, and move to (7, where the angle OCB is made equal
to 90"*. CD is the desired line, and BC=FO.
Otlierwise, at A and B erect perpendiculars; take BF= AB;
produce EF, and at O and H, beyond 0, erect perpendiculars mak-
ing G (7 = HI) = FB, CD will be the desired line, and BC = FG.
42. To Pass an Obstacle by Angular Deflections.
General Case. Angle anything less tJian 90".
At B (Fig. 5) on the obstructed line deflect an angle a to one
side and measure BC, taking C so that after deflecting 2a to the
other side GD v/i\\ clear the obstruction. Make CD = BC And
deflect an angle a to the same side as at £; DE will lie in AB
produced. Draw (7£f perpendicular to BD\ then
BD = BU+HD = 2BC cosa. , . , . (6)
PRELIMINARY SURVEYS.
21
Fig. 6.
Example.— Suppose a = W 10', BC=CD = 520 ft.
BD = 2 X 520 X 0.96959 = 1008.37 feet.
Special Case. Angle 60 degrees.
In this case the triangle BDF(Fig, 6) is equilateral and BF =
BD = DF.
Should it be inconvenient to run to I> we may stop at C, having
measured BG, At G deflect 60° and measure GE; at E again de-
Fio. 6.
fleet 60° and make EF= BG. At ii^ a final deflection of 60° in the
opposite sense will put the telescope in the desired line, FO, and
BF=BG+GE. (5a)
43. To Pass an Obstruction, such as a River, when the Pre-
ceding Methods are Inapplicable.
First Case. Point beyond obstruction visible.
In Fig. 7 let BG be required.
Fio. 7.
At B erect and measure the perpendicular BD ; set instrument
at D and measure angle BDG = a ; then
BG=BDtsina (6)
22 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Or, if a trigonometric table is not at band, make GDE-=. 90** and
fix the point E where DE intersects AB ; measuring EB there
results-, from similar triangles,
CB_BD
BD " EB'
BD*
whence CB =-^ (6a)
Otherwise, if a right angle at B is not convenient, measure
angles GBD = 6, BJDG = a, and side BB. Then c = 180"- (a+ft).
From triangle BDC^
BC = BD^ (66)
sm c ^ '
Example.— a = 56% J = 70% BD = 400 feet.
By (66), BG = 400 ^^^. = 409.8 feet.
' sin 54
Second Case. Point beyond obstruction invisible.
At B (Fig. 8) measure angle b and line BE; move to E and
measure angle p, and set hubs on line EG so the line BG will pass
Fio. 8.
between them. Angle z = EGB = 180 -{b + y). Then from tri-
angle BEG
BG^BE^^ (7)
sm 2 ^ '
Produce EB to D, where DG will be sure to clear obstruction;
measure BD,
From triangle BDG^
tan \{a - x) __ BG - BD
tan^(fi + ic) BG + BD
But a + a; = &, hence
tani(«-ir) = ^g-j-|g.teni6. ... (8)
I^ftELtMIirAftY StftVEtS. J^3
The sum and difference of a and a? are now known, so both may
be readily found.
At D set off the angle a with the transit, and have the chainmen
stretch a cord between the hubs set on line EC at C. Now signal
the flagman to move his rod along this cord until the vertical wire
cuts it at C. Set a hub here and place the transit over it. Sight
to D or E, reverse telescope and deflect into CH,
Article 4. — The Level Party.
44. The Level Party consists generally of two members, the
leveler and a rodman ; sometimes an axeman is added to keep the
lodman supplied with pegs for turning-points and in clearing the
line of sight for the level. As the party follows the transit little
or no clearing will be needed. The instruments used are a level,
A rod, and a hand-axe or hatchet.
46. The Leveler makes all necessary observations with his
instrument, keeping a neat, accurate record of readings and ele«
vations ; also the positions and elevations of benches and turning-
points. He should work out elevations of stations while the rod-
man is going from one station to the next ; he mast see that the
rodman gives him readings at points where the longitudinal slope
changes suddenly, recording the plus. He must plot his profile
at night, or at such times as the chief of party is likely to need it.
The rodman 's readings at turning-points should be checked.
46. The Rodman holds his rod at each station, calling out the
number. If stakes are set only at even stalious, be must hold his
rod midway between stakes, tbe point being found by pacing tbe
distance. Target-readings need only be taken at turning-points
and benches, and tbe rodman should keep n record of these in
his "peg-book," checking the calculations of leveler for heights
of instrument and elevations of turning-points. At any marked
surface change he will hold his rod, calling out the plus to leveler.
He must assist the leveler In plotting up the notes.
A. Adjustments of the Le¥eL
47. To Adjust the Line of Collimation Is to bring the inter-
section of the cross-wires into the optica! axis of tbe telescope.
Set up and level the instrument, then bring the vertical wire
into coincidence with a plumb line or vertical edge of a building,
24 A FIELD-MAKtJAL FOH RAILROAD ENGINEERS.
at the mean length of sight, and note if the verlicjil wire is truly
parallel thereto. If it is not, loosen the cupsttuj- headed screws
holding cross wire ring and turn slightly so that the wire is
parallel to the vertical line.
Loosen the wye-clips and bring the vertical wire into coin-
cidence with the line and. clamp the instrument. Rotate the
telescope in the wyes 180** and note if the wire coincides with the
line. If not, correct one half the error by loosening one and
tightening the opposite, of the capstan-headed screws that hold
the cross-wire riug in place, remembering that the image of
the cross wires is inverted by the eyepiece.
Turn the telescope until the horizontal wire is parallel to the
plumb-line or edge of building, and make the same lest and
correction. Repeat for both wires. The horizontal wire is the
one on which the accuracy of leveling depends, but it is wise to
have both adjusted. Their intersection should remain on a point
during a complete rotation of the telescope in the wyes.
48. To Adjust the Level-bubble is to bring the axis of the
level- tube into the same vertical plane with the line of coUimation,
and to make the bubble stand at the center when the line of sight
is horizontal.
Since the axis of the telescope coincides with the line joining
the center of the wye-rings (which requires these to be of the
same size), it is sufficient to make the axis of the bubble parallel
to this line.
{a) With the telescope over one diagonal pair of leveling-
screws and the clips loosened, bring the bubble to the center of
its run ; then turn the telescope, in the wyes, a little to either side
of the vertical plane through the telescope and note if the bubble
remains at the center. If not, correct the error by means of the
screw at end of the level-tube case arranged for b\teral movement,
iiepeat until the tube may be rotated half an inch or more to
either side of vertical without movement of the bubble. Thia
adjustment is made merely to prevent error from failure to set
level-tube vertically beneath telescope.
{b) With the wye-clips opened well out, again bring the bubble
to the center of its run ; remove the telescope from wyes and
turn it end for end, then carefully replace it in the wyes. Should
the bubble fail to remain at the center, bring it half -way back by
raising the lower or depressing the higher end of tube at the
points of attachment to telescope. Relevel and repeat as a test.
PRELIMINARY SURVEYS.
25
49. To A<^ust the Wyes is to make the axis of the telescope
perpendicular to the vertictil axis. With the wye-clips closed
place the telescope over one pair of leveling-screws aud bring the
bubble to the center of its run ; then turn the telescope half-way
rouud ou its vertical axis, so that its euds have changed places.
If there is any error,. correct by bringing the bubble Jialfway back
to center by means of the screws connecting wyes with level-bar.
Repeat until the bubble remains in the center during a complete
revolution.
B. Theory of Leveling,
50. When the level has been adjusted the line of collimation
N\\\ describe a plane parallel to the horizontal plane tangent to
he earth's surface at the point where the instrument is placed.
Ji. level surface, such as the surface of still water, will coincide
with this plane only at the point of tangency, and will depart
farther and farther therefrom as the point considered recedes
from the instrument. For short sights this difference may be
neglected in railroad work, as will presently be shown, but for
long sights a correction must be applied.
The effect of curvature is to make objects appear lower than
rhey really are, while the refraction of a beam of light, due to
tlie greater density of the layers of air nearest the earth's surface,
has a contrary effect. Experience shows the average error due
to refraction to be about one seventh of that due to curvature.
51. The Error due to Curvature at any point is the deviation
of a tangent line from true level, as j^ p
the point recedes from the point of "
tangency.
Let be the center of the earth, T
the point of tangency, and iVthe point
where the error due to curvature is
desired. Let the notation be as shown
in Fig. 9. From the right triangle
OTP, we have
From which
Fia. 9.
t^
2i2 -f c *
Now, since c is always very small compared with 2i2, the
quotient resulting from the division of f^ by 2R will not differ
26 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
n 1 «
3 <«
'1 R'
sensibly from that obtAined by dividiug by 2B + e. Therefore
we write
.= 4. ........ w
For t = 1 mile, B = 8963 miles, c = about 8 inches. Hence
for any other distance in miles we have, for c,
c = 8 Xt' inches (9a)
The correction for refraction is about -^^j, hence we have,
from (9),
or, closely enough,
C = .85c (10)
Example —What is the correction for a half-mile sight?
For one eighth of a mile?
By (9a), c = 8 X HY = 2" for first case,
and c = 8 X (J)' = 0".125 for second case.
By (10) the final correction is
c = 0.85 X 3 = 1".7 for first case,
c = 0.85 X 0.125 = 0.106" for second case.
52. The Difference of Elevation between two points not so
far apart but that a rod may be read on each from some inter-
mediate point may be readily found from these rod-readings.
In Fig. 10 let the instrument be at i, A and B the points
whose difference of elevation is desired. Let r = AD, r' — BC.
Since the line of sight, DC, is horizontal, the difference of
Fio. 10.
elevation will evidently be r* — r. When the distance from
/ to -4 equals tliat from I to B the errors due to curvature
evidently balance.
PRELIMINARY SURVETS.
27
When the points are so situated that the rod cannot be read
on both from one intermediate position of the instrument, an
D
_. F
A
J
C
^^
r
4
^^^
-^
ij^^^^5^ ^^
Fia. 11.
auxiliary point or points must be used and readings taken on
tliese points in pairs. Thus in Fig. 11 suppose the difference
of elevation of ^ and 5 required :
With the instrument ""at / read on A and some intermediate
point E, Cousideriug the backsights as plus and foresights as
minus, the difference of elevation of A and E is AB — FE,
Again, with the instrument at /' the difference of elevation of E
and B is QE-^ CB, The sum of these differences equals the dif-
ference of elevation of A and B, and may be written {AD -|- OE)
— {EF-{- CB), or, in general, the sum of t7ie backsighU less the sum
of the foresights equals the difference of elevation,
C. Field-work,
63. A Datum is a level surface so taken that it shall lie below
the lowest point likely to be reached by the profile, to which the
surface elevations are referred. It is often spoken of as the
datum-line or datum-plane, and is the zero of elevations.
64. A Bench-mark is a permanent mark, such as a copper or
other bolt let into the top of a solidly fixed stone, whose height
above the datum is known; it may be simply a mark on a stone,
OH a tack driven into the projecting root of a tree, upon which
the rod may be read. In any case it must be so situated that it
cannot change its elevation nor is likely to be disturbed within
the time for which it is intended to be used as a standard of
reference.
The elevation should be marked on some object adjacent to
the bench, with the letters B. M. indicating the nature of the
point.
28 A FIELD-MANUAL FOU RAILROAD ENGINEERS,
56. The Field-work consists in finding the elevation of a
number of points on the line established by transit party suffi-
cient to give, when plotted, a fairly correct outline of the surface
as seen in profile.
A bench-mark is taken at the beginning of the line, and its dis-
tance above mean sea-level or other datum is known or assumed.
The level is set with one pair of leveling screws in the line to be
run (in order that any change in the position of the bubble may
be easily corrected), and the rod is read on the bench. This read-
ing plus the elevation of bench gives the height of instrument
{H, I.) above the datum.
Readings are taken at every hundred feet along the line, or
oftener if the surface changes greatly, until a point is reached
beyond which it is desired to move the level. A peg is driven
firmly into the ground and the rod i-ead on this ; the height of
instrument less the rod reading will give its elevation, as it will
for the intermediate points. This poipt is a temporary bench,
and is called a turning-point. It should be marked by a guard-
stake if it is desired to use it again. The instrument is now car-
ried beyond the turning-point, set up, and the whole process
repeated. Benches and turning points should be read to hun-
dredths or thousandths of a foot, intermediate points to tenths.
Turning-points are marked or T. P. in the notes, and their
positions, as also the bench-marks, noted by both leveler and
rodman in their note-books.
56. The Level Notes may be kept in any convenient form
that is easily understood. The following is used more exten-
sively, perhaps, than any other:
Sta.
B.S.
H.I.
F.S.
Elev.
Remarks.
B.M.
5.618
ii05.613
....
200.0
j B. xM. on root of L. O. tree ec to
1 right of line.
2.8
203.8
1
0.8
204.8
2
5.7
199 9
8
7.8
197.8
4
99
195.7
L120
196.810
10.4;>3
195.190
On peg at 4-1-80' - 20^ to left of line,
by small P. 0. tree.
5
6.d
190.0
6
4.5
191.8
Here the elevation of the datum was taken 200. 00 feet below
the first bench-mark. The instrument was set up near Station 2,
PRELIMINARY SURVEYS. 29
iod a reading of 5.613 taken on the bench; this was written in
the B. 8. column, and when added to the elevation of the bench
gives the height of instrument, 205.613. A reading of 2.3 was
taken on Sta. 0, recorded in the F.8. column, and when sub-
tracted from the H.L yields an elevation of 203.3. The eleva-
tions of other points were determined in the same way. A little
beyond Station 4 the rodman drove a peg and held the rod on it,
yichling a reading of 10.423 and an elevation of 195.190. The
instrument was then moved to a point near Station 7 and a read-
ing of 1.120 taken on the peg; this added to 195.190 made the
new H. L 196.310, and the process continued with this H, L
In most cases it will be sufficient to read benches and turning-
points to hundredths and intermediate points to tenths.
It will be seen from the notes that any error in a turning point
causes the same error in all succeeding points. To guard against
this the rodman is required to keep a " peg-book," in which the
heights of instrument and elevations of turning-points are re-
corded, and which must check with the leveler's record.
67. Wind and sunshine affect the accuracy of the work with
tlie level, as is also the case with the transit. For very great
accuracy a calm, cloudy day is the best, but the railroad engineer
<»annot always choose the best times for his work, and must take
^uch precautions as may be possible while he exercises the great-
est care to prevent and detect errors. The adjustments should
l)e tested at least once a week, even when the greatest care has
been taken, for unequal expansion and other causes may con-
spire to cause them to change.
By making foresights and backsights to turning-points about
equal the error due to curvature will be eliminated; the readings
of rodman at these points should also be checked. The rodman
should hold his rod vertical, which is sometimes accomplished
by meims of a level attached to rod; or the leveler can tell by his
vertical wire when the rod is in the same vertical plane with the
instrument, and by causing the rodman to wave his rod back and
forth slowly, after clamping the target, he can tell if the hori-
zontal wire just bisects the target at its highest position.
68. The Rod should be graduated to feet and tenths, reading
by target at turning-points and benches; intermediate readings
are made by the leveler at his instrument. Strength and dura-
bility are essential qualities. The Philadelphia rod seeou to
30 A FIELD-MAKUAL FOR RAILROAD ENGINEERS.
answer tlie purpose as well as any other now manufactured: the
Troy rod may be used in the same manner as the Philadelphia
rod, but is lighter and less able to stand rough usage.
Article 5. The Topographic ParTIt.
59. The Topographic Party follows the level and secures all
the data necessary for making an accurate contour-map of a strip
of country extending as far each side of the preliminary as may
be needed for the intelligent projection of the location -line.
This distance may vary from 50 to 300 or 400 feet, its width de-
pending on the difficulties to be encountered and the degree of
precision with which the preliminary approximates to the final
location-line. The lateral slope of surface is obtained at the
stations of preliminary by means of the hand-level and tape, by
the slope-level or clinometer, by cross section rods, or by the
transit and stadia. Strictly speaking the topography includes all
the surface features, but for railroad work the surface elevations,
streams, and nature of surface are the most imporUint; it may
be necessary to note the positions of roads, buildings, etc., and
should always be done when practicable without undue loss of
time. A pocket-compass will be of use in observing the bear-
ings of lines.
60. There are two methods of recording the data obtained;
one by means of notes and sketches in a book, the other by
drawing the contours directly on the field sheet as ihe data are
obtained. Station elevations can be taken direct from the leveler's
notes, and constitute the base on which the contour elevaliotife
rest.
Suppose the hand-level to be used and the notes kept in a book,
to be afterwards transferred to the map. Starting with the
known center elevation, the topographer notes the height of his
eye above the ground and calculates the height of center above
or below the next contour; from this the reading of the rod when
held on this contour is found, being the height of station above
contour plus the height of eye. He directs the slopeman in or
out on a line at right angles to preliminary until this reading is
given by the hand-level; the distance out is then measured and
recorded, just as in setting slope-stakes, and the slopeman di-
rected into position on the next contour, in the same manner.
Thus if 5-foot contour.intervals are employed, and the station
1
PRELIMINARY SURVEYS.
31
cievaliou is 321.6 feet ftnd the height of eye 5.3 feet, we shall have
fw the reading at ihe 320-fo()t contour 5.3 + (321.6 - 320)= 6.9.
Motion the slopeman down the slope until his rod reads 6.9 and
measure the distance out, suppose 21 feet. The 315-foot contour
will be 5 feet lower, giving a reading of 11.9, which may be
found in like manner at, siiy, 80 feet out. As the rod reads only
to about 12 feet the lo^wgrapher must move out to this last point,
and with the reading 5.3+5= 10.3 find the 310-foot contour in
ihe same way. On the up-hill side the 325-foot contour will be
found with a reading of 5.3 - (325 — 321.6)= 1.9 feet, and other
contours in like manner.
The notes may be written thus
Sta.
Left.
Center Elev.
Right
824
305 310 315 320
193' 125* 80 ' 21
321.6
825 830 335 340
27* 56* -80' 112
The number above the line is the contour elevation, the num-
ber below its distance out from center.
If preferred the elevation can be taken at regular distances out
and recorded as above; the position of the contour will then be
found by interpolation when mapping the work.
61. If the topography is to be plotted in as the work progresses
the topographer must have a light drawing-board with a pocket
and flap on buck for holding the sheets on which the transit-line
has been plotted the night before ; the station elevations are
marked on the line and the contour positions spotted in as ob-
tained by slopemen, after which the contours are sketched in.
Points where contours cross transit-line are found in the same
manner as side points. The size of the sheets will depend on the
taste of topographer and size of drawing-board; 17x24 to 19x28
inches are good sizes.
The topographer will soon learn to guess at the position his
contours will occupy at the next station ahead, and will sketch
them in lightly, to be erased and corrected when necessary. It is
often sufficient to take lateral readings at every second or third
station.
62, If the Slope level is used, the inclination of the surface is
obcained; then by th^ use of a scale constructed to show the
32 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
horizoDtal distance apart of contours, for tbe given contourin-
terval, for slopes varying from 1* to 20**, the position of contours
can at ouce be spotted on the map. Wellington recommends the
use of t!)e altazimuth as permitting the employment of either
method at ^ill— the altazimuth being merely a haud-levcl with
a clinometer attached.
63. Oross-section Rods are measuring- rods 10 or 12 feet long
carrying a level-bubble. By placing one end at the center,
bringing the rod horizontal, and noting the, height of the end of
rod on the down-hill side, the slope may readily be obtained and
the contours worked in as before. For very rough, broken
ground this method may be preferable to either of the othera.
64. If the Transit and Stadia are employed, very elaborate
topography may be taken with very little fieki work, but the ob
servations require considerable reduction. With a suitable topo
. graphic protractor and the slide-rule mentioned In 33, the large
number of points that may be obtained from each setting of the
transit may be readily plotted and their elevations marked on the
plot, after which the contour-lines can be worked in, and other
features mapped. For small vertical angles no horizontal reduc^
tion is needed.
While not generally favored by railroad engineers in the past,
this method is probably the most rapid and economical of any so
far employed in topographic work.
Article 6. Preliminary Estimates.
66. After completing the field-work of the preliminary survey
the party is usually disbanded, only the transitman, leveler, and
topographer being retained to assist the chief of party to complete
the map, profile, and estimate of cost.
66. The Map may be drawn to any suitable scale, but less than
400 feet to the inch is not to be recommended where it must be
used in projecting location. The transit-line is laid down first
and the topography worked in afterwards from the field-map or
topographer's notes. If it is wanted on a continuous sheet, the
transit- line must first be drawn on a succession of small sheets,
which are added as the plotting progresses, a new sheet being
slipped under the edge of the preceding and tacked down whea
PRELIMINARY SURVEYS. 33
required. The overlapping edge is marked by a number of short
lines extending over onto the sheet beneath, to enable one to re-
place in the proper position. When the line has been plotted the
sheets are pasted together and the whole shifted so as to bring the
transit-line over the continuous sheet. Angular points are then
pricked through and the line drawn on the continuous sheet.
Ordinarily it will answer to have the map drawn on a succession
of small- sheets, to be joined together as required.
The plotting had best be done by bearings, though it may be
Jone from the deflection angles, provided care is used to check
frequently by bearings. Otherwise an error in one angle will
throw all the remaining portion of the line out of position.
If more than one preliminary was run, they should all be shown
on the same sheet whenever possible.
67. The Profile will be drawn by the leveler on profile-paper,
and shows a develoi^ed vertical projection of the line. The scale
will depend on the paper used. There are three scales in general
use. styled respectively Plates ** A," "B," and "C." There is
also a metric profile-paper. Plate * * A " has the vertical exagger-
ated 20 to 1 as compared with the horizontal and is the best to
use where much rockwork is expected. The vertical exaggera-
tion of Plate '• B" is less than of Plate "A"; this plate is most
used for ordinary eartliwork.
A strip of color laid on below the surface-line, and fading out
at the lower edge, adds greatly to the appearance of the profile.
The tentative grade- line and points of change should be drawn in
red.
68. Preliminary Estimates of quantities are made by assuming
a grade-line and drawing it on the profile; then the cuts and fills
are taken from the profile, and the corresponding quantities ob-
tained from Table XIX for the base the road is intended to have
when completed. The nature of the work, whether ordinary
earth or rock, can. of course, be only roughly estimated.
Bridging is estimated from the profile where piling or framed
bents may be used, but where piers and long spans are needed
special surveys with soundings are required, Culverts, drains,
cattle guards, cross-ties, and rails for main line and sidings,
switcli stands, buildings, right of way, clearing, and other factors
entering into the question of cost must all be considered and
allowed for in making up the estimate.
34 A FIELD-MANUAL FOR RAILKOAD ENGINEERS.
EDgiDeering expenses and unforeseen outlays that are sure to
arise should have a liberal allowance.
69. The Report of the chief of party should set forth the ad-
vantages and probable cost of each of the several lines run
when there is more than one. Ou this report frequently depends
whether or not the line is to be located, and it should be clear
and exhaustive, though plainly and concisely worded. The map
and profile form an integral part of the report and show from
what data the estimates were derived.
CHAPTER in.
LOCATION,
Article 7. Projecting Location.
70. After the preliminary has been mapped and the topography
worked in, the engineer proceeds to make a paj^er location for his
guidance in the tield. The solution of the varied and complex
problems that confront him are more or less interdependent.
The guiding principle, applicable to all departments of engineer-
ing, that the best structure is tJuit which for the least cost best an-
swers the purpose fai* which it was intended, should control, even
thougii the resulting structure be inferior, in point of scientific
design, to some other. The best road as regards construction and
grades may be a failure because of excessive first cost, while
the cheapest construclion will entail such heavy operating ex-
penses that it may be equally unprofitable. The alignment must
be as free from curves as possible, while heavy grades are at the^
same time excluded; these two requirements conflict and must
be as well adjusted as possible. The amount of earthwork, of
bridging and other structures must be kept down to the lowest
limits.
71. Starting at the' summit of the most difficult portion of the
route, assume a starting-point and elevation; with the dividers set
at such a distance to the scale of the map as will give a fall of one
contour space— or half space— for the assumed grade, step down
the slope in such a way that the dividers fal^ each time on the
next lower contour, or half-space, according to the fall assumed in
setting dividers. If curve compensation is allowed, the dividers
must be reset for each curve, for the same fall, since the grade
will be slackened on curves. The points at which the dividers
fall are lightly spotted on the map ami connected by a grade
contour, which represents the surface line having the required
gradient. This line will be too broken to be used as a location-
35
o6 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
line, so we have then to draw on the map a succession of curves
aud tangents that will approximate sufficiently close to it. at the
same time that a proper balance is maintained between earthwork
and curvature.
Having lightly plotted the proposed line, the elevations are
transferred to profile- paper, thus giving a profile of the line.
With a fine thread stretched along the profile, to represent the
grade-line, adjust the cuts and fills to suit the nature of the work.
Ill general, fills are cheaper than cuts both in construction and
maintenance; and especially is this true where a shallow surface
luyer of earth is underlaid by rock. It may happen that the
material from excavation must be used in embankment, whieii
the cuts aud fills must be made to balance by shifting the gi*ade-
line until this appears to be the case on the profile.
At the stream crossings the grade-line must be kept safely
above high-water mark, so that sufficient waterway is provided,
and allowance made therefor.
After locating the most difficult portions pass on to the easier
work, returning later on to study the effect this will have on the
part first located. It may be necessary to go over the projection
several times before you can be reasonably sure that the best loca-
tion has been projected; even then the study of the line in the
field will cause many of the details to be altered, sometimes
materially;
Long grades are to be preferred to short ones, but questions of
economy may necessitate the latter in order to lighten work; care
must be taken that the grades are not so badly *' chopped" that
they interfere with the easy riding of the train.
In projecting the line it will generally be best to strike the
curves first and draw the tangents afterwards, though it some-
limes happens that long tangents will control the curves; when
tin's is the case the tangents are drawn to intersection and the
c'.irves afterwards put in.
When transition-curves are employed, a slight offset should be
made at the beginning and end of curves to allow for their inser-
tion in the field. These offsets will be so small that it is ijsetess
to attempt to show them to scale.
72. A Curve-protractor will be of material assistance in find-
ing the degree of curve required to unite two tangents that have
been laid down on the map. It consists of a transparent, semi-
circular protractor having a series of cuiTes from 30' up to 8**
LOCATION. 37
plainly cut upon it. The curves are on both sides^ those on the
reverse side having their concavities turned iu an opposite sense
from those on the face. The scale is usually 400 feet to the inch,
and in any case the map and protractor must be drawn to the
same scale. Sometimes a set of cardboard or hard-rubber curves
are used, but they are inferior to the curve-protractor. To use
it. simply prolong tangents to intersection 'and theu place the
protractor so that the curve admitting of the best grade is tan-
gent to the two straight lines. Mark the points of tangency,
which will be the beginning and end of curve. When the curve
is required to pass through a given point the proper curve may
be immediately found by trial, whereas the calculations would
require some little lime.
Reversed curves should never be allowed on main lines. Suffi-
cient taugent should be interposed to allow space for easing off
the superelevation of outside rails, or for the insertion of tran-
sition-curves when these are to be employed.
73. The Field Corps is substantially that required on the pre-
liminary survey, and the methods of work pretty much the same,
except Ihut curves must now be run iu, and this necessitates more
clearing. If first and second location- lines are to be run (and it is
real economy to run both), it will not be necessary to have the
stationing continuous on the first, so the pluses arising from
** backing up" need only be noted and eliminated when the final
location-line is run. If transition -curves nro to be inserted, they
need not be run the first time, the proper offset being made at
vhe P. T. or P. C of the circular curves, which latter are to be run.
On the final location-line the stationing must be continuous,
beginning with zerp. The stakes are marked as on the pre-
liminary survey, and all hubs that are likely to be used again
must be referenced in, the reference-hubs being set well out of
the way of disturbance by the plow or scraper.
The leveler should make bench-marks every 1000 or 2000 feet,
to be used in running check-levels and in giving grades later on.
From the paper location the notes should be made up in the
office, to serve as a guide in the field; however, noatteuipt should
be nfade to adhere rigidly to them, since slight errors in the
mapping will affect the projected line, while in the field the line
may be shifted here and there so as to fit the ground more snugly
and accord more closely with what the nature of the earthwork
demands.
38 A FIELD-MANUAL FOR HAILROAD ENGlKEEftS.
The highest skill of the eogineer is required to secure the best
locatioD-line, and he should have all the time he needs. Uiidue
haste on location — as on recouuoissance and prellmlnaTy — is
almost sure to result in increased cost of construction.
Abticle 8. Simple Curves.
A, Definitions and Formulas,
74. The Circular Curves that are usually employed to unite
straight reaches of the railroad may be simple, compound, or re-
versed. The use of reversed curves should, however, be limited
to turnouts and cross overs.
a. A Simple Curve is the arc of a circle.
b. A Compound Curve consists of two simple curves, of differ-
ent radii, both on the same side of n common taugeut.
c. A Reversed Curve is made up of two curves of contrary
flexure having the same or different radii, and a common tangent.
d. The Point of Curve (P. C.) is the end of tangent and begin
ning of curve, as at A, Fig. 12.
Fio. 12.
6. The Point of Tangent (P.T.) is the end of curve and be-
ginning of tangent, as at B of Fig. 12.
/. The Point of Intersection (P.I.) is the point where the
tangent at the P. (7. and P.T. intersect when produced. {D of
Fig. 12.)
g. The Intersection Angle {D is the angle at the PL be-
tween the tangents meeting there, and equals the angle at the
center.
h. The Tangent Distance (T) is the length of the produced
tangent measured from the P. C. or P.T. to the P.I. The term
LOCATION.
39
tangetit is applied to a»y straiglit portion of the lioe, but the letter
T will be used to designate the produced portion only.
i. The Mid-ordinate (if) is the portion of the radius inter
cepted between the arc and chord when it cuts the chord at its
middle point.
j. The External (E) is the part of the radius produced to the
P.I.y intercepted between curve and the P.l.
k. The Long Chord {L,G.) is the chord joining the P,C. and
P. T, Frequently the term is applied to any chord longer than
the unit chord.
L The Radias will be denoted by R,
m. The Point of Compound Cunre (P. G. G. ) is the point of
common tangency of the two branches of a compound curve.
(See Fig. 13.)
/'
\p.c.c.
p.l.
^
-^
>.
>
>
y^t
P.R.C.
Fio. 18.
n. The Point of Reversed Curve (P.R,G.) is the point of
common tangency of the two branches of a reversed curve.
o. The Degree of Curve (Z>) is the angle at the center sub-
tended by the unit chord. In the United States this chord is 100
feet, in England 66 feet, and where the metric system is em-
ployed it is taken at 20 meters. Any convenient chord length
may be taken, but for uniformity American engineers have
adopted the chord of 100 feet, and unless otherwise stated it is
always so understood when we speak of the degree of curve.
Half the degree of curve is called the deflection-angle, since
it is the angle to be deflected from the tangent to the chord.
If there were any practical method of measuring around tJie
curve instead of along the chord, an accurate and convenient
ratio for expressing the radius in terms of the degree would be
had. Thus if D is the angle at the center subtended by the arc
of unit length, we have, where a is this unit arc,
40 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
2nR= a—jj'
Hence
i? =
a 360
27t D'
When a equals 100 ft. this becomes
100 860
i? =
2;r
(11)
(11')
R varies inversely as Z), so that knowing the radius for a V
curve, we should have only to divide this by D to get the radius
for a D** curve.
Since the clioi*d is employed instead of the arc, we determine
R by means of the following problem
75. Oiven the Chord (7, and Degree of Curve 2), to Find the
Radius R.
In Fig. 14, AB \^ the chord (7, OE a pei-pendicular from the
center upon AB
From the right triangle AEO we have
R sin \D = IC.
Whence
When (7 is too ft.,
R = -ri^ = iCcosec 4Z).
sm \I> ^ ■
50
R = - — rr, = 50 qos^c ID.
sm ID *
(13)
(m
LOCATION, 41
Comparing results given by formula (12') with those given by
ill'), we have for a few curves:
De^rree of Curve. R by (12'). R by (11'). Difference.
1 5729.65 5729.58 0.07
2 2864.93 2864.79 0.14
8 1910.08 1909.86 0.22
5 1146.28 1145.92 0.36
7 819.02 818.51 0.51
10 573.69 572.96 0.73
14 410.28 409.26 1.02
20 287.94 286.48 1.46
The difference is seen to be about one half a foot for a 7**
curve, one foot for a 14" curve, and one and one-half feet for a
20** curve.
Up to a 7** curve the difference is inconsiderable, and we may
stake out curves with 100-foot chords. From 7 to 14 degrees 50-
foot chords may be used. Therefore
^="slp=^^''^'^''*^- • • • • • (12'«>
For curves from 14** to 28** we should use 25-foot chorda,
for which
i? = ^^ = 12.5cosecJ2). .... (m)
Above 28" shorter chords— say 10 feet— should be used, if the
curve cannot be struck from the center. In this case
Table I of radii was computed by formulas (12'), (12'a), and
(12'&).
In practice it is customary to take tbe radius of a 1** curve ns
5730 feet and to assume the radii to vary inversely as the degree ;
thus for a 4" curve the radius would be Ii=: ^jifl = 1432.5 feet,
while by Table I it is 1432.69 feet— a difference of only .19 foot ;
for a 12" curve R=m^ = 477.5 feet, while by Table I it is
477.68 feet. 'I'he effect of taking 5730 instead of 5729.65 for the
radius of a 1" curve is to reduce the error resulting from the
assumption that It equals 5730 divided by the degree of curve.
42 A ^-iKLD-MANt/AL J?Oft RAILROAD EKQlKEEIlS.
76. The Length of Ounre (L) is found by dividing the angle
at the center (wliich equals the intersection angle) by the deirree
of curve, the result being in chains and decimals of a chain. The
number of P.C,-{-L will give the station number of P. T,
Example.— The P. C, of a 4* curve having / = 26** 30' is at sta.
104 + 12.5. Find L and the number of the P. T, Here
X = ?^ = 6.625 chahis.
104.125 + 6.625 = 110.75 ; hence the number of P. T. is
110 + 75.
77. Use of the Table of Functions of a One-degree Curve. —
In the location of railway curves geometrical accuracy will
frequently be of less importance than rapidity of lield-work, so
long as errors are kept within certain limits.
On tangents slight errors of alignment may readily be detected
by the unaided eye, but on curves these are not so apparent.
Moreover it is not likely that the trackmen will keep them up in
the exact position of their location.
To simplify and shorten the field computations engineers make
use of a table of functions of a V curve, and assume these func-
tions for other curves to vary invereely as their degree, or directly
as their radii. Table IX gives values of the tangent distances,
long chords, mid-ordinates, and externals for a V curve, the
radius of which is taken as 5730 feet. To find these functions
for other curves, divide the tabular values by the degree of curve.
The error resulting from this assumption will, in any practical
case, amount to no more than a few tenths or hundredths of a
foot.
Table IX may also be used as a metric curve table, the tabular
values being taken as meters instead of feet. If the unit metric
chord is 20 meters long, this may be taken as one fifth of the
tabular unit chord; so to use the table multiply the metric degree
by 5 and enter the table with the result as a value of 2).
For instance, a 2° metric curve having / = 40" would have a
mid -ordinate equal to ^ — -^ = 34.56 meters.
-© X o
For the approximate radius of a metric curve divide 5730 by 5
times the degree. Thus a 4° metric curve would have i?= -^ =.
4X6
LOCATION. 43
:=: 386.5 metei's. For the exact mdius make use of formula (12).
Thus for a 4° curve having 20-meter chords R = — — ^ = 286.54
meters, a difference of ouly .04 meters.
If a metric curve is to be retraced with a 100-ft. chain, we
convert the metric degree to the degi-ee referred to 100-ft. chords
by the relation that a 100-ft. chain = 1.524 chains of 20 meters
each; a 20-meter chain = 65.618 ft.; one foot = 0.3048 meters;
one meter = 3.2809 ft.
It will sometimes be a sufficiently close approximation to lake
the 20 meter chain as two thirds of a 100-ft. chain; this will make
the metric curve nearly two thirds of the degree the same curve
would have when laid out with a 100-ft. chain, and the curve with
106-ft. chord? nearly three halves of the degree as laid out with
the 20-meter chain. Thus a 4° metric curve would be equivalent
to a 6' curve laid out with a 100-ft. chain.
In the problems that follow two methods of solution will be
given when practicable—the first being rigid, while the second
As based on the use of Table IX. To shorten the formulas the
subscript 1 will be written after the letters T, L,C., M, and E
when these are the functions of a 1° curve. Thus Tx "4- 28' means
r.he tangent distance for a 1** curve when /= 28**, L.Cx ^ 16**
/;he long chord for a 1° curve when /= 16", etc.
78. Tables of Natural and Logarithmic Circular Functions. —
Many engineers prefer to work altogether by tables of natural
sines, cosines, etc., and time may often be saved by their use.
Nevertheless logarithmic tables are of frequent advantage, even in
the field, and the more important ones, such as Ihe logarithmic
sines, cosines, tangents, and cotangents, together with the loga-
rithms of numbers, are given in the back of the book along with
the tables of natural functions.
79. Given R and G to Find Z>.
From equation (12),
smW = i^ (13)
80. Given / and R (or D) to Find T.
If D is given, find R by (12); then in Fig. 15 from triangle
OAB we get
y=i?tani7. (14)
44 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
By Tablb IX.— Find the tabular value of T for the given
angle /; then
T
(14a)
Example.— /= SS** 40', Z) = 4"; required T,
By (14). r = 1433.69 tau \T 50' = 460.91 feet.
By (14a). r= -^'^
= 460.85 feet, a result differing from the
value found by the rigid method by only 0.06 foot.
81. Given I and 7" to Find i? or D
From (14),
T
R =
tan il
= TcoHl
Then by Table I the degree inay be found.
By Table IX.
(15)
(15a)
82. Given /and D to Find the Long Chord L.C.
First find R by (12) or (12'), or by Table I ; then from the
triangle OAF of Fig. 15,
AF =R sin il.
.-. AG = 2AF= L. C, = 2R sin ^7. . . . (16)
LOCATION.
45
By Table IX. — Find the tabular L, (7. for the given angle /;
then
z.a =
(16a)
83. Given the Radios B and any Chord C to Find the
Ordinate to the Curve at any Point.
First Method. —In Fig. 16 let HE be the chord C\ HK= a
and KE = 6, the segments into which it is divided by the ordi-
nate y. Draw the radius through K\ call the portion between
chord and curve /. By geometry,
from which
ah
y =
2.R^y''
But y^ is small compared with 2i?, and hence we write
ah
^■
"2i?*
(«)
Now y does not differ sensibly from y' in the cases met with in
pnictice, so we write
ah
y =
2/2'
(6)
46 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
If we write B = -yr-, formula {b) becomes
abP
^^ Too ~ *"' 100 ~ ^' *^^ substitute in (c), giving
or very nearly
y = lmnD (17)
y is given in feet when m and n are in chains and decimals of
a chain.
At the mid-point Fym^n, and y = M.
.-. M^in'D (18;
Caution.— Formulas (17) and (18), while very convenient for
field use in passing obstructions, are liable to error when very
long chords or large values of D are used, since they give results
that are too small.
If we write the arcs HN^ NB for a and &, we shall get i-esults
that are too large, yet about as near the true values as by taking
m and n to be the segments of the chord. To illustrate we will
find a few values of M and compare with the true values taken
from Table V.
Degree Length Mid-ord. Mid-ord. Mid-ord.
of of by by by
Curve. Arc. M=l(HF)^D, M=yiHQ)*D. Table V.
2 2 Stations. 1.75 1.75 1.75
2 6 *' 15.69 15.75 15.69
5 2 '* 4.37 4.38 4.36
5 6 " 38.51 39.38 39.06
8 2 ** 6.96 7.00 6.97
8 4 *' 27.29 28.00 27.75
8 5 " 42.02 48.75 43.20
8 6 *' 59.43 63.00 61.98
From this it appears we may use formula (18)— and (17) as
well — taking either the segments of the arc or chord for curves
not exceeding 4" wilh arcs up lo 600 ft. ; for curves from 4' to 6"
LOCATlOHr. 47
they may be used up to 600-ft. arcs, while for cui-ves between
6** and 8** not more ihau 400 feet of arc may be taken.
Srcond Method.— First determine the mid-ordinate. In
triangle OEF,
0F= VH" -id
then
M= FO = R - i^B" " iG' (19)
To find ordinate -4(7 distant d from the mid-point ot EH, draw
OB=d parallel to HE; draw AB at right angles to HE. Then
BA = i/B* - d*.
Therefore
C^ = y= Vi? - d»- i/iJ^-JCT*. . . . (20)
Third Method.— If the chord C is short, we may regard the
arc as an arc of a parabola, for which it is known that ordi-
nates vary as the product of the segments into which they divide
the chord. The mid-ordinate being known, we have
y __ af>
M - (J60«
.obM
.-. y=4-g5- (21)
From formula (h) we have for y = lf, a = 5 = ^C,
^- -2^-85 (^)
The mid-ordinate for any other chord C is
M,
= 85-
Hence
Jfi
C"«
M
"" (7»*
.\ Mx--
iC'V
If C" =
: \C. this gives
\cj
J/,
= iM. .
48 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
This last relation affords an easy method of staking out n curve
when the niid-ordinate of a given chord has been determined.
Fust erect the ordinate if at the mid point of the chord; then
join the ends of chord with the extremity of the ordinate just
measured; the lengths of these chords do not differ much from
JC; at Iheir mid-points erect ordinates equal to i-Sf, giving points
on the curve. Proceed in like manner for other points until a
sufficient number have been located.
84. Given R and / to Find the XSxternal B,
In Fig. n S=GB=OB- 00,
But OB = R sec J/ and 00 = i?.
.*. i? = i2(secii~l) = i2ex8ec J/. . . . (24)
By Table IX.— Find E for a 1" curve for an intersection
angle /; then
E,
E =
D'
(24a)
85. Given T and / to Find E. ,
In Fig. 17 draw BC perpendicular to AB, and produce AO to
b/
intersect BCt^i C. BCis parnllel to AO, and the triangles AOO
and CGB&re similar; hence BC = BO = E. In the right triangle
ABC, angle BAG^\BAF= \L Therefore
(25)
E= rtan J/. . .
EzERCiBS.— Derive equaitiou (25) from (24).
LOCATION.
49
86. Given if and /to Find ^.
From trigonometry,
sec J/ =
Insert this iu (24) and we get
cos J/*
E=R
1 ~ cos J7
cos 4/
(«)
But from Fig. 17. if = i2(l - cos \I), Substitute in (a) :
M
E-.
cos \I
= M sec J/.
87. Given jSTand /to Find R
From (24),
B E
i? = -
sec J/ — 1 ex sec J/
88. Given /and i? to Find /.
From (25),
E
^E
cos J/
vers \I'
T =
= J^cot J/
(27)
(28
tan J/
89. Given the Chord G and Degree of Curve D to Find
the Chord Deflection Offset d,
Iu Fig. 18 extend EA to if, making AH = EA = ^45; join
FiQ. 18.
£r and B and draw ^-fiT to the mid point of HB. Then
HK= EB= Csin 1/).
.-. d=2 HB = 2(7 sin i/)
(29)
50. A FIELD-MANUAL FOR ICAILUOAD EXGINEEUS.
When = 100',
(f = 200 sin |2) (29')
\n
If we write sin i-Z> = ^ from (12) in formula (29), there results
<« = -^- • . . (30)
For curves up to T» G = 100'; hence
^^-R-' • • (80)
For curves from 7° to 14'', C = 50'; therefore
^ = ^ (80")
For R write -^-, and (30'), for G = 100, becomes
.= SZ, = ,.««„ ,8,,
and for G = 50, (30") becomes
d = J-^2> = .43632> = .873.g.. . . . (81')
Example.— Find d for a 6** curve, G = 100 feet.
By (29'), cf = 200 X 0.05234 = 10.47 feet.
By (80'), d = g^ = 10.47 feet.
By (81), d = 1.745 X 6 = 10.47 feet.
90. Given the Chord G and Degree of Curve D to Find the
Tangential Deflection Offset /.
Ill Fig. 18 make EF (tangent at E) equal to EA, and join F
with A. Draw EO to tlie mid-point of FA. Angle AEO =
GEF = \D\ hence, from the figure,
AG=zQF= G&iniD.
\ t = 2GsiulD, (82)
LOCATION. 51
When G = 100 feet,
< = 200 siu JD (33')
Since JD is small, we may write, without material error,
in
sin iD = i sin ^D; then, writing sin JZ) = ^, as in 89, we get
' = M (^>
Making G = 100 ft. aud writing R = ^ gives
When C = 50 feet, (33) yields
t = .218Z> = .436 X ^. ..... (33")
Example.— Find ^ for a 6** curve, 'C = 100 ft.
By (320 i = 200 sin 1^ 30' = 5.24 ft.
By (33'), = .873 X 6 = 5.24 ft.
91. To Find the Subtangential Deflection Ofiset t' for a
Subchord G'
First Method.— By formula (13) fiud the angle at the center
subtended by the subchord G'\ call this angle U, From (32),
i' = 2C7'sinii)' (34)
Second Method.— In Fig. 19, wiih Ens center strike the arcs
FO and AH, taking EF - G' and
EA = G ; prolong EO to B. Now
assuming tbat the chords G' and G
are proportional to their central
Angles we have
AB t / N
-C-^G" ' • (^).
From the similar sectors EFO Fio. 19.
and EAB, since EB = C,
Ji_G'
AW^ a
W
52 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Multiplying (a) and (b) together, term by term.
Whence
-(?)■•
f=i[^\ (35)
ExAMPiiB.--Find t' for a 7' curve when (7 = 60 ft.
Here
By (34),
By (32'),
By (35),
fU)
ly = ^ X 7* (very nearly) = 4** 12'.
^ = 2 X 60 X 0.01832 = 2.20 ft.
t = 6.11 ft.
92. To Find the Tangent OflFset z.
In Fig. 20. EB=e is the required offset. Let AS = n chains =
lOOn feet. AE=FB, the half-chord
having the mid-ordinate AF = EB ;
hence we have, by formula (18),
2 = |»«2). . . . (86)
In this formula we may take n to
be either the length of AE or the arc
AB, in chains. If taken equal to AE
the offsets will be slightly too small,
while if taken equal to AB they will
be a little too large. The use of the
formula is limited to small values of
n and D, as was pointed out in 83.
(See Caution.)
Formula (36) is easy of application and of frequent use in
locating curves by offsets from the tangents. For curves up to
4" n may be as great as 3, but for sharper curves it should
be less.
Example.— Find six offsets to a 4" curve at points 50 ft. apart.
measured around the curve.
FiG.ao.
LOCATION. 53
jBy successive applications of (36) we have
for» = l, 2 = JxiX4= 0.88 feet
n = l, « = J X 1X4= 8 50 "
» = f « = i X f X 4 = 7.88 "
n = 2, « = jx4x4 = 14,00 "
n = i « = JXV-X4 = 21.88 "
n = 3, « = JX9X4 = 31.60 "
The last value of z is in error by about 0.2 ft., but for setting
stakes on construction this difference is not material so long as
the alignment beyond this point does not depend on it.. In
setting track-centers the completed road-bed is available and the
stakes may be set with the transit, in the usual way.
93. l^ifiference in Length of a Circular Arc and its Long
Chord.
First Method. — Let the central angle be a degrees. By (13),
Changing degrees to circular measure, a (in ;r meas.) = —
180
t= ^, The length of arc is /2a = ^^. Then
_•
Arc — chord = B-—^ —c (37)
07. o
Second Method. — An easy approximation may be found as
follows :
Referring to Fig. 17. AE = c, GF= M. Let -4(? = ft = ^ 4- aj.
From the right triangle AFQ
From which a? = — ; — (a)
64 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
NeglectiDg the x in deDOiuiuator as small compared with e
gives
*=T <*>
Alfl
Then will 2ft-c = 2a» = =- (88)
From Huygens' approximation to the length of a circular arc
(see WUliamsou's Differential Calculus, p. 66), arc = — 5 — .
8
Therefore
Arc — chord = —^ c = 4(2* — 0). . . (0)
Inserting the value of 25 — c from (38) gives
Arc — chord = -g— (d)
When the arc is not very great we may write c = 100;ii , where
ni is the number of chains contained in the arc AE. From (18),
remembering that ni = 2n,
if=0.218ni«I^.
Inserting these values of c and if in (<f).
8 (.2l8)*ni^i ?* ^ _1_
3 lOOw, " 800"
Example.— Find the difference in length of arc and chord of
a 4** curve when ni = 6 stations.
The central angle is 4x6 = 24°; then, from Table IV,
c = 595.74.
By (37),
24
Arc - chord = 1482.7 X ^ - 595.74 = 4.84 ft
07. «5
By (39),
Ar<
Remark.— Formula (38) is interesting as showing what a com-
. , , 6 X 6 X 6 X4X4 . . . .
Arc - chord = ^^r = 4.82 ft
LOCATION.
55
paratively small iucrease in leugtb of llmt is caused by a consid-
erable latei-al deflection in alignment. For instance, a lateral
deflectiou of 2000 feet is made at the mid-point of a line 40,000
feet long ; what will be the increase in length?
By (38) the increase is ^^^' = 200 feet, giving for the
increased length 40,200 feet.
40,000
B. Locating Simple Cijrves,
94. To Locate a Curve with the Chain by Ofiiidts from
Chords Produced.
In Fig. 21 let the P. C, fall at B. If BC is a full chain, prolong
Fio. 81.
the tangent AB to IT, making Bff= BC; HO will equal t, which
may be calculnted by (32) or (33'). With B as center, strike an
arc with radius BIT, and with H as center and t as mdius strike
an arc , at G, where these arcs intersect, set a stake. Produce
BC to JSr, making CK-BG-CD\ strike the arc KD from C as
center ; make the chord KB = d, calculated from (29'), (30'), or
(31). Set a stake at D and proceed in like manner for the other
points until the P.T. is reached, where FPis made equal to t
Usually the P C. does not fall at a full station ; then EC = t\
which may be found by (34) or (35). Using this value of t\ we
locate C as above. At B make BR = i'. and prolong BG to
L ; make LD = t and set a stake at D. EM will equal d, and
may be located as before.
We may regai-d KD as equal to KL + 1, and, finding, KL,
^ 56 A FIELU-MAKUAL FOR BAILROAD EKGIKEERS.
measure KD and set D without locating R. To do this we have
the similar triangles BHC and CKL, from which
CK BC
and therefore since KG = CD,
In like manner at F we have
PN=t^, and J^P=</
hence
Make EQ = ti\ prolong QF, and we have the tangent at F.
ExAMPLB.— Given the P.C. of a 5"* curve at 106 + 20 and the
angle of intersection 22"*, to locate the curve.
22
Here £ = -r- = 4.4 stations.
5
Therefore the number of the P. T, is
106.20 4- 4.4 = sta. 110 + 60.
BCiu this case is 80 ft., and by (33')
t = 0.873 X 5 = 4.37 ft.
By (35), r = 4.37 X f^)'= 2.80 ft.
Set off HC = 2.80 ft., and at D make
KD = 2.80 X ^ + 4.37 = 7.87 ft.
At E make ME = d = 8.72 by (31). This will be at sta. 109 ;
at 110 set a stake by offsetting 8.72 ft. The last chord is 60 long,
and hence the offset
]^F:=^ 4.37 X ^ + 4.37 X (j^J= 262 + 1.57 = 4.19 ft.
Make EQ = 1.57 ft., and prolong QF, the terminal tangent.
LOCATION.
67
H-
95. To Locate a D Degree Curve by Offsets from Tangent.
Let AMy Fig. 22, be tangent at A, and E, F, 0, etc., points on
the curve. Tlie offsets BE, CF, ^
etc., maybe found from formula lT
(86),
z = J»«i),
either by taking equal intervals, ^,
AB, BO, CM along the tangent or
by taking E, F, G, etc., at regular
stations around the curve and
using the arc length instead of
the tangent.
When the arc AO is large, or
strict accuracy is required, we
proceed to find the offsets at
regular stations and the lengths
of AB, AG, etc. First find E
from (12) or (12'); then from triangle OEL,
Fig. S2.
BE= AL = R(l - COB D) = B vers D,
AB= LE= B sin D.
lu like manner
CF = AH = R{1 - cos 2D) = R vers 2A
AC^HF = B sin 22),
and so on for any number of stations.
Should^ fall at a plus station, we first find the angle i>i at the
center, then
BE-B vers Z>, ,
AB-B sin />, ,
CF - B vers (Di -f 2>),
AC= 72 sin (Z>, -f /)),
etc. = etc.
The ordinates BE, CF, etc., are evidently equal to the mid-
ordinates for long cl^ords 2LE, 2HF, etc.; hence we can. if
A, E, F, and O, fall at full stations, take them direct from
Tabic V; then take the long chords from Table IV and dividing
these by 2, get the required coordinates.
58 A field-majNUal for railroad engineers.
ExAMPLB. — Locale three stalious of a 4° curve by offsets every
50 ft. on curve.
Referring to Tabic V, the required offsets are 0.87, 3.49, 7.85,
13.94, 21.77, and 31.31. By Table IV the distances measured
along tangent are 50.0, 99.94, 149.76, 199.39. 248.78, and 297.87.
With these values we can set out the curve either way from A.
Had we used formula (36) we should have had for the values
of the offsets 0.87, 3.50, 7.88, 14.00, 21.87, and 31.50.
96. To Locate a Curve by Ofibets from a given Liong
Chord.
Let FK, Fig. 23, be the given chord. We may compute the
offsets yi , y« . . . Jtf by the methods of 83 — of which formula (17^
is the most convenient, within the limits of its applicability —
and setting off these ordinates, locate the curve.
Or w^ may setoff the mid-ordinate M= R^GvaFOA at A,
and at C set off y, = Jf — i? vers D, making
GEMWhe
AC = HL = E sin D.
M- R vers 27), and AE = R sin 2i>.
Another Method is to find the angle KOFhi the center, and
by Table IX deiermiue BA^ M\ then by Tables V and IV
LOCATION. 69
determine BL, 5.V. LH, and ^'G. Then tiC =^ M - BL, which
sel off at C, uud other points in like manner.
Example.— Given the P.O. of a 4** curve at station 160 + 75,
the angle between tangent and chord = 9', i-equired the offsets
necessary to locate the curve.
Here 7=2x9 = 18^
18
.-. X = T = 4.50 stations.
4
Hence the RT, falls at 160.75 + 4.50 - sta. 165 -f- 25: The
mid-point on curve B falls at sta. 163. By Table IX,
3,= !?^ = 17.64 ft.
4
By Table V the mid -ordinate for two stations of a 4"* curve is
BL = 3.49.
Hence HG = 17.64 - 3.49 = 14.15.
By Table IV, HL = AC= 99.94 ft.
Measure AG = 99.94 ft., and set off' GH— 14.15 ft., and drive a
stake at E. In like manner find
(?i?=3.70 and ^Z?= 199.3J ft.
The points P and Q are also located by means of the coordi-
Yiates just determined.
If B had fallen at an odd station, the curve could have been
located in the same manner, ^and P being 100 ft. from B^ Q and
q 300, etc.
97. To Locate a Curye with Transit and Chain when the
Degree D or Radios R is Known.
If R is given, determine D by (13); then, since the angle in
the circumference of a circle is half the angle at the center sub-
tended by the same chord, we may locate points- on the curve by
successive deflections from the tangent.
In Fig. 24 let the P. G. be at A, at which point set the transit,
and with the vernier plates clumped at zero place the telescope
in tangent either by sighting the P.L or by backsighting to some
point in the tangent Deflect from the tangent half the angle at
the center for the sub-chord or chord, and direct the head chain-
man into line while the rear chainman holds his end of the chain
60 A FIELD-MAKUAL POH RAILROAD ENGINEERS.
at the trausit, the cbaiu being kept taut. The stakeman drives a
stake at the point where the head cbaiuman's flag rested, and the
rear chain man advances to this point. Deflect ^D from the chord
AB just run, and while the rear chainman holds his end of the
chain at B direct the head chainman into line at C. Other points
are located by deflecting an additional ^D for each chord length
measured, until a point E is reached to which it is desirable to
Fio. 24.
move the transit. The angle FAM should not exceed about 15".
Move the transit to E, backsight to Ay and deflect FEA = EAF,
when the telescope will be in tangent, and the curve can be con-
tinued until it is again necessary to move the transit. At the
P. T. put the telei^cope in tangent by backsighting to the point
last occupied by transit and deflecting the tangential angle as at
E, The line may now be continued.
98. The Index-angle is read on the vernier-plate, and is the
angle between the tangent to the curve at the P. 0. and any other
line passing through a point on the curve when the telescope is
directed along this line. It is most frequently taken as the angle
between the initial and any subsequent tangent to the curve.
Thus at E the index-angle equals EFP = 2FAE. At any point
on the curve the index-reading in tangent may be found by the
following rule, which may be easily deduced from a figure:
From double the index- angle that fixed the point subtract Vie index-
angle in tangent at the last point; tJie remainder is iJie index-angle
required.
99. Subdeflection-angles may be found by (13) rigidly, or
approximately (and with sufficient accuracy except when D is very
large) by assuming the central angles to be proportional to their
chords. Thus on a 4*" curve the central angle for a sub-chord of
25 ft. would be T, and the subdeflection angle 30'.
LOCATION. 61
Example -Locate a 4** curve to left when the P.C. is at sta.
81-f-25and/=B2'86'.
Here L ^ — f- = 8.15 chains.
4
Hence the P. T. will fail ai 81.25 + 8.15 = sta. 89 + 40. The
first sub-chord is 75 ft. lon^, hnd the first deflection-angle will be
found by (18).
.-. ^5 = 1** 30'.
By the approximate rule, since ^D = 2"*,
2 ""100'
whence i^ = 2 X f = 1** 30' as before.
With transit at P,C. deflect 1* 30' from tangent, measure 75
feet, and set sta. 82. Then a deflection of 3** 30' will determine
83, 5** 30' sta. 84, V 30' sta. 85. Now remove transit to 85. and
with vernier at T 30' backsight to 81 + 25. Reverse telescope
and set vernier at 15° 00', when the telescope will be in tangent.
An index angle of 17" will fix 86, and so on.
The last chord will be only 40 feet long, for which the sub-
deflection -angle is yVtf ^^ 2°> ^^^^ ^^' ^^'' '^^^ index-angle fixing
the PT. is therefore 23" 48'.
To get in tangent at 89 + 40 backsight to sta. 85, with vernier
at 23° 48' ; then by the rule of 98 the index-reading is (23° 48) X
2 — 15° = 32° 36' = /. Set the vernier at this reading and run
tangent.
Caution. —It is not good practice to set more than 4 or 5 sta-
tions on curve from any' one point. Mr. Shunk gives the limit-
ing angle to be deflected from tangent as 20°, and says 15° should
rarely be exceeded. (Field Engineer, p. 82.)
100. The Transit Notes may be conveniently kept in the form
below, which shows the notes for the last example.
When possible the tangents should be run to intersection, the
angle I measured, and the tangent distance calculated. Then
62 A FIELD-MANUAL FOR BAILROAD ENOINEERS
Station.
1"
^1
ii
.
Remarks.
00
-1-40
QP.T.
0«48'
23043/
82«86'
N27«»88'E
N27'»30'E
89
230 0'
88
21° (y
87
19« 0'
86
17«» 0'
85
O
7° 30'
15* 0'
84
5° 30'
88
2«> 0'
8** 30'
88
1«30'
1«30'
4'» C.L.; P.I. set.
+!»
O PC. 4*C.L.
0«» 0'
O* 0'
0» 0'
J = Si' 36'; T ^
418.9 ft.
81
N(K)«»12'E
NeO'lCE
measure along tangents and set P.C. and P,T. from the P.I.
Wbeu tbe curve is run in, the position of the P.T. thus found
should agree with the one set from the P.I. If the error is
greater than the circumstances of the case permit, the curve
must be rerun and tangents remeasured.
101. Another Form of Notes, and in some respects a better one
than the above, is given below. The index-readings are com
puted as though the entire curve were run from the P.C. The
notes for ihe last example would appear as below ;
Station.
II
-J
II
Si
p
h
r
Remarks.
90
+40
0P.5.
0«>48'
le-is'
32«>36'
N27'»36'E
N27«»30'E
89
15<»30'
N8
13«30'
87
11«80'
8()
9«30'
85
7*>30'
81
5*>3C'
bQ
20 0'
S'SO'
8-J
1»30'
1*>30'
!• curve left;
-f25
OP.C7.4«'C.L.
0« 0'
0" 0'
PI set. i=32<»36';
r= 418.9 ft.
81
N 60*12' E
N60»10'E
The computations are all made before beginning the work, and
the notes have the advantage of permitting the tracing of the
curve either way from the instrument without additional rompu-
LOCATION. 63
tations. Suppose the traiisitmaD to have run the curve from the
P.C. tosta. 85, lo which point he removes the instrument. He
there set« the vernier at 0"— the ttngle on limb when telescope
was in tangent at i he P.C. — then sighting theP. (7. he reverses
the telescope and deflects to 9' 30', which will fix sta. 86. Had
the tangent at 85 been desired, a reading of 7" SC— the angle that
located that point — would have put the telescope in the plane de-
sired. A reading of IT 30 fixes 87, and so on to the F.T.
Removing to the P,T., the plates are clamped at T 30', and a
backsight to sta. 85 taken ; then deflecting to 16** 18', the tele-
scope is in tangent at the P. T. Had it been desirable to set 84
from 85, a reading of 5° 30' would fix that point ; others may
be found in the same manner.
Any convenient form of notes, which are intelligible to another
engineer who may have to retrace the curve, may be used, but it
is desirable that some general form should be employed. Either
of the preceding forms seems to meet ordinary requirements.
C. Obstacles.
102. To Pass an Obstacle on a Curve.
FmsT. Suppose Uie obstacle to he one obstructing vision at one
station only.
In Fig. 25 suppose transit set at A, and B and C located from
that point, but the next full station, 12, to be invisible from A,
Fig. 25.
Set a plus station at E, as near the obstruction as may be conven
ient, then set FlOO feet from E. Next make FG= 100- CE,
and locate O with the corresponding deflection-angle. Other
stakes may be set beyond G, or the transit may be removed to
that point and the curve beyond traced.
Second. Suppose the line of sight qhsi^ured for more than me
station, as in Fig. 26,
64 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
If trausit is at Ay deflect an angle HAB that will clear all o\
structioDS, aud at the same time cause B to fall at a full statioa.
Then by Table IV, Table IX, or by formula (16) calculate the
long chord AB ; measure ^2? and move transit to B ; then defect
Fig. 26,
the angle ^BC=jR4-ff when the telescope will be in tangent.
The curve may now be run both ways from B,
If it happen that some stations, as E and F in the figure, are
still invisible, they may be located by offsets from chord or tan-
gent.
Example.— Let the curve be a 3* curve to right ; angle HAB
= 7** 30', the deflection-angle for 5 stations. By Table IV the
long chord is 498.63 feet, which can now be measured and a hub
set at B ; then making angle CBA = 7'' 30', the telescope will bo
in tangent and the curve can be traced either way.
103. To Locate a Curve when the P. G. is Inaccessible.
^ In Fig. 27 let the P.C. at B be in-
Q
FiQ. 27.
accessible ; it is desired to reach a
— point H on accessible ground.
First Method. — Assume a
point jET on the curve such that a
line AH from an accessible point
A, on tangent, will clear the ob-
stacle ; for convenience H should
be at a full station. The arc BH
aud central angle, which equals
ECFy are then known. Calculate
BC=Thj (14) or (14a) ; then
since AB is known, AC, = AB-^
BCy is known.
Now in triangle il (7^, from trig-
onometry,
tani(7i-a) ^ AG - C H
tan 4(7* -fa) ~ AG+GEC
LOCATIOK. 65
But (h + a) = c; hence
ten4(A-a) = ^J-p-^tanlc (40)
Then i(A + a) + i(A — a) = h, the larger angle, and
4(A -fa) — J(A — a) = a, the smaller angle. AH may be
found by the law of sines, or by drawing CE perpendicular
to AH, when
AH = AG cos a + CH cos h (41)
Example.— The P.G. of a 4° curve is at sta. 141 -f 25, and it
is desired to reach the point ^from sla. 139 on tangent.
Suppose H be assumed to fall at sta. 147 ; the curve length ic
X = 147 - 141.25 = 5.75 chains. Then angle c = 5.75 X 4 =
23" (y. By Table IX the tangent distance for a V curve is
Tx i- 23'' = 1165.8 ft.
By (14a), T = ^-^^ = 291.45 ft.
Now AC= 291.45 + 225 = 516.45 ft.,
and
AG+ CH= 516.45 + 291.45 = 807.90,
while
AC- (7fl^= 225ft.;
hence, by (40),
tan i{7i - a) = -^7r=-^ X 0.20345 = 0.05666 = tan 3" 15'.
"07.9
Therefore
and
A = 11** 80' + 3° 15' = 14** 45',
a = 11'' 80' - 3'' 15' = 8' 16'.
By (41),
AH= 516.4£» X 0.98965 + 291.45 X 0.96705 = 793.0 ft.
At A deflect 8" 15' from tangent, measure 793.0 ft. and set a
hub ; move to this point, backsight to .1 and deflect 14" 45' into
tangent, then trace in the curve.
66 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Second Method. — If F, any assumed point in tnngcnt, is
visible from J., AF may be measured by some indirect method;
then AF ^ AB = T, The tangent for a 1" curve having same
iutersection-angle, KFG, is Ti = TxD; find this value of Tx in
Table IX and take out the corresponding- value of 7. With
transit at F deflect the angle KFG, measure FO = FB = T, and
set hub at O, The station number of G is found by dividing the
central angle, = KFG, by the degree of curve Z>. Move to G and
trace the curve.
Example.— Let AF measure 490.5 ft. from sta. 139 of the last
example. Then^J5 = 325 ft., and BF= 490.5 - 225 := 265.5 ft.
265.5 X 4 = 1062 ft., which by Table IX is the value of T, for
/= 21^ Set transit at F, deflect 21% and measure FG = 265.5 ft.
L= -7-=^ 5.25 chains;
4
hence G will fall at 141.25 + 5.25 = sta. 146 + 50. Move to Q
and run the curve both ways.
Third Method.— In Fig. 28 let the inaccessible P.C. be at B,
and let it be required to reach E from a point G on the curve
prolonged backwards from B.
At a given point A on tangent cal-
culate the tangent oifsel by (86) or
the methods of 95, then set this off at
right angles to AB ; set the transit at
C and turn oS ACL = 90<» - COB,
when the telescopy will be in tangent
at C. COB may be found from Table
IX by multiplying AC by the degree
of curve and taking half the iutersec-
tion-angle corresponding to the mid-
ordinate that equals this product. Now deflect and measure
ECL, then by (16) or (16«) calculate CE, which mejisure. Move
to E and deflect LEG = ECL and the telescope will be in
tangent. The central angle BOE = 2LEC - BOG, from which
the arc 5^ and number of sta. E may be found.
Example. — Take the same example ns in the last two cases.
A is at sta. 139, B at 141 -I- 25; hence AB = 2X5 stations.
Fio. 28.
By (36). z = AC=lX{'^ 2CV' x 4 - 17.72 ft.
LOCATIOK.
67
Or by Table IX the angle correspondiDg to the long chord
(2 X 2.25) X 4 ^ 1800 ft. is 18'' 4', for which the mid-ordinale is
71 06
71.06 ft. For our 4" curve the mid-ordiuate will be —^ =: 17.77
ft., whicb equals AG and agrees closely enough with the value
for z above.
Make angle BAG =^90\ and measure -4C = 17.72 ft. Move
to G and sight to A, then make angle ACL = 90" — (9*2') =
80'' 58'. Suppose an angle LCE = 16** 1' to clear the obstacle.
By formula (16),
GE = 2R sin (16*' 1') = 2 X 1432.7 X 0.27592 = 790.6 ft.
Measure along GE 790.6 ft. and set a hub; move to E and run
the curve.
GE might have been found by means of Table IX, for the long
chord of a 1' curve having i = 2LC^ = 32" 2' is 3162.0 ft.;
divide this by 4 and there results GE= 790.5 ft.
104. T9 Pass to Tangent when the P.T. is Inaccessible.
This is just the reverse of the preceding problem, and may be
accomplished by reversing the processes described above.
When the P,T., however, falls in or beyond a river or lake
obstructing the ordiunry methods of indirect measurement, the
case merits a f^pecial solution.
FiRsr MuTHOD —In Fig. 29 let the transit be at A, and B the
P.T. From tiie known station
numbers of -4 and B the length of
curve and angle / may be found;
then, by (14), Aa= R tan J/, or,
T^'4-P
hy{Ua\AC= -^-^.
Move to Cand deflect the angle
/; set a stake F, and one at some
other accessible point E\ measure
angle .ECF = c. Move to F and
measure the angle EFG and the
side EF\ then in triangle EGF
angle e = 180" - (c +/); by trigo-
nometry
CF=^^.EF. (42,
sm c ^
QS A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Siuce BC=zAC, there results BF= CF- ^C;'aDd as the sta-
tion Dumber at B is known, that at F becomes known, and the line
may be continued.
If B is not the P,T., measure back the distance FB, set transit
at B, and continue the curve.
Example.— Let the P.T. of a 2° C.L. fall at sta. 205 + 50— an
inaccessible point; suppose A at sta. 200, angle c = 40°,/= 80%
£'2^=310 ft.
Here
i=5.50x2 = ir 0', and d = 60\
By (14a), T = ^5?^ =. 275.87 ft.
From (42), applying logarithms.
log (7F= 2.49136 -f 9.93753 - 9.80807 = 2.62082.
Whence CF= 417.7 ft. Then BF^ 417.7 - 275.87 = 141.8 ft. ;
therefore the number of i^ will be 206 4-91.8.
Second Method.— In Fig. 30, with the transit at any point A
on ihe curve, assume a long chord AB
and calculate the angle CAB\ deflect
this augle from the tangent AG^ and set
a point ^ beyond obstruction ; set also
a stake at C in tangent.
Move to E and measure AEG and
side EG. Compute AE from the trian-
gle AEG. If this is greater or less
than the kngth of the long chord AB,
take their difference BE and set a hub
at B. With the transit at B tmce out
the curve.
Example. — Given A at sta. 210 of a
3"* C. L., angle a = 12^ b = 92% EC
= 181 ft. Then c = 76% and by solving
the triangle AEG. AE= 844.7 ft. By Table IX the long chord of
Fig. 30.
a V curve for /= 24*^ is 2382.6 ft. ; therefore AB =
2382.6
= 794.2
ft. Now will ^5 = 844.7 — 794.2 = 50.5 ft., which is the dis-
tance along EA that transit must be moved back from E,
LOCATION.
69
106. Given the Perpendicular p from a Point to a Tangent,
to Find the Point on Tangent at which to Begin a Curve of
Given Radius which will Pass through the Given Point.
First Solution.— In Fig. 81 let P be the point, BP the per-
peudicular. We have to find
BA=x, ^^
From P draw PC parallel to
AB ; then in triangle OPG
ip = «* + (5 - py.
From which
X = i^2Rp - pK
(43)
Second Solution.— Consider
p = AC &s the mid -ordinate for
a long chord = 2x ; then pX I>
= the mid-ordinate for a 1" curve
for a central angle equal 2a.
The coiTesponding long chord may be taken from Table IX.
Then
Fio. 31.
i x.q,
2 D •
{48a)
ExAMPLE.-Given ;? = 80 ft., 2> = 4* (5 = 1433.7), to find a?.
By (43), X = V85,962 - 900 = 291.65 feet.
By the second method,
30x4 = 120,
the mid-ordinate for a 1* curve corresponding to an angle o'
23'* 29', for which the long chord is 2332.6. Now. by (43a),
fi 4
106. In Fig. 31, Given x and p to Find the Radius of a
. Curve Tangent to AB at A and Passing through P.
From (43),
i? =
_ aj* + p*
(44)
70 A FIELD-MANUAL FOR KAILROAD EKGINEfiRS.
107. Given the Location of a Point P referred to the P. L
to Find the Radius of a Curve through P which will Unite
the Given Tangents.
Fio. 32.
In Fig. 32 suppose BC — Z, BP = vi known, and angle a cal-
culated ; or PC nnd a may be measured on Ihe field.
From triangle CAO,
6 = 90'' - (a + J/), and CO = i? sec JZ
Now from triangle PCO,
s»n y = ^ sin ft.
Inserting values of PO and GO,
sin y =
___ R sec \I
B
. sill ft = sec J2 . sin 6 =
sin ft
cosfT'
(45)
: ?i eqiiation from which the unknown B has disappeared. Nexi,
from the same triangle, since x = 180** — (ft + y),
sm X
"When / = 90*, it can easily be shown that
B = l-{-m + V2im, .
(46)
(47)
LOCATION.
71
108. To Locate a Tangent to a Ounre from an Outside
Point.
First Method. —In Fig. 83 let P be the point and AHB the
E H
rurye. Run a trial-line PA cutting the curve in A and B,
Measure PA and AB ; or measure PA and angle a between the
chord AB and tangent AL, Then
AB = %AC = 2B sin a,
OC = R cos a.
By geometry, PE = \^PA X P^B, P£7 being the required tan-
gent. From the figure,
GO
CP
PE'
tan » =
tan m -
At P deflect the angle I = m— n from PA and run the tangent.
Second Method.— In Table IX find the long chord for t
central angle 2a ; then
AB:=z2AC =
CH=^.
and CO=:B- CH,
We may now proceed as before.
72
A FIELD-HAXUAL FOR RAILROAD EKGIXEERS.
1
109. To Run a Tangent to Two Located Curves of Contrary
Flexure.
First Case.— In Fig. 34 let FK and LE be the curves, and
KL = 'p measured on the ground.
Fig. 34.
Let FE = ^ be the required tangent.
Draw OxH parallel and OiH perpendicular \XiFE\ from the
triangle Oi ifOa , since FH = Hi ,
whence
«= i/2(/?»-f /?,)!> +p».
(48)
Also.
cos a =
(49)
The arcs FK and LE may be found from tbe angle a and the
known curvatures, after which tbe points^ and Em&y be set.
If ^ is given and p required, it may easily be found from (48).
Second Case, p not kiunon.
Set the transit at a point A on one curve and note the bearing
of ibe laugent to the curve at that point (see Fig. 34); the bearing
of the radius 0%A differs from this by 90**, Run a line ABC of
one or more courses to intersect the other curve at C, Note the
bearings and lengths of these courses and the bearing in tangent
at C, from which calculate the luariiig of C<fx 7^, and R^ being
known, the latitudes antl uepnriuics arc next calcuhitcd. Let OiJV
LOCATION.
73
be the sum of the northiDgs or southings, Oii^Tthesumof the
eastings or westings ; from the triangle Oi OtN,
tan& =
OtN'
and
OiO, = VOxN^ 4- 0^N\
As before, FE is the required tangent and OsS* perpendicular,
while Oi^ is parallel thereto.
cos a =
Rx+Rt
Angle FO^N= 6 - a is the bearing of O^F, while AOtF=
e — b-\-a is the angle of retreat from the known jwint A to F,
where the tangent may be run. The length of t= OiHis
t = 0,0i sin a.
D. Change of Location,
110. To Locate a Curve Parallel to a Given Curve.
Let p be the perpendicular between parallel tangents, and sup-
pose ABC located (see Fig. 35).
If there are no restrictions as to the
position of the points By F, and G
on the second curve, we may cal-
culate the new degree of curve Di
for a radius /?, = i? -f p, by (13),
and trace the curve from any
point, as E, Thus
iA =
50
R^
50
R + P'
If, however, points on the radii
through A» B, and G are wanted,
they are gotten by using the same degree of curve D and com«
puiiug the length of chord FE. From similar triangles,
Fio. 85.
EF
Ri ''
AB
R ''
100
R'
^ H-i A FIELD-MANUAL FOR KAILROAD EKGlNEfiRS.
whence
J?i7» = 100 ~ = 100 :?i^ = 100(1 4- 1). . . (50J
Had EFO been the located curve, with radius R, we should
have had
^5 = 100
(51)
111. To Change the P.C. of a Iiocated Curve so that P.T.
will Fall in a Given Tangent Parallel to Terminal Tangent of
Located Curve.
Let AB, Fig. 86, be the \o,
cated curve; FE, the tangent
ill which the P.T, rnubt fall.
Let the distance between tau-
geuts be HE = p.
Draw BE and 00' parallel to
AF; evidently AC = 00'^BE\
0' being the new position of
center.
In triangle BEE,
BB-AC- -j?-^ = p cosec /.
sin/ ^
(52)
Set the new P. C. by measurement from A, and run the curve
CE. A.ny system of straight lines and curves may be treated as
above, provided 1 is the angle between initial and terminal
tangents and p as before.
Example.— A located 2*' 80' curve, having / = 25', ends in a
tangent 25 ft. outside of desired tangent. Find the change in
position of P.C.
By (52),
AC = 26x 2.86620 = 59.16 ft.
LOCATION.
75
112. To Find the Change in Raditu and Position of P. (7. if
P.T. ia Required to fall on the same Radial liine but on a
Tangent distant p from, and parallel to, Terminal Tangent to
Ijocated Curve.
In Fig. 37 let AB be the located and CE the required curve.
Draw the parallel chords AB and
CE. Draw CA'and^-F perpendicular .A^. C_K
to AB. The &ng\e8 FBE:=CAH-iI^
From the figure,
CH= AG sin ^I,
BF - BE cos \I = p cos \L
Equating, p -
-4(7 sin J/ = p cos J/,
whence
Fio. 37.
4(7 = p cot J/.
In the triangle OPOu dP = AC, OP = B - Bi, and
(B—Bi) iSLuI=zAC = p cot i/,
or B- Bi = AC cot I =zp cot J/, cot /.
Therefore
i?, = 12 - -4Ccot / = i2 - p cot J/, cot 7. .
From trigonometry,
^ , -. sin / , ^ , cos /
cot i/= , and cot 7 = -; — =.
1 — cos 7 sm I
(68)
(64)
Insertiug these values in (54) gives
sin 7 cos 7
Bt=B--p.
= B''P
cos 7
1 — cos 7 ' sin 7
From trigonometry, ex sec 7 =
• '• Bi = 7J —
1 — cos I
vera 7
'B-p-
cos 7
vers 7*
cos 7'
P
e* sec /'
(64)
76 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Example. — A 2° SO' curve strikes 25 ft. inside a tiingent in
which the P. T. must faU. Find the necessary change in radius
and position of P.C. when / = 26^,
By (53) the change in P. G. is
By (54').
AC-26X 4.51071 = 112.77 ft.
25
i?, = 2292.01
.10338
= 2050.38 ft.
By Table I we find this to be the radius of a 2* 47' 41" curve.
113. Given a Located Curve uniting Two Tangents to
Find the Change in Position of P.O. or in Radius for a Given
Change in the Intersection-angle.
First Case. — Radius unchanged.
In Fig. 38 let BCE = / be the origi-
nal intersection -angle, FCE = /' the
new angle. From ihe figure,
AO =^ AG- OG,
AG = Ii (tan J/ - tan J7'). (55)
By Table IX.— From the table, foi
angle /,
T =
Fig. 38.
For/'
r =
Then
D *
AG= r- T'.
Second Case.— P.O. unchanged.
Here the tangent T for the two curves is the same, and
therefore
iJ, tanii' = /?tanli;
. (56)
Whence
ijj = /?tt.ni/.cot jr.
LOCATION. 77
By Tabub IX,
whence D. = 'ElAll . jy = ^» ^ \
1 14. To Find the Change in i? and P. (7. for a Given Change
in /, the P.T. remaining unchanged
/O,
Fig. 39..
In Fig. 39, from the triangles OBQ and OxBH,
00 =z R cos /
and OiH= Ri cos /i.
Now QA = HF\ hence
i?i - i?i cos /, = i2 — i? cos L
Whence
i?.=5[z:i£i4=fil?!if (67)
1 — cos i, vers /i • v /
Also, FA^BQ^BR- BQ.
Inserting values of BE and BOy there results
2?*^ = i?, sin /i - i? sin /. (58)
116. Qiven a liOcated Curve to Find the Change in R for
a Given Change in 7\ I remaining unchanged.
In Fig. 40, from the triangles OAC and OiEC» since
EA = EC- AG,
78 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Ml tan il-- Ri&u 11= EA = T' - T,
Whence jB. = i?+ (T' - T) cot J/. (59)
_E A c
Fig. 40.
By Table IX,^EA being known, T' = T + EA. Then, by
(15a),
__ T,^r
"" — T' — '
If the change in vertex of curve is wanted, there results, from
(25),
^ = (7(? = rtau i/, E' - CH= T' ton \L
Therefore GB = E' ^ E = (T' ^ T) tan iL . . . (60)
GHcau be found from Table IX after finding Di as above.
If Ri is given and EA wanted, (59) yields
EA = T' - T = (Rt - R) tan J/-
116. To Find the Radius of a Curve having the Same P,C,
as a Given Curve, but ending in
a Parallel Tangent.
In Pig. 41 let the perpendicular
distance between tiingents be p, and
AB be the located curve; AOi = Ri
is required. ;
First Method. — Draw OH at
right angles to OiE; then
OiE= OJI+HQ+QE,
or
R, = (R, - i?) cos /+ i? + p.
Fig. 41.
From which 7?, = 7? +
1 — cos 1 vers /
(61)
LOCATION. I'd
Second Method.—^, B, and E lie on the same straight line,
since / is the same for both curves. In triangle BQE angle
EBQ = J7, and
^^=8l^ = ^"^'^*^-
From Table IX, AB = h^i^lJl,
AE = AB 4- BE is Ibe long chord for curve of degree Dx\
therefore
'" AE '
If desired, R may be found by (12') or Table I.
Third Method. — Draw FL parallel to OiE\ then
CF = -——. = » cosec 7.
sini
From Table IX. AG^ —ff^'
AF=i AC-i- CF, the tangent distance for second curve ; hence
2>,=-
AF
Remark. — If transit is set up at -B, it will be well to set E
by measurement from B, to serve as a check when the curve is
run in from A.
80 A FIELD-MANUAL FOR RAILRDAD ENGINEERS.
Article 9. Compound Curves.
>l. Location Problems.
117. QiTen Two Unequal Tangents, their Intersection-angle,
and One Radius, to Find the Other Radius of a Compound
Curve uniting Tangents.
In Fig. 42, AH= Ti and Bn= T^ are the known tangents,
AOi = lit the known radius. BOn = R^ and the angles /i and /»
must be found before curve can be located.
Fio. 4:4.
Extend first branch to F, so that tangent FL is parallel to BH.
Draw UK Aud BG perpendicular to FL ; draw FB and extend
to ^; it will pass through the P.C.C., because the central angles
EG, Fund EO^B are equal Then
To = ^X = ^1 tan \L
In triangle LHK, since LH = To— Ti,
8 = KL^iTo- Ti)cosl
p = HK=BG = (To~ r,)sin/.
Now in triangle BGF Angle BFG = i/j, and
l^FG=T,'[-S''T^,
LOCATION. 81
1
taiii/, = ^.
Draw OsJf parallel to FL ; then
(J2i-i?,)8mJ, = t
whence
H^^Rx^ -T-^ = ijj - ^cosec /a. . . (68)
sin /,
Had B% been required, the equation would have been
i?i =i?, + iooflec/9.
Evidently, L^I-* /,.
In the field the points E and B may be located by running in
the curve from A as stailing-point, or run the chord
from A, and at i?^ deflect angle AFB = J/— J/, = J/, , measure
FB-l9ec i/« And BE = 2Rt sin i/,.
Example.— A 2* curve has the RC, at sta. 110, Ti = 590 ft.,
7\ = 511 8 ft., 7 = 30** 50'. Locate the curve.
By Table IX, r« = 1580/2 = 7d0 ft.
By formulas above,
« = 200 X 0.85866 = 171.78 ft.,
i? = 200 X 0.51254 = 102.51 ft.,
i = 790 + 171.73 - 511.8 = 449.93,
tan J7, = ^^ = 0.22784 = tan 12" 50'.
Then /» = 30** 50' - 25' 40' = 5* 10'.
44 ft 07
E, = 2864.93 - ^f^' = 1833 feet.
82 A FIILD-HAKUAL FOR RAILBOAD EKQINSERS.
By Table 1 this is seen to be the radius of a 8* 7i' curve.
The length of first branch is 258.3 feet, and of the second 821.3
feet; hence the P.C.C. falls at 112 + 58.3, while the P.T. is at
sta. 120 4- 79.6.
118. aWen the Long Chord from P.C. to RT. of a Com-
pound Curve, the Angles it makes with the Tangents and
One Radius, to Find the Other Radius and the Central Angles.
In Fig. 42 AB is known, as also the angles HAB — a and
RBA = h. Two angles and one side of the triangle HAB are
known, and the sides HA — Tx and HB = T^ may be found,
after which the solution is the same as in the last problem.
A solution may be reached in a different manner. I = a + h^
EAF = iJ = \{a 4- h), and BAF =z\{a + h) - a = J(6 - a).
AF = 2i?i sin \I. In triangle BAF two sides and the included
angle are now known, so BF an* angle BFA may be found;
OFB =^ il^ = il " BFA.
Then JKF = 2i?, sin J7a ,
and EB= EF - BF becomes known.
Then EB = 2i2a sin J/, = 2R, sin J/, - BF,
BF
1
whence 2?s
Evidently Ix = I - h
Rx
2 sin i/a •
(64)
119. Given the Radii and Central Angles of a Compound
Curve to Find the Tangent Lengths, the Long Chord from
P.C. to P.T,, and the Angles it makes with Tangents.
In Fig. 43 draw AE and BE from the
P.C. and P.T. to the P.C.C., then
calculate AE and BE by (16) or by
Table IX. In triangle AEB angle
AEB = 180 - {{h + /«). Two sides
and the included angle being known,
the triangle AEB may be solved for
AB and the angles ABE and BAE\
then
BAF=^BAE+\Tx,
Fio.43. ABF:r=ABE+ i/..
The angle AFB of triangle ABF now becomes known and, as
L0CATI02S:. 83
AB is known, the sides AF = I\ and BF = jTj may be com-
puted.
120. Given the Iiong Chord from F.C. to P. T. of a Com-
pound Curve and the Angles it makes with Tangents to
Find the Radii when the Common Tangent is Parallel to Long
Chord.
In Fig. 43 let QHhQ parallel to AB, and OAB = a, HBA = h
known. Then
BAE = BAG = QEA = Ja.
and ABS = EBH = HEB = J6,
Also, AEB = 180" - l(a + ft).
In triangle AEB, remembering that
sin [180 - i(a + b)] = dn K« + *),
ABsinjb
and
4^= .
sin i{a -f 6)'
^^^^^sin^a
sin i(a + by
Since AOiE = a and i^Os^ = ft, the radii Bi and i?s may be
found from formula (16), or (16a).
By (16), ^»-3-n,-p~27i^|«.3inj(«^jy . . . m)
' "" sin lb" 2 sin Jft . sin \{a + b)' ' ' ' ^^'
Example. —Required Ri and R^ , or Di and 2>t , when AB =
900 feet, a = 12% ft = 15^
By (65), i?i =2407.0 ft.
By (66), i?, = 1543.7 ft.
FTom Table I, i>. = 2** 22' 50" and 2), = 3' 42' 44".
84 A FIELD-MANUAL FOR RAILROAD ENGINBBES.
B. Obstacles.
121. To Locate a Point on aie Second Branch of a Oom-
pound Curve when the P.C.G. is Inaccessible.
Ordinarilj the second branch is located by setting transit at the
P. C, C and ruuniug the curve from that point. An obstacle on
either curve may then be passed by the methods given for simple
curves.
When the RC.C. is iuaccessible,
,Ap locate the first branch from IheP.C
and the second branch from the
P. T,, if this latter point is known.
When this is not the case proceed
by one of the following methods:
First. By meana of a long
chord.
In Fig. 44 let ^ be the P.C.a.
A some known point on first
branch, EF a tangent at B, and
AB parallel to FE. The station
numbers of A and B being
known, the arc AB and angle a
are readily found ; then
F!L = /?9 vers ft = /?i vers a.
whence
next.
vers ft =
i?i vers a
(67)
AB = Rx sin a -f- 2?, sin ft.
EL =
(68)
Defiect FAB = a from tangent at A i measure out AB ; set the
tnuisit at B and locate the second branch.
By Table IX. —Take the mid-ordinate in table for an inter-
section-angle 2a ; then
A •
Then EL x A is the mid-ordinate for a 1* curve having
/ = 2ft, from which ft becomes known. From the table now find
AL and LB, the half-chords for angles 2a and 2ft, and proceed as
before.
Second Method.— £;V ^n^n* of tangents,
JVom Fig. 44, AF= FE = ff. tan Ja.
LOCATION.
85
Set transit at F, deflect OFE = a, and by some indirect method
measure to an accessible point H.
and
BH=:FH^FE,
EH
tan \h = -5-, from formula (14).
Angle h is now known and equals OHE, which deflect from
BH\ then measure HB = EH, and with transit at B locate the
second branch of curve.
Or by Table IX.— Find AF = FE, the Ungent distance for
I =z a; then having EH measured, take Ti = EH X !>% and find
the corresponding angle, which equals b ; then proceed to locate
curve as above.
Example.— Let ^4 be at sta. 126, P. C. C. at 128 + 25; the degree
of first branch 4**, and of second 6**.
By the first method EL = 17.635 for a = 9^ and b = 11** 2'.
nearly. AL = 224.1 ft., BL = 182.75 ft., and therefore AB =
406.85 ft. Angle 6 = ir 2' corresponds to 183.9 ft. around 6-
curve; hence the P.T. number is 130 + 08.9.
By the second method AF = 112.74 ft. Suppose FH = 264 ft.,
then EH= 151.26 ft., which multiplied by 6 gives 907.56 ft,
corresponding to / = 18**. The arc EB is now 300 ft., making
^ fall at sta. 131 + 25.
C. Change of Location.
122. Having a Simple Curve Located to Find the P. C.G. so
that a Curve of Given Radius shall connect with a Given
Tangent Parallel to Tangent to
Located Curve.
Let NAB, Fig. 45, ,be the located
curve, HF the tangent in which the
second branch must end. The dis-
tance BO = p between tangents is
kno\yn from measurement. If angle
a can be found, the arc BA becomes
known and the point A can be located
from B. Draw O^L from the center
of second branch perpendicular to
Pio.i5. OxB. In triangle OiO^L, OiOi =
Bi - -Rj, and OiL = 5i - (1?, + p) ; therefore
0%
f ^ \
^
•^
L
Rt
F
*-ip-^
86 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
cosa =
Ri-Rt
1 -
Ri - R,'
(69)
Then a divided by Di gives arc BA.
If desired, Bff may be found from the right triangle BEG, in
which tlie side BO = p and angle OHB = Ja are known —
A, ff, and B lying in the same straight line ; then
BH=z
P
= p cosec |a.
(70)
sin ^
Or BA and 5-4 may be found from Table IX, after which
Bff=BA-^HA.
Example.— A 3*" curve ends in a tangent at sta. 160 + 50,
35 ft. outside of desired tangent. Find the point of compound-
ing with a 4' 50' curve.
From Table I, R for 8** cui-ve equals 1910.08 ft., and for
4" 50' curve 1185.78 ft.
35
Then, by (69), cos a = 1 - ;^ = 0.95168.
From table of cosines angle a is found to be 17° 58'. Dividing
this by 8 gives 5.961 stations for the arc BA. Hence the P,C,C.
number is 160.50 - 5.961 = sta.154 + 53.9, and the new P.Jl is
at sta. 158 + 23.9.
123. Given a I«ooated Oomponnd Curve ending in a
Tangent Parallel to, and a Oiven Distance from, a Tangent
in which the Curve is required to end. To Find the Necea-
sary Change in P. C C.
First Case,— Terminal branch having shorter raditis.
In Fig. 46 let ABC be the located
curve. AEF the one required ; angle
BOiC= a linown, and also MN = p.
If angle EOM = b can be found, the
angle of retreat from ^ to ^ will equal
6 -a.
Draw Oi'K and OiL perpendicular
to ON, which is parallel to OiC,
Fia.40,
Then OK=(R^ R,) cos 6,
LOCATIOK. 87^
Now £J/'= A, - -Si = iJ, - MN. from which KL=:MN^ p.
Hence
ifi — Ri) COB h^(R — Ri) cos a - p.
From which
cos ft = cos a — p ^ p (71)
xl — /f I
Divide 6 — a by 2>, the curvature of first branch, and move
back that number of stations from B to the new P. G,G. at E,
Join 0,0y'\ evidently FG = 0,0i\ aud angle KOt'Oi = CF'G' ;
00i*0i = 90** - i(b - a), OOi'iT = 90** - h. Hence
CFG = KO.'O, = [90 - 4(6 - a)] - (90 - &) = 1(6 + a). (72)
From triangle GQF^
Or, from triangle OOx'Ox, *
FG = Oi'O, = 2(5 - Rx) sin K* - «)•
Had ^i^Fbeen the original curve, b would have been Icnown
and a required.
From (71), cos a = cos 6 + _ ^-, (74)
H — JKi
CFand angle GFMtire given by formulas (78) and (72).
Example. — A 2" curve compounds with a 4" curve at sta.
82 4- 30; a = 20^ 30', p = 40 feet. Find number of new P.G.G.
aud distance between P.T.s,
From (71), cos b = 0.93667 - ^^^ ^1432. 7 = ^'^S^^'
This yields b = 24° 40', and 5 - a = 4° 10'.
The change in P.G.G. is ^-r— = 2.083 stations; the P.G.G.
number is therefore 82.30 - 2.083 = sta. 80 + 21.7.
88 A field-makual fou railroad engineers.
By (72), CFG = J(24' 40' + 20' 30') = IT 35'.
By (73), 2?^ = 40 X 2.60399 =104.3 feet.
Second Cabs. — The terminal branch having longer radiva.
Let CAB, Fig. 47, be the located
curve with FX\C. at A, and let
FK be the tangent in which the
curve is required to end.
The distance BK = p, the radii
OA = li, OxA = Ri , and angle
AOiB = a being known, it will
be sufficient to find angle EOx'F
in order to get the angle of ad-
vance, AOE -=1 a — h. Draw OL
and OxN perpendicular to 0/F
and OxB, From the triangles
Fi«. 47.
Oi'Oif and OxOL,
But
(2?i - 2?) cos h = Oi'iV+ (i?i - E) cos a.
OiN—KB — p\ therefore
(iJi -2?)co86=^ + (i?i — i?)co8a.
Whence
cos 6 = cos a +
ifi-^'
6( — * (
Then —j- will be length of curve from A to E.
(75)
Angle ^i^B = N0,0,' = 00,0/ - i^OiO.
But OOxOx = 90" - \{a - 5) and i\rOiO = 90 - a.
•. KFB = PO'' - \{a - 5)] - [90 - a] = J(a + ft).
From triangle KFB,
FB =
= p . cosec i(a + 6).
sin i(a + 6)
Or, from triangle dOOi', since 0,0/ = FB,
-TO = 2(/?, - i?) sin i(a - d).
(76)
LOCATION.
89
If AEF hekd been the located curve, b would have been given
rind a required. From formula (75),
cos a = cos b —
Hi - B'
(77)
Example.— A 5** curve compounds at sta. 60 with a 2** curve,
and the P.T. is at sta. 80. What will be the number of F.C.C.
if the P,T. fall in a tangent 81 feet inside of terminal tangent?
Here a = 40'.
By (75), cos b = 0.76604 + -~^ = 0.81316.
Hence b = 85° 36' and a - 6 = 4' 24', corresponding to 88 feet
around the 6' curve. The number of the new P. G. (7. is therefore
60 -f 88.
angle KFB = 4(40" C + 36' 86') = 37' 48',
and
FB = Six 1.68157 = 182.16 feet.
124. Given a Iiocated Oompoand Curve to Find Necessary
Change in P.C.C. and Radius of Second Branch to make the
P.T. fall in a Tangent Parallel to First Terminal Tangent
and in a Point on the Same Radial Line.
First Ca.sk,— Second branch Tiamng shorter radius.
In Fig. 48, OB=B. OiB=Bi angle
a aud HO = p are known. OiE=B%
and angle b must be found ; then
?^ = BE will be the change in
p.ac.
Produce first branch to K, where
OiTis parallel to OiC. Since BOK
= BOi C, B, K, aud C lie in the same
straight line; and since EO%F ^
EOKy E, F, and K lie in the same
straight line. Therefore
KOG=:la, and KFH^\b.
FzG. 48.
90 A FIELD-MAKUAL FOR RAILROAD EKOIKEERS.
Prom triaugles KFH &Bd KCO,
, .. HK OK OH ^ . , p
But
FR=i OiL = (R-Ri) Bin a.
. •. tao \h = lau \a +
(2?--B,}8ina* '
Prom triangles OOxL and OO^M,
(R - i?a) sin ft = (1? - /?,) sin a.
^ sin a
(78)
"Whence
i?, = i? - (2? - i?,)
sin b'
(79,
Had ^i^F been the first curve located, h and J?8 would be
kuown, a aud i?i required.
From the figure, reasoning as before,
and
^°^ = ^°^^- (i{-^.)siut '
^ sm a
(80)
(81)
Second CA8B.--^Sigc<?;i(i branch Mving longer radius.
Fio. 49.
In Fig. 49 let AB be the located curve, i^Fthe curve required
OA = R, 0,A =. i?», O^S = i?„ FB = p.
LOCATION. 91
:^
f?a and angle b are wanted, angle a being known.
We can show, as in first case, that
OM=i KF= LB = (2?i - i?) sin a;
and hence
^ ^. HK EL p
Or inserting values,
P
tan lb — tan Ja ~ -n ^r"* — •
• . ■ (iii — 22) sin a
d — b
Angle b now becomes known and — jr— = AE in chains, which
is the change in position of P. C. C.
From triangles 00,Jlf and OO^M,
(Ri - B) sin b = (i2, - i?) sin a
.'.B, = B + (B.--Bf^^ (83)
Had the new tangent fallen outside the old one, we should have
bad
tanja = tan46+^-^-^^^^, . . . (84)
and
A = ;?+(«. -fi)«^*. (85)
126. Having^ a Located Oompound OtOrve, to Find the
Change in F.G.C. and Radius of Second Branch in order to
Cause P.T. to Fall at a New Point in Terminal Tangent.
First Case.— Second branch having shorter radiue.
92 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
In Fig. 50 let NAB be the located curve, and C the point where
P.T, is required to fall. Let BG — k, 0A=^ i?, OiB = Rx, and
angle Ox OH = a be known; angle b and R% are required.
Extend first branch to Fy making OF parallel to OxB. A, B,
and F lie on a straight line, for angles AOxB and ^OJ^are equal;
likewise E, G, and F lie on the same straight line.
From triangles OBF &nd OGF,
^^^ OB GB ^, k
coti6=^-.^=cotia-^
But OF = HM = (i? - -Bi)(l - cos a) = (R- i?,) vers a.
k
cot \h = cot Ja —
(i^— iJi) vers a*
From triangles OOi^and OO^L, since OiP = k,
(B - /?,) sin b = (B-Rx) sin a - *.
Whence
(86)
R^ = R-
k-jR - Rx) sin a
sin 5
(87)
Then b— a divided by D gives arc AM With radius i?, locate
the curve i?(7 from C or ^.
LOCATION.
93
Had NEC been the located curve, R, R% , and b would have
been known, Bi and a required. In this case
cot ia = cot Jd +
/?, = 2?-
{B— Ri)Yerab'
A;-f(i?-g,) sin ft
(88)
Sbcond Casb. — Terminal brancli having longer radius
In Fig. 51 let NAB be the located and NSC the required curve.
Let CB = A; be known. Then, as in the first case,
,,^ OC OB k
.*. cot i6 = cot Ja — '
(i^,--K) versa'
and (J?, - i?) sin a = (/?,- 2?) sin 6 + A: ;
whence
J? - » -L (-g»-^)8ina-A ;
m
. (91)
di A FIELD-MAKUAL tOR RAILROAD ENGINEERS.
Had NEC been located and NAB required, the equations would
have been
and
cot Ja = cot \b -f
Bi=zB +
'-^(H.
k
- B)Yerab*
(i?.-
22) sin 6 + A;
sin a
. (92)
. (93)
In either of these two cases if k is unknown and the new radius
given or assumed, the desired angle and the value of k may be
fouud from the foregouig equations. Or, knowing the new angle,
the new radius and value of k may be found from the sjiiiie
equations.
126. To Replace a Ourve of Oiven Radius, which nnitefi
Two Tangents with Known Intersection-angle, by a Three*
centered Compound Curve.
In Fig. 52 let OA = /? be the radius of located curve
Oa(7= OiA = Bi the radius of terminal portions of the three-
ceiiiered curve, and the other notation as shown in the figure.
Braw OaOa', and draw ^OiSTpei-peudicular thereto. From tri-
ingles O^OiB and O^OH,
O^H = (i?. — -Ri) sin J/i = (ij, - B) sin \L . . . (a)
Suppose Bt and Bi to be assumed ; then equation (a) yields
''^*^' = fclr«^»*^-
(94)
LOCATION. 96
Then AO^' E = CO^O = \{I - I,). . . . (95)
Suppose AO^'E, CO^Oy and R^ to have been assumed. From
(95) find /i ; then, from equation (a),
-B» = i?a - (iia - i?)-?^ (96)
sm J/i '
Example.— Given a 4** curve, / = 38*, and the terminal
brunches composed of a 2* curve for two stations, to find Bi and
Dx for the central portion. -
Here /i = 38° - 2(2 X 2)' = SO'.
From Table I, i?a = 2865 ft., iJ = 1432.7 ft.
Whence i?a — -R = 1432.3 ft.
Log 1432.3 = 3.15603
•* sin 19" 0' = 9.51264
2.66867
" sin 15** C = 9.41300
.-. log 1801.7 = 3.25567
Therefore Bx = 2865 - 1801.7 = 1063.3 ft., and, by Table I,
Bi = 5" 23'. 4, nearly enough.
127. To Substitute a Curve of Given Radius for a Tangent
uniting Two Curves.
In Fig. 53 let the tangent 5C=<, OBz=B, O^C = B^, and
OtA = i?a be known.
Angles a, b, and e must be found in order to substitute curve
AE for the system ABCE,
Draw OF parallel to BG, then O^F =B^'- B, and, from triangle
00,F,
tand = ^ ^^ , ^^'^
t
00, = -T— T = «.cosecd= 4/(i?» --B)« + t». . (98)
sin a
96 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Now iu triangle OOiO^ three sides are known and the angles
c and e maj be computed. Thus if « is the half -sum of the sides,
^ r 00, X o,o»
Angle e may be found in like manner, then b = 180'— (« + <0»
and a = c — 6.
Points A and E may now be located and the curve traced.
Example.— A 3° and a 5° curve are united by a tangent 60O
feet long. Keplace by a 2** curve.
Here R^-R= 1910 - 1146 = 764 feet.
By (97). tan d = ^ = 0.65444 = tan 33** 12'
By (98), 00, = 913.1 feet.
In triangle OOiOa, 00, = 913.1, 0x0^ = 954.9, and 00, =
1718.7 feet. Solving for e and c,
tf = 183'36', c = 23'0'. Then 6 = 18" 12', a = 9" 48.
Article 10. Track Problems.
128. Reversed Curves should never be employed on main
lines because of the shock due to sudden reversal of curvature
and superelevation of outside rail. A short tangent should be
interposed between the two curves, which may ordinarily be
done by changing the end points of the curve, or slightly altering
tb§ radius. If, Jipiypvgr, tn^nslUpn curves are employed to ease
LOCATION.
97
off both curves, there would sceiu to be no objection to the use of
curves of contrary flexure, provided the track may be kept
always in perfect condition. In yards, crossovers, and where
connection is made with existing track, reversed curves may be
employed, and are often imperative.
129. Having a liocated Curve Intersected by a Straight
Une, to Connect them by Another Curve.
Either the radius of the joining curve may be given, or else the
point on first curve at which the junction must be made. The
angle between a tangent to located curve at the point of meeting
and the straight line must be measured. Four possible cases
occur.
FiKST Case. — Joining eurve tangent to located curve internally
and on same Me of cvtting line ag center.
In Fig. 64 let GF be joining curve, with center Oi and radius
Bi, Let radius of located curve OF = B. Draw OiO and OH
perpendicular to the cutting line produced, and OiK parallel to
AH, If Bi is known, we must determine angle b, a having been
R
O5
*^ i Q \a B
♦'■d-y
/ "a
4
Fio. 64.
measured; then b — a gives the length of arc from A to -F where
the P.CG. is to be located. In the triangle EOOi we have
0K= OH- Bi and OOi = B - i?i.
U cos a — Bi
Then
cos 6 =
D
B-Bt
= arc AF.
(99)
Had Fbeen given, we should have b = a-\-AOF, and, from (99),
iJi =
B (cos a — cos b) __B (cos a — cos b)
1 — cos b
vers b
(100)
98 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Example. — A l"* curve is cut by a tangeut that makes au angle
of 64' 32' with tangent to curve. Unite by means of a 4* curve.
By (99), cos 6 = 0.24000 = cos 76* 07', and therefore 6 - a =
!!• 35', making AF, of figure, 11 58 stations.
Second Case. — Joining curve tangent internally to located
curve hut on opposite side of cutting line from center of located
curve.
In Fig. 54 let arc ME, with center Oi and radius i?a , be the
joining curve. From the figure,
coscf =
R cos a + 2?«
(101)
Then arc ^^ = a - li divided by D, and c = ISO** - d.
Had the point j^ been given and i^a required, it would have
been, from (101)
i?(co8 d — cos a)
iJa =
(108)
1 + cos d
Example.— Take the same example as in first case- Ilere,
By (101), coBd = 0.9068 = cos 24* 56'.
Then 64' 32' - 24' 56' = 39' 36',
equivalent to 39.600 stations around curve from A to E,
Third Case,— Joining curve tangent externally to located curve,
toUh center on same side of cutting Une.
Lf v*0.
Fio.M.
In Fig. 65 let arc 5(7,with crnter Oi and radius 7?, ,. be the join-
ing curve. Draw OxE parallel lu CF, and OiC'aml OF pcrpeu-
diculur thereto.
LOCATIOK. 99
From the figure,
(R + Bi)co8b = Rcosa - A;
.-. COS 5= B+Bi ^ ^
Then d = 180 — 5, and AOB = b — a. The curve may now be
traced on the ground.
If -4 C is wanted, we have AC = (/? -f 7?,) sin 5 — B sin a.
If the point B is fixed and Bi required, there results, from (103),
^^^ g(C0S«-C08ft) ^^^^
1 -|- cos
Example.— Take the example given for the first and second
cases.
By (103),
. 5730x0.43-1432.5 .^.. _,...,
^^«' = 5730+1432.5 = ^'^^ = «^ ^^ ^^
J - a = 8r 44' - 64** 32' = IT 12', equivalent to 17.2
stations on located curve from A to B. Angle d = 180** — 81" 44'
= 98* 16', equivalent to 24.567 stations from 5 to C on the
4** curve.
FouHTH Case. —Joining curve tangent externally to located curve,
with center on opposite side of cutting line.
Let Oa, Fig 55, be center of joining curve, Bt its radius.
From the figure,
{B + 7f«) cos (J = i? cos « + Tfa.
B cos a 4- Bi ,.^^,
■--'^"'= B+H, ■ <^«^>
If if is fixed and if, required, (105) yields
P __ i?(cos c — cos a ) _ B(co8 c — cos a) .
' "" 1 — cos c ~~ versin c * '
ExAMFLB. — Take same example as in preceding cases.
By (105), cos c = 0.54403 = cos 57' 02'.
Then « - c = 64" 32' - 57" 2' = 7" 30',
r
100 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
calling for a distance of 7.50 stations from ^ to Jf around
1° curve. From M to Hon 4** curve is 14.258 stations.
130. ToIiOcateaT
A Y is made up of a system of tracks so arranged as to admit
of turning an entire train. Three of the most used arrangements
are given below.
FiHST Casb. — One branch of Y a straiglii line.
This is only the special case of the last problem in which the
cutting line becomes tangent to both curves. In Fig. 56, if any
A
F
c
^
'■'V--
^^--^ '
'V,
>v; \
'"n /^
E
R
_../
/
■^
U l,.^ es
"y{
>?
^1
O
l"-^
.
Fia. 56.
one of the points -4, B, or C is given, the others may be located
by finding the angles c and 6. Draw OiE parallel to CA ; then
in triangle OOxE
(i? + ^i)cos6 = n- Bx,
cos 6 =
B-\-Bx
(107)
This follows at once from (103) by making angle a — 0. Then
angle c = 180 — ft. If AB were a located curve and the point
5 given, formula (107) would furnish us a value for i?..
AnotlieT solution is to prod\ice the tangent at B to cut AG Hi F;
tlien AF= FC = BF, Join 2^ with and 0» ; it can easily be
seen that angle OFOx = 90% and, by geometry.
Therefore
and
BF= Y^X Bi.
ton i« = -g- =1/5;.
(108)
(109)
(liO)
LOCATIOK.
101
Example.— Let AB be a 8* curve, £(7 a 6* curve, the point A
at station 180.
By (107),
"^ ' = iSSrfSi = ^-^^^^ = ''' ''' ^^•
The number of B is 180 + 23.511 = 203 + 51.1. Angle e =
109** 28', equivalent to 18.244 stations on the 6** curve.
Second Case.— I%« three brancTiee curbed and convex iotoardi
eac?i other.
Given the three radii and any
one of the points A, B, or O,
Pig. 57, we have only to find the
angles at the center, then divide
these angles by the degrees of
the respective curves to get their
lengths and locate the three
branches.
In the triangle 00x0%, letting
00 1 = I, 0,0a = m, OOt = w,
»= ta + wi-f-n) = /? + /?, +i?„
we shall have, by trigonometry.
Pio.67.
cos 4a - i/ ^" - Q _ Ab + STT^S.
(111)
Angles b and c may be found in like manner.
The angles may be found otherwise by letting fall a perpen-
dicular from one vertex upon the opposite side, as OE perpen-
dicular to OiOs. Then from the relation
OiOi : OiO + OOt = OOi - OOt : OxE - 0%E
determine O^E and OiE\ then the right triangles O^OE and
OiOE yield values of cosine a and cosine c, after which & may
readily be obtained.
Third Case. — One branch concave to the other two.
In Fig. 58 the triangle OOj Oa may be solved for the angles at
O, Oi , and 0« ; for if the radii are given, the sides OOi = R — Ri,
00% = i? — -Ra, and 0,0i = i?i -|- iJ« are known and the solution
102 A FIELD-MAKUAL FOR RAILROAD EKGIKEERS.
is the same as for second case. Then b is the central angle for
curve AB, a' = 180 — a, the central angle for AG, and c' =
180 - c, the central angle for curve BO.
Example.— If A is at sta. 830 on the 1** curve AB, AC an
8" curve, connect with a 6" curve CB. Here we have
OaO =^730- 717 = 5013, OiO = 5730 - 955 = 4775,
and 0,0, = 955 + 717 = 1672.
Solving this triangle, we get e = 88^ 20', b - 19* 28', and
a = 72** 12'. The number of B is therefore 820 + 19.467 =
889 + 46.7 ; the length of CB is .?1^ = 15.278 stations, and
107 8
of ^Ois -^^ = 13.475 stations,
o
131. To Locate a Reversed Curve between Parallel
Tangents.
First Case. — Badii equal.
(a) The equal radii R and distance p between tangents known.
In Fig. 59 draw 0^ parallel to -4 G^ to meet Oi 5 produced.
From triangle OEOx,
cos«=M^ = l-^. ("^)
and 0^ =r 2JS sin a (118)
r
LOCATION.
103
From triangle ABO,
AB = ~^— = p cosec \a = ^OE* + p*. . (lU)
Fxo. 08.
(5) ^6^ and p known, R required.
Here AB = \/AQ^ -j- p« = *. Draw OJJ to the mid-point of
AC. Triangles iL OiT and ^^(? are similar and AH ^ \k.
Therefore
1*
whence
4p
(116)
Example. — Connect two parallel tracks, 30 ft. c. to c. by a 7*
reversed curve. From Table I, i? = 819 feet, and, bj (112),
cos a = 1 - -^ = 0.98167 = cos 10* 59'.
By (113), OE = 1688 X .19052 = 812.1 feet.
By (114), AB = i^(312.1)» + (30)« = 313.6 feet.
If p = 80, 0^ = 812.1, or AB = 318.5 Lad been given, we
should have had, by (116)
«.2^.„.,«.
104 A FIELD-MANUAL FOR RAILEOAD ENGINKEBS.
Second Case.— Badii unequal.
(a) Suppose the radii B = OA and Ri •= 0,5 (Fig. 59) to be
known We must find central angle a and AB = k. From the
triangle OOiE,
Then AB will be given by (114).
{b) Suppose AB = k, p and B known, to find Bi and angle a.
Triangle ABG yields
sinla = -|- (117)
0, LB is similar to A OB. Hence
Bi^_ ±
LB" p'
But AC = 2B sin ia, and Z-B = i(k - ^C7) = 1(7«. Inserting
this value of LB and solving for Ri,
* = ^ <"«>
From similar triangles,
Inserting me value of Ci = -^^-^ from (118) and solving for
Bi , we get
i?i = j^ - jB. (119)
Example.— -4J? = 800\ p = 30', i? = 819 ft., to find angle
a and Bi.
By (117), sin ia = ^ = 0.10000 = sin 5' 44'.
Therefore angle a = 11* 28'.
By {IW. Bx = ^^^ - 819 = 681 ft., an S'' 26' curve.
LOCATION.
105
132. To Connect Two Parallel Tracks by a OroMOver com-
posed oi two D" Ourres with a Oiven Length of Tangent
betixreen Points of Contrary Flexure.
In Fig/60 let AFQB be the re-
quired crossover, FG=ly EB=p^
and OA = OB = R known;
angle a and AE = x are re-
quired.
Draw OM parallel to AE to
meet (/B produced ; draw also
00 parallel and equal to FQ;
join nud O'. From triangle
OO'O,
I
tany=^^,
. . (idO)
2i?
Fio. 80.
00' = — - = 2R sec y = V^R^ + ^.
cosy 9 w
Then in triangle OO^M,
Oa ""21? sec 1
coay.
(121)
(122)
Now knowing y and t,
a = f - y (128)
Next, X = OMr= Oa sin 2 = 21? sec y sin «. . (124)
Example.— Given i) = 7" 80', p = 62 ft., I = 100 ft., to locate
crossover when A is at sta. 86 + 20.
By (120).
log tan y = 2 - 3.18441 = 8.81659 = log tan 8" 44'.
By (121),
log Oa - 8.18441 - 9.99908 = 3.18633 = log 1632.
By (122),
log cos 2 = 8.16643 - 8.18533 = 9.98110 = log cos 16* 47'.
By (123),
a = 16* 47' - 8* 44' = 13" 8'.
By (124),
log X = 3.18583 + 9.46053 = 2.64586 = log 442.4.
106 A FltLD-MAKtJAti FOR ftAItltOAD fiNGlKEfitlS.
133. To Find the Radius of th« Ravened Ounre AFE, Fig.
61, Oiven Angles / end 1\ and
From the figure,
i?taiil7'= CF.
Adding,
B(\AXk \1 + tan i/') = J?C7=x *.
Fio. 61.
Whence
-B = :
(125)
tan i/ -I- tan i/'
Example.— Given / = 10% i'= 20*, EC = 700 feet, to find R,
700
By (125), i? =
0.08749 + 0.17688
= 2658 ft., a 2' 9f curve.
134. To Locate a Reversed Onrve between Fixed Points.
In Fig. 62 let AB — k, and angles 1 and 1' be known. We
Fig. 88.
have to find R and the angles a and h.
Draw (yo parallel to, and 00 and (yF perpendicular to, AB,
Angle AOQ = / and BOF = /'. Then Oi^ = i? cos / and OF
= RcosI'. Hence
aG' = i?(cos/+co8J0.
In triangle 000, 00' = 2R. Therefore
i?(co8 /-r cos 2') cos /-f cos/'
cos X =
2R 2
an expression from which R has disappeared.
(126)
LOCATION.
107
We now have a = 7 + a? and & = /'+«.
To find i?we have AE+ EF -{- FB = k,
or i? siu i + 3i? Bmx + B sin I' = k.
Whence i? = . , . . t , ,, , .. . . (127)
sin / -f- sill /' -f- 2 SID a; ^ ^
J.n^^A^ expression for i? can be found by drawing ^iVand BL
perpendicular to 00', and BN parallel thereto. Then, since
:ifBAN=:x,
Bsina -\- Bsinb = k cos x.
• • ^ - sina + sind <^^^
Example. — Take the example of the last problem,
A; = 700, J =10% /' = 20\
By (126),
cos x = J(0.98481 + 0.93969) = 0.96225 = cos 15* 48'.
We now have a = 25** 48' and b = 35** 48'.
700 X 0.96225
By (128), B = ^
= 660.2 ft., an 8* 41' curve.
0.43523 + 0.58496
136. To Connect Two Divergent Tangents by a Reversed
•Ourve.
First Case. — Advancing towards the P.I.
Given the mdii B and Bi , the angle / and AO = k, to find the
liugles a and b (Fig. 63).
>
1
"X/
0,
— fL^A
''^^^^-vt!*
H
JL i !Q
Fia. 68.
Draw 00 parallel to the tangent BG to meet OiB produced
Then EF - BO = AF- AE.
Therefore BO =^ IRco^l — k sin /,
108 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
From triangle OOiO,
. Rx+BO 7?, + 1? cos/- A; sin J ^,-^,
^"^' = -s+ir-= irvs. • • <^^^)
Then a = MOxN =5-7, O.Jf being parallel to OA.
SecoND Case. — Receding from the P.J.
In Fig. 63 we have BC — ki , angle /, 7?, and 7?, given, to find
angles a and h.
Produce OA to meet 0,X drawn parallel to GA. AL equals
OiJf = OMco&L
OxH = Ri " BB = Ri - ki tan 7.
.\ AL = 0,M = {Ri - ki tan 7) coa /.
Hence
OZ = 7? -f (7?i - A;, tan 7) cos 7 = 7? + 72, cos 7 - *, sin /.
From triangle OOiL,
cos a =
PL _ /? -f 7?. cos 7 ~ k. slh 7
oo; ~" 72 + iS •
ft = a + 7.
(130)
Evidently,
136. To Change the P.R.C. so that Second Branch of
Curve shall End in a Tangent Parallel to Terminal Tangent
and Distant p therefrom.
In Fig. 64 let MAB be the located curve, EN = p. We must
determine the angle COA, after which the desired curve ACB
may be located.
Draw EOi and XO, parnllel to T^Fnnd NQ.
EL = OxK = p.
LOCATION.
109
Fiom triangles OOi'ffmd 00,Z,
(R + Hi) cos 6 = (2J + i?i) cos a - p.
cos b = cos a —
Angle AOC = b ^ a.
Ji+Bi"
(181)
137. To Find the Radius of a Curved Track.
Measure any chord AB = 21, and mid-ordinate CE = M,
Fio. (J5.
Tbeu in the right triangle OAE (Fig. 66),
-R = :
2M
(182)
CHAPTER IV
TRAN3ITI0N-CUBVE8,
Article 11.— Theory op the Tranbition-cubvb.
138. Elevation of Outer Rail on Curves. —To counteract the
effect of ceutrifugal force on curves the outer rail must be
elevated above the iuner one. It is shov^n in mechanics that the
ceutrifugal force is
F-
32.16-8'
where W is the weight, « the velocity in feet per second, 82. Itt
an average value of the acceleration of gravity in feet per secoud
per second, and R the.radlus iu feet.
In Fig. 66 let tlie vertical HL represent W, the horizontal KH
the ceutrifugal force, A.B the plane of the rails, and CB = 6
the superelevation of outer rail.
From similar triangles,
Equate this value of F to that given
above and solve for e, giving
Fia.66. ^-32.162?- • • ^^^>
The gauge AB should be greater on curves than on tangents
to allow for flange clearance and the effect of a rigid wheel-base.
AC = 4.9 feet is about the right value for the horizontal distance
between centera of rail- heads for standard gauge. In formula
(133) « is in feet per second, but the train velocity is usually given
in miles per hour. Let V = velocity in miles i^er hour, then the
110
TRAK81T10K-CURVES. Ill
22
velocity in feet per second will be « = — - 7. lusertfng these
15
values in (133) gives
4.9 X 484 F» F« , ..^..
" = 82.16X22522 =35' "^^"^5^- ' * ' <^^^
This elevation will be required from the P.C. to the P.T., but
obviously it cannot be introduced suddenly, so that for easy
riding the rate of increase of e should be uniform. From (134) it
is seen that e varies inversely with if, which requires that when
e = 0, M = infinity. Hence R must decrease from infinity to
the radius of the circular curve, while e increases from to its
maximum value.
139. The True Transition-curve should satisfy formula (134),
but so far no such curve has been found that will at the same
time admit of the same ease of location as the simple circular
curve. According to Rankine the first use of any other than
the circular curve was made by Gravatt about 1828 or 1829,
the curve employed being the curve of sines. Another method
described by Rankine is attributed to William Froude about
1842 ; this curve was worked up in the Engineering News by
A. M. Wellington in 1890. Other approximations are the RaU-
road Spiral, developed by W. H. Seailes in 1882, and the cubic
parabola,. described by C. D. Jameson and E. W. Crellin in the
Railroad and Engineering Journal^ 1889.
In 1880 Elliot Holbrook described in the Railroad Gazette the
true transition-curve applicable to small angles and short lengths
of the curve. In 1893 C. L. Crandall published formulae and
tables applicable to large central angles for both the offset and
deflection methods.
140. The Notation here employed will be explained with
reference to Pig. 67. The curve CBB'C is the circular curve
offset at (7and G' from the tangents by the amounts CiJand C'W.
AGB and BG'A' are the transition-curves. A is theP.r.C,
or point of transition -curve, G the P.O., B the P.G.j, ^' the
P.TG.x, G' the P.T., and A' the P.T.i. The co-ordinates of G
are AIT = x\ EG = y'\ of C, ar' and HG=F\ of B, AM = Xi
and 2dB = y\. The length of curve from P. T. G. to any point P
is I, and the whole length from P.T. G. to P C, is /».
112 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
141. Equation of Transition-curve.— Since the rate of
change of e must be uniform, (134) may be written
e^H^
8p'
(185)
Fia. 67.
in which k is the rate of rise of outer rail along curve, and p the
varying radius of curvature. From the calculus pd<f> = d/,
whence
^ ~ dip'
Insert this in (135) and solve for d<p.
^k
d<t> = y;}dl = 2mldl.
(186)
(137)
2m is dependent upon V and k, and is constant for any one
curve.
Integrating (137),
= mi«, . (188)
the constant of integration being zero, for I is zero when is
zero.
TRANSITION-CURVEa 1 13
From the elementary tiiangle drawn at the point P of transit
tion-curve. Fig. 67, dl being tangent,
f = «'" f-
Expanding sin (p by trigonometry,
hi which 3!=:3X2X1, 5! = 5X4X3X3X1. etc
Substituting for <t> its value from (138),
Integrating,
Irat Tvt^* s 0, where ^ is in circular measure. To obtain <p in
Q^gree3, <f> =^ ^'t^-^ = ^^. Inserting this in (139),
^OU O/.o
^^i(_V_ ^^ , 0°' 0"^ , ^
^ V171.89 79 X W ^ 8151 X 10^ 153245 X 10»« "T" • • -y »
or y -IG» (140)
in which G may be found from Table XIV with 0** as argument.
Interpolation must be resorted to for values of not given in the
table, or y computed by the formula.
From the elementary triangle at P, Fig. 67
dx
■^ = cos 0.
Expanding by trigonometry.
114 A FIELD-MANUAL FOR RAILROAD ENG^iNEERS.
Substituting mP for (p and integrating,
^=V"'lO" + ^16"-936() + ---) ^^^^>
Replacing ml* by <p reduced to degrees,
_X 0^^' , 0'* 0°' , >
^"~\^ "32828"'" 2328X10* 33114 X 10»» "^ * * 7
or a; = i - ii?. (142)
iZr varies with 0% and may be taken from Table XIV with 0*
as argument.
142. The Transition-curve Angle /j is the value 4> assumes
attheP.C.i. From (188),
7» = mli* (148)
From (187) and (136),
^_ _L =
d<f> " 2ml ^'
At the P. Ci p = if and may be taken equal to -^, so that
whence
J^ _ 5730
1 7)**
This value of «m In (148^ gives
^'^h'^im (^^^>
Reducing this to circular measure by writing ^»=^»**Igo=57^
gives
143. The Oodrdinates of any point on the curve are given by
(140) and (142). The length of the transitioD onrve beiu^j- knowu
TRANSITION-CURVES. 115
or assumed, ^i and Xi (the coordinates of the P.Ci) may be
found from these equations by the help of Table XIV; the
coordinates of the R 0, (see Fig. 67) will be
F= y, — 72(1 — cos 7i) = yi — JB vers Ji, , . (147)
«' = «, -i?8in/i (148)
144. Deflection-angles.— With the transit at the P.T.G. (or
P.T.i in backing up) the tangent of deflectionaugles may be
found from the relation tan 5 = ?. Dividing (189) by (141),
X
tan 5 = -^ + .009523w«i« + .000167^'^^" + (149)
o
From trigonometry the expansion of the angle in terms of its
tangent is
6 = tan (5 - J tan^ d + J tan* 5 - etc. ... (a)
In (149) write m^ = <f> and substitute in (a) : *
<5 = ^ - .OO28230« - .000068^» (150)
o
From (188) and (143),
/."^"?7""'* ^'
in which -- = n. Prom (ft), <p = /inS4md this in (150) gives
It
8 = ^n* - .002823/iW - .0000687/n". . , . (c)
o
Both S and ii are in circular measure ; to reduce to degrt'cs
multiply by t^. This gives, neglecting terms involving higlier
powers of Ii than the third,
6** = 4^ n«- .00000086 ii»n« (151)*
o
The second term is quite small, ftnd in most cases may be en-
tirely neglected in puu :ice
116 A FIELD-MANUAL FOR RAILROAD ENGIKEERS.
With the instrument at any intermediate point a^V the deflec-
tion-angle for any point xy, measured from initial tangent, will be
tan S = ^ 2 ^. = i{w^* 4- ml"^ + rrdl") +t4?(w"^ + »»'^"*)
+ ^{mHH" + m»«"»)-|- t«t(w»/*^"^ + mHn"*+m*i^n)+ . . . , (152)
in which powers of mt^ higher than the third have been neglected.
Substitute the value of tan S from (152) in (a), write mP = =
/,7i', ml"^ = <p" = iif*"*, by (6), and reduce circular measure to
degrees, giving
S^ = 4-(»' + ^"* + ^^") - a small correction. (153)
3
For instrument at P.T.C, 7i" = ; then (153) yields
(^0*) = "^^' ~ correction,
o
or
(V) = ^\-5o (154)
(154) is the same as (151), as it should be.
For the transit at the quarter-point of transition-curve
n" = r = T-' = 1 J then (153) yields
*i Ci 4
(*i°) = ^»' + A + i»») - correction.
or
(«i*) = ^'^ - Si (16^)
For transit at midpoint of tiansition-curre n" = |, and, from
(158).
(8i*) = ^(»' + i + in) - correction,
or
(*0=3\-«i (168)
TRANSITION-CURVBa
For transit at three-quarter point vl* as | and
( Y) = T<^* + A + W - corwclioii.
117
or
For transit at P.(7.i n" = 1 and
(1»7)
(«,*») = :^V + 1 + n) - oorrecdoD,
or
(«.•) = ^-^A, - A.
(168)
With the transit at the P.T,G*i it will frequently be most con-
venient to measure the deflections from the tangent to the circular
curve at that point. Sometimes this will also be the case for the
transit at the P.C.,.
By reference to Fig. 68 it will be seen that for the transit at B
the deflection from the tangent BC which serves to fix any point
on the curve, as .6, is given by the equation
or, in general,
(V) = y^o + P,.
(159)
118 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Table XV gives the values of A and B for the fi^^ positions of
instrument for wbich equations (154) to (159), inclusive, were de.
duced. The value of A must be multiplied by ~ , but B is taken
o
direct from the table in thousandths of a degree.
If deflection-angles are wanted for other positions of the instru<
ment, or for other points on the curve, they may be computed
from equation (153).
146. Tables. — Three tables are given for use with transition-
curves.
Table XIV was computed for use with formulas (140) and (142)
in determining (7 and ^ ; being assumed and G and E com-
puted.
Table XV gives A and B for computing the deflection-angles
by (154), (155), (156). (157), (158), and (159) for 20 equidistant
stations on the transition-curve. For points not given in the
table A and B must be interpolated. Linear interpolation will
suffice in most cases, though when /i** is quite large second differ-
ences may be preferable for A. B is given in the table in thou-
sandths of a degree.
Table XVI was calculated by assuming U in lengths varying
by increments of 20 feet, then computing /,* by (146), yx by (139),
Xx by (141), F by (147), and ic' by (148). y, and Xx will also be
given more directly by (140) and (142) with the aid of Table XIV.
The excess in length of transition-curve, measured from P, T.G.
to the point on offset at P.C, over xf is tabulated as «; T is
found by trial such that when inserted in (141) or (142) the same
value of a;' will be obtained as in (148). This may be done by as-
suming V a little less than -^ , then computing a;'. More than two
trials will rarely be needed to find a sufficiently close value of V\
then « = Z' — a?', y' is found by (139) after finding l\ or <f)'
may be found from (b) of 144, and used in (140) in connection
with Table XIV. ^. - V is the length from O (Fig. 67) to the
P.G.x\ the difference in length between this and the length of
circular curve from P,G. to P. d is tabulated as «* ; that is, tf' =
{Ix — V) — arc. Then e-\-e' =^ Ix — (j^ -\- circular arc).
For values of h intermediate between those given in. the table
linear interi^olntion will suffice, though second differences may
be used for l^and y, if preferred.
TRANSITION -CU RVES.
119
146. To Unite the Two Branches of a Compound Curve by
a Transition-curve.
The same objections hold to compound curves as to simple
curves uniting with a tangent ; i.e., where there is a sudden
change of curvature there should be a sudden cbange of super-
elevation of outer rail, which of course is not allowable. Instead
of compounding the curves, we may offset them at the P. C, C.
and unite them by means of a portion of a transition-curve tangent
to each of the simple curves.
In Fig. 69 AB and CELM&tc the simple curves that are to be
united by the transition-curve ANE. Extend the transition-curve
Pig. 60.
to (?, where its radius of curvature becomes infinite, and let 08
be its tangent. Call the length of transition-curve from G to A
li , from G to E Is, and from F to A h. E and A are points
of tangency of simple and transition curves. Then Z, = i, — 1%.
The coordinates of A arc Q8=Xi, 8A =y, ; and of V {WV
perpendicular to G8), GW=Xi\ WV = F^ of E, OP = x^ ,
EP = 1/9; of L {LH perpendicular to 08), OH = Xz\ HL = Fz,
Let BC=F^,
The radius of curvature of transition-curve is inversely pro-
portional to its length from G ; hence the curvature is propor-
tional to the length of curve; therefore ^3 : ^1 = Dj : i>i , whence
h = h
A'
(160)
120 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Then i. = it - ^. = h{l - §•) = ^i^^ST^'- • • (1«1)
By (138) or (148). /! " (^ j *
Equating the value of Js from this equation to that resulting
from (146) gives
/.-4^y=^-^* im
WV= Fx and HL = Ft may be taken from Table XVI with
/i and ;, as arguments. Then 0, W= i?i +1^» OtH^Rt+Fn
Draw 0,r parallel to QS, then 0,r = IFiT; hence
0»T=^(R, + F,) - (-Bi +1^»X
and OiT »«/-»*'.
Therefore
OiOt = (a?/ - a?s') cosec a = Vo^ + Ti^. .
(164)
Then OT = CO, - 50,.
or 2?; = iJ, - (i?i + OiO,). (165)
The lengths of ^B and CE are
^5= ^'';:^" 100, aw)
027= '*'~^'' 100. (167)
The excess of transition curve length over AB+ CEia
e. = l,-{I^ + ^^)m. . . . (168)
TRANSITION-CURVES.
121
It AB and CE are quite sharp, we must take account of the
arc excess, so that we have then
«, = ;,- M il^I± 4- ^^^^) 100 4- arc excess |. (168')
'llie arc excess may be taken from the second column of
Table IV, which gives the arc length for one station; this multi-
plied by the number of stations gives the curve length, which
may replace the values within the brackets in (168').
147. Ijength of Transition-curve to be Taken. — In practice
the rate of change of superelevation of outer rail may vary from
1200" *^ 400* ^*^^ ^^® ^*^ * ' ^^^^ evidently klj must equal the
ijuperelevation of outer rail for circular curve ; or, by (185),
F«
Ml
5730
"Writing R = — ^i and solving for 1%
Ix
For k.
1
1200'
17190* •
Ix = 0.07 F*2).
(169)
(169')
For k =5 ;
For k =
400'
U = 0.085F»i>. (169")
I, = 0.023 F«2). (169'")
The following table gives values of ^i in feet per degree of
circular curve for a few values of Fand k.
k
.30 Miles
per Hour.
85 Miles
per Hour
40 Miles
per Hour.
45 Miles
per Hour.
50 Miles
per Hour.
55 Miles
per Hour.
1
-laoo
63
86
113
142
176
212
1
600
8»
48
56
71
87
106
1
m
81
29
37
47
68
70
I
122 A FIELD-MANUAL FOR HAILROAD ENGINEERS.
When only a sliort tangent intervenes between two curves
shorter ti-ansition curves must be taken, requiring larger values
of k, so that overlapping may be prevented.
For illustration suppose a 5** curve to be eased off with a tran-
sition-curve, the highest train-speed being 45 miles per hour and
* = sL- Sy tlie table the value of li will be 71 X 5 = 355 feet,
so that we should probably take a 360-ft. transition-curve, re-
quiring an offset of 4.7 feet by Table XYI.
Article 12 —Field- work.
A Field Formulas.
148. For the cases most frequently presenting themselves in
practice the foregoing formulas may be simplified so as to admit
of the rapid location of points on the transition-curve with all the
accuracy needed on location, though it is best to use the exact
formulas and tables in setting track-centers on the finished road-
bed. When the transition -curve angle is quite large it will be
better to use the accurate methods on location also, but fur the
more common cases the following formulas will answer.
149. Simplified Formulas.— In (189) and (140) neglect, as
small, all the terms following the first, giving
y = '^' = |^=.OO5818Z0- ^^^^^'
In (141) and (142) retain only the first two terms
in which the last term is small for shoit transition -curves and
may often be neglected, x being taken equal to /.
The values of vn and I\ remain as before :
_ 1 _ D ,.^
…[truncated]