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TFZ05
1901
''^'^"-'-^--'-'--"1
Rieliard Sachse, Esq.
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I
FIELD-MANUAL
FOR
RAILROAD ENGINEERS
BT
J. C. NAGLE, M.A., M.C.E.,
professor of Civil Btiqineering in the AgHculturai
and Mechantca4 College of Texas,
SECOND EDITION, REVISED.
SIXTH THOUSAND.
•^ • «•«
*-«
• • •'
• • • a
^ ^ t
* J k A
', *
*o
9 •-
• i i
:p:W YORK:
JOHN WILEY & SONS.
London: CHAPMAN & HALL. Limited
1907
Copyright, 1897
BY
J. C. NAGLE.
75476
'^
KOBKRT DRUMMOKD, BLBOTROTTPBR AND PRINTBR, NSW TCIiK.
PREFACE.
Ea9B 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. Algebniic 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 fornmlas in
connection witli 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 practical)]^
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 appn)ximate 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 \.o «\xq\\> cXi^x^^^ ^^^
IV PREFACE.
sharp curves — and, with tables of Radii, -Long Chords, Mid-
ordimates, 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 may 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. V
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 ^vith
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 Cavbart's
Field Bookfoi' Civil Engineers and were furnished by G inn & Co.
Electrotypes of Tables II, X, XII, XIII, XIX, XX, XXIV, XXV,
XXVI, and also XVI — this last being from Crandall's book,
The 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. Naole.
CoLLsoB 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
XXVI II and XXIX have been added in order to increase the use-
fulness of the book, and are from electrotypes of tables 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
lirith this as argument the quantity can be at once read off. For
•enter heights greater than those given in Table XVII we may
refer to books devoted exclusively to earthwork computations.
OoKJJDoa 8vATi0V| Texas, Januaiy, 1899.
CONTENTS.
CHAPTER L
BECONNOISSANCOSB.
Article 1. Objects of Reconnoissance— How Made,
section page
1. Relative Importance of the Work of Reconnoissance and Location.. 1
2. Object of Reconnoissance 2
8. The Instruments 2
4. Useof Maps 4
ft. Making the Reconnoissance 4
CHAPTER n.
PRBLIMINART SURYETS.
Abticlb 8. Objects; The Field Corps; Dtttibs of the Chief.
0. Objects of Preliminary Surveys 6
7. The Exploration-line ^ 6
8. Data Sought in Making Preliminary Survejrs 7
9. The Field Corps 7
10. The Chief of Party, Dutiesof 7
Article 3. The Transit Partt.
A. duties of the members.
11. Composition of the Transit Party 8
12. The Transitman 8
18-17. Other Members of the Party. 8
18. Instruments 9
B. TRANSIT ADJUSTMENTS — ^THB VERNIER.
19. Kind of Transit 8
90. To Adjust the Plate Levels IC
91. Parallax IC
22. To Adjust the Line of Collimation. V^
98. To Adjust the Standar4s "^
YIU CONTENTS.
SECTION PAOB
24. To Adjust the Level on Telescope 12
25. Direct and Retrograde Verniers 13
26. The Least Count of a Vernier Id
27. To Read a Vernier 14
O. ACCESSORIES.
(1*) The Oreuiienter,
28. Description and Method of Using Gradienter 14
(2?) The Stadia, or Telemeter,
29. Principle of the Stadia 15
80. Formula for Line of Sight Horizontal 15
31. Formulas for Line of Sight Inclined 16
82. The Instrumentol Constant, To Find 17
83. Reducing the Notes 17
D. FIELD-WORK«
84. Station Numbers 18
85. Hubs or Plugs 18
86. Reference-points 18
87. Alignment IS
88. Form of Transit Notes 19
39. Stadia Methods for Preliminary Surreys 19
E. OBSTACLES IN TANGENT.
41. To Pass an Obstacle by Means of Parallel Lines 20
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 the Leveler 23
46. Work of the Rodman 28
ADJUSTMENTS OF THE LEVEL.
47. To Adjust the Line of Collimation 28
48. To Adjust the Level-bubble 84
49. To Adjnn the Wyes S5
B. THEORY OF LEVEUNO.
60. True and Aooarent Level 26
51. The Error Due to Curvature 25
52. The Difference of Elevation of Two Points 26
O. FIELD-WORK.
63. The Datum 27
64 Bench-marks 27
66. Work in the Field 86
CONTENTS. IX
8KCTION PAGB
56. Tbe Level Notes 28
57. Precautions when Using Level 29
68. The Rod 29
Arttole 5. Thb Topographic Partt.
59. Instruments Used; Area to be Mapped 80
60. Methods of Recording Data 30
61. Topographers' Field-sheets 31
62. Use of the Slope-level 31
63. Cross section Rods 82
64. The Transit and Stadia in Topographical Surveying 32
Article 6. Preliminary Estimates.
66. Map of Preliminary Lines . . 82
67. TheProflle 33
68. Preliminary Estimates of Quantities ,. 33
69. Report of the Locating Engineer 34
CHAPTER ra.
LOCATION.
Article 7. Projecting Location.
70. Problems Involved in the Paper Location 35
71. Hints Regarding Methods of Projecting the Line 85
72. The Curve-protractor 86
73. Work in the Field 37
Article 8. Simple Curves.
A. definitions and formulas.
74. Definitions 88
75. To Find the Radius 2?, 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 A iZ and C Being Known 43
80. To Find the Tangent Distance T, / and B Being Known 43
81. To Find B, Given J and r 44
82. Given/and A to Find the Long Chord L.C?.... 44
83. Ordlnates from Chord 45
84-^. To Find the External £ 48
87. To Find i?, 17 and I Given ... 49
88. To Find r, .Band i Given 49
89. To Find the Deflection Offset from Chord Produced 49
90. To Find the Tangent Deflection Offset 50
91. The Sub-tangential Deflection Offset Vw
92. To Find the Tangent Offset 2 ^8^
08. Differenoe in Length of Arc and Long Chord « ^^
CONTENTS.
B. LOGATINO 8IKPLE 0UBVE8.
SECTIOlf PAOB
94. To Locate a Curve with the Chain by Offsets from Chords Produced 66
95. To Locate a Curve by Offsets from Tangent 57
96. To Locate a Curve by Offsets from a Long Chord 58
97. To Locate a Curve with Transit and Chain 59
98. The Index -angle 60
99. Subdeflection-angles 60
100-101. Transit Notes 61
O. .OBSTACLES.
102. To Pass an Obstacle on a Curve 68
103. To Locate a Curve .when the P. C. is Inaccessible 64
104. To Pass to Tangent when the P.T. is Inaccessible 67
105-107. To Pass a Curve through a Given Point 68
108. To Locate a Tangent to a Curve from an Outside Point 71
109. To Run a Tangent to Two Curves of Contrary Flexure 78
D. CHANGE OF LOCATION.
110. To Locate a Curve Parallel to a Qiven Curve 73
111. To Change P,C. in Order to Make P. T. Fall in a Parallel Tangent. . . 74
112. To Change R and P.C. to make P.T. Fall in Parallel Tangent, on
Same Radial Line 75
113. To Find Change in P.C. or R for a Given Change in J 76
114. Required the Change in P.C. and JB for a Qiven Change in J, the
P. r. Unchanged 77
115. To Find New Radius for a Given Change in 3* 77
116. To Find New JB to Connect P. C. with a Parallel Tangent 78
Article 9. Compound Curves.
A. location problems.
117. Given Both Tangents and One Radius, to Find the Other Radius ... 80
118. Given One Radius, the Long Chord and the Angles it Makes with
Tangents, to Find the Other Radius and Central Angles 82
119. Given 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.C is Inaccessible 84
C. CHANGE OF 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. 7. Fall hi a Par-
allelTangent 86
194. To Change P.C.C. and Second Radius so P.T. shall Fall in a Par-
allel Tangent, on Same Radial Line 89
CONTENTS. XI
TXON PAOB
To Change P.C.C, and Second Radius to Cause P.T, to Fall at a
New Point in Same Tangent 91
To Substitute a Three-centered Compound Curve for a Simple One. 94
To Substitute a Curve for a Tangent Uniting Two Curves 95
Article 10. Track Problems.
Reversed Curves, Where to Use 96
To Connect a Located Curve with an Intersecting Tangent 97
To Locate a Y 100
A Reversed Curve between Parallel Tangents 103
A Crossover between Parallel Tracks when a Fixed Length of Tan-
gent is Inserted 105
A Reversed Curve with Unequal Angles 106
A Reversed Curve between Fixed Points 106
To Connect Two Divergent Tangents by a Reversed Curve 107
To Change P.R.C. so P.T. shall Fall in a Parallel Tangent 108
To Find the Radius of a Curved Track 109
CHAPTER IV.
TRANSITION-CURVES,
Article 11. Theory of the Transition-ccjrvb.
Elevation of Outer Rail on Curves 110
Requirements of the True Transition-curve Ill
Notation Employed ill
Equation of Transition-curve 110
Transition-curve Angle, / 114
Codrdinates of Points 114
Deflection -angles * 115
Explanation of Transition-curve Tables 118
To Unite the Branches of a Compound Curve by a Transition-
curve 119
Length of Transition-curve to be Taken 121
Article 12. Field-work.
, A. field formxtlas.
When to Use the Simplified Formulas 122
Simplified Formulas for Transition-curves 122
Offsets 124
Compound Curves 125
B.. setting out transition-curves.
Location by Offsets "NSS*
Location by Deflection angles "^^^
Form of Transit Notes tor Transition-curves • ^^^
XU CONTENTS.
Article 13. Transition curve Problems,
section paoe
156. Tangent Distances and External for Equal Offsets 129
157. Tangent Distances, Offsets Unequal 130
158. Transition-curves Inserted without Changing the Vertex of Cir-
cular Curve 181
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 183
161. Transition-curve Inserted at P.C.C. by Changing Radius of Second
Branch 136
162. 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.
fbogs and 8witche&
Article 14. Turnouts.
A. turnouts from straioht lines.^
166. Definitions 143
167. To Find the Lead, 2, and Radius, JB, in Terms of the Frog Number,
N, and Gauge, g 144
168. Given i? and flr, to Find iyr, J, 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 of
Main Frog, Given i^„ iV, and iV • 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 P5int 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
179. To Find Theoretic I^ength of Switch-rail - 158
180. To Unite Main Track with a Concentric Siding 160
CONTENTS. XIll
C. 1UE tJTUB LEAD.
SECTION ^ PAGE
181. Definitions Ifi-i
18-2. Given N, t, and g, to Find the Stub Lead :62
183. Turnout Table and Explanation ... 163
184. To Stake Out a Turnout 165
185. Curving Rails 166
Article 15. Crossovers.
186. Crossover between Parallel Straight Tracks, a Tangent betvtreen
Frog-points 166
187. A Crossover in the Form of a Reversed Curve 169
188. A Crossover with Fixed Length of Intermediate Tangent 168
189. A Crossover between Curved Main Tracks 168
Article 16. Crossino-frous and Crossino-sufs.
A. crossing-frogs.
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 17i
B. CROSSING-SLIPS.
195. Length and Radii of Slip-rails, Both Tracks Straight 172
196. Length and Radii of Slip-rails, One Track Curved 173
197. Length and Radii of Slip- rails, Both Tracks Curved;. . . .* 173
CHAPTER VI.
CONBTIIUCTION.
Article 17. Definitions ; Qeneral Considerations ; Vertical
Curves ; Elevation of Outer Rail.
•
199. The Division Engineer 176
JMO. The Resident Engineer 176
201-204. Definitions 177
205. To Find the Qr^de-point, Longitudinal Slope Uniform 178
206 Vertical Curves ITji
207 Elevation of Outer Rail on Curves 182
208. EUtsing Qrade on Curves 183
Article 18. Earthwork.'
A. setting slope-stakes.
209. The Distance Out for Level Sections 188
210. To Find Position of Slope-stakes for Surface Inclined 184
211. Cross-section Notes v^
212. Irregular Sections • "^KV
U3. Staking Out Openings "^^
XIV CONTEJSTS.
SECTION ' PAGB
214. Manner of Marking Stakes 187
215. Shrinkage— Growth 187
216. Borrow-pits, Drainage of, etc 188
B. AREAS OF SBCTIONS.
218. Area of Three-level Section 188
219. Area of Five-level Section 186
220. General Formula for Areas '. 190
221. Explanation of Table of Areas of Level Sections and the Three-
level Correction 191
O. VOLUME OF EARTHWORK.
228. Where Cross-sections should be Taken 19S
228. Volume by Averaging End Areas 198
224. The Prismoidal Formula 193
225. Fonn of Record 196
226. Tlie 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 299
230. Earthwork on Curves 199
231. Overhaul 201
Article 19. Grade and Ballast Stakes, Culverts, Bridges,
AND Tunnels.
232. Grade and Center Stakes 209
233. Ballast-stakes 802
285. Openings of, for Culverts, Trestles, etc 202
236. Bridge Piers and Abutments ; . , 203
237. Tunnels 804
Article 80. Monthly and Final Estimates.
238. Monthly Estimates 806
239. Measurements for Earthwork 206
240. Classification of Earthwork 206
211. The Progress Profile 207
24'^. Masonry Estimates 807
218. Bridge Estimates 807
244. Track Material 807
245. Blank Estimate Sheets 808
246. Monthly Payments -. 808
247. Extras 808
248. Final Estimate 808
249. Acceptance • 800
TABLES.
Table Showing Length of JTransition-curve to be Taken ]?1
Table of Values of 9 - Vflft for Stub Lead 163
Turnout Table 164
Table of Corrections for Vertical Curves 181
CONTENTS. XV
PAOB
Table of Elevation of Outer Rail on Curves 182
Table of Prlsmoidal Corrections for Level Sections 196
I. Radii of Curves 212
II. Minutes in Decimals of a Degree 215
III. Tangential OfiFsets ^ 216
IV. Long Chords and Actual Arcs 217
V. Mid-ordinates to Long Chords 218
VL 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 298
XI. Natural Secants and Cosecants 307
Xn. Natural Tangents and Cotangents 320
Xni. Natural Versines and Exsecants 332
XIV. Coordinates for Transition-curves 855
XV. Deflection-angles for Transition-curves. 356
XVI. Transition-curve Table 358
XVII. Areas 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
XXIL 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 1' Arc of Latitud« and Longitude 389
XX Vn. Trigonometric and Miscellaneous Formulas 390
XXVIII. Square Roots and Cube Roots of Numbers from .1 to 28 895
XXIX. Squares, Cubes, Square Roots, and Cube Roots, of Numbers
from 1 to 1000 /* 396
A FIELD-MANUAL FOR RAILROAD
ENGINEERS.
CHAPTER I.
RECONNOISSANCE.
Articlb 1. Objects of Rbconnoissance— How Made.
1. The question of the selection of the proper route for a line
of railway is essentially an economic one, involving not only the
cost of construction, hut 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 preliminary
estimate may decide those whose duty it is to do so : the problem
confronting him is how to secure the best line, answering a given
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 grade of engineering skill — a fact too sel-
dom recognized by those selecting engineers for this work. The
acquirement of the necessary skill can result otAy ^xoxsv \w^^
practice and close observation, coupled w\l\i >^ie tC^VVVv^j \.o\vK^
2 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
grasp and weigh all the complex features of the question. 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 benefit of the beginner who has to
do with the location and construction a few definitions and hints
relating to reconnoissance will be given before going on to the
special problems arising in the work of the railroad engineer.
2. The ReconnoiBsance 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 impoitance, but because he is not usually allowed
sufficient 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
between 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 they make with each other.
RECONNOISSANCE. 3
(b) The Hand-level enables one to obtain differences of ele-
vation between points not far apart.
(c) 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*
<J = 60000(logfl--logA)(l + ?i^zi?), . . (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, T and t the temperatures of the two stations
in Fahrenheit degrees.
If the sum of the temperatures, T-\-t, is taken as lOS"*, formula
(1) reduces to
d = 63000 (log 5^- log ;i) (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 tbe 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 tern*
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 Odometer registers the number of revolutions of a wheel
to which it is attached, and tbe number of revolutions multiplied
Uy the circumference of the wheel gives the space passed over.
* See Plyniptou^s Aneroid Barometer, p. TSft^tot 1ot\x\\]^W\.
4 A FIELD-HANUAL 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 approach 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 whetlier a suitable line may be
secured for the grades and curves previously assumed. With his
pocket-compass lie takes tlie bearings of lines, and by means of
the hand-level and aneroid determines differences of elevation.
EECONNOISSANCB. • 5
Distances are estimated by the eye, paced, and the count taken
from 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
bim, or by the use of the pedometer if his 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 respect. New and unexpected conditions some-
times deceive even the most practiced eye, but under ordinary
conditions almost any one can train his eye to estimate horizontal
distances fairly well. Vertical heights are more deceptive, pos-
sibly 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 difficult 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 traffic. 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 II.
PRELIMINARY SURVEYS.
Article 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 Ezploration-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 flag
set by tlie 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 rod man, 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
witli the compass.
The exploration-lip« will moye than pay for itself in showing
e
PRELIMINARY SURVEYS. 7
what routes it will be unnecessary to make preliminaries orer,
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 party 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.
(b) The Level Party.
(r) The ToPoaRAPHic 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 \^\sk^<^^
and direct the ^ransitman in the piopex c-q\xx^^. ^<^ ^wsXW^^^
8 A FIELD-MANUAL FOB RAILROAD ENGINEERS.
a record — or direct the transitman 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 8. 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 ad-
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 deflections 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 Ohalnman carries a range-pole or " flag," and
drags the chain, which he must see is straight and horisontal
PBELIMINABY SUBVEY8. 9
when setting a point for a stake. He directs the stakeman where
to drive his stake, calling out the number after the rear chainman
has read and called out the number on his stake; he keeps the
axemen in line hy setting his flag and going ahead, directing them
(irhere 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 IV X 2" 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
faci^ 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 x>oint where flag rested marked with a tack. About ten or
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 find 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 \\. Sa
often convenient to have a clamp and tangenX. xciON«ta«\iX. \»\.'i^^-
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 doubles 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-wa.y 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, re focus the object-glass until this movement disappears.
22. To Adjust the Line of CoUimation is to make the line
joining the intersection of cross-wires and optical center of objec-
tive de}5cribe a plane perpendicular to the horizontal axis of instru-
ment.
FiKST 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
piano 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-
lie«kdcd screws holding cross- wire ring and move the vertical wire
PRELIMINARY SURVEYS. 11
over one fourth tlie apparent error — since there were two reversals
— remembering that the image of the cross-wires is inverted, while
that of the object appears in its true position. Test by repetition.
Second Method. — If the limb graduations can be relied on
they may be used in adjusting the vertical wire. With the instru-
ment level sight a well-defined point, then revolve 180° by vernier-
plate, reading both verniers; reverse telescope, and note if line of
sight cuts the point. If not, correct one half the apparent error by
moving diaphragm ; then test by repetition.
The manufacturers adjust the object-glass slide so that the ob-
jective travels in the telescope axis, and this adjustment is not
liable to serious derangement. It is well, however, to sometimes
test by adjusting the line of collimation for both near and distant
objects. If not correct for both, move the ring which guides the
rear end of object-glass slide until the adjustment is correct for
both x>ositions.
Next make the vertical wire vertical by noting if it coincides
throughout its length with a plumb-line, or by observing if it de-
viates from a point, on which the intersection has been fixed, when
the telescope is elevated or depressed. Any error is corrected by
turning the ring after slightly loosening the screws holding it.
The horizontal wire should also be adjusted so that the inter-
section of the cross- wires will be in the axis of the telescope ; if
the transit is to be used as a leveling instrument this adjustment
is essential.
Drive a stake close to the instrument, and with the telescope
clamped as nearly horizontal as can be conveniently done read a
rod held on top of the stake ; about 800 feet distant, and in line
with first stake and instrument, drive a second stake and read the
rod on it. Revolve 180** on vertical axis, reverse the telescope and
bring the horizontal wire to the former reading when the rod is
held on first stake ; if the reading on the second stake is not the
same as before, correct one half the apparent error by moving the
cross- wire ring. Repeat as a test. The vertical wire should again
be tested lest the movement of the ring may have caused it to
change.
23. To Adjust the Standards is to make the plane described
by the line of collimation vertical. Set up the transit about as far
in front of some high building, or other tall object, as the highest
point that can be sighted is above the base. lje\c\\\\^\\i^VcN^.vc\fc'v>X»
ftud fix the inteTsectjon oi thectoss-wlrQaoixXAxQVV^^^^^^^'^'^^
12 A 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
cut 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 us 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 fir&v 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
Fio. 1.
of A and B. Suppose line of sight to cut the rods at ^and Z>,
we must find DO so that the target may be set at the proper read-
PKELTMIKARY SURVEYS. 13
ing to make the line of sight horizontal. Let OF— a, FO = 5,
EA =r, DB = r\ GB = k. Draw DH parallel to GA and 0Q\
then EH=r + k-r\
From similar triangles
Set the target at a reading OB = OD + ^, sight to Gy 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 ;
-5 = value of one space on vernier ;
n = number of spaces on vernier.
Then for the direct vernier
nv = {n — l)l;
from which we get the least count,
n
Tot the retrograde vernier
n« = (n + Vf,^
14 A FIELD-MANUAL FOR RAILROAD ENGIKEER8.
from which the least count is
n
the same result as found for the direct vernier.
So, to find the least count : Diiide the value of one limb space by
tlie 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 -^jf 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 Oradienter.
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 0.75 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.
PRELIHINAEY SUBVEY8.
15
(2°) The Stadia^ or Telemeter,
29. The Stadia is an instrument for determining the distance
of a point from tlie 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,
A __ r^
d^'D'
From optics,
1+1 = L
d^D f
in which/ is the focal length of objective.
Eliminating d from these two equations,
i>=/+{r.
16 A FIELD-HAKUAL FOR RAILROAD SKGIKEBR8.
Let c be the mean distance of objective from center of instm-
ment. Adding this to D gives, for the distance of the rod frmn
the center of the instrument,
l = e+f+j^r.
(2)
£——'•-"«'—
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
Fig. 3.
let r = CB be the reading on rod held vertical ;
r' = FE, the reading perpendicular to line of sight ;
II = AOf the horizontal distance from Ato B\
V = BG, 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 melj
differ more than 15' to 17'. Then, since FBC = n,
r^ = r cos n.
PRELIMINARY SURVEYS. 17
Bj(2'X AB^a + kr".
Hence AB = a + kr cos n.
From triangle ABQ
H = AB cos n
.•. £r = a cos » + Ar cos' n (3)
F= AB sin n;
. •, F = a sin n + A;r sin » cos ».
But 2 sin n cos n = sin 2n.
Hence F = a sin t* + \kr sin Sn. . . • . • (4)
32. The Instrumental Constant a [=c +/ of (2)] may be
found by measuring tbe 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 (8) 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^a-\-kr,
. *. 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 (3) 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-
dtution Diagram, by John Wiley & Sons, gives values of JJand F
graphically. Colby's SHde-rule, manufactured by Mahn & Co.,
St. Louis, gives values of V for distances in feet, yarda, at\Skfc\Kt^
to tenths of a foot, and can be used witVi gTe«A. T«i.\\^\X.i «
18 Jl field-maxual for railroad engineers.
D. Fie/d-work.
34. Station Numbers should begin with zero for the initial
stake, and are marked on rear side of stake, from the top down-
ward, the number of the preliminary, A, B, G, etc., 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 with the surfaca
The flag is held on the top and carefully aligned, the position of
the point being marked by a tack. A special tack with concave
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 thd
point whose position they serve to locate. They should be driver,
beyond reach of disturbance, and are used in replacing a die
located hub.
These need rarely be used on preliminary.
37. Alignment. — It is not intended that the preliminary amV
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 smiall angle until the obstacle
has been passed, then making a deflection in the opposite sense.
Bearings of tangents are taken with the needle, to serve as a
^heck 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.
PRELIKIKARY SURVEYS.
19
38. The Transit Notes may be kept in the form below, which
shows both pages of the note-book. The notes ran from the
bottom np, 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.
Calculated
Ck>urse.
Magnetic
Ck>urse.
Remarks and Sketches.
68
670
66
66
64
630
62
61
20»0'L.
6»2'R.
N. f 48' W.
N. IS* 12' E.
N. 1»46'W.
N. IS* 15' E.
1
o
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-
graphic 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 purjwse intended as the more ex.^«\«k\N^ \sifc^<^
osually employed.
20 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
The transit need onlj be set at alternate stations (which maj be
anj 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 Idnes. — In
Fig. 4, is the obstruction, AB the obstructed line. At B set
E F
Q H
J
W--
i
\
3 M
3
Fio. 4.
transit ; turn 90° and measure BF long enough to clear obstruc*
tion. Set transit at F, make BFO = 90", and measure FO.
Move to G and backsight to F, making FGC = 90". Measure
OG=FB, and move to C, where the angle OCD is made equal
to 90°. CD is the desired line, and BG=FO,
Otherwise, at Jl and B erect perpendiculars; take BF=AE\
produce EF, and at O and H, beyond 0, erect perpendiculars mak-
ing (?C = HD = FB, CD will be the desired line, and BG = FG.
42. To Pass an Obstacle by Angular Deflections.
Qeneral Case. Angle anything less tlian 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 (72> will clear the obstruction. Make CD = BC And
deflect an angle a to the same side asat ^; BE will lie in AB
produced. Draw CH perpendicular to BD; then
BD = BU+HD = 2BC coso, , , , . (5)
FAELIUINABI SUBTBT8.
ExAMPLK.— Suppose a = 14° 10', BC = CD = 520 ft.
£i) = 3 X 520 X 0.96969 = 1008.87 feet.
Special Cabb. Angle 60 degree*.
In this esse tlie triangle BDF(Fig. 6) is equilateral and BF=
BD = DF.
Slionld it be iDconTenient to run to Z) ne ma; stop at C, having
At (7 deflect 60° and measure GE; at E again de-
PlO. B.
fleet 60° and make EF=BC. At ? a Snal deflection ot 60° in the
opposite sense will put the telescope in the desired line, FO, and
BF-BC+CE. (5o)
43. To Paai an Obstmctlon, luch ai a River, when the Fr«-
OAdlng Methods are Inapplicable.
FmsT Case. Point beyond obttmetion vitfNe.
In Fig. 7 let BC be required
!
At B erect and measure the perpendicular BD ; set Inattavawi.^
a,t D and measure angle BDG^ a ; thea
22 A PIELD-lfAHUAL FUlt RAILROAD BNOIKBBitS.
Or, if a tri^Donietric table ia not at hand, iDake CD S= 90° and
Gi tlie poiDt £1 where X>E interMcte AB ; meaanring SB tbere
results, from similar triaDgles,
whence CB =-^ (6o)
OiheruiiM, if a right angle at £ ia not convenient, measure
angles CBS = b, BDG = a, and side BD. Then e = 180°- (o+6).
From triangle BDC,
BG=BD^^ {»)
Example.— a = S6°, 6 = 70°, BI> = 400 feeL
Bj (6ft), BC = 400 11^-^ = 40B.8 feet.
Second Case. Point beyond ebslntrtion invisible.
At B (Fig. 8) meaaitre angle b and line BE ; move to E and
measnre angle y, and sat hnbs on line BXJ so tti« tine BC will pus
between them. Angle « = SCB = 180 - (6 + y). Then from tri-
angle BBC
BC=BE^^ (7)
Produce EB to D, where DC will be sure to clear obBtructionj
measure BD.
From triangle BDC,
WnjKa -X) _ BC~BD
tan H« + i) SC + BD'
@ut o -^ z = fi, hence
M(«-i) =
PRELIMINARY SURVEYS. 2ii
The sum and difference of a and x are now known, so both maj
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 (7. Set a hub here and place the transit over it. Sight
to D OT 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
I odman supplied with pegs for turning-points and in clearing the
I ine of sight for the level. As the party follows the transit little
or no clearing will be needed. The instruments used are a levels
A rod, and a hand-axe or hatchet,
45. 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 must see that the
rodman gives him readings at points where the longitudinal slope
changes suddenly, recording the plus. Ho 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 bis rod at each station, calling out the
number. If stakes are set only at even stations, he must hold his
rod midway between stakes, the point being found by pacing the
distance. Target-readings need only be taken at turning-points
and benches, and the rodman should keep a 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 Level.
47. To Adjust the Line of Collimation is to bring the inter-
section of the cross-wires into the optical axis of the telescope.
Set up and level the instrument, then bnng \Xi^ ^^tW^-eX. nsSx^
into coincJdei?ce yvith a plumb liue or verUcaV e^^<& ol ^XsvaM^s^^-"
24 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
at the mean length of sight, and note if the vertical wire is truly
parallel thereto. If it is not, loosen the capstan -headed screws
holdiug cross wire liug and turn slightly so that the wire is
parallel to the vertical Hue.
Loosen the wye-clips and bring the vertical wire into coin-
cidence with the Hue and clamp the instrument. Rotate the
telescope in the wyes 180** and note if the wire coincides with the
Hue. If not, correct one half the error by loosening one and
tighteuing the opposite, of the capstan-headed screws that hold
the cross-wire riug in place, rememberiug 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 test and
correction. Repeat for both wires. The hodzontal 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
duriug 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 coUimatiou,
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 Hue.
{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, iu 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 hteral movement.
Repeat until the tube may be rotated half an inch or more to
either side of vertical without movement of the bubble. This
adjustment is made merely to prevent error from failure to set
level -tube vertically beneath telescope.
(6) 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 A4ju8t the Wyes is to make the axis of the telescope
perpendicular to ihe vertical axis. With the wye-clips closed
place the telescope over one pair of leveling-screws and briDg the
bubble to the center of its run ; then turn the telescope half-way
round on its vertical axis, so that its ends have changed places.
If there is any error, correct by bringing the bubble half way back
to center hy 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.
60. When the level has been adjusted the line of collimatlon
'^111 describe a plane parallel to the horizontal plane tangent to
A he earth's surface at the point where the instrument is placed.
a level surface, such as the surface of still water, will coincide
«vith 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
they really are, while the refraction of a beam of light, due to
the 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.
61. The Error due to Curvature at any point is the deviation
of a tangent line from true level, as
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
(i? + c)« = iJ« + iK
From which
Fig. 9.
c =
2i? -f c •
Now, since c is always very small compared "wivVi *5i."R^ XJa^
quotient resulting from the divisiou of t* by ^R\«\W noV ^-jSks.
26 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
sensibly from that obtained by dividing by 2B + e. Therefore
we write
«=i w
For i = 1 mile, B = 3968 miles, e = about 8 inches. Hence
for any other distance in miles we have, for e,
c = 8 X <* inches (9a)
The correction for refraction is about -j^c, hence we have,
from (9),
^ 1 6 8 <»
7 7 IB
or, closely enough,
= .S5c (10)
Example. — What is the correction for a half-mile sight?
For one eighth of a mile?
By (9a), c = 8 X (D* = 2" for first case,
and c = 8 X (i)' = 0".125 for second case.
By (10) the final correction is
c = 0.85 X 2 = 1".7 for first case,
c = 0.85 X 0.125 = 0.106" for second case.
62. The Di£ference of Elevation between two points not so
far apart but that a rod may be read on each from some inter-
mediate poiut may be readily found from these rod-readings.
In Fig. 10 let the instrument be at /. A and B the points
whose diflference of elevation is desired. Let r = AD, r' = BC.
Since the line of sight, DO, is horizontal, the difference of
Fig. 10.
elevation will evidently be r* — r. When the distance from
I to A equals that from / to J? the errors due to curvature
eyid^ntlj balance
PRELIMINARY SURVEYS.
27
When the points are so situated that the rod canYiot be read
on both from one intermediate position of the instrument, an
Fio. 11.
auxiliary point or points must be used and readings taken on
these points in pairs. Thus in Fig. 11 suppose the difference
of elevation ot A and B required :
With the instrument 'at / read on A and some intermediate
point £1, Considering the backsights as plus and foresights as
minus, the difference of elevation of A and E is AD — FE.
Again, with the instrument at /' the difference of elevation of E
and B is GE— CB, The sum of these differences equals the dif-
ference of elevation of A and B, and may be written {AD + QE!)
^ {EF-\- CB), or, in general, the sum of the backsights less the sum
ef (he foresights equals the difference of elevation,
C. Fie/d-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,
or 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 ad|ace.\>A. \.^
the bench, with the letters B. M. indicating l\i^ \i^V\x\^ o\ >(Xx^
point.
28 A FIELD-MANUAL FOR BAILROAD ENGIHEEBS.
66. 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 talsen 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 tbe 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
(U, I.) above the datum.
Headings are taken at eveiy 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 read on this ; the height of
instrument less the rod reading will give its elevation, as it will
for the intermediate points. This point 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 o or T. P. in the notes, and their
positions, as also the bench-marks, noted by both leveler and
rodman in their note-books.
66. 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. L
205.613
F.S.
Kiev.
B.M.
5.613
• . • •
200.0
1
2
• • •
• • • •
• * • ■
• • • •
• •
- • • •
2.3
0.8
5.7
7.8
9 9
203.3
204.8
199.9
197.8
195.7
l.liM
196.310
10.4-.i3
195.190
5
6
• • ■ •
• • • •
• • • •
6.3
4.5
190.0
191.8
Remarks.
B. M. on root of
right of line.
L. O. tree W to
j On peg at 4 + 30' - 20^1© left of Hue,
I by small P. O. tree.
Here the elevation of the datum was taken 20Q 00 feet below
the first bench-mark. The instrument was set up near Statiou 2.
PRELIMINARY SURVEYS. 29
ind 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.I, 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,
yielding 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 ff. 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.
57. Wind and sunshine aflfect the accuracy of the work with
the level, as is also the case with the transit. For very great
accuracy a calm, cloudy day is the best, but the railroad engineer
cannot always choose the best times for his work, and must take
wch precautions as may be possible while he exercises the great-
est care to prevent and detect errors. The adjustments should
be tested at least once a week, even when the greatest care has
been taken, for unequal expansion and other causes may con-
spire to caiise 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 means of a level attached to rod; or the leveler can tell by his
vertical wire when the rod is iu the same vertical plane with the
instiiiment, 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. S\Ye\i^\>a. «ww^ ^\\\^-
bUity are essential quaJities. The Plu\ade\\>\\\xw to^ «fe^\a^ vck
30 A FIELD-MAKUAL FOR RAILROAD ENGINEERS.
answer the purpose as well as any other now manufactured; the
Troy rod may be used iu the same manner as the Philadelphia
rod, but is lighter and less able to stand rough usage.
Article 5. The Topographic Party.
69. The Topographic Party follows the level and secures all
the data necessary for making an accumte 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 800 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 surfaqe elevations,
streams, and nature of surface are the most important; 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 the data are
obtained. Station elevations can be taken direct from the levelefa
notes, and constitute the base on which the contour elevations
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 tlie 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 iu 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
PRELIMINARY SURVEYS.
31
elevation is 321.6 feet and tiie height of eye 5.8 feet, we shall have
for the reading at the 320-foot 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, say, 80 feet out. As the rod reads only
to about 12 feet the topographer must move out to this last point,
and with the reading 5.3 -f- 5= 10.3 find the 310- foot contour in
the 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
305 310 815 320
193' 125* 80 ' 21
321.6
325 330 335 840
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 back for holding the sheets on which the transit-line
has been plotted the night before ; the station elevations ai'e
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 inclinaUon ot \\\e ^wxl^^^Vs^
obcained; then by the use of a scale coualvucVed V.o s\xo^ \>afc
B2 A FIELD-MANUAL FOR RAILROAD ENGINEERS. ^
horizontal distance apart of contours, for the given contourin-
terval, for slopes varying from V to 20°, the position of contours
can at once be spotted on the map. Wellington recommends the
use of the allaziniuth as permitting the employment of either
method at will— the altazimuth being merely a hand-level with
a clinometer attached.
63. CrosB-section Rods are measuring- rods 10 or 12 feet long
carrying a level-bubble. By placing one end at the center,
bringing the rod hoiizontal, 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 others.
64. If the Transit and Stadia are employed, very elaborate
topography may be taken with very little fieW-work, but the ob
servations require considerable reduction. With a suitable tope
graphic protractor and the slide-rule mentioned in 33, the large
number of points that may be obtained from each setting of the
ti-ansit 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.
65. 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
trausit-line ihust £ist he drawn on a succesaloM of small sheets,
which are added as the piotting pvogxesaea, ». lie^ ^ViefeX. \j^\w%
slipped under the edge of the preceding and \&ckft^ dorwu^Ywasi
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
done 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 developed 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 '* G." 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 earthwork.
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,
switch stands, buildings, right of way, clearing, and other factors
entering into the question of cost must all be co\i«A«t^^ ^\^^
allowed for in making up the estimate.
34 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
EngineeriDg 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
wlic'B there is more than one. On this report frequently depends
wht'tlier 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 frou)
what data the estimates were derived.
CHAPTER in.
LOCATION,
Article 7. Projecting Location.
70. After the prelimiuary has been mapped and the topography
worked in, the engineer proceeds lo make a paper location for his
guidance in the field. 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 Vie best atructtire is that which for the least cost heat an-
swers the purpose for which it was intended, should control, even
though 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 construction will entail such heavy operating ex-
penses that it may be equally unprofitable. The alignment must
be as fi-ee 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 difl^cult 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 fall 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 and connected by a grade
contour, which represents the surface-line having the required
gradient. This line will be too broken lo \>e \\s.fe^ «j^ ^ \oc»^^sstt.-
6Q A FIELD-MANUAL FOR RAILROAD EKGIKEEBS.
line, so we have then to draw on the map a succession of curves
and 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.
In general, fills are cheaper than cuts both in construction and
maintenance; and especially is this true where a shallow surface
layer of earth is underlaid by rock. It may happen that the
material from excavation must be used in embankment, when
the cuts and fills must be made to balance by shifting the grade-
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 Hue it will generally be best to strike the
curves first and draw the tangents afterwards, though it some-
times happens that long tangents will control the curves; when
this is the case the tangents are drawn to intersection and the
curves 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 useless
to attempt to show them to scale.
72. A Curve-protractor will be of matenal assistance in find-
Ing- tbe degree of curve required to unite two tangents thai have
/feen laid down ou the map. It consists oi a Ua.u«^\fewX, ««wvv
c'rcular protractor hnviDg a series ot curves Itom ^^ w^v^ck^''
LOCATION. 37
plainly cut upon it. The curves are on both sides, those on the
reverse side having their concavities turned in 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 then place the
protractor so that the curve adihitting of the best grade is tan-
gent to the two straight lines. Mark the points of taugeucy,
which will be the beginning and end of curve. When the curve
is required to pass through a given point tlie proper curve may
be immediately found by trial, whereas the calculations would
require some little time.
Reversed curves should never be allowed on main lines. Suffi-
cient tangent 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 that curves must now be run in, and this necessitates more
clearing. If first and second locution-lines are to be run (and it is
real economy to run both), it will not be necessary to have the
fltationing 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 are to be inserted, they
need not be run the fii-st time, the proper offset being made at
vhe P. T. or P. (7. of the circular curves, which latter are to be run.
On the final location-line the stationing must be continuous,
beginning with zero. 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, noalteuipt should
be made to adhere ligidly to them, since slight errors in the
mapping will affect the projecled line, while in the field the line
may be shifted here and there so as to fit the ground moi*e snugly
and accord more closely with what the nature of lli^. ^'Nc>ia:H*<«S^
4einaudiL
I
38 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
The highe»t skill of the eDgiueer is required to secure the best
locatiou-Iinc, and be should have all the time he needs. Undue
haste on location — as on recoimoissance and prelimlnaTy — is
almost sure to result in increased cost of construction.
Article 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 a common tangent.
c. A Reversed Curve is made up of two curves of contrary
flexure having the same or different rudii, 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.
Fig. 12.
e. 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 theP.C. and P.T. intersect when produced. {D 0%
Fig. 12.)
g. The Intersection Angle (/> is the angle at tbe P.L be-
tween the tangents meeting there, and equals the angle at the
center.
74. The Tangent Distance (7') is tiie length of the produoe<^
MOK^u; mousured from tUg P, G^ qx P-2\ W t.be P.^, ^tiQ iom^
LOCATION.
39
tangent is applied to any straiglit portion of the line, but the letter
y will be used to designate the produced portion only.
i. The Mid-ordinate {M) 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./., intercepted between curve aud the P.l.
k. The Long Chord (L,G.) is the chord joining the P. (7. aud
P.T, Frequently the term is applied to any chord longer than
the unit chord.
I. The Radius will be denoted by R.
m. The Point of Compound Curve (P. G. G. ) is the point of
common tangency of the two branches of a compound curve.
(See Fig. 13.)
n. The Point of Reversed Curve (P. E.G.) is the point of
common tangency of the two branches of a reversed curve.
o. The Degree of Curve {D) 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 speuk 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 the
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 &v\b\fi\i^^^\s^ >^2kR. atti
gt ^}x\\, l^^th, wc hare, wU«re a U \U\a uviW w^^
40 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
27tR = a .
360
Hence
„ a 360
(11)
When a equals 100 ft. this becomes
„ 100 360
(110
R varies inversely as D, so that knowing the radius for a 1*
curve, we should have only to divide this by i> to get the radius
for a D° curve.
Since the chord is employed instead of the arc, we determine
i? by means of the following problem
75. Given the Chord C, and Degree of Curve D, to Find the
Radius H.
In Fig. 14, AB IS the chord (7, OE a perpendicular from the
center upon J. J?
From the right triangle AEO we have
li sin iD = ja
Whence
R - } , ^ = iCcosec ID,
sm {D ^ '
When (7 is 100 ft.,
50
• • •
(12)
^ "^ ^uTP ~ ^ **'**' *^* * • •< • (W
LOCATIOK. 41
CompariDg results given by formula (12') with those given by
Ul'), we have for a few curves:
Degree of Curve. R by (120. R by (11'). Difference.
1 5729.65 5729.58 0.07
2 2864.93 2864.79 0.14
3 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 T curve the difference is inconsidernble. and we niny
stake out curves with 100-foot chords. From 7 to 14 degrees 50-
foot chords may be used. Therefore
For curves from 14° to 28° we should use 25-foot chords,
for which
i?==4^ = 12.5cosecJ2) (12' b)
am ^B * ^
Above 28° shorter chords— say 10 feet— should be used, if the
curve cannot be struck from the center. In this case
-^=^T;rVn = 5cosecJtyi> (12'c)
sin ffj^-i^
Table I of radii was computed by formulas (12'), (12'a), and
a2'b).
In practice it is customary to take the radius of a 1° curve as
5730 feet and to assume the radii to vary inversely as the degree ;
thus for a 4° curve the radius would be jK = ^^ = 1432.5 feet,
while by Table I it is 1432.69 feet— a difference of only .19 foot ;
for a 13° curve /i= -»}|a = 477.5 feet, while by Table I it is
477.68 feet. The effect of taking 5780 inslead of 5729.05 for the
radius of a 1° curve is to reduce Ihe error resulling from \Jaft
(ynumptioa that E equals 5730 divided by tU^ (l^g;ce«.QV oj^^^
42 A FIELD-MANUAL FOR RAILROAD EKGIKEERS.
76. The Length of Curve (L) is fouDd by dividiDg the angle
at tbe center (which equals the intersection angle) by tlie degree
of curve, tbe result being in chains and decimals of a chain. The
number of P.C.-^L will give tbe station number of P.T,
Example.— The P. G. 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.
4
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 locatiou of railway curves geonictncal accuracy will
frequently be of less importance than rapidity of tield-work, so
long as errors are kept within certain limits.
Ou 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 as.sume these func-
tions for other curves to vary in vei-sely as their degree, or directly
as their radii. Table IX gives values of the tangent distances,
long chords, mi d-ordi nates, and externals for a 1° curve, the
radius of whicli 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 longj 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 D.
For instance, a 2° metric curve having / = 40" would have a
mid-ordinate equal to ]r-^z = 34.56 meters.
-* X 5
For the Jipproxiuiate radius of a metric curve divide 5730 by 6
5780
times tUe degree, TUua a 4" oietriQ cuvvi; wguW Uftve /?:= rrri
LOCATION. 43
— 286.5 meters. For the exact radius make use of formula (12).
Thus for a 4° curve having 20-meter chords R = — — ttx = 286,54
° sm 2
meters, a difference of only .04 meters.
If a metric curve is to be retraced with a 100-ft. chain, we
convert the metric degree to the degree 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 take
the 20 meter chnin 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-fl. chain, and the curve with
100-ft. chords nearly three halves of the degree as laid out with
the 20-meter chain. Thus a 4° metric curve would be equivalent
^.o 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
rs 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 Ti ^ 28' means
r.he tangent distance for a V curve when 7=28', L.G.\ ^ 16"
Ahe 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 the logarithmic
sines, cosines, tangents, and cotangents, together with the loga-
rithms of numl)ers, are given in the back of the book along with
the tables of natural functions.
79. Given R and C to Find D.
From equation (12),
sin ID = ^ (13)
80. Given / and Ji (or D) to Find T.
If D is given, find i? by (12); then in Fig. 15 from triangle
44 A FIELD-MANUAL FOR RAILROAD ENGIKEERS.
By Table IX.— Find the tabular value of T for the given
angle /; then
(14a)
Example.— /= SS*' 40', Z) = 4"; required T,
By (14), T= 1432.69 tan 17° 50' = 460.91 feet
1848 4
By (14a). T= -^^ = 460.85 feet, a result differing from the
value found by the rigid method by only 0.06 foot.
81. Given /and Tto Find Ror D
From (14),
T
i? =
tan II
= TcoHL .
• •
(15)
Then by Table I the degree may be found.
By Table IX.
^=4
T
{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=Jiiim il.
.\ AQ = 'iAF= L.C. = 3iJ sin 4X .
{1«J
LOCATIOK.
45
By Table IX. — Find the tabular L,C, for the given angle /;
then
L.G.=
L' C7,i
(16a)
83. Given the Radius R and any Chord G 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 = b, the segments into which it is divided by the ordi-
nate y. Draw the radius through K; call the portion between
chord and curve y'. By geometry,
from which
(2i? - yOy = ab,
ab
y =
.»•
But y' is small compared with 27?, and hence we write
ab
y' =
2i2
{a)
Now y does not differ sensibly from y' in the cases met with in
practice, so we write
y z= — >$»>
46 A FIELD-MANUAL FOR RAILROAD ENOIKEEKS.
5730
If we write E = -jj-f formula (b) becomes
abP ,.
^"2X5780 ^^
■^^ 100 ~ *"' 100 ~ ^' *°^ substitute in (c), giving
y = jjTgQ mnn = 0.8739iini>,
or very nearly
y = imnD (17)
y is given in feet when m and n are in chains and decimals ot
a chain.
At the mid-point F, m = n, and y = M.
.'. Jf=in«D (18)
Caution. — Formulas (17) and (18), while very convenient fot
field use in passing obstructions, are liable to eiTor 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, NE for a and 6, we shall get results
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=yiHG)*D. Table V.
2 2 stations. 1.75 1.75 1.75
15.69 15.75 15.69
4.37 4.88 4.86
88.51 89.88 89.06
6.96 7.00 6.97
27.29 28.00 27.75
42.02 43.75 48.20
59.43 63.00 61.98
2 6
5 2
5 6
8 2
8 4
8 6
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° with a res up to 600 ft. ; for curves from 4* to 6*
LOCATION. 47
they may be used up to 500-ft. arcs, while for curves between
6** aud 8° uot more thau 400 feet of arc may be takeu.
Second Method.— First determine the mid-ordinate. In
triangle OEF,
OF- 4/jR«- jCf»;
then
M:=FQ = R- 4/i2«- JO'' (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 = \/B* - dK
Therefore
C7^ = y= |/i? -cJ«- 4/i2«-J(7». . . . (20)
Third Method. — If the chord G 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-ordinnte being known, we have
y ab
.-. y = 4-^ (21)
Prom formula (h) we have for y = Jf, a = ft = iC
jf=M)! = j^
%R SB ^ ^
The mid-ordinate for any other chord C is
Jf,:
Hence
M^
C7'«
M
"■ (7«*
•
• •
Jf.=
-(f)'
If (7' =
: ^0, this gives
(28)
Ml = Jitfl ••••••••* ^^» 1
48 A FIELD-MANUAL FOR RAILROAD ENGINBERS.
This lust relatiou affords nn easy method of ^taking out a curve
wlieu the mid-ordiuate of a given chord has been determioed.
First erect the ordinate M at the mid point of the chord; then
join the cuds of chord with the extremity of the ordinate just
measured; the leugths of these chords do not differ much from
iC; at their mid-points erect ordiuaies equal to JJf, giving points
on the curve. Proceed in like manner for other points until a
sufficient number have been located.
84. Given li and / to Find the External B,
In Fig. 17 K- OB=OB- OG.
But OB=n^c\l and 00 = /?.
.*. E = Ii{seciI-l) = RexaeciL ... (24)
By Table IX.— Find E for a 1' curve for an intersection
angle 1 ; then
/?.
(24a)
E=^,
D
86. Given T and / to Find E.
In Fig. 17 draw EC perpendicular to AB, and produce AQ Xo
. b/
intersect ^Cat C. BCis parallel to AO, and the triangles AOO
and CGBhra similar; hence BC= B0 = E. In the right triangle
ABC, angle BAG= \BAF= \L Therefore
E= Ttan J/. . .
EzERCiss.— Derive equation (25) from (24).
(«5)
LOCATION*.
49
86. Given ifand /to Find 27.
From trigonometry,
aec J/ =
Insert this in (24) and we get
cos jr*
E=B
1 — cos |/
COB II '
But from Fig. 17, Jf = 22(1 - cos II). Substitute in (a) :
M
(a)
Ezz
cosf/
= if8ec|/. •
• • # • •
m
87. Given ^and /to Find i?.
From (24),
B E
i? = -
, ^ jgr coa K
sec J/ — 1 exsecj/ vers J/
. • .
(27)
88. Given /and i^to Find 2.
From (25),
T =
= 17 cot }/.
• • • •
(28
tan^/
89. Given the Chord C and Degree of Curve D to Find
the Chord Deflection Offset d.
In Fig. 18 extend EA to H, making ii^T = i^^ = AB ; join
H
Fio. 18.
H and £ and draw AK to the mid point of HB. Then
HK= KB= CsiniZ).
.-. d= HB = 20bIu\D. • . .
V^iSN
50 A FIELD-MANUAL FOR It/IILUOAD ENGIlfBEBS.
When C = 100',
tf = 200 sin JD (2^)
in
If we write sin Ji> = ^ from (12) in formula (29), there results
d^-^ (30)
For curves up to T, = 100'; hence
10000
d=-^ (80)
For curves from 7° to 14°, C = OC; therefore
d = ^ (80")
For R write -^, aud (30'), for C = 100, becomes
'^=S^ = ^'«^' ...... (81)
and for C = 50. (80") becomes
d = ?^2> - .4363D = .878 . ^. . . . (81')
07o0 ^
Example.— Piud d for a 6" curve, C = 100 feet.
By (2^), tf = 200 X 0.05234 = 10.47 feet.
By (30'). d = ^^ = 10.47 feet.
By (31), d = 1.745 X 6 = 10.47 feet.
90. Given the Chord C and Degree of Curve D to Find the
Tangential Deflection Ofifset t
111 Fig. 18 make EF (tangent at E) equal to EA, and join F
with A. Draw EG to the mid-point of FA. Angle AEO =
OEF = \D; hence, from the tigure,
AQ = OF= CsiniD.
\ t = 2C%\u\D (82)
LOCATIOK. 51
When C = 100 feet,
t = 200 sin in (32')
Siuce JD is small, we may write, without material error,
sin JZ> = J sin ^D; then, writing siu Ji) = ^, as in 89, we get
c
i
Making = 100 ft. and writing H = -yr- gives
10000
' = 21^5730^ = «-8'8i). .... m
When C = 50 feet, (33) yields
t = .2182) = .436 X ^ (38")
Example.— Find t for a 6" curve, (7 = 100 ft.
By ^320 t = 200 sin 1** 30' = 5.24 ft.
By (33'), = .873 X 6 = 5.24 ft.
91. To Find the Subtangential Deflection OSaett' for a
Subchord C
First Method.— By formula (13) find the angle at the center
subtended by the subchord C; call this angle ly. From (32),
f = 2C' sin J2>' (34)
Second Method.— In Fig. 19, with ^as center strike the arcs
FO and AE, taking EF = C and
EA = C; prolong EO to B. Now
assuming that the chords C and C
are proportional to their central
Angles we have
/> — C' * * '
From the similar sectors EFQ Fio. 19.
and -^ilS, since EB = C,
AB" t
^^
52 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Multiplying (a) and {b) together, term by term,
Whence
0'" i' ' C
=<?)■■
(85)
Example.— Find t' for a 7* curve when C7 = 60 ft.
Here
By (34),
By (m.
By (36),
60
2^ = j^ X 7' (very nearly) = 4' 12'.
f = 2 X 60 X 0.01832 = 2.20 ft.
t = 6.11 ft.
r = 6.11 X
60
100
V= 2.20 ft.
7
92. To Find the Tangent Offset z.
In Fig. 20, EB=z is the required offset. Let AE!= n chains =
lOOn feet. AE= FB, the half-chord
having the mid-ordinate AF -=. EB;
hence we have, by formula (18),
2 = In^D, ... (36)
In this formula we may take n to
be either the length of ili^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 2), as was pointed out in 83.
(See Caution.)
Formula (36) is easy of application and of frequent use in
locating curves by offsets f mm the tangents. For curves up to
4" n may be as great as 8, but for sharper curves it should
be less.
Example. — Find six offsets to a 4** curve at i)oints 50 ft. apart,
measured around the curve.
Fio. ao.
LOCATION. 63
By successive applicatioDs of (86) we have
forn = i 2 = iXiX4= 0.88 feet
n = 1, « = J X 1X4= 3 50 "
n = I, « = i X f X 4 = 7.88 "
n = 2, « = JX4X4 = 14.00 "
n = |, e = JxV-X4 = 21.88
n = 3, e = jx9x4 = 31.50
«<
((
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. Difference in Length of a Oircular Arc and its Long
Chord.
First Method. — Let the central angle be a degrees. By (13),
jra*
Changing degrees to circular measure, a (in ;r meas.) = - —
_o o
: jr=-^. The length of «rc is Ra = R^jr-^, Then
07. o 07.O
Arc — chord = iJr=-o —c (87)
07. o
Second Method. — An easy approximation may be found as
follows :
Referring to Fig. 17. AE = c, QF=: M. Let ^G^ = 6 = ^ + aj.
From <he right triangle AFQ
(^•r-f
+ Jf«.
3f»
From which x-= — ; — . ..«••%» Vp^
54 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Keglectiug the x in deuomiuator as small compared with e
gives
« = T (»)
Then will 2*-<j = ap = ^^ (88)
From Huygens' approximation to the length of a circular arc
(see WUllamsou's Differential Calculus, p. 66), arc = — = — .
Therefore
Qr ^
Arc — chord = — ^ 6 = |(26 — c). . . (o)
Inserting the value of 2d — c from (38) gives
8Jf«
Arc — chord = -g— (d)
When the arc is not very great we may write c = lOOwi , where
Hi is the number of chains contained in the arc AE, From (18),
remembering that Ui = 2n,
M = 0.218»i«I>.
Inserting these values of c and Jf in ((f),
Arc - chord = | ^:^^^ = ^n.'Z/.. nearly. . (89)
Example.— Find the difference in length of arc and chord of
a 4" curve when Wi = 6 stations.
The central angle is 4 X 6 = 24°; then, from Table IV,
e = 595.74.
By (37),
Arc - chord = 1482.7 X ^ - 595.74 = 4.84 ft
By (89),
A i^ 6X6X6X4X4 ^^.
Arc - chord = ^r = 4.83 ft
oOU
Rem A RK. —Formula (38) is interesting as showing what a com-
LOCATION.
55
paratively small iDcrease in leugtb of line is caused by a consid-
erable lateral deflection in alignment. For instance, a lateral
deflection of 2000 feet is made at the mid-point of a line 40,000
feet long ; what will be the increase in length?
By (88) the increase is ~|^' = 200 feet, giving for the
increased length 40,200 feet.
40,000
B. Locating Simple Curves,
94. To Locate a Curve with the Chain by Offiiets from
Chords Produced.
In Fig. 21 let the P. C. fall at ^. If ^(7 is a full chaUi, prolong
Pio. 81.
ihe tangent AB to H, making PF= BC\ EC will equal /, which
may be calculated by (32) or (S^). With B as center, strike an
arc with radius BE, and with E as center and t as mdius strike
an arc , at G, where these arcs intersect, set a stake. Produce
BC to K, making CK-BC-GI>\ strike the arc KB from C as
center ; make the chord KB = d^ calculated from (29'), (30'), or
(31 ). Set a stake at B and proceed in like manner for the othec
points until the P. T, is reached, where VP is made equal to fe
Usually the F,C, does not fall at a full station; then BG = i\
which may be found by (34) or (35). Using- this vahie of*', w«
locate G as above. kXB' make BB — t', and prolong BG to,
L ; make LB = t and set a stake at B. EMyviW equal d, and'
may be located as l)efore.
"We may regaixi KB as equal to KE-^^t^ «L\\dc« tkji^Nxi^^ "^I**-.
56 A FIFLD-MANUAL FOR BAILBOAD ENGINEERS.
measure KD and set D without locatiug B, To do this we have
the similar triangles BEG and CKL, from which
CK BC
and therefore, since KO = CD,
B(f
In like manner at ^we have
PiV'=<^. and FP^U*
hence
NF=PN+ W.
Make EQ = tx\ prolong QF, and wc have the tangent at F.
Example.— Given the P. C. of a 5' curve at 106 + 20 and the
angle of intersection 22*", to locate the curve.
22
Here C = -^ = 4.4 stations.
Therefore the number of the P. T. is
106.20 + 4.4 = sta. 110 + 60.
BC in this case is 80 ft., and by (33')
t = 0.873 X 5 = 4.87 ft.
(80 \'
^ J = 2.80 ft
Set off EG = 2.80 ft., and at D make
100
JTD = 2.80 X ^ + 4.87 = 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
NF= 4.37 X ^ + 4.37 X f j-^j = 2.62 + 1.57 = 4.19 ft
Make EQ = 1.57 ft., and prolong QF, the terminal tangent
LOCATION.
67
96. To Iiocate a D Degree Curve by Offiiets from Tangent.
Let AM, Fig. 22, be taugeut at A, aud E, F, Q, etc., points on
the curve. The offseU BE, CF, ^ B
etc., may be found from formula t.
(86).
z = WD,
either by taking equal intervals, ^
AB, B(Jt CM along the tangent or
by taking E, F, (?, etc., at regular
stations around the cui*ve and
using the arc length instead of
the tangent.
When the arc AO i& large, or
strict accuracy is required, we
pi-oceed to find the offsets at
regular stations and the lengths
of AB, AG, etc. First find B
from (12) or (12'); then from triangle OEL,
Fio. 22.
BE= AL = B{\ - cos D) = B vers D,
AB= LE= B sin D.
In like manner
CF= AH= B(l - cos 2D) = B vers 22>,
AC=HF = BBm2D,
and 80 on for any number of stations.
Should A fall at a plus station, we first find the angle 2>i at the
center, then
BE= B vers Di ,
AB= B §in Pi ,
CF = B vers (2>, + 2»,
AC= i2 8in(A +DI
etc. = etc.
The ordinates BE, CF, etc., are evidently equal to the mid-
ordinates for long chords 2LE, 2HF, etc.; hence we can, if
A, E, f, and O, fall at full stations, take them direct from
Table V; then take the long chords from Table IV aiid dv?\^\i^
these by 2. get the required coordinates.
58 A FIELD-MA19UAL FOR RAILROAD ENGINEERS.
Example. — Locate ihree stalious of a 4° curve by offsets every
50 ft. on curve.
Referring to Tabic V, the required offsets are 0.87, 8.49, 7.85,
13.94, 21.77, and 81.81. By Table IV the distances measured
along tangent are 50.0, 99.94, 149.76, 199.39, 248.78, and 297.87.
Wiib ibese values we can set out tbe curve cither way from A,
Had we used formula (36) we should have hud for the values
of the offsets 0.87, 3.50, 7.88, 14.00, 21.87, and 31.50.
96. To Locate a Ourve by Ofiaeta from a given Long
Chord.
Let FK, Fig. 23, be tbe given chord. We may compute the
offsets yi , y« . . . Jf by the met hods of 83— of which formula (17»,
y = ImnD,
is the most convenient, within the limits of its applicability—
mid setting off these ordiuates, locate the curve.
Or we may set off the mid-ordinate M= E'^cybFOA at A,
and nt C sei oft y^ = M — Ji vers D, making
AC= IIL = lismD.
OEvfiUhe
yi = M - R vers 22>, aud AE = 72 sin 22>.
Another Method is to find the angle KOF2X the center, and
by Table IX determine BA = M \ then by Tables V and IV
LOCATION. 59
determine BL, BN, LH, and ^'0. Then EC = M - BL, which
set off ut Cf and other points in like manner.
Example.— Given the RC. of a 4* curve at station 160 + 75,
the angle betvreen tangent and chord = 9', required 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 + 26. The
mid-point on curve B falls at sta. 163. By Table IX,
M = I?^ = 17.64 ft.
By Table V the mid-ordinate for two stations of a 4" curve is
BL = 8.49.
Hence EC = 17.64 - 3.49 = 14.15.
By Table IV. EL = AG= 99.94 ft.
Measure AG = 99.94 ft., and set off CE= 14.15 ft., and drive a
stake at E, In like manner find
(?^=3.70 and ^i?= 199.39 ft.
The points P and Q are also located by means of the coordi-
hates just determined.
If B had fallen at an odd station, the curve could have been
located in the same manner, E and R being 100 ft. from B, G and
Q 200, etc.
97. To Locate a Curve with Transit and Chain when the
Degree 1) or Radius R is Known.
If B 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 clamped at zero place the telescope
in tangent either by sighting the P I. 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 dViA^xk.-
man into line while the rear chainman \io\d& Yi\^ eiiidi ol \Xi<^ OciSiSa^
60 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
at the trausit, the cbaiu being kept taut. The stakemau drives a
stake at the point wliere the head chainmau*s flag rested, and the
rear chuinman advances to this point. Deflect iD 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 jwints
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 FAE should not exceed about 15^
Move the transit to E, backsight to A, 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 telescope 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 lo the curve at the P.C. 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 thai fixed the point stibtraei the index-
angle in tangent at the last point; tlie remainder is the index-angle
required.
99. Subdeflection-angles may be found by (18) 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
26 ft. would be T, and the subdeflect ion angle 30'.
LOCATION. 61
Example -Locate a i° curve to left when the P. 0, is at sta.
81 H- 25 and /= 3:3' S&,
Here L -*. — f- = 8,15 chains.
4
Hence the P. T. will fail al 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).
.-. i6 = VdO\
By the approximate rule, since (2) = 2*,
2 "lOO*
whence J5 = 2 X f = 1* SO* as before.
With transit at P,C, deflect V 30' from tangent, measure 75
feet, and set sta. 82. Then a deflection of S*" 30' will determine
83. 5** 30* sta. 84, 7' 30' sta. 85. Now remove transit to 85, and
with vernier at 7* 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 ^jf of 2^ that is, 48'. The index-angle fixing
the P. T, 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' = I. 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
1)e1ow, which shows the notes for the last example.
When possible the tangents sliould be ruulo \\\\^\^«icNXav\,>>\^
angle I measured, and the tangent dlsvance c%\e\3\«\vi^. 'W^q.
62 A FIELD-MANUAL FOR BAILROAD EKGINEEBS
1
a
o .
•s «
— a
1?
Is
Station.
efleci
angl
Inde
readi
« be
a <
alcul
Coui
Remarks.
Q
« .
90
-HO
op.r.
0«>48'
23«'48'
82<»86'
NgT^aCE
Nar^acE
89
280 0'
88
21° (y
87
19« 0'
86
IT" 0'
85
7<'80'
16<» C
84
s-ao'
83
2o 0'
a-so'
88
1»80'
1»80'
4» C.L.; P.I. set.
+26
O P.O. 4«'C.L.
0° 0'
o» c
0° C
I =r 82« 86'; T =
418.9 ft.
81
N 60O12' E
NewicE
J
measure along tangents and set P.O. and P,T, from the P.L
When the curve is run in, the position of theP.T. thus found
should agree with the one set from the P,L 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 the last example would appear as below :
Station.
action-
ogle.
otal
ngle.
ulated
urse.
Remarks.
-2
E-i^
¥
h
90
+40
OP.3.
0°48'
16° 18'
38° 36'
N27°36'E
N27°80'E
89
15° 30'
88
13° 30'
87
tl°30'
86
9° 30'
85
7°.W
84
5°3C'
83
go 0'
3° 30'
82
I03O'
1°30'
4° curve left;
+25
0P.a4<»C.L.
0« 0'
0° 0'
P./ Ret.i=82°36';
2'= 418.9 ft.
81
N60°12'E
N60°10'E
72ie computations are all made before begmnm^ ll\ft ^ork, and
tl/e notes have the advantage of perm\U\iig t\ie Xx^iem^ ol nVjl^
curve either way from the instrument v.'U\\o\i\. «Ld6\\.\o\\\\\ comvAi-
LOCATION. 63
tatioDS. Suppose the traDsitman to bave run the curve from the
P. C, to 8ta. 85, to which poiut he removes ihe instrument. He
there sets the vernier at 0" — the angle on limb when telescope
was in tangent at the P, C, — then sighting the P. C. he reverses
the telescope and deflects to 9"* 80', which will fix sta. 86. Had
the tangent at 85 been desired, a reading of T SO'— the angle that
located that point — would have put the telescope in the plane de-
sired. A reading of 11' 80 fixes 87, and so on to the RT.
Removing to the P.T., the plates are clamped at 7" 80', and a
backsight to sta. 85 taken ; then deflecting to W 18', the tele-
scope is in tangent at the P, T, Had it been desirable to set 84
from 85, a reading of 5** 80' 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 de&iirable that some general form should be employed. Either
of the preceding forms seems to meet ordinary requirements.
C, Obstac/es.
102. To Pass an Obstacle on a Curve.
First. Suppose Vie obstacle to be 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, H^ to be invisible from A,
Fig. 25.
Set a plus sfntion at Ey as near the obstruction ns may be conven*
lent, then set FlOO feet from E. Next make FO = 100 - GE,
and locate O with the corresponding deflection-angle. Other
stakes may be set beyond O, or the transit may be removed to
that point and the curve beyond traced.
Second. Suppose the line of sight obscured for more thflrtv oa*
station, as in Fig, 26.
64 A FtELD-UAKtTAL FOR RAILROAD ENQINBKRS.
If tranmt U at A, deUcct Hn angle BAB that will dear all ob
structlODS, aud at the same time cauae B lo fall at a full at^iiMi.
Then by Table IV, Tnble IX, or by forinuls (16) calculate the
long chord JS ; measure Jfi and move trandt toB; then deflect
the hogle ABC = BAS vihca the telescope will be In taogent
The curve may now be run both ways from B.
If it happen that some stations, sa Eand Fin the figure, are
still iDTisible, they may be located by offsets from chord or tan-
gent.
ExAMPLB.— Let the curve be a 3* curve to right ; angle RAB
= T 30', the deflection-angle for S statioua. By Table IT the
long cliorU Is 498.C3 feet, which can now be measured and a hub
set at i? ; then making angle CBA = r Sff, the telescope will he
In Inngent and the curve can be traced either way.
103. To Locate a Onrve when the P.O. U Inacceailble.
^ lu Fig. 27 let the P.O. at .B he In-
accessible ; it Ib desired to reach i
. point II on acceadble ground.
First Hetbod. — Assume ■
point ff on the curve such that a
line AH from an accessible point
A, on tangent, will clear the ob-
stacle ; for convenience B should
be at a full station. The arc BB
and central angle, which equnla
BCF, are then known. Calculate
JSC = r by (14) or (14a) ; then
since AB is known, AO, = ABi^
BC, Is known.
Now in iiianglej4(7H, from trig-
onometry,
tan { (A - n) ^ AC-CB
Ian J(/i + '0 AC+CB"
LOCATION. 65
But {h + a) = e; hence
tau l(7t - a) = ^^>7 ^.// ^an Jc (40)
Then l(k + a) + J(7t — a) = /i, the larger angle, and
l(?i -{- a) — 1(/a — a) = a, the smaller angle, AH may be
found by the law of sines, or by drawing CE perpendicular
to AH, when
AH = AC cos a + CH cos h (41)
Example.— The P.O. of a 4** curve is at sta. 141 + 25, and it
is desired to reach the point JJfrom sta. 189 on tangent.
Suppose H be assumed to fall at sta. 147 ; the curve length is
Z = 147 - 141.25 = 5.75 chaius. Then angle c = 5.75 X 4 =
23" (y. By Table IX the tangent distance for a V curve is
Ti "4- 23'' = 1165.8 ft.
By (14a), T = ^^^ = 291.45 ft.
Now AC= 291.45 + 225 = 516.45 ft.,
and
AC+ GH=: 516.45 + 291.45 = 807.90,
wbJie
40-=- PH=mft.;
hence, by (40),
Urn iih - fl) = -^^- X 0.20345 = O.m^ = tan S" 1&.
oui.9
Therefore
h = 11" 30' + 3" 15' = 14" 45',
and
a = IV 30' - 3" 16' = 8' 16'.
By (41),
AH= 516.45 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 A and deflect 14" 45' into
tiiujf^ent, tlien trace in the curve.
66 A FIELD-MANUAL FOR HAlLROAD £KOtN£EttS.
Second Method. — If F, any assumed point in tangent, is
visible from A, AF may be measured by some indirect method;
then AF — AB — T, The tangent for a 1* curve having same
intersection-angle, KFG, is Ti = TxD; find this value of Ti iu
Table IX aud take out the corresponding value of 7. With
ti-ansit at F deflect the angle KFQ, 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 AB = 225 ft., aud 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.
X = -7-= 5.25 chains;
4
hence (? will fall at 141.25 + 5.25 = sta. 146 + 50. Move to Q
aud run the curve both ways.
Third Method. — In Fig. 28 let the inaccessible P.O. be at B,
and let it be required to reach E from a point C on the cmve
prolonged backwards from B,
At a given point A on tangent cal-
culate the tangent offset by (36) 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 telescope 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 intersec-
tion-angle corresponding to the mid-
ordinate that equals this product. Now deflect and measure
ECL, then by (16) or (16a) calculate CE, which measure. Move
to E aud deflect LEC = ECL and the telescope will be in
tangent. The central angle BOE = 2LEC - BOC, from which
the arc BE a.nd number of sta. J^may be found.
Example. — Take the same example as in the last two cases.
A is at sta. 139. B at 141 + 25; hence AB = 2.25 stations.
Fig. 28.
Bjr{8dX M = AC=iX (2.25)« X 4 = 17.72 ft.
LOOATIOK.
67
Or by Table IX the angle corresponding to the long chord
(3 X 2.25) X 4 -= 1800 ft. is 18" 4', for which the mid-ordinale is
71.06 ft. For our 4" curve the raid-ordiuate will be — ;-- = 17.77
4
ft., which equals AG and agrees closely enough with the value
for 2 above.
Make angle BAC—W, and measure ^r= 17.72 ft. Move
to G and sight to A, then make angle ACL = m^ —(9** 2') =
80' 58'. Suppose an angle LGE— 16" 1' to clear the obstacle.
By formula (16),
CE = 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 = 2LGE=S2^2' is 3162.0 ft.;
divide this by 4 and there results GE = 790.5 ft.
104. To 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 ordinary methods of indirect measurement, the
case merits a special solution.
First Method —In Fig. 29 let the transit be at -4, and B the
P.T. Prom the known station
numbers of A and B the length of
curve and angle / may be found;
then, by (14), AG = R tan J/, or,
hyii4a),AG=-^.
Move to G and deflect the angle
/; set a stake F, and one at some
other accessible point E, measure
angle EGF=c. Move to F &nd
measure the angle EFG and the
side EF, then in triangle KGF
angle e = 180" - (c +jy, by trigo-
Dometry
Fio. 20.
sm c
v*a.
68 - A PIELD-HANUAL FOR BAILROAD EXGINEEBS.
Since SC = AG, Ibere rtsiills BF= CF~ AC; and as the lit-
tloD aumberat Biskuowu, iliat ali^'lMcoinesknonii, and the line
niny be continued.
If B is not the P.T., measure buck Ibe distance FB, Bet treusll
at B, and continue tlie curve.
Example.— Let the P. T. of a 2° C. L. fall at sla. 805 + 50— nn
inaccesaible point; suppose A at sta. 300, angle o = 40°,/ = 60°,
£F=310ft.
i = 5.50 X 2 = ir 0, and
^_ 651.74 _„„„^
= 60%
From (42), applying logaiitbms,
log CF= 2.49188 + 9.03753 - 9.80807 = S.6
Whence CP" = 4l7.7ft. Then Bi^=4I7.7-275.e7 = U1.8 ft;
therefore the number of Fwill be 206+01.8.
Second Method.— In Fig. 80, wiib ilie tranait at anj poiut A
on Ihe curve, assume a long chord AB
and calculate the angle CAB; deOeci
this angle from (he tangent j4(',and set
a point £ beyond obstrucliou; set also
a stake at C iu tangent.
Move to ff and measure AEC and
side AC Compute .dfi from Ihe trian-
gle AEC. If this is greater or less
than Ihc Icngtb of the long chord AB,
take their difference BE and set a huh
at B With the tmusit nl B trace out
the curve.
£\AMi LE.— Given A at sta. 210 of a
3 C L angle a = 12°. h = B3°, EG
*^o ^ = 181 ft Then .: = 70°. and by solving
the tiiangle J AC. Ah = 844.7 ft. By Tublo IX the long ebord of
2382.6
a r curve tor 7=24° is 2383.8 tt-: iberefore,l^= —3— = 794.9
ft. Now vrill EB = 814.7 - 79-1.2
tance ukmg SA that tninsil must hi
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. 31 let P be the point, BP ihe per-
pendicular. We have to find
^.1 = X.
From P draw PC parallel to
AB ; theu in triangle OPG
i? = ir« + (i? - py.
From which
X = i^2Bp - p\
(43)
Second Solution. — Consider
p = AG as the mid-ordiuate for
a long chord = 2x ; then pX B
= the mid-ordinate for a 1** curve
for a central angle equal 2a.
The corresponding long chord may be taken from Table IX.
Then
Fio. 31.
(43a)
Example.— Given ;? = 30 ft., 2) = 4" (i? = 1432.7), to find x.
By (43), X =y85,962 - 900 = 291.65 feet.
By the second method,
30 X 4 = 120,
the mid-ordiuate for a 1' curve corresponding to an angle of
23° 29', for which the long chord is 2332.6. Now. by (43a),
1 ^ 2332.6 001 a 4. .
a? = j^ X — 3 — = 291.6 feet.
a 4
106. In Fig. 31, Given x and p to Find the Radius of a
Ourve Tangent to AB 2Li A and Passing through P.
From (43),
a'^ + p^
2p
* •
V^fc^
70 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
107. Given the Location of a Point P referred to the f
to Find the Radius of a Curve through P which will Ub
the Oiven Tangents.
Fio. 8S.
In Fig. 82 suppose BC — h BP = vi known, and angle a
culated ; or PC and a may be measured on the field.
From triangle GAO^
h=zW -{a + J/), and 00 = i? sec J/.
Now from triangle PGO,
sin y = -pQ sin *.
Inserting values oi PO and CO,
i? sec |/ . . 1 , . . sin &
m y = — -"— . sin 6 = sec iZ . sm h = ;-=-, .
B cos J/
sm
an equation from which the unknown B has disiipiHJared. N<
from the same triangle, since x = 180° — (6 + y),
R = 'i" * . PC
sm X
When 7= 90°, it can easily Im shown that
/i = i -J. tn 4- V2im, ,,,.,,
LOCATION.
71
108. To Locate a Tangent to a Curve from an Outside
Point.
First Method.— In Fig. 33 let P be the point and AHB the
' E H
curve. 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 = 2AC = %R sin a,
OG = R cos a.
By geometry, PE = VPA X PB, PE being the required tan-
gent. From the figure,
CO
CP
tan m =
PE'
At P deflect the angle 1 = m—n from PA and run the tangent.
Second Method. — In Table IX find the long chord for &
central angle 2a ; then
AB = ^AG =
x/. C7.I
CiEr =
and aO = B-(JH.
We tnay now proceed as before^
..■<
\
N
72 A FIELD-MANUAL FOR RAILUOAD EN6IKEERS.
109. To Run a Tangent to Two Ijocated Oorves of Oontrary
Flexure.
First Case.— In Fig. 34 let FK and LE be the curves, and
KL = p measured on the ground.
0.1
i^^-TT-St-^
>-
B
1
^
\
\
\
\
%
./
^
•-——--•-■5
Ri
0,
N
'*—
l»t
H
Fio. 84.
Let FE^ t be the required tangent.
Draw OiH parallel and O^H perpendicular tOjPj^; from the
triangle OiHO^ , since Fff = Bi ,
whence
^= i/2(i?, +i?,)i)+i>».
(48)
Also,
cos a =
(49)
The arcs FK and £i^ may be found from tbe angle a and the
known curvatures, after which the points ^ and Em&y be set.
If t is given and p required, it may easily be found from (48).
Second Case, p not known.
Set the transit at a point A on one curve ^nd note the bearing
of tbe tangent to the curve at tbat point (see F^g. 84); tbe bearing
of tbe radius Os J. differs from this by 90°. Run a line ^^C7 of
one or more courses to intersect tbe otber curve at C Note the
bearings and lengths of tbese courses and the bearing in tangenl
at (7, from which calculate tbe bearing of COi, i?i and /?« being-
kuowQ, tl^Q latitudes and depttrtvues ar^ ycxt calculated* I^et 0%^
LOCATION.
73
be the sum of the northings or southings, OiiVthe sumc^ tZte
eastings or westings ; from the triangle OiO%N,
tan ft =
and
OiOa = V O^N'' -\- 0^N\
As before, FE is the required tangent and O9H" perpendicular,,
while Oi£r is parallel thereto.
cos a =
Angle FO^N=: ft - a is the bearing of O^F, while AOiF =
c — ft + a is the angle of retreat from the known point A to F,
where the tangent may be run. The length of t= OiHia
t =z Ofi^ 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 E, F, and G
on the second curve, we may cal-
culate the new degree of curve Di
for a radius 22i = i? + p, by (13),
and trace the curve from any
point, as E, Thus
-4 in 50 50
"^"^ ^^' = Wr R + ^'
Fig. 35.
If, however, points on the radii
through A, B, and C are wanted,
they are gotten by using the same degree of curve D and cpm-
puiiug the length of chord FK, From similar triangles,
KF _ AB _ 100
^1 " ^ "" »'
74 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
whence
^z^=ioo~ = ioo^i^ = ioo(i + |). . m
Had EFO been the located curve, with i-adius E, we should
have had
^5=100
jg — p
(51)
111. To Change the P.C. of a Located Curve so that P.T,
will Fall in a Given Tangent Parallel to Terminal Tangent of
Located Curve.
Let AB, Fig. 36, be the lo-
cated curve ; FB, the tangent
in which the P,T, must fall.
Let the distance between tau-
gents be HE = p.
Draw BE and 00' parallel to
AF ; e viden tly ^ C = 00' =BB,
O being the new position of
center.
In triangle BEE,
BE- AG- ^j = p cosec J. . . . . (53)
Set the new P. C. by measurement from A, and run the curve
CE. Any system of stmight lines and curves may be treated as
above, provided I is the angle between initial and terminal
tangents and p as before.
Example.-— A located 2** 30' curve, having / = 25", ends in a
tangent 25 ft. outside of desired tangent. Find the change in
position of P. (7.
P7(^^X
^C; = 25 X 2.3Q620 = 59.16 ft.
LOCATION.
75
112. To Find the Change in Radius and Position of P.O. if
\T, is Required to fall on the same Radial Iiine but on a
tangent distant p from, and parallel to, Terminal Tangent to
ijocated Curve.
In Fig. 37 let AB be the located and GE the required curve.
Draw the parallel chords AB and
IE. DrawCfl'andjBi^perpendicular _A c k L
xiAB, The angles FBE= CAH=iI,
From the figure,
CH= AC sin ^I,
BF r= BEcoaiI = p cos iZ
Equating, p
AC lAn ^I = p cos il, Q
whence Fig. 37.
AO = p cot JZ
(58)
In the triangle OPOi, 0,P = AC, OP^R-' Bi, and
• {B — Ri) tan /= ^C = p cot J/,
r R- Ri= ACcotI=:p cot J7. cot /.
Therefore
Ri = R- AC cot I = R-p cot J/, cot L .
From trigonometry,
. (54)
. , -. sm / , ^ _ cos /
cot J/= ;; and cot / = -; — =,
' 1 — cos 7 sm i
Inserting these values in (54) gives
n n sin / cos I _ COS / „ cos /
S,=zR-^p. ^ — — y . -^—y - R -p -: R -p-
^ — COS / ' sin /
From trigoiiiometry, ex sec /
1 — cos/
yers f
cos/'
vers/*
,-. Ifi = R-
ex aec X
...... K^\
76 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Example. — A 2" 30' curve strikes 25 ft. inside a ttingent io
^bich the P.T. must fall. Find the uecessaiy change in radius
and position of P.G. when / = 25®.
By (53) the change in P. G. is
AG^25X 4.51071 = 112.77 ft.
By (54'),
i?, = 2292.01 -
25
A
.10338
= 2050.38 ft.
By Table I we find this to be the radius of a 2" 47' 41" curve.
113. Given a Iiocated Ourve uniting Two Tangents to
Find the Change in Position of P. G. or in Radius for a Given
Change in the Intersection-angle.
First Case. — Radius unchanged.
In Fig. 38 let BGE = /be the origi-
nal intersection -angle, FGE = /' the
new angle. From the figure,
AG =^ AG- GG,
or
AG = R (tan K - tan y). (55)
By Table IX.— Froni the table, foi
angle /,
T =
Fio. 38.
For/'
r =
D
Then
AG^ r- r.
Second Case.— P.O. unchanged.
Here the tangent T for the two curves is the same, and
therefore
Ji, tanil' = 12 tan y-,
WJietice
£x^ i? tan iX ^ CoX \r
• < ^
LOCATION. , 77
Bt Table IX, ^
1 14. To Find the Change in B and P. C, for a Given Change
ti /, the P,T. remaining unchanged
Fig. 89.
In Fig. 89, from the triangles OBO and OiBH,
00 = B cos I ,
and OiH=BiCoaIi,
Now QA = EF; hence
Bi — i?i cos /i = i? — i? cos J.
Whence
B. = b\^L^J^ ^ e"-^ (67)
1 — COS ii vers Ii ^ '
Also, FA = HO = Bn- BO.
Inserting values of BE and BO, there results
FA = Bi sin /, - i? sin /. (58)
116. Given a Located Curve to Find the Change in B for
I Given Change in 1\ I remaining unchanged.
In Fig. 40, from the triangles GAG and OvBO, ««yc^
^A=^ EC- AC,
t8 A FIELD-MANUAL ^OR llAILROAt) ENGINEERS.
Whence
Bi tan U - /? tan J/ =
A
EA = r
T) cot 17.
c
- T.
(S9)
Fio. 40.
By Table IX.—EA being known, T' = T+ EA. Then, by
(15a),
2>i =
T'
If the change in vertex of curve is wanted, there results, from
(25).
E= CO = rian il, E' = CH = T' Urn iL
Therefore OB = E' - E = {T' ^ T) tan iZ . . . (60)
OHctLu be found from Table IX after finding Di as above.
If Hi is giveu and EA wauled, (59) yields
EA = T' - T=(Rx - R) tan JI
116. To Find the Radius of a Curve having the Same P.C.
^ C F ^ ^ Given Curve, but ending in
a Parallel Tangent.
In Fig. 41 let the perpendicular
distnuce between tangents be p, and
ABbe the located curve; AOi = Ri
is required.
First Method. — Draw OH Ai
right angles to OiE; then
0,E= OJI+ H0+ OB,
or
Rx = (i?, - i?) cos 7+ i? + p
Fio. 41.
From which Rx — R •\-
1 — COS / vers /
(«1)
tOCAtlON. to
Second Method.—^, B, aud E lie on the suine straight line,
since / is the same for both curves. lu triangle BOE angle
JSBO = y, aud
BE = . , , = p cosec 1/.
sm il
From Table IX, AB= -^"^^
D
AE ■= AB -\- BE is the long chord for curve of degree Di;
therefore
If desired, R may be found by (12') or Table I.
Third Method. — Draw FL parallel io OiE\ then
CF= -^, = pcosec/.
siu /
From Table IX, ^ C7 = ^^^^.
AF= AC-\-CF, the tangent distance for second curve ; hence
D, = ^' ^^'
^ii^
Remark. — If transit is set up at jB, 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 RAILROAD ENQINEERS.
Article 9. Compound Curves.
A. Location Problems,
117. Given Two Unequal Tangents, their Intersection-angle^
and One Radius, to Find the Other Radius of a Compound
Curve uniting Tangents.
In Fig. 42, All = Ti and Bll = T9 are the known tangents,
AOi = Ri the known radius. BOi = /?« and the angles /i and J%
must be found before curve can be located.
Fio. 4!i.
Extend first branch to /^, so that tangent FL is parallel to BE.
Draw HK find BO per[>fntlicular to FL ; draw FB and extend
to ^; it will pass through the P.C.C., because the central angles
EOiFdma EOiB are equal Then
To = AL = Ri tan il.
I.
In triangle ZUK, since LH = To - Ti ,
= ^1 = in- r,)cos/.
p = IIK^- no = (To - Tx) shiZ
Now in triangle DGF iing\e BFO = \U, and
l^FG-Ts,-^%-Tx,
LOCATION. 81
tani/a = ? (62)
Draw Oailf parallel to FL ; then
(i?i — J?s) sin Js = ;,
Whence
i?« = i?i - -7-T = ^i — ^cosec /,. . . (63)
sin /a
Had Hi been required, the equation would have been
i?i = i?a + ' COSeC Ja.
Evidently, Ix=^I- 7a.
In the field the points E and B may be located by running in
the curve from A as starting-point, or run the chord
from A, and at F deflect angle AFB = \I — J/a = |/, , measure
FB-lsec Ua and i?^ = 2/?a sin Ua-
Example.— A 2** curve has the P.O. at sta. 110, Ti = 590 ft.,
Ta = 511.8 ft., 7 = 30" 50'. Locate the curve.
By Table IX, To = 1580/2 = 790 ft.
By formulas above,
« = 200 X 0.85866 = 171.73 ft.,
1? = 200 X 0.51254 = 102.51 ft.,
^ = 790 + 171.73 - 511.8 = 449.93,
tan i/a = ^^lll" = 0.22T84 = tan n'' 50'.
44V7. y«5
Then /, = 80'' 50' - 25° 40' = 5' 10'.
440 07
i?a = 2864.93 - - ~ = 1833 feet.
.4ooiu
82 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
By Table I this is seen to be the radius of a 3*^ 7i' curve.
The length of first branch is 258.3 feet, and of the second 821.3
feet; hence the P.(7.C. falls at 112 + 58.3, while the P,T, is at
sta. 120 H- 79.6.
118. Given the Long Chord from P,G. to P,T, 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
HBA = b. Two angles and one side of the triangle HAB are
known, and the sides HA = Ti and EB = 7a may be found,
after which the solution is the same as iu the last problem.
A solution may be reached in a different manner. I = a + b,
HAF = U = 2(« H- *), and BAF z=z \{a -\- h) - a = \Q) - a).
AF = 2i?i sin J/. In triangle BAF two sides and the included
angle are now known, so BF and angle BFA may be found;
OFB =- l/a = i/ - BFA.
Then EF = 2/?, sin J/, ,
and EB = EF — BF becomes known.
Then EB = 2i?a sin 47, = 2i?i sin 47, - BF,
BF
whence 7?> = 7?i — ^-—. — ;-=- (64)
2 sm \u
Evidently Ii = I - I^
119. Given the Radii and Central Angles of a Compound
Curve to Find the Tangent Lengths, the Long Chord from
F.C. to P. 2., and the Angles it makes with Tangents.
In Fig. 43 draw AE and BE from the
P.C, and RT, to the PaC, then
calculate AE and BE by (16) or by
Table IX. In triangle AEB angle
AEB = 180 - i(7, H- 7a). Two sides
and the iucluded augle being known,
the triangle AEB may be solved for
AB and the angles ABB and BAE;
then
BAF= BAE+iL,
r'<»- «• ABFr= ABE + i7,.
The angle AFB of triaugle AHF now becomes known and.
LOCATION. 83
AB is known, the sides AF = Ti and BF = T9 may be com-
puted.
120. Given the Long Chord from P.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. 48 let QHhe parallel to AB, and OAB = a, HBA = b
known. Then
BAE = BAG = OEA = \a,
and ABE = EBH = HEB = \b,
Also, AEB = 180** - l(a + h).
In triangle J.2Z[S, remembering that
sin [180 - i(a + 6)] = sin \{a + J),
and
sin i(a + by
BE= ^^^^^^^
sin J(a + b)'
Since AOiE = a and i^Oai? = 6, the radii Bi and i?s may be
found from formula (16), or (I60).
By (16). iJ. = i4J = -_i^^iyi_. ... (65)
' ' sm Ja 2 sm ^a . sin J(a -|- 6) ^ ^
^ \BE AB^in^a
^' "" sin 16 ~ 2 sin Jft . sin \{a + b)' ' ' ' ^^^^
ExAMPLR. — Required Bi and B^ , or 2)i and 2>9 , when ^5 =
900 feet, a = 12", b = 15^
By (65), i?i = 2407 ft.
By (66), i?a = 1543.7 ft.
^hjm Table I I), = 2** 22' 50" and D^ = ^'^ ^K ^^' *
84 A FIELD-MANUAL FOR RAILROAD ENGIKEERS.
B. Obstacles.
121. To liocate a Point on i.ne Second Branch of a Com-
pound Curve when the P.G.G. is Inaccessible.
Ordinarily the second branch is located by setting transit at the
P. C. C and running the ciirve from that point. An obstacle on
either curve may then be passed by the methods given for simple
curves.
When the P,C,C. is inaccessible,
locate the first branch from the P. (7.
and the second branch from tbe
P. T,t if this latter point is known.
When this is not the case proceed
by one of the following metbods:
First. By means of a long
chord.
In Fig. 44 let ^ be the PC. C,
A some known point on first
bi-anch, EF a tangent at B, and
AB parallel to FE. The station
numbers of A and E being
known, the arc AE and angle a
Fig. 44.
are readily found ; tben
EL = i?a vers b = Ri vers a.
whence
next,
vers 6 =
Rx vers a
(67)
^5=ifi sina + ifasinft (68)
Deflect FAB = a from tangent at A \ measure out AB ; set tbe
transit at B and locate tbe second brancb.
By Table IX. —Take the mid-ordinate in table for an Inter-
section-angle 2a ; then
EL =
A
Then EL X B^ is the mid-ordinate for a 1° curve having
7=2^, from which b becomes known. From the table now find
AL and LB, the half-chords for angles 2a and 26, and proceed as
before.
Second Method.—^ tneans of tangents.
J*rom Fig. 44, AF = FE = Ri tan {a.
LOCATION.
85
Set traDsit at F, deflect OFE = a, and by some indirect method
measure to an accessible point H,
and
EH= FH-FB,
EH
tan \b = ■=^, from formula (14).
/la
Angle b 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 tangent distance for
I = a; then having ^ZT measured, take Ti = EH X -Da and find
the corresponding angle, which equals b ; then proceed to locate
curve as above.
Example.— Let -4 be at sta. 126, P. C. (7. 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 = 11° 2* corresponds to 183.9 ft. around 6°
curve; hence the P.T. number is 130 -f- 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
B fall at sta. 131 + 25.
C. Change of Location,
122. Having a Simple Curve Located to Find the P.C.C. 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 bniuch must end. The dis-
tance BG = p between tangents is
known from measurement. If angle
a can be found, the arc BA becomes
known and the j^oiut A can be located
from B. Draw O^L from the center
of second branch perpendicular to
Pro. 45. 0,5. In triangle 0,OaZ, O^O-k^
Bi — Bt, and OiX = A — (R^ + p) ; lYv^ixelox^
86 A FIELD-MANUAL FOK RAILROAD ENGINEERS.
if I — i?9 — p ^
p
Ux --B.'
(69)
Then a divided by Di gives arc BA.
If desired, BHmsij be found from the right triangle BHOt in
which the side BG = p and angle ORB = Ja are known—
A, H, and J? lying in the same straight line ; then
BE = . , = p cosec Ja.
(70)
sin {a
Or J?^ and HA may be found from Table IX, after which
BH=BA-nA,
Example. — A 8" curve ends in a tangent at sta. 160 + 50,
85 ft. outside of desired tangent. Find the point of compound-
ing with a 4' 50' curve.
From Table I, R for 3" curve equals 1910.08 ft, and for
4° 50' curve 1185.78 ft.
Then, by (69), cos a = 1 —
85
724.8
= 0.95168.
From table of cosines angle a is found to be 17" 58'. Dividing
this by 3 gives 5.961 stations for the arc BA, Hence the P.CC.
number is 160.50 - 5.961 = sta. 154 + 53.9, and the new P.T. is
at sta. 158 + 28.9.
123. Given a Located Oompound Curve ending In a
Tangent Parallel to, and a Given Distance from, a Tangent
in which the Curve is required to end. To Find the Neces-
sary Change in P. C. C.
First Case. — Terminal branch Jiaving shorter radius.
In Fig. 46 let ABC be the located
curve, AEF the one required ; angle
BOiC= a linown, and also MIf = p.
If angle EOM = b can be found, the
angle of retreat from B to E will equal
6 — a.
Draw O/A" and OiL perpendicular
to ON, which is parallel to OiC,
Fio. 46.
Then OK = (R- i?,) cos 6,
0L = {R-ROcwa.
LOCATION. 87
Now LM^ Rx-KL = R,'^ MN, from which KL = MIf=: p.
Hence
{B — jRi) cos 6 = (if — Bi) cos a — p.
From which
cos 6 = cos a — - ^ -- (71)
Divide 6 — a by i>, the curvature of first branch, and move
back that number of stations from B to the new P. C, C. at E,
Join 0.0,'; evidently FC = OxOi\ and angle KOx'O^ = OZ^O ;
00/0, = 90° - \{h - a), OOx'K = 90'' - b. Hence
C2?V3f = KO.'Oi = [90 - 4(* - a)] - (90 - 6) = 1(6 + a). (72)
From triangle COF,
Or, from triangle OOi'Oi ,
FC = Oi'Oi = 2(if - i?,) sin \fp - a).
ELad AEF been the original curve, b would have been known
and a required.
From (71), cos a = cos 6 + ^ ^_ (74)
B — B\ '
CFanA angle CFMnre given by formulas (73) and (72).
Example. — A 2° curve compounds with a 4** curve at sta.
82 -f 30; « = 20" 30', p = 40 feet. Find number of new P.C.C.
aud distance between P.T.s.
40
From (71), cos b = 0.93667 - ^3^4.^ J^^^gg^ = 0.90874.
This yields 6 = 24° 40', and 6 - a = 4° 10'.
The change in P. CO. is -^-— = 2.083 stations; the P.O.C,
number is therefore 82.30 - 2.083 = sla. SO -V '^^.•'\-
88 A FIELD-MANUAL FOK RAILROAD BNGIKEEB8.
By (72).
By (73),
CFQ = 4(24" 40' -h 20' SC) = 22° 85'.
2^ = 40 X 2.60399 = 104.2 feet.
Second Gabb. — The terminal branch hating longer radiut.
Let CAB, Fig. 47, be the located
curve with P.C.C, at A, aud let
FK be the tangent in which the
curve is required to end.
The distance BK = p, the radii
OA = B, OiA = Bi , and angle
AOiB = a being known, it will
be sufficient to find angle EOi'F
in order to get the angle of ad-
vance, AOE = a — b. Draw OL
and OiN perpendicular to Oi'F
and OiB. From tlie triangles
Fio. 47.
0,'OJf and 0,0X.
(Bi - B)cosb= OiN+ {Bi - 12) cos a.
But
Ox'N=KB = p)
therefore
(iJ, - -B) cos 6 = p + (JRi — i?) cos a.
Whence
cos b = COB a -\-
Bi-B'
(76)
Then -^ will be length of curve from Ato E,
Angle KFB = NO.Ox' = 00,0,' - NOxO.
But 00,0,' = 90" - l(a - 6) and i\rO,0 = 90 -a.
•. KFB = [90° - J(a - b)] - [90 - a] = \{a + h).
From triangle KFB,
FB^
P
siu \{a + b)
Or, from triangle OxOOx\ since OxOi — FB,
FB = 2(/;, - i?) sin l(a - 6).
= p . cosec J(a + 6). . • . (7(J)
LOCATION.
89
If AEFhtid been the located curve, b would have been given
and a required. From formula (75),
cos a = cos b —
Hx^Ii'
(77)
Example. — A 5** curve compounds at sta. 60 with a 2" curve,
and the RT. is at sta. 80. What will be the number of F.G.C.
if the P,T, fall in a tangent 81 feet inside of terminal tangent?
Here a = 40".
81
By (75), cos b = 0.76604 + j^ = 0.81316.
Hence b = 85** 36' and a - 6 = 4' 24', corresponding to 88 feet
around the 5° curve. The number of the new P. C. G. is therefore
60 4-88.
angle KFB = 1(40"' 0' + 35*' 86') = 87'' 48',
and
FB = Six 1.63157 = 132.16 feet.
124. Qiven a Located Compound Ourve to Find Necessary
Change in P.C.G. 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 Ziine.
First Cask,— Second branch having shorter radius.
In Fig. 48, OB=R, OiB=Ri angle
a and HO = p are known. OiE=E^
and angle b must be found ; then
-^^ = BE will be the change in
ROM.
Produce first branch to -ff, where
OK is parallel to 0, G. Since BOK
= BOi G, B, K, and G lie in the same
straight line; and since EOiF ~ oI^A-JlAI
EOK, E, F, and K lie in the same ^
straight line. Therefore
KCG^ia, and EFH=ib.
^f^.^
90 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
From triaugles KFHviud KCG,
HK GK OH
tanl6 = ^^ = g^ + ^=tania + ^
But
FH=z OiL = (i? - i?,) sin a.
.'. tan 16= tan }a +
(i?~ i?i)8ina
(78)
From triangles OOi L and OOtM,
(R - i?a) sin 5 = (i? - i2i) sin a.
Whence
B^^B-'iR- i?,)
sin a
sin 5
(W,
Had ^-EF been the fii-st curve located, b and B% would be
known, « and Ri required.
From the figure, reasoning as before,
(80)
and
tan la = tan i* - (^ , ^,) sip h
i?, =i?-(i2-i?.)^ (81)
^ sin a
Second CABK.^Second branch having longer radhu.
Fio. 49.
In Fig. 49 let AB be the located curve, i^^the curve required,
OA = R, OxA =^ Ri , 0,K = R^, FB = p.
LOCATION. 91
/?2 and angle b are wanted, angle a being known.
We can show, as in first case, that
HFK = lb, HBL = \a,
OM:=KF=LB = (ifi - R) sin a;
and hence
^ .. HK EL p
Or inserting values,
tau U = tan > - ^^— ^^j^ (82)
r
Angle b now becomes known and — jr— = ^i^in chains, which
18 the change in position of P.G.G.
From triangles OOiif and OO^M,
(JR, - R) sin b = (Rx — R) sin a
>•. 5, = i?+(i?, -i?)?|5_? (88)
Had the new tangent fallen outside the old one, we should have
had
tania = tanl6+(-^-^^^^^, ... (84)
and
i?. = 72 +(/?,- if)®4^ (85)
sm a ^ '
126. Having a Located Compound Ourye, to Find the
Ohang« in P.G.G, and Radius of Second Branch in order to
OatiM P.T. to Fall at a New Point in Terminal Tan^«xLt«
PlBST Oji»^. ^ Second l>ranch /lacing bIiqtUt radius.
92 A FIELD-MAKUAL FOR KAILBOAD EVGINEEBS.
In Fig. 50 let NAB be tbe locatod curve, and C the poiut wbere
P.T. is required to fall. Let 5C = A, 0^ = * OiB = /?,, and
uogle Oi OH = a be known; angle b and R% are required.
Fio. sa
Extend first brunch to F, making OF parallel to OiB. A, B,
and F lie on a siraigbt line, for angles ^Oi^and AOFbXt equal;
likewise E, (-. and F lie on tbe same straight line.
From triangles G'BFand QCF,
^^. OB CB ^ , *
But OF = EM = {li - 22,X1 - cos a) = (fi - i?,) vers a.
.'. cotjft = cot Ja —
(/i— i?i) versa
From triangles 00|/7and OOtL, since OiP = A^
(i? - /?,) sin 6 = (if- i?,) sin a - *.
Whence
ff. = B+
k -{R - Jg|) sin g
sin b
(»
Then 6 — a divided by D gives arc A E With radius Jff|1oca
the curve BC from (7 or E,
LOCATIOK.
93
Had NEO been tbe located curve, R, R% , and b would have
been known, Ri and a required. lu tbis case
cot }a = cot ^ -f
(R - i?a) vers h'
(88)
ig, = i?-, ^ + (i?-g.)8in5 ^g^j
sin a
Second Casb. — Terminal branch Tuiting longer radius
In fig. 51 let NAB be tbe located and iV'^Ctbe required curve.
H L R,
Fig. 51.
B
Let CB = A; be known. Tben, as in tbe first case.
OG _0B k
cotie>_^^_ — -— .
.'. col|d = cotJa
{Ux - R)yeraa'
(90)
and
(R, - R)Bii\a=(fiu - -K)siu64-A;;
wbence
^^^ (fl.-fi)sina-*
smd ^ '
94 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Had NBO been located and NAB required, the equations would
have been
cot \a = cot Jd +
(R% - R) vers h
. . . (W)
and
Ri=R +
(Rt - IQ sin 5 + A;
sin a
• • •
(98)
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
found from the foregoing equations. Or, knowing the new angle,
the new radius and .value of k may be found from the same
equations.
126. To Replace a Curve of Oiv«n Radius, which unites
Two Tangents with Known Intersection-angle, by a Three-
centered Compound Curve.
In Fig. 52 let OA = R he the radius of located curve.
Oa(7= Oi'A = Ri the radius of terminal portions of the three-
centered curve, and the other notation as shown in the figure.
Draw OaOa', and draw i^O// perpendicular thereto. From tri-
angles O^OiH and O^OU,
O^U = (/?a - /2.) sin \L =^{R^-R) sin il, . . . (a)
Suppose Ri and Ri to be assumed ; then equation (a) yields
sin J/i = -j^ j^ sin J/.
Ri - Ri
(W)
LOCATION. 95
Then AO^' E r=: CO^G = i(I - Ir). . . . (95)
Suppose AOiB, CO9O, and i?a to have been assumed. From
(95) find 7i ; then, from equation (a),
Bi=B,- (M, - R)^^^ (96)
sm J/|
Example.— Given a 4° curve, / = 38*, and the terminal
branches composed of a 2*" curve for two stations, to find Bi and
Di for the central portion.
Here /i = 38" - 2(2 X 2)* = 30*.
Prom Table I, JR. = 2865 ft., B= 1432.7 ft.
Whence i?, - JR = 1432.3 ft.
•
Log 1432.3 = 3.15603
•* sin 19" 0' = 9.51264
2.66867
" sm 15" (/ = 9.41300
.-. log 1801.7 = 3.25567
Therefore Bi = 2865 - 1801.7 = 1063.3 ft., and, by Table I,
Bi = 5" 23'. 4, nearly enough.
127. To Substitute a Curve of Qiven Radius for a Tangent
nniting Two Curves.
In Fig. 53 let the tangent BG = t, OB = B, Ofi = B^ , and
0%A = i?a be known.
Angles a, &, and c 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.
tan d = ^ _^ (97)
t
00, = -j^ =t.coaec d= V{R^- Kf'\-X>^, • V5sfe>>
9G A FIELD-MAKUAL FOR RAILROAD ENGINEERS.
Now iu triangle OOiOt three sides are known and the angles
c and e may be computed. Thus if s is the half-sum of the sides,
cos ic = i/i^iZli^ .
Fio. 58.
Angle 6 may be found in like manner, then b = 180* — (« + ^
and a = e — b.
Points A and 27 may now be located and the curve traced.
Example. —A 3** and a 5° curve are united by a tangent 500
feet long. Replace by a 2° curve.
Here E,-R= 1910 - 1146 = 764 feet.
By (97), tan c? = ?^ = 0.65444 = tan 88* 12'
By (98), OOi = 918.1 feet.
In triangle 00, O9, OOi = 918.1, 0,0, = 954.9, and OOt^
1718.7 feet. Solving for e and c,
tf = 183'86', c = 23''0'. Then 6 = 18° 12', a = 9° 48'.
Article 10, Track Problems.
128. Reversed O^^rves should never 1>e employed on maiD
lines because of the shock due to pudden reversal of curvature
and superelevation of outside mil. 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
the radius. If, however, transition curves arc employed to ease
LOCATION.
.97
off both curves, there would seem to be no objection lo the use of
curves of contrary flexure, provided the track may be kept
alwfyrs 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 Located Curve Intersected by a Straight
lane, to Connect them by Another Curve.
£ithcrthe 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.
First Case. — Joining curve tangent to located curve internally
and on same side of cutting line as center.
In Fig. 54 let OF be joining curve, with center Oi and radius
Ri. Let radius of located curve OF = R. Draw OiO and OH
perpendicular to the cutting line produced, and OiK parallel to
AH, If Ri is known, we must determine angle b, a having been
Fio. M.
measured ; then h — a gives the length of arc from Aio F where
the P.C.G. is to be located. In the triangle KOOi we have
0K= OH- Ri and OOi = R - Ru
R cos a — Ri
Then
cos 6 =
& — a
H
R — Ri
= arc AF.
(99)
Had Fbeen given, we should have b = a-VA0F,tvw^A'c««!^^5*^>
n _ ii (cos a — COS b> __ R (^co& a — eo^V> ^ V^?si^^
1 — cos h Nfe\^b
98 A FIELD-MANUAL FOR RAILROAD ENGIKEERS.
Example. — A 1" curve is cut by a tangent that makes an angle
of 64° 32' with tangent to curve. Unite by means of a 4* curve.
By (99), cos b = 0.24000 = cos 76' 07', and therefore b - a =
IV 35', making AF, of figure, 11.58 stations.
Second Case. — Joining curve tangent internally to located
'curve but on opposite side of cutting line from center of located
curve.
In Fig. 54 let arc ME^ with center Oa and radius i?a , be the
joining curve. From the figure,
cos d =
i? cos fl -f i?a
5-i?a '
(101)
Then arc AE = a - d divided by A and c = 180° - d.
Had the point E been given and B-t required, it would have
been, from (101)
i?(cos d — cos a)
i?a =
(108)
1 + COS d
Example. — Take the same example as in first case. Here,
By (101), cos d = 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,
with center on same side of cutting line.
Fio. 55.
la Fjg. 651etarcBC,Yf\i\i center Oi and radius Ex , be the join-
IBS' curve. Dmw OiE parallel to C¥, and OiC md OF v«^cV«o^-
dicular thereto.
LOCATION. 99
I^m the figure,
(i? + Ri) cosb = Rcoaa — i?i;
.-. cos 6 = ^^^^ (103)
Then d = 180 — 5, and AOB = b — a. The curve may now be
traced on the ground.
If AC is wanted, we have AC = (i? + Ri) s\nb — R sin a. .
If the points Is fixed and Ri required, there results, from (103),
^^^ g(co3a-co8 6) ^^^^
1 -j- COS 6
ExA3iFLB. — Take the example given for the first and second
cases
By (103),
_ 6730X0.43^1432.5 _ o 144 - cos 81" 44'
^""^ ^ - 5730 + 1432.5 " "'^^ ^ ^"^ ^^ ^^
ft - a = Sr 44' - 64" 32' = 17" 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 B to C on the
4" curve.
Fourth Case. — Joining curve tangent externally to located curve,
teith center on opposite side of cutting line.
Let O2, Fig 55, be center of joining curve, i?a its radius.
B^rom the figure,
(R + i2«) cos c = i? cos a + i?a.
R cos a + i?a .^^^,
.'. cosc= jg_|_^^ (105)
If if is fixed and R^ required, (105) yields
_ i?(cos c — cos a) R(coa c — cos a) ,^^^.
1 — cos c versm c
ExAiiFLB. — Take same example as in preceding cases.
Qy (105), cos c = 0.54403 = cos 57" 02'.
Then a - c = 64" 32' - 57* 2' = T W,
100 A FIELD-MANUAL FOR RAILROAD EJTGINEEB8.
calliug for a distance of 7.50 stations from ^ to if aroun
V curve. From ilf to 2? on 4° curve is 14.268 stations.
130. To Locate a T
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.
FiKST Case. — One branch of Ya straight 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
Fig. 56.
one of the points A, B, or is given, the others may be located
by finding the angles c and 6. Draw OiE parallel to CA ; then
in triangle OOxE
(R + Bi) cos 6 = i? - Bx.
. *. cos h =
B-Bx
B + Bx
(107)
This follows at once from (103) by making angle a = 0. Then
angle c = 180 — b. If AB were a located curve and the point
5 given, formula (107) would furnish us a value for B%.
Another solution is to produce the tangent at B to cut AC at F;
then AF = FC = BF Join i^with and*0, ; it can easily be
seen that angle OFOx = 90°, and, by geometry.
BF= VRX Bx.
(108)
BF /Bx
Therefore tan \b = — =4/ . • •
• • •
. (109)
and
tan
*<' = ^=i^l • • ^""'
LOCATION.
101
ExABCFLBL— Let AB te a S* curve, BGk 6° curve, the point A
Ht station 180.
7^7(107). V .
cos ^ = ioiA i 7 ^'^ = 0. 33817 = cos W 32'.
* • t
The number of B is 180 + 23.511 = 203 + 51.1. Angle c =
lOO"* 28', equivalent to 18.244 stations on the 6** cur\e
Second Case. — The three branches curved and conves^ t-^wards
each otiier.
Given the three radii and any
one of the points A, B, or G,
Fig. 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 OOiO^, letting
OOi =. I, O1O2 = m, 00a = n,
€ = i{l + m + n) = B + Ri+B^,
we shall have, by trigonometry.
Fig. 67.
cos^t
y m. n "f (R^ R.)(R, 4- Ro\
(R+ R,)(Ri + R,)
(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 Oi Oa. Then from the relation
OiOa : OiO + 00a = OOi - 00a : OiE - O^E
determine Oa^ and OiE\ then the right triangles O^OE and
OiO^ yield values of cosine a and cosine c, after which b may
readily be obtained.
Third Case. — One branch concave to the oilier two.
In Fig. 58 the trian^-le OOj O2 may be soVved ioT Wife ^x^^^"^ v^»
O. O,, and Oa ; for if the radii are given, iVie s\v\fes OOv =^ K — ^\%
0Os=Ii- Ha, and Ox 0, = R, -^ B^ are kno^u «Ai^ \\ife ^c>\\>s:v3«^
102 A FIELD-MANUAL FOR RAILROA^D ENGINEERS.
is the same as for second case. Tbcti b is the central angle f«
curve AB, a' = 180 — o, the ^ceati;al^ angle for AO, and e'
180 — c, the central angle for curve BC.
Example.— If A is at sta. 820 on the V curve AB, AC an
8° curve, connect with a 6° curve CB, Here we have
OaO = 5730 - 717 = 5013. OiO = 5730 - 955 = 4775,
and O^Ot = 955 + 717 = 1673.
Solving this triangle, we get c = 88** 20', b = 19** 28', and
a = 72" 12'. The number of B is therefore 820 + 19.467 =
839 + 46.7 ; the length of CB is .?1^ = 15.278 stations, and
o
107 8
of ^Cis -^ = 13.475 stations.
o
131. To Locate a Reversed Curve between Parallel
Tangents.
First Case. — Radii equal.
(a) The equal radii R and distance p between tangents known.
In Fig. 59 draw OJ^ parallel to J. G^ to meet O1.B produced.
From triangle OEOi,
2R - p . p ,..^.
• and 0E=2Rs\aa (118)
LOCATIOK.
103
From triangle ABO,
AB = -^-^ z=z p coseo \a = f/aP"+p. . (114)
Fig. 50,
(J)) AQ and p known, R required.
Here AB - i/AQ^^fp^ = k. Draw OH to the mid-point of
-4C7. Triangles -dOJET and ABO are similar and AH =^ \k.
Therefore
"Whence
A.'
4p*
(115)
Example. — Connect two parallel tracks, 30 ft. c. to c. by a 7'
reversed curve. From Table I, i? = 819 feet, and, by (112),
cos a = 1 —
30
1688
= 0.98167 = cos 10** 59'.
By (113), OE = 1638 X .19052 = 312.1 feet.
By (114), AB = V(312.1)» + (30)« = 313.5 feet.
If p = 30, OE = 812.1. or AB = 313.5 had been given, we
should have had, by (115)
i?=M= 819 feet.
104 A FIELD-JiANUAL FOB iU.ILROAD ENOINBKBS.
Second Case. — Radii unequal.
(a) Suppose the radii R = OA and Ri - OxB (Fig. 59) to be
known We must find central angle a and AB = k. From the
triangle OOxE,
^^'^- R+R, -^-:btx- • • ^^^^
Then AB will be given by (114).
{b) Suppose AB = k, p and R known, to find Ri and angle a.
Triangle ABG yields
sin ia = -|- . . (117)
OiLB is similar to AOB, Hence
is" !)•
But ^a = 2R sin ia, and Z5 = i(A; - u4C/) = iCi. Inserting
this value of LB and solving for i2i,
i?. = -^. (118)
From similar triangles,
Ri Or
R - k-Gi'
Inserting me value of (7i = -^-^ from (118) and solving for
rC
Ri , we get
i?, = ^ - 12. (11»)
ExAMPLE.—^^ = 300', p = 30', R = 819 ft., to find angle
a Hud Ri.
By (117), sin ^a = .-^ = 0.10000 = sin S** 44'.
Therefore angle a = 11** 28'.
Ev (11?^. R, = ^-^11 _ 819 - 681 ft., an 8" IJ5' curve.
'
LOCATION. 105
132. To Oonnect Two Parallel Tracks by a Crossover com-
posed of two 2)° Curves with a Given Length of Tangent
between Points of Contrary Fleznre.
In Fig. 60 let AFQB be the re-
quired crossover, FO = lf EB=p,
and OA = OB = S known; H «- ^/^
angle a and AE = x are re- -
quired.
Draw OM parallel to AE to
meet O^B produced ; draw also - „
00 parallel and equal to FO;
join O and O'. From triangle
oao,
tany=^, .... (120)
21?
0(y = -=^ = 2i?sec y = V^R' + P. . . (121)
cosy
Then in triangle OO'M,
C0S2 =
Oa 2i? sec
i^k'^""^"- • ^'''^
Now knowing y and 0,
a-z-y (123)
Next, X = OM- oa sin « = 21? sec y sin 2. . (124)
Example.— Given B = T 30', p = 62 ft., / = 100 ft., to locate
crossover when A is at sta. 86 + 20.
By (120).
log tan y = 2 - 3.18441 = 8.81559 = log tan 3° 44'.
By (121),
log oa = 8.18441 - 9.09908 = 3.18533 = log 1582.
By (122),
log cos 2 = 3.16643 - 3.18533 = 9.98110 = log cos 16'' 47'.
By (123),
a = W 47' - 3** 44' = 13** 3'.
By (im
Jog X = S. 18533 -f 9.46053 = 2.^45^^ = \o^ ^Aa A.
106 A FIELD-MANUAL FOR RAILROAD EKOIKEEBS.
133. To Find the Radius of the Reversed Ounre AFE, Fig.
^0' 61, Given Angles / and i', and
BG = *.
From the figure*
R tan \I = BF,
i?tanl/'= OF.
Adding,
J2(tan \1 + tan if) = BC^l
Fig. 61.
Whence
i? =
(125)
tan i/ + tan i/'
Example.— Given / = 10", i'= 20*, 5C = 700 feet, to find R
700
By (125), 2? =
0.08749 + 0.17688
= 2658 ft., a 2** 9J' curve.
134. To Locate a Reversed Curve between Fixed Points.
In Fig. 62 let AB = Ar, and angles 1 and /' be known. We
Fig. 63.
have to find R and the angles a and b.
Draw O'G parallel to, and OG and CF perpendicular to, >^,1
Angle AOG = I and B^F = I'. Then OE = R cos / and ^^
= i^ cos/'. Hence
OG = if(cos / + cos r).
In triangle 00' G, 00 = 27?. Therefore
_ 7?(cos I -r- cos i') _ cos / >f cos I' ^^ad)
cos a; _ g-^^ _ 2 ' • *
an expression from which R has disappeared.
LOCATION.
107
We now have a = / + a? and b = F + w.
To find Ryre have AE+ EF -\- FB = k,
^r iJsin /+ 3i?sin x + l?sin /' = k.
Whence
i? = -
k
(127)
sin / + sin /' -f 2 sin x"^
Another expresmn for R can be found by drawing ^iVand BL
perpendicular to 0(y, and BN parallel thereto. Then, since
^BA]Sr=x,
Bsina-\-Ii8inb = k cos x,
k cos X
i? =
sin a -f sin 6' '
Example. — Take the example of the last problem,
A; = 700, 7=10% /' = 20".
By (126),
cos a; = i(0. 98481 + 0.93969) = 0.96225 = cos 15" 48'.
We now have a = 25" 48' and b = 35" 48'.
700 X 0.96225
(128)
By (128), i? =
= 660.2 ft., an 8" 41' curve.
0.43523 + 0.58496
135. To Oonnect Two Divergent Tangents by a Reversed
IJurve.
First Case. — Advancing towards the P.I,
Given the radii B and Bi , the angle / and AO = A;, to find the
ingles a and b (Fig. 63).
r^-^
Fio. 68.
Draw 00 parallel to the tangent BG to meet OiB produced.
Then EF= BG = AF- AE.
Therefore BO = B coa I - k sin I.
108 A FIELD-MANUAL FOR RAILROAD EKOIKEERS.
From triangle OOiO,
cos h —
_ Ri+BO Hi+EcosI- k sin /
B+Mt B-^Bi
. (129)
Then a = JfOiiV =5-7, Oi if being parallel to OA.
Second CASE,^Beceding from the P.I.
In Fig. 63 we have BC = ki , angle /, i?, and Bi given, to find
angles a and b.
Produce OA to meet OiL drawn parallel to CA. AL equals
OiM= OiHco8l.
OxH = Bt ^ EB= Bx - kitanl.
.'. AL — OxM=i (Bi — ki tan /) cos 7.
Hence
OL = B + (Bx - kx tan 7) cos 7 = if + Bx cos 7 - A;i sin 7
From triangle OOiL,
cos a =
OL* /? + 7?, cos 7- k, sin 7
00, ~ i^ + Bi
Evidently, b = a + L
. . . (180)
136. To Change the P.B.C. so that Second Branch of
Curve shall Bnd in a Tangent Parallel to Temunal Tangent
and Distant p therefrom.
In Fig. 64 let MAB be the located curve, EN = p. We must
Fio. 64.
determine the angle CO A, after which the desired curve ACS
mixy be located.
i>raw HOx' and 70, pnnillol to EFixud NO.
UL = 0,K = p.
LOCATION.
109
)m triangles OOi'H&nd OOiL,
{B + Bi) cos b = {R-j- Bi) cos a
P
-p.
.*. cos 6 = cosa —
B+B»
(181)
gle AOC= h ^ a.
7. To Find the Radius of a Curved Track.
asure any chord AB = 21, and mid-ordinate CE = M,
Fio. 65.
I in the right tmngle OAE (Fig. 65),
i?>- (B - Jf )» = i«.
. B =
2M
(182)
CHAPTER IV
TRANSITION-CXmVSS.
Article 11.— Theory of the TRANSinoN-cuRyB.
138. Elevation of Outer Rail on Onrvee.— To counteract the
effect of centrifugal force on curves the outer rail must \k
elevated above the inner one. It is shown in mechanics that th«
centrifugal force is
F=:
82.16jR'
where W is the weight, v the velocity in feet per second, 83.1ft
on average value of the acceleration of gravity in feet per second
per second, and R the radius in feet.
In Fig 66 let the vertical EL represent W, the horizontal KH
the centrifugal force, AB 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
ACt^
. 088)
=
d2.16£*
The guuge AB should be greater on curves than on tangents
to allow for jQange cleamuce and the effect of a rigid wheel-base.
AC = 4.9 feet is about the right value for the horizontal distance
between centers of mil- heads for staudanl gauge. In formula
(138) « is in feet per second, but the tniin velocity is usually given
in miles per hour. Let V = velocity in miles per hour, then the
UO
TRAK81TI0N-CURVES. Ill
22
^^locity in feet per second will be v = —V. Inserting these
^^lues in (133) gives
' = 82.16 X225i?=3i^'°"*^^y- ' * ' <^^>
This elevation will be required from the P.O. to the P. 21, but
obviously it cannot be introduced suddenly, so that for easy
■f iding the rate of increase of e should be uniform. From (134) it
is seen that e varies inversely with /?, which requires that when
« = 0, R = 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 Kankine 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 Rail-
road Spiral^ developed by W. H. Searles in 1882, and the cubic
parabola, described by C. D. Jameson and E. W. CrelJin in the
Railroad and Engineering Journal^ 1889.
In 1880 Ellis 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 Fig. 67. The curve CBB'G* is the circular curve
offset at (7 and C from the tangents by the amounts CHsiJid C'H',
AQB and BQ'A! are the transition-curves. A is theP.^T.C.,
or point of transition-curve, G the P. (7., B the P,G.u B' the
P.TG.u G' the P.r., and A' the P.T.i. The co-ordinates of O
are AH = aj', HO = y'; of (7, ar' and EG = F; of B, AM = Xx
and MB = yi. The length of curve from P. T, G. to any point P
(s I, and the whole length from P.T.C. to P.d \al\.
]12 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
141. Bquation of TraiiBition-ciirve. — Since the rate o^
change of e must be uniform, (134) may be written
79
^ = ^ = 8?'
(^35)
Fig. 67.
in which k is the rate of rise of outer rail along curve, and p tbe
varying radius of curvature. From the calculus pd<f> = dl,
■whence
(186)
Insert this in (135) and solve for d<p.
dip = yjdl = 2mldl. (137)
2m is dependent upon V and k, and is constant for any one
curve.
Integrating (137),
<p = ml\ (188)
the constant of integration being zero, for I is zero when <p is
zero.
TRANSlTION-(;URVEa 113
From the elementary triungle drawn at the point P of transit
t.joQ.curve» Fig. 67, dl being tangent.
Expanding sin by trigonometry*
in which 8! = 8 X 2 X 1, 6' == 6 X 4 X 8 X 2 X 1> etc
Substituting for <t> its value from (188),
dy = dl\mV - -g- + "120" "• 5040 + ' • 7
^tegrating,
tot ml* s {pt where is in circular measure. To obtain (f> in
itegrecs, :^ ^^'tq^ = i^- Inserting this in (139),
^ " V171.89 79 X tO» '*■ 8151 X 10* 153245 X 10" "T" ' • -y'
or y = i(7, ....*. . (140)
in which G may be found from YaWe 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
Expanding by trigonometry.
114 A FIELD-MANUAL FOR RAILROAD ENGIKEEB8.
Substitaliug ml* for <p and iutegrotiog,
__/, tnU* m*l* mH'* \ .....
*- *^^" "lO"^ 216 "■ 9860 "^•* 7 ^ ^
Replacing inl*hy <p reduced to degrees,
— ( ^" 0** 0°« \
*~ Y " 3^"^ 2328 X 10» " 33114X 10""^ • * 7
or x = l-lB, (142)
^yaries with ip'*, and may be taken from Table XIY with (p'
as argument.
142. The Transition-cimre Angle /i is the value assumes
at the P. a 1. From (138),
/. = mil* (143)
From (137) and (136),
cU__ J
dip ~ 2ml ~ ^'
5780
At the P.C.i p = B and may be taken equal to -=r^, so that
1 _ j._ 5780
whence
2lxR~ 11460^,
This value of m in (143) gives
^ ~ 07. » ~ 11 AM\r. (^'*'
^'-2iJ- 11460 (*^
Reducing this to circular measure by writing -^i=/i°qQjv= — ^—x
loO 57.80
gives
V = 28.65^ =f^i 0«)
143. The Oodrdinates of any point on the curve are given by
(140) and (142). The length of the transition-curve being known
TaAN8ITI0K-C U R VES. 115
cr assumed, yi and Xi (the coordinates of the P,G.i) may be
iound from these equations by the help of Table XIV; the
<X)Ordinates of the P. C. (see Fig. 67) will be
F^ yi — 22(1 — cos /,) = yi — Jf vers Ji, . . (147)
oj' = 0?, - i? sin /i (148)
144. Deflection-angles.— With the transit at the P.T.G. (or
P.T.I in backing up) the tangent of deilection-augles may be
found from the relation tan 5 = ?. Dividing (139) by (141),
X
ml*
tan 5 = ^ + .009523m»^« + .000167w»i" + (149)
o
From trigonometry the expansion of the angle in terms of its
tangent Is
« = tan 5 - t tan» S + 1 tan» ^ — etc. ... (a)
In (149) write ml^ = and substitute in (a) :
5 = -f - .002823</>» - .0000680* (150)
o
From (188) and (148),
I
ia which -- = n. From (6), (p = /jn^ and this in (150) gives
d = ^*n« - .002823/i%« - . 0000687, »n". . . . (c)
Both S and 7i are in circular measure ; to reduce to degrees
multiply by j^x. This gives, neglecting terms involving higher
powers of It than the third,
d** = 4^ n*- .00000086 7x»n» (151)
d
The second term is quite small, and in most cases may be en-
tirely neglected in practice.
116 A FIELD-MANUAL FOR RAILROAD ENQINEER8.
With the instrumeut at auy intermediate poiut oj'V the deflec-
tiou -angle for any poiut xy, measured from initial tangent, will be
tan 8 = ^- V^, = \(ml'' + fnl"^ + rnlV) +Ti«(^'^ + »w'^"*)
X — xf
-\- ^{mHH" + m»W"»)+ T*ir(^^^'^"* + mm"*-^mHH"^)-\- .... (152)
in which powers of mP higher than the third have been neglected.
Substitute the value of tan 8 from (158) in (a), write m^ = =
/,w'*, ml"^ = (J>" = iiw"*, by (h), and reduce circular measure to
degrees, giving
8^ = -^{n* + n"* + nn") - a small correction. (153]
o
For instrument at P. T, C, , 7*" = ; then (153) yields
(8^) = -^w' — correction,
o
or
(V) = ^\-^o (154J
(154) is the same as (151), as it should be.
For the transit at the quarter-point of transition-curre
n" = ^' = i^* = ^; then (153) yields
i\ VI 4
(<^i") = ^(^^ + A + i^) - correction.
or
(^j-) = -^"^j - 5j (\m
For transit at mid-point of transition-curve n" = J, and, from
(153).
(V) = '^C^' + i + i^) - correction.
or
a^c^ = ^'ii* - ^j aw)
TRANSITION-CURVES.
117
For trausit at three-quarter point vl* s= { and
oofiecllon»
(V) = T^«' + * + }»)
or
For transit at P.<7.t ii' ss 1 and
/.'
(157)
or
(«.') = -^(n* + 1 + «)
/.
— oorrectioiDt
(5i^) = ^^. -^..
(158)
With the transit at the P.T.Ci it will frequently be most con*
?euient to mctisurc the dcflectious f rem the tangent to the circular
curve at that point. Sometimes this will also be the case for the
tfausit at the P. C. , .
By reference to Fig. 68 it will be seen that for the transit at B
A
P.T.C. J JT^
the deflection from the tnngcnt BG which serves to fix any point
on the curve, as .6, is given by the equation
CfT, in general,
(V) =? y ^o + Bi , * . \^5S3iv
118 A FIELD-MA19UAL FOR RAILROAD ENGINEERS.
Table XV gives the values of A and B for the five positions of
iDStrument for which equalions (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 theinstrU'
ment, or for other points on the curve, they may be computed
from equation (158).
145. Tables. — Three tables are given for use with transition-
curves.
Table XIV was computed for use with formulas (140) and (142)
in determining G and E ; being assumed and 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
sufllce in most cases, though when 7i° 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 /i* by (146), yx by (139),
aji by (141), F by (147), and a;' by (!48). yx and jr, 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. 21(7.
to the point on offset at P. C, over ^ is tabulated as 0; Z' is
found by trial such that when inserted in (141) or (142) the same
value of 7! will be obtained as in (148). This may be done by as-
suming V a little less than ^ , then computing x'. More than two
trials will rarely be needed to find a sufficiently close value of V\
then e = V - x\ y' is found by (139) after findhig l\ or 0'
may be found from (6) of 144, and used in (140) in connection
with Table XIV. U - V is the length from O (Pig. 67) to the
P.G.i\ the difference in length between this and the length of
circular curve from P.O. to P. d is tabulated as ef ; that is, tf' =
{Ix — V) — arc. Then e-\- e' = li — (jj* -f circular arc).
For values of Ix intermediate between those given in the table
linear interpolation will suffice, though second differences may
he used for i^and yi if preferred.
TRANSITION -CURVES.
119
146. To Unite the Two Branches of a Compound Curve by
n Transition-curve.
The same objections bold to compound curves as to simple
ounres uniting with a tangent ; i.e., where there is a sudden
change of curvature there should be a sudden change of super-
elevation of outer rail, which of course is not allowable. Instead
of compounding the curves, we may offset them at the P. (7. (7.
and unite them by means of a portion of a transition-curve tangent
to each of the simple curves.
In Fig. 69 AB and CELMsLre the simple curves that are to be
united by the transition-curve ANE, Extend the transition-curve
to G, where its radius of curvature becomes infinite, and let G8
be its tangent. Call the length of transition- curve from G^ to ^
^1 , from Q to E h, and from E to A h. E and A are points
of tangency of simple and transition curves. Then /, = ^, — It.
The coordinates of A arc G8= Xi, 8 A =y, ; and of V{WV
perpendicular to (?/S), GW=Xi\ WV = F^; of E, GP=Xz,
EP=yt\ oi L {LH perpendicular to G8), GH = a?,', 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 ^ : ^ = i>» ; i)i , whence
<t^\
120 A FIELD-MANUAL FOR RAILROAD ENQINEEB8.
Then l,z l^^h = l^U - ^\ = li^^i^. . . (1«1)
By (188) or (148), ^ " (fT j '
Equating the value of 7s from this equation to that resulting
from (146) glyes
^•-M^y=^* (».<«)
WV = Fx and HL = F^ may be taken from Table XVI with
U and It as arguments. Then OiW—Rx+Fx OzH— J?, +i^«.
Draw Ox r parallel to Q8, then OiT = TTif; henoe
OtT = (i?, + J^O - (A + ^iX
and OiT^Xi'^Xt'.
Therefore
^^=' (ig, + W)-(ig.+P,) ' • • • <^^>
0»0, = (a?,' - xt') cosec a = Vo^ + TO?. . (ie4)
Then (75 = CO, - P0„
or i^'a = -Ri - (i?i + ftO,> (16(9
The lengths of ^-B and CE are
^^^ ^'""^''' lOO, (leO)
oi7= ''° ^^^'' 100 aeT)
The excess of transition-curve length over .45+ CBis
e,^h- ( ^'^^ "" + "" ^/'" jlOO. , , . (168)
TRANSITION-CURVES.
121
If AB and CE are quite sharp, we must take account of the
arc excess, so that we have then
4j, = /, - n iL_JL + ^5__^j 100 + arc excess 1.
(168')
The 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
mn&y replace the values within the brackets in (168').
147. Iiength of Transition- curve to be Taken. — In practice
the rate of change of superelevation of outer rail may vary from
j^QAQ *^ 400- Call the rate k ; then evidently kit must equal the
superelevation of outer rail for circular curve ; or, by (185),
78
kit =
dR'
5730
Writing R = ^Tr* *^^ solving for ^i
D
li =
VD
For k^
1200'
For * =
1
For * =
1
400*
17190A;
^1 =0.07F«2>.
(169)
(169')
li = 0.035 F«2>. (169")
i, =0.023 F«2). (169'")
The following table gives values of U in feet per degree of
circular curve for a few values of Fand k.
h
30 Miles
per Hour.
35 Miles
per Hour
40 Miles
per Hour.
45 Miles
per Hour.
50 Miles
per Hour.
55 Miles
per Hour
1
ISOO
68
86
112
142
176
212
1
eoo
82
48
56
73
87
106
1
21
29
87
47
58
\
70
\
122 A FIELD-KAKUAL FOB &AILHOAD EKOIKEER8.
When only a short tangent interrenes between two curves
shorter transition-carYes must be taken, requiring laiger values
of kf so that OTerlapping may be prerented.
For illustration suppose a 5° curre to be eased oft with a tran-
sition -curve, the highest train-speed being 45 miles per hoar and
k = —- . By the table the value of h will be 71 X 5 = 355 feet,
so that we should probably take a 360-ft. tranntion-curvc, re-
quiring an offset of 4.7 feet by Table XYL
AbTICLE 12. — FlELD-WOBK.
A Fkfd FomutloB,
148. For the cases most frequently presenting themselves iii
practice the foregoing formulas may be simplified so as to admii
of the rapid location of points on the transition-cunre vrith all tbo
accuracy needed on location, though it is best to use the exact
…[truncated]