Document text
Google
This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project
to make the world's books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.
Marks, notations and other maiginalia present in the original volume will appear in this file - a reminder of this book's long journey from the
publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing tliis resource, we liave taken steps to
prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for
personal, non-commercial purposes.
+ Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.
+ Maintain attributionTht GoogXt "watermark" you see on each file is essential for in forming people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liabili^ can be quite severe.
About Google Book Search
Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web
at|http: //books .google .com/I
r
TTf-
ENQINEEMHQ
9- as'
^y/s'
r
A FIELD MANUAL
FOR
RAILROAD ENGINEERS
A FIELD MANUAL
FOR
RAILROAD ENGINEERS
y^ BY
V,
JrC. NAGLE, M.A., C.E., M.C.E.
Member American Society of Civil Engineers; Formerly Chairman of the
Board of Water Engineers for the State of Texas
THIRD EDITION, THOROUGHLY REVISED
TOTAL ISSUE, SEVENTEEN THOUSAND
NEW YORK
JOHN WILEY & SONS, Inc.
London: CHAPMAN & HALL, Limited
Copyright 1897, 1917,
BY
J. C. NAGLE
PRE8» OF
BIMUNWORTH A CO.
' ,' BOOK MANUFACTURERS
' '. 12/24 BROOKLYN, N. Y.
^- PREFACE TO THE THIRD EDITION.
(S
I
^0 Nearly twenty years ago the first edition of this manual
was issued. Except for minor corrections, and the addition
of Tables XXVIII and XXIX, in the second edition, less than
two years later, no changes have since been made. For several
years the author has had a general revision in contemplation,
but other and more pressing calls have prevented this. As
finally made, the revision covers more than was intended when
it was begun, but falls short of what it should be, and has been
done in a fragmentary manner. It is hoped, however, that
the usefulness of the book has been increased.
Considerable new matter has been added to Chapter V.
^\ In order to conform to present practice, formulas have been
^ derived for the conditions of straight frogs and switch points.
Some engineers prefer the more approximate treatment, in which
r^ both switch-rails and frogs are assumed to be curved, and
^ certainly the formulas are simpler and easier of application.
•: For those who preffer it, the old method has been retained
in large part. One may take his choice of methods.
A brief treatment of the mass-curve in connection with
. overhaul has been inserted, and the subject of grades has been
'- amplified. Scattered through the text numerous other changes,
^ additions, and omissions have been made.
A number of new tables have been incorporated in the text,
and to the tables in the back of the book have been added
Table la. Radii of Metric Curves; Table IXa, Correction
Factors for Use with Table IX; and Table XXX, The True
Meridian and the Magnetic Declination. Table XIX has been
extended to show volumes per 100-foot lengths of Prismoids
for heights varying by tenths of a foot, up to fifteen feet. Soon
after the first edition appeared, considerably more than this
was suggested by Mr. R. A. Thompson, now a member of the
m
IV PBEFACE.
Engineering Board of the Interstate Commerce Commission,
but the author side-stepped then — ^and before the new table was
half finished wished that he had continued to do so.
Many helpful suggestions have been derived from the follow-
ing works: "Modem Location of Standard Turnouts," by
Kurtz; "Railroad Surveying," by Pickles and Wiley; "Rail-
road Curves and Earth-work," by Allen; "Field Engineering,"
by Searles and Ives; "Track Fonbuke and Tables," by Roberts;
"Railroad Construction," by Webb; "American Railway Engi-
neering Association Manual," Edition of 1911. Acknowledg-
ment for suggestions is here made.
Acknowledgments are also due many engineers who have
suggested additions from time to time, many of which have
been incorporated. Had all been included this would no longer
be a handbook.
Special acknowledgments are due Mr. C. M. Kurtz for many
suggestions in addition to those derived from his excellent
handbook. Table IXa has been inserted at his instance, and
a part of it is his work. Table XXX is also due to his sug-
gestion. The section on land-Une ties is his, and hkewise the
problem shown in Fig. 131. The treatment in connection
therewith was suggested by him. Mr. A. C. Love, Professor
of Railway Engineering in the Agricultural and Mechanical
College of Texas, also made many useful suggestions, par-
ticularly in connection with transition-curves and the mass-
curve. Formula (172)i was suppUed by him.
J. C. Nagle.
Austin, Texas,
February, 1917.
PREFACE TO THE FIRST EDITION,
Ea9E of reference and unifQrmitj 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 sectire a systematic
arrangement of tbe 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-explanator)r, so that
the explanations necessary in connection with the problems have
been reduced to a minimum. Algebraic equations stand each in
a distinct line, thus rendering them more easily read.
A knowledge of the elements of geometry and trigonometry has
f>een assumed, and only in the derivation of a few formulas in
connection with the theory of transition-curves will any higher
mathematics be needed. But these formulas may be accepted by
the reader who is unfamiliar with the calculus without in any
way affecting his ability to understand their applications or to
follow subsequent reasoning.
One can most readily turn to what he wants in a book after hav-
ing become familiar with its contents in the classroom. Keeping
this in mind this book has been written so that it may be used as
a text as well as for reference in the field. Wherever practicable
solutions to problems have been given in a rigid, general form,
followed by illustrative examples, so that the student need not
lose sight of the principle involved while following the solution
for a particular case. Wherever approximate solutions seemed
preferable they have also been given and their limritations 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 carve with reference to short chords foi
V
vi PBEFACE.
sharp carves — and, with tables of Radii, Long Chords, Mid«
ordinates, etc., based on appropriate equations — ^the errors result-
ing from assuming the radius to vary inversely with the degree
of curve will generally be found to be quite small.
Chapter I gives briefly the general method of making Re-
connoissance; Chapter II treats of Preliminary Surveys; while
Chapter III relates to Location.
Chapter IV, on Transition-curves, follows the method adopted
by Professor Crandall, and enables one to locate the transition-
curve with rigid accuracy where such is necessary. Approximate
methods are also given by means of which the curve may be as
easily located as any of the more limited easement curves ordi-
narily met with.
Chapter V, on Frogs and Switches, contains all that is necessary
for their location. The formulas have been arranged to give the
desired quantities in terms of the frog number whenever the re-
sulting equations would be easier of application than the trigono-
metric ones usually given. The turnout tables are unusually full
tod 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 oh 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.
PEEFAOE. Vil
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 ^ith
the definition of the degree of curve requiring curves sharper
than 7** to be ran 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 Carbart's
Field Book for Civil Engineers and were furnished by Ginn & Co.
Electrotypes of Tables II, X, XII, XIII, XIX, XX, XXIV. XXV,
XXVI, and also XVI — this last being from Crandall's book,
2%e Transition Curve — were furnished by John Wiley & Sons.
Of the otbers, 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
over 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 readin|f
proof.
J. C. NAaiiB.
CoLLEOB Station, Tszas, May, 1807.
CONTENTS.
CHAPTER I.
RECONNOISSANCE.
Abticlb 1. Objects of Rxconnoibbanch — How Madi.
SaCnOR PAGB
1. RelatiTe Importance of the Work of Reoonnoiaaanoe and Location. 1
2. Object of Reconnoissance 2
3. The Instruments 2
4. Use of Maps 4
6. Making the Reoonnoissanoe 4
CHAPTER II.
PRELIMINARY SURVEYS.
Abticuq 2. Objects; Tbb Fxbld Cobps; Dutiks or tbjb Chodf.
6. Objects of Preliminary Surveys 8
7. The Ezploration-Iine 8
8. Data Sought in Making Preliminary Surveys. 0
9. The Field Corps 9
10. The Chief of Party, Duties of 9
ABncLB 8. Ths Tbanbit Pabtt.
A. DUnXS OF THX MBMBBBS.
11. Composition of the Transit Party 10
12. The Transitman 10
13-17. Other Members of the Party 10
18. Instruments 11
B. TBANaXT ADjrSTUXNTB — THB YUUOBB.
19. Kind of Transit 11
20. To Adjust the Plate Levels. 12
21. ParaUax 12
22. To Adjust the Line of Collimation 12
23. To Adjust the Standards 13
24. To Adjust the Level on Telescope 14
25. Direct and Retrograde Vemiers 16
26. The Least Count of a Vernier 15
27. To Read a Verbier 16
ix
CONTENTS.
C. ACCB880BU8.
(!•) The Oradienter,
SECTION rAGB
28. DeBcription and Method of Using Gradienter 16
(2*X Th$ Stadia, or Ttlemder,
29. Principle of the Stadia 17
30. Formula for Line of Sight Horisontal 17
31. Formulas for Line of Sight Inclined 18
32. The Instrumental Constant, To Find 19
33. Reducing the Notes 19
D. FIBLD-WOBK
34. Station Numbers 20
36^ Hubs or Plugs 20
36. Reference^points. 20
37. Alignment , 20
38. Form of Transit Notes ^1
39. Stadia Methods for Preliminary Surveys , V, 21
X. GBBTACLSa IN TANGENT.
41. To Pass an Obstacle by Means of Parallel Lines. i . 22
42. To Pass an Obstacle by Angular Deflections 22
43. To Measure aoross a River 23
Abticle 4. Thb Lbvel Pabtt.
44. M-ake-up and Instruments •....••. 25
45. Woric of the Leveler , 20
46. Work of the Rodman » 20
ADJVBTMBNT8 OF IBB LBYKi.
47. To Adjust the Line of Collimation 25
48. To Adjust the Level-bubble 26
49: To Adjust the Wyes. ......... .> * 27
B. THBOBT or USVBtma.
50. True and Apparent Level 27
51. The Error Due to Curvature .' , 27
52. The Difference of Elevation of Two Points 28
O. nSLB WOBS.
53. The Datum. 20
54. Bench-marks 29
55. Work in the Field ,..\ '. 30
56. The Level Notes ', 30
57. Precautions when Using Level 31
58. The Rod. .....,.,,.,.......,,,,,. 31
CONTENTS* Zl
Asncxji 5. Tbb Topograpuo Pabtt.
BKCnON rAGB
59. Instrumeats Used; Area to be Mapped 32
60. Methods of Recording Data 32
61. Topographers' Field-sheets 33
62. Use ot the Slope-level 33
63. Cross-section Rods 34
64. The Transit and Stadia in Topographical Surveyinc 34
Abticlb 6. Pbbumikabt Ebtxmatbs.
66. Map of Preliminary Lines 34
66a. Land Line Ties 35
67. The Profile. . . ; 86
68. Preliminary Estimates of Qtiantitiee 37
69. Report of the Locating Engineer 87
CHAPTER III.
LOCATION.
Abtkcui 7. Pboibctimo Looatioii.
70. Problems Inyolved in the Paper Location 39
71. Hints Regarding Methods of Projecting the Line 33
72. The Curve-protractor 39
73. Work in the Field 40
Article 8. Simplb Curves.
A. DEFINITIONS Avh FORMULAS.
74. Definitions • * 41
75. To Find the Radius R, the Degree of Curve Being Known 43
75a. Degree of Metric Curve Defined 44
76. To Find the Length of Curve 45
77. The Functions of a One-degree Curve. 45
79. To Find D, B, and C Being Known 47
80. To Find the Tangent Distance T, I and R Being Known 47
81. To Find R, Given I And T 48
82. Given / and £>. to Find the Long Chord L.C 48
88. Ordinates from Chord 49
84-86. To Find the External B 52
87. To Find /?.£ and / Given 53
88. To Find T, £ and Z Given : 53
89. To Find the Deflection Offset from Chord Produced 53
do. To Find the Tangent Deflection Offset 54
91. The Sub-tangential Deflection Offset , 55
92. To Find the Tangent Offset t 56
93* Difference in Length of Arc and Long Chord 57
> ••
Xn CONTENTS.
B. LOCATING SIMPLE CUBVES.
SECTION PAOB
94. To Loonte a Curve with the Chain by Offsets from Chords Produced 69
95. To Locate a Curve by Offsets from Tangent 61
96. To Locate a Curve by Offsets from a Long Chord 62
97. To Locate a Curve with Transit and Chain 63
98. The Index-angle 64
09. Subdeflection-angles 64
100-101. Transit Notes 65
C. OBBTACLBS.
102. To Pass an Obstacle on a Curve 67
103. To Locate a curve when the P.C. is Inaccessible 68
104. To Pass to Tangent when the P.T. is Inaccessible 71
105-107. To Pass a Curve through a Given Point 78
108. To Locate a Tangent to a Curve from an Outside Point 76
109. To Run a Tangent to Two Curves of Contrary Flexure 76
D. CHANQB OF LOCATION.
110. To Locate a Ciu-ve Parallel to a Given Curve 77
111. To Change P.C. in Order to Make P.T. Fall in a Parallel Tangent. . 78
112. To Change R and P.C. to Make P.T. Fall in Parallel Tangent, on
Same Radial Line 70
113. To Find Change in P.C. or R for a Given Change in / 80
114. Required the Change in P.C. and R for a Given Change in J, the
P.T. Unchanged 81
115. To Find New Radius for a Given Change in 7 81
116. To Find New R to Connect P.C. with a ParaUel Tangent 82
Abticlb 9. Compound Curves.
117. Given Both Tangents and One Radius, to Find the Other Radius. 84
118. Given One Radius, the Long Chord and the Angles it Makes with
Tangents, to Find the Other Radius and Central Angles 86
119. Given the Radii and Central Angles, to Find the Tangents, the Long
Chord, and the Angles it Makes with Tangents 86
120. Given the Long Chord and Angles Made with Tangents, to Find
Both Radii when Common Tangent is Parallel to Long Chord. . 87
B. OBSTACLBB
121. To Locate Second Branch when P.C, is Inaccessible 88
. C. . CHANGE. OF. location
122. To Compound a Simple Curve so P.T. shall Fall in a Parallel Tan-
gent" . .' 80
123. To Find Change in P.C.C. Necessary to Make P.T. Fall in a Par-
allel Tangent 90
124. To Change P.CiC. and Second Radius so P.T. shall Fall in a Par^
allel Tangent, on Same Radial Line 98
coMltsNTS. xiii
raenoM pagb
125. To Change P.C.C. and Second Radius to Cauae P.T. to Fall at a
New Point in Same Tangent 95
126. To Substitute a Three-oentered Compound Curve for a Simple Onje 98
127. To Substitute a Curve for a Tangent Uniting Two Curves 99
Abticle 10. Tback Fbobljbms.
128. ReverMd Curves, Where to Use 100
129. To Connect a Located Curve with an Intersecting Tangent 101
130. To Locate a Y 104
131. A Reversed Curve between Parallel Tangents 106
132. A Crossover between Parallel Tracks when a Fixed Length of Tan-
gent is Inserted 109
133. A Reversed Curve with Unequal Angles 110
134. A Reversed Curve between Fixed Points 110
135. To Connect Two Divergent Tangents by a Reversed Curve Ill
136. To Change P.R.C. so P.T. shall Fall in a Parallel Tangent 112
137. To Find the Radius of a Curved Track 133
CHAPTER IV.
TRANSITION-CURVES.
Abtxoud 11. Theory of the Transition-cubyb.
138. Elevation of Outer Rail on Curves 114
139. Requirements of the True Transition-curve 116
140. Notation Employed- / .. .. ^ , 115
141. Equation of Transition-curve .. « 116
142. Transition-curve Angle, 1 118
143. Co5rdinates of Points 118
144. Deflection-angles 119
145. Explanation of Transition-curve Tables 122
146. To Unite the Branches of a Compound Curve by a Transition-
curve 123
147. Length of Transttion-o\irve to be Taken 125
Article 12. Field-work,
a. field fobmulas.
148. When to Use the Simplified Formulas 128
149. Simplified Formulas for Transition-curves 128
160. Offsets 130
151. Compound Curves 131
B. BETTENO OUT TRANSITION-CXTBYEB.
153. Location by Offsets 132
154. Location by Defleotion-angles 133
155. Form of Transit Notes for Transition-curves 135
XIV GQNTSNTB.
Abtxclb 13. Transition Cubyb Problems,
section paos
156. Tangent Distances and External for Equal Offsets 137
157- Tangent Distances, Offsets Unequal 138
158. T^ansition-c^ryes Inserted without Changing the Vertex of
C^cular Curve 139
159. Transition-Kjurves Inserted with Least Deviation from Old Track. 141
160. Transition-curves Inserted at Ends of Long Circular Curve» Cen-
tral Portion Undisturbed 141
161. Transition-curve Inserted at P.C.C, by Changing Radius of Second
Branch , 144
162. To Insert Transition-curves at the Ends of Two Circular Curves
United by a Common Tangent 146
163. To Unite a Tangent and Circular Curve when the Offset Cannot
be Directly Measured ,.»... .,...,. 147
164. Inserting Transition-curves in Old Track . ,.,,.. ^ 148
165. Remarks on Tabular Interpolations • 148
CHAPTER V.
TURNOUTS, FROGS AND SWITCHES.
Article 14. Turnouts.
A. turnouts rBOM btraioht lines.
166. Definitions, , 151
167. To Find the Radius and Lead for a Turnout from Straight Track
for Straight Frogs and Switch-rails. 155
16S. To Find the Coordinates of Center and Quarter-points of Lead
Rail 156
17a Practical Leads. 158
171. To Find Increase of Tangent at Heel of Switch-rail, and New
Radius, for a given Increase in Lead 158
172. To Find Increase of Tangent at Toe of Frog, and New Radius for
a Given Increase in Lead 159
173. Tables of Theoretical and Practical Leads for Straight Switch-
rails and Frogs 160
174. Ordinates from Long Chord to Outer Rail of Turnout 163
176. The Frog Tangent, * 163
176. Field Work for Turnout from Straight Track 164
176|. To Locate Curve beyond Frog Tangent 165
177. Three-throw and Tandem Switches 165
178. To Find Angle of Crotch-frog for Tandem or Three-throw Switch. 166
179. To Unite Straight Main Line with Parallel Siding. Using a Given
Length of Tangent in Turnout 167
180. Frog Tangent continued to P.O. of Side Track Curve 168
181. Ladder Tracks 169
turnouts raoM ourvbsJ
182. Degree of Turuout from Curved Main Track 171
188. Turnout from Inside of Curved Maia Traek to Unite with
^. Intors^ctiug Straight Trsek. 173
CONTSNTS. XV
BSCnOH TAQU
184. Turnout from Outedde of Curved Main Track to Unite with
Intersecting Straight Track « 175
Article 15. Turnout Formulas vor Curved Switch-rails and Frogs.
a. turnouts from straight lines.
186. To Find the Lead, 2| and Radius, /2, in Terms of th^ Frog Numberi
N, and Gauge, g 178
187. To Find Theoretical Length of Switch-rail 179
188. To Find Lead and Number of Crotch-frog for a Double Turnout
to Opposite Sides of Main Track 180
189. To Find Turnout Radius and Lead of Crotch-frog in Terms of
Crotch-frog Number 181
190. To Find Radius of Curve from Point of Middle Frog to Point
of Main Frog, Given Nx, iV, and N' 181
191. Double Turnout to Same Side of Main Track 183
192. To Find Radius of Curve between Frog-points for a Double
Turnout to Same Side of Main Track 184
193. To Unite Main Track with Siding, Reversing Point Opposite Frog. 185
R. TURNOUTS FROM CURVES.
194. To Find Lead and Radius for Turnout to Concave Side of Main
Line 186
195. To Find Lead and Radius, Turnout to Convex Side 189
196. To Unite Main Track with a Concentric Siding 191
C. THE STUD LEAD
197. Definitions , 193
198. Given N, i, and g, to Find the Stub Lead 193
199. Cuning Rails 194
200. Turnout Table and Explanation 195
Article 16. Crossovers. Intersecting Tracks. Wtbs.
201. Crossover between Parallel Straight Tracks, with Tangent between
Frog-points. Frog Numbers Equal 100
202. Crossover between Parallel Straight Tracks. Frog Numbers
Unequal 197
203. Crossover between Concentric Curved Tracks. 198
204. To Connect two Intersecting Straight Tracks which Make a Small
T Intersection Angle with Each Other 198
205. A Simple Curve Uniting Two Intersecting Straight Tracks when
the Intersection Angle is Large 200
206. To Unite a Straight Track with an Intersecting Curved Track . . . 201
207. To Unite Two Intersecting Curved Tracks 203
908. To Locate a Y when One Leg is a Straight Track, the Radii
and Frog Numbers Given. ; 205
XVI CONTENTS.
SBcnoN pAoa
209. To Locate a Y when the Three Branches are Curved and Conrez
Toward Each Other 206
210. To Locate a Y when One Branch is Concave Toward the Other Two 208
Article 17. Cbobsing-fboos a.nd Cbobsino-slips
▲. CBOBBINO-FROaS
212. Length of Rail Intercepted between Two Intersecting Straight
Tracks 209
213. Angles of a Set of Crossing Frogs,. One Track Curved 209
214. Angles of a Set of Crossing-frogs, Both Tracks Curved 210
B. CROBSING-BLIPB.
216. Length and Radii of Slip-rails, Both Tracks Straight 211
217. Length and Radii of Slip-rails, One Track Curved 211
218. Length and Radii of Slip-rails, Both Tracks Curved 212
219. Diagonals for Crossing-frog^ets 214
CHAPTER VI.
CONSTRUCTION.
Article 18. Definitions; General Conbidebationb; VebticaiJ
Curves; Elevation of Outer Rail.
221. The Division Engineer and Resident Engineer 215
222-225. Definitions 216
226. To Find the Grade-point, Longitudinal Slope Uniform 217
227. Vertical Curves 217
228. Elevation of Outer Rail on Curves 221
Article 19. Velocity Heads, Grades, Distance, and
Curvature.
229. Velocity Head of Trains 222
230. Virtual Grades 224
231. Grades, Distance and Curvature — Effect of 226
232. Ruling Grades 225
233. Minor Grades 226
234. Helper Grades, or Pusher Grades 226
235. Distance 226
236. Curvature 227
237. Curve Compensation 227
{Article 20. Earthwork.
A. sbttinq blopb-btakbb.
238. The Distance Out for Level Sections 228
239. To Find Position of Slope-stakes for Surface Inclined 229
CONTENTS, XVU
BBCnOK rAQB
240. Croes-eeotion Notes 231
241. Irregular Sections 232
242. Staking Out Openings 232
243. Manner of Marking Stakes 232
244. Shrinkage — Growth 232
24& Borrow-pits, Drainage of , etc 233
B. ARBAB OF SECTIONS.
247. Area of Three-level Section. ,i 233
248. Area of Five-level Section : 234
240. General Formula for Areas 235
250. Ezi>lana'ion of Table of Areas of Level Sections and the Three*
level Correction 236
a VOLUMl} OF BABTHWOBX.
251. Where Cross-sections Should be Taken 237
252. Volume by Averaging End Areas 237
253. The Prismoidal Formula 238
254. Form of Record 240
265. The Prismoidal Correction 240
256. Computation of Volumes when Passing from Cut to Fill 243
257. Use of Tables of Volumes in Making Preliminary Estimates 244
258. Side Ditches 244
259. Earthwork on Curves 244
Abtioub 21. Haul and Ovbbhaul. Thb Mass Cubvb.
260. Haul Defined 246
261. Overhaul Defined" 246
262. Amount of Overhaul 246
263. Cross Haul 248
264. The Mass Curve 248
265. The Swell of Earthwork. Shrinkage Allowance 252
266. Application of Mass Curve when Swell is Considered 252
267. Limit of Economic Haul 253
268. Application of Mass Diagram to a Continuous Profile 253
200. Properties of the Mass Diagram 256
270. Another Method of Constructing the Mass Curve 256
AsncLB 22. Clbabino and Gbubbino, Drainage Areas and
Run-off Formulas; Culverts and Bridoes; Tunnels.
271. Clearing and Grubbing 257
272. Drainage Areas 268
273. Formulas for Area of Waterway 250
274. To Stake Out Culvert Limits. , 261
275. To Stake Out Openings for Trestle Bridges 263
276. Pile Bents for Trestle Bridges 264
277. Cut-offs for Piles 264
xvin COm'ENTS.
nonoN pAoi^
27S. Staking out Foundations 265
279. Bridge Piers and Abutments. Bridges on Curves 266
280. Relation of Trestles to Embankments 267
281. Pilling in Trestles 267
282. Tunnels. 26»
283. Tunnel Cross-sections 269
284. Shafts 270
285. Ventilation of Tunnels 270
286. Tunnel Portals 27X
Abticle 23. Finishing Eabthwobk. Tback Latino.
287. Center Stakes 274^
288. Grade Stakes , 276
289. Sodding 276
290. Setting Track Centers and Orade Stakes 277
Abticlb 24. Monthly and Final Estimates.
291. Monthly Estimates 278
292. Measurements of Earthwork for Monthly Estimates 27S
293. Caasaificaxion of Earthwork 278
294. Progress Profile 279
296. Masonry Measurements 279
296. Bridge Measurements ; ; 279
297. Track Material 280
298. Estimate Sheets 280
299. ReservatioD in Payments for Monthly Estimates 280
300. Extras 281
301. Pinal Estimate 281
J02. Acceptance 281
TABLES.
Table Showing Length of Transition-curve to be Taken for Various
Values of 7 and * 125
Table of Minimum Lengths of Transition-curves 127
Tftble of Theoretical Switch-leads for Straight Frogs and Switch-rails. . 161
Table of Practical Switch-leads for Straight Frogs and Switch-rails 162
Table of Values of ^ - Vot for Stub Lead I94
Turnout Table for Curved Frogs and Switch-rails I95
Table of Distances between Frog-points of Crossovers 197
Table of Corrections for Vertical Curves 220
Table of Elevation of Outer Rails on Curves 221
Table of Velocity Heads for Various Train Velocities 223-
Table of Prismoidal Corrections for Level Sections 241
TaWe of Areas of Right of Way 100 Feet Wide 268
I. Radu of Curves 285
lo. Radii of Metric Curves. , 288
II. Minutes in Decimals of a Degree 289
CONTENTS. XIX
TABLB PAGB
III. Tangential Offsets , 290
lY. Long Chords and Actual Arcs 291
y. Mid-ordinates to Long Chords 292
VI. Logarithms of Numbers 294
VII. Logarithmic Sines and Cosines 312
VIII. Logarithmic Tangents and Cotangents 327
IX. Functions of a One-degree Curve 342
IXa. Correction Factors for Use with Table IX 372
X. Natural Sines and Cosines 373
XI. Natural Secants and Cosecants 382
XII. Natural Tangents and Cotangents 395
XIII. Natural Versines and Ezsecants 407
XIV. Coordinates for Transition-curves 430
XV. Deflection-angles for Transition-curves 431
XVI. Transition-curve Table 433
XVII. Areas of Level Sections 446
XVIII. Corrections for Three-level Ground 460
XIX. Cubic Yards per 100 ft. in Terms of Center Height 461
XX. Cubic Yards per 100 ft. in Terms of Sectional Area 465
XXI. Rise per Mile of Various Grades 460
XXII. Slopes for Topography 470
XXIII. Material Reqxiired for One Mile of Track 470
XXIV. Mutual Conversion of Feet and Inches into Meters and
Centimeters 471
XXV. Mutual Conversion of Miles and Kilometers 472
XXVI. Length of 1' Arc of Latitude and Longitude 472
XXVII. Trigonometric and Miscellaneous Formulas 473
XXVIII. Square Roots and Cube Roots of Numbers from .1 to 28. . . . 478
XXIX. Squares, Cubes, Square Roots, and Cube Roots, of Numbers
from 1 to 1000 479
XXX. The True Meridian and the Magnetic Declination 487
Index ••• 490
A FIELD-MANUAL FOR RAILROAD
ENGINEERS.
CHAPTER I.
BECONNOlSSANCfX,
Articlb 1. Objects or RECoimoiBSAKCB—- 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, but of maintenance and operation, and a
consideration of the immediate and future traffic likely to pass
over the completed road.
Tlie 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 7iow to secure the best line, answering a given
purpose, for Hie 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 only from long
practice and close observation, coupled with the ability to fully
2 A FIELD-HAKUAL FOB RAILROAD EKGINEERS.
grasp and weigh all the complex features of the question. A
passing refereuce 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 reconuoissance will be given before going on to the
fpcdal problems arising in the work of the railroad engineer.
2. The Reconnoissance is A rapid, general survey of the area
through which the proposed railroad must pass, made only with
such instruments as can be easily carried, and which should ena-
ble the engineer to restrict the more accurate instrumental work
that follows to one or two general lines. The time required for
this part of the work will in general be only a small fractiou 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-
estimateB its iniportaace^ but because he is not usually allowed
sufficient time in which to study thoroughly the area under con
sideratlon.
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 ail 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 IiiBtnimeiits needed will rarely be more than a pocket-
oompaas, 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.
(6ij Th« Hiu&d-l«vdl enables one to obtain dlfleronces vi ele-
raUoQ between points not far apsrk
(c) The Aneroid Barometer gives approximate heights of the
mercnrj column, and serves to rougblj 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 aneroidSp which have bemi previousl/
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 differenoe of elevation by the formula*
d = 60000(log^-logA)(l + ?y^Z^), . . (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, l^and t the temperatures of the two stations
in Fahrenheit degrees.
If the sum of the temperatures, T-^t,ia taken as lOS"*, formula
(1) reduces to
d = 63000(log^-logA). . . , . . (1')
ExAHFLB. — The reading of the barometer at the foot of a
mountain is 28.8 inches, and at the top 26.7 inches. Required
the height of the mountain.
By (1'). d = 63000 (log 38.8 - 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 tom^
peratnre.
(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 the number of revolutions multiplied
Uy the circumference of the wheel gives the space passed over.
*i«M»«i*««
* See Flyiuptoo*« Aneroid Barometer, p. 38, for formula (1).
4 A FIELD-HAHTTAL TOB RAILBOAD ENGINEERS.
4. The Map. — Before beginning the recotinoissance 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
unce 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 biaximum 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
frround.
5. Having tentatively decided on the limiting grades and cur-
vature to be employed, the engineer goes carefully over the
ground, examining the entire area that seems likely to afford
passage, in order to determine whether a suitable line may be
secured for the grades and curves previously assumed. With his
pocket-compass he takes the bearings of lines, and by means of
the hand-level and aneroid determines differences of elevation.
BBGOlJTNOISgAKOE. 5
IHstanoes are estimated by the eye* paced, and the count talKen
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
him, 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-
aibly because we have less practice in this line, and the mind
seems naturally to exaggerate the vertical as compared with the
horizontal ; practice, however, will enable us to make allowance
for the natural tendency to overestimate heights and slopes.
The ground should be gone over in both directions, for the ap-
pearance may be quite different when approached from different
quarters. Ruling points, such as a pass in the mountains, the
crossing of a large stream, or a town or city through which the
road must be built, serve to reduce the problem to a number of
special ones, each having its own solution.
In a mountainous region offering a limited number of possible
routes, but heavy construction work, it may often happen that
the location of a line is a much less 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 factora 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.
The recQnnolssance should disclose which of the three follow^
6 A FIELD-MANTAL FOB RAILROAD ENOlNEERS,
ing general classes the prelixninary survey will probably come
under:
(a) VaUey lines, as will be the case for a railroad between
two towns situated in the same valley, and on the same stream.
The located line will follow the meanderings of the stream more
or less closely and may sometimes be located entirely on one
side of the stream, or it may cross and recross the stream. The
hydraulic slope of the stream may not be uniform and the
located line may have to be carried on the slopes of the hiUs
some distance back from the stream in places. Valley locations
generally present fewer difficulties than either of the others
named below.
(6) Cro8S*c(mnirylinea have alwBy a one or more summits which
are controlling factors. The lowest available, practicable ones
should be selected unless these should involve such an increase
in length of line as to make a higher one preferable. If the
line follows a stream up one side of a range of hills it will most
probably be so located as to follow another down the opposite
side, for low points in a divide indicate that the som-ces of
streams on opposite sides of the range approach each other
most doeely at these points.
(c) Mountain lines usually require " development '* — that is an
increase in the length of line — in order to keep within the nil-
ing grade. Sometimes this is accomplished by following up a
lateral valley, on a continuous ascending grade if we are work-
ing toward^ the summit, and vice versa if we are receding from
the summit, and introducing a sharp curve, crossing or roimd-
ing the lateral valley, then working back on the opposite side,
still on the same character of gradient as was used on the first
side of the lateral valley.
id) Switch-back^. — Sometimes the natural ground slope is sc
steep that a " switch-back '' must be , used. This method per*
mits the located line to be carried up the face of a steep slope.
It involves slow and expensive operation of trains, however,
because, after climbing as far as the topography permits, the
train must be stopped, a switch thrown behind it, and the
train backed up-grade as far a^ this branch of the switch-back
extends, after which another switch is thrown in front of Hie
train and forward movement made along another branch of the
switdi-back. This procedure is followed for as many branches
BECONNOISSANCB. 7
as there are in the switch-back, each one requiring the train to
come to ar full stop.
(e) Loops. — Sometimes the located line must be made to
double back and cross itself at a higher or a lower level, accord-
ing as an ascent or a descent is being made, and not infrequently
timnels must be resorted to in making such loops.
CHAPTER IL
PBEUMJNART SURVEYS.
Article 2. Objects; The Field Corps ; Duties of ths Chief.
6. The 01](|ect8 of the preliminary surveys are to secure all the
data necessary to determine which one of the routes selected on
reconnoissance is the most feasible, all things considered, and the
approximate cost of construction. In rough country it will be
economical to make two, or even three, surveys over the route se-
lected for location before beginning to place the line in the position
it is finally to occupy. The first of these is often omitted, and is
called an *' exploration-line " ; it will frequently save the making
of the more expensive *' preliminary " over one or more of the
routes.
7. The Exploration-line may be made with either transit or
compass, and consists of a rapidly ran 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 the flagman. As the progress of the level party will be
slower than that of the compass party, it will be economical to add
an extra rodman, and sometimes a recorder. The compassman
may sketch in the features adjacent to the line while waiting for
his chainmen, who may be either in front of or behind the com-
pass.
The stadia method of surveying — ^to be spoken of later — would
seem to offer exceptional advantages for this work — only three or
four men being needed in addition to the chief. With it, by set-
ting the transit over alternate stations, very rapid progress may be
made, and obstacles avoided with as much or greater ease than
with the compass.
The exploration-line will more than pay for itself in showing
8
PBELIMIN'ART SURVBTS. 9
what roates 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) Thb Transit Party.
(6) The Level Party.
(e) The Topographic Party.
10. The Chief of Party receives his orders from the chief en-
gineer, or such other officer as may be in charge, directs the mo-
tions of the surveying corps, and is responsible for their conduct
and progress. He provides accommodations and supplies, pays all
expenses, taking receipts or vouchers for all outlays — in dupli-
cate when required. In the less thickly populated sections he
must provide tents, wagons, cook, and all necessary camping outfit
and supplies. He must direct the field operations in person, keep-
ing in advance of the transit, establish turning-points or hubs,
and direct the transitman in the proper course. He should keep
10 A FIELD-MANUAL FOB RAILROAD EKGIKEEBS.
areoord— or direot the transitman and topogtapber to'do b6-^
the character of earthwork likelj to be enconntered, the plad
where drains, culverts, bridges, cattle-guards, etc., are needed
the nature of material for embankment, piling, etc., adjacent i
the line ; the probable amount of clearing and grubbing, and a!
other features likely to affect the cost of construction. He sbouj
see that the names of property owners and residents along t1
line and the positions and bearings of property lines, wh<l
possible, are noted. *
He should have authority to discharge assistants — except trans J
man, leveler, and topographer — whose services are unsatisfactorj
and in many cases it will be best for him to have entire contra
engaging or discharging any member of the corps as circumstance
may require. .
Artiolb 8. The Transit Party. '
A. Duties of the Members.
11. The Tranait Party should consist of a transitman, hea
ohainman, rear chainman, rear flagman, stakeman, and as maii(
axemen as may be required — rarely less than two even for opei
country. '
12. The Transitman cares for his instrument, keeping it in ad
justment; directs the chainmen into line; notes the angle betwee]
successive tangents as read on plates; notes also the bearings o
tangents, of highways, streams, and property lines (on location)
with the plus at which the line crosses them. If there is n<
topographic party he must make sketches, on the right-hand pag<
of note- book, of the surface features adjacent to the line; th<
red line down the middle of page represents the transit line
whether straight, broken, or curved, to which the sketches aW
adjusted. He must see that the axemen keep in line, in ordei
that no unnecessary chopping may be done. Large trees nee^
rarely be felled on preliminary, even when a given general coursC|
has to be followed, for small angles may be turned to avoid them,,
the deflections to right being made to approximately balance thod0
to left. I
When the chief of party is absent the transitman is ranking
man, and will take temporary charge.
13. The Head Ohainman carries a range-pole or '* flag," and
draga the chain, which he mast see is straight and horizontal
PBELIMIlSrA.EY SUaVBTS. H
l^IioD setting a point for a stake. He directs the stakeman wbeM
^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 by setting his flag and going ahead, directing them
|irhere to cat by keeping them in line with the flag and transit.
^The speed of the party is depend^it on the rapidity and accuracy
^witb which he can set his flag in position, by ranging with stakes
^Iready set between him and transit, and in seeing that the
axemen make all their work count.
14. The Rear Ohainman must be careful to hold his end ai
^e 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
i^ee that the stakeman has marked it correctly; he must make a
aote of pluses for roads, fences, streams, etc., to be given to the
transitman later on.
16. The Stakeman must keep himself supplied with stakes
ftibout IV X 2" X 24", marking the number on them plainly, and
giving them as directed by the head chainman.
r If sawed stakes are not provided, he must cut the stakes and
face them for the numbers. He must keep on hand a number of
plugs or **hubs/* to be driven flush with the ground and having
Che point where flag rested marked with a tack. About ten oi
iwelve inches to the left of and facing the hub a guard stake is
driven, on which is marked the station number, and which enables
^ne to find the hub at any time.
Q 16. The Axemen do all necessary clearing and chopping in
^rder that the transit and level parties may have a clear sightway,
jimd yet restrict the work of clearing to a minimum. One of them
^ay be detailed to keep the stakeman supplied with stakes.
"^ 17. The Rear Flagman holds his flag on tlie last turning-
point for the transitman to use in back-sighting.
18. The Instmineuts used by the party are the transit (or
compass), one-hundred-foot chain or tape, range-poles, and the
necessary axes and hatchet for axemen and stakeman.
B. Transit Adjustments — The Vernier.
19. For railroad work the transit is usually plain, but it is
often convenient to have a clamp and tangent movement to tele-
12 A FIELD-HAKUAL FOB BAILROAD ENOIKBERS,
8cop6| a vertical circle, a level on telescope, stadia wires, and a
gradienter; the solar attachment will rarely he needed.
»
20. To Adjust the Plate IjeveLi.— The axis of the instrument
is set at right angles to the plates hj 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-vraj back by means of the capstan-headed screws at
the ends of the tube, Relevel and repeat until the bubble remains
at the centre after reversal. Do the same for the other bubble.
Both bubbles should remain at the centres of their tubes during a
complete reversal.
21. Parallax is an apparent movement of the cross- wires with
respect to the object sighted when the eye is moved from side to
side of the eyepiece, and shows that the image does not fall in the
plane of the cross-wires. In precise measurements it should be
removed before making an observation with the telescope. To do
this, first bring the cross-wires clearly into view when the object,
glass is turned towards the sky, then, when sighting an object,
note if there is any relative movement of cross- wires and image
when the eye is moved from side to side at the eyepiece ; if there
is, refocus the object-glass until this movement disappears.
22. To Adjust the Line of Oollimation is to make the line
joining the intersection of cross-wires and optical center of objec-
tive describe a pjane perpendicular to the horizontal axis of instru-
ment.
First Method. ^Level the instrument and clamp the move-
ments on vertical axis. Sight some well-defined object distant
about the length of an average sight, and in the same horizontal
plane as telescope. Reverse the telescope on its horizontal axis,
and fix a point about as far from instrument as first point, and in
the same horizontal plane. Revolve the instrument on its vertical
axis and sight the first point; then reverse the telescope and note
if line of sight cuts the second point. If not, loosen the capstan*
heitded screws holding cross- wire ring and move the vertical wire
PBBLIMIKABT SUBVBYS. 13
orer one fourth the apparent error — since tliere 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 instra*
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 positions.
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.
Brlye 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 300 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. Level the instrument
imd fix the intersection of the cross- wires on the highest point that
14 A FIELD-MAKUAL TOB BAILBOAD EKOIKEEBS.
ean be easily siglited. Depress the telescope and fix a point neai
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 thej
cut it the standards are in adjustment. If thej do not, bring
them half'iffay 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 first peg and pro*
ceed as before until the condition is satisfied. Or we may proceed
as follows :
In Fig. 1 let the transit be at 0, and A and B be the pegs. AG
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 E and D,
we must find JDG bo that the target may be set at the proper read-
PEBLIMIKA.BY SURVBYS. 16
ing to make the line of sight horizontal. Let OF:sz a, FO = b,
EA = r, DB- r\ CB = k. Draw DH parallel to CA and 00;
ihenEH=:r + k-r\
From similar triangles
)
Set the target at a reading GB = OD + /, sight to G, and the
line of sight will be horizontal. Bring the bubble to the center of
its run while the telescope is in this position, and the adjust-
ment is complete.
If desired, a correction for the curvature of the earth and re-
fraction may be introduced, but for short sights this is a useless
refinement.
26. The Vernier is an auxiliary scale for measuring smaller
divisions than those graduated on the limb. There are two
classes, the direct-reading and the retrograde, according as the
fractional parts of limb readings are taken on that side of the
«ero 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 Lmmst 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 yemier.
Let I =r value of one space on limb ;
o = value of one space on vernier ;
n = number of spaces on vernier.
Then for the direct vernier
nv = ln~l)l;
from which we get the least count,
n
For the retrograde vernier
nv = (n + ly.
16 A FIELD-MANUAL FOS BAILBOAD EK6IKBEB8.
from which the least count is
n
the same result as found for the direct yernier.
So, to find the least count : Divide the^cUiie of one limb space bg
the number of spaces on the vernier.
For example : If the limb of a transit is divided to half-degrees
and the number of spaces on the vernier is 80, the least count
will be J divided by 30, or ^ of a degree — that is, 1 minute.
27. To Read a Vernier, take the number of the last division on
limb back of the vernier zero, then look along the vernier until a
line is found to coincide with a line on the limb ; add the number
of this vernier line, multiplied bj the least count, to the scale
reading, and the result will be the required reading.
C, Accessories.
(!•) The Gh-adienter.
28. The Qradienter 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 fro^n 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.
PRELIKIKABY SUBYEYS.
17
(2') The Stadia, or Telemeter.
29. The Stadia is an instrument for determining the distance
of a point from the observer bj noting the space intercepted on a
rod bj a given visual angle, as determined bj 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
cats 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. I«ine of Sight Horizontal.— In Fig. 2 let a and h 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 A = a&,
r = AB, d = distance of cross-wires from objective, J) ^ distance
of rod from objective.
From similar triangles,
From optics,
1 + 1=1
in which/ is the focal length of objective.
Eliminating d from these two equations.
18 A FIELD-MANUAL FOB BAILBOAD EKGIKEEB8.
Let c be the mean distance of objective from cehter of instm-
ment. Adding this to D gives, for the distance of the rod from
the center of the instrument,
i=-o+f+j-r.
(2)
f
J- may be made constant, when (2) becomes
l=za + kr (2')
31. liine 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 = CD be the reading on rod held vertical ;
r' = FE, the reading perpendicular to line of sight ;
11= AO, the horizontal distance from Aio B ;
V = BO, the difference of elevation between A and B ;
n = BAOf the angle of inclination of line of sight.
Assume angles AFB and AEB = 00*, from which they rarely
differ more than 15' to 17'. Then, since FBC = n,
r' = f» cos n.
PBBUHIN'ART SUBTSTS. 19
By (y), AB==a + A;/.
Hence AB =za-^kr cos ti.
From triangle ABG
Hss AB eoBn
•*. jET = a cos n + kr cos* n • • (3)
F=Jl-Bsinn;
,*, V=^ a sin n + kr sin ;i cos n.
Bat 2 sinn cos n = sin S;]*.
Hence F = a sin n -{- ^Ar sin 2n (4)
32. The InstmmeBtal Constant a [^e+f of (2)] maj be
fonnd by measuring the distance from center of instrument to
mean position of objeetiTe, which equals c ; then focusing on a
very distant object, preferably a star, and measuring from center
vf objective to plane of cross- wires, which equals/. The sum of
these distances is a in formulas (3) and (4).
If the stadia wires are fixed, k may be found by measuring for-
ward OB 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-^b ^a-\-kT.
,\ k^^, a. constant ratio,
r
If the stadia wires are adjustable, we may so adjust k that any
desired reading may be had for a given length of base. A con-
venient value of k is 100, which corresponds to an intercept of
1 foot on the rod at 100 feet from a point a feet in front of the
instrument, 2 feet at 200 feet in front, etc.
33. A Stadia Table based on formulas (8) and (4) is published
by the D. Van Nostrand Company in Winslow's Stadia Surveying^
and can be used more rapidly than the formulas. Johnson's Re-
liueHon JHagram, by John Wiley & Sons, gives values oi H and V
graphically. Stadia slide-rules of various types are no^ on the
market which admit of rapid determination of Fand of a correc-
tion to be applied to the apparent distance AB to determine H,
20 A FIELD-MAXUAL FOB BAILBOAD EKGINEEBB.
D. Field-work.
84. 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 tbe preliminary, A, J5, C, stc., being marked
on^tbe forward side. Tbe 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 a«e short,
flat-topped stakes driven into the ground flush with the surfaea
The flag is held on the top fuid carefully aligned, the position of
the point being marked by a tack. A special tack with concave
bead offers & 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 th^
point whose position they serve to locate. They should be driveiv
beyond reach of disturbance, and are used in replacing a dis
located hub.
These need rarely be used on preliminary.
37. Alignment. — It is not intended that the preliminary ant\
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 obstruct
tions may be avoided by turning a small 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
check on the angle read on the plates.
In easy country not requiring a topographic party large angles
should not be turned, a succession of small ones with short inter-
vening tangents being substituted in order to make the prelimi-
nary profile approximate more closely to the location profile.
These short tangents may conveniently be the long chords of the
curve that is te follow.
<* Such a tack Is manufactured by the A. S. Aloe Co., St. Louli.
PBSLIKIKABY SUBYBYS.
21
38. The Traniit NotM may be kept in the form below, which
shows both pages of the note-book. The notes ran from the
bottom up, the right-hand page being reserved for sketches ; the
red line np the middle of the page represents the transit line,
whether straight or broken, to which the sketches must be
adjusted.
8ta.
08
670
66
65
64
630
62
61
Angle.
ao^CL.
e^a'R.
Calculated
Course.
N. !• 48' W.
N. 18« ly E.
Msgrnetio
Course.
N. !• W W.
N. 18« 15' E.
Remarks and Sketches.
0
O
39. Stadia Methods for Preliminary Surveys. — Preliminary
lines are usually run with the transit, but the compass will
answer nearly aa well in moat cfises, 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 inap 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 tho
profile could readily be taken. With a skilled observer and two
to four rodmen the progress may be more rapid, and fully aa
good for the purpose intended as the more expensive method
nsnally employed,
22 A FIBLD-MAKUAL 70S BAILBOAD EKGINEISRS.
The transit need onlj be set at alternate stttioitB (wliioh maj be
any length within the reading limits of the wireB)» the bearings to
other stations and points off the line being taken with the needle.
The horizontal angle should also be x^ead 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
f611owing 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 Unes. — In
Fig. 4, 0 is the obstruotion, AB the obatructed line. At J? set
Flo. 4.
transit ; turn 90** and measure BF long enough to clear obstruc-
tion. Set transit at F, make BFO = 90", and measure FG,
Move to O and backsight to F, making FOC = 90". Measure
OC=:FB, and move to (7« where the angle OCB is made equal
to 90°. CD is the desired line, and BC = FG.
Othenme^ 9X A and B erect perpendiculars; take BFs^AS;
produce FF, and at G and H, beyond 0, erect perpendiculars mak-
ing (? (7 = -HZ> = FB, CD will be the desired line, and BO = FG.
42. To Pass an Obstacle by Angular Deflections.
Genbral Case. Angle anything leas than 90*.
At B (Fig. 5) on the obstructed line deflect an angle a to one
side and measure BG^ taking C so that after deflecting 2a to the
other side OD will clear the obstruction. Make CD = BC and
deflect an angle a to the same side as at B; DB will lie in AB
produced. Draw CIT perpendicular to J92>; then
BD = BH+HD=:2BG coo <!. .... <Q
FBELIHINABT SUBTBTB.
BzAKPLS.— Suppose a = 14° 10', BC=CD = Smn.
fli) = a X 680 X 0.96958 = 1008.87 fe«t.
Special Case. Angle 60 degrees.
la thiB caae the triuigle BDF(Fig. 6) ifl equilateral and BF=s
BD = DF.
Shoald it be inconvenient to run to i> ne may stop at C, liaving
measured BC. At C deflect 60° and measnre CB; at E agftlu de-
flect 60° and make BF= BC. At F», final deflection ot 60° in the
opposite sense will put the telescope in the desired line, FO, and
BF=BC+CK (5<i)
43. To Paas an Obrtmction, snch a* a River, wben the Pre-
ceding Methods are Inapplicable.
First Case. Point l>eyond i^trueU'on vtnife.
In Fig. 7 let BG be required.
the perpendicular BS ; set InstruUieDt
angle BDG- a ; thea
BO = BDUma (6)
34 i. FIBLD-IUNUAI. FOB RAILBOAD BttQIHEBBS.
Or, if % trigonometric table la not «t huid, make CI)E= 90° and
fix the point S where i?j? inteisecta AB ; meaanring EB then
results, from similar triangles,
CB_BB
BD ~ EB'
whence CB='^ {60)
0(herv)Ue, if a right Angle at £ is not convenient, measure
angles CBD = 6, BBC = a, and side BD. Then e = 180°- (a+6).
From triangle BBC,
BC = BD^^ <*'
ExAKPLB.— a = 06*, a = TO', BB = 400 feet.
B7(«>). BC=400||^=409.8feet.
Second Case. Point beyond lAitraetion inzMbU.
At B (Fig. 8) measure angle ( and line BE; move to B and
'e angle y, and set hubs on line EG so the liue BC will pass
Flo. 8.
between them. Angle i = JCB = 180-(6+y). Tbenfromtri-
angle BEO
c^be'^^.
BC^BE-r^- (7)
Produce EB to B, where BC will be sure to clear obstruction;
From triangle BBC,
tan i(a - .f) _ BC - BB
unj(a + ij BC + aiJ'
Bat a 4- x = b. hence
tan i(o - t) = ^^-^^ tan *6
PRELIMINARY SURYBY8. 25
The sum and difference of a and x are now known, so both may
be readily f oand.
M D set off the angle a with the transit, and have the chainmen
stretch a cord between the hubs set on line EG 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 i? or J^, reverse telescope and deflect itxto CH,
' If ft = 90** the problem is greatly simplified.
Article 4, — ^Thb Level Pabty.
44. The Level Party consists generally of two members, the
leveler and a rodman ; sometimes an axeman is added to keep the
lodman supplied with pegs for turning-points and in, clearing the
line of sight for the level. As the party follows the transit little
or no clearing will be needed. The instruments used are a level,
A rod, and a hand-axe or hatchet.
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. He must plot his profile
at night, or at such times as the chief of party is likely to need it.
The rodman 's readings at turning-points should be checked.
46. The Rodman holds bis rod at each station, calling out the
number. If stakes are set only at even stations, be must hold his
rod midway between stakes, the point being found by pacing tbe
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 AcljQit the Line of Oollimatiofl is to bring the inter*
section of the cross-wires into the optical axis of the telescope.
Set up and level the instrument, then bring the vertical wire
into coincidence with a plumb-line or vertical edge of a building,
r
26 A FIELD-HAKUAL FOR RAILROAD ENGINEERS.
at the mean length of sight, and note if the yertical wire is trizly
parallel thereto. If it is not, loosen the capstan -headed, screws
holding cross-wire ring and turn slightly so that the wire is
parallel to the vertical line.
Loosen the wye-clips and bring the yertical wire into coin*
eidence with the line and clamp the instrument. Rotate the
telescope in the wyes 180** and note if the wire coincides with the
line. If not, correct one half the error by loosening one and
tightening the opposite, of the capstan-headed screws that hold
the cross-wire ring in place, remembering that the image of
the cross wires Is inverted by the eyepiece.
Turn the telescope until the horizontal wire Is parallel to the
plumb-line or edge of building, and make the same test and
correction. Repeat for both wires. The horizontal wire is the
one on which the accuracy of leveling depends, but it is wise to
have both adjusted. Their intersection should remain on a point
during a complete rotation of the telescope in the wyes.
48. To A<3Just the Level-bubble is to bring the axis of the
level tube into the same yertical plane with the line of collimation,
and to make the bubble staud at the center when the line of sight
is horizontal.
Since the axis of the telescope coincides with the line joining
the center of the wye-rings (which requires these to be of the
same size), it is sufficient to make the axis of the bubble parallel
to this line.
(a) With the telescope over one diagonal pair of leveling-
screws and the clips loosened, bring the bubble to the center of
its run ; then turn the telescope, in the wyes, a little to either side
of the vertical plane through the telescope and note if the bubble
remains at the center. If not, correct the error by means of the
screw at end of the level-tube case arranged for lateral movement.
Repejit until the tube may be rotated half an inch or more to
either side of vertical witbout movement of the bubble. This
adjustment is made merely to prevent error from failure to set
level-tube vertically beneath telescope.
(b) With the wye-clips opened well out, again bring the bubble
to the center of its run ; remove the telescope from wyes and
turn it end for end, then carefully replace it in the wyes. Should
the bubble fail to remain at the center, bring it halfway back by
raising the lower or depressing the higher end of tube at (he
points of attachment to telescope. Relevel and I'epeat as a lesl.
PBBLIHIKABT dURVETS.
27
49. To Adjust thd Wyes is to. make the axis of the telescope
perpendicular to the vertical axis. With the wye-clips closed
place the telescope over one pair of leveling-screws and bring the
bubble to the center of its run ; then turn the telescope half-way
rouod on its vertical axis» so that its ends have changed places.
If there is any error, correct by bringing the bubble Jialf-icay back
to center by means of the screws connecting wyes with level-bar.
Repeat until the bubble remains in the center during a complete
revolution.
B. Theory of Leveling.
60. When the level has been adjusted the line of collimation
^ill describe a plane pamllel to the horizontal plane tangent to
he earth's surface at the point where the instrument iis placed.
A level surface, such as the surface of still water, will coincide
with this plane only at the point of tangency, and will depart
farther and farther therefrom as the point considered recedes
from the instrument. For short sights this difference may be
iieglected iu 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.
51, The Error due to Owrvature at any point is the deviation
of a tangent line from true level, as
the point recedes from the point of
tangency.
Let 0 be the center of the eaiTth, f
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)« = 2? + t\
From which
Fig. 9.
c =
2^ -f <j '
Now, since c is always very small compared with 212, the
quotient resulting from the division of t* by 22? will not differ
28 A FIELD-MA17UAL FOB BAILBOAD ENGINEERS.
sensibly from that obtained by dividing by 2B + e. Therefore
"we write
"=4 w
For t = 1 mile, B = 3963 miles, c = about 8 inches. Hence
for any other distance in miles we have, for e,
e = S X t^ inches. ...... (9a)
The correction for refraction is about -^c, hence we have,
from (9),
/T 1 6 8 <•
7 7 7 B
or, closely enough,
C = .85c (10)
Example —What is the correction for a half-mile sight ?
For one eighth of a mile?
By (9a), <J = 8 X (J)* = 2" for first case,
and c = 8 X (i)' = 0".125 for second case.
By (10) the final correction is
6 = 0.85 X 2 = 1".7 for first case.
c = 0.86 X 0.125 = 0.106" for second case.
62. The Difiference of Elevation between two points not so
far npart but that a rod may be rend on each from some inter-
mediate point may be readily found from these rod-readings.
In Fig. 10 let the instrument be at /, A and B the points
whose difference of elevation is desired. Let r = AD, r' — BO,
Since the line of sight, DO, Is horizontal, the diflference of
D 0
Pia. 10.
elevation will evidently be / — r. When the distance from
J to ^ equals that from / to i? the errors due to curvature
evidently balance.
PRBLIHIITABT SVRYBTS.
29
When the points are so situated that the rod cannot be read
on both from one hitermediate 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 of A and B required :
Wiih the instrument 'at I read on A and some intermediate
poiut E, 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 OE— CB, The sura of these differences equals the dif-
ference of elevation of A and By and may be written (AI) + OE)
^ {EF-{- CB), or, in geueral, the sum of the backsights less the sum
of the foresights equals Hie difference of elevation,
C. Field-work,
63. A Datum is a level surface so taken that it shall lie below
the lowest point likely to be reached by the profile, to which the
surface elevations are referred. It is often spoken of as the
datum-line or datum-plane, and is the zero of elevations.
64. A Bench-mark is a permanent mark, such as a copper or
other bolt let into the top of a solidly fixed stone, whose height
above the datum is known ; it may be simply a mark on a stone.
Of a tack driven into the projecting root of a tree, upon which
the rod may be read. In any case it must be so situated that it
cannot change its elevation nor is likely to be disturbed within
the time for which it is intended to be used as a standard of
reference.
The elevation should be marked on some object adjacent to
the bench, with the letters B. M. indicating the nature of the
point.
80 A FIELD-MANUAL FOB BAILROAB SNGIKEERS.
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 taken at the beginning of the line, and its dis-
tance above mean sea-level or otber datum is known or assumed.
The level is set with one pair of leveling screws in the line to be
run (in order that any change in the position of the bubble may
be easily corrected), and the rod is read on the bench. This read-
ing plus the elevation of bench gives the height of instrument
{ff, I.) above the datum.
Readings are taken at every hundred feet along the line, or
oftener if the surface changes greatly, until a point is reached
beyond which it is desired to move the level. A peg is driven
firmly into the ground and the rod 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 pi'ocess
repeated. Benches and turning points should be read to hun-
dredths or thousandths of a foot, intermediate points to tenths.
Turning-points are marked © or T. P. in the notes, and their
positions, as also the bench marks, noted by both levcler 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
F. 8.
Elev.
Remarks.
B.Bf.
5.61S
205.613
• • • •
200.0
J B. M. on root of L. O. tree BO' to
right of line.
0
• • • •
• • • •
2.8
208.8
■
1
• • • •
• • • •
0.8
204.8
2
• • • •
• • • •
5.7
199.9
3
• • '* •
• ■ • «
7,8
197.8
4
• • * •
■ ■ • •
9.9
195.7
O
1.120
196.310
10.423
195.190
j On peg at 4-f 80^ - 20^ to left of line,
1 by small P. 0. tree.
!S
• • • •
• • • •
6.3
190.0
6
• • • •
• • • •
4.5
191.8 \
Here the elevation of the datum was taken 20Q 00 feet below
the first bench-mark. The instrument was set up near Station 2,
PBELIMINARY SUBYBYB. 31
And a reading of 5.618 taken on the bench; this was written in:
the B' S. column, and when added to the elevation of the bench
gives the height of instrument, 205.618. A reading of 2.3 was
taken on Sta. 0, recorded in the F,8. columu, and when sub-
tracted from the H,I, yields an elevation of 208.8. 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 10.428 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 JET. /. 196.810, and the process continued with this H. L
In most cases it will be sufficient to read benches and turning*
points to hundredths and intermediate points to tenths.
It will be seen from the notes that any error in a turning-point
causes the same error in all succeeding points. To guard against
this the rodman is required to keep a ** peg-book," in which the
heights of instrument and elevations of turning-points are re«
'corded, and which must check with^the leveler's record.
67. Wind and sunshine alfect 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
'Yuch 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-
spii-e to cause them to change.
By making foresights and backsights to turning-points about
equal the error due to curvature will be eliminated; the readings
of rodman at these points should also be checked. The rodman
should hold bis 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 in the same vertical plane with the
instniment, and by causing the rodman to wave his rod back and
forth slowly, after clamping the target, he can tell if the hori-
zontal wire just bisects the target at its highest position.
68. The Rod should be graduated to feet and tenths, reading
by target at turning-points and benches; intermediate readings
are made by the leveler at his instrument. Strength and dura-
bility are essential qualities. The Philadelphia rod seems to
32 A FIELD-MAKUAIi IfOB BAILBOAD EKQINBERS.
answer the purpose as well as any other now manufactured; the
Troy rod may be used in the same manner as the Philadelphia
rod, but is lighter and less able to stand rough usage.
Article 5. The Topographic Party.
69. The Topographic Party follows the level and secures al)
the data necessary for making an accurate contour-map of a strip
of country extending as far each side of the preliminary as may
be needed for the intelligent projection of the location-line.
This distance may vary from 50 to 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 surface 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.
00. 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 leveler'a
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 the next contour; from this the reading of the rod when
held on this contour is found, being the height of station above
contour plus the height of eye. He directs the slopeman in or
out on a line at right angles to preliminary until this reading is
given by the band-level; the distance out is then measured and
recorded, just as in setting slope-stakes, and the slopeman dM
reeled into position on the next contour, in the same manner.
Thus if 6-foot contour.intervals are employed, and the station
PBELIMIKART SUBYBYS.
33
elevation is 321 .6 feet and the height of eye 6.3 feet, we shall haTd
for the reading at the 820-foot contour 5.8 -f (821.6 ~ 820)=r 6.9.
Motion the slopeman down the slope until his rod reads 6.9 and
measure the distance out, suppose 21 feet. The Slo-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 5.8+5= 10.8 find the 810-foot contour in
the same way; On the up-hill side the 325-foot contour will be
found with a. reading of 5.8 - (826 - 821.6) = 1.9 feet, and other
contours in like manner.
The notes may be written thus
8ta.
Left.
Center Elev.
Hight.
824
805 810 816 820
193* 125' 80 ' 21
821.6
825 330 385 340
27' 56* 80' 112
The number above the line is the contour elevation, the num-
ber below its distance out from center.
If preferred the elevation can be taken at regular distances out
and recorded as above; the position of the contour will then be
found by interpolation when mapping the work.
61. If the topography is to be plotted in as the work progresses
the topographer must have a light drawing-board with a pocket
and flap on buck for holding the sheets on which the transit- line
has been plotted the night before ; the station elevations are
marked on the line and, the contour positions spotted in as ob-
tained by slopemen, after which the contours are sketched in.
Points where contours cross transit-line are found in the same
manner as side points. The size of the sheets will depend on the
taste of topographer and size of drawing-board; 17x24 to 19x28
inches are good sizes.
The topo£^rapher will soon learn to guess at the position his
contours will occupy at the next station ahead, and will sketch
them In lightly, to be erased and corrected when necessary. It is
often sufficient to take lateral readings at every second or third
station.
62« If the Slope level is used, the inclination of the surface is
obtained; then by the use of a scale constructed to show the
34 A FIELD-MAKUiLL FOB BAILBOAD ENGINEERS.
horizontal distance apart of contours, for the given contour-in-
terval, for slopes varying from 1* to 20°, the position of contours
can at once be spotted on the map. Wellington recommends the
use of the altazimuth as permitting the employment of either
method at will— the altazimuth being merely a hand-level with
a clinometer attached.
63. Oross-section Rods are measuring* rods 10 or 12 feet long
carrying a level-bubble. By placing one end at the center,
bringing the rod horizontal, and noting the height of the end of
rod on the down-hill side, the slope may readily be obtained and
the contours worked in as before. For very rough, broken
ground this method may be preferable to either of the othera.
64. If the Transit and Stadia are employed, very elaborate
topography may be taken with very little field-work, but the ob
servations require considerable reduction. With a suitable topo
graphic protractor and the slide-rule mentioned In 33, the large
number of points that may be obtained from each setting of the
transit may be readily plotted and their elevations marked on the
plot, after which the contour-lines can be worked in, and other
features mapped. For small vertical angles no horizontal reduc
tion is needed.
While not generally favored by railroad engineers in the past,
this method is probably the most mpid and economical of any so
far employed in topographic work.
Article 6. Prblihinabt Estimatbb.
66. After completing the field-work of the preliminary survey
the party is usually disbanded, only the transitman, leveler, and
topographer being retained to assist the chief of party to complete
the map, profile, and estimate of cost.
66. The Map may be drawn to any suitable scale, but less than
400 feet to the inch is not to be recommended where it must be
used in projecting location. The trausit-liue is laid down first
and the topography worked in afterwards from the field-map or
topographer's notes. If it is wanted on a continuous sheet, the
transit- line must first be drawn on a succession of small sheets,
which are added as the plotting progresses, a new sheet being
dipped under the edge of the preceding and tacked down whttu
PBELIMINABY SnfiVSY& 35
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.
66a. Land Line Ties should be made on both preliminary
and location surveys at the time the field work is being done,
for the reasons given in the following paragraphs supplied by
Mr. C. M. Kurtz, author of Modem Location of Standard Tum-
cvtsi
*'A matter of primary importance in running a location line
is to properly co-ordinate it, wherever possible, to all property
lines and comers and also state, county, town and section
lines. In those portions of the United States and Canada
where the land has been surveyed into townships, sections,
etc.y it is very necessary to tie the surveyed railroad line to
the lines and comers of the public land surveys.
" This is a very simple matter in some comparatively flat and
long-settled portions of the United States, where the centers
of the coimty roads have been established on section lines,
and the section comers are located at coimty road intersections.
In such locations the survey station and angles at which the
location line crosses the land lines can be observed by the transit
party as it goes forward along the line, and the distances to
the comers can be measured by two extra chainmen. West
of the Rocky Mountains, however, these conditions seldom
occur, for section lines are more often than otherwise not defined
by either fences or roads, and section comers will often prove
to be lost or obhterated. In this case it becomes a very im-
portant fimction of the topographic party to hunt up all. corners
which can be found in the vicinity of the located line, and to
36 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
make the usual tie by survey station, angle of crossing and
distance to the comer. It may be necessary to run a random
section line between comers, intersecting it with the raihroad
line, and then from such survey data obtain the tie to the tme
line by computation.
" If time or conditions will not permit such a survey to be
made, it may suffice to make simple polar co-ordinate ties,
either by a direct or traverse line, according to conditions,
to the comers from some convenient point on the located line.
In general, however, it will be time and money well spent to
have land line ties made, wherever possible, from station inter-
section of section line, angle and distance to nearest comer,
and to have these ties made before right of way purchase or
beginning of construction.
"Where the new railroad line traverses government owned
land it is imperative that ties to the public land survey lines and
comers be recorded on all maps made for filing purposes.
"Land Hne ties enable the location party draftsman to cor-
rectly map the sections, townships, etc., as the location line
is plotted, and later to show proper ownership of the various
parcels of land crossed by the railroad. This is vitally im-
portant in making right of way purchases. The ownership
of unoccupied but privately owned land is best determined
from records in the county assessor's office."
67. The Profile is drawn on specially engraved paper, pref-
erably a thin and translucent paper bearing orange-colored
lines so as to admit of blue printing, or, better still, on profile
tracing linen, also engraved with orange-colored lines. The
scale adopted will depend upon the paper used, but in nearly
all cases the vertical scale is greatly exaggerated as compared
with the horizontal. When distances and elevations are meas-
ured in feet both Plate "A" and Plate "B" are in common
use. Plate " A " has vertical lines spaced one-fourth inch apart,
every tenth line being heavier than the others. It has hori-
zontal lines spaced one-twentieth of an inch apart, every fifth
line being heavier than the intermediate ones, and every fiftieth
line heavier still. For the normal scale of the paper 400 feet hori-
zontal equal 1 inch and 20 feet vertical equal 1 inch. This
makes the exaggeration of vertical to horizontal 20 to 1. This
plate is more generally used than is Plate "B," especially where
PRELIMINABY SURVEYS. 37
rock work is expected. Plate '^B'' has the same general plan
of engraving as Plate ''A'' except that the spacings are 4X30
to the inch, as against 4 X 20 to the inch for Plate "A." Another
form of spacing is 5X25 to the inch, known as Plate "C,"
which, however, is less frequently used. There is also a metric
paper, graduated to millimeters both ways, with every tenth
line heavier than the others, which may be used for a variety
of purposes. It is especially adapted for use when measure-
ments are made in the metric system.
68. Preliminary Bstimates 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 ha?e
when completed. The nature of the work, whether ordinary
aarth 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 considered and
allowed for in making up the estimate.
Engineering expenses and unforeseen outlays that are sure to
mrise should have a liberal allowance.
4
69. The Report of the chief of party should set forth the ad-
▼antsges and probable cost of each of the several lines run
when there is more than one. On this report frequently depends
whether or not the line is to be located, and it should be clear
and exhaustive, though plainly and concisely worded. The map
and profile form an integral part of the report and show from
what data the estimates were derived.
t -—
CHAPTER III.
LOCATION.
Article 7. Peojkcting Location.
70. After the preliminary has been mapped and the topography
worked in, the engineer proceeds to 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 the best structure is tJiat which for the least cost test an-
»v>ers 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 free from curves as possible, while heavy grades are at the
same time excluded; these two requirements conflict and must
be as well adjusted as possible. The amount of earthwork, of
bridging and other structures must be kept down to the lowest
limits.
71. Starting at the summit of the most difiacult 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 fa!^ each time on the
next lower contour, or half-space, according to the fall assumed in
setting dividers. If cur^e compensotion is allowed, the divid^TB
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 to be used as a location-
line, so we have then to draw on the map a succession of curves
ai^ 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 ai-e
transferred to profile-paper, thus giving a profile of the line.
38
LOCATION. 89
Wiih a fine tbraad stretched along the profile, to represent the
grade-line, ad just the cuts and fills to suit the nature of the work.
In general, fills are cheaper thau 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 Id embankment, when
the cuts and fills must be made to balance by shifting the grade-
line until this appears to be the case ou 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 tbat the grades are not so badly " chopped" that
they interfere with the easy riding of the train.
In projecting the line it will generally be best to strike- the
curves first and draw the tangents afterwards, though it some-
times happenis 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 material assistance in find-
ing the degree of curve required to unite two tangents that have
been laid down on the map. It consists of a trauspai-eut, semi-
circular protractor having a series of curves from SCV up to 8*
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
40 ' A FIBLD-MANTTAL FOR BAILROAD ENGINEERS.
it, simply prolong tangents to intersection and then place the
protractor so that the curve admitting of the best grade is tan-
gent to the two straight lines. Mark the points of tangency,
which will be the beginning and end of curve. When the curve
is required to pass through a given point the proper curve may
be immediately found by trial, whereas the calculations would
require some little time.
Reversed curves should never be allowed on main lines. Suffi-
cient tangent should be interposed tt> allow. space for easing ofi
the superelevation of outside rails, or for the insertion of tran-
sition-curves when these are to be employed.
73. The Field Oorps 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 location- lines are to be run (and it is
real economy to run both), it will not be necessary to have the
6tationing 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 first time, the proper offset being made at
vhe P. T. or P.C. of the circular curvesi which latter are to be run.
On the final location-line the stationing must be continuous,
beginning with zero. The stakes are niarj^ed 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 disiurbance by the plow or scraper.
The leveler should make bench-marks every 1000 or 200Q 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 iu the field; however, noatteuipt should
be made to adhere rigidly to them, since slight errors in the
mapping will affect the projected line, while in the field the line
may be shifted here and there so as to fit the ground more snugly
And accord more closely with what the nature of the earthwork
demands.
The highest skill of the engineer is required to secure the best
location-line, and he should have all the time he needa. Undue
haste on location — as on reconnoissance and preliminary — ia
almost sure to result in increased cost of construction.
LOCATION.
41
Articlb 8. Simple Curves.
4. Definitions and Formulas.
74. The Oircular Ourves 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 Ourve is the arc of a circle.
b. A Oompound Ciitve 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 tbe same or different radii, and a common taugent.
d. The Point of Ourve (P. C.) is the end of tangent and begin-
ning of curve, as at A, Fig. 12.
c
\ )
/,_/^
jr
Ex
V
M ^^
^
c ^
H
^yR
B
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 tbe
tangent at theP.C. and P,T. intersect when produced. (D of
Fig. 12.)
g. The Intersection Angle (i^ is the angle at the P.I. be-
tween the tangents meeting there, and equals tbe angle at the
center.
h. The Tangent Distance (T) is the length of the produced
tangent measui-ed from the P. C, or P,T. to the P./. The term
tangent is applied to any straight portion of the line, but the letter
T will be used to designate the produced portion only.
t. 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 Bxtemal (E) is the part of the radius produced to the
P./., intercepted between curve and the P.I,
42 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
k. The Long Chord (L. G.) is the chord joining the P. G, and
P. T, Frequently the term is applied to any chord longer than
the unit chord.
L The Radios will be denoted by R.
m. The Point of Compound Curve (P. (7. G, ) is the point of
common tangency of the two bmnches of a compound curve.
(See Fig. 13.)
P.C
n. The Point of Reversed Curve {P.R.G.) is the point of
common tangency of the two branches of a reversed curve.
o. The Degree of Curve (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 speak of the degree of curve.
Half the degree of curve is called the deflection-angle, since
it is the angle to be deflected from the tangent to the chord.
If there were any practical method of measuring around tJie
&urf)$ instead of along the chord, an accurate and convenient
ratio for expressing the radius in terms of the degree would be
hod. Thus if D is the angle at the center subtended by the «re
of unit length, we have, where a is this unit arc,
360
D'
a 360
2i^-:5- • • • w
When a equals 100 ft. this becomes
2iri?s=a.-
Hence
S^
i?=l
,100 860 5729.65
2t D
J)
(11')
LOCATION.
43
2? varies inversely as 2), so that knowing the radius for a 1*
curve, we should have only to divide this by B to get the radius
for a D** curve.
Since the chord is employed instead of the arc, we determine
It by means of the following problem •
76. Given the Chord C7, and Degree of Curve D, to Find the
Radius ^. .
In Fig. 14, AB \% the chord C, OE a perpendicular from the
center upou AB
From the right triangle AEO we have
i?sinii) = \C.
Whence
When C is 100 ft.,
2?=-T
\0
sin^i>
= i(7cosec§2>. . . (12)
JB = -.
50
sm^
P^ = 50 cosec i2). . . (12')
Comparing results given by formula (12') with those given by
<11'), we have for a few curves :
Degree of Curve. J? by (120.
1 5729.65
2 2864.93
3 1910.08
6 1146.28
7 819.02
10 573.69
14 410.28
20 287.94
fiby(ll').
Difference
5729.58
0.07
2864.79
0.14
1909.86
0.22
1145.92
0.86
818.51
0.51
572.96
0.78
409.26
1.02
286.48
1.46
44 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
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 inconsiderable, and we may
stake out curves with 100-foot chords. From 7 to 14 degrees 50-
foot chords may be used. Therefore
25
For curves from 14° to 28° we should use 25-foot chords,
for which
B = g^ = 12.6 cosec JD. . . . . (12'«)
Above 28° shorter chords— say 10 feet— should be used, if the
curve cannot be struck from the center. In this case
B = -; — T-f. = 5 cosec ^2) (12'c)
sin T^-U
Table I of radii was computed by formulas (12'), (12'a), and
fl2'6).
Ill 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 -B = ^J^ = 1482.5 feet,
while by Table I it is 1432.69 feet— a difference of only .19 foot ;
for a 12° curve -B = i|}^ = 477.5 feet, while by Table I it is
477.68 feet. The effect of taking 6730 instead of 5729.65 for the
radius of a 1° curve is to reduce the error resulting from the
assumption that R equals 5730 divided by the degree of curve.
76a. The Radii of Metric Curves are based on the use of
20 meter chords, equivalent to 65.6174 feet, since one meter
equals 3.280869 feet. Successive stakes are given even num-*
bers, as 224, 226, 228, etc., in order that any given number,
multiplied by ten, will give the length of the line in meters
from the beginning to the station in question. If numbered
continuously the station number must be multiplied by 20
in order to obtain the length in meters from the origin.
The French designate the degree of curve by the angle be-
tween the tangent and the chord instead of by the angle at
the center, or by haK of what we call the degree of curve. There-
LOCATION. 45
fore, a curve having a central angle of 4**, for one chord length,
would be called a 2® curve.
If Rm is the radius of a metric curve, dm its degree, as
defined above, then
Rm — 10 -i-sm dm ^lOcosecdm* . • . (12a)
Other functions may be computed, in meters, by using
Bm for R and dm for §D in the formulas which follow, based
upon the definition of the degree of curve as the angle at the
center subtended by one 100-ft. chord, two 50-ft. chords, four
25-ft. chords, etc., depending upon the sharpness of cm^ature.
Approximate values may be found from our own tables . Thus,
numerically, Rm (in meters) =0.2JK for D=2dm in Table I.
Also L.C.m=0,2 L.C., Mm^0.2M., J^ot=0.2 E and Tm = 0.2T
for the values found from Table IX, when the table is entered
with D=2dm as argument.
If the equivalent value of the degree of curve in the units
we use is wanted, reduce Rm to feet and in Table I find the
corresponding value of D.
Table la gives values of metric radii for curves up to 10
degrees, corresponding to a 30° 20' curve as defined in Section
76, if 10-ft. chords are used.
76. The Length of Curve (L) is found by dividing the angle
at the center (which equals the intersection angle) by the
degree of ciurve, the result being in chains and decimals of a
chain. The number of P,C»-\-L will give the station number
of P.T.
Example.— The P.C, of a 4° curve having 7*26** 30' is at
8ta. 104+12.5. Find L and the number of the P.T, Here
L = —-i- = 6.625 chains.
4
104.126+6.625 = 110.75; hence the number of P.T. is 110+75
77. Use of the Table of Functions of a One-degree Curve. —
In the location of railway curves geometrical accuracy will
frequently be of less importance than rapidity of field-work,
80 long as errors are kept within reasonable limits.
On tangents shght errors of alignment may readily be de-
tected by the unaided eye, but on curves these are not so ap-
parent. Moreover, it is not Hkely that the trackmen wiU
keep curves up in the precise position of their location.
r
46 A FIELD-MANUAL FOR BAILROAD ENGINEERS.
Table IX oontains the most frequently used functioxis of
a 1° curve, and its use serves to greatly simplify and shorten
the computations of these fimctions for other curves. Its
ordinary use for this purpose requires the assumption that the
functions vary inversely as the degree of curve, or directly as
the radius. This assumption is not so very violent if the
chord length used is made to vary with the degree of curve as
advised in Section 76- Thus 100-ft. chords may be used for
curves not exceeding 7 degrees; 50-ft. chords for curves between
7 and 14 degrees; 25-ft. chords for curves between 14 and
28 degrees; and 10-ft. chords for all sharper curves. Table I
gives the values of the radii of chords up to 20 degrees, which
radii were computed on this basis.
In computing Table IX the radius of a 1° curve was assumed
to be 5730 feet instead of the more precise value of 5729.65
feet. This, not only for simpUcity of computation, but also
because the approximate values of the functions found for
other curves by the use of the table, agree more closely with
the true values than would be the case if the value 5729.65
had been used in computing the tabular quantities.
Table IXa contains correction factors, by the use of which
the precise values of the L.C.^ ilf, Ey and T of any curve
may be found from the values given in Table IX when the
chord length employed in locating the curve is known.
Let n= number of chord lengths used per degree of curve,
(n = l when c«100 feet; n=2 when c=50 feet; w=«4 when
c=25 feet, and n = 10 when c = 10 feet). Let (/^/)=fiin }/,'
vers } J, exsec §/ or tan J/ according as the precise value of the
L.C.J My EjQtTva wanted for the particular value of c used.
Let A; be a factor defined by its use in the equation below:
100
^2n
from which 1 + A; =
1 D'
5730 sin^-
2 n
and, therefore,
•T^,« T n \£ IP ^^ fr Tabular function.^ , ,.
Irue X/.C;., M, E, or x «-T=r *— ^-(l+W.
Degree of curve ^
LOCATION. 47
Example. — Given a 10° curve nm with 100-ft. chords through
an angle / = 80°, to find the true T and M.
By Table IX, 7=^^^:2=480.8 feet, M= ^^^ = 134.06 feet.
By Table IXa the correction factor is +0.00120, and there-
fore the true T = 480.8(1.0012)= 481.38, and the true M =
134.06(1.0012) = 134.22.
Had 50-ft. chords been used, as advised in Section 76| the
correction factor would have been only +0.00024, or one-fifth
as much as when 100-ft. chords are used.
Table IX may also be used for finding the functions to
employ when running metric curves, the tabular values being
taken as meters instead of feet. The unit metric chord is
20 meters, and the degree as defined in Section 75a is half the
angle at the center for this chord, so we must enter Table IX
with 5X2c2m = 10(im as argument. Thus for a 2° metric curve
having 7=40°, ^=^| =17.28 meters.
In the examples given in some of the sections which follow,
the subscript 1 is written after the letters which designate
the function referred to, in or^er to save space. Thus Ti < 28°
means the tangent distance for a 1° curve when 7 »28 degrees.
78. Tables of Natural and Logarithmic Oircular Fnnctiong. —
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 ad vantage, even in
the field, and the more important ones, such as the logarithmic
sines, cosines, tangents, and cotangents, together with the loga-
rithms of numbers, are given In the back of the book along with
the tables of natural functions.
79. Given R and C to Find 2). '
F«rom equation (12),
siii-J2> = ^ (18)
80. Given / and R (or 2?) to Find T.
If D is given, find R by (12); then in Fig. 15 from triangle
OAB we get
T=Riwi\L (14)
48 ^ FIELD-MANUAL FOB BAILBOAD EKGINEEBS.
Bt Tabus IX. —Find the tabular value of T lot the giyen
angle/; then
7»— ll
"JD*
(14a)
ExAMPMB«— /= 35° 40', Z) = 4''; required T,
By (14), T = 1432.69 tau IT 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 BorD
From (14),
R-^ r-^=rCOtl/.
tan J/
Then by Table I the degree may be found.
By Table IX.
r.
(16)
D =
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=B sin J/.
.% AQ = 2AF =:L.a= 2E sin ^Z . . . (lo)
LOCATIOK.
49
By Table IX. —Find the tabular L, C, for the given angle /;
then
x.a =
(16a)
83. Given the Radius B and any Chord G to Find the
Ordinate to thie 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? - y'y = ab,
, ah
y =
But ff is small compared with 2R, and hence we write
ah
lf^
22?
(«)
Now y does not differ sensibly from ^ in the cases met with in
practice, so we write
ab
^ = 2B ^*>
50 A FIELD-MANUAL FOR RAILROAD EKOINEBKS.
If we write iJ = -yr-, formula (b) becomes
dbP
' • ^""3X5780 ^^^
Let — = wi, .^ = n, and substitute in (<;), giving
10000 ^ ^Q^o _.
y = ttjqq «*«^ = 0.878mnA
or very nearly
p = |mnl> (17)
y is given In feet when m and n are in chains and decimals of
a ciiain.
At the mid-point F, in = n, and y^M,
.-. Jf=Jn»D (18)
Caution. — Formulas (17) and (18), while very convenient foi
field use in passing obstructions, are liable to error when very
long chords or large values of D are used, since they give results
that are too small.
If we write the arcs HNy NB for a and b, 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
Curve.
Length
Ai-c.
Mid-ord.
Midord.
lf=|(^C?)»I>.
Mid-ord.
by
Table V.
2 Stations.
1.75
1.75
1.75
6 '*
15.69
15.75
15.69
2 "
4.37
4.38
4.86
6 "
88.61
89.88
39.06
2 "
6.96
7.00
6.97
4 '/
27.29
28.00
27.75
5 •'
42.02
43.75
43.20
6 "
59.43
63.00
61.93
2
5
5
8
8...
8
8
From this it appears we may use formula (18)— and (17) as
ivell — taking either the segments of the arc or chord for curves
not exceeding 4"" with arcs up to 600 ft. ; for curves from 4" to 6*
LOCATION. 51
they may be .U8«d up to 600-ft. arcs, while fur curves between
6" and 8° uot more than 400 feet of arc maj be lakco.
Skcond Method. •- First determine the mid«ordinate. In
triangle OEF,
OF- ^W - JC* ;
then ,
M^FQ = R'- i^J^'-iO' (W)
To find ordinate ^C distant d from the mid-point ot Mfft draw
OB = d parallel to HE; draw AB at rigbt angle's to 1IE» Then
BA = i^Ii* - d*.
Therefore
C74 = y= V^* -d«- V-B*-i(7*. . . . (20)
Third 3Iethod.— If the chord C is short, we may regard the
arc as an arc of a parabola, for which it is known that ordi-
Dates vary as the product of tbe segments into which they divide
the chord. The mid-ordinnte being known, we have
P aft
M (JCO«
.-. y-^-Qi (21)
From formula (b) we have for y = Jf, a = ft = ^(7,
jif=(i£L^^£! (22)
The mid ordinate for any other chord C is
Hence
.'.M.^m{^\ (2«)
UO'^ \C, this gives
52 A FIELD-MANUAL FOB RAILBOAD ENOIKEERS.
This last relation affords an easy method of staking out a curve
when the mid-oixlinate of a given chord has been determined.
Fii-st erect the ordinate M at the mid- point of the chord; then
join the ends of chord with the extremity of the ordinate just
measured; the lengths of these chords do notdi^er much from
\C\ at their mid-points erect ordinates equal to \M, giving points
on the curve. Proceed in like manner for other points until a
sufficient number have been located.
84. Given R and / to Find the Ezt<«mal B,
In Fig. 17 E- QB= OB- 00,
But OB=R sec J/ and 00 = B,
.-. E = B(secil-l) = BexaeciL . . . (24)
By TABTiE IX.— Find B for a 1* curve for an intersection
angle /; then
(24a)
^=F-
85. Given T and / to Find E.
In Fig. 17 draw BO perpendicular to AB, and produce AO to
b/
intersect BCnX 0, BO is parallel to AO, and the triangles AGO
and OOB&re similar; hence BO ^ BO = E. In the right triangle
ABO, tingle BAO=ziBAF:=zil Therefore
^=2* tan J/. . •
ExKtciSB.— Derive equation (25) from (24).
(26)
LOCATIOir.
53
86. GivenifandJtoFindJZ:
From trigonometry,
sec }/ =
Insert tbis in (24) and we get
cos J/*
* «
COS J/ '
But from Pig. 17. Jf = 22(1 - cos J/). Substitute in (a) :
IB =
M
= if sec i J.
cos)/
87. GlTeni^andJtoFimdl?.
From (24),
B _ Jg • cos )7
sec J/ — 1 "* exsecf? "" vers \i
m
J? = — :=; — r = — ^^-rrfcS-K^::::^^. . . (27)
88. Given /and ^ to Find 7.
From (25)*
r=
E
= Jg cot }/.
(28
tan}/
89. Given the Chord 0 and Degree of Curve B to Find
the Chord Deflection Offset d.
In Fig. 18 extend EA to H, making AH =EA = AB; join
H
Fie. 18.
i7 and B and draw ^iT to the mid-point of HB.
. .:. d-EB = main iB..
Then
(29)
54 i. FIELD-MANUAL FOB ttAILBOAD EKGIKEBSS.
When (7 = 100',
ci = 200 sin JD. ......... (gy)
If we write sin 12) = ^ from (12) in formula (29), there results
^=:« <«>)
For curves up. to 7', 0 = 100'; hence
10000
tf=-~^ (90')
For curves from l"* to 14% 0 = 50'; therefore
^ = ^. . (80")
For R write ^^, and (80'), for C = XOO, becomes
and for C ^ CK). (aO^') becomes
d = ^^Z> =: .4863P s: .873 . J. . . . (810
Example.— Find d for a 6** curve, (7 = 100 feet.
By (29'), d = 200 X 0.05234 = 10.47 feet.
By (30'), d = ^ = 10.47 feet.
By (81). d = 1.745 X 6 = 10.47 feet.
90. Given the Chord 0 and Degree of Curve D to Find the
Tangential Deflection Offset t
In Fig. 18 make EF (tangent at E) equal to EA, and join J^
with A, Draw EQ to the mid-point of FA. Angle AEO =
OEF = ^D; hence, from the figure,
AG^QF^CBiniJX
\ t^2CaniD. (83)
4 *
LOCATION. 56
When 0 = 100 feet,
< = 300 sin J2> (32')
Since ^D is small, we may write, without material error,
irt
sin \D = i sin ^D; then, writing sin ^D = ^, as in 89, we get
Jti
' = & (33)
5780
Making C = 100 ft. and writing li = -y-- gives
When (7 = 50 feet, (88) yields
t = .2182? = .436 X ^ (33")
Example.— Fmd t for a 6** curve, C = 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 0£fset i' for a
Subchord C
First Method.— By formula (13) find th© angle at the center
subtended by the subchord C ; call this angle D'. From (32),
f = 2C' sin iD' (34)
Bbookd MsTHOD.^In Fig. 19, with ^as center strike the arcs
FG and AH, taking EF = C and A
JSA = 0; prolong EO to B. Now
assuming that the chords C and 0
are proportional to their central
Angles we have v'^^Cr— --j2— ^-^^
AB t
■• • «
From the similar sectors EVQ Pio.19.
and EAB, since EB =: (7,
AB'^ t
»)
56 A FIELD-MANUAL FOB BAILROAD EKGIKEEBS.
MultipIyiDg (a) and {b) together, term by term.
Whence
0_l C'
='(?)■•
(85)
Example. —Find t' for a 7* curve when 0 s= 00 ft
Here
AA
-^ = 1^ X 7' (very nearly) = 4' 13'.
By (34), f
By (32'), t
By (36), t'
100
= 2 X 60 X 0.01832 = 2.20 ft.
= 6.11 ft.
= 6.11 X (^)'= 2.20 ft.
92. To Find the Tangent Oflfaet «.
In Fig. 20, EBz=z\b the required offset. Let AS = n chains =
lOOn feet. AE—FB, the half-chord
^^ "'^-^* ^ having the midordinate AF = EB ;
hence we have, by formula (18),
z = Jn«2).
(86)
In this formula we may take n to
be either the length of AS or the arc
AB, in chains. If taken equal to AB
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 />, as was pointed out in 83.
(See Caution.)
Formula (36) is easy of application and of frequent use in
Locating curves by offsets from the tangents. For curves up to
4** n may be as great as 3, but for sharper curves it should
be less. '
Example. — Find six offsets to a 4** curve at points 50 ft. apart,
measured around the curve.
Fio. 20.
«<
<«
LOCATION. 57
TBiy successive applications of (36) we have
forn = 4, e = |xiX4= 0.88 feet
n = 1, « = 1 X 1X4= 8.50
» = |, « = iX fX4= 7.88
» = 2, «=}X4X4 = 14.00 "
n=:|, « = JXV-X4 = 21.88 *'
» = 3, «=:}X9X4 = 31.60 "
Tlie last value of e is in error by about 0.2 ft , but for setting
stakes on construction tliis difference is not material so long as
the alignment be3i^d 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 Circular Arc and its Long
Chord.
First Method. — ^Let the central angle be a degrees. By (18),
Bin K = A
If/1*
Changing degrees to circular measure, a (in ;r meas.) = -—
ISO
p K7-Q- The length of arc is Ra = ^jnrTi' Then
Arc — chord = B-=-^ —e (87)
57. o
Second Method. — An easy approximation may be found as
follows :
Referring to Fig. 17, AE = e, 0F= Jf. Let il(? = d = - + a?.
From the right triangle AFQ
iH'-i
+ M^,
From which x = — ; — • • (a)
58 A FIELD-HAKUAI^ FOB RAILROAD ENGINEERS.
Neglecting the ic in denominator aa small compared with c
gives
*=T <*)
Then will 2ft - <j = Sa' = ^^. ...... (88)
c
From Huygens* opproximation to the length of a circular arc
(see WUliamson's Differential Calculus, p. 66), arc = — 5 — ^
o
Therefore
Arc — chord = — = c = |(26 — c). . . («)
V
Inserting the value of 26 — c from (38) gives
8ilf«
Arc — chord = -^7- . (d)
When the arc is not very great we may write e = 100»i , where
Ui is the number of chains coutaiued in the arc AE, From (18)^
remembering that Ui = 2/i,
Jf=0.218n,»2).
Inserting these values of c and 3f In ((f),
Arc - chord = | ^^^ISgil^ ^ ^„..^, „,,,.,y. , (g^j
Example.— Find the difference in length of arc and chord of
a 4° curve when ni = 6 stations.
The central angle is 4 X 6 = 24^ then, from Table IV,
6 =:r 595.74.
By (87),
Arc - chord = 1432.7 X ^ - 595.74 = 4.84 ft.
By (89),
A 1.^ 6X6X6X4X4 . ^^ ..
Arc - chord = ^rj: = 4.82 ft
' Remark— Formula (38) is interesting as showing what'a com-
LOCATION.
59
parstiyely small iDCi-ease in length of line is caused by a consid*
erable lateral deflection 1q 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.
B, Locating Simple Curves,
94. To Locate a Curve with the Chain by O&ets from
Ohorda Produced.
In Fig. 21 let the R C. fall at R If 5(7 is a full chain, prolong
Fia. 81.
the tangent AB to H, making 5^= BC; EC will equal U which
may be calculated by (32) or (Sy). With B as center, strike an
arc with radius BH, and with H ns center and i as radius strike
an arc , at (7, where these arcs intersect, set a stake. Produce
BO to K, making CK = BO = GD; strike the arc ED from C as
center ; make the chord KD = d, calculated from (29'), (30'), or
(31). Set a stake at D and proceed in like manner for the other
points until the P. T. is reached, where FP is made equal to t.
Usually the P.G. does not fall at a full station ; then HO s f,
which may be found by (34) or (35). Using this value of <', we
locate C OS above. At B make BR = i', and prolong BO to
L ; make LD = t and set a stake at 2>. EM will equal d, and
may be located as before.
We ntny regard KD as equal to KL 4* t, and, 4ndiog| KL.
60 A FIFXD-MANUAL FOR RAILROAD ENGINEERS.
measure KD and set D without locatiDg R, To do this we have
the similar triangles BEG and CKL, from which
CK BO'
and therefore since KG = GD,
^^--^BO'
In like manner at F we have
PN=t^, and FP^U^
hence
l!rFz=PN-\-U\
Make EQ = ti\ prolong QFy and we 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.
. f
22
Here £=-=- = 4.4 stations.
0
Therefore the number of the P, T, is
106.20 + 4.4 = sta. 110 + 60.
BG in this case is 80 ft., and by (83')
t = 0.873 X 5 = 4.87 ft.
By (35), i' = 4.37 X (^Y= 2.80 ft.
Set off EG = 2.80 ft., and at D make
KD = 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 (fo^)'= ^-^^ + 1-57 = 4,19 ft.
Make EQ = 1.57 ft., and prolong QF, the terminal tangent
LOCATION.
61
H
96, To Locate a D Degree Curve by Oflbets from Tangent.
Let AM, Fig. 22, be tangent at A, and E, F, O, etc., points on
the curve. The offseU BE, GF, f^ B
etc., may be found from formula l
(86),
either by taking equal intervals, j^
ABt BO, CM along the tangent or
by taking E, F, O, etc., at regular
stations around the curve and
using the arc length instead of
the tangent.
When the arc -4(7 is large, or
strict accuracy is required, we
proceed to fuid the offsets at
regular stations and the lengths
of AB, AO, etc. First find R
from (12) or (12'); then from triangle OEL,
Fio. 23.
BE- AL- iJ(l - cofe 2)) = ^ vers D,
AB — LE— R sin 2).
In like manner
GFz= An= R{X - cos 22)) = R vers 2A
AC^HF = RH\n2D,
and so on for any number of stations.
Should A fall at a plus station, we first find the angle Dx at the
center, then
BE= RveraDi /
AB— R sin J), ,
CF - R vers (2)i + 2>),
AG=i i?sin {Dx + D\
etc. = etc.
The ordinates BE, CF, etc., are evidently equal to the mid-
ordinatea 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 and dividing
these by 2, get the required coordinates.
63 A FIELD-MAiqUAL FOB RAILROAD ENGINEERS.
ExAicpLB.— Locate three stations of a 4** curve by offsets every
50 ft. on curve.
Referring to Table V, the required offsets are 0.87, 8.49, 7.86,
18.94, 21.77, and 81.31. By Table lY the distances measured
along tangent are 60.0, 99.94, 149.76, 199.39. 248.78, and 297.87.
With these values we can set out the curve either way from A.
Had we used formula (36) we should have had for the values
of the offsets 0.87, 3.50, 7.88, 14.00, 21.87, and 31.50.
96. To Locate a Ourve by Ofisets from a given lionj
Ohord«
Let FKt Fig. 23. be the given cboi-d. "We may compute the
offsets yi , y, . . . if by the methods of 83— of which formula (17),
is the most convenient, within the limits of its applicability—
and setting off these ordinates, locate the curve.
Or we may set off the mid-ordinate if = iJ^ers FOA at A,
and at C set off y, = if — i2 vers J), making
ACz=HL=z R sin D.
OE will be
yx = M- R vers 2A and AE = R sin 2jD.
Akotber Method is to find the angle KOF^X the center, and
' Table IX determine BAr=: M\ then by Tables V and IV
LOCATION. 63
detenninc BL, BN, LH, and IfG. Then ffOi= M^ BL; which
set off at C, and other points in like manner.
EzAMFLB.— Given the P.O. of a 4** curve at station 160 -f 75,
the angle between 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 + 36. Tha
mid-poiut on curve B falls at sta. 163. By Table IX,
J, = !?^ = 17.64 ft.
By Table Y the mid-ordinate for two stations of a 4^ curve is
BL = 3.49.
Hence HC = 17.64 - 3.49 = 14.15.
By Table IV. EL = AG^ 99.94 ft.
Measure AO =r 99.94 ft., and set off Cir=s 14.15 ft., and drive a
stake at H, In like manner find
6^27=3.70 and 4i&= 199.39 ft.
The points P and Q are also located by means of the coordi-
Kates just determined.
If B had fallen at an odd station, the curve could have been
located in the same manner, H and P being; 100 ft. from B, Q and
q aoo, etc.
97. To Ijocate a Curve with Transit and Chain when the
Degree D ox Radius E is Known.
If R is given, determine D by (13); then, since the angle in
tbe 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. 0. 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.L or by backsighting to some
point in the tangent Deflect from the tangent half the angle at
the center for ihe sub-chord or chord, and direct the head chain-
man into line while the rear chainman holds his end of the chain.
64 A FIELD-MANUAL FOB RAILROAD ENGINEERS.
at the transit, the chaiu being kept taut. The stakeman driyes a
stake at the point where the head chainman's flag rested, and the
rear chain man advances to this point. Deflect ^D from the chord
AB just run, and while the rear chainman holds his end of the
chain at B direct the head chainman into line at C. Other points
are located by deflecting an additional iZ> for each chord length
measured, until a point E is reached to which it is desirable to
Fio. 24.
move the transit. The angle ^^^ 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 to the curve at the P. 0. and any other
line passing through a point on the curve when the telescope is
directed along this line. It is most frequently taken as the angle
between the initial and any subsequent tangent to the curve.
Thus at E the index-angle equals EFP — 2FAE. At any point
on the curve the index-reading in tangent may be found by the
following rule, which may be easily deduced from a figure:
From double the index-angle Viat fixed t?ie point subtract the index-
angle in tangent at tJie last point; tJie remainder is the index-angle
required,
99. Subdeflection-angles may be found by (13) rigidly, or
approximately (and with suflacient 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 1**, and the subdeflection angle 80'.
LOCATIOK. 65
Example — -Locate a 4*" curve to left when the P.C7. is at sta.
81 -f- 25 and /= 82* 56'.
Here L ^ — f- = 8,15 chains.
Hence the P. T. will fai'l ai 81.35 + 8.15 = sta. 89 + 40. The
first sub-chord is 75 ft. long^, and the first deflection-angle will be
found by (18).
.-. j5 = r 30'.
By the approximate nile, since \B = 2*9
2 '"100'
whence J5 = 2 X f = 1* 30' as before.
With transit at P. (7. deflect 1** 80' from' tangent, measure 75
feet, and set sta. 82. Then a deflection of 3° 30' will determine
83, 5** 30' sta. 84, 7** 30' sta. 85. Now remove transit to 85, and
with vernier at T 30' backsight to 81 + 25. Reverse telescope
and set vernier at 15** 00', when the telescope will be in tangent.
An index angle of 17'' will fix 86, and so on.
The last chord will be only 40 feet long, for which the sub-
deflection-angle is y^JV o^ 2^ t^&i IS, 48'. The mdex-angle fixing
the P.r. 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* 86' = /. 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. Shune gives the limit-
ing angle to be deflected from tangent as 20'', and says 15* should
rarely be exceeded. {FieJd Engineer, p. 82.)
100. The Transit Notes may be conveniently kept in the form
lielow, which shows the notes for the last example.
When possible the tangents should be run to intersection, the
angle 1 measured, and the tangent distance calculated. Then
66 A FIELD-MANUAL FOB BAILROAD EKGIKEEBS
•
a
o .
•n ®
M^
— p
ated
•se
Station.
Defiec
angl
Inde
readJ
Inde:
Tang
Calcul
Com
a h
Remarks,
90
-HP
op.r.
0«4B'
23»48'
82<>86'
N27<»a6'E
Nar^acE
89
23«» 0'
"»
an
.
«!*» 0'
87
19*» 0'
.
86
17» (y
■
K>
o
7»30'
15* (y
84
s^acv
as
2« 0'
8«30'
■ -
82
1«»30'
1«>80'
4«» C.L.; P./. set.
4-26
0P.C.4«C.L.
O' 0'
0« C
0" 0'
I = 820 36'; T =
418.9 ft.
81
N 60«12' E
NeO'lO'E
measure along tangents and set P.O. and P.T, from the P.H
When the curve is run in, the position of the P. T. thus found
should agroe 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 giveli below. The index-readings are com-
puted as though the entire curve were run from the P.O. The
notes for ihe last example would appear as below :
Station.
Deflection -
angle.
Index-
reading.
Total
Angle.
Calculated
Course.
Magnetic
Course.
Bemarks.
90
+40
0P.5.
0<»48'
16«18'
32»36'
N27»36'E
N27»30'E
89
16«'30'
88
IS^SO'
87
11*30'
86
9" 30'
85
0
7''80'
64
5«3C'
8,1
2* O'l L03O'
82
1«80'
1*30'
4» cuKve left;
+25
OP.a4'»C.L.
O" 0'
0* 0'
PJ. set. 2=32*36';
!r= 418.9 ft.
81
N60'»12'E
N eo^io' E
The computations are all made before beginning the work, aod
the notes have the advantage of permiitiug the tracing of the
curve either way from the instrument without additional compa-
LOCATION. 67
tatioDs. Suppose the trausitman to have run the curve from the
JP, C. to sta. 85, to which point he removes the instrument. He
there sets the vernier at C — the angle on limb when telescope
^as in tangent at the P. C7,--theu sighting the P. C he reverses
the telescope and deflects to 9° 3C', which will fix sta. 86. Had
the tangent at 85 been desired, a reading of T 8(y— the angle that
located that point — would have put the telescope in the plane de-
sired. A reading of ir 80' fixes 87, and so on to the P.T.
Bemoving to the P.T., the plates are clamped at 7° 30', 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
he found in the same manner.
Any convenient form of notes, which are intelligible to another
engineer who may have to retrace the curve, may be used, but it
is desirable that some general foim should be employed. Either
of the preceding forms seems to meet ordinary requirements.
C Obstacles.
102. To Pass an Obstacle on a Curve.
First. Suppose Vie obstacle to be one obeiructing tnsion at otu
station only.
In Fig. 25 suppose transit set at A, and B and C located from
that point, but the next full station, 27, to be invisible from A,
Fie. 25.
Set a plus stntion at E, as near the obstruction as may be conven-
ient, then set J^'lOO feet from K Next make FO = 100 - CE,
and locate O with the corresponding deflection-angle. Other
stakes may be set beyond G, or the transit may be removed to
that point and the curve beyond traced.
Second. Suppose the line of sight obscured for more than one
station, as in Fig. 26.
68 A FIELD-MANUAL FOB BAILEOAD EKOIKBERS.
If transit is at A, deflect an angle HAB that will clear all ob-
structions, and at the same time cause B to fall at a full station.
Then by Table IV, Table IX, or by formula (16) calculate the
long chord AB ; measure ABund move transit to B ; then deflect
Fig. 26.
the angle ABC= BAH when the telescope will be in tangent.
The curve may now be run both ways from B,
If it happen that some stations, as E and F in the figure, arc
still invisible, they may be located by offsets from chord or tan-
gent.
ExAMPLB.— Let the curve be a 3* curve to right ; angle HAB
= T** 30', the deflection-angle for 5 stations. By Table IV the
long chord is 498.63 feet, which can now be measured and a hub
set at B ; then making angle CBA = 7"* 30', the telescope will be
in tangent and the curve can be traced either way.
103. To Locate a Curve when the P. C. is Inaccessible.
In Fig. 27 let the P. G. at B be in-
accessible ; it is desired to reach a
point H on accessible ground.
First Method. — Assume a
point H on the curve such that a
line AH from an accessible point
A, on tangent, will clear the ob-
stacle ; for convenience H should
be at a full station. The arc BE
and central angle, which equuls
HCFf are then known. Calculate
BC= T hj (14) or (14a) ; then
since AB is known, AC, = .4J?-f
PC, is known.
Now in triangle A CH, from trig-
onometry,
tan|(A^g) ^AC-OH
taui(^ + a) "AO+OE
Fio. 27.
LOCATIOK. 69
But (h + a) = e; hence
tan 1(7* - a) = ^^,^^^ tan ic (40)
Then |(A + o) + i(h -^ a) ^ h, the larger angle, and
4(7* + fl) — i(^* — a) = a, the smaller angle. AH may be
found by the law of sines, or by drawing CE perpendicular
to AH, when
AS'= u4(7cosa+ ^^008 7* (41)
Example.— The P.C. of a 4'' curve is at sta. 141 + 25, and it
ia desired to reach the point iZ'from sta. 139 on tangent.
Suppose H be assumed to fall at sta. 147 ; the curve length ia
i = 147 — 141.25 = 5.75 chains. Then angle c = 5.75 X 4 =
23'' (/. By Table IX the tangent distance for a 1** curve is
T'l ^ 23" = 1165.8 ft.
By (14a), T = ^^^ = 291.45 ft.
Now AC = 291.45 + 225 = 516.45 ft.,
and
AC+ CH== 516.45 + 291.45 = 807.90,
while
AC- CH=z 225 ft.;
hence, by (40),
tan i(7* - a) = -^^ X 0.20345 = 0.05666 = tan r 15'.
o07,9
Therefore
A = 11" 30^ + 3" 15' = 14" 45',
and
a = 11' 30' - 8' 15' = 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 seta
hub ; move to this point, backsight to A and deflect 14* 45' into
tangent, then trace in tbe curve.
70 A FIELD-MANUAL FOB BAILROAl? ENGINEERS.
Second Method. — If F, auy assumed point io tangent, is
visible from Ay AF may be measured by some indirect method;
then AF — AB = T, The tangent for a 1** curve having same
interaection-angle, KFQ, is Tt = TxD; find this value of Ti in
Table IX and take out the corresponding value of 7. With
ti-ansit at F deflect the angle KFO, measure FG t= 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 2). 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., and BF= 490.5 - 225 := 265.5 ft.
265.5 X 4 = 1062 ft., which by Table IX is the value of 1\ for
Z= 21'. Set transit at F. deflect 21% and measure FG = 265.5 ft.
X = 2i-= 6.25 Chains;
4
hence (7 will fall at 141.25 + 5.25 = sta. 146 + 50. Move to G
and run the curve both ways.
Third Method.— In Fig. 28 let the inaccessible P. (7. be at B,
and let it be required to reach E from a point G on the curve
prolonged backwards from B,
At a given, point A on tangent cal-
culate the tangent offset by (86) or
the methods of 95, then set this off at
right angles to AB ; set the transit at
C and turn oft ACL z=^ 90* - COB,
when the telescope will be in tangent
at C COB may be found from Table
IX by multiplying AG 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
FCL, then by (16) or (16a) calculate CE, which measure. Move
to E and deflect LEG = EGL and the telescope will be in
tangent. The central angle BOE = 2LEG - BOG, from which
the arc 5^ and number of sta. ^may be found.
ExAMFLE. — Take the same example as in the last two cases.
A is at sta. 189, B at 141 + 25; hence AB = 2.25 stations.
By (36), 9 = AC=iX (2.25)» X 4 = 17.72 ft.
Fig. 28.
LOCATIOK.
71
Or by Tiible IX the angle corresponding to the long chord
(2 X 3.25) X 4 ^ 1800 ft. is 18" 4', for which the mid-ordinatc is
71 06
71.06 ft. For our 4" curve the mid-ordiuate will be ' = 17.77
ft., which equals J.C and agrees closely enough with the value
for z above.
Hake angle BAG^ 90^ and measure AC = 17.72 ft. Move
to G and sight to A, then make angle ACL = 90" — (9" 2*) =
80" 58'. Suppose an angle LCE = 16" 1' to clear the obstacle-
By formula (16).
CE = 2R sin (16" 1') = 2 X 1432.7 X 0.27592 = 790.6 ft.
Measure along CE 790.6 ft. and set a hub; move to E and run
the curve.
CE might have been found by means of Table IX, for the long
chord of a 1" curve having i = 2LC^ = 32" 2' is 3162.0 ft;
divide this by 4 and there resulu CE =: 790.5 ft.
1.04. To Pass to Tangent when the P,T, is Inaccessible.
This is just the reverse of the preceding problem, and may be
accomplished by revereing the processes described above.
When the P.T.. however, falls in or beyond a river or lake
obstructing the ordiuaiy methods of indirect measurement, the
case meiits a special solution.
First Method —In Fig. 29 let the transit be at -4, and B the
P.T. From the known station
numbers of A and B the length of
cui-ve and angle / may be found;
then, by (14), AG= R tan J/, or,
by (14a), ^(7=-^.
Move to C and deflect the angle
/: set a stake F, and one at some
other accessible point E\ measure
angle EGF^c, Move to i''' and
measure the angle EFG and the
side EF\ then in triangle ECF
angle e = 180" - (c +/); by trigo-
nometry
Fio. 20.
sino
(48)
73 A FIELD-MANDAL POE RAILROAD ENQINEEE8.
Biace BC=AO, there reaiilu BF= GF— AC; and as tbe sta-
tion number at B Is known, that at V becomes known, and the lioe
may he continued.
If Sis not theP.r.. measure back the distance FB, set transit
at B, and contiuue the curve.
Example.— Let the P. r. of a 2° C.L. fall at ata. 205 + 6C— an
Inaccessible point; supposed at sto. 200, angle e = 40°,/ = 80'
EF=%Vil\.
Here J = 5.50 K 3 = 11° 0', and « = 60'.
•a,,l-\A„\ T ""'■" _ or-K or, *,
. From (42), applying logarilhms,
log CF= 2.49136 + 9,93758 - 9.80807 = 2.02082.
Whence C'J' = 417.7 ft. Then Bi'*= 417.7 -275,87 = 141.8 ft.;
therefore the number of ,P'H'ill be 206 + 91.8.
Sbcomd Method.— In Fig. 30, with the transit aX any point A
on the curve, assume a long chord AB
and calculate the angle fJAB; deflect
this angle ftom the tangent AG, and set
a point ,£ beyond oltslrucliou; set'also
a Blnke at G in tangent.
Move to ff nnd measure AUG and
, side EG. Compute AE from the trian-
gle AEG. If tills is gi-eater or less
than the length of the long chord AB,
take (iielr difference BE mid set a hub
at B. With the transit at B trace out
the curve.
EsAHrLE.— Given A al sta. 210 of a
8° C- L,, angle a = 12°. 6 = 93*, EC
Fio. 30. „ 181 ft Tiieu 5 _ 70- and by solving
the tilangle AEG. AE= 844,7 ft. By Table IX the long eliord of
a r curve tor 7= 24° is 2383.6 ft,; therefore JB= ^?^ = 794.2
ft. Now will EB = 844.7 - 794.3 = 50.5 ft., which ia the dis-
tance along EA that transit must be moved bat^ from E.
LOCATION.
73
105. Given the Perpendicular p from a Point to a Tangent,
to Find the Point on Tangent at which to Begin a Cnrve of
Oiven Raditu which will Pass through the Given Point.
First Solution. — ^In Fig. 81 let P be the point, BPthe per-
pendicular. We have to find
BA = X.
From P draw PC parallel to
AB; then in triaogle OPG
J2» = ip« + (ij - py.
From which
X = V2i2p - p«. . (43)
Second Solution.— Consider
p = AC fxa the mid-ordiuate for
a long chord = 2x ; then pX I>
= the mid-ordinate for a 1** curve
for a central angle equal 3a.
The corresponding long chord may be taken from Table IX.
Then
Fio. 31.
(43a)
Example.— Given p = 30 ft., D = 4* (-B = 1482.7), to find x.
By (43). X = ^"85,962 - 900 = 291.65 feet.
By the second method.
80 X 4 = 120,
the mid-ordinate 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 ..^ ^ . ^
ns =z ^ K — - — = 291.6 feet.
a 4
106. In Fig. 31, Given x and p to Find the Radius of a
Onnre Tangent to AB at A and Passing through P.
From (48),
li — — s — -
2p
(44)
74 A FIELD-MANUAL FOB RAILROAD ENGINEERS.
107. Oivan the Location of a Point P raferred to the P. i.
to Find the Radius of a Otirve through P which wiU Unite
the Given Tangents.
FlO. 82L
In Fig. 82 suppose BC = h BP = m known, and angle a cal-
ciliated ; or PC and a may be measured on the field.
From triangle GAO,
J = 90** - (o + J/), and CO = R sec J/.
Now from triangle PCO,
sin y = ^ sin h.
Inserting values of PO and CO,
i2 sec J/ ,1.x sin 6 , ._
sm y = rr— '— . sm ft = sec 4/ . sm J = ^7, . (45)
M * cos J/ '
an equation from which the unknown R has disappeared. Next,
irom the same triangle, since x = 180* — (6 -|- y),
jj = ?!5_* . PC.
sm iv
(46)
"When / = 90", it can easily be shown that
R= l + m-{' V2lm (47)
LOCATIOK.
75
108. To Locate a Tangent to a Onrve from an Outside
Point.
FiBBT Method.— In Fig. 38 let P be the point and AHB the
' E H
Fio. 88.
curve. Run a trial-line PA cutting the curve in A and B,
Pleasure PA and AB ; or measure PA and angle a between the
chord AB and tangent AL. Then
AB = 2A0 = 2R sin a,
OC = B cos a.
By geometry, PB = VPA X PB, PE being the required tan-
gent. From the figure,
CO
CP
R_
PE'
tan n =
tan m =
At P deflect the angle I = m— n from PA and run the tangent.
Second Method. — In Table IX find the long chord for a
central angle 2a ; then
AB = 2AC
Jj* C7.|
C£r =
and
CO=:B-CH,
We may now proceed as before.
76 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
109. To Run a Tangent to Two Located Curves of Oontrary
Flexure.
First Cabb.— In Fig. 34 let FK and LE be the curyes, and
KL =-p measured on the ground.
M
Fio. 34.
Let FE= t be the required tangent.
Draw OiH parallel and O^JI perpendicular to FE; from the
triangle OiHOi , since FH = Hi ,
(R^ + R^-^py = (Hi + i2.)» -f t* ;
whence
i = i^2(Iii + Ii,)p + p». ... (48)
Also,
cos a =
7?i +igi»
Mi-i-Ei+p
(49)
The arcs FK and LE may be found from the angle a and the
known curvatures, after which the points i^ and Emay be set.
If i 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 and note the bearing
of the tangent to the curve at that point (see Fig. 34); the bearing
of the radius O^A differs from this by 90**. Run a line ABC of
one or more courses to intersect the other curve at G. Note the
bearings and lengths of these courses and the bearing in tangeut
at C, from which calculate the bearing of COi. Hi and H^ being
•known, the latitudes and departures are next calculated. Let O^y
LOOATIOK.
77
l>e the sum of the northings or southings, OiiVthe sum of the
eastings or westings ; from the triangle OiO^N,
tan 5 =
OxN
and
0x0^ = VOxN"" + 0-,N\
As before, FE is the required tangent and Oaff perpendicular,
-while OiH\& parallel thereto.
cos a =
0x0^
Angle F0ilf= 6 - a is the bearing of O^F, while AOtF^z
c — 5 4" ^ ^s the angle of retreat from the known point ^ to ^^
where the tangent may be run. The length of < = Oj^is
t = O^Qi sin a.
D. Change of Locat/on.
110. To Locate a Curve Parallel to a Given Curve.
Jjctp be the perpendicular between parallel tangents, and sup-
pose ABC located (see Fig. 85).
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 lii = i? -f- p, by (13),
and trace the curve from any
point, as E. Thus
50
•0^
A'R
•171 50
>0
Fio. 85.
If, however, points on the radii
through A, B, and C are w^anted,
they are gotten by using the same degree of curve D and com-:
puling the length of chord FE. From similar triaugles,
EF AB 100
Hi
li
R'
78 A FIELD-MANUAL FOR BAILROAD EKGINEEBS.
'wlience
EF=^ 100 ~ = 100 ^i^ = 10o(l + I). . (5(5
Had EFO been the located curve, with radius i?, we shouM
have had
^B = 100
(51)
111. To Change the P. (7. of a Located Curve bo that P,T.
will Fall in a Giveh Tangent Parallel to Terminal Tangent of
Located Curve.
Let AB, Fig. 86, be the lo-
cated curve; FE, the tangent
iu which the P,2\ must fall.
Let tlie distance between tan-
gents be HE = p.
Draw 5^ and 00' parallel to
AF ; eviden tly .4 C = 00' ^BE,
O being the new position of
center.
o;i::-..>-6'
Fio. 86.
In triangle BEE,
BE- AO^ -A-, = p cosec /.
sin/
(52)
Set the new P. C by measurement from A, and run the curve
CE, Any system of straight lines and curves may be treated as
above, provided 1 is the angle between initial and terminal
tangents and p as before.
EzAMFLS.— A located 2" 80' curve, having / = 25*, ends in a
tangent 25 ft. outside of desired tangent. Find the change in
position of P. (7.
By (52),
^C7 = 25 X 2.86620 = 50.16 ft
liOCATIOK.
79
112. To Find the Change In Radius and Poaition of P.(7. if
P,T, ia Required to fall on the same Radial Une but on a
Tangent distant p from, and parallel to, Terminal Tangent to
Located Curve.
In Fig. 37 let AB be the located and CE the required curve.
Draw the parallsl chords AB and
CE. Draw CHund SFperpeudicular
U>AB. Tbeangles FBE= CAH-\I^
From the figure,
BP — BE cos \I = p cos \L
Equatiug,
J C7 sin J/ = p cos i/,
whence Fig. 87.
4C7 = i)C0ti/.
(53)
In the triangle OPOi , 0,P = AC, OP^R- Bx, and
(J? — Bi) isinl= AC = p cot J/,
or Br- Bi = AC cot I = p cot J/, cot /.
Therefore
Bi = 12 - ^C7cot / = /? - p cot i/.cot /. •
From trigonometry,
(54)
cot i/=
sin /
1 — cos /
and cot / =
c6s/
sinT
Insertiug these values iu (54) ^ives
n n sin / cos / ♦ cos / _, cos /
1 — cos I sin I
1 — cos /
vers r
lYom trigonometry,
ex sec / =
vers/
cos I'
.'. i?i = 22 -
P
ex sec /*
(54)
so A FIELD-MAKUAL FOB BAILBOAD EKQIKEEBS.
EzAMFLB.— A 2** 30' curve strikes 25 ft. inside a tangent in
which the P, T, must fall. Find the necessary change in radius
and position of P,G. when / = 2o®.
By (58) the change in P. C, is
ulC = 25 X 4.51071 = 112.77 ft.
By (54'),
Rx = 2292.01 -
25
.10338
= 2050.38 ft.
By Table I we find this to be the radius of a 2' 47' 41" curve.
, 113. Qiven a Iiocated Curve uniting Two Tangents to
^ind the Change in Position of P,G. or in Radius for a Given
Ohange in the Intersection-angle.
First Case. — Radius unchanged,
E^ In Fig. 38 let BOB = / be the origi-
, y nal intersection-angle, FGE = /' the
'! '> new angle. From the figure.
or
AO = R (tan J/ - tan i/O. (55)
Fio. 88.
By Table IX.— From the table, for
angle /,
Fori'
r =
D
r' =
Then
AG^T- T*.
Second Casb.— P.C. unchanged.
Here the tangent T for the two curves is the same, and
therefore
i?, tan J/' = 2? tan J/;
.... (56>
Whence
Bi = i? tan J/- cot jr.
LOCATION. 8 1
Bt Table IX,
whence ^ = H^\ i? = ^4^.
1 14. To Find the Change in i? and P. (7. for a Qiven Ohange
in /, the P.T. remaining unchanged
^0,
Fig. 89.
Id Fig. 89, from the triangles OBQ and OxBH,
00 — B cos /
and OjJ5r= ^i cos Ji.
Now OA = EF; hence
2?i — jBi cos /i = i? — i? cos J.
Whence
1 — cos Ii vers ii ^ '
Also, F^ = HO ^BH - -5(7.
Inserting values of BE and £(?> there results
FA = Bi sin Ii - i? sin 7. (58)
115. Oiven a Located Curve to Find the Change In B for
a Given Change in 1\ I remaining unchanged.
In Fig. 40, from the triangles OAO and OiEO, sinco
M = EC^AC,
82 A FIELD-MANUAL FOR RAILROAD ENGIKEERS.
Whence
Hi tan U- i?tan |/= JK4 = T'
S^ =2?+ (2" - T)coHI.
A 0
7
-T.
(»)
Fig. 40.
By Table IX. —^^1 being known, T' ^ T+ EA. Then, by
(15a),
If the change in vertex of curve is wanted, there results, from
(25),
E- GO = T\Aii il E' =z GB = T' tfiu iL
Therefore QH= E' - E-(T' - T) tan i/. . . . (60)
(7-5" can be found from Table IX after finding Di as above.
If Ri is given and EA wanted, (59) yields
EA = T' - T^iRi" R) tan JJ.
116. To Find the Radius of a Curve having the Same P.G.
f^ C F ^^ a CMven Ourve, but ending in
a Parallel Tangent.
In Pig. 41 let the perpendicular
distance between tangents be p, and
AB be the located curve; AOi =: Ri
is required.
First Method. — Draw OB at
right angles to OiE; then
0,E= OiB+BQ+ OE,
01
Ri=:(Ri " R)coaI+R + jK
FiQ. 41.
From which i2i = i? +
_P___.
1 — cos /
B +
P
vers
-J. . , . (61)
LOCATIOK. 83
Second Method. — A, B, and E lie on the same straight line,
since / is the same for both cuvycs. In triangle BOB angle
EBO = J/, and
^^=-£j} = p<^^i^'
From Table IX, AB = ^' ^"J^ ^
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 to OiE; then
CF = -^-^ = p cosec /.
sin/ '^
From Table IX, A0=^^'^^
D '
AF^ AC-\- CF, the tangent distance for second curve ; hence
i>.=^'^^'
AF
Remark. — If transit is set up at B^ it will be well to set E
by measurement from B, to serve as a check when the curve is
ran in from A,
84 A FIELD-MANUAL FOR RAILROAD ENGINEERS.
Article 9. Compound Curves.
A, Location Problems,
117. Qiven Twa tJnequal Tangents, their Intersection-angle,
and One Radius, to Find the Other Radius of a Compound
Curve ni#ing Ttogents.
In Fig. 42, Ali= Tx and BR— Tt are the known tangents,
AOx = Ri the known radius. BO^ = i?a and the angles /i and I»
must he found before curve can be located.
Fio. 42,
Extend first branch to F, so that tangent FL is parallel to BE.
Draw UK and BO perpendicular to FL ; draw FB and extend
to J^; it will pass through the P.C.C, because the central angles
i^OiJ^and EOiB are equal. Then
To = uiZ = i2i tan il.
In triangle LHK, since LE=To--Ti,
$-EL = iTo- Ti) cos/,
p = EK=BO = (To- Ti)siul.
Now in triangle BG^-F angle BFO = K,, and
l:=.FG = T. + »^T^,
LOCATIOK. 85
tant/. = f («8)
Draw Oaif parallel to FL ; then
(Rx — R^ sin 7. s (
whence
I
sin Ji ^
Ha<} A been required, the equation would haye been
Rx^R%-\-l cosec 1%,
Evidently, /» = /--/,.
In the field the points E and B may be located by running in
tbe curve from A as starting-point, or run the chord
^i?'=2i?isinii' .
from J., and at i^ deflect angle AFB = i^— \I% = \I\ , measure
FB = I sec i/a and ^i? = 22?, sin i/j.
ExAMPLK.— A 2* curve has the P.C7. at sta. 110, Ti = 590 ft.,
Ti = 511.8 ft., 1 = 30** 50'. Locate the curve.
By "nible IX, To = 1580/2 = 790 ft.
By formulas above,
# = 200 X 0.85866 = 171.73 ft.
;? = 200 X 0.51254 = 102.51 ft.,
Z = 790 + 171.73 - 511.8 = 449.93,
tan 4/« = ^^ = 0.22784 = tan 12" 50^.
Then /» = 30* 50' - 25** 40' = 5* Itf.
i?, = 2864.93 - ^^ = 1888 feet.
86 A FIELD-MAKUAL FOR BAtLROAD ENQXNSSRS.
By Table I this is seen to be the radius of a S** 71' curve.
^ ^he 16Dgth of first branch is 258.8 feet, and of the second 821.8
feet; hence the P.C.C, falls at 113 + 58.8, while the P.T. is at
sta. 120 + 79.6.
118. Qiven the Long Chord from P. (7. to RT. of a Com-
pound Curve, the Angles it makes with the Tangent* and
One Radius, to Find the Other Radius and the Central Angles.
In Fig. 4^ 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 HB = T^ may be found,
after whU^h the solution is this same as in the last problem.
A solution may be reached in a different manner. / = a + ft,
HAF = i/ = i(» + 6), and BAF = •}(« -|- J) - a = J(6 — a).
AF = 2i?i sin |/. In triangle BAF two sides and the included
angle are now known, so BF and angle BFA may be found,-
QFB =z il, = il - BFA.
Then ffF ^ ^Bi uin 11% ,
and EB== EF- BF becomes known.
Then EB = 2i?a sin J/a = 2i?i sin }J, - BF,
whence ij. = ft _ ^_^ (64)
Evidently L = I - I^
119. Given the Radii and Central Angles of a Compound
Curve to Find the Tangent Lengths, the Long Chord from
P. (7. to F,T,i and the Angles it makes with Tangents.
In Fig. 43 draw AE and BE from the
P. a and P.r. to the P.aC7., then
calculate AE and BE by (16) or by
Table IX. In triangle AEB angle
AEB = 180 - i(/, + /,). Two sides
^ and tlie included angle being known,
the triangle AEB may bo solved for
AB and the angles ABB and BAB\
then
BAF- BAE'\-\Ix,
^»«- ^ ABF = ABE + \I%.
The angle AFB of triangle ABF now becomes known and, as
A Q
F
i
-^^
^
1
LOCATION, 87
AB is known, the sides AJ^ ss Ti and BF = Tt may be com-
puted.
120. Oiv^n tli9 I«ong Ohord from F,C. to P,T, of a Oom«
pound Oimre -and the Angles it makes with Tangents to
Find the Radii when the Common Tangent is Parallel to Long
Chord.
In Fig. 43 let OMhe parallel to AB, and GAB = a, SB A = b
known. Then
BAE = MAG = GEA = ia,
and ABS = EBH == HEB = Jd.
Also, AEB = 180* ^ l(a + &).
In triangle J.^^, remembering that
sin [180 - i(a + b)] = sin i(a + b),
ABBia^b
AB^
sin \{a + 5)'
and
BE= -^^s***^*
8in4(a + 5y
Since AOiE = a and iSTOa^ = 5, the radii Bi and 2?i may be
found from formula (16), or (I6a).
By<l«>' * = s^ =
^J? sin ib
2 sin )a . sin }(a -f- 5)
. . • . («5)
_ ffi^ _ AB sin jg
^ "" sin Jd " 2 sin 16 . sin i(a + 6)' ' ' " ^^^^
Example.— Required i?i and B» , or i>i and Dt , when iL9 ^
900 feet, a = 12% 6 = 15%
By (65), Bi = 2407.0 ft.
By (66), Bt = 1543.7 ft.
Prom Table I, i), = 2* 22' 50" and Z>, = 8' 42' 44".
88 A FIELD-MANUAL, FOR RAILROAD ENGINEERS.
A<
B, Obstacles.
121. To Locate a Point on vtkB Second Branch of a Oom-
ponnd 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 curve from that point. An obstacle on
cither 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.O.
and the second branch from the
P.T., if this latter point is known.
"When this is not the case proceed
by one of the following methods:
FiKST.
…[truncated]