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( A

FIELD-MANUAL

FOR

KAILROAD ENGINEERS,

BY

J. C. NAGLE, M.A., M.C.E.,

professor nf Civil Knaineenna if the Agricultural and Medianicat Culleye of Texas.

SECOXD KlUTIOX, HE VISED. SECOXI) T1JOLSAJ5-D.

KEW YORK:

JOHN AVILEY & SONS.

London: CHAPMAN & HALL, Limited.

1903.

Copyright, 1897

BY

J. C. NAGLE.

ROBERT DRUMMOND, ELECTROTYPER AND PRINTER, NEW YGnK.

PREFACE.

Ease of reference and uniformity of notation are essential in a

book that is to be consulted in the field. With this in mind an

effort has been made in the following pages to secure a systematic

arrangement of the subject-matter and uniformity of terms and

- notation. Except for a few cases Greek letters have been avoided

"' and a single letter is used to designate an angle. In so far as

"j)racticable each figure is intended to be self-explanatory, so that

^the explanations necessary in connection with the problems have

-> been reduced to a minimum. Algebraic equations stand each in

ra distinct line, thus rendering them more easily read.

t A knowledge of the elements of geometry and trigonometry has

^been assumed, and only in the derivation of a few formulas in

^^connection with the theory of transition-curves will any higher

mathematics be needed. But these formulas may be accepted by

the reader who is unfamiliar with the calculus without in any

way affecting his ability to understand their applications or to

follow sabst>quent reasoning.

rj 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,

5' followed by illustrative examples, so that the student need not

viose sight of the principle involved while following the solution

for a particular case. Wherever approximate solutions seemed

preferable they have also been given and their limitations pointed

'^ut.

Free use has been made of the Table of Functions of a One- degree Curve, thus reducing the labor of field computations. By defining the degree of curve with reference to short chords for

I I 45577 '"

IV PREFACE^

sharp curves and, with tables of Radii, Long Chords, Mid- ordinates, etc., based on appropriate equations the errors result- ing from assuming the radius to vary inversely with the degree of curve will generally be found to be quite small.

Chapter I gives briefly the general method of making Re- connoissance; Chapter II treats of Preliminary Surveys; while Chapter III relates to Location.

Chapter IV, on Transition-curves, follows the method adopted by Professor Crandall, and enables one to locate the transition- curve with rigid accuracy where such is necessary. Approximate methods are also given by means of which the curve may be as easily located as any of the more limited easement curves ordi- narily met with.

Chapter V, on Frogs and Switches, contains all that is necessary for their location. The formulas have been arranged to give the desired quantities in terms of the frog number whenever the re- sulting equations would be easier of application than the trigono- metric ones usually given. The turnout tables are unusually full and give not only the theoretical lead but the stub lead as well, from which the practical lead can be at once found when the length of switch-rail is known.

Chapter VI, on Construction, tells how to set slope-stakes, and gives simple methods for computing areas and volumes either directly or by the use of tables. A short table of prismoidal corrections is given for end sections level, and also a formula for three-level sections, by means of which a suitable table may be computed if desired.

The tables at the end of this book have been arranged with a view to ease of reference, for, whatever the character of the text, the chief value of a field-book must depend upon the ease with which the tables may be consulted and upon their extent and accuracy. Table IX Functions of a One-degree Curve sepa- rates the logarithmic functions on the one side from the natural functions on the other and will be of assistance in locating these tables. Table XVI Transition-curve Table reading lengthwise of the page, likewise serves to separate the trigonometric tables from the miscellaneous tables that follow.

Some engineers object to the use of logarithmic tables in the field, but for them the natural functions are at hand ; while for those who prefer logarithms the five-place tables of logarithmic sines, cosines, etc., will be found easy to consult and interpolate between.

PREFACE, V

All trigonometric tables are five-place, and others were carried to as many decimal places as their cliaracter demanded.

Tables I, III, IV, and V have been computed to agree ^vith the definition of the degree of curve requiring curves sharper than 7 to be run with chords less than 100 feet in length, as described in the text. Tables XVII and XV 111 were also com- puted expressly for this book.

Tables VI and XXVll are from electrotyj^es fram Ca' iiart's Field Book for Cioil FJngineers and were furnished by (linn k Co. Electrotypes of Tables II. X, Xll, XIII, XIX, XX, XXIV, XXV, XXVI, and also XVI this last being from Crundall's book, 'I'he Transition Curve were furnished by John Wiley & Sons.

Of the others, some were arranged from standard tables and others adapted in part and extended to increase their usefulness.

It will be noticed that vertical lines have been omitted wher- ever practicable, thus rendering it easier to refer to the tables.

Acknowledgments are due my associate. Professor D. W.

Jpence, for aid in making the tabular computations and in reading

proof.

J. C. Nagle.

College Station, Texas, May, 1897.

PREFACE TO THE SECOND EDITION.

In this edition some of the typographical and other minor errors that appeared in the first edition have been eliminated. Tables XXVIII and XXIX have been added in order to increase the use- fulness of the book, and are from electrotypes of tables in Traut- v/ine's Pocket Book. A suggestion has been made by one who has had occasion to use the tables quite freely that Table XIX be extended so as to give quantities for variations of one tenth of a foot in center heights, but such extension would have increased the size of the book unduly. When closer approximations are wanted than are given by Table XIX the area for the given center height can be taken from Table XVII and by entering Table XX irith this as argument the quantity can be at once read off. For enter heights greater than those given in Table XVII we may refer to books devoted exclusively to earthwork computations.

J. C. N.

College Station, Texas, Jauuary, 1899.

CONTENTS.

CHAPTER I.

RECONNOISSANCE. Article 1. Objects of Reconnoissance How Made.

SECTION PAGE

1. Relative Importance of tlie Work of Reconnoissance and Location.. 1

2. Object of Reconnoissance 2

3. The Instruments 2

4. Use of Maps , 4

5. Making the Reconnoissance 4

CHAPTER II.

PRELIMINARY SURVEYS.

Article 2. Objects; The Field Corps; Duties of the Chief.

6. Objects of Preliminary Surveys 6

7. The Exploration-line .- ^

8. Data Sought in Making Preliminary Surveys 7

9. The Field Corps , 7

10. The Chief of Party, Duties of 7

Article 3. The Transit Party, a. duties of the members.

11. Composition of the Transit Party fi

12. The Transitman 8

13-17. Other Members of the Party

18. Instruments S

B. TRANSIT ADJUSTMENTS THE VERNIER.

19. Kind of Transit S

20. To Adjust the Plate Levels IC

21. Parallax IC

22. To Adjust the Line of Collimation IC

23. To Adjust the Standards 11

vii

Vlll CONTENTS.

SECTION PAGE

24. To Adjust the Level on Telescope 12

25. Direct and Retiograde Verniers 13

26. The Least Count of a Vernier c....o 13

27. To Read a Vernier 14

C. ACCESSORIES.

(1) The Gradienter.

28. Description and Method ofUsing Gradienter 14

(2) The Stadia, or Telemeter.

29. Principle of the Stadia 15

30. Formula for Line of Sight Horizontal 15

31 . Formulas for Line of Sight Inclined 16

32. The Instrumental Constant, To Find .^ IT

33. Reducing the Notes 17

D. FIELD-WORK.

34. station Numbers , 18

35. Hubs or Plugs 18

36. Reference-points . . 18

37. Alignment 18

38. Form of Transit Notes , 19

39. Stadia Methods for Preliminary Surveys 19

E. OBSTACLES IN TANGENT.

41. To Pass an Obstacle by Means of Parallel Lines 20

42. To Pass an Obstacle by Angular Deflections 20

43. To Measure across a River 21

Article 4. The Level Party.

44. Make-up and Instruments 23

45. Work of the Leveler 23

46. Work of the Rodman 23

adjustments of the level.

47. To Adjust the Line of Collimation ... 23

48. To Adjust the Level-bubble 24

49. To AdjntJt the Wyes 25

B. THEORY OF LEVELING.

50. True and AuDarent Level 25

51. The Error Due to Curvature 25

52. The Difference of Elevation of Two Points 26

C. FIELD-WORK.

53. The Datum 27

54 Bench-marks 27

55 Work in the Field 28

\

CONTENTS. ix

SECTION , PAGE

56. Tbe Level Notes 28

57. Precautions when Using Level 29

58. TheRod /v;..^.. .,... 29

Article 5, The Topographic Party.

59. Instruments Used ; Area to be Mapped 30

80. Methods of Recording Data 30

61 . Topographers' Field-sheets o . 31

62. Use of the Slope-level 31

63. Cross-section Rods 32

64. The Transit and Stadia in Topographical Surveying 32

Article 6. Preliminary Estimates.

66. Map of Preliminary Lines . . 32

67. The Profile 33

68. Preliminary Estimates of Quantities 33

69. Report of the Locating Engineer 34

CHAPTER III.

LOCATION.

Article 7. Projecting Location.

70. Problems Involved in the Paper Location 35

71. Hints Regarding Methods of Projecting the Line 35

72. Tbe Curve-protractor 36

r^'i. Work in the Field 37

Article 8. Simple Curves.

A. definitions and formulas.

74. Definitions 38

75. To Find the Radius R, the Degree of Curve Being Known 40

76. To Find the Length of Curve 42

77. The Functions of a One-degree Cui've 42

79. To Find Z), -R and C Being Known 43

80. To Find the Tangent Distance T, I and B Being Known 43

81. To Find R, Given I and T 44

82. Given i and D, to Find the Long Chord i.C 44

83. Ordinates from Chord 45

84-86. To Find the External E 48

87. To Find 7?, E and 2 Given 49

88. To Find r, S and i Given 49

89. To Find the Deflection Offset from Chord Produced 49

90. To Find the Tangent Deflection Offset 50

91 . The Sub-tangential Deflection Offset 51

92. To Find the Tangent Offset z 52

33. Difference in Length of Arc and Long Chord 53

X CONTENTS.

B LOCATING SIMPLE CURVES. SECTION PAGE

94. To Locate a Curve vvitli the Chain by Offsets from Chords ProduceU 55

95. To Locate a Curve by Offsets from Tangent 57

96. To Locate a Curve by Offsets from a Long Chord 58

97. To Locate a Curve with Transit and Chain 59

98. The Index-angle 60

99. Subdeflection-angles 60

100- lOL Transit Notes 61

C. OBSTACLES.

102. To Pass an Obstacle on a Curve 63

103. To Locate a Curve when the P. C. is Inaccessible , 64

104. To Pass to Tangent when the P. T. is Inaccessible 67

105-107. To Pass a Curve through a Given Point 69

108. To Locate a Tangent to a Curve from an Outside Point . 71

109. To Run a Tangent to Two Curves of Contrary Flexure 78

D. CHANGE OF LOCATION.

110. To Locate a Curve Parallel to a Given Curve 73

111. To Change P.O. in Order to Make P.T. Fall in a Parallel Tangent. . 74 112 To Change R and P.C to make P.T. Fall in Parallel Tangent, on

Same Radial Line 75

113. To Find Change in P.C. or R for a Given Change in 7 76

114. Required the Change in P.C. and R for a Given Change in 7, the

P.T' Unchanged . ... . .. 77

115. To Find Nevv Radius for a Given Change in T 77

116. To Find New 72 to Connect P.C. with a Parallel Tangent 78

Article 9. Compound Curves.

A. location problems.

117. Given Both Tangents and One Radius, to Find the Other Radius ... 80

118. Given One Radius, the Long Chord and the Angles it Makes with

Tangents, to Find the Other Radius and Central Angles . 82

119. Given the Radii and Central Angles, to Find the Tangents, the Long

Chord, and the Angles it Makes with Tangents ,.... 82

120. Given the Long Chord and Angles Made with Tangents, to Find

Both Radii when Common Tangent is Parallel to Long Chord S3

B. obstacles.

121. To Locate Second Branch when P. C. is Inaccessible 84

C. CHANGE OF LOCATION.

122. To Compound a Simple Curve so P.T. shall Fall in a Parallel Tan-

gent.. . 85

123. To Find Change in P.CC. Necessary to xMake P.T. Fall in a Par-

allel Tangent 86

124. To Change P.CC. and Second Radius so P.T. shall Fall in a Par-

allel Tangent, on Same Radial Line , 89

CONTENTS. XI

SECTION PAGE

125. To Chanjje P.C.C. and Second Radius to Cause P. T, to Fall at a

New Point in Same Tangent 91

126. To Substitute a Three-centered Compound Curve for a Simple One. 94

127. To Substitute a Curve for a Tangent Uniting Two Curves 95

Article 10. Track Problems.

128. Reversed Curves, Where to Use 96

129. To Connect a Located Curve with an Intersecting Tangent 97

130. To Locate a Y 100

131. A Reversed Curve between Parallel Tangents 102

132. A Crossover between Parallel Tracks when a Fixed Length of Tan-

gent is Inserted , 105

133. A Reversed Curve with Unequal Angles 106

134. A Reversed Curve between Fixed Points 106

135. To Connect Two Divergent Tangents by a Reversed Curve 107

136. To Change P.jR. a so P.T. shall Fall in a Parallel Tangent.. 108

137. To Find the Radius of a Curved Track 109

CHAPTER IV.

TRANSITION-CURVES.

Article 11. Theory of the Transition-ccrve.

138. Elevation of Outer Rail on Curves 110

139 Requirements of the True Transition-curve Ill

140. Notation Employed ill

141. Equation of Transition-curve 112

142. Transition-curve Angle, / 114

143. Coordinates of Points 114

144. Deflection-angles 115

145. Explanation of Transition-curve Tables 118

146. To Unite the Brandies of a Compound Curve by a Transition-

curve 119

147. Length of Transition-curve to be Taken 121

Article 12. Field-work.

A. field formulas.

148. When to Use the Simplified Formulas 122

149. Simplified Formulas for Transition-curves 122

1.50. Offsets 124

151. Compound^^ Curves 125

B. setting out transition-curves.

153. Location by Offsets 125

154. Location i;y Deflection-angles 126

155. Form of Transit Notes for Tran.sition-curves 128

. i

Xll CONTENTS.

Article 13. Transition curve Problems.

SECTION PAGE

156. Tangent Distances and Exterual for Equal Offsets 129

157. Tangent Distances, Offsets Unequal 130

158. Transition-curves Inserted without Changing the Vertex of Cir-

cular Curve 131

159. Transition-curves Inserted with Least Deviation from Old Track.... 133

160. Transition-curves Inserted at Ends of Long Circular Curve, Cen-

tral Portion Undisturbed 133

161. Transition-curve Inserted at P.C.C. by Changing Radius of Second

Branch 136

162. To Insert Transition-curves at the Ends of Two Circular Curves

United by a Common Tangent 138

163. To Unite a Tangent and Circular Curve when the Offset Cannot be

Directly Measured . 139

164. Inserting Transition-curves in Old Track 140

165. Remarks on Tabular Interpolations 140

CHAPTER V.

FROGS AND SWITCHES.

Article 14. Turnouts.

A. turnouts from straight lines.^

166. Definitions 143

167. To Find the Lead, I, and Radius, R, in Terms of the Frog Number,

N, and Gauge, g 144

168. Given R and g, to Find N, I, and Frog-angle, F 146

169. To Find Theoretic Length of Switch-rail 146

170. To Find Lead and Number of Crotch-frog for a Double Turnout to

Opposite Sides of Main Track 147

171. To Find Turnout Radius and Lead of Crotch-frog in Terms of

Crotch-frog Number 148

172. To Find Radius of Curve from Point of Middle Frog to Point of

Main Frog, Given i\r,, N, andN' 148

173. Double Turnout to Same Side of Main Track 150

174. To Find Radius of Curve between Frog-points for a Double Turn-

out to Same Side of Main Track 151

175. To Unite Main Track with Siding. Reversing Point Opposite Frog . . 152

176. To Lay Out a Ladder-track .. 153

B. turnouts from curves.

177. To Find Lead and Radius for Turnout to Concave Side of Main

Line 154

178. To Find Lead and Radius, Turnout to Convex Side 157

179. To Find Theoretic Length of Switch-rail (17)

y is given in feet when m and n are in chains and decimals of a chain. At the mid-point F, m = n, and y = M.

.-.31= pi^D (18)

Caution. Formulas (17) and {\S), while ver}' convenient for field use in passing obstructions, are liable to error when very long chords or large values of I) are used, since Ihey give results that are too small.

If we write the arcs HN, NE for a and b, we shall get results that are too large, 3'et about Jis 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 J/ and compare with the true values taken from Table V.

Decree Length Mid-ord. Mid-ord. Mid-ord.

of of by by by

Curve. Arc. M^i{HF)-^D. M=l{HGyW. Table V.

2 2 stations. 1.75 1.75 1.75

2 G " 15.69 15.75 15.69

5 2 " 4.37 4.38 4.36

5 6 " 38.51 39.38 39.06

8 2 " 6.96 7.00 6.97

8 4 " 27.29 28.00 27.75

8 5 " 42.02 43.75 43.20

8 6 " 59.43 63.00 61.93

From this it appears we may use formula (18) and (17) as well taking eitlici- llie segments of the arc or chord for curve.'! not exceeding 4 with arcs up to 600 ft.; for curves from 4 to 6"

LOCATION. 47

tbej may be used up to 500-ft- arcs, while for curve Ijetwedu 6 !ind 8' uot more Ibrm 400 feet of arc may be takeu.

SF.co^'D Method First determine the mid-ordinate. In triangle OEF,

0F= \^W - \(T-\ then

M=FG = R- VR' - \G' (19)

To find ordinate J.C distant d from the mid-point ot EH, draw OB=d parallel to HE; draw AB at right angles to HE. Then

BA = x^R' Therefore

CA = y= VR' - d^ -> \/R' - \0. . . . (20)

Third Method. If the chord C is short, we mav resrard the arc as an arc of a parabola, for which it is kuo-A-n that ordi- nal es vary as the product of the segments into which they divide the chord. The mid ordinate being known, we have

..y=4-^. (21)

From formula (b) we have for y = M, a = b = ^C,

The mid-ordinate for any other chord C" is

Hence

M~ C

.-. M, = m{^J (23)

If C = ^C, this gives

M,=iJtr. (23)

48 A FIELD-MANUAL FOR RAILROAD ENGINEERS.

This last relation affords an easy method of staking out a curve when the mid-ordiuate of a given chord has been determined. First erect the ordinate iLT at the midpoint of the chord; then join the ends of chord with the extremity of the ordinate just measured; the lengths of these chords do not differ much from IC; 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 i? and / to Find the External E. In Fig. n E=GB=OB- OG. But OB=R sec \I and OG = R.

. . E=.R{sec \l-\) = R ex sec i/. . . . (24)

By Table IX. Find E for a 1 curve for an intersection angle I; then

E =

D

(24a)

85. Given T and / to Find E.

In Fig. 17 draw BG perpendicular to AB, and produce AG \,o

b/

intersect 5Cat C. BG\% parallel to AG, and the triangles AGO and GBCfiva similar; hence BC = BG = E. In the right triangle ABC, angle BAG=\BAF= \I. Therefore

^ = r tan \L

(25)

Exercise. Derive equation (25) from (24).

LOCATION.

49

86. Given 3/ and /to Find E. FroDi trigonometry,

IT 1

sec 4/ = r>.

' COS \I

Insert this in (24) and we get

.1 cos 47

E=R

cos |/

(a)

But from Fig. 17, M = R{\ - cos \I), Substitute in {a) :

M

E =

cos \I

= M sec 4/.

(26)

87. Given ^and /to Find i?.

From (24),

E E

R =

sec |/ 1 ex sec ^/

88. Given /and ^ to Find /. From (25),

_, cos 4/

= E 5_

vers |/

* s

(27)

(28

89. Given the Chord C and Degree of Curve D to Find the Chord Deflection Offset d. In Fig. 18 extend EA to H, making AH = EA = AB; join

"O Fig. 18. U and 5 and draw AK to the mid-point of HB. Then

/Zir = KB= C sin i/). .-. d= IJB = 20 sin W

(29)

50 A FIELD-MANUAL FOR RAILROAD ENGINEEHS.

When C = 100',

d = 200 sin ID (29')

If we write sin 1^ = ^ from (12) in formula (29), there results

^ = -^ (30)

For curves up to 7^ C = 100'; hence

^ 10000

^=-E- (3^')

For curves from 7" to U\ C = 50'; therefore

.

2 X 5730 When C = 50 feet, (33) yields

t = .218Z> = .436 X

D

(33)

(33')

(33")

Example. Fiud t for a 6 curve, G = 100 ft. By r32') t = 200 sin 1 30' = 5.24 ft.

By (33'), = .873 X 6 = 5.24 ft.

91. To Find the Subtangential Deflection Offset i' for a Subchord C"

First Method. By formula (13) fiud the angle at the center subtended by the subchord C; call this angle D'. From (32),

i' = 2C' sin^D'. . (34)

Second Method. In Fig. 19, with ^as center strike the arcs FG and AH, taking EF = C and EA= C; prolong EG to B. Now assuming that the chords C and G are proportional to their central angles we have

From the similar sectors EFO and EAB, since EB = G,

Fia, 19.

AB

C'

t' '

(*)

^o

>'Z A FIELD-MANUAL FOR RAILROAD ENGINEERS.

MiiltiplyiDg (a) and (b) together, term by term,

Whence

C_l C^ C'~ t' ' C

'-{?)

(35)

Example. Find t' for a 7 curve when (7 = 60 ft.

Here

ly

By (34),

t'

By (32'),

t

By (35),

t'

= ^ X 7 (very nearly) = 4' 12'.

= 2 X 60 X 0.01832 = 2.20 ft. = 6.11 ft.

92. To Find the Tangent Offset z.

In Fig. 20, EBz is the required offset. Let AE= n chains =

lOOn feet. AE=FB, the half-chord having the mid-ordinate AF = EB , hence we have, by formula (18),

2 = In-I). . . . (36)

In this formula we may take n to he either the length of AE ov the arc AB, in chains. If taken equal to AE the offsets will be slightly too small, while if taken equal to AB they will be a little too large. The use of the formula is limited to small values of n and B, 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 71 may be as great as 3, but for sharper curves it should

be less.

Example. Find six offsets to a 4 curve at points 50 ft. apart,

measured around the curve.

Fig. 20.

LOCATION.

53

By successive applicatious of (36) we have

for 71 n n n n n

1

= 1, 2 =

3

2'

9

5

= |X i X4 =

= 3, 2 =

I X 1 X 4

i X I X 4

i X 4x4

I X -f - X 4

i X 9X4

0.88 feet

3 50

7.88 14.00 21.88 31.50

The last vahie of z is iu error by about 0.2 ft , but for seltiug stakes on coii.structiou this difference is not material so long as the alignment beyond this point does not depend on it. In setting irack-cenlers the completed road-bed is available and the stakes may be set with the transit, in the usual way.

93. Diflference in Length of a Circular Arc and its Long Chord. First Method. Let the central angle be a degrees. By (13), c

sin \a =

^R

Changing degrees to circular measure, a (in n meas.) =

: . The length of arc is Ra = R^ir^- Then 57. o 57. o

It a 180

a

Arc chord = Rzrrr^ ^

5

Then will 26 - c = 2^ = (38)

From Huygens' approximation to the length of a circular arc

(see Williamsou's Differential Calculus, p. 66), arc = ^ .

o

Therefore

Arc chord = c = |(25 c). . . (c)

o

Inserting the value of 2o> c from (38) gives

8i/'- Arc chord = ^ {d)

When the arc is not very great we may write c = 100/h , where 7ii is the number of chains contained in the arc AE. From (18), remembering that rii = 2n,

M = 0.21Snr'J). Inserting these values of c and M\n (d),

Arc - chord = | ^^^^""-^ = ^^n,WK nearly. . (39)

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, c = 595.74.

By (37),

Arc - chord = 1432.7 X ^ - 595.74 = 4.34 ft.

By (39).

. , - 6X6X6X4X4 .._.

Arc - chord = ^;r = 4.32 ft

Remark. Formula (38) is interesting as showing what a com-

LOCATION.

55

paratively small iucrease iu length of liuo is caused by a consid- erable lateral deflection in alignment. For instance, a lateral deflection of 2000 feet is made at the mid-point of a line 40,000 feet long ; what will be the increase in length?

By (38) the increase is )^^ ^^^J^ = 200 feet, giving for the

40,000

increased length 40,200 feet.

B. Locating Simple Curves.

94. To Locate a Curve with the Chain by Oft*ets from Chords Produced. In Fig. 21 let the P. C. fall at B. If BC is a full chain, prolong

Fig. 21.

the tangent AB to if, making 55"= BC , HO will equal t, which may be calculated by (32) or (33'). With 5 as center, strike an arc with radius BH, and with H as center and t as radius strike an arc , at G, where these arcs intersect, set a stake. Produce BC to K, making CK = BC = CD \ 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.C. does not fall at a full station ; then HC = t', which may be found by (34) or (35). Using this value of t\ we locate as above. At B make BB = t', and prolong RG to L ; make LB = t and set a stake at Z>. EM will equal d, and may be located as before.

We may regard KD as equal to KL -{- t, and, finding, KL,

56 A FIFLD-MANUAL FOR RAILROAD ENGIXEERS.

measure KD and set D without locatiug R. To do this we have the similar triangles BHC and CKL, from which

KL t'

CK BC and therefore, since KG = CD,

In like manner at ^ we have

PN= t^, and FP = W hence

Make EQ^ = ti , prolong QF, and wc have the tangent at F. Example. Given the P.C. of a 5" curve at 106 + 20 and the angle of intersection 22, to locate the curve.

22 Here L = -^ = 4.4 stations.

5

Therefore the number of the P. T. is

106.20 + 4.4 = sta. 110 -f 60.

BCin this case is 80 ft., and by (33')

t = 0.873 X 5 = 4.37 ft.

By (35), ^=4.37X (1^^=2.80 ft.

Set off HC - 2.80 ft., and at B make

KB = 2.80 X ^ + 4.37 = 7.87 ft.

At E make ME = d = 8.72 by (31). This will be at sta. 109 ; at 110 set a stake by offsetting 8.72 ft. The last chord is 60 long, and hence the offset

NF= 4.37 X Y^ + 4.37 X ' ^^^)'= 2.62 + 1.57 = 4.19 ft. Make .fi;^ = 1.57 ft , and prolong QF, the terminal tnngcnt.

LOCATION".

57

H-

95. To Locate a D Degree Curve by Oflfsets from Tangent.

Let AM, Fig. 22, be tangent at A, and E, F, G, etc., points on tbe curve. Tbe offsets BE, CF, ^ B

etc., may be found from formula l (36),

eitber by taking equal intervals, ^ AB, BC, CM along tbe tangent or by taking E, F, O, etc., at regular stations around tbe curve and using tbe arc lengtb instead of tbe tangent.

Wben tbe arc AO is large, or strict accuracy is required, we proceed to find tbe offsets at regular stations and tbe lengtbs of AB, AG, etc. First find R from (12) or (12'); tben from triangle OEL,

Fig. 23.

BE= AL = R{1 - cos D) = R vers D,

AB = LE= R sin D. lu like manner

OF = AH = R{\ - cos 22)) = R vers 2D, AC= HF = R sin 2D,

and so on for any number of stations.

Sbould A fall at a plus station, we first find tbe angle Bi at tbe center, tben

BE = R vers Bi ,

AB= R sin P, , CF = R vers {Br + D), AC= i?sin(A + B), etc. = etc.

The ordinates BE, CF, etc., are evidently equal to tbe mid- ordinates for long chords 2LE, 2IIF, etc.; hence we can, if A, E, F, and 0, 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.

58 A FIELD-MAJsUAL FOR RAILROAD ENGINEERS.

Example. Locate three stations of a 4' curve by offsets every 50 ft. on curve.

Referring to Table V, the required offsets are 0.87, 3.49, 7.85, 13.94, 21.77, and 31.31. By Table lY the distances measured along tangent are 50.0, 99.94, 149.76, 199.39, 248.78, and 297.87. With these values we can set out the curve either way from A.

Had we used formula (36) we should have had for the values of the offsets 0.87, 3.50, 7.88, 14.00, 21.87, and 31.50.

96. To Locate a Curve by Offsets from a given Long Chord.

Let FK, Fig. 23, be the given chord. We may compute the offsets yi,y^. . .Mhy the methods of 83 of which formula (17),

y = ImnD,

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 J/ = i^^ersi^O^ at A, and at C set off ^2 = iW 7? vers D, making

AC = HL = R sin D.

GE will be

yi - M - B vers 2 A find AE = 7? sin 22).

Anotheii Method is to find the angle A'Oi^at the center, and by Table IX delennine BA - M ; then by Tables V and IV

LOCATION. 59

delermino BL, BN, Lll, and J^^G. Then HC = M - BL, which scL oil" lit C, aud other points in like manner.

Example. Given the P.C. of a 4 curve at station 160 + 75, the angle between tangent and chord = 9, required the offsets necessai-y to locate the curve.

Here 7=2x9 = 18.

18 ,'. L -r = 4.50 stations. 4

Hence the P.T. falls at 160.75 + 4.50 r= sta. 165 + 25. The mid-point on curve B falls at sta. 163. By Table IX,

ri

M='-^ = 17.64 ft. 4

By Table V 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, IlL = AC = 99.94 ft.

Measure AG = 99.94 ft., and set off CII= 14.15 ft., and drive a stake at //. In like manner lind

GE=d.lO and ^1^= 199.39 ft.

The points P and Q are also located by means of the coordi- hates just determined.

If B had fallen at an odd station, the curve could have been located in the same manner, 7/ and P being 100 ft. from B, G and Q 200, etc.

97. To Locate a Curve with Transit and Chain when the Degree D or Radius E is Known.

If R is given, determine 7> by (13); then, since the angle in the circumference of a circle is half the angle at the center sub- tended by the same chord, we may locate points on the curve by successive deflections from the tangent.

In Fig. 24 let the P C. be at A, at which point set the transit, and with the vernier plates clamped at zero place the telescope in tangent either by sighting the P.I. or by backsighting to some point in the tangent Deflect from the tangent half the angle at the center for the sub-chord or chord, and direct the head chain man into line while the real chainnian holds his end of the chain

60 A FIELD-MANTAL FOR RAILROAD ENCtINEERS.

at Ihe transit, the cbaiu beiug kepi taut. The stakeuiau drives a stake at the point where the head chainman's Q.i\g rested, aud the rear chainman advances to this point. Deflect iD from the chord AB just run, aud 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 ID for each chord length measured, until a point E is reached to which it is desirable to

Fig. '-M.

move the transit. The angle FAE shouM 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.C. and any other line passing through a point on the curve when the telescope is directed along this line. It is most frequently taken as the angle between the initial and any subsequent tangent to the curve. Thus at E the index-angle equals EFP = 2FAE. At any point on the curve the index-reading in tangent may be found by the following rule, which may be easily deduced from a figure:

From double ihe index-angle that fixed the point subtract the index- angle in tangent at the last point; the remainder is the index-angle required.

99. Subdeflection-angles may be found by (13) ligidly, or approximately (and with sufficient accuracy except when D is very large) by assuming the central angles to be proportional to their chords. Thus on a 4 curve the central angle for a sub-chord of 25 ft. would be V, and the subdefleclioiianirle 30'.

LOCATIOK. 61

Example. Locaie a V curve to left when the P.C. is at sla. 81 -f- 25 and I=H2' 86'.

32.6 Here L - ," = 8.15 chains.

4

Hence the P. T. will fa.l ac 81.25 -j- 8.15 = sta. 89 + 40. The rst sub-chord found by (13).

first sub-chord is 75 ft. lon^, and the tirst deflection-angle will be

^'^* = im7-2"

i(5 = V 30'.

By the approximate rule, since ^D = 2",

45 _ 75 2 100'

whence i5 = 2 X I = T 30' as before.

With transit at P.C. deflect 1 30' from tangent, measure 75 feet, and set stu. 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 7 30' backsight to 81 + 25. Reverse telescope and set vernier at 15 00', when the telescope will be in tangent. An index angle of 17' will fix 86, and so on.

The last chord will be only 40 feet long, for which the sub- deflection-angle is {^ of 2^ that is, 48'. The index-angle fixing the P.T. is therefore 23 48'.

To get in tangent at 89 -f 40 backsight to sta. 85, with vernier at 23 48' ; then by the rule of 98 the index-reading is (23 48') X 2 15' = 32' 36' = /. Set the vernier at this reading and run tangent.

Caution It is not good practice to set more than 4 or 5 sta- tions on curve from any one point. Mr. Siiunk gives the limit- ing angle to be deflected from tangent as 20% and sa^s 15 should rarel}' be exceeded. {Field Engineer, p. 82.)

100. The Transit Notes may be conveniently kept in the form below, which shows the notes for the last example.

When possible the tangents should be run to intersection, the angle 1 measured, and the t:ingent distance calculated. Then

62 A FIELD-MANUAL FOR RAILROAD ENGINEERS

a

O -

, ii)

a

'Z a>

z

X c

y. ^

cs j2

Oj 03

Station.

C TO

i-iH

II

Remarks.

90

+40

QP.T.

048'

23 48'

32 36'

N 3736' E

N 2730' E

89

23 0'

88

21 0'

87

19 0'

86

17 0'

85

O

7 30'

15 C

84

.5 30'

83

2 0'

3 30'

82

130'

130'

4 C.L.; P.I. set.

-f25

OP.C.4C.L.

0 0'

0 0'

0 0'

I = 32 36'; T ~ 418.9 ft.

81

N 6012' E

N6010'E

measure along tangents and set P.C. and P.T. from the P. I. When the curve is run in, the position of the P.T. thus found should agree with the one set from the P.I. If the error is greater than the circumstances of the case permit, the curve must be rerun and tanircnts remeasured.

101. Another Form of Notes, and in some respects a better one than the above, is given below. Tl^e index-readings are com- puted as though the entire curve v^-ere run from the P.C. The notes for ihe last example would appear as below :

Station.

otal ngle.

3 5

Remarks.

M p

E-)

For transit at mid-point of transition-curve n" = |, and, from (153),

7 (^i) = -5-(^' 4- i + i^) - correction,

or

(5^) = ^-^A^ -B^ (156)

TRANSITION-CURVES.

117

For transit at three-quarter poiut n" = | and

or

(^D = -t(^*' + T5 + f^) - correction.

(5j) = ^A^ B^ (157)

For transit at P.C.i 7i" = 1 and

//

or

(5x) = -^(w' + 1 + w) - correction, o

/i'

(5.-) = :^^, -5,.

(158)

With the transit at the P.T.d it will frequently be most con- venient to measure the deflections from the tangent to the circular curve at that point. Sometimes this will also be the case for the transit at the P.C.i.

Bv reference to Fig. 68 it will be seen that for the transit at B

the deflection from the tangent BC which serves to fix any point on the curve, as .6, is given by the equation

or, in general,

(