CURV E S
R A I L R OA D
EA R T H W OR K
C
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F RA NK A L LEN, S E
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M E M B ER A M ER I C A N S OC I ET Y O F C I V I L E N G IN EER S
P R O F E SS O R O F RAI L R O AD E NG I N EE R I N G I N T H E M A S S A CH U S ET T S
I N S T I T U TE O F TECH N O L O G Y
M C G RA W— HI LL B O O K C OM P A NY
NE W Y OR K : 2 3 9
LONDON : 6
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W E ST 39T H ST R EET
ERI E ST E C 4
8 B OU V
1 92 0
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I NC
o
PRE FACE
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for t h e u s e of th e s tu den t s i n th e
I t h a s bee n u s ed i n litho g r a phed s heets f or a
a u thor s cl a s s e s
n u mber of ye a r s i n v ery n e a rly the pre s e n t for m a n d h a s g i ve n
A n e ffort
s a t i s f a ctio n su fficie n t to s u gg e s t p u tti n g it i n pri n t
h as bee n ma de to h a ve the d e mo n s tr a tio n s s i mple a n d d i rect
a n d s peci a l c a re h a s bee n g i ve n to the a rr a n g e me n t a n d the
typo g r a phy i n order to s ec u re cle a rn e s s a n d co n c i s e n e ss o f
ma the ma tic a l s t a te me n t M u ch o f the ma teri a l i n th e e a rlier
p a rt of the book i s n eces sa rily s i mil a r t o th a t fo u n d i n on e or
more of s e ver a l excelle n t field books a ltho u gh the me thod s of
de mo n s tra tio n a r e i n ma n y c a s e s n ew Thi s w ill b e fou n d tru e
e s peci a lly i n C o mpo u n d Cu r ve s for which s i mple t re a tme n t
h a s bee n fo u n d qu ite po s s ible
Ne w ma teri a l will be fo u n d i n
”
“
the ch a pters on Tu r n o u ts a n d on Y Tra ck s a n d Cro ss i n gs
The S pir a l E a s e me n t Cu r ve i s tre a ted ori g i n a lly The ch a pter s
o n E a rthwork a r e e s s e n ti a lly n e w
they i n cl u de S ta k i n g O u t ;
C o mp u ta tio n directly a n d w i t h T a b l e s a n d D i a g r a ms ; a l s o
H a u l tre a ted ordi n a rily a n d by M a ss D i a g r a m
M o s t of the
ma teri a l rel a ti n g t o E a rthwork i s n ot el s ewhere re a dily a va il able
for s tu de n ts u s e
The book h a s been wr itte n e spe c i a lly t o meet the n e ed s of
s t u de n t s i n e n g in eeri n g colle g e s b u t i t i s prob a ble th a t it will
be fo u n d u s ef u l by ma n y e n g i n ee r s i n pra c t ice T h e s i z e of
p a g e a llow s it to be u s ed a s a pocket book i n the field It i s
di ffi cu lt to a v oid typo gr a phi c a l a n d cleric a l error s ; the a u thor
will co n s ider it a fa vor if he is n otified of a n y errors fo u n d to
Tm s b o ok w a s pr e p a r e d
’
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C
B
O S TON
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Sep te mber , 1899
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F R ANK A LL E
N
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PRE FACE T O FIFTH E DITI ON
re vi s io n of this edi tio n h a s bee n exte ns i ve F ew p a g e s
de a li n g with cu r ves h a v e es c a ped s o me ch a n g e I n co ns ider
a ble p a rt i t h as bee n a ma tter of refi ni n g or cle a ri n g u p poi n ts
s how n by t e a chi n g t o a dmi t of i mpro v e me n t
A co n s ider a ble
a mo u n t of n ew ma teri a l h a s bee n a dded a n d a few less i mpor
t a n t proble ms omitted by re a rr a n g e me n t a n d co n de n s a tio n i n
pl a ce s the s iz e of th e book h a s n ot bee n a ppreci a bly i n cre a s ed
The ch a pter on T u r n o u t s h a s bee n a lmo s t co mpletely rewr itte n
r a ilro a d pr a ctice i n T u r n o u ts h a s pro g re ss ed ma teri a lly i n l a te
ye a rs a n d co mplete re vi s io n of this ch a pter s ee med a d vi s a ble
The ch a pter on C o n n ecti n g Tr a cks a n d Cro ss i n gs h a s bee n ma
t er i a lly exte n ded
The ch a pte r on S pir a ls h a s l a rg ely bee n
rewritte n a n d a d a pt ed t o the u s e of the S pir a l of th e A mer ic a n
R a ilw a y E n gi n eeri n g As s oci a tio n the merits of which a ppe a l
to the a u thor a s ide fro m the o ffici a l s a n ctio n which es t a bli s hes
it a s s t a n d a rd A few b u t n ot ma n y i mporta n t ch a n g es h a ve
bee n ma de i n the ch a pters on E a rth work
I t i s s til l tr u e th a t while t hi s text w as prep a red p r i ma rily for
s tu de n ts n e v erthele ss this book h a s pro v ed to be well a d a p te d
t o the req u ire me n ts of t h e pr a ctici n g r a ilro a d e n g i n eer or
other e n g i n eer w h o h a s t o de a l wi th cu r ve s or with e a rthwork
co mp u t a tio n
Ja n u a ry 19 14
C F R A N K ALLE N
T HE
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TO SI XTH E DI TI ON
PRE FACE
HI G H W
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A Y pr ctice follow
a
s,
ma n y w a ys , r a ilro a d pr a ctice
in
l a yi n g o u t cu r ve s a n d co mp u t i n g e a rthwork b u t there a r e
s o me fe a t u re s of differe n ce
a n d the s u bject of C irc ul a r A rc s
whi ch recei v ed ori g i n a l tre a tme n t i n the l a s t edi tio n h a s bee n
c a rr ied f u rther
I n the co mp u t a tio n of e a rthwork s o me method s n ew to text
books h a v e bee n a dded ; the s e h a v e co me to the a u thor fro m
”
the pr a ctice i n va l u a tio n work
”
ha u l
Ma n y p a g es h a v e
A few p a g e s h a v e bee n a dded on
h a d perfecti n g ch a n g e s a n d s i mpler tre a t me n t o f s o me s u bjects
h a s bee n fo u n d worth w hile
A pril 192 0
C F R A N K ALLE N
in
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CONT E NTS
C HAP TER I
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R E C O N N O I S S A NC E
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S E CTI ON
1—2
.
Oper a tio n s i n loc a tio n Reco n n oi s s a n ce
F e a t u re s of topo g r a phy
Na t u re of ex a mi a tio n
E lev a tio s how t a ke n
P u rpo s e s of reco n oi s s a n ce
Import a n ce of reco n n oi s s a n ce
Pocket i n s tr u me n t s
.
n
5 —6
.
7 -8
.
.
n
n
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,
.
C HAP TER II
PR E L IM I N A R Y S UR VE Y
.
.
Gr a de s
Na t u re of preli mi n a ry
Importa n ce of l ow g r a de s P u s her g r a de s
P u rpo s e s of preli mi n a ry s u r v ey
M ethod s
Na t u re
B a cki n g u p ; a ltern a te li n e s
Note s
.
.
.
.
.
8— 9
.
10—1 1
.
Org a n i z a tion of pa rty Loc a ti n g en g i n eer
Tra n s it ma n ; a l s o for m of n otes He a d
B a ck fl a g
Re a r ch a i n ma n
Axe ma n
S t a k e ma n
Le v eler ; a l s o for m of n ote s
Rod ma n
Topo g ra pher Preli mi n a ry by s ta di a
11
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-
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—13
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1 4—15
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16— 17
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C HAPTER III
L O C A T I O N S UR VE Y
.
3 1—33
.
Na t u re of
.
loc a tio n F ir s t method S eco d method
Lo g t a n g e n t s
T a n g e n t fro m broke n l i n e of preli mi a ry
M ethod of s t a ki n g o u t loc a tio n
.
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n
n
n
C HAPTER IV
S I M L E C UR VE S
.
P
37 - 39
.
efi n itio n s M e a s u re me n ts Deg ree of c u rv e
F or m u l a s for de g ree a n d ra di u s
Ta n g e n t di s t a n ce T Al s o a pproxi ma te method
D
.
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.
43
.
.
13—14
.
Con ten ts
VI
.
S E CT I O N
xtern a l di s ta n ce E M iddle ordi n a te M Chord C
48 F or mu l a s for R a d D i n ter ms of T E M C I
49—5 1 S b c hord c
Le g th of ou r v e L
S u b a gle d
5 2—5 3 F ield w o k of fi di g P C a n d P T with ex a mple
5 4 55 M ethod of de fl ectio a g le s
56 57 De fl ectio n a g le s for s i mple c u r v e s
58—59 E x a mple
C a u tio n
60 Whe n e tire c r v e c a n n ot be l a id fro m P C
6 1 Whe n tr a n s it i s o c u r v e a n d P C n o t v i s i b le
62 Whe e n tire c u r v e i s V i s ible fro m P T
6 3 M etric c u r v e s
64 F or m of tr a s it book for s i mple c u r v e s
65 Circ u l a r a rc s with exa mple s
66 6 7 M ethod s of o fis e t s fro m the t a g e t a d fieldwork
4 0 41
68 M ethod of d e fl ectio n di s ta ce s
42
6 9 —7 0 Off s et s bet w ee two c u rv e s a n d for s e er a l s t a tio s
43
44
7 1 7 2 D e fl ectio n di s t a ce s for c r v e s with s b chord s
45
7 3 Appro x i ma te s ol u tio for ri g ht tri a n g le s
45
7 4 F ieldwork for de fl ectio di s t a n ce s
46
D e fl ectio n di s t a n ce s w ith s hort s b chord
7 5 7 6 Ca u tio n
47 4 9
Ordi n a te s a t a y poi n t
7 7 84 Mi ddle ordi n a te s
85 F i n d a s erie s of poi n t s by middle ordi n a te s
50 52
86 88 S u bs tit u te n e w c u r v e s to e d i n p a r a llel t a n g e n t s
C u rv e to j oi n t a n g e n t s a d p a s s thro g h g i v en poi t 5 2 5 3
5 3—5 4
9 1 93 F i n d where g i e c u r v e a n d g i v e li n e i n ter s ect
54—5 5
Ta n g e n t fro m c u rv e to g i v e n poi t
56
Ta n g e t to two c u r v es
56 57
96 99 Ob s t a cle s i n r u n n i n g c u r v e s
45 — 47
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E
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-
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C HAP TER V
.
C O M OUN C UR VE S
D
P
.
e itio n s F ieldwork Da t a
G i v e R1 R 1 1 I ; req u ired I Tl T
G i e n T R I I ; req u ired T1 B r I :
G i e n TI R t I t I ; requ ired T R I
R 1 I ; req u ired T1 I t I
G i en T R
G i v e T R L I ; req u ired I 1 T1 R;
R1
G i e T1 T R I ; req u ired I t I
G i e Tl R 1 R I ; req u ir ed I I 1 T
D fin
n
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,
,
,
8,
,
8 ,
, ,
, ,
,
,
,
,
,
, ,
, ,
, ,
3 ,
n
8 ,
8 ,
,
v
n
,
, ,
, ,
v
n
,
,
v
v
v
,
,
,
,
,
,
,
,
,
,
s,
, ,
,
,
Con ten ts
VI1
.
i e
req u ired I T R
Gi v e n T1 T R 1 I ; req i red I 1 1 R
lo g chord a g le s a n d R ; req ired I z I I R;
Gi ve
lo g chord a g le s a n d R1 ; req ired I t I I R
Giv e
S u b s tit u te for s i mple c u r v e a co mpo u d c r v e to e d
i p a r a llel t a n g e n t
G i v e n s i mple c u rv e ; req u ired r a di u s of s eco n d c u rv e
to e n d i n p a r a llel t a n g en t
of s eco d c u rv e
G i v e n s i mple c u r v e ; req u ired
to e n d i n p a r a llel t a g e t
Ch a n g e P C C to e n d i p a ra llel t a g e n t
G v n T1, R 1, I 1, I ;
8,
,
n
n
n ,
n
, ,
u
,
n
,
8
,
8
s,
u
s
,
n
,
8 ,
u
,
n
,
s,
,
,
s ,
,
u
,
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n
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n
n
115—118
.
n
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n
C HAP TER VI
R EVE R S E C U RVE S
.
D
1 1 9—1 24
.
125 —126
.
127
.
12 8
.
e er s ed c u r ve s between pa r a llel t a n g e n t s
66—68
G i e TI R I R 2 I ; req ired I I I 2 T2
68 6 9
F i n d co m mo r a di s to co n n ect t a g e n t s n o t p a r a llel
69
G i ve n u eq a l r a d ii a n d t a g e n t s n o t p a r a llel ; r e
q u ired cen tr a l a g le s
R
v
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v
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—
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u
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n
,
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C HAP TER V I I
PA R A O L I C C U R E S
.
V
B
Us e of p a r a bolic c u r v e s
.
ropertie s of the p a r a bol a 7 1
L a y o u t p a r a bol a by off s et s fro m t a n g e n t
7 2—7 3
La y ou t p a r a bol a by middle ord i n a te s
74
Vertica l c u r ve s ; method s ; le n g th s
7 4 78
.
P
-
C HAP TER V III
T U R N OU T s
139
.
Defi n itio n s
.
142—143
.
144—146
.
147
.
148—149
.
.
N u mber of fro g
i d fro g a g le fro m n u mber of frog
S plit s witch ; de s criptio
Ra di u s a d le a d le n g th s of clo s u re r a il s
82—83
Co o rdi a te s to c u r v ed r a il s
Al s o pr a ctic a l le a d s
84—85
Method s of l a yi g o u t li n e beyo d frog
86
T u r o u t s ; co ordi a te s of poi t wh ere c u rv e pro
d ce d b a ckw a rd beco me s p a r a ll el to ma i n tr a ck
87 88
M ethod s of co ecti g p a r a llel tr a ck s by t u r o t s 8 8 90
F n
14 1
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n
n
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n
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u
150 —155
.
-
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nn
n
n
u
-
Con ten ts
.
t b s witch tu r n ou t s
S tu b s witch t u r o u t s for c u rv ed tr a ck s
Spli t s witch t u r n o u t s for c u r ved tr a ck s
R a di u s of t u rn o u t beyo n d frog fro m c u rv ed ma i n
tra ck to p a ra llel tr a ck
La dder a n d body tr a ck s
C ro s s o v er bet w een c u r ved p a ra llel tr a ck s
C ro s s o v er betwee n s tr a i g ht tr a ck s n o t p a r a llel
r a dii n ot eq u a l
Three throw or ta n dem split s wi tch
S u
n
.
-
,
,
-
C HAP TER IX
C O NN ECT I N T RA C S A N C R O SS I NG S
.
K
G
D
.
tr a ck s defin ition
Y tr a ck s co n n ecti g br a n ch tr a ck s
C ro s s i g of two c u r v ed tr a ck s
Cro s s i g of t a g e n t a n d c u r v e
C ro s s i g of two s tr a i g ht tr a ck s ; s lip s witch
Tu rn o t co n ecti n g two s tr a i g ht tr a ck s cro s s i g
T u rn o u t fro m s tra i g ht ma i tra ck to s tra i g ht
br a ch tr a ck
Tu rn o u t fro m c u r ved ma i n tr a ck to s tr a i g ht bra ch
tr a ck
Tu r o t fro m s tr a i g ht ma i tra ck to c u rv ed bra n ch
tr a ck
Tu rn o u t co n n ecti n g two ma i n tra ck s o e s tra i ght
the other c u r ved
Tu rn o u t con n ecti n g two c u r ved ma in tra ck s
Y
,
n
n
n
n
n
u
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n
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n
n
n
n
u
n
n
,
C HAP TER X
S I R AL EAS E M E N T C UR VE
,
.
P
.
lev a tio n of o u ter r a il ; n ece s s ity for spir a l
1 15
E q u a tio s for c u bic p a r a bol a a n d c u bic spir a l
116—1 17
P ropertie s of s pir a l with f u n d a men t a l for mu l a s 118—119
’
Am Ry E g A ss n s pira l ; de s cri p tio n ; for mu l a s 120 —12 1
Ta n g e n t di s t a n ce s circle with spira l s ; exa mple 122—123
124
G i v e n D t ; req u ired p q s
124—125
G i v e n D p ; req u ired other d a t a
126
F ieldwork for s pir a l s a n d c u r ve
E
n
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c,
c,
o
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Con ten ts
1x
.
PAGE
12 7
yi n g ou t s pir a l by o ff s ets fro m t a n g en t
12 8—12 9
La yi n g o u t Spir a l ; tr a s it a t i n ter medi a te poi t
130
Expl a n a tio n of cert a i n A R E A s pir a l for mu l a s
1 3 1 13 2
S pir a l s for co mpo u n d c u r v e s
1 32
Le n g th s of s pir a l s
S u b s tit u te s i mple c u r v e with s pir a l s for t a n g e n t
co ecti n g t wo s i mple c u r v e s
S u b s tit u te c u r v e with s pir a l s for s i mple c u r v e
La
n
n
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-
nn
202—204
.
C HAP TER XI
EA R THW O R
S ETT I NG S T A E S
h t t ke
D t ; w
d how m rked
.
K
a
a
a
s a
F OR
s an
K
.
13 7
a
Method of fi n di n g rod re a di n g for g r a de
13 8- 13 9
139
or fil l a t cen ter
140—142
S ide s t a ke s s ectio le el ; s ectio n n o t lev el
143—145
K eepi n g n ote s ; for m of ote book
t to fill 146 147
P a s s fro m
C ro s s s ection s where t a ke n
147
Open i g i n e mb a k me t Ge n er a l le v el ote s
Le v el three le v el five le vel irre g u l a r s ectio n s
148
Cu t
n
,
v
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ou
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,
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n
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-
,
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,
C HAP TER XII
C O M U T ING EAR THW O R
M ETH O S
.
D
OF
P
K
.
ri n cipa l metho d s u s ed Av er a g i g en d a re a s
K i d s of cro s s s ectio n s s pecified
Le v el cro s s s ectio n
Three le v el s ectio
F i v e le v el s ectio n
Irreg u l a r s ectio n
Irre g u l a r s ectio n ; r u le of th u mb
Other irre g u l a r s ectio n s
U s e of pl a i meter
P ri s moid a l for mu l a
P ri s m
oi d a l for mu l a for pri s ms wedg es pyr a mid s
Na t u re of re g u l r s ectio n of e a rthwork
P roof of pri s moid a l for mu l a where u pper s u rfa ce i s
wa rped
P ri s moid a l correctio n ; formu l a s
Correctio n i n s peci a l c a s e s
C orrectio n for pyr a mid
Correctio n for five le v el s ectio s
Correctio n for irreg u l a r s ectio n s
P
n
.
n
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-
-
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-
n
-
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.
n
,
a
-
n
,
.
Con ten ts
X
.
C HAP TER XIII
S EC AL P R O L E M S N EA R TH W O R
.
I
P
B
I
K
.
Correctio n
for c u r v a t u re
Correctio n where chord s a r e le s s th a n 100 feet
Correctio of irr eg u l a r s ectio n s
O pe i n g i n e mb a k men t B orro w pit s
Tr u n c a ted tri a n g l a r pri s m Tr u c a ted r e cta n g u
l a r pri s m
A s s e mbled pri s ms
Additio a l hei g ht s
Co mp u te fro m hori z o t a l pl a e belo w fi n i s hed s u r
.
.
n
n
n
-
.
u
n
.
n
.
n
n
erie s of s ectio n s a lo g a li e
Co mp u te s ectio n fro m l o w hori z o n t a l li n e
S ectio n s on s teep Side s lope
S
n
n
C HAP TER XIV
EARTHW O R K TA L E S
.
B
.
or mu l a for u s e i L a d K t a ble s
A rr a n g e me n t of t a ble ; expl a n a tio n ; exa mple
Ta ble for pri s moid a l correctio n s ; exa mple
E q u i v a len t le v el s ectio s fro m t a ble s
Ta ble s of tri an g u l a r pri s ms I dex to t a ble s
Arr a n g e me n t of t a ble s for tri a g u l a r pri s ms ; ex
a mple
Applic a tion to irreg u l a r s ectio n s
F
n
n
.
n
.
n
n
C HAPTER XV
K D I A GR A M S
EA
HW
.
RT
OR
.
M ethod of di ag r a ms with di s c u s s io n
Co mp u t a tio n s a n d ta ble for di a g r a m of pri s moid a l
— 186
1 87 188
correctio n
D i a g r a m for pri s moid a l correctio a n d expl a n a tio n
of co n s tr u ctio n
E xp l a n a tio n a n d exa mple of u s e
Ta ble for di a g r a m for tri a n g u l a r pri s ms
Co mp u ta tio n s a n d t a ble for di a g r a m of three le vel
19 1 194
s ectio n s
195
Check s u po n co mp u t a tio n s
—
n
-
-
283 —284
.
Con ten ts
xi
.
P AGE
o n s tr u ctio n of di a g r a m : a l s o c u r ve of lev el s ec
tio n
U s e of di a g r a m for three le el s ectio n s
Co mme t o n r a pidity by u s e of di a g ra ms
P ri s moi d a l correctio for irre g u l a r s ection s by a i d
C
-
v
n
n
C H AP TER XVI
HA U L
,
.
.
e itio n a d me a s u re of h a u l
Le g th of h a u l how fo u d
F or mu l a for ce ter of g r a v ity of a s ectio n
F or mu l a ded u ced
F o rmu l a m odi fied for u s e with t a ble s or di a g ra ms
F o r s ectio le s s th a n 100 feet
D fin
n
n
n
,
n
.
n
297
.
F or
C HAP TER XVII
M ASS D I A GR A M
.
.
D efi n itio n of ma s s di a g r a m
.
T a ble a n d method of
P ropertie s of ma s s di a g r a m
Gr a phic a l me a s u re of h a u l e xp l a i r ed
fl
Applic a tio to ma s s di a g r a m
B orro w a d w a s te s t u di ed by ma s s di a g r a m
P ro fit a ble le n g th of h a u l
E x a mpl e of u s e o f di a g r a m
E ffect of s hri k a g e o ma s s di g r a m
D i s c u s s io n of o v erh a u l
Tre a tme t of o erh a u l by ma s s di a g ra m
F u rther ill u s tr a tio n of u s e of ma s s di a g r a m
.
n
.
n
n
n
DI
AGR AM S
n
v
a
2 05 —206
R AI LR OAD CUR VE S A ND E AR THWOR K
CH APT ER I
.
oper a t io n s of loc a ti n g a r a ilro a d
pr a cticed i n thi s co u n try a r e three i n n u mber
1
.
“
Th e
.
co mmo n ly
as
,
I
.
R E CO NN O I SSA N CE
.
L I M I N AR Y SU RVE Y
III L OC A TI O N S URVE Y
II
.
PR E
.
I
.
.
.
RE C ONN OI SS ANCE
c
c e i s a r pid
.
r vey or r a ther a critic a l
ex a min a t io n of cou n try withou t the u s e of the ordi n a ry i n s tr u
me n ts of s u r veyin g C ert a i n i n s tr u me n ts howe ver a r e u s ed
the A n eroid B a ro met er for i n s t a n ce It i s v ery co mmo n ly the
c a s e th a t the ter mi n i of the r a ilro a d a r e fixed a n d ofte n i n ter
medi a te poi n t s a l s o It i s de s ir a ble th a t n o u n n ece ss a ry re
s trictio n s a s to i n ter medi a te poi n t s s ho u ld be i mpo s ed on the
e n gi n eer t o preve n t h i s s electi n g w h a t h e fin ds to be the bes t
li n e a n d for thi s re a s o n it i s a d v i s a ble th a t the reco n n oi ss a n ce
s ho u l d where po s s ible precede the dr a wi n g of the ch a r ter
The fir s t s tep i n reco n n oi s sa n ce s ho u ld be to procu re the
3
be s t a va il a ble ma p s of the co u n try a s t u dy of the s e will g en
e r a ll y f u r n i s h to the e n g i n eer a g u ide a s to the ro u te s o r s ectio n
of co u n try th a t s ho u ld be ex a mi n ed
If ma p s of the U n ited
S ta te s G eolo g ic a l S u r vey a r e a t h a n d with co n t ou r li n e s a n d
other topo gr a phy c a ref u lly s hown the reco n n oi ss a n ce ca n be
l a rgely deter mi n ed u po n the s e ma p s L i n e s cle a rly i mpr a c
t i ca bl e will be t hrow n o u t the ma xi mu m g r a de clo s el y deter
mi n ed a n d the field ex a mi n a tio n s red u ced to a mi n imu m No
2
.
T h e Re on n oi s s a n
a
su
,
,
,
,
,
,
.
.
,
.
‘
,
,
.
,
.
.
,
,
.
,
,
1R a i
2
i
t
lroa d Ca roes a n d E a rthwork
u i df b efa cce
s hO
‘‘
p te d fin a l l y fro m a n y
.
ch ma p bu t a
c a refu l field ex a mi n a tio n s hou ld be ma de o ver the ro u te s i n di
ca t e d on the co n to u r ma p s
The ex a mi n a tio n i n g en er a l
s ho u ld co v er the g e n er a l s ectio n of co u n try
r a ther th a n be
con fin ed to a s i n gle li n e betwee n the ter min i A s tr a ight lin e
a n d a s t r a i g ht g r a de fro m on e ter mi n u s to the other i s de s ir a ble
b u t thi s i s s eldo m po s s ible a n d i s i n g e n er a l fa r fro m po s s ible
If a s i n g le li n e o n ly i s ex a mi n ed a n d thi s i s fou n d to be n e a rly
s tr a i ght thro u gho u t
a n d with s a ti s f a ctory g r a de s
it ma y be
tho u g ht u n n ece s s a ry to c a rry the exa min a tio n f u r ther It will
frequ e n tly howe v er be fo u n d a d va n t a geo u s to de v i a te co n
s id e r a b l y fro m a s tr a i g ht li n e i n order to s ec u re s a ti s f a ctory
I
n
m
a
n
r
a
de
s
y c a s e s it will be n ece s s a ry to wi n d a bo u t more
g
o r le s s thro u g h the co u n try i n order to s ec u re the be s t l i n e
Where a h i g h hill or a mo u n t a i n lie s directly between the
poin ts it ma y be expected th a t a li n e a r ou n d th e hil l a n d
s o mewh a t re mote fro m a direct li n e will pro v e more f a v or a ble
th a n a n y other Un le s s a re a s o n a bly direct li n e i s fo u n d the
ex a mi n a tio n to be s a ti s fa ctory sho u ld embr a ce a ll the s ectio n of
i n ter ve n i n g co u n try an d a ll fe a s ible li n e s s ho u ld be ex a mi n ed
4 There a r e two fe a t u re s of topo g r a phy th a t a r e likely to
pro v e of e s peci a l i n tere s t i n reco n n oi ss a n ce r i dg e l i n es a n d
r ou t e
'
‘
‘
su
,
.
,
,
,
.
‘
,
,
.
,
,
,
.
,
,
.
.
,
,
,
.
,
,
,
.
,
.
,
va l l ey l i n es
.
A r i dg e l i n e a lo n g t h e whole o f i ts co u rs e 18 hi gher th a n the
i
a
s
n
ro
d
i
mm
edi
tely
dj
ce
n
t
to
it
o
e
a
ch
s
ide
Th
t
the
a
u
n
a
a
g
i
m
s a l s o c a lled
ro
d
lope
dow
w
rd
fro
it
to
both
ide
It
s
n
a
s
s
u
n
s
g
,
.
.
a
w a ter s h ed l i n e
.
the gro u n d i m
medi a tely a dj a ce n t t o it o n e a ch s ide The grou n d s lope s
Va lley lin e s ma y be c a lled wa ter
u pw a rd fro m i t to both s ide s
co u r s e l i n es
A p a ss i s a pl a ce on a rid g e l i n e lo wer th a n a n y n ei ghbori n g
poi n ts on the s a me rid ge Very i mport a n t poi n ts to be deter
mi n ed i n reco n n oi s s a n ce a r e the p a s s e s where the rid g e li ne s
a r e to be c o ss ed ; a l s o the poi n ts where the va lley s a r e to be
r
cro s s ed ; a n d c a refu l a tte n tio n s ho u ld be g i ve n t o the s e poi n t s
I n cro ss i n g a va lley thro u g h which a l a rg e s tre a m flow s it ma y
be of gre a t i mporta n ce to fin d a g ood brid ge cro ss i n g I n s o me
c a s e s where there a r e s erio u s difficu ltie s i n cro s s i n g a rid g e a
tu n n el ma y be n ece s s a ry Where s u ch s tr u ctu re s either
A va ll ey l i n e, to the co n tra ry , i s lower t h a n
.
.
.
.
.
,
.
,
.
,
R econ n a i s s a n ce
brid g e s
3
.
tu n n el s a r e to be b u ilt f a v or a ble po i n ts for th eir
co n s tru cti o n s hou ld be s elected a n d the re s t of the li n e be com
I n ma n y p a rt s of the Un ited S t a te s a t the
pe ll e d to co n for m
pre s e n t ti me the n ece s s ity for a v oidin g g ra de cro s s i n gs c a u s e s
the cro s s i n s of ro a d s a n d s treet s to beco me g o v ern i n g poi n ts
of a s g re a t i mport a n ce a s rid g e s a n d va lley s
There a r e s e v er a l pu rpo s e s of reco n n oi s s a n ce firs t t o
5
fin d whether t h ere i s a n y s a ti s f a ctory li n e betwee n the propo s ed
ter mi n i ; s eco n d to e s ta bli s h which of s e ver a l li n e s i s be s t ;
third to deter min e a pproxi ma tely the ma xi mu m gr a de n e ces
fou rth to report u po n the ch a r a cter or
s a ry to be u s ed ;
n
n
d
a
eolo
ic
l
for
tio
the
co
u
try
the
prob
ble
co
t
m
n
o
f
a
s
o
f
a
a
g
g
co n s tr u ctio n depe n di n g s o mewh a t u po n th a t ; fifth to ma ke
n ote of the exi s ti n g re s o u rce s of the co u n try i t s ma n u f a ctu res
mi n e s a gr ic u ltu r a l or n a tu ra l prod u cts a n d the c a p a bilitie s for
impro ve me n t a n d de velop me n t o f the cou n try re s u lti n g fro m
the i n trod u ctio n of the r a ilro a d T h e report u po n r e con n ois
I t is
s a n ce s ho u ld i n cl u de i n forma tio n u po n a ll the s e poi n t s
for the deter mi n a tio n of the third po i n t me n tio n ed t he ra te of
ma xi mu m g r a de th a t the b a ro meter i s u s ed
Ob s erv i n g the
ele va tio n s of g ov ern i n g poi n ts a n d kn owi n g the di s t a n ce s be
twee n tho s e poi n t s it i s po s s ible t o for m a g ood j u d g men t a s to
wh a t r a te of ma xi mu m g ra de to a s su me
6 The E le v a t i on s a r e u s u a lly t a ke n by th e A n er o i d B a r ome
ter
T a ble s for co n v ertin g b a ro meter re a di n gs i n to ele va tio n s
a bo v e s e a le v el a r e re a dily a va il a ble a n d i n co n v e n ie n t form fo r
field u s e ( S ee T a ble X L A lle n s F ield a n d Office T a ble s )
D i s ta n ce s ma y be deter mi n ed with s u fficie n t a ccu r a c y i n
ma n y c a s e s fro m the ma p where a g ood on e exi s t s
W here
thi s method i s i mpo s s ible or s ee ms u n de s ira ble the dis ta n ce
ma y be de ter mi n ed i n o n e of s e ver a l differe n t w a y s
Whe n
the trip i s ma de by w a g o n it i s c u s to ma ry t o u s e a n Odometer
a n i n s tr u me n t which me a s u re s a n d record s the n u mber of
re vol u tio n s of the wheel to which it i s a tt a ched a n d thu s t h e
di sta n ce tr a veled by the w a go n There a r e differe n t forms of
odo meter I n i ts mo s t co mmo n for m it depe n d s u po n a ba n g
i n g wei g ht or pe n du l u m which i s s u ppo s ed t o hold i t s po s itio n
h a n g i n g vertic a l while the wheel t u r n s The i n s tru me n t i s
a tt a ched to the wheel betwee n the s poke s a n d a s n e a r to the
h u b a s pr a ctic a ble
A t low s peeds it re g is t ers a cc u r a tely ; a s t h e
or
,
,
.
,
g
.
:
.
,
,
,
,
,
,
,
,
,
,
.
.
,
.
,
,
,
.
.
.
-
’
.
.
,
.
,
,
.
,
,
,
.
l
.
,
,
,
,
.
.
R a i l r oa d Cu r ves
4
E a r t h work
and
.
peed i s i ncre a s ed a poin t i s re a ched where the cen trifu g a l force
n e u tr a li z e s o r o v erco me s the force o f
g r a v ity u po n the pe n d u
l u m a n d the i n s tr u me n t f a ils to re g i s ter a cc u r a tely or perh a p s
a t hi g h s peeds to re g is ter a t a ll
If thi s for m of odo meter i s
u s ed
a cle a r u n ders t a n di n g s ho u ld be h a d o f the co n di tio n s
u n der which it f a il s to correctly r e g i s ter
A theoretic a l di scu s
s io n mi ght clo s ely e s t a bli s h the poi n t a t which the ce n trif u a l
g
force will b a l a n ce the force of gr a vity The wheel s triki n g
a ga i n s t s to n e s i n a ro u g h ro a d will cre a te di stu rb a n ce s i n the
a ctio n of the pe n d u l u m s o th a t t h e odo meter will f a il to re i s ter
g
a cc u r a tely a t s peed s le s s th a n th a t deter mi n ed u po n the a bo v e
a s s u mptio n
A cyclo me t er ma n u f a c tu red for a u t o mobile u s e is co n
a n d s o me a s u re s
n e cte d both with the wheel a n d the a xle
n
a n d a xle
iti
ely
the
rel
ti
betwee
the
wheel
ve motio n
a
v
s
o
p
Ma n y
a n d thi s o u g ht to be reli a ble for re g i s teri n g a cc u r a tely
en g i n eers prefer to cou n t the re volu tio n s of the wheel the m
s elv e s tyi n g a r a g to the wheel to ma ke a co ns pic u o u s ma rk
fo r co u nti n g
Whe n th e t rip i s ma de on foot p a ci n g wi ll g i v e s a ti s f a c tory
res u l ts A n i n str u me n t c a lled the P edometer re g is ters the
res u lts of p a ci n g A s ordi n a rily co n s tr u cted the g r a d u a tio n s
re a d to qu a rter mile s a n d it i s po s s ible to e s ti ma te to on e
te n th th a t di s t a n ce P edo meter s a r e a l s o ma de which re gi s ter
p a ce s In pri n c iple t h e pedo meter depe n d s u po n the fa ct th a t
with e a ch s tep a cert a i n s hock or ja r i s produ ced a s the heel
a n d e a ch s hock c a u s e s the i n s tr u me n t to
s trike s the g ro u n d
re g i s ter Tho s e re g i s terin g miles a r e a dj u s ta ble to the le n g th
of p a ce of the we a rer
If the trip i s ma de on hor s eb a c k it i s fou n d po ss ible to g et
— a ited hor s e by fir s t deter mi n i n
h
i
t
ood
re
lt
with
e
dy
s
u
s
a
s
a
s
g
g
g
r a te of tr a v el a n d figu ri n g dis ta n ce by the ti me co n s u med i n
tr a veli n g E xcelle n t re s u lts a r e s a id to h a v e bee n s ec u red i n
thi s w a y
It i s cu s to ma ry for e n gi n eers n Ot to u s e a co mp a s s in
7
r e co rin oi s s a n ce
a ltho u g h thi s i s s o meti me s do n e i n order to
tra ce the lin e tra ver s ed u po n the ma p a n d with gre a ter a ccu
r a cy A pocket le vel will be fo u n d u s ef u l The s killf u l u s e of
pocket i n s tru me n t s will a l mo s t cert a i n ly be fo u n d of gre a t va lu e
to t h e e n g i n eer of reco n n oi s s a n ce
s
,
,
,
.
,
.
.
,
.
,
,
,
,
.
~
,
.
,
.
.
,
,
.
,
,
.
,
,
.
.
,
,
.
.
.
.
,
.
.
.
CH AP T E R II
II
.
I I N A RY S URVEY
P RE L M
.
T h e Pr el i mi n a r
.
y S rv e y i b ed po
re su l ts of the
reco n n oi s s a n ce It i s a s u r vey ma de with the or d i n a ry i n s tr u
me n t s of su r v eyi n g
I ts p u rpo s e i s to fix a n d ma rk u po n the
g ro u n d a fir s t tri a l li n e a pproxi ma ti n g a s clo s ely to the proper
fi n a l li n e a s the diffic u l ty of the co u n try a n d the experie n ce of
the e n g i n eer will a llow f u rther th a n this to collect d a t a s u ch
th a t thi s su r vey s h a ll s er ve a s a b a s i s u pon which the fin a l
L oc a tio n ma y i n telli g e n tl y be ma de
I n order t o a pproxi ma te
clo s ely i n the tri a l li n e i t i s e s s en ti a l th a t th e ma xi mu m g r a de
s ho u ld be deter mi n ed or e s ti ma ted a s correctly a s po ss ible a n d
th e li n e fixed with d u e re g a rd thereto
It will be of v a l u e to dev ote s o me a tte n tio n here to a n ex
”
“
pl a n a tion a bo u t Gr a de s a n d M a xi mu m Gr a de s
The ide a l li n e i n ra ilro a d loca tio n i s a s tr a ig ht
10
Gra de s
Thi s i s s eldo m if e v er re a li z ed Whe n the two
a n d le v el li ne
termi n i a r e a t differe n t ele va tio n s a li n e s tr a i ght a n d of u n i
for m gr a de beco mes the ide a l I t i s co mmo n l y i mpo s s ible to
I n oper a ti n g
s ec u re a li n e of u n ifor m g r a de betwee n ter mi n i
s o me
a n e n g i n e di vi s io n will be a bo u t 10 0 mile s
a r a ilro a d
ti me s le s s ofte n more I n loc a ti n g a n y 100 mile s of r a ilro a d
it i s a l mo s t certa in th a t a u n ifor m g ra de c a n n ot be ma i n t a i n ed
M ore co mmo n ly there will be a s u cce ss io n of hills p a rt of the
li n e u p gr a de p a rt down g ra de S o meti me s there will be a
co n ti n u o u s u p gr a de b u t n ot a t a u n ifor m r a te With a u n i
for m gr a de a loco moti v e e n g in e will be co n s t a n tly exerti n g i ts
ma ximu m p u ll or doi n g i ts ma xi mu m work i n h a u li n g the
lo n ge s t tr a i n it i s c a p a ble of h a u li n g ; there will be n o power
w a s ted i n h a u li n g a li ght tr a i n o ver low or le vel g r a de s u po n
which a he a vier tr a i n co u ld be h a u led Where the g r a de s a r e
or ri s i n g irre g ul a rly it
n o t u n ifor m b u t a r e ri s i n g or f a lli n g
will be fo u n d t h a t the topo gr a phy on s o me p a rtic u la r 5 or 10
9
.
u
s
as
u
n
th e
.
.
,
.
,
,
.
.
-
.
.
.
,
,
.
,
.
.
,
,
,
.
,
.
'
,
.
,
,
.
.
,
.
,
,
6
,
Pr eli mi n a ry S u r vey
7
.
mile s i s of s u ch a ch a r a cter th a t th e gr a de here mu s t be s t eepe r
th a n i s re a lly n ece s sa ry a n ywhere el s e on the li n e ; or there
ma y be two or three s tretche s of g r a de where a bo u t the s a me
r a te of gr a de i s n ece s s a ry s teeper th a n el s ewhere req u ired
The s teep g ra de thu s fou n d n ecess a ry a t s o me s peci a l poi n t or
“
”
poi n ts on the li n e of r a ilro a d i s c a lled the M a xi mu m Gr a de
”
”
“
“
or L i mi t i n g Gr a de it bei n g the gr a de
R u li n g Gr a de
or
th a t li mit s the wei g ht of tr a i n tha t a n e n gi n e ca n h a u l o v er the
whole di vi s io n I t s hou ld the n be the e fior t t o ma ke the r a te
bec a u s e the lower the
o f ma xi mu m gr ade a s low a s po s s ible
r a te of the ma xi mu m g r a de the he a vier the tr a i n a g i ve n loco
moti ve ca n h a u l a n d bec a u s e it co s t s n ot v ery mu ch more to
The ma xi mu m g ra de
h a u l a he a v y tr a i n th a n a li ght on e
determin ed by the reco n n oi s s a n ce s ho u ld be u s ed a s the b a s i s
for the preli mi n a ry su r vey H ow will thi s a ffect the li n e ?
Whe n ever a hill i s e n co u n t ered if the ma xi mu m gr a de be
it ma y be po s s ible to c a rry the li n e s tra i gh t a n d o ver
s teep
the hill if the ma xi mu m g r a de b e low it ma y be n ece s s a ry to
deflect the li n e a n d c a rry it a ro u n d the hill Whe n th e maxi
mq gr a de h a s been o n ce properly de t er mi n ed if a n y s a vi n g
ca n be a cco mpli s hed by u s i n g it r a ther th a n a g r a de le ss s teep
the ma xi mu m g r a de s ho u ld be u s ed I t i s po s s ible t h a t th e
tra i n lo a d s will n ot be u n ifor m thro u gho u t the di vi s io n It
will be a d va n ta g eo u s to s pe n d a s ma ll s u m of mo n ey to keep
a n y gr a de lower t h a n the ma xi mu m l n Vi ew of t h e p os s i bi l i ty
tha t a t thi s p a rtic u l a r poi n t the tr a i n lo a d will be he a vier t h a n
e ls ewhere on the di vi s io n A n y s a v i n g ma de will i n g en era l
be of on e or more o f three ki n d s
”
A mo u n t or
of exca va tio n or e mb a n kme n t ;
a
qu a n tity
b D i s ta n ce ;
0
C ur va tu re
11
I n s o me c a s e s a s a tisf a c tory g r a de a l ow g r a de for a
ma xi mu m ca n be ma i n t a i n ed thro u g ho u t a di v i s i o n of 100
mile s i n le n g th with the exceptio n of 2 or 3 mile s a t on e poin t
o n ly S o g re a t i s the va l u e of a low ma xi mu m gr a de th a t a l l
ki n ds of expedie n ts will be s o u g ht for to p a s s the difficu lty
withou t i n cre a s i n g the r a te of ma xi mu m g r a de , which we kn ow
wi ll a pply to the whole di v i s io n
12
S o meti me s by i n cre a s i n g th e le n gth of li n e we a r e a ble
to re a ch a gi v e n ele va tio n with a lower r a te of g r a d e
S o me
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R a i l r oa d Cu r ve s a n d E a r tirwor k
8
.
i e he a v y a n d expen s i v e cu ts a n d fills ma y s erv e the pu r
po s e S o meti me s a ll su ch de v ice s f a il a n d there s till rema i n s
n ece s s a ry a n i n cre a s e of g r a de a t thi s on e poi n t bu t a t thi s
poi n t o n ly I n s u ch c a s e it is n ow c u s to ma ry to a dopt the
hi gher r a te of gr a de for the s e 2 or 3 mile s a n d O pera te the m by
“
u s i n g a n extr a or a dditio n a l e n g i n e
I n thi s c a s e the
r u lin g
“
for the di vi s io n of 100 mile s i s properly the ma xi mu m
g r a de
”
pre va ili n g o ver the di v i s io n ge n er a lly the hi gher g r a de
g r a de
for a few mile s o n ly bei n g kn ow n a s a n A u xili a ry Gra de or
”
“
more co mmo n ly a P u s her Gr a de
The tra in which i s h a u led
o v er the e n gin e di v i s io n i s helped o ver the a u xili a ry or pu s her
”
“
n
n
f
r
a
de
by
the
u
se
o
a
a
dditio
a l e n g i n e c a lled a
P u s her
g
“
Where the u s e of a s hort P u s her Gra de will a llow the u s e
”
of a low ma xi mu m g ra de there i s e vid en t eco n o my i n i ts
u se
The critic a l di s cu s s io n of the i mport a n ce or va l u e of
s a v i n g di s t a n ce c u r va t u re ri s e a n d f a ll a n d ma xi mu m g r a de
i s n ot withi n th e s cope of thi s book a n d the re a der i s referred
”
to Welli n gto n s E co n o mic Theory of R a ilwa y L oc a tio n
The P reli mi n a ry S u r vey follows the g e n er a l lin e ma rked
13
ou t by the reco n n oi ss a n ce b u t thi s r a pid ex a mi n a tio n of co u n
try ma y n ot h a ve fu lly deter mi n ed which of two or more li n e s
In thi s
i s the be st the a dv a n t a g e s ma y be s o n e a rly b a l a n ced
c a s e tw o or more preli mi n a ry su rveys mu s t be ma de for com
p a ri s o n Whe n the reco n n oi s s a n ce h a s fu lly deter mi n ed the
n
r e s till left for the preli mi n a ry
e
er
l
ro
te
cert
i
det
il
n
u
a
a
s
a
a
g
It ma y b e n ece ss a r y to r u n two li n e s
s u r v ey to deter mi n e
on e on e a ch s ide of a s ma l l s tre a m a n d po ss ibly a li n e cro s s i n g
it s ev era l ti me s The reco n n oi ss a n ce wo u ld ofte n fa il to s ettle
It i s de s ir a ble th a t the preli mi n a ry
mi n o r poi n t s like thi s
b
u t it i s
s u r v ey s ho u ld clo s ely a pproxi ma te to the fi n a l li n e
;
n ot i mport a n t th a t it s ho u ld f u lly coi n cide a n ywhere
”
“
preli mi n a ry i s to pro v ide a
A n i mport a n t p u rpo s e of the
ma p which s h a ll s how e n o u g h of the topo gra phy of the co u n try
s o th a t the L oc a tio n proper ma y be proj ected u po n thi s ma p
Worki n g fro m the lin e of s u r v ey a s a b a s e li n e me a su reme n ts
s ho u ld be t a ke n s u fficie n t to Show s tre a ms a n d v a rio u s n a t u r a l
object s a s well a s the co n t ou rs of the su rf a ce
The P reli mi n a ry Su r vey s er ve s s e ver a l p u rpo s e s
14
F i r s t To fix a ccu r a tely the ma xi mu m g r a de for u se in
Loc a t io n
t m s
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Pr e li mi n a ry S u r vey
8
.
To de t er mi n e which of s ever a l li n e s i s b e s t
Th i r d To pro vide a ma p a s a b a s i s u po n which th e L o c a tio n
ca n properly be ma de
F o u r th To ma ke a clo s e e s ti ma te o f the co s t of the work
F ifth To s ec u re i n cert a i n c a s e s le ga l r i g ht s by fili n g pl a n s
It s hou ld be u n ders tood t h a t t h e preli mi n a ry su rvey
15
i s i n g e he r a l s i mply a me a n s to a n e n d a n d r a p idity a n d
eco n o my a r e d e s ira ble It i s a n i n s tr u men ta l s u rv ey M e a s
u r e me n ts of di s t a n ce a r e t a ke n u s u a lly with the ch a i n a ltho u g h
A n g le s a r e t a ke n g e n er a lly with a
a t a pe i s s o meti me s u s ed
tr a n s it ; s o me a d voc a te the u s e of a co mp a s s The lin e i s
or d in a rily r u n a s a broke n li n e with a n gle s b u t i s occ a s io n a lly
r u n with c u r v e s co n n ecti n g the s tr a i g ht s tretche s g e n er al ly for
the re a s o n th a t a ma p o f s u ch a li n e i s a va il a ble for fil in g a n d
certa in le ga l ri g hts re su lt from s u ch a fili n g With a co mp a s s
a n d i n p a ss i n g s ma ll ob s t a cle s a
n o b a ck s i g ht n eed be t a ke n
co mp a s s will s a ve ti me on thi s a cco u n t A tr a n s it li n e ca n be
c a rried p a s t a n ob s t a cle re a dily by a z i gz a g li n e C o mmon
pr a ctice a mo n g e n g i n eer s f a v ors the u s e of the tr a n s it r a ther
”
“
th a n the co mp a s s S t a ke s a r e s e t a t e very S t a tio n 100 feet
a p a rt a n d the s t a ke s a r e ma rked o n the f a c e the firs t 0 the
n ext I the n 2 a n d s o to the e n d of the li n e
A s t a ke s et 102 5
feet fro m the be g in n i n g wo u ld be ma rked IO 25
L e ve l s a r e t a ke n on the g ro u n d a t the s ide of the s t a ke s a n d
a s mu ch ofte n er a s there i s a n y ch a n g e i n the i n cli n a tio n o f the
n
A
r
ro
u
d
ll
the
s
u
rf
a
ce
hei
ht
s
a
e pl a tted on a profile a n d
g
g
the gr a de li n e a dj u sted
16
The li n e s ho u ld be r u n fro m a g o ver n i n g poi n t tow a rds
co u n try a llowi n g a choice of loc a tio n th a t i s fro m a p as s or
fro m a n i mporta n t brid ge cro ss i n g towa rd s co u n try o fferi n g n o
There i s a n a d va n ta g e i n ru n n i n g fro m a
gre a t diffic u ltie s
s u mmit dow n hill
howe v er t o the a bo v e co n s ider a
s u bject
tio n s I n ru n n i n g from a s u mm it dow n a t a pr e s cr i be d r a t e
of gr a de a n experie n ced e n g i n eer will c a rry the li n e s o th a t a t
the e n d of a d a y s work the le vel s will s how the li n e to b e
a bo u t where it o u g ht to be
F or thi s p u rpo s e the le v el s mu s t
be worked u p a n d the profile pl a tted to d a te a t the clo s e of
e a ch d a y A n y s lig ht ch a n g e of li n e fo u n d n ece ss a ry ca n the n
be ma de e a rly the n ext mor n i n g A method s o meti me s a dopted
in w
orki n g dow n fro m a s u mmit is for the loc a ti n g e n g i n eer t o
S econ d
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R a i lr oa d Cu r ves a n d E a r th w ork
10
.
pl a t h i s gr a de li n e on the profile d a ily i n a d va n ce a n d the n
d u ri n g the d a y pl a t a poi n t on h i s profile whe n e ver he ca n
co n v e n ien tly ge t on e fro m h i s le veler a n d th u s fin d whethe r
h i s li n e i s too hi g h o r too low
17
Occ a s io n a lly the re s u lt of two or three d a y s work will
yield a li n e extre mely u n s a ti s f a ctory e n o u g h s o th a t the work
of the s e two or three d a y s will be a b a n do n ed The p a rty
”
“
backs u p a n d t a ke s a fre s h s t a rt fro m s o me co n ve n ie n t
poi n t I n s u ch c a s e the c u s to m i s n ot to te a r ou t s e vera l
p a g e s of n ote book b u t i n s te a d to s i mply dr a w a lin e a cro s s
”
“
A t s o me f u t u re
the p a g e a n d ma rk the p a ge A b a n do n ed
ti me the a b a n do n ed n ote s ma y co n vey u s efu l i n for ma tion to
the e ff ect th a t thi s li n e w a s a tte mpted a n d fo u n d u n a va il a ble
I n g e n er a l a ll n ote s worth t a ki n g a r e w orth s a v i n g
S o meti me s a fter a li n e h a s bee n r u n thro u g h a s ectio n of
cou n try there i s l a ter fo u n d a s horter or better l i n e
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6
(I )
In th e fi gu re
ed for ill u s tra tio n the firs t l i n e A Lin e
i s repre s e n ted by AEBC D u po n which the s t a tio n s a r e ma rke d
co n ti n u o u s ly fro m A to D 3 50 s ta tio n s The n e w li n e 8
6 0 a n d the s t a tio n i n g i s held
Li n e s t a rt s fro m E St a IOQ
co n ti n u ou s fro m O to where it co n n ects with the A Li n e a t
A Li n e a n d i s a l s o
The poi n t C i s St a
2 7 of the
C
It i s n ot c u s to ma ry to re s ta ke
Li e
8
St a 3 0 7
I3 of the
the li n e fro m C to D i n a ccord a n ce with 8 Li e s ta tio n i n g
I n s te a d of thi s a n ote i s ma de i n the n ote books a s follows
B Li n e
I3
27
3 07
A Li n e
St a 3 12
S o me e n g i n eers ma ke the n ote i n the followi n g for m
8 6 ft
St a 3 07 to 3 I3
The firs t for m i s prefer a ble bei n g more direct a n d less li a ble
to c a u s e co n fu s io n
“
u s
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n
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n
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-
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Pr e li mi n a ry S u r vey
ho u ld be k ep t cle a rly a n d n icely i n a n o t e
book — n ev er on s ma ll piece s of p a per The d a te a n d the
n a me s of me mbers of the p a rty s ho u ld be e n tered e a ch d a y i n
the u pper left h a n d cor n er of the p a ge A n o ffice copy s hou ld
be ma de a s s oo n a s opportu n ity offers both for s a fet y a n d con
Th e o r i g i n a l n ot es s h o u l d a l wa ys be p r es er ve d ; they
ve n i e n ce
wou ld be a d mi ss ible a s e v ide n ce i n a co u rt of l a w where a copy
wo u ld be rejected When two or more s ep a r a te or a lter n a te
lin e s a r e r u n they ma y be de s i gn a ted
Li n e C
8
Li n e A Li n e
C Li n e
8 Li n e
A Li n e
18
A l l n ote s
11
.
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s
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-
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"
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"
”
“
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,
,
The Or ga n i za t i on of P a rt y ma y be a s follows
1 L oc a ti n g E n g i n eer
T ra n s itma n
2
3 H e a d C h a i n ma n
St a ke ma n
Tr a n s i t P a r ty
5 R e a r C h a i n ma n
6 B a ck F l a g
7 A xe me n ( o n e o r more )
8: L e v eler
L e v el P a r t y
9 R odma n ( sbmet i mes two )
10 Topo g r a pher
Topo gr a phi c a l P a rt y
11 A s s i s t a n t
12 C ook
13 Te a ms ter
19
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c
the chief of p a rty a n d i s
re spo n s ible for the b u s i n e s s ma n a g e men t of the c a mp a n d
p a r ty a s well a s for the co n du ct of th e s u r vey H e deter
m i n e s where the li n e s h a ll r u n keepi n g a he a d of the tr a n s it
a n d e s t a bli shi n g poi n ts a s fore s i g ht s o r tu r n i n g poi n t s for the
tr a n s itma n I n Ope n co u n try the extr a a xe ma n ca n a s s i st by
holdi n g the fl ag a t tu rn i n g poi n t s a n d th u s a llowi n g th e l oca t
i n g e n g i n eer t o pu s h o n a n d pick ou t other poi n ts i n a d va n ce
The loc a tin g en gi n eer keeps a speci a l n ote book or me mor a n d u m
book i n it he n ote s on the gro u n d the qu a lity of ma teri a l rock
e a rth or wh a te ver it ma y be ; t a ke s n ote s to deter mi n e the
le n gt h s a n d po s itio n s of brid g e s c u l verts a n d other s tru ct u re s
s h ow s the loc a li tie s of ti mber b u ildi n g s to n e s borrow pit s a n d
20
T h e Lo a t i n g E n g i n ee r i s
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R a i lr oa d Cu r ve s
12
an
d E a r th wor k
.
othe r ma teri a ls va lu a ble for the exec u tio n of the work ; In fa ct
ma ke s n ote s of a ll ma tter s n ot properly a tte n ded to by th e
tr a n s i t le veli n g or topo gr a phy p a rty The r a pid a n d fa ithf u l
pro sec u tio n of the work depe n d u pon the loc a ti n g e n g in eer
a n d the p a rty o u g ht to deri v e i n s pir a tio n fro m the e n er y a n d
g
V i g or of their chief who s ho u ld be the le a der i n the work
In
ope n a n d e a s y co u n try the loc a ti n g e n g i n eer ma y i n s till l i fe
i n to the p a rty by hi ms elf t a ki n g the pl a ce of the he a d ch a i n
ma n occ a s io n a lly I n co u n try of s o me di ffic u lty hi s ti me will
be fa r better e mploye d i n pro s pecti n g for the be s t li n e
21
The Tr a n s i tma n doe s the tr a n s it work r a n ge s i n the
li n e fro m the i n s tr u me n t me a s u res the a n g le s a n d keep s the
n ote s of the tr a n s it s u r v ey
The followi n g i s a g ood form for
the left h a n d p a g e of the n ote book
,
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Note s of t opo g r a phy a n d re ma rks a r e e n tered on
the ri ght
h a n d p a ge which for co n ve n ie n ce i s di vided i n to s ma ll s q u a re s
by bl u e lin es with a red li n e ru n n i n g u p a n d down thro u g h the
middle
The s ta tio n s r u n fro m bo tt o m to top of p a g e Th e be a ri n g i s
t a ke n a t e a ch s etti n g a n d recorded ju s t a bove the corre s po n di n g
poi n t i n the n ote book or oppo s ite a p a rt of the li n e r a ther
th a n oppo s ite the poi n t Ordi na rily the tr a n s it ma n t a ke s the
be a ri n gs of a l l fen ce s a n d ro a d s cro ss ed by the li n e fi n d s the
a n d recor d s the m i n their
s ta tio n s fro m the re a r ch a i n ma n
proper pl a ce a n d direction on the ri ght h a n d p a g e of the n ot e
book S ectio n li n e s of the Un ited S ta tes L a n d Su r vey s s hou ld be
,
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-
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14
R a i lr oa d Cu r ve s
and
E arth w ork
.
be dri ve n with the fl a t s ide tow a rds the i n s tru men t a n d ma rked
on th e fro n t with the n u mber o f the s t a tio n
I n ter medi a te
s ta ke s s ho u ld be ma rked with t h e n u mber of the l a s t s ta tio n
the a dditio n a l di s ta n ce i n feet a n d te n th s a s IO
The
s t a t i o n i n g i s n o t i n ter r u pted a n d t a ke n u p a n ew a t e a ch tu r n i n
g
poi n t b u t is co n ti n u o u s fro m be g i n n i n g to en d of the s u rv ey
A t e a ch tu r n i n g poi n t a pl u g s ho u ld be dri ve n n e a rly fl u s h with
the g ro u n d a n d a witn e s s s t a ke dri v e n i n a n i n cl in ed po s i t io n
a t a di s t a n ce o f a bo u t 1 5 i n che s fro m t h e pl u
the
ide
a
n
d
a
t
s
g
towa rd s which the a d va n ce li n e deflects a n d ma rked W a n d
u n der it the s t a tio n of the pl u
g
24
The R ea r Ch a i n ma n hold s the re a r e n d of the t a pe
o v er the s ta ke l a s t s et bu t doe s n ot hold a ga i n s t the s ta ke to
loo s en it H e c a ll s C h a i n e a ch ti me whe n the n e w s t a ke i s
re a ched bei n g c a ref u l n ot to o v ers tep the di s t a n ce He s ho u ld
s t a n d be s ide the li n e ( n ot o n it ) whe n me a s u ri n g a n d t a ke p a i n s
n ot to ob s tru ct the V iew of the tr a n s it ma n
He checks a n d i s
re spo n s ible for the correct n u mberi n g of Sta kes a n d for a l l
di s t a n ce s les s th a n 100 feet a s the he a d ch a i n ma n a lw a y s hold s
the e n d of the ta pe The s t a tio n s where the li n e cro s s e s fen ce s
ro a d s a n d s tre a ms s ho u ld be s et down i n a s ma ll n ote book a n d
reported to the tra n s it ma n a t the e a rlies t co n v e n ien t opp or t u
The re a r ch a i n ma n i s re s po n s ible for the t a pe
n i ty
The B a c k F l a g hold s the fl a g a s a b a cks i ght a t the
25
poi n t l a s t occ u pied by the tr a n s it The o n ly s i g n a l s n ece ss a ry
“
for h i m to u n ders ta n d fro m the tra n s it ma n a r e pl u m
b the
”
fl ag
The fl a g s ho u ld a lw a y s be i n po s itio n
and
a l l ri g ht
The
a n d the tr a n s it ma n s ho u ld n ot be del a yed a n i n s t a n t
b a ck fl a g s ho u ld be re a dy to co me u p the i n s t a n t h e recei ve s
”
“
the a ll ri ght s i gn a l fro m the tr a n s it ma n The d u tie s a r e
s i mple bu t freq u e n tly a r e n ot well perfor med
The Axe ma n c u ts a n d cle a rs thro u gh fores t or br u s h
26
A g ood a xe ma n s ho u ld be a ble to keep the li n e well s o a s to
I n ope n co u n try he prep a re s the
cu t n othi n g u n n ece s s a ry
s ta ke s re a dy for the s t a kema n or a ss i s t s the loc a ti n g e n g i n eer
a s fo r e fl a g
27
T h e Le v eler h a n dle s the le vel a n d g e n er a lly keep s the
which ma y ha ve the followi n g for m for the left h a n d
n ote s
p a g e The ri gh t h a n d p a ge i s for re ma rk s a n d de s criptio n s o f
t u r n i n g poi n ts a n d ben ch ma rk s It i s de s i r a ble th a t t u rn i n g
,
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-
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Pr e li mi n a r y S u r vey
15
.
poi n t s s ho u ld where po s s ible be de s cribed a n d th a t a ll be n ch
R e a di n gs o n t u r n i n g
ma rk s s ho u ld be u s ed a s t u r n i n g poi n t s
poi n t s s ho u ld be recorded to h u n dredth s or to tho u s a n dth s of
de pe n de n t u po n the j u d g me n t of the C hief E n g i n eer
a foot
S u rf a ce re a di n gs s ho u ld be ma de to the n e a re s t te n th a n d ele
A s elf re a di n g rod h a s
va t i o n s s e t dow n to n e a re s t te n th o n ly
A t a r g et rod i s
a d va n t a g e s o v er a t a r g et r o d for s hort s i g ht s
po s s ibly better for lo n g s i gh ts a n d for t u rn i n g poi n ts The
”
Phil a delphi a R o d
i s bot h a t a r g et rod a n d a s e lf re a di n g
r od
a n d i s th u s well a d a pted fo r r a ilro a d u s e
B e n ch ma rk s
s ho u ld be t a ke n a t di s t a n ce s of fro m 1000 to 15 0 0 feet ,depe n di n g
u po n the co u n try
A ll be n ch ma rk s a s s oo n a s c a lcu l a ted
s ho u ld be e n tered to g ether o n a s peci a l p a g e n e a r the e n d o f
the book The le veler s ho u ld tes t h i s lev el freq u e n tly to s e e
th a t i t i s i n a dj u s tme n t The le v eler a n d I od ma n s ho u ld
to gether bri n g the n ote s to d a te e very e ve n i n g a n d pl a t th e
profile to corre spo n d
The profile of the preli mi n a ry li n e s hou ld s how
,
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‘
-
"
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-
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-
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-
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-
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-
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y
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.
a
S u rf a ce li n e ( i n bl a ck )
.
Gr a de li n e ( i n red )
b
.
0
.
d
.
6
.
f
.
.
Gr a de ele v a tio n s a t e a ch ch a n g e i n g r a de ( i n red )
.
R a te of g r a de , per 10 0 ( i n red ) ;
ri s e
fa ll
S t a tio n a n d deflec t io n a t e a ch a n g le I n the li n e ( i n bl a ck )
Note s of ro a ds ditche s s tre a ms brid g e s etc ( i n bl a ck )
.
,
,
,
,
.
.
.
The Rod ma n c a rrie s t h e rod a n d hold s it v ertic a l u po n
the gro u n d a t e a ch s t a tio n a nd a t s uch i n termedi a te poi n ts a s
ma rk a n y i mporta n t ch a n g e of s lope of the g ro u n d
The s u r
fa ce oi s tre a ms a n d po n d s s ho u ld be ta ke n whe n me t a n d a t
fre qu e n t i n ter va l s where po s s ible if they co n ti n u e n e a r the li n e
28
.
.
,
,
.
16
Ra i l r oa d Cu r ves
an
d E a r th work
.
e el s s ho u ld a ls o be t a ke n o f hi gh w a ter ma rks where ver
tr a ce s of the s e a r e vi s ible The rod ma n c a rrie s a sma ll n ote
book i n which he en ters the rod re a di n gs a t a l l tu r n i n g poi n t s
I n co u n try which i s ope n b u t n o t le v el the tra n s it p a rty i s
li a ble to ou tr u n the le vel p a rty In su ch c a s e s gre a t er s peed
will be s ec u red by the u se of two rod men
29
The T op og raphe r i s or s ho u ld be on e of the mo s t va l
u a b l e me mber s o f the p a rty
I n ti me s p a s t it h a s n o t a lw a y s
bee n fo u n d n ece ss a ry to h a v e a topo g ra pher or if e mployed
h i s d u ty h a s bee n to s ketch i n the e n er a l fe a t u re s n ece ss a ry to
g
ma ke a n a ttr a cti v e ma p a n d repre s e n t hills a n d b u ildi n gs s u ffi
ci e n t l y well with refere n ce to the li n e t o s how
i n a g e n er a l
w a y the re a s o n for the loc a tio n a dopted
S o meti me s the chief
o f the p a rty h a s for thi s p u rpo s e t a ke n the topo r a phy
At
g
pre s e n t the be s t pr a ctice f a v ors the ta ki n g of a ccu ra te d a t a by
the topo gr a phy p a rty
T h e topo gr a pher ( with on e or two a s s i s t a n t s s ho u ld t a ke the
)
s t a tio n a n d h e a r i n g
( or a n gle ) of e very fe n ce or s treet li n e
cro s s ed by the s u r vey ( u n le ss t a ke n by the tra n s it p a rty ) ; a ls o
ta ke me a s u re me n t s a n d be a ri ngs for pl a tti n g a ll fe n ce s a n d
bu ildi n gs n e a r e n o u g h to i n flu e n ce the po s itio n of the L oc a tio n ;
a l s o s ketch
a s well a s ma y be
fe n ces bu ildi n gs a n d other
topo gr a phic a l fe a tu re s of i n teres t which a r e t oo re mote t o r e
qu ire ex a ct loc a tio n ; a n d fi n a lly e s ta bli s h the po s itio n of
co n t ou r li n e s s tre a ms a n d po n d s withi n li mits s u ch th a t th e
L oc a tio n ma y be properly deter mi n ed i n the co n to u red ma p
The work of loc a ti n g co n to u rs i s u s u a lly a cco mpli s hed by the
u s e of h a n d le vel a n d t a pe
s t a n ce s c a ref u lly p a ced ma y i n
di
(
ma n y c a s e s be s u fficie n tly
The le vel p a rty h a s de
“
t e r mi n e d the ele va tio n s of the g ro u n d a t e a ch
s t a tio n
s e t by
The s e ele va tio n s a r e g i ve n the topo g r a phers
t h e tr a n s it p a rty
to s er ve a s be n ch ma rk s for u s e i n loc a ti n g co n to u rs I t i s cu s
t o ma r y to fix o n the g ro u n d the poi n t s where the co n to u r s cro s s
the ce n ter li n e where they cro s s lin e s a t ri ght a n gle s to the
ce n ter li n e a t e a ch s ta tio n a n d occ a s io n a lly a dditio n a l poi n ts ;
the n to s ketch the co n to u rs by eye betwee n the s e poi n ts C ro s s
s ectio n s h e e ts i n block s or i n book form a r e u s ed for thi s p u r
po s e The u s u a l co n to u r i n ter va l i s 5 feet
The topo g r a pher
A poi n t o n a co n to u r i s fo u n d a s follows
by ta pe
s ta n ds a t the s t a tio n s t a ke ; a m e a s u re me n t i s t a ke n
L v
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«
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-
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‘
,
Pr el i mi n a ry S u rvey
17
.
rod of th e di s ta n ce fro m the topo gr a ph er s feet to h i s eye
thi s a dded to the s u rf a ce hei ght a t ce n ter s t a ke ( a s obt a i n ed
”
“
fro m th e le vel p a rty ) g i v e s the hei ght of eye a bo v e d a t u m
The di ffere n ce betwee n th is hei ght of eye a n d the ele va tio n
of the co n to u r g i ve s the proper rod re a di n g for fixi n g a poi n t
o n the co n to u r a n d the rod i s c a rried v ertic a lly a lo n g the g ro u n d
The poi n t th u s fo u n d i s the n
u n til thi s re a di n g i s obt a i n ed
loc a ted The topo g r a pher u s e s thi s poi n t a lre a dy fix e d a s a
”
t u r n i n g poi n t fin d s a n ew h i s hei ght of eye a n d proceed s to
It i s more co n v e n ie n t a t ti me s
find a poi n t o n the n ext co n to u r
to c a rry on the proce s s i n re vers e order ; th a t i s to hold the rod
o n the g ro u n d a t the s t a tio n a n d let the topo g r a pher pl a ce h i m
“
”
The hei ght of eye
s elf where h i s feet a r e on the co n to u r
mu s t be the di s t a n ce fro m topog r a pher s feet to eye a dded to the
ele va tio n of co n to u r The rod re a di n g a t the s t a tio n will be the
“
di ffere n ce betwee n thi s hei ght of eye a n d the ele va tio n of
the gro u n d a t the s ta tio n
The h a n d le v el i s s o mewh a t l a cki n g i n preci s io n bu t by ma k
i n g a fre s h s t a rt a t e a ch s t a tio n a s a be n ch ma rk c u mu l a ti ve
errors a r e a v oided a n d f a ir re s u lts s ecu red by c a refu l work
I n s te a d of a h a n d le vel s o me topo gr a phers u s e a cli n o meter
a n d t a ke a n d record s ide l
pes a s a b a s i s for co n to u r li n e s
Topo gr a phy ca n be t a ke n r a pidly a n d well by s ta di a s u r vey or
by pl a n e t a ble Thi s i s s el d o m do n e a s ma n y e n g i n eers a r e
n o t s u fficie n tly f a mili a r with their u s e
M u ch more a cc u r a te
re s u lts ma y be re a ched by pl a n e t a ble a n d a p a rty of three
well s killed i n pl a n e t a ble work will a cco mpli s h more th a n a
p a rty of three with h a n d le vel
30
S o me e n g i n eers a d v oc a te ma ki n g a g e n er a l t op ogr a phi
ca l s u r vey o f the ro u te by s t a di a
i n s te a d o f the s u r vey a bo ve
“
s
de cribed I n thi s c a s e n o s t a ki n g ou t by s ta tio n s wo u ld be
do n e A l l poi n ts occ u pied by the t r a n s it s ho u ld be ma rked by
pl u gs properly refere n ced which ca n be u s ed to a id i n ma rki n g
the Loc a tio n on the g ro u n d a fter it i s determi n ed on the co n to u r
ma p Thi s method h a s bee n u s ed a n u mber of ti me s a n d i s
cl a i med to g i ve econ o mic a l a n d s a ti s f a ctory re s u lt s it i s prob
a ble th a t i t will h a v e co n s t a n tly i n cre a s i n g u s e i n the f u tu re
a n d ma y pro v e the be s t method i n a l a r e s h a r e o f c as e s
g
’
or
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’
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”
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‘
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CH A PTER I I I
LOC A T I ON S UR VE Y
’
III
.
.
.
c
the fin a l fitti n g o f th e li n e to
t h e gr o u n d
I n L oc a tio n c u r ve s a r e u s ed to co n n ect the s tr a i g ht
”
“
li n e s or t a n g e n ts a n d the a li g n me n t i s la id ou t co mplete
re a dy for co n s tru ctio n
The p a rty i s mu ch the s a me a s i n the preli mi n a ry a n d the
d u tie s s u b s t a n ti a lly the s a me M ore work de vol ve s u po n the
tr a n s itma n on a cco u n t of the c u rve s a n d it i s g ood pr a ctice to
”
n ote keeper
to the p a rty ; he ta ke s s o me of the tr a n
a dd a
s i t ma n s work a n d g re a ter Speed for the e n tire p a rty i s s ec u red
M ore s kill i s u s ef u l i n the he a d ch a i n ma n i n p u tti n g hi ms elf i n
po s itio n on c u r ve s H e ca n re a dily r a n ge hi ms elf o n t a n ge n t
The form of n ote s will be s how n l a ter The profile i s the s a me
except th a t it s hows for a li gn me n t n otes the P C a n d P T o f
c u r ve s a n d a ls o the de gr ee a n d cen tr a l a n gle a n d whether to
the rig ht or lef t
It i s well to con n ect fre qu e n tly loc a tio n s ta ke s with p r el i mi
n a ry s t a ke s whe n co n v e n ie n t a s a check o n t h e work
I n ma ki n g the loc a tio n s u rv ey two di s ti n ct method s a r e i n
u s e a mo n g e n g i n eer s
32
F i r s t M et hod of Loc a ti o n — Us e preli mi n a ry s u rv ey a n d
p r eli mi n a ry profile a s gu ide s i n re a di n g the co u n try a n d loc a te
the li n e u po n the gro u n d Experie n ce will e n a ble a n e n gi n ee r
to g e t v ery s a ti sf a ctory re s u lts i n thi s w a y i n n e a rly a l l c a s es
The be s t e n g i n eers i n loc a ti n g i n thi s w a y a s a r u le l a y the
t a n g e n ts firs t a n d co n n ect the c u r ves a fterwa rds
33
S e c o n d M et h od — Us e preli mi n a ry li n e preli mi n a ry pro
fil e a n d e s peci a lly the co n to u r li n e s o n the preli mi n a ry ma p ;
S o me
ma ke a p a per loc a tio n a n d r u n this i n on the g ro u n d
f
t
o
s
e
m
n
f
to
i
e
their
loc
a
ti
e
n
i
n
eer
a
co
plete
r
o
s
o
a
s
v
a
g
g
g
g
Thi s is g oi n g too fa r It i s s u fficie n t to fix
n ote s to r u n by
31
T he Lo a t i on S u rv ey i s
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-
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18
,
Su r vey
L oca ti on
19
.
the ma p the loc a tio n of t a n ge n t s a n d s pe c ify the de gr ee of
c u rve The s eco n d method i s mu ch more de s ir a ble bu t the
firs t method h a s s till s o me u s e It i s well a ccepted a mo n g
e n g i n eers th a t n o re vers ed cu rv e s hou ld be u s ed ; 20 0 feet of
t a n g e n t a t le a s t Sho u ld i n ter ve n e Neither s ho u ld a n y c u r ve
be very s hort s a y les s th a n 3 00 fee t i n len gth
A mo s t di ffic u lt ma tter i s the l a yi n g o f a lo n g t a n g e n t
34
L a ck of perfect a dj u s t me n t a n d
s o th a t it s h a ll be s tr a i g ht
”
“
co n s tr u ctio n of i n s tr u me n t will c a u s e a s wi n g i n th e t a n
The bes t wa y i s to r u n for a di s ta n t fore s i ght A n other
ge n t
w a y i s t oh a v e the tra n s it a s well a dj u s ted a s po s s ible a n d e ve n
the n ch a n g e e n d s e very ti me i n re ver s i n g s o th a t errors s h a ll
n ot a ccu rh dl a t e
It will be n oticed th a t the preli mi n a ry i s r u n
itho u t c u r ve s bec a u s e more eco n o mic a l i n ti me s o meti me s
in w
cu r ve s a r e r u n howev er either bec a u s e the li n e ca n be r u n
clo s er to i ts pr oper po s itio n or s o meti me s i n order to a l low of
fili n g pl a n s wi th the Un ited S ta te s Or s ep a r a te S t a te s
I n L oc a tio n , a s i n gle t a n g e n t ofte n t a ke s the pl a ce of a
35
broken li n e i n the preli min a ry a n d it beco me s i mporta n t to
deter mi n e the directio n of the t a n g en t with refere n ce to s o me
p a rt of the broke n li n e Thi s i s re a dily do n e by fin di n g the
co Or di n a t es o f a n y g i v e n poi n t with refere n ce to th a t p a rt of
the broke n li n e a s s u med te mpor a rily a s a meridi a n The
co u r s e of e a ch li n e i s c a lc u l a ted
a n d the coordi n a te s of a n y
poi n t thu s fou n d It s i mplifie s the c a lc u l a tio n to u s e s o me
p a rt Of the preli mi n a ry a s a n a s su med meridi a n r a ther th a n to
u s e the a ctu a l be a in gs of the li n e s
co Or d i n a t eS of two
The
r
poi n ts on the propo s ed t a n g e n t a llow the directio n of the
t a n gen t to be deter mi n ed with refere n ce to a n y p a rt Of the
preli mi n a ry Whe n the a n gle s a r e s ma ll a n a pproxi ma tio n
s u fi cie n tly clo s e will be s ec u red by a ss u mi n g i n a l l c a s e s th a t
f
the Co s i n e of the a n gle 1s 1 000000 a n d th a t the Si n es a r e directly
proportio n a l to the a n gles the ms elv e s I n a dditio n to thi s t a ke
the di s ta nce s a t the n e a re s t e v e n foot a n d the c a lcu l a tio n
beco me s mu ch s i mplified
”
“
36
The l ocated li n e or L oc a tio n a s i t i s Ofte n c a lled i s
s t a k ed ou t ordi n a rily by ce n ter s t a ke s which ma rk a s u cce s s io n
of s tra i ght li n e s c on n ected b y c u r ve s to w
hich the s tr a i ght li n e s
a re ta n g e n t
The s tr a ight li n es a r e b y ge n er a l u s a g e c a l le d
on
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T a n gen t s
.
,
,
CH AP T ER IV
.
S I MPLE C URVE S
.
The cu r ve s mo s t g en era lly i n u s e a r e Ci rc u l a r cu r ve s a l
tho u g h p a r a bolic a n d other cu r ve s a r e s o meti me s u s ed C ircu la r
c u r ve s ma y be cl a ss ed a s S i mp le Compou n d Re v er s ed or S p ira l
A S i mple Cu r ve i s a circ u l a r a r e exte n di n g fro m on e t a n
The poi n t where the c u rv e le a ve s the firs t
g e n t to the n ext
”
t a n g e n t i s c a lled the P C me a n i n g the poi n t of c u r va t u re
a n d the poi n t where the c u r ve j oi n s the s eco n d t a n g e n t i s
me a n i n g the poi n t o f t a n g e n cy
c a lled the P
The P C
a r e ofte n c a lled the T a n g e n t P oi n ts
and P T
Ii the ta n
g e n t s be prod u ced they will meet i n a poi n t of i n ters ectio n
”
“
c a lled the Ve rt ex V The di s ta n ce fro m the v ertex to the
”
“
The d is
T
P C or P T i s c a lled the
T a n g e n t Di s ta n c e
ta n ce fro m the vertex to the cu r ve ( me a s u red towa rds the
ce n ter ) i s c a lled the Ext e r n a l Di s ta n c e E The li n e joi n i n g
the middle of the Ch or d C with th e middle Of the c u rv e s u b
te n ded by th i s chord i s c a lled the M iddl e Ordi n a t e M The
r a di u s of the c u r ve i s c a ll ed the Ra d i u s R The a n gle of
deflectio n betwee n the ta n g en ts i s c a lled the In t e r secti on An g le
The a n g le a t the ce n ter s u bte n ded by a chord of 100 fee t i s
I
c a lled the De g ree O f Cu rv e D A chord of le ss th a n 100 feet
i s c a lled a s u b c h ord c ; i t s ce n tr a l a n g le a s u b a n g le d
The me a s u reme n t s on a c u r ve a r e ma de :
38
u b c ord
n fe t i mes a fu ll chord O f
fro
by
a
s
s
o
m
P
O
a
(
( )
h
100 ft ) to the n ext f u ll s t a tio n the n
and
n f u ll s ta tio n s
by
chord
of
feet
e
ch
betwee
10
0
a
s
b
( )
fi n a lly
a s u b chord ( s o me
fro
the
l
e
by
a s t s ta tio n on the c u r v
m
0
( )
ti me s a f u ll chord of 100 ft ) to P T The total di s t a n ce fro m
P C to P T mea s u r ed i n th i s wa y i s the Le n gt h O f Cu rv e L
The De g re e of Cu rve is defi n ed a s the a n gle s u bte n ded
39
by a ch o r d of 10 0 feet r a ther th a n by a n a r c of 100 fee t
37
.
,
.
,
,
,
.
,
.
.
,
,
.
.
.
.
'
.
.
.
,
.
,
.
.
.
.
,
.
.
,
,
,
.
,
,
,
“
.
,
.
.
,
-
-
,
,
.
.
-
.
.
,
.
,
,
,
.
.
.
.
.
.
-
,
,
.
.
.
,
20
.
R a i lr oa d Cu r ve s a n d E a r th wor k
22
.
S o me e n gi n eers u s e s horter chord s for s h a rp c u r ve s , a s 1
°
to
2 5 ft
ft 8 to
50 ft ; 1 6 to
Va l u es of R a n d D a r e re a dily co n vertible F or th is p u rpo s e
u s e T a ble I
A lle n [
ra ther th a n for mu l a ( 1 ) or
whe n a o
c u r a te res u lts a r e requ ired I n proble ms l a ter where either R
or D i s g i ve n both will i n g e n er a l be a ss u med to be g i ve n
A pproxi ma te va l u e s ca n be fo u n d witho u t t a ble s by
The
r a di u s of a 1 c u r ve 5 7 30 s ho u ld be re me mber ed P recis e
re s u lt s a r e i n g e n er a l n ece s s a r y
°
100
°
.
.
.
.
.
,
,
,
.
,
,
.
,
°
.
43
.
.
,
,
Pr obl em
Gi ven I , a l s o R or D
.
.
R eq u i r ed T
.
AOB = NVB = I
A0 : 0 8 = R
‘
T
Av = vs
T = R ta n } I
( 5)
.
E x a mple
.
Gi ven D
R eq u i r ed T9
T a ble I ,
.
I
9
60
°
.
R 9 l og 2
30 2 4' l og t a n 2
°
T9
l og
Note th a t l og R 9 i s t a ke n direc t ly fro m T a ble I
44
.
Appr oxi ma te M e t h od
.
.
T1 = R 1 t a n % I ; T = R a t a n § I
Ta
71
£
2
R1
Da
2
1
.
1
1
__
Da
( a pprox )
.
( a pprox )
.
T a ble
III
f
1
o
a
u
s
2
u
s
v
f
r
a
lle
i
e
l
e
o
o
rio
l
e
u
T
V
s
va
s
n
A
v
f
1
,
g
,
T a ble IV , A lle n , g ive s a correctio n to be a dded a fter d i vid
.
e
.
i n g by D “
.
S i mpl e Cu r ve s
E xa mple
As before
.
Gi ven D
.
9
R equ i r ed T9
23
.
60
I
°
.
T a ble I I I ,
9
(
.
( ppro x )
a
Ta ble I V c orre c t io n
‘
.
,
,
.
.
38
( ex a c t)
the sa me a s befo r e
45
Pr obl e m
.
Gi ven I , a l s o R or D
.
R eq u i r ed E
.
.
Us i n g pre v io u s fi gu re ,
VH
R exs ec } I
E
.
Ta ble X XXIII s hows defi n itio n of exs ec a n t
T a ble XIX g i ve s n a t u r a l ex s ec
T a ble XV g i ve s lo ga rithmic exs ec
App r o xi ma te M eth o d
.
.
.
.
.
.
.
B y method u s ed for
( a pprox )
.
T a ble V g i ves v a lu e s for R I
.
46
.
P r ob l em
Gi ven 1
.
.
; l
a s o R or
R eq u i r ed M
D
.
.
FH = M :
R ve r s % I
Ta ble XXX I I I s h ows defi n i tio n Of vers in e
.
T a ble
XIX g i e
t u r a l ver s
T a ble X V g i v e s lo ga rith mic vers
T a ble II g i ve s cert a i n middle ordi n a tes
v s na
.
.
.
.
.
47
.
P r ob le m
.
Gi ven I , a l s o R or D
C
2 R s in
.
.
R equ i r ed ch o r d A8
Ta bl e V
.
C
.
I
III gi ve va l e for cert i lo g ch ords
.
s
u
s
a n
n
.
R a i l r oa d Cu r ve s
48
.
Tr a n s po s i n g , we fin d a dditio n a l for mu l a s , a s follows
.
fro m ( 5 )
R :
(7)
R
(9 )
R
( 10 )
R
T co t é I
v
ers t I
2 Mn % I
x
ppro
(
)
4
( )
49
E a r th work
and
.
bl em
P ro
a
.
( 6)
Da
ppro
x
(
)
8
( )
Da
ppr
o
x
)
(
a
.
a
.
Gi ve n s u b- a n g l e d , a ls o R or D
.
.
Requ i r ed s u b ch o r d c
-
.
c
2 R s i n 4d
A pproxi ma t e M ethod
1 00
'
c
100
Th e
.
2 R s in
M
sin
D
d
-
gD
D
ppr
o
x
)
(
a
.
preci s e for mu l a i s s eldo m if e ver u s ed
Pr oble m
’
.
.
Gi ve n s u b- c h o r d 0 , a ls o R or D
R eq u i r e d s u b- a n g l e d
.
cD
100
l e
va u
18
more frequ e n tly n eeded a n d
d
c
2
1 00 2
D
.
S i mpl e Cu r ve s
.
A mo di fic a t io n of thi s for mu l a i s a s foll ows
6 x
for a n y va lu e D ,
Q
x D “ ( re s u lt i n mi n u te s )
‘
c x
2
T hi s gi v e s
v
a
ery s i mple a n d r a pid method of fi n di n g the
l e of Q i n mi n u tes a n d the formu l a s ho u ld be re me mbered
va u
,
2
E xa mple
.
Gi ven s u b h or d
D
-c
6
°
R eq u i r ed s u b a n g l e d
-
( 2 0)
B y ( 2 1)
D
3 1 85
3 13 5
3 82 2
12 7 4
19 1 1
60 '
d
d
°
4 0 8’
60
’
2 04
’
°
°
2 0
'
4
2
111
.
B y ( 2 2)
D
9 5 55
11466
mi n u te s
1
2
.
'
°
2 04
2
M eth od
is
ofte n prefer a ble to I or I I
.
.
.
R a i lroa d Cu rves a n d E a r th work
51
.
P r oble m
.
Gi ven [ a n d D
R eq u i r ed L
.
.
.
The L e n g th of Cu r ve L i s the di s ta n c e a ro u n d the c u rv e
me a s u red a s s t a ted i n 3 8 or L 6 1 100 n
62
n
t
n
i
a
a
Whe
the
f
will
be
co
t a i n ed
a
P
s
u
ll
s
t
i
n
a
C
t
o
D
( )
i n I a cert a in n u mber of ti me s n a n d there will re ma i n a s u b
1 00 n
02
a n gle d z s u bte n ded by i t s ch ord 6 2 a n d L
“
,
:
,
.
.
,
.
,
.
,
I
+
n
D
I
—
a
100
100 n
L (a
02
I)
ppr ox
pprox )
.
Whe n the P G i s a t a s u b s ta ti o n a n d P T a t a fu ll s ta
t i o n the s a me re a s o n i n g hold s a n d
( b)
.
-
.
,
.
.
,
L
100
I
—
pprox
)
(
a
.
n d P T a r e a t s u b s ta ti o n s the s a me
h
a
n
Whe
bot
C
P
( )
formu l a hold s
1
pprox
100
a
L
)
(
0
-
.
.
.
,
-
.
Tr a n s po s i n g
_
,
f£
pprox
)
(
a
.
1 00 1
L
The s e formu l a s ( 23 ) ( 24)
for mu l a s i n co mmo n u s e
E x a mple
Gi ve n 7 c u r ve
tho u gh a pproxi ma t e
,
are
.
°
.
39
I :
.
R eq u i r ed L
°
°
39 37
I
D
E
x a mple
.
G i ve n D a n d L
Gi ve n 8
a ls o ,
°
cu r v e
.
.
93
P C
.
86
.
L :
D
R eq u i r ed I
.
.
P T
.
.
7
8
60
’
I
:
'
4
0
57
°
.
the
S i mp l e Cu r ves
52
F i el d -w or k of fi n d i n g P C
.
.
a nd P
.
27
.
.
T
.
ru n n i n g i n the li n e it i s co mmo n pr a ctice to co n ti n u e the
s t a tio n n
a s fa r a s V to s e t a pl u g a n d ma rk a wit n e s s s t a ke
with the s t a tio n of V a s t hu s obt a i n ed The a n g le I i s the n
”
”
“
“
me a s u red a n d repe a ted a s a check
Ha vi n g g i ve n I o n ly a n i n fi n ite n u mber of c u r ve s ci u ld b e
It i s therefore n ece s s a ry t o a ss u me a dditio n a l d a t a to
u s ed
deter mi n e wh a t cu r ve to u s e It i s co mmo n to proceed a s
follo ws
In
,
,
.
.
i
,
,
,
.
'
.
A
s
u
m
e
either
1
directly
s
D
)
( )
(
a
.
c a lcu l a te D
n d c a lc u l a te D
3
T
a
( )
E an d
(2)
.
.
It i s of te n difficu lt to determi n e o ff h a n d wh a t de g ree of c u rve
will be s t fit the g ro u n d F req u e n tly the va l u e of E ca n be
re a dily deter mi n ed on the g ro u n d The deter mi n a tio n of D
fro m E i s re a dily ma de u s i n g the a pprox i ma te for mu l a
“
-
,
.
.
,
,
D,
E1
Ea
t a i n a bl e )
S i mil a rly , we
va l u
e of T
a ,
ma y be li mited to a g i ve n ( or a s ce r
and
fro m thi s re a dily fin d D
11
a
Ta
This proce s s i s to deter mi n e wh a t va l u e of D will fit the
r h a lf
d
o
u
s
ro
d
it
co
e
ie
t
e
er
lly
to
e
the
e
ree
n
i
n
n
n
n
u
n
d
v
a
a
s
g
g
g
'
4
n
n
e
1
0
de gree n e a re s t t o th a t c a lc u l a ted ( S o me e g i eers u s
’
’
etc )
etc r a ther th a n 1 3 0 o r 3
100 a n d 3
Whe n the D i s th u s det ermi n ed a ll co mpu t a tio n s mu s t b e
s trictly b a s ed o n thi s va l u e o f
i
u l a ted a n ew
F
ro
m
the
d
t
fi
n
a
lly
a
dopted
T
s
l
c
a
a
c
a
b
( )
n g a t V the P T i s s e t by l a yi n g
The
i
r
e
t
till
bei
n
u
m
n
s
t
s
c
( )
o ff T
I t i s eco n o mic a l to s e t P T before P 0
m
u
n
d
s
e
t
The
s
t
tio
n
f
i
s
c
a
lc
l
a
ted
a
P
0
fro
the
a
o
P
d
O
( )
n e a re s t s t a tio n s t a ke
u ri n g b a ck fro m V )
or
by
m
e
a
s
(
n d s t a tio n of P T
a
The
le
th
c
r
e
c
lc
l
ted
e
n
i
s
a
u
a
f
u
o
v
L
g
( )
th u s deter mi n ed ( n ot by a ddi n g T to s t a tio n of V )
Whether D E or T s h a ll be a s s u med depe n d s u po n the
C u r ve s a r e ofte n r u n ou t
s peci a l re qu ire me n t s i n e a ch c a s e
fro m P 0 witho u t fi n di n g or u s i n g V b u t the be s t e n g i n eeri n g
u s a g e s ee ms to be i n f a v or of s etti n g V whe n e ver thi s i s a t a l l
pr a ctic a ble a n d fro m thi s fi n di n g the P C a n d P T
0
,
,
,
°
.
°
°
°
.
.
,
,
,
.
.
.
.
.
,
.
.
.
’
.
.
.
.
.
.
.
,
.
,
,
.
.
.
,
‘
,
,
.
.
.
.
.
R a i lroa d Cu r ves a n d E a r t hwor k
53 ; E
xa mp l e
.
.
G i ve n a l i n e, a s sh own i n s ketch
.
R eq u i r ed
S i mp l e
a
T a n g en ts
P T i s to be a t le a s t 3 00 ft
o
C u r ve to
c n n ect
.
fro m e n d of li n e
Us e s ma lle s t de gree or h a lf de gree co n s i s te n t with th is
F i n d d e g ree of c u r ve a n d s t a tio n s of P C a n d P T
.
.
.
.
.
T a ble III
TI
.
.
T ( a pprox )
.
92
1 2 5 8 g5
TI
T a ble I V COI I
.
.
.
u se
5
°
c u r ve
20 5
.
‘‘
.
.
230
°
.
07
I
:
22
°
'
14
T
V
46
T r:
2
P C
.
44
.
L r:
4
P T 2 : 48
.
.
I t wi ll be n oticed t ha t the s t a tio n
of the P T i s fou n d by
a ddi n g L to the s t a tio n s of the P 0
by
ddi
n
T
to
the
a
n
o
t
g
(
s t a tio n of V )
.
.
.
.
.
S imil a rly
Ta ble V
1 7 ft
Gi ven E
.
17
E1
.
E
°
u se 6
30
I :
V:
46
L
T
I
P G
44
10 2
’
c u rve
70
c orr
.
.
09
T:
.
.
L :
U der the co ditio
pre s cribed a bo ve whe n T i s gi ve n the
de gree or h a lf de gree n ext l a rger mu s t be u s ed i n order to
s ec u re a t l ea s t the req u ired di s t a n ce ( to e n d of li n e a bo v e )
Whe n E i s g i ve n th e n e a re s t h a lf de gree i s g e n era lly u s ed
n
n
,
ns
,
,
,
,
.
,
.
R a i lr oa d Cu r ves a n d E ar thwork
30
.
Pr oble m
To fin d the To ta l D efl ecti ons for a Si mpl e
Cu r ve h a v in g g i ven the D eg r e e
56
.
.
“
.
I
.
Wh en the cu r ve beg i n s a n d en ds a t even s ta ti o n s
The di sta n ce fro m sta tio n to s t a tion is 100 feet
tio n a n gle s a r e r equ i red
.
.
The
.
cu t e a n gle between a t a n ge n t a n d a chord is e qu a l to
o n e h a lf the ce n tr a l a n g le s u b t e n ded by th a t chord
An
a
A| :
100
The a cu te a n gle betwee n two chords which h a s i ts vert ex i n
th e circ u mfere n ce i s eq u a l to on e h a lf th e a r e i n cl u ded betwee n
t ho s e chord s
1 00 a n d I A2
I
2
i
m
il
rly
D
S
a
%
.
:
2
3 —
2 A3 = % D
an d
‘
B
:
,
100 \ a n d
Thi s a n gle 4, D i s c a lled by H e n ck a n d Se a rle s t h e Defle ct io n
S hu n k a n d T r a u t wi n e c a ll
An g le a n d will be s o c a lled here
”
it the T a n g e n ti a l A n g l e
The weight of e n g i n eeri n g opin io n
”
D eflectio n A n g le
a ppe a r s t o be l a r g ely i n f a v or of the
”
will be a s follows
Th e
Tota l D efl ecti on s
‘
.
,
.
.
vA2 = VA 1 + i D
V A3 :
VA2 + % D
VAB will be fo u n d by s u cce s s i ve I n cre me n t s of % D
VAB
ta t i on
IL
VBA
1
Q
.
Thi s f u r n i s he s a
check
on
.
the co mpu
.
Wh en th e c u r ve beg i n s a n d en ds wi th a s u b ch or d
-
VA | = % d
-
VA 2 = VA I + § D
VA3 = VA2 + i D
.
‘
S i mpl e Cu r ve s
VAE i s fo u n d by a ddi n g
31
.
d2 to pre vio u s
o l d efle ctio n
”
Thi s f u rn i s he s check
The tot a l d efl e c
VAB VBA 41
tio n s s ho u ld be c a lc u l a ted by s u cce s s i ve i n cre me n t s ; the fi n a l
ch eck u po n g I the n checks a ll the i n ter medi a te tota l
deflectio n s T h eex a mple on n ext p a g e will illu s tra te th is
”
t ta
.
.
.
.
.
57
.
c r ve h a vi g g i ve
F ield -w ork of l a yi n g ou t a s i mple
n
u
n
th e
po s itio n a n d s ta tion O f P C a n d P T
( a ) S e t the tr a n s it a t P C ( A)
e t the vern ier a t 0
b
S
)
(
e
6
S
t
s
a
s
o
n
or
o
n
cro
s
h
ir
V
n
N
a
d re ve r s e)
( )
(
e
m
S
t
off
s
o
e t i me s gD ) for poi n t I
d
d
1
)
(
g (
( e) M e a s u re dis t a n ce cl ( s o meti me s 100 ) a n d fix I
( f ) S et ofi tot a l deflectio n for poi n t 2
1 00 a n d fix 2 e tc
( g ) M e a su re di s t a n ce I Q
h
n
h
W
e
tot a l deflectio n to B is figu red s ee t h a t it } I
( )
th u s checki n g c a lcu l a tio n s
i ) S ee t h a t the proper c a lc u l a ted di s t a n ce 6 2 a n d th e tota l
deflectio n t o B a g ree wit h the a ctu a l me a su re me n t s o n th e
g ro u n d checki n g th e fie l d work
( k) M o ve t r a n s it to P T ( B )
( l ) T u rn verni er b a ck to 0 a n d beyon d 0 to 5 1
( m) S i g ht On A
( n ) Tu r n vern ier to 0
( o ) S ig ht t ow a rd s V ( or re vers e a n d s i ght towa rd s P) a n d s ee
th a t the li n e checks o n V or P
It s ho u ld be Ob s er ved th a t three c h ec ks on th e work a r e
Obt a i n ed
1
The c a lc u l a tio n of th e tota l deflectio n s i s checked if to t a l
deflectio n to 8 } I
2 The c h a i n i n g i s checked if the fi n a l s u b chord me a su r ed
on the gro u n d
c a lc u l a ted di s t a n ce
3 : The tr a n s it work i s checked if t h e to t a l deflec t io n t o 8
bri n g s the l i n e a cc u r a tely on B
The check i n I i s e ffecti ve o n ly whe n th e to ta l deflectio n for
e a ch poi n t i s fo u n d by a ddi n g th e proper a n gle to th a t for the
precedi n g poi n t
The check i n 3 a s s u re s the g e n er a l a ccu r a cy of the tr a n s i t
work bu t doe s n ot pre ve n t a n error i n l a yi n g off t h e to ta l
deflectio n a t a n i n termedi a te poi n t on the c u r ve
.
.
.
.
.
.
.
.
.
.
.
.
-
,
.
.
,
.
-
,
.
.
.
.
,
.
‘
.
.
,
.
.
:
1
.
-
.
.
.
.
.
,
.
,
R a i lr oa d Cu r ve s a n d E a r thwork
32
E xa mple
.
.
Gi ven Notes of Cu r ve
P T
.
.
13
0
R C
.
10 +
.
11 0 1
cu
m r‘
flc
R equi r ed th e
to s t a
6
tota l d e e ti on s
26
.
3
6
0
d1
t o 11
2
3
45
v
.
,
°
3 4 7 I t o 12
3
3
°
°
6 47
6
'
t o 13
0
i
Sx
2
8 08 t o 13
°
’
16'
1 6 16 ’
°
1
°
iI
’
8
8 0
59
Ca u t i on
—
c he ck
.
If a cu rv e of n e a rly 1 80
I i s to be l a id ou t fro m A I t i s
e vide n t th a t it wo u ld be di ffi c u lt or i mpo s s ible t o s et the l a s t
”
“
poi n t a cc u r a tely a s the i n ter s ectio n wo u ld be b a d I t i s
u n de s ir a ble to u s e a tot a l deflectio n g re a ter th a n
I t ma y be i mpo s s ible to s ee the e n tire cu r ve fro m th e R C
°
,
.
,
.
at A
.
It will therefore frequ e n tly h a ppe n th a t fro m on e ca u s e or
a n other the e n tire c u r ve c a n n ot be l a id ou t fro m the P G a n d
it will be n ece s s a ry t o u s e a modifi c a tio n of t h e meth od de
s c ribed a bo ve
,
,
,
.
S i mple Cu r ves
60
.
F i eld -w or k
f m th e P C
ro
.
33
.
Wh en th e e n ti r e cu r ve ca n n ot be l a id o u t
.
.
F irs t M é t h od
.
c
u rv e a s fa r a s C a s before
( )
tr
a n s it poi n t a t s o me co n v e n ie n t poi n t a s C ( e v e n
e
t
S
b
( )
s t a tio n prefer ably ) a n d mo v e tr a n s it to C
T
a n d beyon d 0 by t h e va lu e of
u r n v er n ier b a ck to
6
( )
a n g le VAC
S i ght on A
T
r
e e t h a t t r a n s i t li n e i s on a u xili a ry
n ver n ier to
S
u
6
( )
t a n g e n t NC M ( VAC NCA bei n g me a su red by 5 a r e AC)
off
n e w deflectio n a n gle ( 5d or 4D )
e
t
S
(f )
poi
t
n d proceed a s i n ordi n a ry c a s e s
n
4
a
e
t
S
(g)
a
L a y ou t
.
,
,
.
°
.
.
»
1
.
.
.
,
S eco n d M ethod
(a )
( b)
c
( )
.
S e t poi n t C a s before , a n d mo ve tr a n s i t to C
.
S e t ver n ier a t 0
i ht on A
for the poi n t 4 VA4
S e t o ff the proper tot a l deflectio n
NCA MC 4
VA4 e a ch me a s u red by 4 a r e AC 4
( d ) R e vers e tr a n s it a n d s e t poi n t 4
”
“
( 6 ) S et off a n d u s e the proper tot a l deflectio n s for th e
re ma i n i n g poi n ts
The s eco n d method i s i n s o me re s pect s more s i mple a s the
n ote s a n d c a lc u l a tio n s a n d a l s o s etti n g Off a n gle s a r e the s a me
a s if n o a dditio n a l s et t i n g wer ema de
B y the firs t method the
deflectio n a n gle s to be l a id Off will i n g e n er a l be e v e n mi n u te s
O fte n de gree s or h a lf de gree s a n d a r e th u s e a s ier to l a y off I t
i s a ma tter of p er s o n a l choice which of the two mé t h o ds s h a ll
be u s ed It will be di s a s tro u s to a ttempt a n i n correct co mbi n a
tio n of p a rts of the t wo method s
°
and s g
.
:
.
.
,
.
.
,
,
,
.
,
,
,
.
,
.
.
34
R a i lr oa d Cu r ves a n d E a r th work
F ield w ork
Wh e n th e tr a ns i t i s i n th e c u r ve a n d th e
-
P C i s n ot v i s i bl e
.
.
(a )
C o mp u te
.
.
,
.
deflectio n a n gle s
,
P C
.
check on
to P T ;
.
.
1
.
2
( s a me a s i n
( b) S e t vern ier a t deflect i o n a n g le co mp u ted for poi n t ( e g 2 )
.
.
ed a s b a cks i g ht
e t li n e o f s i ht o n b a ck s i ht
0
S
cl
p
2
n
a
d
a
m
)
(
g
g
( )
If v er n ier be ma de to re a d
the li n e of s ight wi ll the n be i n
directio n o f P C ( s i n ce a n gle LA? 2 4A)
( d ) S et Off d e fl ectio n a n gle s co mp u ted for 5 etc
6 2 F i eld wo r k
Wh e n en t i r e cu r ve i s vi s i bl e fr o m P T
us
.
.
.
.
.
,
.
.
-
.
.
( a ) C o mpu te deflectio n a n gle s , P C
m
i
s
a
e
n
a
s
(
.
S et tr a n s i t a t P T
.
check o n
to P T
.
.
.
( b)
(6)
with vern ier a t 0 a n d s i ght on P C
S e t Off co mp u ted a n g le s for 1 2 3 4 5
(d)
S e t o ff
.
°
.
.
,
5
,
,
,
.
.
i ht a t V for check on tr a n s it work
a nd s g
.
Thi s method i s prefer a ble to t h a t g i ve n i n 5 7 It s a ves the
tra n s i t s etti n g a t P C The l on g s i ghts a r e t a ke n firs t before
error s of ch a i n i n g h a ve a cc u mu l a ted a n d before the tra n s it h a s
s ttled or w a rped i n the su n li g ht
The l a s t poi n t on c u r v e i s
s e t a t a s ma ll a n g le with the t a n g e n t s o th a t the i n ter s ectio n i s
g ood a n d a n y a cc u mu l a ted er r ors of ch a i n i n g will n ot mu ch
The method i s a lre a dy a ccepted pr a ctice
a ffect the li n e
.
.
e
,
.
.
,
.
.
S i mpl e Cu r ves
63
.
M etr i c Cu rv es
35
.
.
Metr i c S ys tem a ch a i n of
100 meter s i s too lo n g , a n d a ch a i n of 10 meter s i s t o o s hort
S o me e n g i n eer s h a v e u s ed the 3 0 meter c h a i n s o me the 2 5
meter ch a i n , b u t l a tely the 2 0 meter ch a i n h a s bee n g e n er a lly
U n der thi s s y s te m a
S ta
a dopted a s the mo s t s a ti s f a ctory
”
Ordi n a rily , e very s eco n d s ta tio n on ly i s
i s 1 0 meters
ti on
On c u r ve s ,
St a 2 , St a 4 , etc
s et , a n d the s e a r e ma rked St a 0
ch o r ds o f 2 0 meter s a r e u s ed Us a g e a mo n g e n g i n eer s va rie s a s
to wh a t is me a n t by the D eg r ee of Cu r ve u n der the metric
There a r e t w o d i s ti n ct s ys te ms u s ed , a s s how n below
s y s te m
I n R a ilro a d L oc a t io n u n der th e
“
.
-
,
-
.
.
.
.
.
.
,
.
.
.
I T h e D eg r ee of Cu r ve i s the a n g l e a t t h e c en ter su bte n ded
by a chord of 1 ch a i n of 2 0 meter s
II
The D eg r ee of Cu r ve i s the deflecti o n a n g l e for a chord
o f 1 ch a i n of 2 0 meter s ( or o n e h a lf t h e a n g le a t the c e n t er ) ;
II Or v er y clo s ely the D eg r ee of Cu r ve i s th e a n g l e a t th e
cen ter s u bte n ded by a chord of 1 0 meter s ( e qu a l t o 1 s t a t io n
le n gth )
T a ble s
F or s e v er a l re a s o n s t h e l a t t er s y s te m i s f a v ored he r e
u po n thi s b a s i s h a v e bee n c a l c u l a t ed
g i vi n g cert a i n d a t a for
metric c u r ve s S u ch ta bles a r e t o be fo u n d i n Alle n s F i eld
a n d Office T a ble s
I n ma n y co u n trie s where the me tric s y s te m i s u s ed it i s n ot
cu s to ma ry to u s e the D eg r ee of Cu r ve a s i n dic a ted here I n
M exico where the metric s y s t e m i s a dopted a s the o n ly le g a l
s t a n d a rd v ery ma n y of t h e r a ilro a d s h a v e bee n b u ilt by com
pa n i es i n corpor a ted in thi s co u n try a n d u n der th e directio n of
e n g i n eers tr a i n ed here The u s a g e i n dic a ted a b ove h a s bee n
the re s u lt of the s e co n d i tio n s If the metric s y s te m s h a ll i n
th e fu t u re beco me the o n ly le g a l s y s te m i n the Un ited S t a te s
a s n ow s ee ms po s s ible on e o f the s y s te ms o u tli n ed a bo v e will
prob a bly preva il
I n forei gn co u n trie s where th e D eg r ee of Cu r ve i s n ot u s ed i t
i s c u s to ma ry a s pre vi o u s l y s t a ted to de s i gn a te th e c u rv e by
i ts r a di u s R a n d t o us e e ve n fi g u re s a s a r a diu s of 1000 feet
or 2 0 00 fee t o r 1000 meter s or 2 000 meter s
A s the r a di u s i s
s eldo m me a s u r ed on the g ro u n d the o n ly co n v e n ie n ce i n e v e n
figu re s i s i n pl a tti n g while there i s a co n s t a n t ly re cu rri n g in con
ve n i en ce i n l a y i n g off the a n g le s
.
.
.
'
.
,
,
.
.
,
’
.
.
,
.
,
,
,
,
.
.
,
,
.
,
,
,
,
,
,
,
.
,
,
«
,
.
R a i lr oa d Cu rves a n d E a r th work
64
.
For
m of T r a n s i t Book ( le ft h a n d pa g e )
.
-
.
( D at e )
f
N
a
m
e
o
a
r
t
y
s
P
)
(
St a t i o n
Po mt s
De scri p of
Tota l
Cu rve
Deflect
.
.
N4 6 00
0
9 +
P T
.
,
.
V
0 +
R C
N23
V i s n ot a poi n t o n the c u r ve
.
°
l5 E
’
Ne vert hele ss , it i s c u s t o ma ry
to record the s t a tio n fo u n d by c h a in i n g a lo n g the ta n g e n t
The ri ght h a n d p a ge i s u s ed for su r vey n ote s of cro s s i n gs o f
fe n ce s a n d va rio u s s imil a r d a ta It s ee ms u n n ece s s a ry to s how
a s a mple here
.
-
.
.
R a i lr oa d Cu r ve s a n d E a r thwork
38
.
The deflect i o n a n gle s will be ( to n e a re s t g mi n u te ) 1 52
’
3 44
5
9
F o r ch a i n i n g the
7 29’
le n g th of chord i s n ece s s a ry a n d ma y be co mpu ted by for mu l a
Where the r a di u s i s l a rg e n a tu ra l s i n e s ma y n o t g i ve
s a ti s f actory re s u lt s a n d it m a y be n ece s s a ry to u s e the a u xili a ry
t a ble s of l og s i n e s
A s i mpler method i s to u s e A lle n s T a ble XX A which gi v e s
fo r R
1 the differe n ce be t wee n a r c a n d c hord for va rio u s ce n
tra l a n gles
°
°
°
°
°
,
,
,
.
.
’
,
.
F or
c e tra l a gle 3 45
n
°
n
diff
’
T a ble XX
.
.
,
A
.
600
R
0 00 7
.
Ar c
Chord
The P T of th e c i rcu l a r a r c s ho u ld be s et wi th the requ ire d
m
n
n
reci
io
by
lo
chord
fro
s
P C a n d the s e ver a l chord s me a s
g
p
”
check
u r e d with a de g ree o f preci s io n s u ffi cie n t to s ecu re a
a g a i n s t ma teri a l error
.
.
.
.
.
II ( a ) Us e a s er i es of eq u a l ch o r ds of con ven i en t l en g th
fol l o wed by a s u b ch or d to th e P T
n
n
s
o
p
te
deflectio
le
to
corre
po
d
m
u
n
a
s
C
b
g
( )
m
c
o
p
u t e a r c le n g th s to corre s po n d
C
( )
m
b
n
o
p
e
chord
le
C
u
t
s
u
d
g th
( )
’
600
43
1
I
8
R
Ex a mp le
Gi ve n a s befo r e
Ta ke chord le n gth of 40 i t
deflect io n a n gle for chord of 40 ft
Le t i l
.
-
.
,
.
.
.
-
.
°
:
.
.
.
.
Then s i n
2°
a
600
corre s po n di n g ce n tr a l
’
di fi
F or c e n t r a l a n gle 3 49
and
ang
'
"
3
4
1 5
7
°
:
’
=
9
4
3
le d1
°
Ta ble XX
°
.
.
,
A
.
R
a rc
4 le n gt h s of a r c
en
fro m p 3 7
for R 600
tire a r c
.
su b-a r c
600
0 06 009 5 :
.
s u b-a r c for R :
1
S i mple
Fro m p 38
Ta ble XX
.
.
0 0 60 09 50
,
3
,
Cu r ves
s u b -a r c
39
.
for R
1
°
3 51
26
’
3 5"
F or
c e tr a l
n
a ng
le 3 2 7 ' di ff
°
.
R
I II
even
.
f
Us e u n i or m defle cti o n
( )
a
a n g l es
mi n u te, excep t f or fi n a l s u b—ch o r d
to
so
me
c
on ven i en t
.
( b) C o mp u te chord le n g th s to corre s po n d
m
r c le n g th s to corre s po n d
o
p
u
te
a
0
C
( )
.
.
Exa mp le
.
f
Gi ven a s be o r e
F o r 5 eq u a l a rc s
A ss u me i 1
F or
“
°
2 00
I
.
"
29
1 8 43
°
:
1
il
the n
'
cen tr a l a n gle 4
d i ff
Ch ord le n gth for 4
4 00
T a ble XX
.
°
°
1 8 43
’ 2
”
9
16
fi n a l su b gle dz
F or ce n tr a l a n gle 2 43 ’
2 43 2 9 ”
°
°
°
-a n
diff
'
T a ble XX
.
F or
c e t ra l a gle 2 43 2 9
n
n
’
"
A
,
600
R
°
.
Ta ble XX
are
.
600
R
2 8 5 3 3 24
.
di ff
fi n a l s u b—c hord
.
1
.
A
600
4 x ce n t r a l a n gle 4
°
,
len gt h
a rc
I
ce n tr a l a n gle
'
.
2 x 600 x s i n 2
:
600
”
52 2 1
R
°
R
’
°
2 i1
°
°
’
.
R a i lroa d Cu r v es
39 A
a nd
E a r thw or k
.
A co n v e n i e n t for mu l a
for the differe n ce betwee n chord a n d
a r e i s the followi n g which th o u g h a pproxi ma te
i s e ss e n ti a lly
correct whe n the va l u e of the chord c i s n ot l a rg e i n co mp a ri s o n
with R
L et
l
le n gth of a r c
c le n g th of corre s po n di n g chord
,
,
.
.
The n l
c
03
13
24 R 2
24 R 2
both a pp rox 1ma te
For s u ch va l u e s a s
or c 5 0 a n d R 1000 or R 2 000
the co mp u t a tio n i s a t o n ce s i mple
F or other va l u e s the co mp u t a tio ns a r e co n v e n ie n tly ma de
o n a s li de r u le
It ma y re a dily be
No proof of this for mu l a i s g i v e n here
pro v ed a lo n g the li n e s of 1 88 p 1 19 ma kin g u s e of formu l a
p 42 a s a for mu l a of the circle
I n a cu r v ed s treet it i s n ot u n co mmo n to de s cribe the a li g n
me n t by g i vi n g the r a di u s R of the ce n ter li n e a n d a l s o the
dist a n ce ( or s t a tio n i n g ) a lo n g the ce n ter li n e me a s u red a lo n g
the a r c ( r a ther th a n by a s eries of chord s a s 111 r a i lro a d work)
It is a ls o n ece s s a ry to k n ow the len gths I, a lo n g the property
li n e on the o u ts ide of the c u r ve a n d the le n g th I a lo n g the i n
s ide li ne
:
:
,
.
,
.
.
.
,
.
,
.
,
,
,
,
,
.
,
,
.
L et A
l,
I,
I,
w,
ce n tr a l a n g le s u bte n di n g p a rt or whole
corres po n di n g le n g th on ce n ter li n e
o u ts ide l i n e
i n s ide l i n e
width fro m ce n ter to o u ts ide li n e
i n s ide li n e
tot a l width of s treet
w
w;
R a n g le A
l
w, ) a n g le A
l, = ( R
w ) a n g le A
l = (R
w, a n gle A
l
l,
w a n gle A
l= l
.
.
,
o
,
.
c
,
,
,
c u r ve
.
S i mp le Cu r ves
39 B
.
The v a lu e s of w, a n d w a r e u s u a lly n ot l a rg e a n d co mmo n ly
e ve n n u mbers The co mp u t a tio n s of di ffere n ce s therefore a r e
more s i mply ma de th a n co mp u t a tio n s of tot a l v a l u e s
This i s tr u e whether A s u bte n d s a n a r c of 100 ft or a s u b a r c
o r the f u ll a r e fro m P C to P T
S i mil a rly for a n y chord o n the ce n ter l i n e
,
,
,
,
.
.
.
.
.
.
2 R si n % A
61 :
2 ( R + 10 1) S i n % A
c,
2 ( R — w3 ) s i n % A
cc :
2 w, s i n % A
cs
2 w, s i n % A
we
R
61
,
.
cc :
:
,
R
00
R
S o meti me s
o meti me s the other s e t of formu l as will
pro v e more co n v e n i e n t
W here there a r e ma n y poi n ts to be s e t e a ch s ide li n e a s well
a s the ce n ter s ho u ld be s et by tr a n s it by de fl ectio n a n g le s
The followi n g t a ble s ho ws the n ece ss a ry d a t a a n d a co n ve n ie n t
for m of n ote s
on e,
s
,
.
,
,
.
,
.
De s c r ipt io n
C ho rd s
C u rve
i t
.
I
0
°
To R gh
P T
.
Le ft
:
,
,
Rig ht
C e nte r
'
19 42 2 0
"
39 24 40
"
5
1 3 00
17
°
'
°
'
42 00
.
50 08
11 28 40
c
T:
5 44 20
70 1
P C
8
.
.
70 3
Whe n the c u rv e i s s hort a n d a few poi n ts o nl y n eed be set
o n the o u ts ide a n d i ns ide li n e s the s e poi n ts ma y be s e t by fixi n g
dis t a n ce by the proper chord le n gths a n d li n e by me a s u ri n g w,
or w fro m the a ppropri a te poi n t o n the ce n ter li n e
,
,
,
,
.
R a i lr oa d Cu r ve s a n d E a r thwork
40
66
.
Pr oble m
Gi ven D a n d s t a tio n s of P C
.
.
.
.
and
P T
.
.
R eq u i r ed to l a y o u t the c u rv e by the meth od
of Offs e ts fro m t h e T a n g e n t
cu r ve AG
L e t AG’ be t a n g e n t t o
F in d
E AE = % d
’
Whe n AE
:
_
_
G”FG
a 1
d beco me s % D
the n
100 ,
.
%D
d + D
.
a 2
(x3 ,
etc
.
D r a w EH t a n g e n t a t E
. .
A l s o FN t a n g e n t a t F
.
The a for e a ch chord i s fo u n d by t a ki n g th e cen tr a l a n gle to
th e beg i n n i n g o f the chord pl u s the d efl ecti on a n gle for the c hord
'
.
”
F EH
(1 2
d
ag
AE
’:
”
C FM
HEF
lD
NFG
0 , cos a l
EE :
c, s i n (1 1
100 cos 0t2
”
FF :
10 0 s i n
100 cos a s
”
GO :
100 s i n a g
'
EE + FF
" :
EF
”
FG :
’
FF
’
"
’
”
GG
FF +
’:
GG
,
et c
.
the co mpu t a tio n s i n di c a ted a bo ve a lwa ys u s e n a tu r a l
s i n e s a n d co s i n e s
’=
AG
R s i n AOG
F or a check
'
R vers AOG
GG
where O i s a t ce n ter of c u r ve
n
d
a
n
s
s
l
o
u
e
s
v
co
p
t
tio
i
edi
tely
bo
e
i
e
a
a
m
s
m
n
m u a
F or th e
g
vers i n e s
F or
,
.
,
:
.
,
.
Cu r ves
S i mple
41
.
Th es e check co mpu t a t io n s i n vol ve the r a di u s ( or degr ee )
a n d the ce n tr a l a n g le ; the pre v io u s co mp u t a tio n s i n v ol ve th e
u s e of 0 a l s o
s i n ce the for mu l a
”
pproxi ma te for mu l a perfect pre c i s io n i n th e che ck
c a n n ot be expected
If a ch e ck pe rf ectly preci s e i s requ ired u s e f ormu l a ( 18 )
2 R s i n 4 d i n s te a d o f for mu l a ( 2 0 ) a n d c a rry a ll i n termedi
c
a te work to the n ece s s a ry de g ree of preci s io n
Thi s method of O ffs ets fro m the T a n g e n t i s a preci s e me thod
a n d a llow s of a n y de s ired de g r e e o f preci s io n i n field work
veg etc , i s by
A n other method o f fi n di n g the a n g le s o n
dra wi n g perpe n d i c u l a rs to the chord s a t K L a n d M
is
an
”
a
,
.
,
.
,
-
.
,
,
,
T he n
Ea
a l
d
%
052
051
5 61 + % D
d
D ( as b e for e )
052
D
etc
,
.
.
.
c h a b ei g fo d by ddi g a i cre me t to
n
l e of a
va u
un
n
a
n
n
pre vio u s
n
l
.
A 180
AOG-
as :
wh i ch g i ve s a
l e s o f a co mpu ted
I f AE EF FG a r e p a rts o f a co mpo u n d c u r v e the s a me
g e n er a l method s a r e a pplic a ble excep t th a t the chec ks of
R s i n AOG a n d R ver s AOG a r e n o t t he n a va il a ble
,
check
%D
,
”
on a l l va u
.
,
,
,
.
F i el d-w or k
C a lc u l a te
.
l o EE FF GG’
’
’
S et E F G by me a s u re me n ts AE’ E’ F’ F' G'
S e t E by di s ta n ce AE ( Cg ) a n d EE’
AE , E F , F G ;
’
’
’
’
’
’
a s
’
,
,
’
,
,
,
,
S et F
EF ( 10 0 ) a n d FF’
Set G
FG ( 1 0 0 )
a nd
.
’
GG
,
.
R a i lr oa d
42
68
Pr obl em
.
Cu r ves a n d E a r thwork
Gi ve n D a n d th e s ta t i o n s of P C
.
.
.
.
a nd P
.
T
.
R eq u i r ed t o l a y ou t th e cu r ve by th e meth od
of Defle ct i on Di s ta n ces
.
Wh en th e c u r ve beg i n s a n d en ds a t even s ta ti o ns
.
I n the c u r ve AB, l et
AN be a t a n g e n t
chord
0
EE perp t o AE’
a
AE
any
’
.
ta n g e n t deflectio n
the
“
chord d eflectio n
,
FF
1:
AO
’ 2
BB
E0
R
D ra w
O M p e r pe n
d i cu l a r to AE:
Th en
'
OI Cl
’
FF
2 a ; AF z AE
'
produ ced
Whe n AE i s a fu ll sta tio n of 100 feet a l oo
,
Fi el d-w or k
.
The P C a n d P T a r e a s s u med to h a ve bee n s et
( a ) ca lc u l a te 0 1 0 0
( b) S et poi n t E di s t a n t 100 ft fro m A a n d di s t a n t 0 100 fro m
’
100 ft
AE E
’
’
n d fin d
1
n
P
a
0
rod
u
ce
to
0
a
d
i
s
t
t
A
E
E
F
0
F
F
)
(
(
2 (1 1 00 fro m F’ ( EF= 1 00
m
u
n
P
roceed
s
i
il
a
rly
t
i
l
i
re
a
ched
P
d
8
s
(
( )
’
l
t
n
n
T
a
t
tio
precedi
off
A
s
a
8
P
G
(1
e
F
1 00
y
g
( )
(
)
.
.
.
.
.
.
.
.
.
.
.
’
( FG B
’
i
s
t
a
n
e
n
t
to
the
c
u
r
e
a
t
P
G
B
v
B
f
g
(
( )
.
R a i lroa d Cu r ves a n d E a r thwork
44
71
P r obl e m
.
.
o n s of P C a n d P
R e q u i r ed to l a y ou t th e Cu r ve by D efl ecti o n
D i s ta n ces
Gi ven D
.
a nd
th e s ta ti
.
.
.
.
Wh en th e cu r ve beg i n s a n d e n ds wi th a s u b ch or d
-
.
I
i n iti a l s u b chord
fi n a l s u b chord
ta n g de fl for c
L et AE
-
HB
.
H H
II
a!
In g e n er a l
72
.
c,
-
E’ E
by ( 26)
0;
u
u
ai
.
.
u
;
01
10 02
2
2 17
'
2
a l oo
0 42
100
“100
s
2
1
3 00
a;
(21 00
a f Z ( 11 0 0
it i s better t o u s e ( 29) th a n a ,
Ex a mp le
.
Gi ven P T
.
.
R C
6
0
R
“ M
.
R eq u i r ed a l l d a t a n ece s s a ry to l a y o u t c u r ve by
ti o n D i s ta n ces
”
.
C a lc u l a te witho u t T a ble s
R a di u s 1
6
°
.
R e s u lt to Th
foot
.
c u r ve
°
955
1002
9 55
2 x 9 55
4 50
2 (2100
(17 5
X
( 7 42
x
68 0
57 3
T a ble II g i ve s (1 100
.
107 0
l
e
u
a
s
v
pre
c
i
e
(
)
955
D efl e c
S i mple Cu rve s
45
.
Th e di s t a n ce AF i s s li ghtly s hor ter th a n AF
’
.
It i
e
n er a ll y
g
s
I f de s ired
’
ficie
t
to
t
ke
the
poi
n
t
E
by i n s pectio n s i mply
n
a
su f
for thi s or a n y other p u rpo s e a s i mple a pproxi ma te s ol u tio n of
ri ght tri a n gle s i s a s follow s
.
,
73
P r obl e m
.
Gi ven th e h yp o te n u s e
.
( or ba s e) a n d a lti tu de
.
R eq u i r ed th e di f er en ce between ba s e a n d h y
r
o
t
e
n
u
s
e
o
,
p
2 _
c
(
c
a
-
a
2
h
)(
c
l
--
a
)
-
: _
i n the fi g u re ,
0
a
.
2
h2
2
h
h
2
26
(
a ppr ox
.
h
1
2
'
<
a p pr ox
2 a
3
( 01
.
Where ver h i s s ma ll i n co mp a ri s o n with a or c the a ppr oxi
ma tio n i s g ood for ordi n a ry p u rpo s e s
,
.
Exa mp le
0
.
a
a
The preci s e for mu l a g i ve s
74
F i e l d w or k f or
( ) Ca lc u l a te a m
a
t
71
-
.
a .,
,
.
a,
.
R e me mber
th a t t a n g e n t d efl ec
io n s a r e a s the s q u a r es of the chord s
”
A lle n a s
t a n ge n t o ffs et
61 10 0 i s fo u n d i n T a ble II
F
n
n
s
a
n
m
a
n
d
n
i
d
the
poi
t
E
di
t
t
fro
di
fr
b
E
s
t
a
t
c
o
m
A
a
;
)
(
.
,
.
,
.
’
,
,
A
.
'
2 :
A
E
E
(
’
a
u
n
n
l
ff
rect
xili
ry
t
e
t
E
a
a
a
t
a
o
A
A
a i)
( )
g
( y
’
m
F
a
u
a
n
n
u
n
d
i
n
ro
xili
ry
t
e
t
prod
ced
po
d
a
fi
t F
A
E
g
( )
E
c
.
.
,
’
( FF
a 1 00 ;
’
EF
100 ; EF F
F ro m chord EF pr od u ced , fin d poi n t G
( e)
.
’
( GG
’
2 (1 100 ; FG
FG
m
S
i
il rly for e ch f ll t tio
f
( )
a
A t l a s t e ve n
(g)
8t
a
,
u
s a
n , u se
2 a mo, et c
.
t tio n on c u r ve H , erect a n a u xili a ry ta n
s a
"
o
f
f
M
66
( y
,
a l oe ;
~
”
GG H
ro m G H pro du ced fin d 8
a,
"
”B
i
F
n
a
n
n
t
i
d
t
e
t
a
8
HH
HH
a
g
( )
(
, ;
T he v a l u e s of a m a t a , s ho u ld be c a lc u l a ted t o th e n e a re s t
foo
t
r ho
(h)
F
”
,
,
.
,
,
,
R a i lr oa d Cu r ve s a n d E a r thwork
46
75
Ca u t i on
.
Th e t a n g e n t deflectio n s v a ry a s th
.
.
q
u a r es of
e
s
the chor ds n ot directly as th e chord s
C u rv e s ma y be l a id ou t by thi s method withou t a tr a n s it by
”
the u s e of pl u mb li n e or fl a g for s ig hti n g i n po in ts a n d with
fa i r de gree of a ccu ra cy
F or c a lcu l a ti n g a l oe a . a , i t i s s u fficie n t i n mo s t c a s e s t o u s e
,
.
,
.
,
th e a
pprox va lu e R
.
,
,
57 3 0
a
'
A
D,
c u rve ma y b e th u s l a id ou t
withou t the u s e of tr a n s it or t a ble s
F or ma n y a pproxi ma te p u rpo s e s i t i s well a n d u s efu l to
”
r e me mber th a t the
it
chord deflectio n for 1 c u r ve i s
n e a rly
A he a d
a n d for other de g ree s i n direct proportio n
ch a i n ma n ma y thu s pu t hims elf n ea r l y i n l i n e withou t the a i d
o f the tr a n s it ma n
”
The method of D eflectio n D i s ta n ce s is n ot well a d a p t ed
for co mmo n u s e b u t will ofte n be of v a l u e i n e merg e n cie s
.
'
°
.
.
,
.
.
,
76
.
P r ob l em
.
G i ve n D a n d s t a t io n s of P C
.
.
and P
R eq u i r ed to l a y ou t the c u r ve by
D i s ta n ces
s
Ca u ti on
ma ll
.
T
.
fl ec ti o n
De
when the fi r s t s u b ch or d i s
-
.
I t will n ot be s a ti s f a ct ory i n
.
the c u rv e fro m thi s s hort chord
bes t be s hown by exa mple
.
thi s c a s e t o prod u ce
Th e method to be u s ed ca n
.
Le t P 0 2 41
Fi eld -w or k
M eth o d 1
( )
a
90
.
.
.
e t sta
S
‘
.
42 u s i ng 0
10 a n d a m
10 2
a 100
1005
.
fro m 42 ) offs etti n g a m) from ta n ge n t
u ced a n d 2 a i oo offs et
c
by
chord
prod
4
4
t
t
a
S
e
s
( )
M eth o d 2
n
u
s
s
a
a
u
poi
c
r
e
prod
ced
b
ckw
rd
i
n
o
n
u
v
S
e
t
a
t
g
(a )
( b) S et s t a
.
43 ( 1 00 ft
.
.
.
.
.
,
C
90 a n d 039 0
(1 1 00
92
10 02
bo ve
u ced a n d 2 (1 100 o ffs et
by
chord
prod
s
4
e
t
t
a
c
S
3
( )
A s l i g h t a p p r ox i ma t i on i s i n vol ved i n e a ch of the s e metho ds
M ethod 1 i n vol ves le s s l a bor
et s ta
S
b
)
(
.
42 , u s i n g c
10 a n d a m a s a
.
.
.
.
.
S i mpl e Cu r ve s
77
.
Or d i n a te s
Pr obl e m
.
47
.
.
Gi ven D a n d two p o i n ts on a cu r ve
.
the M i ddl e Or d i n a te fro m the
c hord joi n i n g the t wo poi n ts
R eq u i r ed
.
M
By
for 100 ft
.
chord
a ng
M
78
.
Pr oble m
.
R ver s D
:
le a t ce n ter betwee n a n y t wo poi n t s
M
.
R v er s
M
betwee n poi n ts 2 s t a tio n le n g th s a p a rt
Let A
R v ers g I
:
D
.
.
R ve r s AA
'
:
.
Gi ve n R a n d c
/
.
R eq u i r ed M
.
HL = M = R
T a ble XXI A lle n g i ve s s qu a re s a n d s q u a re roots for cer ta i n
n u mber s
If the n u mbers to be s qu a red ca n be fou n d i n thi s
t a ble u s e
Otherwi s e u s e lo ga rith ms a n d
,
.
,
79
.
P r oble m
.
Gi ve n R a n d C
.
R eq u i r ed the Or di n ate a t a n y g i ven p oi n t Q
M e a s u re LQ
s
KN
q
.
—
Q)
.
48
R a i lr oa d Cu rves
E a r t h wor k
and
.
Whe n 0 1 00 ft or le s s a n
a pproxi ma te for mu l a will ge n er a ll y
s u ffice
P r oble m
Gi ve n R a n d 0
R eq u i r ed M ( appro x)
80
.
,
.
.
.
.
.
H
HL z AH
R
2
AH
?
2 R
Where AB i s s ma ll co mp a red with R
,
£
; ( a pprox
AH
2
02
(
8R
81
.
Exa mp le
R eq u i r ed M
5 7 30
9
)
rox )
a PP
‘
1 00 , D
G i ve n C
.
.
9
°
.
6 3 6 37
8
10000
.
509 3 6
4 9064 0
P rec i s e v a l u e
45842 4
M
3 2 2 160
3 056 16
-
1 6 544
T a ble XXVI I Alle n g i ve s middle ordin a tes for c u rvi n g r a ils
o f cert a i n le n g th s
.
,
,
.
82
.
P roble m
.
Gi ven R a n d 0
R e q u i r ed
.
Or d i n a te
at
App r ox i ma te M eth od
I
.
M e a s u re LO
q
M
( a pprox )
HL
.
.
‘
a ny
g i ven p oi n t
Q
S i mpl e Cu r ves
49
‘
c e HK
S in
( pprox )
a
(1
M ( a pprox )
.
.
M
KQ
Wh e
n
q
l
E Q
.
as i n
KK
’
! a n d KQ
t
i
fig u r e KK’
,
3
( 5)
:
2
M ( a ppr ox )
.
2
Wh e n
q
_
6
1
M ( a ppr ox )
.
4
2
Whe n
3
q
c
M ( a ppro x )
.
4
2
T h e c u rv e th u s fo u n d i s
a
ccu ra tely a p a r a bol a b t f
di s ta n ce s this p r a cti ca lly c o i n cide s with a circle
83
.
II
App r oxi ma te M eth od
.
27 ?
q
KQ
.
M e a s u re LO a n d Q B
.
71
T
hort
'
’
2
KQ
or s
u
,
( ppr ox ) f r o m
a
)(5 ) (
C
~
.
q
w
a ppr ox
.
)
( a pprox )
( 36)
.
S o me t i me s on e , s o me t i me s th e other of t he s e meth ods will be
prefer a ble
84
.
.
m
Exa m
e
.
Gi ve n 0
1 00 , D
’
9
°
fro m T a ble s
M
.
Or di n a te a t p oi n t 3 0 ft di s ta n t
en ter to wa r d e n d of ch or d
R equ i r ed ,
fr om c
I
.
30 ft
.
.
II
.
.
RI
.
7 0 7 40
M
Ordi n a te
P rec is e re su l t for d a t a a bo ve
AQ
80
BQ
20
57 30
.
3 2 660
R9
2 5468
2 R9
7 19 2 0
63 6 7 0
82 50
R a i lro a d Cu r ve s a n d E a r th work
50
85
P r obl em
.
.
Gi ven R a n d c
.
.
R equ i r ed a s er i es of p oi n ts o n th e
M:
HL
—
0
2
c u r ve
.
( a pprox )
.
8
21
2
1— ( a pprox
.
AH
( a pprox )
e tc , as
fa r a s d es u a bl e
.
2
6
2
M
r ox
a
pp
(
4
4
RS
.
4
.
This
me thod i s u s ef u l for ma n y g e n er a l p u rpo s e s , for ordi
n a te s i n be n di n g r a il s a mo n g others
.
86
Pr obl em
.
Gi ven a S i mp l e Cu r ve jo i n i n g two ta n g en ts
.
.
R eq u i r ed the P C of a n ew cu r ve of the s a me
.
r a di u s
g e nt
AB
’
BE
BB
V
The n AA
A l s o B’ BE
'
.
perpen dic u l a r di s ta n ce be twee n t a n gen ts
p
’ :
88
.
.
’
n
88
.
re q u ired cu r ve
’
’
Joi
which s h a ll en d i n a p a r a l l el ta n
AB be the g i v e n cu r ve
L et
.
'
s in
88
OO :
'
'
V VB
V
’
I
I z: p
AA
’
JS 111 I
’
3
7
( )
Whe n the p ropo s ed ta n g e n t
i s o u ts i d e the ori g i n a l t a n g e n t
0
0
the di s ta n ce AA i s to be a dded
Whe n i n s i de it i s t o be s u btr a cte d
t o th e s ta tio n o f the P C
,
'
.
.
,
.
R a i lr oa d Cu r ves
52
88
P r obl em
.
an
d E a r th work
.
Gi ve n a S i mp l e Cu r ve j o i n i n g t wo ta n g en ts
of a n ew c u r ve
R eq u i r ed th e r a d i u s a n d P C
.
.
.
to en d i n a p a r a l l e l ta n g en t w i th th e n ew
P T di r ec tl y opp os i teth e o l d P T
.
.
.
.
Le t AB be
the g i ve n c u r ve of
r a di u s
R
.
the requ ired cu r ve of
’
u
ra di s R
AB
’
’
.
88
’
19
2
.
D r a w perpe n dicu l a r O’ N
NM
and are
The n C M
’
’
BM
BO
BM
BM
’
’
O M:
’
'
exsec NOO
ON
(R
—
R )
e xs e c
AA
AA
’
p
OM
I
p ;
O’ N
’
BB
’
'
’
’
'
R
ONt a n NOO’
’
(R
R
R ) ta n
’
P
( 41)
exs ec I
I
Wh en th e n e w t a n ge n t i s o u ts ide the ori g i n a l ta n g e n t ( a s i n
’
d
R a n AA i s a dded to th e s ta tio n of the P C
the figu re ) R
’
Whe n the n e w t a n g e n t i s i n s i de the ori g i n a l ta n g e n t R R
p
’
n
a
d
A
A i s s u btr a c te d fro m s ta tio n of P C
R
R
exs ec I
’
,
.
,
'
.
89
P r obl em
.
.
To fin d th e
S i mp l e Cu r ve th a t s h a l l joi n
two g i ven ta n g en ts a n d p a ss
o
o i nt
With the tr a n s it a t V the
g i ve n poi n t K ca n of te n be
be s t fixed by a n gle BVK a n d
di s ta n ce VK If th e poi n t K
be fixed by other me as u re
me n ts the s e g e n era lly ca n
re a dily be red u ced to th e
a n g le BVK a n d d is ta n ce VK
th r u g h a g i ven p
.
,
.
,
.
.
.
,
S i mpl e
90
.
P robl em
Cu r ve s
53
.
Gi ve n th e two ta n g en ts i n ter s ecti n g a t V, th e
xed by a n g l e
a n g l e I , a n d th e p oi n t K
.
fi
B a n d di s ta n ce VK
BVK
b
.
f cu r ve to jo i n th e
R eq u i r ed t h e r a di u s R
o
two ta n g en ts a n d p a s s thr ou g h
K
.
In th e t ri a n g le VOK we h a ve g i ve n
b a n d OVK
VK
R
F u rth er
co s
‘
I
i
VO : OK
R
R
cos i l
s i n OVK
s i n VKO
s
m VKO
C OS
B)
(i f
0 0 8 41
”
s i n VKO
I + fl)
F ro m d a ta thu s fo u n d , the t ri a n gle VOK ma y be s ol ved for R
I n s ol vi n g thi s tri a n g le the a n gle VOK i s ofte n v ery s ma ll
.
.
A
li ght error i n the va l u e of this s ma ll a n gle ma y occ a s io n a
l a rg e error i n the va l u e of R I n thi s c a s e u s e the followin g
S ec on d M e th od of fi n di n g R a fter VOK h a s bee n fou n d
s
.
.
F i nd
.
AOK :
Th e n
% I + VOK
A l s o DVK:
R v er s AOK
LK
b s i n DVK
b s i n DVK
ers AOK
v
91
.
P r obl em
.
Gi ven R , I , B( BVK)
R eq u i r ed b ( VK)
.
.
I n th e t ri a n gle VOK
OK
R
OVK:
90
OV
S ol ve tri a n gle for b
.
A ls o fin d VOK a n d s ta tio n of K if de s i red
.
I +B
R a i lr oa d Cu rve s
54
92
and
E a r thw ork
P r oble m
.
wh er e
a
cu r ve
between
Fin d
fin d th e p o i n t
T0
.
s tr a i g h t
.
c
li n e
s ta ti on s
i n ter s e ts
a
.
where the s t ra i ght li n e V ’ K
c u ts V B a t V
M e a s u re KV’ B
Us e V’ a s a n a u xili a ry v ertex
F i nd 17 fro m V B by
S ol ve by precedi n g proble m
’
.
.
.
’
.
93
.
Appr oxi ma te M et h od
S e t t h e middle poi n t
.
H by method of ordi a te
n
s
.
If the a r c H8 is s e n s ibly a s tra i ght li n e fin d the i n ters ec tio n
,
of HB a n d C D
.
Otherwi s e s e t
the poi n t G by method of ord i n a te s a n d g et
,
i n ter sect i o n of HG a n d C D
.
A dditio n a l po i n ts on the a r c ma y be s et if n ece s s a ry , a n d th e
proce s s co n ti n u ed u n til the requ ired preci s io n i s s ec u re d
The poi n t s H a n d G ca n be s et witho u t the u s e of a tra n s it
with s u fficie n t a cc u ra cy for ma n y p u rpo s e s a pl u mb li n e or fl a g
”
“
bein g u s ed i n s ighti n g i n
.
,
.
94
.
Pr ob l em
.
Gi ve n
a
S i mp l e Cu r ve a n d
th e cu r ve
R e q u i r ed
p oi n t
a
a
p oi n t
o u ts i de
.
ta n g en t to th e
cu r ve
f o m th t
r
a
.
be the g i ve n
’
c u r ve
P the poi n t ou t
s ide the c u r v e
BLa t a n g e n t a t B
L e t BDE
.
.
.
M e a s u re LBP, a l s o 8 P
.
S i mple Cu r ves
I n t h e t n a n gl e BPO
55
.
we h a v e g i ve n PBO
BP, BO:
,
S ol v e the tri a n gle for BOP a n d OP
.
Th e n
cos
DOP
BOD
BOP
DOP
F ro m BOD fin d s t a tio n of D fro m kn own po i n t B
.
It s ho u ld b e n oted th a t if l og OP i s fo u n d thi s ca n b e u s e d
O the r s i mil a r
a g a i n witho u t looki n g o u t the n u mber for O P.
c a s e s will occ u r el s ewhere i n c a lcu l a tio n
3
Whe n for a n y re a s o n it i s difficu lt or i n co n ve n ie n t t o me a s u r e
BP directly the a n g le s CBP BC P a n d th e di s ta n ce BC ma y b e
me a s u red a n d BP c a lc u l a ted
,
.
'
'
,
,
.
94 A
.
T e n ta t i v e M et h od
F i eld-w or k
F ro m the
.
.
t
tio
n (B
n e a re s t t o th e r e qu i r e d po i n t D
( )
)
fin d by the a pproxi ma te method where BP c u t s the c u r ve a t C
n e a re s t s t a tio n
u ce PC to B
If
E
be
the
prod
(
h
n C D a n d wi t h
u
m
e
D
with
D
s
li
htly
re
a
ter
t
a
A
s
s
B
5
g
g
( )
u r ve
tra n s it a t P C s e t the poi n t D ( tra n s it poi n t) tru ly o
r ve
n d l a y off a t a n g e n t to
M
o
e
the
tr
n
i
to
D
a
c
a
s
t
v
( )
at D
T hi s will very n e a rly s trike P
a
s a
'
,
.
,
.
,
.
.
.
,
.
.
If the t a n g en t s trike s a w a y fro m P a t Q me a su re Q DP
a n d mo v e the poi n t D ( a he a d o r b a ck a s the c a s e ma y be ) a di s
t a n ce 0 du e to a n a n gle a t the ce n ter d Q DP The t a n ge n t
fro m thi s n e w poi n t o u ght to s trike P a l mo s t ex a c tly
I n a l a rg e n u mber of c a s e s the poi n t D will be fo u n d on the
firs t a tte mpt s u fficie n tly clo s e for the re qu ired p u rpos e
If a ta n ge n t betwee n two c u r ve s i s req u ired s i mil a r methods
by a pproxi ma tio n will be fo u n d a va il a ble
( d)
,
,
,
.
.
.
.
,
.
R a i lroa d Cu r ves a n d E a r thwork
56
95
.
Pr obl e m
.
Gi ve n two S i mp l e Cu r ve s
.
.
R eq u i r ed a ta n g e n t to bo th Cu r ves
F i n d co n v e n ie n t poi n ts A a n d B o n the g i ve n
L et AK a n d BL be t a n g e n t s
.
c u r ves
.
.
z
M e a s u r e l i n e AB a n d a n gl es BAK a n d ABL
.
‘
L e t AO
R , a n d BP
R,
( both g i ve n )
.
r li n e OP a n d a n g le s A
S ol v e ABPO fo
DP a n d BPO
.
cos
R,
COP
AOG
R,
an d 0 7 0
AOP ;
COP
3
180
°
COP
BPD
0 90
BPO
.
Whe n a ta n g e n t is to co n n ec t two tra ck s alr ea dy l a id it ma y
be dete r mi n ed by a process s i mil a r to 9 4 A by te n t a ti ve me thod
,
.
l e s on Cli rv es
Ob s ta c
96
.
.
W h e n V i s i n a cc e s s i bl e
.
M e a s u re VLM, VML, LM
.
I:
VLM
VML
LV a n d VM a r e re a dily c a lc u l a ted ,
and
AL a n d MB deter mi n ed
.
ome c a s e s the b e s t w a y i s t o
a s su me the po s itio n of P C a n d fu n
and
ou t the cu r v e a s a tri a l line
fi n a lly fin d the po s itio n of P C cor
mu l a
the
ethod
of
for
m
r e ctl y
by
In
s
.
.
,
.
‘
.
S i mpl e Cu r ves
97
Wh en th e P C i s i n a cc e s s i ble
.
.
.
57
.
.
t blis h s o me poi n t D ( a n e ve n
i s prefer a ble ) by method of
s ta tio n
”
“ off et
o r otherwi s e
s fro m T a n g e n t
s
M o ve tr a n s it to B ( P T ) a n d r u n
ou t c u r v e s t a rti n g fro m D a n d chec ki n g
on ta n g e n t V B
Es a
.
.
.
.
98
Wh en t h e P T i s i n a cc e s s i ble
.
.
.
.
m
W ith i n s tr u me n t s till a t V s et s o e
,
co n ve n ie n t poi n t D mo ve tr a n s it to
P C a n d r u n i n c u r v e to D a n d the n
p a s s the ob s t a cle a t B a s a n y ob s t a cle
on a t a n g e n t wo u ld be p a ss ed
,
.
.
,
,
.
Whe n Obs ta c le s on t h e Cu r v e occ u r s o a s to pre v e n t
r u n n i n g i n the c u rve n o g e n er a l r u le s ca n well be gi ve n
S o meti me s r es ett i n g the tr a n s it i n the c u r ve will s er ve , S o me
ti me s i f on e or two poi n ts o n ly a r e i n vi s ible fro m the tr a n s it
the s e ca n be s e t by “ d efl ec t i o n d i s ta n ces a n d the c u r ve con
”
“
ti n u e d by
wi tho u t re s etti n g the tr a n s it
defl ec ti o n a n g l es
”
offs ets fr o m th e ta n g en t ca n be u s ed to ad va n
S o meti me s
”
“
ta g e S o meti mes poi n ts ca n be s e t by o r di n a tes fro m
chords S o me ti me s the method s how n on p a g e 54 9 2 a s su m
i n g a n a u xili a ry V i s the o n ly on e po ss ible
It s hou ld be h om e i n mi n d th a t it i s s eldo m n eces s a r y th a t
the fu ll s ta ti on s s ho u ld be s e t If it be po ss ible t o s et a n y
poi n ts who s e s t a tio n s a r e kn own a n d which a r e n ot to o fa r
a p a rt thi s i s g e n er a lly s u fficie n t
F i n a lly for p a s s i n g ob s t a cle s a n d for s ol vi n g ma n y proble ms
which occ a s io n a lly occ u r it i s n ece s s a ry t o u n ders ta n d the
va rio u s method s of l a yi ng ou t c u r ve s a n d to be f a mili a r with
the ma the ma tics of cu r ve s ; a n d i n a dditio n to exerci s e a r ea
s o n a ble a mo u n t of i n g e n u ity i n the a pplic a t io n o f the k n owle dge
po ss es s ed
99
.
.
,
,
,
”
,
.
,
.
,
.
,
.
.
,
.
,
,
,
,
.
,
,
CH AP T E R V
.
U
COM PO ND CURVE S
10 0
Wh e n on e c u rve follows a n other the two c u r ve s h a vi n g
.
,
co mmo n ta n g e n t a t the poi n t of j u n ctio n a n d lyi ng u po n the
s a me s ide o f the co mmo n t a n g e n t the t wo c u r v e s form a Com
a
,
,
p ou n d Cu r ve
.
Whe n two s u ch cu rve s lie u po n Oppo s ite s ide s of the co mmo n
t a n g e n t the two c u r ves the n for m a R ever s ed Cu r ve
I n a co mpo u n d c u r v e the poi n t a t the co mmo n t a n g e n t where
“
the two cu r v e s j oi n i s c a lled t h e P C C me a n i n g the poi n t
”
o f co mpo u n d c u r va t u re
I n a re ver s ed c u r v e the poi n t where the c u r v e s joi n i s c a lled
”
“
the
me a n i n g the poi n t of re vers ed c u rva t u re
,
.
,
.
,
.
.
,
.
,
.
F i eld -w or k
.
co mp o u n d c u r ve or a r ever s ed cu r ve
L a yi n g o u t a
.
( a ) S e t u p tr a n s it a t P C
.
.
( b) Ru n i n s i mple c u r ve to P C C
.
( ) M o ve tr a ns it to R C 0
6
( d)
.
( )
or P R C
.
.
.
or P R 0
.
.
.
S e t li n e of s i g ht on c ommo n t a n gen t wi t h ver n ier a t 0
method of
6
.
.
by
60
R u n ou t s eco n d c u rv e a s a s i mple c u r ve
Da ta Us ed i n Comp ou n d Cu r v e For mu l a s
I n the c u r ve of l a rg er r a di u s , OA
R,
I n t h e c u r ve of s horter r a di u s , PB
R
A l s o LVB
°
I
.
.
.
AOG
BPC
I,
AV
I8 ;
VB
T,
.
T, ;
60
104
“
.
R a i lr oa d Cu rves a n d E a r thwork
Pr obl em
.
Gi ve n T, , R, , R, , I
R equ i r ed T, , I , , I .
A
.
.
.
L
D r a w a rc s NP a n d KC
.
D r a w perpe n dicu l a rs MP, LP, S B, UB
.
T he n
AN
R.
KP
LS
KS
OP ver s NOP
V B i VES
PB v ers KPE
1,
s n
T, s i n
I
R , ver s
I
T, s i n
I
R . ver s
I
Rt
Ra
MP
AV
-
105
LP
LK
R . ) v er s
(R,
AM
P r ob l em
.
R , ) s i n I, + R , s i n I
Gi ven T, , R , , I , , I
R eq u i r ed T l , R y , I ]
R:
T. s m I
R.
v
Pr obl em
.
UV
SB
T. cos I
.
.
R . ver s I
er s I ,
Gi ve n T, , T, , R
I
.
R equ i r ed R 1 , I r , L
.
T, s i n I
R , ver s I
T, + T, cos I — R , s i n I
T, + T, cos I
.
R , si n I
Comp ou n d Cu r ves
107
61
.
P r obl em
.
.
Gi ve n T , , R , , R . , I
R eq u i r ed T, , I , , I .
D r a w a rc s NP, KC
’
f
k
.
.
perpe n di cu l a rs
D ra w
AS
.
PM. VU
.
,
The n
LM
BP
KN
MN
LM
LN
KN
LN
KL
MN
LK
OP v ers NO P
(R1
‘
R . ) V0 1 S
e
T,
R , s in I
Pr ob l em
AV s i n VAS
R , ve r s
T, s i n
-
L
AU
T, , R , , I, , I
R, , I,
T, ecs I
.
.
I — I,
R , ver s I
R, — R .
R3
R , ) sin I,
( R,
I
T, s i n I
PM
R equ i r ed T ,
:
I
RI
Gi ven
.
AO v er s AOK
R , v er s I
v r s I,
AS
LS
'
I,
VB
KS
T, sm I
ver s I ,
T.
Pr obl em
.
R , sin I — ( R,
R , ) sin I.
T, ,
T, , R , , I
Gi ven
R equ i r ed R , , I , , L
h] s
R, v s I
er
R , — R.
_
si n I,
.
.
T, s 1n I
R , s i n I — T, cos I
.
T, oos I
T.
62
R a i l r oa d Cu r ve s a n d E a r thwork
r ble m
P o
.
Gi ve n , i n th e
R eq u i r ed
fig
AB VAB, VBA, R ,
u r e,
,
R , , I ,, I , , I
D ra w
.
l o perpen dicu l a rs
NP ;
a rc
.
a s
K B MP S P
,
,
I :
VAB
.
NM
AK
KM
AN
AB s i n VAB
PB co s S PB
AN
—
AB s i n VAB + R ,
—
AB s i n VAB
MP
ta n NPM
OP z
111
.
Proble m
.
VBA
:
cos
R , ver s I
"
KB
SB
AB co s VAB
PB s i n S PB
AB co s VAB
R , s in I
l
ta n
/
f
I—J
NM
MP
]l
ef
R,
R, — R,
G i ven , i n th e
Re
q u i r ed
fig
AB, VAB, VBA, R ,
u r e,
R , , I,, I , , I
D ra w a r c PN
PM, AS
.
.
l o perpe n dicu l a rs
a s
.
I = VAB + VBA
NM = LK
SK
OA v ers AO K
AB s i n VBA
ers I
AB s i n VBA
R;
v
AS
OA s i n
s in
R,
ta n NPM
t a n 4 1,
Il z
OP Z R l — R s
SL
NM
AT
AOK
AB cos VBA
I
AB cos VBA
60
MP
NM
MP
I — Is
MP
s in
I,
Comp o u n d
112
Pr oble m
.
.
Gi ven
Cu r ves
S i mp l e Cu r ve
a
ta n g en t
en di n g
i n a g i ven
.
A s eco n d c u rve of g i v e n r a di u s i s to
g i ve n p a r a llel t a n ge n t
63
.
le a v e thi s a n d en d i n a
'
.
R eq u i r e d th e P C 0
.
.
.
L e t AB be the gi v e n c u r ve of r a di u s R ,
.
C be the P C G
’
.
.
.
the s eco n d c u r ve of r a di u s R
B Ezp di s t a n ce betwee n t a n g e n ts
C8
’
T he n MN2
113
v
ers COB
v
ers COB
ta n g en t to th i s cu r ve
,
.
Rl — R ,
Gi ve n , a S i mp l e C u r ve of
.
E8 z p
.
r a di u s
RI ;
a ls o a
line
n ot
.
o
R eq u i r ed , th e r a d i u s R 2 o
fa
s ec n d cu r ve
to
co n n ect a g i ve n
u r ve a s a P O G
o
t
h
i
s
c
w
i
h
h
v
l
i
n
s
a
n
e
n
i
i
t
o
n
t
t
e
en
e
a
a
t
t
n
g
g
p
.
,
.
.
be the gi ven c u r ve of r a
d ins R 1
LB the g i ve n li n e
C be a poi n t s elected ( a s con
n ece s s a ry )
as
ve n i e n t or
the gi ve n P C 0
C B the req u ired c u r ve o f r a
di n s R 2
L e t AC
.
.
.
.
.
.
ro m C l a y o ff a u xili a ry t a n g en t
C D c u tti n g LB a t K
M e a su re C K a n d a n gle DKB
F
.
CK
The n R 2
t a n § DKB
KB z C K
Thi s fixe s the po s itio n of B the P T th u s a llowi n g a
on the field work
,
-
.
.
.
,
“
check
”
R a i lr oa d Cu r ves a n d E ar th work
64
1 14
.
Gi ve n
a
S i mp l e Cu r ve o
t a n g e n t t o th i s cu r ve
f a di
R1 ;
us
a ls
o a l i n e n ot
.
R eq u i r ed , th e P C C
.
r
.
.
f a s eco n d cu r ve of g i ven r a di u s R 2 to
o
.
o
l ea ve th i s c u r ve a n d jo i n th e g i ve n li n e a s a t a n g e n t
.
L e t AC HE be the g i ve n cu rve
TB the g i ve n li n e
AO ra di u s R I
.
.
.
PC r a di u s R 2
.
C B req u ired s eco n d c u r ve
C requ ired P
.
rom a co n ve n ie n t poi n t H
on
th e gi v e n c u r ve l a y o ff
a u xili a ry t a n g e n t HK c u tti n g
TB a t K
F
.
M e as u re HK a n d a n gle T KH
.
The n
HOL
TKH
SL
85
HK s i n TKH
OH vers HO L
HK s i n T KH
R I vers T KH
LE
p
DB
MB
MD
MN = MB — NB = DB = p
MN
—
L
(Rz
PO vers OPN
VGI S 1 2
‘
R 1)
T h e a n gle AOH i s g i v e n
.
C PB
T KH
COH
AOH
COH
AOC
11
T his s er v e s to fix s t a tio n o f P C 0
.
KB
.
.
at
C
.
KL
LB
HS
HK cos TKH
LB
OH s i n TKH
HK cos TKH
KB= R 1 s i n T KH
.
HKcos T KH
OP s i n OPN
— R
i
s
n
I
R
a
I
,
)
(
C om ou n
p
115
P r obl em
.
ta n g en t
d Cu r ves
65
.
Gi ve n a Co mp o u n d 0 u r no en d i n g i n a g i ven
.
.
R e q u i r ed to ch a n g e th e P
so a s
.
to en d
i n a g i ve n p a r a ll e l ta n g en t, th e r a di i
ma i n i n g u n c ha n g ed
re
.
I Whe n the n ew ta n g e n t lie s
o u ts i deth e o ld ta n g en t a n d the
c u r ve e n ds with c u r ve of l a r g er
.
,
r a di u s
.
Le t ACB be
the g i ven
co
m
po u n d c u r ve
’ ’
AC B the req u ir ed c u r ve
’
P rod u ce C O to P
dr a w a r c
”
C B a n d co n n ect P’ B”
P rod u ce a r c AC to B a n d co n
.
.
’
,
’
.
’
a
,
B
E E
.
.
n e ct OB’ .
D r a w perpe n d ic u l a rs C S D, CT K, B LE , a n d B E
’
The n
EB
”
‘
”
EB
’
’
59
P C’ v
( PC vers C PB
-
l
(KB
’
B
T )
C’ P’ B”
OC
’ v
er s C OB
’
’
OC v ers COB )
’
(R,
p
v
er
s
.
LB
‘
DB”
p
’
’
ers I,
’
v
v
er
s Il
’
R , ) v ers L
(R;
er s I t }
6
( 5)
II Whe n the n ew ta n g en t lie s i n s i d e the o l d ta n g en t
a n d the c u r ve e n d s with the c u r v e of l a r g er r a di u s
116
.
.
,
.
—
Rz — R.
= v er s I z —
’
ve r s I z
66
.
III Whe n the n ew ta n gen t lie s o u ts i de the o ld ta n g e n t
a n d the cu r ve e n d s with c u r v e o f s ma l l e r r a d i u s
W ith a n ew figu r e i t ma y be s how n th a t
117
.
.
,
.
'
L
F, _ R
:
ve r s I ,
( 67 )
’
ve r s I
I'
( v
IV Whe n the n ew ta n g en t l ie s i ns i de the ol d ta n g e n t
a n d the c u r ve en d s with c u r v e of s ma l l er r a d i u s
118
‘
.
.
,
’
.
v
ers I
’—
,
_v
ers I
,
( 6 8)
CH AP T ER VI
.
U
REVER S E D C RVE S
.
It i s co n s idered u n de s ir a ble th a t re ver s ed c u r ve s s hou ld be
u s ed on ma i n li n e s or where tr a in s a r e to be r u n a t a n y con
The ma rked ch a n ge i n directio n i s ob j e ct io n
s i d e r a b l e s peed
a ble
a n d a n e s peci a l di ffic u l ty r e s u l ts fr om there b ei n g n o
opportu n ity t oele va te t h e o u ter ra il a t the P R 0 The u s e of
re ver s ed cu r ve s on li n e s of r a ilro a d i s ther efore v ery g e n era lly
co n demn ed by e n g i n eers F or y a rd s a n d s ta tio n s re v ers ed
cu r ve s ma y ofte n be u s ed to a d va n t a g e a l s o for s treet ra il
w a y s a n d perh a p s for other p u rpo s e s
,
.
'
,
.
.
.
.
,
,
.
,
119
.
P r obl em
.
Gi ven t h e p erp e n d i c u l a r d i s ta n ce between
p a r a ll el t a n g en ts , a n d th e common r a d i u s
o th e r ever s ed cu r ve
f
.
.
R eq u i r ed th e c en tr a l a n g l e o
f ea ch c
Le t AH a n d BD be
a
u r ve
.
the pa r
llel t a n g e n ts
AGB the re v er s ed cu rve
perpen dicu l a r
HB p
di s ta n ce betwee n t a n
g e n ts
D r a w perpe n di c u l a r NM
.
.
.
.
Th e n
ver s AOG
v
1 20
.
Pr obl em
.
ers I
L et
AOG
AN
BM
w
AO
PB
AO
tsp
,
R
Gi ven p , L
R equ i r ed R
.
.
to
v
er s I
66
,
BPC 2 :
68
R a i lr oa d Cu rves a n d E a r th work
124
P r obl e m
.
Gi ven R 1 , R 2 , p
.
R eq u i r ed I r
fro m ( 7 5 )
v
Pr obl e m
.
ers I
.
.
.
,
Gi ven a P 0
of two ta n g en ts n ot
p a r a l l el a l s o th e ta n g en t d i s ta n ce fr o m
P 0 to V a l s o th e a n g l e of i n te r s ec ti on
a l s o th e u n eq u a l r a d i i of a r ev er s ed c u r ve
.
.
u p on
on e
,
.
.
,
,
t o co n n ect th e ta n g en ts
R equ i r ed th e
cen tr a l
.
f th e s i mp l e
a n g l es
o
c
cu r ves , a n d ta n g en t d i s ta n e,
L e t AV
T1
g i ve n ta n g e n t di s ta n ce
AOG
e q u i r ed a n gl e S
BPC
BV
T2
req u ire d t a n g e n t
di s ta n ce
.
A
g i ve n P 0
v
v
AVT
I
AO
R1
r eq u ired c u r ve
AGB
V to P T
.
.
.
ertex
PB r:
1
g
R
VT
s
2
l ve n
J
ra d u
eco n d ta n ge n t
D r a w a r c AL, a l s o perpe n dic u l a rs OL, AS , AK
.
The n LT p perpe n dicu l a r di s ta n ce betwee n p a r a llel t a n
s
L
O
er
v
C
R
R
2
I
g e n ts a n d by ( 7 5 ) p
(
)
2
(R1
(R 1
LT
LK
AS
R 2 ) v er s LOC
AO ver s AOL
AV s i n AVS
R 2 ) v ers
R 1 v ers
v
BV
T2
12
ers 12
I
ers I + TI s m I
R1 v
R1
AK
TB
TI cos 1 + R 1 s i n I
(R 1
VS
4 T1 s i n I
R2
R 2) si n I 2
R e ve rs e d Cu r ves
a
126
P r ob l e m
Gi ven BV i n s tea d of AV, a n d oth er da ta
.
.
69
.
o
a b ve
as
.
R eq u i r ed I I , I 2 , etc
.
D r a w perpe n dic u l a rs PH, BF, BG
.
UH
p
perpe n di c u l a r di s ta n ce betwee n p a ra llel ta n ge n t s
UH
R 2 ) v er s 1 1 :
(RI
TI :
FH
T2 cos I
GB
R 2 vers I
T2 s i n I
R 2 v er s I
T2 s i n I
R 2 s in I
.
R1
Re
(R 1
R 2) sin I I
( 80 )
M a n y proble ms i n re ver s ed c u r ve s ca n be s i mply a n d qu ickly
s ol v ed by u s i n g the a va il a ble d a ta i n a wa y to bri n g the proble m
i n to a s h a pe where it beco me s a c a s e of p a r a llel t a n g e n ts with p
kn own a n d which ca n be s ol ved by
Thi s i s tru e p artic u l a rly of s idi n gs a n d y a rd prob le ms
,
.
127
.
Pr obl em
Gi ven th e l en g th of th e common ta n g en t a n d
.
th e a n g l es
of i n ter s e cti on wi th th e s epa
r a t ed ta n g en ts
.
R eq u i r ed th e co mmo n
c
u r ve t o
r a di u s o
f a r ever s ed
jo i n th e two s ep a r a ted ta n g en ts
L e t VA
VB
.
AGB
co mmo n ta n ge n t
s ep a r a ted t a n g e n t s
req u ired c u r ve
LVAC
IA ;
VAVB
l
AVA, BVB
'
VAVB
l
.
MVB B
2
VAC
IB
Vs C
R ta n % I A + R ta n % I B
( 8 1)
ta u i I B
t a n i IA
An a pproxi ma t e me thod i s a s follow s
F i n d T“ for a 1
T he n
°
c r ve
u
l o Tm ( T a ble III )
a s
.
(
ro x )
a pp
.
R a i lr oa d Cu rves a n d E a rth work
70
1 28
.
v er s ed
c u r ve
l yi n g
on
g en t ;
a ls
a
o
f a n oth er
o I
th e P 0
,
.
g i ve n ta n
th e p o s i ti o n
ta n g e n t
n ot
u
r
v
c
e
f
R eq u i r ed th e c e n tr a l
a l s o th e p o s i t i on of P T
th e r eve r s ed
.
.
.
ACB be the re v ers ed c u rv e
L et
AH, B K , th e g i v e n t a n g e n ts
A, the g i ve n P 0
.
AOC
r1
C PB
I2
.
M e a s u re f r o m Ato s o me co n v e n ie n t poi n t D o n B K let AD= b
M e a s u re a l s o HAD a n d ADK
The n the a n gle betwee n t a n ge n ts ADK HAD I
E xte n d a r c CA to G whe re c u r ve i s p a r a llel to B K
.
.
.
.
D r a w pe r pe n dicu l a rs AE, OG, AU, SCT
T he n
ro m ( 7 6 )
.
AE
UG
AD s i n ADK
OA ver s AOG
b s i n ADK
F
R I ver s I
Ri +Rz
II
A ls o BD
[2
I
BK
EK
ST
AU
OP S i n C PB
(R 1
.
p a r a l l el ; a l s o th e u n
eq u a l r a d i i , R 1 a n d R 2
o
a ls
or a r e
'
I
11 a nd I 2 ;
f
Gi ven ,
o
a n g l es
.
R 2) sin Iz
ED
ED
OA s i n AOG
AD cos ADK
R 1 s in I
b cos ADK
CH AP T ER V11
U
.
P ARAB OLI C C RVE S
.
to j oi n two t a n g en ts p a r a boli c
a rc s h a v e bee n propo s ed a n d u s ed i n or d er t o do a w a y w i th
the su dde n ch a n g e s i n directio n which occ u r where a circ u l a r
c u r ve le a ve s or j o i n s a t a n ge n t P a r a bolic c u rve s h a ve how
e ver fa iled to meet with f a vor for r a ilro a d c u r ve s for s e ver a l
re a s o ns
1 P a r a bolic c u r ve s a r e le ss re a dily l a i d ou t b y i n s tru me n t
th a n a r e circ u l a r c u rve s
It is n ot e a s y to co mp u te a t a n y g i ve n poi n t the r a di u s of
2
cu r va t u re for a p a r a bolic c u r ve i t ma y be n ece ss a r y to do thi s
e ither for c u r vi n g r a il s or for de termi n i n g th e proper eleva tio n
for th e o u ter r a il
”
”
3 The u se of th e
or other
E a s e me n t
or
Tr a n
Spir a l
c u r ve s s ecu re s the des ired res u l t i n a mo re s a ti s f a ctory
s it i o u
1 29
I n s te a d of circ u l a r a rc s
.
,
,
.
.
,
,
.
.
.
.
.
,
.
wa y
,
.
There —a r e howe ver ma n y c a s e s ( i n L a n d s c a pe Ga rde n i n g
or el s ewhere )
where a p a r a bolic c u rv e ma y be u s ef u l ei ther
bec a u s e it i s more gr a cef u l or bec a u s e witho u t i n s tr u me n t it i s
more e a s ily l a id ou t o r for s o me other re a s o n
I t i s s eldo m th a t p a r a bolic c u rve s w ou ld be l a id ou t b y
i n s tru me n t
,
,
,
.
.
130
.
P r op e r t i es of t h e Pa ra b ol a
.
( a ) The loc u s of th e mi ddle po i n ts of a sy s te m of p a ra llel
chord s of a p a ra bol a i s a s tra i gh t li n e p a r a l lel to the a xi s of the
p a ra bol a ( i e a d i a meter )
( b) The locu s of th e i n ter s ectio n of p a i r s of ta n ge n ts i s i n
th e di a meter
n
a
c
The
t
)
g e n t to t h e p a r a bol a a t the vertex of th e di a meter
(
is p a r a llel to the chord b i s ected by thi s di a meter
( d) D i a me ters a r e p a r a llel to the a xis
.
.
.
.
.
.
71
72
R a i lr oa d Cu r ves a n d E a r th work
( e) Th e e qu a tio n of the p a r a bol a
.
the co ordi n a te s me a s u red
u po n t h e di a me t er a n d th e t a n e n t
a t t h e en d of th e d i a meter i s
g
,
R equ i r ed to l a y ou t th e
s ets
Let AV,
e
V B be the g i e
v
f m th e t
a n g en t
ro
p a r a bol a
by
“
f
o
”
.
n
eces s a rily eq u a l )
a n d AHB t h e p a r a bolic c u r v e
J oi n the ch or d AB dr a w VG bi s ecti n g AB
D r a w AX BY, p a r a llel to VG prod u ce AV to Y
The n VG i s a di a meter of the p a r a bol a
AXp a r a llel to VG i s a l s o a di a meter
The eq u a tio n o f the p a r a bol a referred to AXa n d AY a s axe s 18
t a n g n ts ( n o t n
,
.
'
.
,
.
.
.
?
y
4 p ’x
.
In s te a d of s ol vi n g thi s eq u a tio n e n g i n eers co mmo n ly u s e the
proportio n
{231
x2
c
He n e
Next bi s ect
VB t D
a
.
D r a w C D p a r a llel to AX
.
The n
BD2
BV2
CD
C D HV
HV
Pa ra b oli c Cu r ves
S i mil a r l
y ma ke
AN = NF = FV
,
HV
T he n
EF z
In
fo u n d
a
73
.
si
—
HV
mi l a r wa y, a s ma n y po i n ts a s a r e n eede d ma y be
.
Fi el d-wor k
132
.
( )
F i n d G bi s e t i n g AB
.
c
a
F
i
n
d
b
( )
.
H bi s ecti g GV
n
F in
d
.
n
n
d
poi
t
s
a
P
N F di vidin g AG AV p r opor ti on
Q
( )
a t ely ; a l s o R a n d D di vidi n g GB a n d BV proportio n a t el y
Us e s i mple r a ti o s whe n po s s ible ( a s }
0
,
,
,
,
,
,
:
.
,
a ,
c a lc l a t ed d i s ta c e KN
n PN
h
L
ff
t
e
a
o
o
d
,
y
)
(
u
on
Q F l a y off EF
on
RD l a y ofi C D
n
In figu r e opp os i t e ,
ma n y pu r po s e s , or
F or
ma n y c a s e s , i t w i ll gi ve r e s u l ts
in
cie n tly c los e t o proceed witho u t e s ta bli s h i n g P Q R ; the
d ire ctio ns of NK EF C D bei n g gi ve n a pproxi ma t el y by eye
W he n th e a n gle AVG i s s ma ll ( a s i n t h e fi g u re ) it w ill g e n er a lly
be n ece s s a ry t o fin d P Q R a n d fix the di rec t io n s i n which
t o me a s u re NK EF C D
Whe n the a n gle AVG i s l a rg e ( g re a ter
th a n
it will
a n d the di s t a n ce s NK EF C D a r e n ot l a r g e
ofte n be u n n e c e s s a ry to d o thi s No fixed r u le ca n be g i ve n
a s t o whe n a pproxi ma te me thod s s h a ll be u s ed
E xperie n ce
e d u c a t e s the j u d gme n t s o th a t e a c h c a s e i s s e ttled u po n i ts
merits
s u ffi
,
,
,
,
,
,
.
,
,
,
,
,
,
.
,
,
,
,
.
.
.
R a i l r oa d Cu r ve s a n d E ar th work
74
133
P r oble m
.
.
.
Gi ve n t wo ta n g e n ts to a p a r a bo l a , a l s o th e
p os i t i o n s of P 0
.
.
a nd
P T
.
.
R eq u i r ed to l a y o u t th e p a r a bol a by
mi d
d l e or di n a tes ”
.
ordi n a te s a r e ta ke n fro m the middle of th e chord a n d
p a ra llel to GV i n a l l c a s e s
Th e
,
.
F i eld-w or k
.
bli s h H a s i n l a s t proble m
( )
( b) L a y off S E i HV; a l s o TC i HVs
( 0 ) L a y off UW i TC a n d co n ti n u e th u s u n til a s u fficie n t
a
E s ta
.
,
nu
mber of poi n ts i s obt a i n ed
.
The le n g th of c u r ve ca n be co n ve n ie n tly fo u n d o n ly by me a s
u r e me n t o n the g ro u n d
Note the d iffere n ce i n method betwee n 8 5 a n d 13 3
.
.
134
c
Ve rt i a l Cu rv e s
.
.
It i s co n ven ie n t a n d c u s to ma ry to fix th e g r a de l i n e u po n the
profile a s a s u cce s s io n of s tr a i ght li n e s a l s o to ma rk the ele va
tio n a bo ve d a t u m pl a n e of e a ch poi n t where a ch a n g e of g ra de
occu rs ; a l s o to ma rk the r a te s of gra de i n feet per s ta tio n cf
A t e a ch ch a n g e of g r a de a v ertic a l a n gle i s for med
100 feet
To a v oid a s u dde n ch a n g e of directio n it i s cu s to ma ry to i n tro
d u ce a v ertic a l c u r ve a t e v ery s u ch poi n t where the a n gle i s
l a rg e e n o u gh to w a rr a n t it The cu r ve co mmo n ly u s ed for thi s
pu rpo s e i s the p a ra bol a A circle a n d a p a ra bol a wo u ld su b
The p a ra bol a
s t a n t i a lly coi n cide where u s ed for v ertic a l c u r ve s
e ff ect s the tra n s itio n r a ther better theoretic a lly th a n the circle
bu t i t s s electio n for the p u rpo s e is d u e pri n cip a lly to i t s g re a ter
It i s g e n er a lly l a id to exte n d a n
s i mplicity of a pplic a tio n
equ a l n u mber of s t a tio n s on e a ch s ide of the v ertex
.
'
.
.
'
'
.
.
.
,
'
.
.
R a i lr oa d Cu rves a n d E a r th work
76
P r obl em
th e n u mber
’
f g a de g of AV; 9 of V8
h lf
ch i de
of t ti o
Gi ve n th e r a tes o
.
.
s a
r
ns n ,
s
on e a
a
f ve r tex co ver ed by th e ver ti ca l c u r ve
o
,
th e e l eva ti
o n of th e p oi n t A
R eq u i r ed th e
s ta t i
a lso
.
e l eva ti o n ,
at
ea
o n of th e p a r a bo l a AB
,
ch
.
c
D r a w ver ti a l l i n e s
DD D
’
EE E
”
’
,
”
,
VHL, YBM
A l s o hori z o n ta l li n e s
VC,
'
ALM
P rod u ce AV t o Y
offs et DO a t the firs t s ta tio n fro m A
’
A etc
s eco n d
EE
'
.
,
The n
4 al
2
2 a1
G3
‘
2
3 a1
YB
2
a,
n a1
YB
YC
BC
2
fig
M
2
2
n a1
.
I
D u e re g a r d mu s t be g i ve n t o the s i gn s of both g
(86 )
and
g i n t he s e
’
for mu l a s Whether or
F ro m the ele va tio n a t A we ma y n o w fin d the req u ired ele va
tio n s s i n ce we h a v e g i ve n g
a n d we a l s o h a ve
a1
,
,
Pa r a b oli c Cu rves
A meth od be tt er a n d more
DD
”
nv n
= 3g ;
e below
or u s e i s gi v n
n
”
OU
g
”
EE :
2 g —
a2 = 2 g
4 a1
HL
3 g — a3 = 3 g
9 al
’
VL
c o e ie t f
’
g ;
77
.
—
.
a1
”
DO
A ga in ,
’
" —
EE
’
HL
-
”
DO
—
’
”
EE
'
2 g
'
4 a1
al
g
)
g
:
3 a1
—
‘
9 a l — (2 g
ey
z
r ht gr a de th e ele va tio n of a n y s ta t i o n i s fo u n d
fro m t h e precedi n g by a dd i n g a co ns ta n t g
On a v ertic a l cu rve the ele va tio n of e a ch s t a tio n i s fo u n d
fro m th e precedi n g by a dd i n g i n a s i mil a r w a y n ot a co n s ta n t
bu t a va r yi n g i n cre me n t bein g for the
On
a
st a ig
,
.
,
,
,
,
,
.
,
1st s t a t io n fro m A
A
A
c h a gi g by s cce s i e
i
n
u
n
g
a
g
3 al
di ffere n ce s
g
5 a1
e a ch c a s e
of
s
v
2 al
in
.
The A m R y E n g As s n s t a te s a s to le n gth of vertic a l
“
c u r ves th a t on Cl a s s A ro a d s ( ro a d s with l a rg e tr a fii c) r a te s
of ch a n g e of
per s t a tio n on s u mmi ts a n d
per s t a t io n i n
per s t a tio n
s a gs s h ou l d n o t be exceeded
On mi n or ro a d s
”
per s ta tio n i n s a gs ma y be u s ed
With
on s u mmit s a n d
v ery s teep g r a de s howe v er e ve n hi g her r a te s th a n reco mme n ded
by the A ss oci a tio n ma y s o meti me s s ee m n ece s s a ry
”
The ra te o f ch a n g e per s t a tio n corre s po n d s t o 2 a l i n the
fore goi n g formu l a s
Let r
r a te of ch a n g e per st a tio n
137
.
.
.
.
.
,
'
.
.
,
,
,
.
.
.
The n fro m ( 86)
9
_
n
R a i l r oa d Cu r ves a n d E a r thwork
78
F ro m p r a ctic a l
.
co n s idera tio n s the v ertic a l cu rv e will i n
g e n er a l exte n d a n eq u a l n u mber o f f u ll s t a tio n s o n e a ch s ide of
the v ertex
:
.
,
,
.
The n
mu s t be a n eve n n u mber ( n ot odd )
n
The r a te s of g ra de a ro u n d th e c u r ve will be
g
Ea
l
‘r
g
r
g
_
2l r i
’
ch r a te di fferi n g by r fro m the precedi n g
138
Gi ven
St a
_
‘
.
E x a mp le
.
e tc
'
.
.
Gr a de s a s follows
Ele v
Ra te
.
.
5
IO
I22 00
.
IS
g
A ss u me
7;
r
1r
E
g
The n
g
’
g
gr
g
I% r
7
r
2% r
g
T
E nd of c u r v e
9
!
The ele v a tio n for St a 1 5 th u s obt a i n ed a g ree s wi th the elev a
tio n s hown i n the d a ta A ll the i n termed i a te ele va tio ns a r e
”
therefore checked
.
.
.
CH AP T ER VII I
T
139
.
A Tu r n ou t i s
URNOU T S
.
.
tr a ck le a di n g fro m a ma i n or o the r
a
tr a ck
T u rn o u ts ma y be for s e ver a l p u rpo s e s
.
I
c
B r a n h Tr a ck ( for li n e u s ed a s a B r a n
.
11
.
er a l t r a ffic )
r
n
.
f
r p a ss i n g tr a i n s a t s t a t i o n s , s t or i n g c a r s ,
o
(
S i di n g
.
ch R oa d fo ge
lo a di n g or u n lo a di n g a n d va r io u s p u r
po s e s )
,
.
Sp u r Tr a ck
for
p
u rpo s e s other t h a n g e n er a l tr a fl i c
(
a s to a q u a rry or w a reho u s e )
,
.
Cr os s Over
n g fro m on e tr a c k t o a n other
for
p
a
ss
i
(
g e n er a lly p a r a llel )
,
.
T h e e s s e n t i a l p a r t s of a tu rn o u t a r e
1
.
1
.
c
Th e S wi t h
.
2
.
T h e F r og
3
.
.
Th e Gu a r d R a i l
.
S o me de vi ce i s n ece s s a ry to c a u s e a tr a i n t o tu rn fro m th e
ma i n t r a ck
thi s i s c a lled the S wi tch
2
A ga i n , i t i s n ece s s a ry th a t on e r a il of the t u r n o u t tr a ck
s ho u ld cro ss on e r a il of the ma i n tr a ck ; a n d s o me de v ice i s
n ece s s a ry t o a llow t h e fla n g e of the wheel to p a s s thi s cro s s i n g ;
”
thi s dev ice i s c a lled a F r og
3 F in a lly if the fl a n g e of the wheel were a llowed t o be a r
a g a i n s t the poi n t of t h e fro g there i s d a n g er th a t the whe el
mi ght a c c ide n t a lly be tu rn ed to the wro n g s ide of the fro g
poi n t Therefore a Gu a r d R a i l i s s e t oppo s ite to the fro g a n d
thi s pre ve n ts the fla n g e fro m h e a r i n g a ga i n s t the fro g poi n t
”
.
.
'
.
.
,
,
.
,
.
79
R a i l r oa d Cu rves
80
a nd
E a r th work
.
ro g s a r e of va rio u s for ms a n d ma kes b u t a r e mo s tly of thi s
g e n er a l s h a pe a n d th e p a rts a r e n a med a s follows :
P
poi n t
T
to n gu e
L toe
H
heel
M mo u th
0 = thro a t
WW wi n gs
Thi s s hows the s ti ff frog
”
“
The s pri n g fro g i s ofte n u s ed where the t ra ffic on the
ma i n li n e i s l a r g e a n d o n the tu r n o u t s ma ll
I n the s pri n g
’
’
fro g W W i s mov
a ble
AD r e pr e
s e n t s the ma i n li n e
F
,
,
.
.
,
.
,
and
W W i s pu s h e d
’
’
D
ide by the wheel s
o f a tr a i n p a ss i n g
o ver the t u rn o u t
n u mbers
F ro g s a r e cl a ss ified by a s erie s of st a n d a rd
”
“
The A m R y E n g A ss n fixes the n u mber n by di vidi n g
as
.
.
’
.
m
A
KD A
,
.
.
le n g th of to n g u e by wi dth of heel
,
L
A
-- i
n
B
B
Thi s i s st a n d a rd pr a ctice b u t n ot a dopted by a ll ra ilro a d s
”
“
The fr o g a n g l e i s the a n g le betwee n t h e s ide s of the
to n gu e of the fro g APB
.
,
.
1 40
.
Pr oble m
.
Gi ven n
.
R e q u i r ed F r o g A n g l e F
co t
cot
.
PH
F
l AB
=
2
n
F
§
8
9
( )
The fro g i s n ot bro u ght t o a fin e theoretic a l po i n t or ed g e
”
a ct u a l
poi n t pre s e n t pra ctice le a ves
bu t i s left bl u n t a t the
the frog on e h a lf i n ch thick a t the a ctu a l poi n t
”
“
'
.
L et
b
The n
nb
thickne s s a t a ct u al poi n t
d is ta n ce th eoretic a l to a ctu a l poi n t of fro g
.
,
.
Tu rn ou t s
wi t ch co mmo n ly u s ed a t th e pre se n t
”
ti me i s the s plit s witch
F i g A s hows t h e s witch s e t for the
tu r n o u t a n d F i g B for the ma i n l i n e W it h the s plit s witch the
1 41
Th e fo r m of
81
.
.
s
.
.
.
.
,
o u ter r a il of th e ma i n l i n e a n d the i n n er ra i l of the tu r n o u t
c u r v e a r e co n ti n u o u s The s witch r a il s AB a n d C D a r e e a ch
pl a n ed down a t on e e n d to a wed g e poi n t s o a s to lie for a por
tio n of their le n gth clo s e a g a i n s t the s tock r a il a n d s o g u ide the
wheel i n the directio n i n te n ded A n a n gle c a lled the s witch
a n g le i s th u s for med bet w ee n the g a u g e li n e s of the s toc k r a il
The s witch r a il s a r e con
a n d the s w itch r a il a s DC E of F i g B
n e ct e d by s e ver a l tie rod s a n d o n e o f the rod s n e a r the po i n t
i s co n n ected with a n other rod which g oe s to the s witch s ta n d S
to
co
ectio
with
the
i
n terlocki n g tower ) fro m which the
or
a
n
n
n
(
poi n t o f s witch i s thrown either for ma i n tra ck or for tu r n o u t
The joi n t betwee n the fixed e n d of the s witch r a il
a s de s ired
a n d the co n n ecti n g r a il a t B or D i s n ot bolted ti g ht e n o u g h to
prev e n t the s li g ht motio n of the s witch r a il n ece s sa ry The
s witch ra il th u s f a s te n ed a t the e n d B i s n ot s piked a t a l l for i ts
e n tire le n g th a n d a cts a s a hi n g ed piece B oth r a il s th u s mo v e
to g ether a n d thro u gh their e n tire le n gth s lide on fl a t s teel
pl a te s pro v ided for th a t pu rpo s e The fixed ( o r hi n g ed ) en d of
thi s r a il B i s pl a ced fa r e n ou gh fro m th e s tock r a il t o a llow s a t
Thi s i s 6 % i n che s with the le n gth of s witch
i s fa ctor y s piki n g
r a il va ryi n g fro m 1 1 feet to 33 feet i n the s ta n d a rd s of the A m
R y E n g A ss n
Ga u g e of tr a ck i s di s t a n ce fro m i ns ide of r a il
'
to i ns ide of r a il
S t a n d a rd g a u g e i s 4 8M
,
,
.
,
,
,
,
,
.
,
,
.
.
,
,
,
.
,
,
.
.
,
,
.
.
,
'
.
,
’
.
.
.
.
.
R a i lr oa d Cu rves a n d E a r th work
82
.
.
The s wi tch r a il i s n ot pl a n ed to a fin e ed g e bu t i s left wi th
a ppreci a ble thick n e s s freq u e n tl y o n e q u a rter of a n i n ch
The
poi n t i s n ot left re a lly bl u n t b u t i s s h a ped down thro u gh a s hort
di s t a n ce fro m the poi n t s o th a t the wheel fl a n g e s h a ll sa fely
p a s s by
I n the c a s e of the fro g it s ee ms n ece s s a ry to di s ti n g u i s h c a re
f u lly betwee n the theoretic a l poi n t a n d the a ctu a l poi n t With
the s witch there i s n o occ a s io n to co n s ider the theoretica l poi n t ;
the a ctu a l poi n t or the mo va ble e n d of the s witch r a il i s the
o nl y poi n t n eces s a r y to con s ider
I n l a yi n g ou t a t u rn o u t fro m a s tr a i ght tra ck the s witch r a il
i s s tr a i ght ; the fro g i s a l s o s tr a i g ht a circ u l a r c u r v e c a lled the
le a d c u rv e i s i n trod u ced to co n n ect the s e a n d lie t a n ge n t to
the m
,
.
.
.
,
,
.
,
,
,
,
.
Pr oble m
.
Gi ven i n
a
c
tu r n ou t , th e g a u g e of tr a k g ;
o
l e n g th of s w i tch r a i l 1 ; th i ckn e ss a t p i n t
of
w ; h e e l d i s ta n ce betwe e n g a u g e s i d es
d i s ta n ce r m th eo r e ti a l p i n t to
r a i ls t
fo
c
o
f g k ; f g g l F d mb
of f g t i t p o i t b
; th i k
l o l ad
di
R
R q i d
of l d
l p i nt
E f o m p o i t of w i t h t th o t i
l p i t f f og
d l o t
t
of f g
toe of
c
n
e u re
,
n es s
a n
e
an
a
s
n
a s
cu r ve
ea
s
n
,
a n
ro
us
ra
r
ro
ro
ro
c
nu
er
.
a s
e
o
o
e re
ca
o ac ua
o n
o
L et EILF a n d C DF
r
.
be the
r a ils of t u r n o u t
El a n d C D the s witch r ails
ID i s perpe n dic u l a r to
ODE
D r a w p a r a lle ls a n d per
,
.
.
p en di cu l a r s IM, LN, O M,
LP, a s a r c LA
l o
s witch a n g le H El
L et S
‘
heel dis t a n ce Hl
t
.
,
,
l
w
EI
Q D
C D,
thickn e ss of
s wi tc h r a i l a t E
.
84
‘
Cu r ves a n d E a r th wor m
R a i lr oa d
1 44
.
Gi ve n fo r a t u r n o u t, R , l , t, S
.
F
.
R eq u i r ed co or d i n a tes to c u r ved r a i l a t q u a r ter p o i n ts
-
A, B , C
.
C o n s ider ce n ter of cu r ve to be ma rked 0
.
rod u ce c u r ve DI to U where it i s p a r a llel to E H
D r a w per pe n dic u l a rs IH AA B B ’ CC
P
'
UW :
t
EW :
v er s
R +
§
s in
S
—
S)
UOC :
—
’
,
,
S :
.
,
a
d
l z
UQ B = UOA + i ( F — S )
UO D = UOC + i ( F
S)
S)
for
check
a
(
)
u t e r ror o f more th a n
witho
foot
(
)
F
EH
l
EA
':
Z UA d
R + £
s in
7
£
R +
si n
O
—l
':
AA
13 +
BB’ :
B +
2
’:
EB
UOB — d
2
EC
'
R+
:
g
s i n UOC —
l
co :
d
ve r s
UOA+ a
Z
§
m g
ve r s UOB -l— a
ve r s
UOC -l- a
To a v oid c u tti n g ra il s on e or the other of the clo su re
r a il s between heel of s wi t ch a n d t oe of fro g ma y be ma de fu ll feet
wi tho u t fra ctio n s B y le n gthe n i n g the t a n g e n t of the s witch
r a il beyo n d the heel the le a d i s i n cre a s ed ; by le n g the n i n g the
t a n g e n t of the fro g b a ck of the toe the le a d i s decre a s ed The
le a d s fo u n d i n t hi s w a y a r e c a lled pr a ctic a l le a d s
the le a d s
”
pre v io u s ly co n s idered a r e c a lled theoretic a l le a d s
The A m R y E n g A ss n h a s a dopted cert a i n co mbi n a tio n s
”
of s witche s a n d fro gs a s s t a n d a rd a n d c a lc u l a ted a ta ble of
r a dii le a d s ( both theoretic a l a n d pr a ctic a l ) a n d co ordi n a te s of
qu a rt er poi n ts T a ble X XII A a n d XXII B show thes e
145
”
.
,
.
,
.
,
.
’
.
.
.
-
,
,
.
.
Tu rn ou ts
Pr oble m
Gi ve n th e i n c r ea s e
.
85
.
f
c
l ea d
o
n e es s a r y
cti c a l l ea d ; a ls o F S l t k g
R eq u i r ed i n cr ea s e of ta n g en t p a s t h e el of s wi tch
pr a
,
,
,
,
,
to
s ecu r e
.
,
.
theoretic a l le a d ; E’ M pra ctic a l le a d
’ ’
n
d
E AD E a E A C DF be the corre s po n di n g t u rn o u ts
L et EM
EA
EC
’
l
l
2:
'
chords AD C D t a n g e n ts AVN D V
req u ired i n cre a s e
g i ve n i n cre a s e of le a d NC
D r a w p a r a llel AA ;
'
The n E E
of ta n g e n t ;
'
'
:
.
ADV
,
DVN
F
C DV
7
F ollowi n g ( 9 1 )
F or
S ; AA
’
’
EE
( 9 8)
z
g
l
t
’
1) S i n S
“
k Si“ F
2 s i n § ( F + S ) si n % ( F
S)
99
( )
fi n di n g cc—ordi n a tes of q u a rter poi n ts u s e i n s te a d of ( 96 )
,
the followi n g
146
S ) ; CA A
'
S)
F
(
%
R
S ) a n d AC a n d AD coi n cide
“F
’
s in
S
F
}(
I n tri a n gle A AC, A CA
’
,
.
a
z
P r oble m
.
(l
t
’
l ) si n S
R +
Gi ve n th e dec r e a s e o
fl
5
v
L et D E
k and
,
BF
’
th a t
s
’
k
,
,
,
A
9
6
)
(
ea d n e ces s a r y
c a l l ea d s ; a l s o F S l t k g
R eq u i r ed i n c r ea s e of ta n g e n t p a s t t oe of fr og
cu r e p r a cti
ers S
,
to s e
.
.
F ro m the fi gu re i t ma y be f o u n d
m % (F
2 s i n § ( F + S ) s i n 5( F
S)
S)
1
00
)
(
R a i lr oa d Cu rves
86
and
E a r th work
.
I t h a s beco me
the c u s to m to s ta ke ou t the po si tio n of
the fro g poi n t F F ro m thi s poi n t F a g ood tr a ck fore ma n will
8
work b a ckwa rd a n d l a y ou t the
t u r n ou t a ccor d i n g to the s t a n d a rd
pl a n
F o r a n y con t i n u a n ce of t u r n o u t
beyo n d the poi n t o f fro g where
thi s i s a ma tter of fieldwork a very
co mmo n pra ctice i s a s follows
n t o f fro
e
t
n
s
the
tr
it
oppo
ite
the
poi
S
a
s
a
( )
g at T
( b) L a y off o n the tr a n s it ( on t h e proper s ide o f O ) the
va l u e of the fro g a n g le F
( c) S i g ht i n the directio n T H p a r a llel to A8
u
d
T
rn o ff HTC F
( )
( e) The tr a n s it the n s i ghts a lo n g TG with v ern ier a t
A n y c u r ve de s ired ma y the n be l a id o ff co n ve n ie n tly by de
fl ect i on a n g le s a n d thi s cu r ve will co n n ect a t T ( oppo s ite F)
with wh a te ver a rr a n ge me n t of tr a ck exte n d s b a ck wa rd fro m the
poi n t of fro g to the poi n t of s witch Where the li n e i n a d va n ce
o f F i s n e w loc a tio n TC i s the b a s i s for th a t loc a tio n ; TC i s
either co n ti n u ed a s a s tr a i ght l i n e or i t beco me s the t a n g en t to
a de s ired c u r v e a n d the tr a n s it i s a lre a dy s e t on TG with the
v er n ier a t
Whe n the t u r n o u t i s to co n n ect with s o me tra ck
p a r a llel to the ma i n tr a ck the s i mple s t method i s to re s ol ve the
proble m i n to a c a s e of re v er s ed c u r ve s with p a r a llel ta n g e n ts by
the followi n g method s i mil a r to th a t of 12 5 If the c u rve
u s ed beyo n d F i s exte n ded b a ck wa rd tow a rd the poi n t of s witch
u n til it beco me s p a r a llel to the ma i n tr a ck
the ou te r r a i l of
thi s c u r ve will n ot i n g e n er a l be t a n ge n t to the corre s po n di n g
r a il of the ma i n tra ck bu t there wi ll be a n offs et by a s ma ll d is
ta n ce which we ma y c a ll a a n d the re ver s ed c u r ve mu s t be
fig u red for a di s t a n ce betwee n p a ra llels of p a r a ther th a n p
the a ctu a l d i s t a n ce betwee n p a r a ll el tr a cks If there be a t u rn
‘
o u t a t e a ch of the p a r a llel tr a ck s p
2 a s ho u ld be u s ed
Thi s method of tre a t me n t i s n ot dis s i m
il a r to the u s e of p
a n d q i n s pir a l s a n d h a s va l u e i n ma n y c a s e s other th a n tho s e
o f p a r a llel tr a ck s ; s e ver a l c a s e s wil l be tre a ted i n the n ext
ch a pter
The method of fi n di n g a follows
147
.
.
,
.
'
,
,
.
,
’
.
.
,
.
,
.
‘
,
,
,
,
.
,
'
'
,
,
,
,
,
,
,
.
.
,
,
.
.
R a i lr oa d Cu r ves
88
a nd
Ea r thwor k
.
If it be de s ired to
u s e g re a ter preci s io n
a nd
t a ke i n to a cco u n t the f a ct
th a t the frog i s s tra i ght
fro m theoretic a l poi n t F t o
heel G a n d to ma ke the
c u r ve beyo n d the fro g
t a n ge n t t o F6 a t G
149
.
,
,
,
L e t FG
The n
AD
FD z
h
g
h sin F
v
y,
s in
F — h cos F
sin
F
3
R l
g)
R +
—
ers F
a:
n
- -
P r oble m
ver s F
h co s F + n b
10
6
(
)
l h s ia
( 10 7 )
a
--
ver s ed cu r ve ext e n di n g
u ne
ee
e
r
u a
es
an
r
a l s o p erp e n di
ee
a r a l l el
tr a cks ; a l s
r
,
F
’
q
i d LA
F n
F
ro m ( 7 6)
.
/ /
0
be heel s of fro g s F a n d F
’
;
08 2
a ; and
v
PB = R2
131 ;
MC
er s AOB
GOB
az
r
o h
o
o
’
,
h ,
c u l a r di s ta n ce between tr a cks
.
Ls
o
a ls
p , a n d ga u ge g
R e u i r ed a n g l es G0 8 a n d BPH
o
a re
f om h
f f o g b tw n p
F
q l f og
gl
h eel o
Le t G a n d H
f
l f f og t
Gi ven th e r a d i i R 1 , R 2 , of two p a r ts o
.
by ( 10 7 )
“2
“1
RI
R2
AOB
F
and
Tu rn ou ts
89
.
M ore co mmo n ly th e t w o frog s will h a ve th e s a me n u m
be r a n d the r a dii of the re vers ed c u rv e will be the s a me
151
.
.
0
Whe n
F
F
’
//
0
and
R1
R2
R1
.
P roble m
Gi ven F
.
R
BPH z AOB
GOB
1 52
R2
’
F , n, b;
:
F
a ls o p ,
g
.
R eq u i r ed th e l e n gth l ,
o
f
ta n g e n t
two
f og
r
s
th e
be twee n
.
’ be theo
d
F
n
Let
a
F
r e t i ca l poi n t s of fro gs
D r a w KTNL p e r p e n di c
l r t o A8
The n
u a
TN
KL
FNs i n T PN = p
l s ia
l i s the di s t a n c e
NL
’
’
F N cos P NL
KT
g
-
g
p
sin
m
g cos F
11
( 0)
F
fro the theoretic a l poi n t a t F to poin t N
’
oppo s ite the theoreti ca l poi n t a t F
The a bo v e s olu tio n hold s g ood wh a te ver be the t u rn o u t u s ed
’
F or a cro ss o ver betwee n exi s ti n g tra ck s if the di s t a n ce FF
be c a lcu l a t ed both fro g poi n t s ca n be loc a ted a n d th e e n tire
tu r n ou t s ta ked ou t witho u t tra n s it
.
.
,
,
.
fro m ( 3 0)
’:
FF
l
fi wppr ox )
gl
.
The di s t a n ce fro m a ct u a l poi n t of on e fro g t o the a ct u a l po i n t
’
f the other FF 2 n b
.
R a i lr oa d Cu rves a n d E ar thwork
90
ble m
Pro
.
Gi ven F , n , p , g
.
.
R e q u i r e d t h e r a di u s o
fc
th e p a r a l l el ta n g e n ts
u r ve R 2 ,
o n n e ct
to c
.
If P R C be t a ke n a t F the
theoretic a l p oi n t of fro g
Then TPF I
F
.
.
.
,
.
:
,
2 :
US — I S
z
ver s
.
p
—
g
g
ve r s F
2
S eco n d S o l u ti o n
p
—
.
UT = p — g ; PW = R 2 — 1
H
PW
by ( 11 8)
Rg —
155
.
P r obl e m
.
UT 2 n
g
(p
Gi ven g , p , I, F
—
2
g) 2
.
R eq u i r ed R 2
of c u r ve to
o
c n n ect p a r a l l e l t a n g e n ts
.
be the theoretic a l
poi n t of fro g ; l the d is
t a n ce fro m theoretic a l
poi n t of fro g to S oppo s ite
P C of c u rv e
D r a w the p er p e n di cu
la rs S U S M
Le t F
.
.
.
.
,
Then
NT
ML
MT
FS s i n UFS r: NT
NL
PS v er s S PT
su
LM
l si a
p
—
g
Rg—
g
ve r s F
'
ers F
B y t a ki n g F8 or t
h ( the di s t a n ce fro m theoretic a l poi n t to
heel of fro g) for mu l a ( 1 13 ) co vers the ca s e where the re vers in g
cu r v e s t a rts fro m the heel of fro g
v
.
92
R a i lroa d Cu r ve s
DF = C D co t C FD ;
z—
z
DFZ
Fo
’
E
2:
DO
E :
E a r th work
and
.
g cot % F ;
E z
2 gn
g
g
= 2 R9
2
R +9
2
2
E :
R +
g
2
2
E
— Q
R
+
2g
R z
n
2
R +
—
R +
= 2 g n2
2 g
En
The s e for mu l a s i n 15 6 a n d 157 a pply o n ly i n the c a s e of
th e s tu b s witch a n d a r e n ot to be u s ed for s plit s w itch t u r n o u t s
-
158
.
-
,
P r oble m
.
.
Gi ve n th e d eg r ee D of a s tu b s wi tc h t u r n o u t
-
f m
ro
a s t r a i g h t tr a
ck
.
R eq u i r ed th e d eg r ee of cu r ve D ’ f or a
s w i tch
tu r n o u t
f d eg r e e D
o
f m
s tu b
ma i n tr a ck
g , r ema i n i n g th e s a me
ro
F, n ,
a cu r ved
.
It ma y be s hown th a t for a t u r n o u t to the i ns ide of the c u r v e
’
D
D
D ( a pprox )
11
9
(
)
for a tu r n o u t o u t s ide the c u r v e
’
D
D
D m ( a pprox )
’
exc ept th a t
D
D m D a pprox )
when
D
T a ke the c a s e of the t u r n o u t on the i n s ide of a c u rved ma i n
tra ck
Whe n the ma i n tr a ck i s s tr a i ght g the dis ta n ce fro m fro g
”
poi n t to the r a il oppo s ite i s the ta n g e n t de fl ectio n of 7 0
for the o u ter r a il of the tu r n o u t c u r ve w ho s e deg ree i s a ppr ox i
ma tely D
F ro m ( 2 6 B )
s o th a t
:
.
",
:
.
-
.
.
,
,
,
,
.
Whe n the ma i n li n e i s c u r ved g become s the offs e t betwee n
two c u r ve s on e the o u ter r a il of the tu rn o u t c u r ve a n d the
other the o u ter r a il of ma i n tr a ck
A ss u mi n g the chor ds 0 for the o u ter r a il s of the t u rn o u t
,
,
,
.
Tu r n ou ts
93
.
c u r ve s to be eq u a l i n the t w o c a s e s of s tr a i ght ma i n tra ck a n d
c u r ved ma i n tr a ck
Dz)
by ( 2 7 )
(M D
a“
a n d the de g ree o f the t u r n o u t c u rv e mu s t be s u ch th a t
:
.
The va l u e s of c a n d E a r e n e a rly e q u a l s o th a t wh a t i s tr u e of
the chord i n thi s rel a tio n i s als o tr ue of E ( very clo s ely )
Therefore for a g i ve n va l u e of E
D m ( a pprox )
D
D
F u rther more the le n gth o f t u rn o u t c u rv e i s e qu a l t o 0 ( very
‘
.
’
:
.
clo s ely ) ; for the g i ve n le n g th
s i n ce
D
’
0
:
the a n gle I
cD
= D , t h e d 1ffer e n ce 1n a n gl e -
’
—
10 0
F, and
CD ",
CD
0
10 0
10
th a t the fro g a n gle 18 n ot ch a ng ed ( ma teri a lly )
S i mil a r co n s ider a tio n of the tw o c a s e s of t u rn o u t o u ts ide t h e
c u r ve of ma i n t ra ck will s how the expre s s io n s a bo ve to be tr ue
so
.
'
.
159
.
Exa mpl e
R e qu ired the s t u b s wi tch tu r n o u t fro m a 3
-
.
ma i n li n e c u r v e u s i n g a No 9 fro g
.
°
.
T a ble XXII s hows for a No 9 fro g the
deg ree of c u r ve
7
D
The de g ree of ma i n li n e
3
Dm
deg ree of t u r n o u t
10 3 1
D
D
Dm
’
B y preci s e for mu l a
10 3 2
D
I n a s i mil a r wa y for a t u r n o u t on the o u ts ide of the s a me
c u r ve
.
°
°
°
'
’:
—
’
0
D
1 60
.
’
D
A n other le s s ma the ma tic a l , b u t v ery u s ef u l ill u s tr a tio n
thi s : If we co n cei ve the s tr a i gh t ma i n t r a ck a n d the s t u b
s witch t u rn o u t c u r ve to be repre s e n ted by a model wh ere the
”
a
r
s
a
m
e
a de of el a s tic ma t eri a l
r il
be n di ng proce s s
; u s in g a
it wi ll follow th a t if the ma i n tra ck r a ils be be n t i n t o a circu l a r
c u r ve with the tu r n o u t i n s ide the n the r a i ls of th e t u rn o u t
c u r ve wi ll be be n t i n to a s ha rper c u r ve a n d s h a rper b y the
de g ree of c u r ve
i n to which the s tr a i ght tra ck i s be n t
S i mil a rly whe n the s tr a i g ht tr a ck i s be n t i n the oppo s ite d i r e c
tio n the t u r n o u t c u rve will beco me fla tter by the a mou n t o f
is
,
,
.
.
,
R a i lr oa d Cu r ves
94
1 61
Pr oble m
.
E a r thwork
and
Gi ve n F , n , k, g ,
.
.
Dm
.
R eq u i r e d th e sp l i t s wi tch tu r n o u t
-
g i ve n c u r ved ma i n tr a ck
f m th
ro
e
.
Ta ble s X XII A a n d XXII B gi v e for va rio u s n u mbers of
fro g the le n g th of s witch r a il l heel di s ta n ce t le a d E r a di u s
R a n d de gree D of le a d cu r ve le n gth o f fro g fro m toe to the o
r e t i ca l poi n t It
a l s o co ordi n a te s to q u a rter poi n ts
The s e
ta ble s s how the s ta n d a rd s a dopted by the A m R y E n g A s s n
for t u r n ou ts fro m t a n ge n ts
F or t u r n o u t s fro m cu rv ed tr a ck s a pplyi n g the
be n di n g
proce ss I t k E re ma i n u n ch a n g ed i n le n gth ; thi s i s tr u e
a l s o o f the co ordi n a te s a t the q u a rter poi n t s the y va l u e s bei n
g
me a s u red a lo n g the c u r ved ma i n r a il a n d a: va l u e s n or ma l to
thi s r a il s tra i ght r a i ls beco me cu r v ed to the degree of the
c u r ved ma i n tr a ck tr a ck or r a ils a lre a dy cu r v ed a r e ben t i n to
cu r v e s s h a rper th a n before by D m ( or fl a tter by
depe n di n g
u po n which s ide of the ma i n tr a ck t h e c u r ved tr a ck lie s )
’
The d egr e e of le a d cu r ve D D i D m
The fro g re ma i n s s tra i g ht n ece s s a rily the d i s t a n ce k i s s ma ll
for a ll s h a rp le a d cu r v e s a n d the re su lti n g e r ror will be s ma ll
F u rther more the s tr a i g ht fro g i s l a id a s p a rt of the ma i n tr ack
which i s a ss u med to be cu r ved s o th a t a correct ma the ma tica l
tre a tme n t i s i mpr a ctic a ble
The s witch r a il ca n be a n d s hou ld be cu r ved t o the degree D m
It i s better to cu r ve it i n a be n di n g ma chi n e b u t it i s ofte n l a id
s tr a i ght a n d the tr a ffic depe n ded u po n to cu r v e it t o a fit a g a i n s t
the s tock ra il
,
,
,
,
,
,
-
.
.
.
.
.
.
,
”
,
,
,
,
-
,
,
.
i
:
.
,
,
,
.
-
.
,
.
1 62
.
E x a mp le
ber 9 fro g Ta ble XX II A gi ve s
F or a n u m
.
,
’
=
10
h
l :
T a ble XXI I B g i ve s for
D
pr a ctic a l le a d s
9
°
”
E0
the co ordi n ate s a r e
-
I n u s i n g a n u mb er 9 tu r n o u t i n s ide a 2
D
’
9
0
2
°
°
°
cu r ved tr a ck
11 29
’
Th e other li n e a r di me n s io n s re ma i n u n ch a n g ed
.
R a i lr oa d Cu r ve s a n d E arth work
96
1 64
E x a mp le
.
.
.
T u r n o u t fro m cu r ve o u ts i de the ma i n tr a ck
=
4 ;
L et s
4 7 08
g
.
.
P r eci s e M eth od
.
m
E +
g
(p
ta n 9 0
g
m
(p
’
"
19
40
3
l 0)
”
4
6 50
9
16
n
)
g
—
§(F
ta n
°
O)
°
-
’
°
ta n 6 50 ’ 49 "
5p
D2
1 00 ( F + 0 )
L
D2
’:
100 x 13
’
8
App r o xi ma te M eth o d
8
°
°
°
.
A pply the
be n di n g proce s s of p 9 3
I n the c a s e o f a t u r n o u t fro m a s tr a i gh t ma i n tr a ck where
.
.
,
8 and p
n
15
from ( 1 12 ) 132 —
3
g) 2
(p
2 x 64
R2:
13 2 4 9 ; D 2
10 0 x 7
4
F
“
7 09
°
'
( T a ble xx u )
.
for s tr a i ght tr a cks
5
I pm
“
°
8 19
°
4
L
4
0
°
D2
0
as
’
with s tr a ig ht tr a ck
’ preci s e method
1
6
8
)
(
°
preci s e metho d )
.
Tu rn ou ts
97
.
Whe n the s id i n g i s i n s i de
the ma i n tr a ck
I n a s i mil a r f a s hio n it ma y be
u s i n g th i s fi gu re th a t
s how n
1 65
II
.
.
.
,
,
F
ro m tri a n gle OFT
(
t a n .) 0
—
o
l
g
m
f
H
F ro m t ri a n g le PFS
(p
p
2
9
M
0)
100 ( F
0)
W h e n the s idi n g i s ou t
the ma i n tra ck b u t with the
s i de
ce n ter of t u rn o u t c u r ve i n s ide of ma i n
tr a ck
L e t EFS be the o u ter r a il o f ma i n
tra ck
FT t h e i n n ef r a il of t u rn o u t
166
.
111
.
,
.
.
.
F ro m tri a n g le OFT
(p
ta n 9
} 0
?
Rm I
--
1
2
F mm
t r l a n gl e PFS
.
R2
!
QQ
ta n g ( F + 0 )
p
2
With both § 165 a n d § 166 a pproxi ma te re s u lts ma y be
re a ched by u s i n g the be n di n g method of p 9 3 Where the
r a di u s R 2 of th e s eco n d c u r ve i s l a rg e a n d p i s s ma ll the a p
proxi ma te method will be s u ffl ci e n tly clo s e ; where p i s l a rge
E xperie n ce will deter
th e preci s e method will be n ece s s a ry
mi n e i n wh a t c a s e s it will be s u fficie n t to u s e the a pproxi ma te
re s u lt s a n d where preci s e for mu l a s s ho u ld be u s ed
,
.
,
.
,
,
.
,
.
R a i lr oa d Cu r ves a n d E a r thwork
98
1 67
f or
fig
P
.
roble m
as
s h o wn
in
th e r a d i u s R of s t u b
ch cu r ve a l s o th e p e r
s wi t
,
p e n d i cu l a r
twee n
q
Gi ve n
.
tr a cks
u r e,
.
e ual
f og
r
d i s ta n ces
be
a ls
s
o
.
R equ i r ed AOB, BC , C D
.
Fr o m ( 7 1 ) v ers AOB
BC s i n C BE
CE
BC s i n AOB = p ’
or
and
CD
S i n ce
the s t a n d a rd t u r n o u t c u r ve exte n d s o n ly fro m heel of
s witch to toe of fro g a n y co n ve n ie n t c u r v e beyo n d the fro g i s
If a cu r v e of the s a me degree a s th e s tu b s witch
a ppropri a te
cu r ve be u s ed beyo n d the fro g poi n t the a bo ve formu l a s will
a pply ( wh a te v er the s t a n d a rd t u r n o u t c u r v e ma y be ) s i n ce the
o u ter c u r ved r a il exte n ded b a ck co me s ta n g e n t to the r a il of the
ma i n tr a ck The s t u b s witch c u r v e th u s i s v ery co n ve n ie n t to
,
-
.
,
,
-
.
u se
.
If it s ee ms a d vi s a ble t o co n s ider the fro g stra i ght fro m poi n t
a t F to heel a t G i n the fi g u re below
,
L et FG 2
CM :
R
z
g
h
h sin F
2 n 2 (g + h si n F )
Thi s i s th e r a
di n s of the cu r ve
whos e o u ter r a il
i s t a n g e n t to the
r a il of the ma i n
-
tra ck a n d a l s o to the fro g a t i ts heel G
F or a s erie s o f tr a cks like tho s e a bo v e whe n the ma i n tr a ck
i s c u r v ed the co mp u t a tio n s ma y be ma de for s tr a i g ht tr a cks
J u s t how fa r this proce s s
a n d the be n di n g proce s s a pplied
ma y be c a rried w ill be deter m i n ed by experie n ce
.
,
.
.
R a i l r oa d Cu rves a n d E a r th work
1 68
.
P roble m
.
Gi ve n f or tr a cks s h o wn i n
fig
.
u r e th e r a d i u s
R of th e c u r ve beyo n d th e h eel of
f g
a ls
ro
o
’
o
p , p betwee n p a r a l l e l tr a cks ; a l s F , n g
R e u i r ed a n g l e AOK a n d di s ta n ce F’ F”
q
Le
.
.
.
t GK with i t s ce n ter a t 0 be o u ter r a il of the g i ve n cu r ve of
r a di u s R
P rod u ce thi s cu rv e to A whe n i t i s p a r a llel to HM
L e t BC with ce n ter a t P a n d N D with ce n ter a t Q be s i mil a r
cu r ve s prod u ced
”
L e t FG F E F S be s tr a i g ht li n e s fro m theoretic a l poi n t to
heel of fro gs
.
.
,
,
.
’
,
,
.
BP = NQ = R —
g
:
A
H
;
KB :
LN
F i n d a by
—
The n by ( 7 6 )
oy
v
¥2 R
g
+a
ers AOK
KL
( 12 8 )
2
sin
;
AOK
f
L
‘
S i n ce KF
’
”
LF
KL
FF
I
s in
AO K
101
Tit r ri ou ts
'
a
.
1 69
Gi ve ii the r adi a l d i s tance be tween
P roble m
g i ve n
c u r ved ma i n tr a ck a n d a p a r a l l el s idi n g
.
.
.
'
a
.
The two tr a cks a r e to be co n n ected
by a cro s s o ver which s h a ll be a r e
u n e q u a l r a d ii
v ers ed c u r ve o f g i ve n
beyo n d the fro gs
R eq u i r ed th e ce n tr a l a n g l e of e a ch
cu r ve of th e r ever s ed c u r ve
ce n ter li n e of i n n er tr a ck
L e t AC
-
,
.
.
.
AO s
RI
’
’
; RP l
; RQ
R2
’
’
d
an
R 2 a r e the r a dii o f the c u r ve s
beyo n d the fro gs a n d ma y be a s s u med
a s a n y re a s o n a ble v a l u e s
F i n d a l a n d a 2 by a pplyi n g the
”
be n di n g proce s s ( p 9 3 ) a n d the n
.
.
r
105
o
)
(
The n i n the tri a n gle POO fin d
Em I p
’
'
“
R2
,
“
S ol v e for OPQ , POO, Po o, the n ROB
I n pr a ctice thi s proble m mi g ht
Gi ve n
p, g
’
o
r
n
d
F) a
A s s u me n (
n
(
t a ke the followi n g for m
.
or
F ro m the s e v a l u e s of n a n d n
co mp u te a ll d a t a requ ired for
Thi s w ill i n vol ve
a cro s s o v er betwee n s tr a i g ht ma i n tr a ck s
a s s u mi n g va l u e o f D 1 a n d D 2 a n d co mp u ti n g a l a n d a 2 by
1 50
or 15 1
The va l u e s of a 1 a n d a 2 ma y be co mp u ted ei ther for the c a s e
co v ered by ( 105 ) or by
The n a pply the be n di n g proce s s
This will ch a n g e the de g ree s of the tu r n o u t c u r ve s by the
a mo u n t of
b u t the l e n g t h s o f the t u r n o u t c u r v e s will re ma i n
u n ch a n g ed ( a pprox ) a n d the d i s t a n ce s y r a n d y z obt a i n ed
by ( 103 ) or ( 106 ) wil l a ls o re ma i n u n ch a n g ed ( a pprox ) a s will
a l s o the v a l u e s of a a n d a 2
’
-
.
.
.
'
e
.
o
.
,
.
Rai lr oa d Cu rves and E a r th wor k
1 02
.
Probl e
m
.
.
Gi ven two ma i n tr a cks n ot p a r a l l el
th e
n
qu a l f
u ne
h, h , 9 ;
,
f
o
o
o
R2 , o
r e ver s ed cu r v e c n n e cti n g
r
ee
f om h l t h l f f g
f gF
ti o
of
ee
o
ro
one
n
A ls o
’
F
F
n,
l
s
a ls
r og a n g e
,
th e u n equ a l r a d i i R I ,
a ls
’
’
.
a ls
ro s
o
th e tw o
o th e p o s i
.
R eq u i r ed th e a n g l es BPS a n d S OH o th er e
ver s ed cu r v e ; a l s o th e p o s i ti o n of p o i n t B
L et O H = R 1 +
HF
g
R2 +
:
B
P
;
f
%
h
S e t tr a n s it a t t heoretic a l poi n t of g i ve n fro g F
L a y off FL perpe n dicu l a r t o T E
M e a s u re FL a l s o FLE
D r a w perpe n dic u l a r s HD FK OC
a n gle betwee n ma i n tr a ck s
I
Let
I
90
LFK
The n FLE 90
.
.
.
,
.
,
,
:
.
°
°
.
HOA— COA
HOC
I
F
.
I)
DK
h COS ( F
FK
FLcos I ; LK
FLs i n I
FK— h s i n ( F
I )
HO
:
C E= HD +
v
er s ( F — I )
.
CH AP T E R I X
CONNE CT I NG
.
T RACK S AND CROS S I NGS
.
bra n ch le a ve s a ma i n tra ck
Thi s i s c a lled
a n a dditio n a l tr a ck i s l a id co n n ecti n g the t w o
”
”
“
Y tr a ck a n d the co mbi n a tio n of tra cks i s c a lled a Y
a
172
.
In
ma n y c a s e s where a
,
.
“
.
,
P r oble m
.
Gi ve n
a
ma i n
s tr a i g h t
HK, a l s o
tr a ck
th e
of cu r ve beyon d th e
“
fr og A ls o r a di u s R 2 of Y tr a ck be
A l s o s e l ect p r a cti ca bl e
tw ee n th e fr o g s
P C
.
a nd
.
R1
r a di u s
.
.
f F 1 , F 2 , F3
va l u es o
R eq u i r ed
.
HK fr om P C
c
th e di s t a n e
o
t u r n u t to P C
.
cen tr a l
f
o
.
f
a n g l es
“
Y
”
c
tr a k ;
t u r n ou t a n d
o
o
t r a ck to th e p i n t
.
.
f j u n cti on
o
a ls o
th e
of
Y
“
the
s tr a i g ht ma i n tr a ck
g i ve n
.
AB the t u r n o u t
C L the
D ra w
“
Y tra ck
F i n d AH
(l l
; KC :
Then cos AO B
1:
G2 ;
BI
(23
.
.
= l 80
°
—
I‘
by ( 10 7 ) p 88
.
.
ON
OP
HO — KP
R1 + a 1 — R2 — a 2
O B + BL + LP
R 1 + R2 + a 3
(R 1
R2
a3
1 04
) s i n Ic
.
perpe n dic u l a r NP
AOB z I ,
y
.
”
HK = NP = l
C PL = I
”
.
HK be
Let
f
o
Con n ecti n g Tr a ck s
174
.
P r obl em
Gi ve n
.
10 5
.
ma i n tr a ck H E K , a ls o th e
s tr a i g h t
a
Cr os si n g s
and
f g OB
beyo n d th e
a nd
T O,
r a di u s
ce n tr a l
a n g le
AOB, of t u r n ou t c u r ve co n
n ecti n g
w i th
a
.
.
c
th e r a di u s PC of
Y
”
.
.
t a n ce BD
P T
.
.
“
f
o
f
o
.
f mRT
ro
.
Y
.
“
Y
.
c
”
o
.
HK fr om P 0
R eq u i r ed th e d i s ta n ce
.
a ls
A ls o s el ect
tr a ck
p r a cti ca bl e va l u es of F 1 , F 2
tu r n ou t to P C
,
ta n g en t BD
s e on d
“
ro
tr a k ;
.
of
a ls o d i s
f tu r n ou t c u r ve to
o
tr a ck
.
H E K be th
g i ve n
ma i n
tr a ck ABD the tu r n o u t
”
the Y tr a ck
CL
Le t
e
.
R1 ; C P Z R
L e t AO
HK= AC = l ;
HA a l ;
P
z
AOB
II ;
C PL
I2
BD
KC = a 2
DL z a z
D ra w p a r a llel AV
rod u ce D B t o E
F i n d a 1 a n d a 2 by
The n 8 0
ED
P
.
KP ta n
(R 2
m
l o
a s
(R 2
HK
1
a2
a2
co
t
)
R 1 ta n
(12
)
HI
s in
KV
E
31
1
I1
$
1
I:
R 1 ta n
I1
EH
EK
(R2
E V)
AO t a n 4 AOB
) t a n i [2
COt
.
(V B
C PL
m
'
z
5I1
R 1 ta n
.
2
1
I1
18 1 1 1
1
3
1
( 5)
c a s e differe n t fro gs a r e u s ed n e a r D a n d K s o th a t KC a n d
DL a r e n ot eq u a l the fo r mu l a s will be modified
L et
KC
a the s ma ller v a l u e
DL
a , the l a r g er va l u e
F ollowi n g the me t hod of
1 9 1 p 122
In
.
,
,
.
.
,
EK = ( R 2
a,
)
cot
1
i 1
(I f - a “
f
Sln
t
1
“w
11
as
10 6
R a i lroa d Cu r ves
175
.
Pr obl em
.
Ea r th work
and
.
I n the acco mp a n yi n g s ketch where
HBC 5 ma i n tr a ck
tu rn o u t
”
“
Y tr a ck
AD
.
.
LK
HB
Gi ven
OB
l ;
LQ Z R z
AP = R 1 ;
S e l ect P I
F2
F3
.
.
R eq u i r e
d the poi n ts
he n
(l g
DL z:
(1 3
by ( 10 7 )
PH
R1
a 1
R2
02
R2
as
2
DQ
F in d
.
Z CK
CQ
C o n s ideri n g
D and C
( 1 ;
F i n d AH
t
.
B O a s b a s e of a ri ght tri a n g le
a s i ts a ltit u de
HB
OPH a n d PO the hypote n u s e
PH
,
F in d a ls o
:
O
Q
he n
POQ OPQ POO
the n
BOC APQ
D a n d C wi ll the n be e a s i ly dete r min ed
t
,
Rm + R2 + a 2
,
,
the figu re where
.
the ma in tra ck a n d LK is
”
“
tr a ck
th e t u r n o u t AD the
Y
HBC i s
.
,
OB
Gi ven
AP :
S el ect F 1
KQ
R2
R 1 ; BOC :
0
'
F2 ; F3
R eq u i r ed the poi n ts A a n d D
.
F i n d a l , a 2 , (l g by ( 10 7 )
Fin d O
l o
the n
a s
and
N O N the
,
,
n
EP
EOF, EQ
EN 2
POO
HB
EQ P
OQ N
PQ O deter mi n e s po s itio n of L or D
EPQ
deter mi n e s le n gth AD a n d E N
HB fixe s H or A
R a i lr oa d Cu r ves a n d E a r th work
108
177
P r obl e ms Gi ven a
.
cu r ve cr os s i n g a ta n g e n t ,
’
R , g, g ,
a n d a n g l e 0 be tween ta n g e n t a n d c u r ve
.
.
R eq u i r ed fro g a n g le s a t A, B , F,
D r a w AO, B0 , CO, PO, DO ;
D
.
l o MO perpe n dic u l a r to CM
a s
,
.
T he n
MO
R cos C
M°
cos
MOA
cos A
g
I
7 9
g
2
cos
MOD
cos D
)
g
2
00 8 B
cos F
Q
.
2
DOF = MOD
Th e r a il l e n gth DF
Exa mpl e
Gi ven 0
.
R
32
MOF = D
ang
°
-
F
le DOF; a n d BF
D
°
3; g
8 g
BL
FL
.
‘
4
’
R eq u i r ed a n gle D a n d dis ta n ce DF
R 3 log = 2 8553 8 5
.
3 2 2 8 cos = 9 9 2 6 l 90
’
°
.
MO 604 7 48
R 3 = 7 16 7 8
:
.
1g
.
1g
:
.
’
= OF= 7 15 2 8
.
1g
l
l og = 2 7 83 2 6 1
l og _
l og = 2 8 5447 6
l og : 2 85 447 6
.
OF
.
.
°
3 1 55
'
"
2 3 cos = 9
.
’
”
2
4
37 4
3
°
928 7 85
cos
”
2
3 1 55 3
°
Ta ble XX
.
42 ’
DOF= 0
'
°
2 85447 6 z l og 7 15 28 = R
log = 8 090 5 7 9
.
.
.
l og = 8 8 12 = DF
.
.
Con n ecti n g Tr a ck s a n d Cr os s i n g s
178
109
W he n two tr a cks cro ss a t a s ma l l a n gle they a r e ofte n
,
.
co n n ecte d by a s lip s wi tch i n which the o u ter r a il lie s en
t ir e l y withi n the li mits of the cro s s i n g a n d i s co mpo s ed o f two
s witch r a il s a n d a co n n ecti n g c u r ve a s s how n i n the fi g u re below
”
,
.
Pr ob le m
cr o s s i n g of tw o tr a cks th e a n g l e of
b g ; a ls o c l ea r a n ce m
a ls o n
cr o s s i n g fr o g F
fr o m a c tu a l p o i n t of fr o g to p o i n t of sp l i t
s wi tc h
a ls o l a n d t
R eq u i r ed l en g th s a l o n g r a i l be twe en fr og p o i n ts ;
a l s o r a di u s R of cu r ve f o r a s l ip s wi tc h
Gi ven for
.
a
,
,
,
.
,
.
DA
QB
HA
LB = t
Fl E
F4 Q
le n g th of s wi tch ra i l
l:
m
cle a r a n ce requ ired
The n bn di s t a n ce betwee n theoretic a l a n d a ctu a l poi n t s o f
fro gs F1 a n d F4 i n fro g s F2 a n d F3 theoretic a l a n d a ctu a l poi n ts
coi n c i de
.
F1 F3 =
g
bn
—
sin
F1 F2
_
F
'
“
F3 F4 2
‘
F2 F4
I n the s l i p s witch , prod u ce the g a u g e li n e s DA a n d Q B to
V
on
the li n e F2 F3 A ltho u g h the poi n t of s witch h a s a thickn e s s E D
of a bo u t a q u a rter of a n i n ch n o a ppreci a ble error re s u lts if DV
be ca lcu l a ted a ss u mi n g DF2V to be a t r i a n gl e i n which
.
,
,
F2 DV :
Then
S ; DF2V :
90
F
°
5
AV :
R +Q
2
M iddle ordi n a t e for chord AB
A r c AB
.
; F2 0
E1 F2
DV— l
AV
_
ta n % ( F — 2 S )
R +
R
5
v
ers % ( F
a ng
2 S)
le ( F — 2 S )
11 0
R a i lr oa d Cu r ves a n d E a r thwork
P r obl e m
.
.
Gi ven two ma i n tr a cks c r o s s i n g a t a g i ven
c u r ve co n n ect i n g
th e two a n d exten d i n g fr o m h eel of fr o g
to h eel of eq u a l fr og
VF betwee n
R eq u i r ed th e d i s ta n ces VF
th e r a d i u s R of
a n gl e I ;
,
.
’
f fr og s
a ctu a l p oi n ts o
.
P rod u ce gi v e n
c u r ve to I a n d J
where it i s p a r a llel to g i ve n ma i n
tra cks
’
F i n d by ( 10 7 ) a
a
.
.
g
OC Z R -l- (l —
R + a — Q
cv
ta n § I
fro m ( 106 )
CF :
ya
VF
CV—
CF
’
VF
:
If the a n gle a t V is a t a l l s h a rp
a llowa n ce s ho u ld be ma de for the di ff ere n ce betwee n the the
o r et i ca l a n d a c tu a l poi n t of the fro g a t V
,
«
.
Pr oble m
.
c V B a n d th e
s tr a i g h t l i n e A
V of a br a n c h tr a ck i n ter
a nd a t a
s ecti n g i t a t a g i ve n p o i n t V
t
u
u
r
n
o
u
s
r
a
d
i
s
R
o
t
I
l
o
i
v
n
n
l
e
a
e
a
f
g
g
;
c u r ve to co n n ec t br a n ch l i n e a n d h ee l of
fr og a l s o F n h b g
a l s o p o s i ti o n of
R eq u i r ed i n fi g u r e VA V B
Gi ve n
ma i n
s tr a i g h t
a
tr a k
,
,
,
,
,
p oi n t of f r og
,
,
.
,
.
i d a by
F n
AV
AV
’
—
’
VV
E ta n % I +
R ta n % I
VB :
F i n d F fro m
pu t a tio n u s i n g
,
a
s in I
a
ta n I
B by fieldwork
m
co
11 2
Ra i lr oa d Cu r ves
s ect i n g
tu r n
a nd
at an
Ea r th wor k
a n gle
I :
a l s o r a di u s
R , of
o u t cu r ve fr om h eel of fr og to br a n ch
li n e ;
a ls o F , n ,
R eq u i r ed i n th e
fig
h , b, g
u r e,
.
IB, IOA
L e t 0 be the ce n ter of c u r ve of bra n ch li n e
P be the ce n ter of c u r ve of t u r n o u t
D r a w perpe n dic u l a r s PB, OC, PK
Fin d a
by ( 10 7 )
IOC
I
o I ; IC
OC s
o s
Rb s in I
In F i gu re 1
KO = OC
K
2
co s
POK;
PK
OP s i n POK
IB :
IC
PK;
IOA :
POK— I
I n F i g u re 2
—
PK
le z
sin
POK;
IC —
cc
KO
KO ; IOA :
O P cos POK
PO K + 9 0
Other c a s e s will occ u r req u iri n g fi gu re s
°
-
I
differe n t fro m tho s e
s o me o f the m will be s u gg e s ted by the fi gu r e s i n
s hown here
5 181
.
Con n ecti n g Tr a ck s a n d Cr oss i n g s
183
.
P roble m
Gi ve n
.
of
a
s tr a i g h t
c
tr a ck a n d a cu r ved tr a k
c
i n ter s e t i n g a t a g i ve n a n g l e
r a di u s
I ;
a l s o r a di u s
R , of tu r n o u t cu r ve
h ee l offr og t o h ee l of fr og
R eq u i r ed i n th e
pr
—
-
T
113
fig
u r e,
a ls
f m
ro
oF n h b g
,
,
,
,
.
IOA, IB
B
n
be the ce n ter of c u r ve of ma i n tr a ck
P be the ce n ter Of c u rv e Of t u r n o u t
D r a w perpe n dic u l a rs PB OC O K or PK
F i n d a 1 a n d a 2 a t A a n d B by ( 10 7 )
L et 0
,
IOC
,
,
I
OC
cos I ;
OP
( R2
(l l
)
I n F i gu re 1
PK = R ¢ + a 2 — OC
PI —
;
sin
POK;
KO
O P co s POK
I8
2
IC
KO ;
IOA
POK + 90 — I
KO
2
0C
°
i re 2
l n F gu
“ — cos
.
( 131 + G 2 )
3
POK ;
IB :
IC — PK ;
PK
OP s i n POK
lOA :
POK— I
Other c a s e s will occu r req u iri n g fi gu re s
differe n t fro m tho s e
s how n here ; s o me of t he m will be s u gg e s ted by t h e fig u re s i n
18 1
.
R a i l r oa d Cu rves
11 4
184
.
P r oble m
.
E a r th w ork
a nd
Gi ven two cu r ved l i n es of t r a c k o
f
r a di i
R1
R 2 cr o s s i n g ea c h o th er , i n ter s e cti n g a t a n
a n g l e I ; a l s o th e r a di u s R , o
t u r n o ut
r o m h ee l t o h eel o
f fr og a ls o F , n , h , b, g
f
f
R eq u i r ed i n th e
fig
u r e,
API, IOB
Let 0 a n d P be ce n t e r s of ma i n tr a cks
Q be ce n ter of t u r n o u t
OIP
I
-
F i n d a l a t A, a n d a 2 a t B by ( 1 0 7 )
In
tri a n gle
IOP,
IO
R2
IP
R1 ;
O IP
I
QP
Q0
R1
S ol ve for O P, IO P, IPO
I n t ri a n g le OQ P
R2 i
( R,
( Rt
O P co mp u ted
S ol v e for Q O P, Q PO, OQ P
ro m Q PO a n d IPO fin d API
F ro m IOP a n d Q OP fin d IOB
F
,
,
dl
)
02
>
11 6
R a i lr oa d Cu r ves a n d E a r th work
.
p a ss i n g directly fro m ta n ge n t to circu l a r cu r ve there i s
a poi n t ( a t P
where t wo req u ire me n ts co n flict ; the tr a ck
c a n n ot be le vel a cros s a n d a t the s a me ti me h a ve the o u ter
ra il ele va ted I t h a s b ee n the c u s to m to ele va te the o u ter ra il
On the t a n g e n t for perh a p s 100 feet b a ck fro m the P C
Thi s
is u n s a ti s fa ctory
It h a s therefore beco me the be s t pr a ctice
to i n trod u ce a cu r ve of va ry i n g r a di u s i n order to a llow the
tr a i n to p a s s g r a d u a lly fro m the ta n g e n t to the circ u l a r c u rv e
In
,
.
.
.
.
.
,
.
The tr a n s it i o n will be mo s t s a ti s f a c t orily a cco mpli s hed
whe n the ele va tio n e i n cre a s e s u n iformly with the di s ta n ce I
fro m th e T S ( poi n t of s pira l ) where the s pira l e a s e me n t cu rv e
le a ve s the t a n g e n t ; the n 1 i s a co n s ta n t
186
.
.
.
5
2
gv
A ( a co n s t a n t ) o r P l
Rl
S i n ce
g
,
v,
32 2 A
.
A a r e co n s t a n ts , R l
The n
g
2
”
a nd
Rl
C ( a co n s ta n t )
R 0 1.
R
1
z
wher e
t
a
pp
r ox
/
.
1)
D.
Rc
r a di u s of circle
Dc
de gree Of circu l a r cu r ve
lc
3
sc
Then
D
1c
R ds
z.
tot a l le n gth Of spir a l i n feet
141 A
.
the S pira l A n gle or tot a l i n cli n a
tio n of c u r ve to ta n ge n t a t a n y poi n t
“
s
pira l a n gle where s pira l j oi ns c ircle
dl or ds
fro m ( 141)
dx :
1
dl s i n s
an d
dv
dl cos s
.
.
Sp i ra l E a s e men t
Cu r ve
1 17
.
A ll va l u e s of 3 will g e n er a lly be s ma l l , a n d we ma y a ss u me
s in s
dx
s dl
1
:
dy
dl
2
2
l dl
y dy
2 R cl c
2 R ole
I n tegr a ti n g
cos s
a nd
s
:
3
x
,
y
6 R ClC
which i s the equ a tio n of the Cu bic P a r a bol a a c u rve fr e
q u e n tly u s ed a s a n e a s e me n t c u r v e
1 be n o t u s ed the
If howe ver the a pproxi ma tio n co s s
re s u lti n g c u rve will be more n e a rly correct th a n i s the Cu bic
I n thi s c a s e
s in 3
P a r a bol a
8
”
“
,
.
,
:
,
,
.
dx
’2 d ’
s dl
2 R elc
3
1
I n te gra ti n g ,
6 R ,, lc
The re s u ltin g cu r ve we ma y c a ll for the l a c k of a be tte r
”
“
n a me
the Cu bic S pir a l E a s e me n t C u r ve
,
,
.
The Cu bic P a r a bol a i s well a d a pted t o l a yi n g ou t cu r ve s by
”
“
M oder n r a ilro a d pr a ctice f a v ors
Off s ets fro m t h e t a n g e n t
”
“
deflectio n a n gle s a s the method Of work where ver p r a ct i
c a ble In the ca s e Of a n e a s e me n t c u r ve the lo n git u di n a l me a s
u r e me n t s a r e mo s t co n v e n ie n tly m a de a s chord s a lo n g the c u r ve
.
.
,
so
t
h
th a t x
an
is x
ls
repres e n ts a c u rv e more co n ve n ie n t for u e
s
6 R e lc
R
!
6
3
as
well a s more n e a rly correct
Ev
.
ide n tly
c O
propertie s of the two c u rv e s will b e very s i mil a r
The followi n g n ot a tio n i n co n n ectio n with s pir a l s h a s bee n
a dopted by the A m R y E n g A s s n
F or the poi n t O f ch a n ge
fro m t a n ge n t to s pira l T S
fro m s pira l to circu l a r c u rv e S C
fro m circ u l a r c u r v e to s pir a l C S
fro m s pira l t o t a n g e n t S I
Thi s n ota tio n will be a dopted her e
th e
.
’
.
.
.
,
.
.
.
.
,
,
f
,
.
.
.
”
.
.
.
R a i lr oa d Cu r ves a n d E a r thwork
11 8
187
G i ve n , i n a Cu bi c Sp i r a l , l ,
'
.
R eq u i r ed 3 , s o , a n d
o
“
.
R,
c
t ta l d efl e t i o n a n g l es
Z
2
142
BGN
s
and
C FN
8
2 R e l,
2 B.
2 R ol e
Thi s ( 145 ) i s the expre s s io n ( i n the fo r m of le n g th of a r e for
r a di u s 1 ) for t h e ce n tr a l a n g le of the co n n ect i n g circ u la r c u r ve
I n a n other for m
for a le n gth Of on e h a l f the le n g th of s pir a l
-
it is
c
.
(l
c
200
i n feet a n d s o i n de g ree s )
,
( 145 A )
‘
v
If th e circu l a r c u r v e be prod u ced b a ck fro m C to K where it
“
beco me s p a ra llel to AN i ts le n gth in feet will be s i n ce KOC
3
,
CFN
Sc
.
A l s o AL
q
2
(
prox )
ap
4
5
1
B
)
(
.
A ga i n for a n y poi n t B o n the s pira l
s in
BAN
s in i
3
9
pprox
)
(
3
3
pprox
)
(
a
.
a
.
c
2
1
fronn ( 142 )
2 R o z,
i
w n6 110 e
3
a nd ic
—
fie
4
6
1
)
(
3
4
6
1
A
(
A l s o the b a ck de
fl ectio
n
ABG
s —
ACF = 2 i ,
i :
BGN
3 i — i :
BAN
2i
1
4
6
B)
(
R a i lr oa d Cu r ves a n d Ea r th wo r k
12 0
.
h a ve bee n co n s idered
a s me a s u red a lo n g the c u r v e i ts e l f ; b u t me a s u re me n ts i n th e
field a r e n ece ss a rily t a ke n by chords Thi s i s reco gn i z ed i n
defi n i n g the deg ree of a s imple c u r v e 3 9 a s the a n gle a t the
ce n ter s u bte n ded by a ch o r d of 100 ft C o n s i s ten t with thi s
‘
i n the A m R y E n g A s s n S pir a l the le n g th of s pir a l i s
me a s u red by ten eq u a l ch o r ds so th a t the theoretic a l cu rv e i s
bro u g h t i n to h a r mo n y with field pr a ctice This s pir a l will be
referred to here a s the A R E A S pira l a n d a dopted i n pl a ce
The tw o Cu r ves s u b s t a n ti a lly coi n cide u p
o f the Cu bic S pir a l
to the poi n t where 3
a n d the di s c u s s io n o f th e C u bic
S pir a l a pplie s i n a g e n er a l w a y to the A R E A S pir a l a l s o
B eyo n d s
1 5 the A R E A S pir a l h a s i ts t a ble s co mp u ted
s u b s t a n ti a lly witho u t a pproxi ma tio n s ma ki n g it a v ery perfec t
a n d co n v e n ie n t tr a n s i tio n c u r ve e v e n for s h a rp c u r ve s o n s treet
r a ilw a y s
The A R E A Spira l reta i n s the followi n g fe a t u res ch a r a c
t e r i s t i c o f the C u bic S pira l
f
The
de
ree
c
u r ve v a rie s directly with the le n gt h fro m
o
a
g
( )
the T S
D Z
14 1 A
190
I n the Cu bic S pir a l , the le n g ths
.
.
.
.
.
.
,
,
,
.
.
.
.
.
,
.
0
.
.
.
.
.
°
.
o
.
.
.
,
.
.
.
.
.
.
(
.
)
The deflectio n a n gle s v a ry a s the s q u a re s of the le n gth s
fro m the T S
( b)
.
.
4
1
( 6 A)
0
The s pir a l a n gle a t the poi n t where the s pira l joi n s the
circ u l a r c u r ve i s eq u a l to the ce n tra l a n g le of a circ u l ar c u r v e
o f the s a me de g ree a n d of a le n g th o n e h a lf th a t of the s pir a l
( d)
-
.
4
1
5
A)
(
2 00
n t on
pr
ctic
l
p
rpo
e
oi
s the deflectio n a n g le to a n y
u
s
a
a
( )
p
th e s pir a l i s on e thi r d the s pir a l a n g le a t th e poi n t ( u p to a
01
v a l u e Of S
1 46
i
e
F or
-
‘
(
c
3
)
eyo n d 1 5 a n d u p to 45 for v a l u es of s correct va l u e s com
s how t h e followi n g e mpiric a l
n
n
by
the
E
A
ss
A
m
R
t
d
u
e
y
g
p
formu l a to a pply
8
8
B
°
°
o,
’
.
.
.
0 00 2 9 7 3
8
i a n d s a r e i n de g ree s
.
n ds
n
u
s
i
s
c
s
s
e
v
i
e
re
lt
o
g
.
Sp i r a l E a s e men t Cu r ve
W ith t h e
at
1 21
.
S pir a l , the a n g le ma de with
the t a n g e n t
the T S by the firs t chord i s t a ke n a s
.
.
so
3 00
No a ppreci a ble error i s fo u n d to re s u lt if the a n g le s ma de by
cce s s i ve chord s with thi s t a n g e n t a r e t a ke n a s ex a ct mu lti
ple s o f a s follows
su
:
1 , 7 , 19 , 3 7 , 6 1 , 9 1 , 12 7 ,
169 , 2 17 , 2 7 1
It i s e vide n t th a t the s e v a l u e s of 061 042 etc depe n d u po n s
a n d a r e i n depe n de n t of the le n g th of chord u s ed
“
offs et s fro m
F o r co mp u ti n g v a l u e s of x y the method o f
the t a n g e n t
66 i s a dopted a n d cc ordi n a tes x 31 a t e a ch chord
poi n t a r e fo u n d by u s i n g
,
,
o
.
.
o,
o
-
lc
10
F or
s i n (1 1 ,
l
—
C OS a ,
10
°
10
,
C OS 052 , e t c
.
g i ve n va l u e o f s the fi n a l cc ordi n a te s x y will be
a
-
o
o
g i ve n v a l u e of s o
to S C th a t
A
10
1
si n a 2,
'
directly proportio n a l to I
.
lo
,
.
C
—
.
so
th a t
a
y
will be co n s ta n ts of a
;
;
o
It will be tr u e o f the lo n g chord C fro m T S
.
will al s o be a co n s t a n t
Z
co n de n s ed t a ble o f va l u e s of
.
.
f g
32 , i
50
i s g i ve n i n
,
c
T a ble
O
VII B for va l u e s of s di fier i n g by 0
Thi s t a ble will h a ve occ a s io n a l r a ther th a n frequ e n t u s e ; i n
t e r me di a t e va l u e s m
a y be i n terpol a ted wi th s u fficie n t precis io n
}
for ordi n a ry c a s e s ; the l a bor of i n t erpol a ti n g wil l n ot be bu r
°
o
,
d e n s o me
.
F ro m the s e v a l u e s of d o a n d yo, deter mi n ed
bo ve va l u e s
of i
h a ve bee n co mp u ted for s u cce s s i ve val u e s of s u p to 45
a n d the s e a r e t a b u l a ted i n T a ble V I I
All of the co mp u t a tio n s
me n tio n ed a bo v e h a v e bee n ma de by the A m R y E n g A s s n
F or co n v e n ie n t u s e i n the field the deflectio n a n g le t o e a ch
chord poi n t i s n ece ss a ry a n d the a u thor h a s co mpu ted t he s e for
su c c e ss i ve va l u e s of s a n d t a b u l a ted th e
m i n T a ble V II
The deflectio n a n gle s a r e co n st a n t for a g i ve n v a l u e of s a n d
ma y be u s ed for thi s v a l u e of s wh a te ver the le n g th of s pir a l
pro vided the chord le n gth i s ma de o n e te n th the le n gth of Spira l
as
a
,
°
o
o
.
’
.
.
.
.
,
o
.
o
o
,
-
.
R a i l r oa d Cu r ves a n d E a rt h wor k
1 22
.
V a lu e s of p a n d q h a v e bee n co mpu ted by the a u thor by ( 148
a n d ( 148 A ) for va rio u s de g ree s of c u r v e a n d for va rio u s le n g th s
of s pir a l a n d the s e a r e fo u n d i n T a ble VI which g i v e s for e a ch
de gree a n d h a lf de g ree of c u rv e a s er i e s of len gth s of s pira l a n d
for e a ch le n gth va lu e s of s p q oc y C
,
,
,
,
o,
,
191
.
,
,
o,
,
o,
.
Gi ven I , l o, a n d R o or D o
P r oble m
.
.
R eq u i r ed th e Ta n g en t D i s ta n ce To
‘
.
i d q a n d p b y § 189
o r by T a ble VI or by
T a ble VII B
( a ) Whe n the s pir a l s
a t both e n d s o f the cir
cu l a r c u r v e a r e a like
F n
.
'
.
L e t AL= q a n d LK= p
AV: AL+ LV
—
AL+ OL t a n % LO D
Z
L= q + ( R +p ) t
I
c
,
I
an
T. s o+ Tc +p t a n l I
To i s a n g n
4
1
9
)
(
t e t di s t a n ce for circu l a r c u rv e a lo n e for the
where
g i ve n va l u e of I
s ep a r a te
Whe
d
i
ere
t
pir
l
s a r e u s ed a t the e n ds
a
n
ff
n
s
b
( )
va l u e s mu s t be fo u n d for LV a n d D V
,
.
,
.
L et
LK
p;
BD = p .
D ra w a r c D E
.
A l s o pe r pe n dicu l a r s E
V
’
,
VS
.
VS = p z — p .
LV= ( R
ov
n
t
a
p, )
41
I
ta n I
P’
si n I
49
1
A
)
(
149 3 l
R a i lr oa d Cu rves a n d E a r th wor k
1 24
192
P r oble m
Gi ve n D o a n d lo
.
.
R eq u i r ed p ,
q
The A m R y E n g A ss n
for mu l a s for v a lu e s of p a n d q
’
us
.
.
.
.
o th er d a ta for sp i r a l
a nd
,
fro m ( 145 A ) s
.
l oD o
o
1
4
5
A)
(
2 00
e s the followi n g empiric a l
,
p
q
bD o
a lo
el o —
fD
o.
T a ble s of the coe fii ci e n ts a b e f co n den s ed fro m the
A R E A T a ble s a r e g i ve n i n T a ble VII B for v a l u e s of s
differi n g by 30 i n ter medi a te v a l u e s ma y be i n terpol a ted
The deflectio n a n g le s ma y be fo u n d a s before fro m T a ble VII
,
.
.
.
,
,
,
o
.
’
.
.
193
.
Pr oble m
Gi ve n D o a n d p
.
R eq u i r ed to l a y o u t sp i r a l
.
.
5
q
pprox )
(
2
fro m ( 145)
KC
fro m ( 145 R )
9
2
a
.
CN
for s pir a l
cQ
CN
_
0
2
2R
4
CQ
§
l °2
2 Re
8 Re
co
99
3 Q N
3 KL z 3 p
CN
4Q N
4 KL
fro m ( 144)
5
.
co
3
CQ
l
( a pprox )
for c i rcle
Q N ( a pprox )
.
( pprox )
( a pprox )
a
.
.
3
14
( 9 a)
[a
C N = 2 3 T L = 8 TL = 4 KL
.
TL
55
2
5
( a pprox )
.
( 149 D )
Sp i r a l E a s emen t Cu r ve
12 5
.
3 p the le n g th of c u r v e ma y be re a dily determi n ed
F ro m CQ
If circ u l a r c u r ve KC h a s ce n ter a t O , KOC
CQ
‘
01
V
ers S
C FN
so
lo
2L
o
for other deflectio n s i
io
.
3p
i t:
OK
L for circ u l a r c u r ve KC
fro m ( 146)
.
( 1 46 A )
io
The b a ck deflectio n ACF 2 i
T a ble XXXIII will f a cilit a te s o me of the s e co mp u t a tio n s
‘
B y the a bo v e m ethod the v a l u e s of s a n d l ma y be re a ched
with s u b s t a n ti a l a cc u r a cy wi tho u t the u s e of the s pir a l t a ble s
Where clo s e res u l ts a r e n e ce ss a ry p ma y be r e co mp u ted by
T a ble VII B fro m the v a l u es of s a n d l a lre a dy fo u n d by the
If the n ew va l u e of p i s n ot s u ffi cie n tly clo s e
a bo v e for mu l a s
t o the g i v e n v a l u e correct v a l u e s of s a n d t ma y be fo u n d by
tri a l The va l u e of q i s fo u n d by T a ble VII B
The deflectio n a n g le s ma y t hen be t a ke n fro m T a ble V II
W hile the method of 19 3 i s more l a borio u s th a n the more
co mmo n method of § 19 1 it h a s Speci a l va l u e bec a u s e it i s
thoro u ghly el as tic a n d a n y g i v e n le n g th of s pir a l ma y be u s ed
I n a s i mil a r w a y if the v a l u e of p ( to g ether with D ) deter mi n e s
the s pira l to be u s ed the method of 193 beco mes u s ef u l
o
.
.
o
,
o
.
-
,
.
o
o
.
o
,
o
.
.
.
,
.
,
o
,
.
1
App r oxi ma t e M e t hod
Pr oble m
.
.
‘
G i ven D o a n d e i th e r l o o r p
R eq u i r ed s o a n d th e de
flecti o
n a n g l es
.
w i th
A ss u me the lo n g chord KC to be eq u a l t o
RI
193 fin d 3 p
B o a n d 3 p b y ( 2 6)
By
o
.
2
0 .
573 °
57 3 0
fro m B
o u t u s i n g ta bles
D
and
L
by ( 2 6 )
or
fin d L
fro m
~
L
q ( a pprox )
and i
.
Other deflectio n s a r e fo u n d by i
C o mp u t a tio n s
to
4
A
1
6
)
(
i n v ol vi n g the u s e of ( 2 6) ma y be ma de u s i n g
T a ble s X XXII I a n d XXX IV
.
R a i lr oa d Cu r ve s
194
F ield w ork of La yi n g ou t S pir a l
.
S elect o n the g ro u n d the v ertex
( )
a
E a r t h work
and
V
.
.
me a su re I ; or el s e
and
fix on g ro u n d poi n t L oppo s ite the poi n t K where the circu l a r
c u r ve will beco me p a r a llel to ta n g e n t
( b) S elect the le n g th
l of s pir a l to joi n g i v e n
circu l a r cu r ve ; this ma y
be t a ke n fro m T a ble VI
or
co mpu ted by § 19 3
fro m D a n d p
nd
c
F
i
v
a
l
u
e
of
q
( )
fro m T a ble V I or
and s
by 1 93
t
a
t
e
by
d
S
T
A
S
( )
me a su ri n g T fro m ver
tex or by me a su ri n g q fro m poi n t L a s the c a s e ma y be
n s it a t T S r u n i n s p i r a l u s i n g deflectio n a n le s
With
tr
a
6
g
( )
fro m T a ble VII
n s it a t S C
u r n v er n ier to 0
With
tr
a
t
n
d
beyo
n
d
0
a
f
( )
to me a s u re a n gle s
i ( this will be 2 t whe n s i s le ss th a n
,
.
o
o
.
o
.
.
.
o
,
.
.
.
.
°
.
.
o
°
o
o
o
n
n
a
n
d
T
ke
b
ck
i
ht
o
whe
a
a
s
v er n ier re a ds 0
T
S
the
g
(g )
li n e of s ig ht i s o n a u xili a ry t a n g e n t
u n i n circ u l a r c u r v e by d e flectio n a n gle s ; the ce n tr a l
R
h
)
(
2 s
I
a n g le o f circ u l a r c u r v e
n s it a t S T
a
t
r u n i n s eco n d s pir a l
a
n
With
tr
i
o
t
C
S
(
)
( )
”
“
o
n
1
6
C
heck
OS
( )
”
“
check i s n o t s u b s t a n ti a lly perfect r e s et th e
) If the
(Z
po i nt a t C S
I t i s i mport a n t th a t e a ch s pir a l s h a ll be correct thro u gho u t
i ts e n t ire le n g th
I n c a s e the s pir a l a n d circ u l a r c u r v e do n o t
‘
check properly a t the C S the d is crep a n cy s ho u ld be thrown
i n to the circ u l a r c u r v e where i ts effect will be u n i mport a n t
Whe n the circu l a r c u r ve is vi s ible fro m the C S the ge n er a l
method of 6 2 will g i ve the be s t re s u l ts a s follow s
u t fir s t s pir a l fro m T S
L
a
o
to
A
S C
y
( )
n d s pir a l fro m S T to C S
eco
L
a
u
t
s
B
o
y
( )
l
a y o u t circ u l a r c u r v e fro m
tr
a
n
it
t
a
n
d
S
e
t
u
s
a
C
S
C
p
( )
S C to C S a n d check a n g le to S T
°
.
.
,
.
o.
.
.
.
.
.
.
.
-
,
.
.
.
.
,
.
.
.
,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
R a i lroa d Cu r ves a n d E a rt h wor k
1 28
.
It ma y occ a s io n a lly ( a ltho u g h n ot frequ e n tly ) h a ppe n
a n d it
th a t the e n tire s pir a l c a n n ot be l a id o u t fro m the T S
will be n ecess a ry to deter mi n e deflectio n a n gles whe n the
tr a n s it i s a t s o me i n ter medi a te poi n t on the s pir a l I t will be
de s ira ble to occu py s o me re gu l a r chord poi n t
I n a n y Cu bic S pir a l the de g ree of c u r ve D i n cre a s e s u n ifor ml y
with the le n gth ( 14 1 A ) He n ce
the de gree of cu rve a t mu s t be
e qu a l to the d i fier e n ce i n de g ree
betwee n the circ u l a r c u r ve a n d
the s pir a l a t 5 where len gth A I
19 6
.
.
.
,
.
.
,
.
C5
.
S i n ce
the di verg e n ce i n the de g ree of the s pir a l is the s a me
for a g i v e n di s t a n ce whether thi s di ver g e n ce be fro m the t a n
g e n t AL or fro m the c u r ve C K i t will n a tu r a lly follo w fro m the
pri n ciple s e s ta bli s hed i n § 69 th a t the offs et to the s pira l for a
g i ve n di s ta n ce fro m C will be the s a me a s the offs et for the s a me
d i s ta n ce fro m A s i n ce the ch a n g e i n de g ree a t corre s po n di n g
poi n t s i s a lwa y s the s a me whether fro m t a n g e n t or c u rve
The s a me co n cl u s io n will be re a ched by referri n g to § 16 0
“
m
of p 93 where the el a s tic model a n d the be n d
n e a r the botto
”
i s referred to ; thi s be n di n g proce s s bei n g there
i n g proce s s
fo u n d to be correct ( a pprox ) fro m the de mon s tra tio n § 15 8
p 9 2 If thi s pri n ciple be correct it will follow th a t KT T L
which ma y be co n s idered a n extre me c a s e Th a t KT TL i s
de mo n s tr a ted ( i n 149 D ) to be correct i s a n a dditio n a l a s s u ra n ce
o f the correct n e s s o f the pri n ciple s t a ted a bo v e
I t will f u rther follow if E I a n d D 5 a r e eq u a l a n d a t eq u a l
d i s t a n ces fr om A a n d C re s pecti vely th a t the a n g le s E A a n d
F or the offs et di vided by the
D C 5 will be equ a l ( clo s ely )
di s ta n ce g i ves a pproxima tely the s i n e o f the a n gle a n d s i n ce
the s i n e s a r e equ a l the a n gle s a l s o a r e e qu a l ; s imil a r l y the
a n gle s LAT a n d KCT a r e eq u a l
I n other word s the di verg e n ce of a n y g i v e n s pir a l for a g i ve n
di s t a n ce i s the s a me either i n offs et or i n a n gle whether the
di verge n ce be fro m the ta n g en t or from the circ u l a r cu rv e
~
,
,
,
,
.
.
,
,
.
.
.
,
,
.
.
,
‘
,
.
,
‘
,
.
,
,
,
.
Sp i r a l E a s emen t C u rve
129
.
therefore follo w th a t if a t
a n y poi n t B on the s pir a l ABC the tr a n s it
be s et u p a n d the li n e of s i g ht be bro u gh t
197
.
I t will
\
,
\
/
J
c
/
ili a ry t a n ge n t BG a t th a t poi n t the n the defle c tio n
a n g le to a n y forw a rd poi n t on the s pira l will be the s u m o f
”
“
n g le
f
r the di s t a n ce fro m B t o
the
to
l
deflectio
n
a
o
1
t
a
)
(
tha t poi n t du e to the circ u l a r c u r v e HBJ whos e de g ree i s t h e
”
“
de gree of the s pir a l a t B ; a n d ( 2 ) the tot a l deflectio n a n gle
fro m the ori gi n a l t a nge n t for th a t s pira l for the s a me di s t a n ce
recko n ed fro m the T S For a ny b a ck po in t t h e deflectio n
a n gle fro m thi s a u xili a ry t a n g e n t will be the di ffere n ce be t wee n
the s e a n gle s
The proper u s e of the s e deflectio n a n gle s will a llow the li n e
o f s i g ht to be bro u g ht o n the a u xili a ry t a n g e n t a s well a s g i ve
me a ns for s etti n g a ll poi n ts o n the s pir a l
on
th e a u x
,
,
,
,
.
.
,
.
,
.
Ex a mpl e
for wa rd deflectio n a n gle s fro m po i n t 6
on a s pir a l 300 feet lo n g to j oi n 5 cu r v e
.
R e qu ired
°
.
,
7
so
°
The t a n g e n t 8 0 is fo u n d by l a yi n g off fro m chord AB t wi ce
t h e forw a rd deflectio n to poi n t 6 o r 2 x 54 ’
1 48 ’
x5
D a t poi n t 6
3 00
D eflectio n a n g le for 30 ft on 3 c u r ve
27
’
n
l
The tota l a g les wi l be a t poi n t 7 2 7
0 1'
2 8'
,
°
,
°
’
°
°
'
.
,
Th e b a
c
8,
54 '
9,
’
'
1
8
+ 13
10 ,
'
24
06
’
2
2
°
12'
k d efl ecti ons w i ll be a t po i n t 5 ,
27
'
0 1’
2 6’
4,
54’
06 ’
48 '
3,
8 1'
2,
'
10 8
I
24
1,
'
13 5
’
0,
’
102
°
'
°
’
°
’
Th e b a ck defle ti o n fro m p o i n t 6 t o T S a ls o = 0 54 x 2
1
c
37
54
°
.
.
1 24
z:
'
1 38
1 48
’
0
1 80
R a i lroa d Cu r ves a n d E a rthwork
.
The method of deter mi n i n g the a n g le betwee n the t a n
g e n t a n d a n y chord of the s pir a l ma y n ow be re a dily u n der s tood
a n d i s de s cribed i n the P roceedi n s of the A m R
g
y E n g A ss n
a s follows
D i v idi n g the s pir a l i n to t en eq u a l p a rt s the a n gle be t wee n
t h e t a n g e n t a t the T S a n d the chord fro m a s pir a l ( n
1 ) to
the poi n t ( n ) i s the ce n tra l a n gle of the Spir a l fro m the T S to
the poi n t ( n
pl u s the de gree of c u r ve a t the poi n t ( n 1)
ti me s h a lf the dis ta n ce i n s ta tio n s fro m ( n 1 ) to ( n ) pl u s the
deflectio n fro m the t a n g e n t a t the T S to th e chord s u b te n di n g
”
the fir s t t e n th of the s pir a l
198
.
,
’
.
.
.
,
.
.
.
.
,
.
.
se
.
+
S u b s tit u ti n g th e s u cce s s i v e n u mera ls 1 to 10 for n , the S11 0
l es
ces s i ve
va u
a n d 271 —
”
of a
300t h s
“
of s o
are
1 , 7 , 19 , 3 7 , 6 1 , 9 1 , 12 7 , 16 9 , 2 1 7 ,
.
imila r fa s hio n the Am Ry En g A s s n h a s c a lc u l a ted
the forw a rd a n d b a ckwa rd defle c tio n s whe n the tr a n s it i s a t a n
i n ter medi a te s t a tio n on the s pira l a n d T a ble VII A show s the s e
a s mu ltiple s ( by f u ll n u mbers ) of the firs t chord deflectio n
a n gle i ]
I n fi n di n g the n u mbers for thi s Ta ble the a s s u mptio n w a s
ma de th a t the deflectio n a n gle fro m the T S to a n y poi nt i s
o n e third the s pir a l a n g le t o th a t poi n t
which is a pproxi ma te
o n ly where s exceed s
Whe n the tr a n s it i s s et a t a poi n t
’
'
n
P a n d a deflectio n a n gle ( fro m the a u xili a ry t a g e n t a t P ) i s
t a ke n to a n other poi n t P ” the A m R y E n g A ss n s t a te s
The for mu l a s a n d r u le a r e a pproxi ma te a n d s ho u ld n o t be
’
’ to
ed
whe
n
the
ce
n
tr
a
l
a
n
le
fro
m
P exceeds the ce n tr a l
u s
P
g
a n g le fro m the T S by more th a n
Ta ble VII A f u r n i s he s a v ery s i mple method of fi n di n g for
w a rd a n d b a ck deflectio n s whe n it beco me s n ece ss a ry to s e t the
tra n s it a t a n i n ter medi a te poi n t on the s pira l While mu ltiply
i n g i ; ma y be s o mewh a t b u rde n s o me s etti n g u p a t i n ter medi a te
poi n t s will n ot be freq u e n t a n d s i mplicity i s of pri me i mpor ta n c e
In
a
’
s
.
.
.
.
.
.
,
s
o
’
.
.
.
-
.
.
.
,
,
.
1 32
R a i lr oa d Cu rves a n d E a r th wor k
Exa
mpl
e
Gi ven D,
.
D,
lo
200
.
.
ro m T a ble s VI a n d VII fin d deflectio n a n gle s for a c u r ve of
4
7
3 with l
D
2 0 0 where s
3
On 4 ci r cu
l a r c u r ve deflectio n a n g le for 20 chord
0
4 c u r ve deflectio n
s pir a l d eflectio n
’
for poi n t
1 0 24 ’
0 0 1’
0 25
F
°
°
°
°
o
,
°
o
’
°
°
°
2
°
°
’
°
12 ’
°
36
°
2
°
2
°
48
3
°
'
0 48
3
1
4
1
5
2
6
7
8
02
’
05
’
10
’
’
00
15
’
’
24
2’
12
’
2
’
° ‘
9
3 36
°
10
°
4 00
'
29
’
38
’
49
'
60
’
0
°
5 0’
1
°
17’
°
46
'
2
°
15’
2
°
46 '
3
°
17
'
3
°
50 ’
4
°
2 5’
1
2
°
5 00 ’
Th e s e a r e t ot a l d eflectio n a n g le s fro m a u x i li a ry ta n g e n t whe n
the tr a n s it i s on the 4 c u r ve
°
Fi e ld w ork
.
.
K i n g ro u n d fro m t opo gra phy or other pr a c ti c a l
requ ire me n ts the sa me a s for a n y co mpo u n d cu r ve
( b) A ss u me l a n d co mp u te p
( )
Fi x L o r
a
,
.
o
.
x A a n d C tr u e t r a n s it poi n ts o n c u rv e a t di s t a n ce s i
Fi
0
( )
,
f
2
fro m L or K
( d ) S et tr a n s it a t A
( 6 ) B ri n g li n e of s i g ht o n a u xili a ry t a n ge n t a t A
”
“
tot a l deflectio n a n g les t o s pira l a n d r u n i n
(f) S et off
S pir a l
.
.
_
.
.
200
.
Det e r mi n a ti on of Le n gt h of S pi r a l
.
The b a s is u s ed by the A m R y En g A s s n for fixi n g the
proper le n gth of s pira l i s the i n cre a s e per s eco n d of the ele va
tio n of the o u ter r a il Too r a pid a n i n cre a s e it is tho u ght will
c a u s e s o me di s co mfort to p a s s e n g ers The di s c u s s io n i s too ex
te n ded for a pocket book a n d will n ot be a tte mpted here
The A m Ry En g A s s n h a s prep a red a d i a g r a m s hown a s
T a ble VII C which co vers the reco mme n d a tio n of the A ss oci a
t io n for fixi n g the le n g th of s pir a l s
’
.
.
.
,
.
,
.
.
,
’
.
.
.
.
Sp i r a l E a s e men t Cu rve
201
.
P r oble m
Gi ven
.
t wo
ta n g en t
si
ea
mp l e cu r ves wi th
s u bs ti t u te
r a di u s
ch en d
si
a
o
c
mp l e cu r ve of
c n n e t i n g sp i r a l s
w i th
at
.
g i ve n t a n g e n t , co n n ecti n g
a n d C B o f r a d n R ,, a n d R , re s pecti vely
L e t DC
co n n e cti n g
.
R eq u i r ed to
g i ve n
133
.
t
the t wo cu rv e s AD
.
L e t GT be the g i ve n n e w c u rv e of r a diu s R o
.
A s s u me s u it a ble Spir a l s a n d fin d fro m t a ble VI ,
the s e s pir a l s a l s o q1 a n d q2
Joi n OP a n d dr a w perpe n dic u l a r OL
ST
f
r
o
9
2
1
,
GE
pl and
RI
pz
.
.
R‘ R”
The n t a n LOP
t
In the tri a n g le OPQ
5
0Q
’
co s
LO P
there a r e g i ve n
zR
c
Rs
p1 i
QP
S ol v e thi s tri a n gle for OQ P, Q OP, OPQ
T he n C PS
EO D
1 80
°
( OPQ
OPL)
°
( Q OP
LOP)
90
R0
.
Kn owi n g the s t a tio n s of D a n d C, the s t a tio n s of E a n d S a r e
'
re a dily fo u n d a n d a l s o the s t a t io n s of t h e C S
a pplyi n g 9 1 a n d 2 2
.
.
.
and
S C
.
by
R a i lr oa d Cu r ve s a n d E a r th work
1 34
20 2
.
Pr obl e m
Gi ven I a n d R C
.
ci r cu l a r
.
c u r ve GHE a ls o
,
co r r e
sp o n di n g q f or
p
a nd
a
sp i r a l
th e g i ven
to fit
TS
.
.
cu r ve
.
R eq u i r ed th e di s ta n ce
BH 2
11
th r ou g h
wh i ch
c u l a r cu r ve GHE
mu s t be moved i n a l o n g
V0 to a l l o w th e u s e of
th e ci r
thi s
sp i r a l ;
GA
di s ta n ce
P C to T S
.
a ls o
.
.
d
th e
f om
r
.
BH = PO = KG
p
cos
-
d :
Pr obl em
.
i I
AL + LK t a n LKG
q + p ta n§ I
Gi ven I , R , a n d h
R equ i r ed p an d d
( 150 A)
.
.
=
h cos § I
p
q i s fo u n d by T a ble V I I B or by §
d = q + p t a n gL
I n r e r u n n i n g old li n e s to i n trod u
-
c irc l r c rve i to be repl ced by
c e pir l s where a ori gi a l
s
a
n
,
n
pir a l a n d a circu l a r c u r v e
of the s a me de g ree
i t i s cle a r th a t the circ u l a r c u r v e mu s t
n ece s s a rily be s e t i n tow a rd s the ce n ter fro m H by a cert a i n
a mo u n t h
P r a ct ic a l co n s ider a tio n s ma y ofte n fix the dis ta n ce
The method of 1 9 3
h by which the c u r ve mu s t be mo v ed
will be fo u n d of co n s idera ble va l u e i n re v i s io n s of li n e s i n ce
i t a llow s gre a t fl exibility i n the s electio n of s pir a l s
u a
u
s
a
a s
,
.
.
.
R a i lr oa d Cu r ve s a n d E a r th work
1 36
.
Whe n it i s n ece s s a ry to keep the middle poi n t H u n
ch a n ged on a cco u n t o f a bri dg e or he a vy e mb a n kme n t or
otherwi s e it the n beco me s n ece s s a ry to ma ke p a rt of the c u r ve
The mo s t pr a ctic a l method
s h a rper a s CF i n the fi g u re belo w
a ppe a rs to be to a ssu me the a n g le FOH the p a rt of the c u r v e to
re ma i n u n ch a n g ed a l s o a ssu me the va l u e of p a n d co mpu te a ll
oth er n ece s s a ry d a t a
204
.
,
,
,
,
.
,
,
.
P r ob l em
Gi ve n I
.
ci r cu l a r
a nd
c u r ve a l s o p of p r o
a ls o
p os ed sp i r a l
,
,
FOH
a n g le
th e
II
ci r cu l a r
wh i ch i s to
f
o
cu r ve
re
ma i n
ch a n g ed
q u i r ed th e r a
Re
u n
i
.
d i u s R 2 of n e w cu r ve
to
wi th
or i g i n a l
FH ;
a ls
en t
w i th p
a ls
o
co
d
to T S
.
cu r v e
o q co n s i s t
a nd
R
d i s ta n
th e
DA
FOH
mp o u n d
C F,
ce
f mP0
ro
.
.
.
11
DOH
OP ver s NOP
R 2 ) v er s
(R I
I
NM
11 )
v
F i n d q fro m p a n d R 2 by
DA 2
d
ND
LP
KP = p
p
Rl — Ez z
Then
M0
er
p
1
s (2 1
7
193
.
AL
DL
AL
MP
q
Il )
R 2) s in
I
1 1)
2
1
5
A)
(
beco me s c o n ti n u o u s fro m the
firs t s pir a l thro u gh H a n d to i ts co n n ectio n with the s eco n d
s pir a l
A n other pr a ctic a l me t hod wo u ld be to a ssu me R 2 a n d p a n d
c o mp u te 11 q d
B y ma ki n g FOH
,
.
,
,
.
11
0, R 2
CH AP T E R XI
.
S ET T ING S T AKE S F OR E ART HWORK
.
co n n ectio n with E a rthwork i s s t a ki n g
”
a s it i s co mmo n ly c a lled
ou t or
S ett i n g S l op e S t a ke s
There a r e t wo i mpor ta n t p a rts of the work of s ett i n g s lope
s t ak e s
1 S e tt i n g the s ta ke s
205
.
T h e fir s t s t ep i n
.
,
,
.
.
II
.
K eepi n g t h e n o te s
.
T h e d a ta for s e tt i n g th e s t a ke s a r e
The
ro
o
m
e
n d w i th ce n ter s t a ke s s et a t e very s t a t i o n
u
s
g
( )
(
t i me s ofte n er )
d
A
record
f
be
n
c
h
m
rk
s
a
n
f
ele
tio
n
n
d
r
te
o
a
o
va
s
a
a
s of
b
( )
gr a de s e s t a bli s hed
c
The
b
a s e a n d s ide s lope s of the cro ss s ec t io n for e a ch
)
(
cla ss of ma teri a l
a
.
,
.
-
.
pra c ti ce n o te s of a l ign me n t a f u ll profile a n d va rio u s
c on ve n ie n t d a t a a r e co mmo n l y gi ve n i n a ddit io n t o the a bo ve
In
,
,
,
.
206
.
I
.
S etti n g th e S ta kes
Th e
.
work c o i t
n s s s of
( a ) M a rki n g u po n the b a c k o f th e ce n te r s ta kes th e
fil l
or
”
“
cu t
”
i n fee t a n d te n th s , a s
C
F
( b) S e tti n g s i de s ta ke s or s lope s ta ke s a t e a c h s i de of th e
c e n ter lin e a t the po i t where the ide s lope i ters ec ts th s
d ma rki g
f a c e of th e gro d
po th e i er s ide f th e
n
un
s ta
ke th e
cu t
or
,
s
an
“
fill
n
”
at
n
u
n
th a t poi n t
13 7
.
nn
e
o
ur
R a i lr oa d Cu r ve s a n d E a r th work
1 38
20 7
.
( a ) The proce ss o f fi n di n g the cu t or fill a t th e c e n te r
.
t ke i s a s follows
s a
Gi ven f or a n y s ta ti on th e h ei g h t o
e l eva ti on
of g r a de
f
ho
i n s tr u me n t :
hi , a n d th e
.
The n the requ ired r od r ea di n g for g r a de
It i s
e c e s s a ry t o fi gu re hg for e a ch sta ti o n
t n
n o
t
.
kg a t Sta 0
.
H
6
‘
( C
A l s o u se s i mil a r n ot a tio n for r y
g
.
ra te of gr a de ( ri s e per sta ti on )
T he n
ho
g
hm
9
ha
s
ho,
g , etc
z
hi
79
I t will
y
,
rg
2
rg
=
h
+
( oe 9 ) hi
“
—
0
9
1
g , et
0
S i mi l a rl
.
,
c
.
be n ece s s a ry or cert a i n ly de s i ra ble to figu re he a n d
It i s well to fig u re ho a n d r g ( a s a
r o a n ew for e a ch n e w h ,
c he c k) for th e l a s t s ta tio n before e a ch tu rn i n g poin t
,
.
.
.
R a i lr oa d Cu r ve s a n d E a r thwork
140
20 9
1
( )
( b)
.
S ett i n g t h e S ta k e for th e S i de S l ope
.
Wh en th e s u rfa ce i s l evel
b
AB
b a s e of s ec t io n
0
OG
ce n ter heigh t
BN
A
EN
DM
d :
T h en
.
‘
OD :
M s ide s lope
OE
di s ta n c e ou t
d = GB + BN
—
% b + sc
S e tt i n g t h e S t a ke for t h e S i d e S l ope
( 2 ) Wh en th e s u rfa c e i s n ot l evel
H ere the proce s s i s le s s s i mple
.
.
.
b
A8
ba se
6
06
ce n ter heigh t ( or ou t)
.
S etti n g S ta k e s f or E a r th wor k
Bu t h
t
ha t h
ide height rig ht
h,
EK
h;
DH
do
GK
dis ta n c e ou t ri ght
do :
GH
l eft
s
1 41
.
left
d. = 4b
s h.
b
%
s h,
k n own It i s e vi de n t fro m th e figu r
c a n d h1 < c i n t h e c a s ei n dic a ted a n d t heref o re
a n d h , a r e n ot
e
.
,
,
It wo u ld be q u ite po s s ible i n ma n y c a s e s t o t a ke me a su r e
me n ts su ch th a t th e r a t e of Slope of th e li n e s OE a n d O D wo u l d
be kn own a n d th e po s itio n s of E a n d D deter mi n ed by c a lc u l a
tio n fro m su ch d a t a B u t s peed a n d re s u lt s fin a lly correct a r e
the e s s e n ti a l s i n thi s work a n d the s e a r e be s t s ec u red by fin d
i n g h , a n d h a n d the corre s po n di n g d , a n d d u po n th e gro u n d
by a s erie s of a pproxi ma tio n s a s de s cribed below
H a v i n g determi n ed c u s e thi s a s a b a s i s a n d ma ke a n e s ti ma te
a t o n ce a s t o th e prob a ble va l u e of h a t the poi n t where the
s ide s lope w i ll i n ter s e c t the s u rf a ce a n d c a lc u l a te d
s
b
h
4
t o corre s po n d
M e a s u re ou t t hi s di s ta n c e se t the r od a t the poi n t th u s fou n d
t a ke the rod re a di n g on t h e s u rf a ce a n d if the cu t or fill th u s
fou n d fro m the rod re a di n g y ield s a v a l u e of d e qu a l to t h a t
the poi n t i s c orrect Otherwi s e ma ke
a ct u a lly me a s u red o u t
a n e w a n d clo s e a pproxi ma ti o n fro m the be t ter d a t a j u s t oh
t a i n e d a lw a y s s t a rti n g with h a n d c a lc u l a t i n g d
a n d repe a t
the proce s s u n til a poi n t i s re a ched where the cu t or fill fo u n d
fro m the rod re a di n g yield s a di s t a n ce ou t eq u a l to th a t t a ke n
on the g ro u n d
The n s e t the s t a ke a n d ma rk the cu t or fill
corre s po n di n g to h u po n the i n n er s ide a s previ o u s ly s ta ted
P erfor m the s a me oper a tio n i n a s i mil a r w a y to determi n e
n d ma rk thi s s t a ke a l s o u po n th e in n er s ide wi th
sh
a
b
%
a cu t or fil l e q u a l to h o
,
.
,
,
,
.
,
,
,
,
o
,
,
.
,
,
,
'
o
,
.
,
,
.
o,
,
,
,
o,
.
142
R a i lr oa d Cu r ves a n d E a r th wor k
.
It req u ire s a cert a i n a mo u n t of work i n the field to a p pr e ci
a t e f u lly the proce s s here o u tli n ed b u t which i n pr a ctice i s v ery
s i mple
It ma y i mpre s s s o me a s bei n g u n s cie n tific a n d a t first
tri a l a s s low b u t with a little pra ctice it is s u rpri s i n g how
r a pidly a l mo s t by i n s ti n ct the proper poi n t i s re a ched ofte n
withi n the re qu ired l i mits of prec is io n a t the firs t tri a l while
more th a n t w o tri a l s will s el dom be n ece ss a ry excep t i n difficu lt
co u n t ry
,
.
,
,
,
,
,
,
,
.
The i n s tr u me n t a l work i s j u s t the s a me i n pri n ciple a s a t
t h e ce n ter s ta ke
.
H
NE
r,
A
rod re a di n g a t s lope s ta ke ri gh t
,
K N — NE = r g
h ere
rg
i s the s a me fo r ce n ter , ri ght a n d left o f s ec t io n
.
o me c a s e s it ma y be n ece ss a ry to ma ke on e or more
re s etti n g s of the le v el i n order to re a ch the s ide s ta ke s fro m
the ce n t er s t a ke I n thi s c a s e of co u rs e a n e w r g mu s t be
c a lc u l a ted fro m the n e w h , Thi s i n trod u ce s n o n ew pri n ciple
b u t ma ke s the work s lower
In
s
’
,
,
.
,
.
.
lope bo a rd or lev el bo a rd h a s q u ite fre qu e n tly been
I n certa i n s ectio n s of co u n try thi s mi ght
u s ed to a d va n t a g e
be co n s idered a l mo s t i n di s pe n s a ble
It co n s i s ts s i mply of a
lo n g s tra i g ht ed ge o f wood ( perh a ps 1 5 ft lo n g ) with a le vel
mo u n ted i n the u pper s ide
It i s u s ed with a n y s elf re a di n g
rod A rod q u ickly h a n d ma rked will s er ve the pu rpo s e well
Ha vi n g g i ve n the cu t or fill a t the ce n ter or a t a n y poi n t i n the
the le v eli n g for the s ide s t a ke s a n d for a n y a dditio n a l
s ectio n
poi n ts ca n re a dily a n d with s u fficie n t a cc u r a cy be do n e by
”
“
thi s le vel bo a rd a n d the n ece s s ity for t a ki n g n ew tu r n i n g
poi n ts a n d res etti n g the le vel a voided
A
s
-
-
.
.
-
.
,
-
.
.
.
,
,
,
,
,
,
-
,
.
R a i l r oa d Cu r ve s a n d E a r thwork
144
211
For m of Cr os s -S e t i on B ook
c
.
.
l
f
e
t
ha n d pa g e )
(
.
( D a te )
(Na me s o f Pa rty)
B a s e 2 0 ; 1 to l
l 4 ; 1 % t0 1
c
C ro s s -S e t io n
46 9 7 P T
+ 27 2 P
.
.
.
C
.
+ 14 7
-
212
.
S etti n g S ta k e s f or E a r th wor k
i
h
a
n
e
R
h
d
P
a
t
g
)
( g
.
.
1 45
R a i lr oa d
1 46
213
Cu r ves
an
d E a r th work
.
c
Cro s s -s e ti o n s a r e t a ke n a t e v ery fu ll s t a tio n , a t e ver y
.
or P T of c u r ve where ver gra de c u ts the s u rf a ce a n d i n
I n th e fi gu re below
a ddi tio n a t e v ery bre a k i n the s u rf a ce
s ectio n s s h o u l d be t a ke n a t the followin g
sh owin g a profil e
sta t i o ns
P 0
.
.
.
,
.
,
,
.
,
,
f
A t Sta t i o ns
7,
8,
13 ,
14,
0,
2,
1,
3,
9,
10,
15 ,
I6,
I7 P T ,
.
.
4,
6,
s,
12,
11,
I8
.
It i s n ot n ece s s a r y a ctu a ll y t o dri ve s ta ke s i n a l l c a s es
where a cro s s s ect ion i s t a ke n a n d re c orded bu t i n e very c as e
where t hey will a i d ma teri a lly i n c o n stru ctio n s t a ke s s ho u ld be
s et
It i s be s t to err on the s a fe s ide which i s the liber a l s ide
I n p a s s i n g fr om ou t to fill it i s c u s t o ma ry to ta ke f u l l cro ss
s ec t io ns
n ot o n ly a t the poi n t wh e
re the gr a de lin e c u ts t h e
s u rf a ce a t the c en ter li n e of s u r vey b u t a l s o where t h e gr a d e
c u ts th e s u r fa c e a t the o u ts ide of the b a s e both r i gh t a n d l eft
a s i n the fi g u re below wh i ch ill u s tr a te s th e n ote s o n p 14 4 ;
f u ll cro s s s ectio n s a r e t a ke n n ot o nl y a t s ta t io n s 2 7 6 bu t
a l s o a t 2 4 64 a n d 2
87
214
-
,
.
.
,
.
,
,
,
,
,
.
,
-
.
.
,
R a i lr oa d Cu rves a n d E a r th work
.
cro s s sec ti o n bo ok or i n a le vel bo ok c a rri e d for th a t
p u rpo s e Keepi n g t he s e or a n y other n o t es on a s lip of p a per
i s b a d pr a cti c e
of
th e
-
.
.
be mo s t re a dily co mpu ted whe n the
”
“
s ec t io n i s a
th a t i s whe n the s u rfa c e i s level
L evel S e ct i o n
a cro s s th e s ectio n
b u t th i s i s s eldo m the c a s e a n d for p u rpo s e s
of fin a l co mp u t a tio n it i s n ot often a tte mpted t o t a ke me a s u re
me n t s u po n th a t b a s i s
218
E a rthwork
.
ca n
,
,
.
r a ilro a d work the grou n d i s s u fficie n tly
”
re gu l a r to a llow of Th r ee L evel S ect i on s bei n g t a ke n on e
le vel ( ele va tio n ) a t th e ce n ter a n d on e a t e a ch s lope s ta ke a s
to 1
s how n b y t he s e n ote s where B a s e i s 2 0 a n d S lope
219
In g e n er a l , i n
.
,
-
,
,
,
,
lly a pplied o n l y to
e g u l a r s e c tio n s where the wid t h s of b a s e o n e a ch s ide o f the
ce n t er a r e the s a me I n regu l a r three le vel s ectio n s the ca l cu
l a tio n of qua n titie s ca n be ma de qu it e s i mple T o f a cilit a te
the fi n a l e s t i ma tio n of q u a n tities i t i s bes t t o u s e t hree level
s ectio n s a s fa r a s po ss ible
Th e t e r m
Th r ee-L evel S ec ti o n
is
u su a
r
-
.
.
-
,
.
220
.
In
i ci e n t ,
su fi
ma n y c a s e s where three-le v el s e c t io n s a r e n ot
”
“
it ma y be poss ible to u s e F i ve Level S ec ti o n s
-
,
co n s i s ti n g of a le vel a t the ce n ter on e a t e a ch s ide where the
ba s e meets the s ide s lope a n d on e a t e a ch s i d e s lope s ta ke a s
s how n by the followi n g n o t e s
B a s e 2 0 S lope 1 t o 1
,
,
,
,
The term F i ve Level S ecti on i s u su a l l y a ppl ied on ly to
re gu l a r s ectio n s where th e b a s e a n d th e s id e s lope s a r e th e
s a me o n e a ch s ide o f the c e n t e r
“
”
-
.
Where the gro u n d i s very ro u gh level s h a v e to be
t a ke n where v er th e g ro u n d req u ire s a n d the c a l c u l a tio n s mu st
be ma de to s u it the requ ire me n ts of e a ch s peci a l c a s e a ltho u gh
certa i n s y s tema tic me thod s a r e ge n era lly a ppl i c a ble S u ch
“
s ec t io n s a r e c a lled
I r r eg u l a r S ecti on s
221
.
,
,
,
.
.
CH AP T E R
X II
.
ME T HODS OF COMP U T I NG E ART HWORK
.
c a lcu l a ti n g the v ol u me s or q u a n titie s of Ea rth
work the pri n cip a l method s u s ed a r e a s follow s
I I P R I S M O I D A L F OR M U L A
I A V E R A GI N G E ND A R E A S
222
.
”
“
In
,
.
223
.
.
I
Aver a g i n g En d Ar e a s
.
.
.
.
St t i
a
L et A O
a
1
re a of cro s s s e c t io n a t Sta t io n 0
-
‘
6
AI
on
CG
( 6
( 6
‘
6
H
le n gt h of s ectio n Sta 0 to Sta
v ol u me of s ectio n of e a rthwork
t
S
a
o I)
t
0
(
l
,
V:
.
.
.
AO
T he n V
Al
2
1
40 —
1 AI
l ( i n c u bic feet )
5
.
2
7
( i n c u bic y a rd s )
A s ( 1 58) i s c a p a ble of expre ss io n V :
1
40
l
A
+ 1
5
2
i t i s pr a ctic a lly b a s ed on
the a ssu mptio n th a t the v ol u me con
s i s ts of two pri s ms o n e of b a s e A O a n d o n e of b a s e A
1 a n d ea ch
o f a le n gt h or a lti t u de o f
,
,
,
i
R a i lr oa d Cu r ve s a n d E a rth wor k
1 50
.
To u s e thi s method we mu s t fin d the a re a A of e a ch
cro s s s ectio n ; the cro ss s ectio n ma y be
224
.
,
-
( )
a
-
L eve l
225
.
(
( b)
.
a
Th r e e- L eve l
.
( )
c
c
e
el
r
v
L
o
s
s
C
S e t i on
)
b
ba s e
AB
F i ve-L evel
I
r
r
e
u
l
d
r
a
g
)
(
.
.
3
s
ide s lope
“
l
E:
—
AL
c
EM
BM
ce n ter ht
06
a re a o f cro s s s ectio n
A
The n DL EM s o
.
-
2
sc
bc
c
2 26
.
c
n
h
ee
e
el
e
t
i
L
v
o
S
T
r
b
( )
b
c
d,
A
Then
A
.
F
b a s e AB
s
ce n ter hei ght
h,
s ide hei g ht EK
di s t a n ce ou t ME
d,
a re a of cro s s s ectio n
(b
)
so
irs t M ethod
s
.
ide s10 pe
i e hei ght D H
di s ta n ce ou t DL
s d
-
c
AGO
GBE
OGE
OGO
5 ( hi
)
d
( l
2
2
h. )
.
R a i lr oa d Cu r ves a n d E a rt hwork
152
228
( c ) F i v e Le v el S ect i on
.
-
.
.
Us e n ot a tio n the s a me a s before
f
The n
i n a dditio n l et
hei g ht MB
r
A = LG M
heig ht LA
EMGB + D LGA
’
cb
d
f ,
m
2
2
2
Z
d
l
l
s
,
_,
Cb
£
1
ft
2
,
r
dr
z
2
229
.
( d) Ir r egu l a r S ecti on
.
FIG 1
.
.
The Irreg u l a r S ecti o n
a s s how n i n the fi g u re
ma y be
di v ided i n to tr a pe z oids by vertic a l li n e s a s i n F i g 1 or i n t o
tri a n g les by v ertic a l a n d di a g o n a l li n e s a s i n F i g 2
”
“
,
,
.
,
,
FIG 2
.
.
.
.
The tri a n gle s i n F i g 2 ca n be co mp u ted i n g ro u ps of tw o a n d
with the a d va n t a g e of on e le ss n u meric a l co mp u t a tio n th a n is
n ece ss a r y i n F i g 1 proceedi n g a s follow s
I n F i g 2 let h o d o etc a pply to poi n ts i n dic a ted by s u b
s cripts a n d let OG be a t the ce n ter li n e
.
.
.
,
,
,
,
,
.
.
,
A
h
d
%[ 0 ( A
h
d
+
0
L( o
(1
d ia>
l
0)
h0 ( d L
d M)
km( d e
0)
M eth ods of Comp u ti n g E a r thwor k
15 3
.
The pri n ciple i n v olv ed h a s fo u n d express io n i n a r u le
”
o f th u mb
which h a s h a d con s ider a ble u s e i n the R a ilro a d
Va l u a tio n work co n d u cted u n der the directio n of the I n ters t a te
C o mmerce C o mmiss io n
F or the p u rpo s e s of thi s r u le ce r t a i n preli mi n a r i e s s ho u ld be
co mplied with a s follows
( a ) The n ote s mu s t s how va l u es of d to e a ch edg e of b a s e
s i g n for v a l u e s of d t o left of
( b) Us e a rbitr a rily the
ce n ter
s i gn to ri g ht of ce n ter
a
f
c
se
i
n
f
o
r
a
n
l
u
e
o
U
s
v
h below the b a s e g r a de i n
y
g
( )
c u ts ( a s for s ide ditche s )
d
N
f
o
r
n
s
n
n
a
s
ote
poi
t
ori
i
l
rf
ce
ro
d
ho
ld
s
o
u
a
f
u
n
s
u
o
g
)
g
(
a ppe a r i n br a ckets
The r u le 18
1
S t a rt a t a n y poi n t ; u s e e v ery va l u e of h i n order pro
cee di n g clockwi s e a ro u n d the fi g u re
2 M u ltiply e a ch va l u e of h by ( d
db) u s i n g a l g ebr a ic
va l u e s
s e n t s the v a l u e of d for poi n t n ext i n
ere
H
d
repre
(
a d va n ce a n d db for poi n t n ext b a ck
)
3 F i n d th e a l g ebr a ic s u m of the s e a n d di v ide by 2
The re s u lt is th e a re a of the s ectio n
The n ece ss ity for u s i n g va l u es of h or d a lg ebr a ic a lly i s con
fi n ed to th e p u rpo s e s of t his r u le S u ch va l u e s a r e n ot u s ed
a lg ebr a ic a lly i n other p a rts of thi s b o ok
It i s e vide n t th a t th e ru le of th u mb de s cribed a pplie s cor
r e ct l y t o the s ol u tio n of the s e ctio n s how n i n F i
g 2
The r u le ma y be s how n to a ppl y to Tri a n g u l a r S e c t io ns a n d
to T w o L e vel S e ct io ns i n whi ch the c e n ter height is la cki ng
230
.
.
,
.
.
.
.
.
,
.
.
a
.
o
,
.
.
.
.
.
.
.
,
.
M
F IG 3
.
.
I n F i g u re s 3 a n d 4, N n eed n ot be o n ce n ter li n e
I n F ig 3 ,
.
.
dr a w li n e s v erti c a l NM a n d hori z o n ta l HK a n d FM
,
and
,
.
FK
.
I t ma y re a dily be s how n th a t a re a s MS H
FS K
.
Joi n HM
R a i lr oa d Cu rves
1 54
F IG 3
.
The n a re a
and
E a rthwor k
.
FIG 4
.
.
HNF
FNK
HNM
h H( d F
d N)
.
h r (dN
d H)
w h i c h e vide n tly co mplie s with the r u le when a ll v a l u es of d a r e
to the ri g ht of ce n ter I t will a pply eq u a lly i n other c a s es by
u s i n g v a l u e s of d a lg ebr a ic a lly
A n ex a mple will more f u lly ill u s tr a te the u s e of t h is r u le
’
.
.
E
FIG 6
.
.
Note s
4
(
7 0 [c 0
.
.
44
sq
.
ft
.
15 6
R a i lr oa d Cu r ves a n d E a rthwork
.
bee n u s ed for c a l c u l a ti n g
irre gu l a r cro ss s ectio ns i s to pl a t them on cro s s s ectio n p a per
”
I n very irre gu l a r cro ss
a n d g e t the a re a by
P l a n i meter
s ec tio n s t h i s me t h od wo u ld pro v e eco n o mic a l a s co mp a red with
direct co mp u ta tio n by ordi n a ry methods bu t it i s prob a ble th a t
i n a l mo s t e v ery c a s e e q u a l s peed a n d eq u a l preci s io n ca n be
obta i n ed by the u s e o f s u ita ble t a ble s or di a gra ms (to be ex
pl a i n ed l a ter ) ; for thi s re a s o n the u s e of the pl a n i meter i s n ot
reco mme n ded certa i n ly where di a gr a ms a r e a va il a ble
232
.
A n other method which h a s
-
-
,
.
'
,
,
.
Wh a te ver ma y be the for m of s ectio n or wh a te ver the
method of co mp u t a tio n h a v i n g fo u n d the va l u e s of A for e a ch
cro s s s ectio n the v ol u me V i s fou n d for the E n d A re a M ethod
by the for mu l a a bo ve g i ve n
233
.
,
,
-
,
,
.
V:
A0
A1
l
In cu
ds
.
159
.
It i s fo u n d th a t thi s formu l a i s o n ly a pproxi ma tely correct
I ts s i mplicity a n d s u bs ta n ti a l a cc u r a cy i n the ma j ority o f c a s e s
re n der it s o va l u a ble th a t it h as beco me the for mu l a i n mo s t
co mmo n u s e It g i v e s res u l ts i n ge n er a l l a rger th a n the tr u e
s olidity
.
”
,
.
234
II
.
.
,
P r i s moi da l For mu l a
.
A pri smoid i s a s olid
h a vi n g for i ts two e n d s a n y diss i mil a r
p a r a llel pl a n e fi g u re s of the s a me n u mber of s ides a n d a ll the
”
s ide s of the s olid pl a n e fi g u re s a l s o
A n y pri s moid ma y be re s ol ved i n to pri s ms pyr a mid s a n d
wed g e s h a vi n g a s a co mmo n a ltit u de the perpe n dic u l a r di s ta n ce
betwee n t h e two p a r a llel e n d pl a n es
L et A, a n d A I
a re a s of e n d pl a n e s
a re a of middle s ectio n p a r a llel to the e n d
pl a n e s
le n gt h o f pri smoid or perpe n dic u l a r d is
l
t a n ce betwee n e n d pla n es
v ol u me of the pri s moid
V
T he n it ma y be s how n th a t
,
.
,
,
,
.
.
.
:
,
.
:
V
.
A
( O
4a .
A)
Ill e thod s
B
a
or k
a
r
t
w
n
E
C
o
m
p
u
t
i
h
f
g
o
157
.
re a of lower f a ce or b a s e of a pr i s m wed ge
or pyr a mid
,
,
,
.
b
a
re a of u pper f a ce
.
m
middle a re a p a r a llel to u pper a n d lower fa c e s
a
a
ltitu de o f pri s m wedg e or p y ra mid
s
s
olidi ty
,
,
.
.
Th e n th e a re a of the u pp er fa ce b i n terms of l ower ba s e B
will be for
a n d th e
mi ddl e a r ea m will be for
P ris m
m= B
Th e s olidit
y 3 wi ll be for
Pr i s m
s = aB
Wed ge
Q
2
6
P yr a ml d
9
? g
+0
S i n ce a pri s moid i s co mpo s ed o f pri s ms , wed g e s , a n d pyr a mid s ,
the s a me express io n ma y a pply to th e pri smo i d a n d thi s ma y
be pu t i n the g e n er a l form
,
V = (Ao
i
u s n g th e n
4
A)
ota t i o n of th e preced in g p a g e
.
63
1
A)
(
R a i lr oa d Cu r ve s
15 8
236
.
and
E a r th work
.
A re gu l a r s ectio n of e a rthwor k h a v i n g for i t s su rfa
ce a
pl a n e f a ce i s a pris moid M o s t s ectio n s of e a rthwo r k h a ve n ot
their s u rfa ce pl a n e a n d a r e n ot s trictly pris moid s a ltho u gh
t hey a r e s o re g a rded by s o m e writers
I n thi s fi g u re the li n e s E0 0 0 a n d E1 0 1 a r e n ot p a r a llel a n d
therefore the su rf a ce 0 0 0 1 E1 E0 i s n ot a pl a n e The mo s t com
mon a ssu mpti on a s to thi s s u rfa ce i s th a t the li n e s 0 0 0 1 a n d EOEl
a r e ri g ht li n e s a n d th a t the s u rf a ce 0 0 0 1 E1 E0 i s a w a rped s u r
f a ce g e n er a ted b y a right li n e mo vi n g a s a g en era tri x a lwa ys
.
,
,
.
,
.
,
,
I
A,
8,
G,
p a r a l lel t o the pl a n e OOGOBOEO a n d u pon the l i n es 0 0 0 1 a n d
The s u rf a ce th u s
E0 E1 a s directric es a s i n dic a ted i n the fi gu re
”
“
a r a boloid
e
er
ted
w
rped
rf
ce
c
a
lled
hyperbolic
p
n
a
i
a
s
a
a
s
u
a
g
p ri smoid a l for mu l a a pplie s a l s o t o
I t will be s how n th a t the
thi s s olid which i s n ot howe ver properl y a pri s moid
I n the c a s e of a s ectio n who s e e n d s a r e p a r a llel a n d
23 7
tri a n g u l a r i n form it ma y be s hown th a t the pri smoid a l formu l a
a ppli e s whe n o
ne s u rf a ce i s pl a n e
whether the other two s u rf a ce s a r e
pl a n e or w a rped i n the ma n n er a bo ve
o u tli n ed
I n the followi n g fi gu re ABC a n d
DEF a r e p a r a llel a n d the s u rf a ce s
ACFD a n d BC FE ma y be co n s idered
w a rped s u rf a ces a n d AB E D pl a n e
b a s e AB
bo
L et
.
,
.
.
,
,
,
,
.
,
,
,
a nd
b]
.
DE
lt itu d e of ABC a n d h a ltit u de of DEF
di st a n ce betwee n p a r a llel e n d pl a n es
l
a re a of ABC ; a n d A I
a re a of DEF
AO
A ls o u s e n ot a t io n b km A for a s ectio n di s t a n t a: fro m ABC
ho
a
1:
x,
,
,
.
R a i lr oa d Cu r ves
1 60
E a r thwork
a nd
.
r a ilro a d e a rthwork the P ris moid a l F or mu l a i s
ofte n b u rde n s o me i n i t s a pplic a tio n
F o r tri a n g u l a r s ectio ns
three le v el s ectio n s the work ma y be s i m
a n d for re g u l a r
pli fie d by co mp u ti n g the qu a n titie s fi r s t by the i n ex a ct method
”
“
e n d a re as
of
a n d the n a pplyi n g a cor r ecti o n which w e ma y
”
c a ll the Pr i s moi da l Cor r e c ti on
239
.
”
“
In
.
-
,
,
.
olidity by e n d a re a s
s olidity by pri s moid a l formu l a
V,
s
VI,
.
The n
0
pri smoid a l correctio n
I n th e tri a n gu l a r s ectio n
V,
V,
12.
—
5
boho + i bi h o
a
l
12
.
h
3
0
h
3
b
+
1
;
0
0
(
boho + 2 b1h 1 + boll/1 + bl hO)
C :
V.
V,,
l-—
Tz
l
—
—
-h
h
b
h
b
h
b
b
1
o)
l
o
I
1
1
o
o
(
b — b0 010 — h 1 )
which i s the f u n d a me n t a l formu l a for pri smoid a l corr e ctio n
I n the fig u r e oppo s ite for the s o l id OODOGOEOEI GI D1 0 1
,
,
0
i i ai
lz
l
12
D1
(
0
l
oi
)
(
h,
C
01
dl +
(
(11
a n d DO 2
(1 1 +
1
>
>< i
D o)
co
d
)(
a,
co
dl o
dr o)
0
F OP the Solid GoBoEoEl Bl Gl ,
S i mil a rly for the s olid AoGoDoDl G1A1
so for the e n tire s olid 0
1 (01
12
.
D
)( 1
co
D0 )
,
.
M eth ods of Co mp u ti n g E a rthwork
1 61
.
Go
80
Th is for mu l a ca n be u s ed o n ly whe n the W Idth of b a s e
F ro m the method of
i s the s a me a t both e n d s of the s ectio n
i ts deri va tio n it i s e v ide n t th a t for the ri g ht h a lf of a re g u l a r
three level s ectio n
240
.
.
(
W he n
co
1
7
A
6
)
(
d o)
l
100
l
100
12 x 2 7
1
S i n ce
d
)( ,
01
V,
C
D O)
( 61
(
e1
c0
D
)( 1
i n cu
V1,
.
y ds
.
16
( 8)
( 1 69 )
F or a s ectio n of le n g t h l ,
0,
A
9
16
)
(
F or the p u rpo s e s of pri smoid a l correctio n , it I s S i mpler t o u s e
or
meric a l v a l u e s of c a n d D or d a n d n e g lect the s ig n
s i n ce the s e a r e s y s te ma tic a lly u s ed t o repre s e n t cu t o r fill a n d
the correctio n for a n y g i v e n n u meric a l va l u e s of c a n d D i s the
s a me whether the s ectio n be cu t or fill
Therefore whe n ( 0 1 co) ( D 1 D O) i s p os i ti ve, th e a r i th meti
ca l v a l u e oi C i s t o be s u btr a cte d fro m V
Whe n ( 0 1 co) ( D 1 D O) i s n eg a ti ve , the a rithmetic a l va l u e
The l a tter c a s e s eldo m occ u r s i n
of C is t o be a d ded to V
pr a ct ice except where C is v ery s ma ll , perh a ps s ma ll e n ou g h
nu
.
“
,
,
,
,
to
be n eg lect ed
.
.
1 62
241
R a i lr oa d Cu r ves a n d E a r t hw or k
.
I n p a ss i n g fro m ou t t o fill a s i n the fi
l
for the ri gh t h a lf
co
01
(
12
co
l
for the left s ide
l
(
12
F or th e s peci a l c a s e
a s
00
re
fro m ( 167 A )
) (d 1
01
12
gu
.
r
fro m ( 166 )
)
0
) (DI
d r o)
ide hill s e cti o n
the pri smoid al correctio n for ou t will be
{
1
9
Co
) (d 1
r
a,
di m
)
the pris mo i d a l c orrectio n for fill will be
0,
12
h i,
hlo
b
b
2
The q u a n tities of cu t a n d of fill wi ll be kept s ep a r a te a fter
a pplyi n g the correctio ns
,
.
1 64
R a i lr oa d Cu r ve s
E a r thwork
and
2
I n s pectio n of the for mu l a C
(
2
) (D 1
cl
.
D o)
co
1
( 67 )
ma kgs it cle a r th a t the correct io n wi ll be l a r g e whe n the two e n d
s ectio n s di ffer mu ch i n s i z e , a n d s ma ll wh é n the e n d s e ct io n s
Ordi n a rily i n a l a rg e s ectio n both c a n d D a r e
l a rg e F or a n y g i ve n a r ea of s ect i o n i n a reg u l a r three le v el
s ectio n if c i s ma de s ma ller D mu s t be i n cre a s ed i n n e a rly like
me a s u re a n d for mu l a ( 16 7 ) will s how little ch a n g e i n the va l u e
o f 0 e v e n if c be ch a n g ed if the a re a re ma i n s the s a me
F or the p u rpo s e on l y of fi n di n g the pri s moid a l correctio n
there a r e s e vera l a pproxi ma te method s b a s ed on the pri n ciple
a bo ve s t a ted
1 Where the s ect io n i s o n ly s li ghtly irre g u l a r
Ne g lect a l l
i n ter medi a te hei g hts a n d figu re correctio n fro m c a n d D Thi s
i s a v ery s i mple method
Where more c a refu l re su lts s ee m de s ira ble
2 F i n d 6 a n d D for a n
equ i v a le n t le vel s ectio n
th a t i s
a le vel s ectio n of eq u a l a re a to the irre g u l a r s ectio n
Us e the
c a n d D thu s deter mi n ed in comp u ti n g the pri smoid a l corre o
tio n The s e ca n be u s ed with the c a n d D of a re gu l a r three
le vel s ectio n or with the c a n d D of a n other e qu i va le n t le vel
s ectio n
The 0 a n d D of the eq u iva le n t lev el s ectio n ma y be fo u n d
fro m Ta ble s or fro m D i a gr a ms who s e u s e will be s hown i n
l a ter ch a pte r s
3 F i n d a n eq u i va le n t re gu l a r three le vel s ectio n ( n ot le v el )
either by
n
ret
a
i
n
i
n
c
a
n
d
co
m
p
u
ti
a
g D or
g
( )
n i n g D a n d co mpu ti n g 0
ret
i
a
b
( )
The method of doi n g thi s will be ma de s i mple by D i a gra ms
de s cribed i n a l a ter ch a pter
4 P lot the irre g u l a r s ectio n on cro s s —s ectio n p a per a n d dr a w
li n e s t o for m a reg u l a r three le vel s ectio n which wi ll c lo s ely
a pproxi ma te i n form to the irre g u l a r s ectio n a n d fin d c a n d D
While the re s u lts obt a i n ed by a n y of the a bo v e method s a r e
a pproxi ma te the re s u l t i n g error ca n be o n ly a s ma ll fr a ctio n of
the en tire correctio n which i s its elf s ma ll
The method of a vera gi n g e n d a re a s a n d a pplyin g the pr i s moi
d a l correctio n a llows of g re a t r a pidi ty a n d s ecu re s gre a t pre
ci s i on
a n d well meets the req u ire me n t s of moder n r a ilro a d
pra ctice
are n
e a rly eq u a l
.
-
.
,
,
,
,
.
.
.
.
.
.
,
.
,
.
.
,
.
,
.
-
.
,
.
.
,
.
-
,
,
,
’
,
.
,
,
,
.
.
CH AP T E R
X II I
.
S PE CIAL PR OBLEM S
245
Corr ecti on for Cu rva tu r e
.
.
.
I n the c a s e o f a c u rv e , the e n d s of a s ec t io n of e a rt hwork a r e
p a ra llel bu t a r e i n e a ch c a s e n or ma l to the c u rv e I n ca l
cu l a ti n g t h e s olidity of a s ectio n of e a rthwork we h a v e hereto
fore a s s u med the e n d s p a r a llel a n d for cu r v es thi s i s equ i va le n t
t o t a ki n g the m perpe n dic u l a r to the chord of t h e c u r ve be t wee n
the t wo s t a tio n s
The n a s s hown i n F i g 1 ( where IG a n d GT a r e ce n ter li n e
chord s ) the s olidity ( a s a bo v e ) o f the s ectio n s IG a n d GT will
be t oo g re a t b y t h e wed g e s h a ped ma s s RGP a n d to o s ma ll b y
n ot
.
,
,
,
.
-
,
.
,
-
,
Whe n th e cro s s s ectio n s o n e a ch s ide of t h e c e n t e r
a r e eq u a l the s e ma s s e s b a l a n ce e a ch other
W he n t h e cro s s
Q GS
-
.
.
,
ectio n on on e s ide di ffers mu ch i n a re a fro m th a t on the o t he r
the correctio n n ece s s a ry ma y be co n s ider a ble
s
‘
.
16 5
,
R a i lroa d Cu r ve s a n d E a r th work
1 66
.
Q
I n F i g 2 , u s e c , h i , h r , dz, dr , b, s , a s before
Le t D
de g ree of c u r ve M a ke BL AD, a n d
.
.
.
j oi
n
OL
.
ba la n ces
OLBG a n d there re ma i n s
a n u n b a l a n ced a re a OLE
D r a w OKP p a r a l l el to
The n
O DAG
,
.
.
AB
.
the
By
P a pp u s
“
Theore m of
( s ee L a n z a Ap
,
plied M ech an ic s ) If a pl a n e
a re a lyi n g wholly on the s a me
s ide o f a s tr a i ght li n e i n i t s o wn
pl a n e revol ve s a bo u t th a t li n e
a n d thereby g e n er a te s a s olid
of re vol u tio n the v ol u me q f the
s olid th u s g e n er a ted i s eq u a l to
FI G 1
the prod u ct of the re vol vi n g
a r e a a n d of the p a th de s cribed by the ce n ter o f g r a v ity of the
Dl a n e a re a d u ri n g the re vol u tio n
The correctio n for c u r va tu re or th e s olidity de veloped by
thi s tri a n gle O LE ( F i g 2 ) rev ol vi n g a bo u t OG a s
a n a xi s will be i t s a re a x the di s t a n ce de s cribed
by i t s ce n ter of g ra v ity The di s ta n ce
o u t ( hori z o n t a l ) to the
ce n ter of gr a vity fro m
th e a xi s ( ce n ter li n e )
0
will be two third s of
the me a n of th e di s
t a n ce s ou t t o E a n d to
“
,
,
,
.
.
.
,
,
.
.
L, or
and
FIG 2
.
2
dl + dr
3
2
the dis t a n c e de s cribed will be
3
dl 1 dr
X
2
OLE
OK x
2
T h e a re a
“
"
.
R a i lr oa d Cu rves
1 68
E a r t h work
a nd
.
the c a s e of a n irre g u l a r s ectio n a s s hown i n F i g 4 th e
a re a a n d di s t a n ce to ce n ter o f g r a v it y for ex a mple of O H EML
(
)
ma y be fo u n d by a n y method a va il a ble a n d the correction
In
,
.
,
,
,
FI G
.
4
.
figu red a c c ordi n gly The correctio n for cu r va tu re i s i n pre s e n t
r a ilro a d pra ctice more fr equ en tly n e g lected th a n u s ed Ne ver
t h el e s s i t s a mo u n t i s s u fficie n t i n ma n y c a s e s t o f u l ly w a rr a n t
.
,
,
.
,
,
i ts u s e
248
.
.
Ope n i n g i n E mb a n kme n t
.
W here a n ope n in g i s left i n a n e mb a n kme n t there r e ma i n s
,
o u ts ide the re gu l a r s ectio n s th e ma ss D E K H F
.
mu s t be c a lc u l a ted i n 3 pie c e s ADF, B E K H , ABHF
,
b a s e = AB
di s ta n ce o u t r i gh t
di s t a n ce ou t left
b
d,
d;
p
r
z
z
z
p
BH
AF
i;
}
t a ke n p a r a llel to c en te r lin e
hei ghts a t
t
82
olidity ADF
B EKH
83
ABHF
31
s
.
-
Pr obl ems
Sp eci a l
1 60
.
”
m
n
t
h
T heore m of P a pp u s ,
T he n ( a pproxi a tely ) followi g
e
me a n o f tri a n gu l a r s ectio n s AD a n d AF x d i s ta n c e de
s,
s
cri bed b y c e n ter of gr a vity
.
I n the q u a rter co n e AFD, AF
p;
—
Q
=
D
d
z
A
2
T he n a v e r a g e r a di u s R ;
AF + AD
mh
2
A re a o f vertic a l t ri a n gu l a r s ectio n A ;
c
D i s t a n ce fro m A t o ce n t er of g r a v ity of vertic a l s e t i on
Ar c de s cr i bed by ce n t er of gr a vity
x
2 x 6
( cu
x 27
2
fiR ; ( cu
81
S i mil a rly , i n the q u a rter co n e B E K
'
T h e a v era g e r a di u s R ,
.
y ds )
.
.
yds )
.
H
BH + 2 BK + BE
4
( cu ft )
.
.
32
0 009 7 fr 1tr 2 ( cu
f
.
.
yds )
For th e soli d AGBHF
a
re a AF
a
re a BH
2
( hp : + f p ) b
r
4
r
.
R a i lr oa d Cu r ve s a n d E a r th work
170
.
Th e work of
deri v i n g for mu l a s ( 17 6) a n d ( 17 7 ) is a ppr oxi
ma te thro u g ho u t b u t the tot a l qu a n titie s i n v ol ved a r e i n g en
era l n ot l a rge a n d the error res u lti n g wo u ld be u n i mporta n t
There s ee ms to be n o method of a cc u r a tely co mpu t i n g t hi s
s olidity which i s a d a pted to g e n er a l r a ilro a d pr a ctice
,
,
.
.
2 49
.
B or r ow -P i t s
.
I n a dditio n to the ordi n a ry work o f exc a va tio n a n d e mb a n k
me n t for r a ilro a d s , e a rth i s ofte n
borrowed fro m o u ts ide
the li mits of t h e work proper ; a n d i n su ch exc a va tio n s c a lled
‘
borrow-pits , it i s co mmo n to prep a re the work by di vidi n g
”
“
the s u rf a ce i n to s qu a re s rect a n gle s or tri a n gle s ta ki n g le vel s
a ga i n
a fter the
a t e very cor n er u po n the ori g i n a l s u rf a ce ;
exc a va tio n of the borrow pit i s co mplete d the poi n ts a r e repro
d u ce a a n d le vel s t a ke n a s eco n d ti me The exc a v a tio n i s th u s
di vided i n to a s erie s of vertic a l pri s ms h a v i n g s qu a re r e ct a n gu
The s e pri s ms a r e co mmo n ly
l a r or tri a n gu l a r cro s s s ectio n s
tr u n c a ted top a n d botto m The le n gth s or a ltitu de s of the
vertic a l ed g e s of the s e pri s ms a r e g i ve n by the di ffere n ce i n
lev el s t a ke n
l s t o n the ori g i n a l s u rf a ce a n d
2 d a fter the exc a v a tio n i s co mple t ed
Thi s method of me a s u re me n t i s very g e n era lly u s ed a n d for
ma n y p u rpo s e s
,
,
,
,
-
,
.
,
-
.
,
.
.
,
,
,
.
,
,
.
250
.
c
T r u n a t e d T r i a n gu l a r P r i s ms
Let A
.
re a of ri ght s ectio n EFO of a
tr u n c a ted pri s m the b a s e ABC
bei n g a rig h t s ectio n
Dei g ht AH
a
,
111
he
BE
h3
CK
a
a
ltitu de of tri a n gle EFD dr oppe d
fro m E to FD
V
V
ol u me of pri sm
ABC KHE
s,
s
olidity
ABCFDE
pyr a mid FDEHK
Ra i lr oa d Cu rves a n d Ea r thwork
17 2
.
T h en u s i n g met hod of e n d a re as ,
AEHD
BGKC
2
b
h1 + h4
2
V:
I“
A
+b
h 2 + h3
hz
h3
I“
(
ft )
cu .
.
( cu
27
y ds )
.
1
1
8
(
)
.
We ma y fin d V correct by th e pri s moid a l for mu l a i f we
a pply the pri s moid a l correctio n
The pri s moid a l correctio n
0 ( o r i n thi s c a s e AD
0 s i n ce D O
D1
The
C
BC
for mu l a therefore re ma i n s u n ch a n g ed It i s e vide n t fro m thi s
the n th a t the s ol u tio n hold s good a n d the for mu l a i s correct
n ot o n ly whe n the s u rf a ce EHKG i s a pl a n e b u t a l s o whe n it i s
a w a r ped s u rf a ce g e n er a ted by a ri ght li n e mo v i n g a lwa y s p a r
a llel t o the pl a n e AD HE a n d u po n EG a n d HK a s directrice s
S o me e n gi n eers prefer t o cro s s sectio n i n recta n gle s of b a s e
I n thi s c a s e
15 ’ x
,
,
.
,
.
,
,
,
,
,
,
.
-
’
15
x
18 ’
M
hi
10
“
7“
4
27
hl
h ?»
l
"
712
133
"
l 134
”
d
(
cu
.
y ds )
.
)
Other co n ve n ie n t di me n s io n s wil l s u gg e s t t he ms el ve s , a s
10
'
x
or
'
20 x
or
’
20 x 2 7
B y thi s method the co mp u t a tio n s a r e re n dered
’
li gh tly more
co n ve n ie n t ; bu t the s i z e of the cros s s ectio n a n d the s h a pe
whether s qu a re or rect a n gu l a r s ho u ld depe n d on the topo g
r a ph y
The firs t e s s e n t i a l i s a ccu ra cy i n re s u lts the s eco n d
i s s i mplicity a n d eco n o my i n field work a n d e a s e of co mp u ta tio n
s ho u ld be s u bordi n a te to both of the s e co n s ider a tio n s
-
s
,
,
,
,
.
-
,
.
Pr oblems
Sp e ci a l
252
.
As s emb l e d P r i s ms
17 3
.
.
of a n a s s embl y of pri s ms of e qu a l b a s e i t is n ot
n ece s s a ry to s ep a r a tely c a lc u l a t e e a ch pri s m bu t th e s ol i d i ty of
a n u mber o f pri s ms ma y be c a lc u l a ted i n on e ope r a t io n
I n t h e pri s m B
I n th e c a s e
,
,
.
,
V8
VC
4
_
A
Z
as
b4
( 4
be
,
4
F ro m i n s pectio n i t
etc
.
will be s ee n t a ki n g A a s th e c ommon
a re a of b a s e of a s i n gle pri s m a n d t a ki n g t h e su m of the
s oliditie s t h a t th e hei g ht s a 2 a 5 e n ter i n to the ca lc u l a t i o n of
,
,
,
,
pri s m on ly ; a 3 a , i n to tw o pri s ms e a c h ; bl be on e on ly ;
be b5 i n to three pri s ms b3 b4 i n to fo u r pr i s ms ; a n d s i mil a rl y
thro u gho u t
Let
su m o f hei g h ts c o mmo n to on e pr i s m
t1
on e
,
,
,
,
,
.
t2
u.
u
u
u
t w o p r i s ms
u
hree
f ou r
ta
t
t4
The n the tota l vol u me
V:
A
A
M
( cu
t 1 + 2 62 + 8 l 3 + 4 l 4
.
ft )
.
174
R a i lr oa d Cu rves a n d E a r th work
253
.
Add i t i on a l He i g hts
.
.
Whe n the s u rf a ce of the gro u n d i s ro u gh it i s n ot u n u su a l
to t a ke a dditio n a l hei ghts the u s e of which i n g e n er a l i n v ol ve s
a ppreci a ble l a bor i n co mp u t a tio n
it bei n g n ece s s a ry co mmo n ly
to di vide the s olid i n to tri a n g u l a r pri s ms a s s u gge s ted by the
fi gu re s j u st below which i n cl u de the c a s e of a tra pez oid
,
,
,
,
,
,
.
co mpu ta tio n s ma y be s i mplified i n the two s peci a l c a s es
which follow
Whe
n the a dditio n a l h ei g h t h i s
a
( )
i n the ce n ter of the rect a n g le
Here the s olid i s co mpo s ed of a n
a s s e mbly of 4 tri a n g u l a r pri s ms who s e
c
.
ri ght s ectio n s a r e of e qu a l a re a
if
The v ol u me o f th e a s s embled pri s ms
A
—
h 1 — h 2 — h3 — h 4)
the to t a l vol u me i s t h a t du e to the fo u r cor ner hei g hts pl u s
the v olu me of a pyr a mid of equ a l a re a of b a s e a n d who s e a lti
tu de i s the differe n ce betwee n the ce n ter height a n d the me a n
o f t h e fo u r cor n er h ei g ht s
or
.
17 6
R a i lr oa d Cu r ves a n d E a r thwork
.
The co mmo n pr a ctice i n the ca se of borro w pi ts i s th a t
2 49
s ta ted i n
Whe n the orig i n a l s u rf a ce a n d the s u rf a ce to
which the exc a va tio n i s ma de a r e both so mewh a t ro u g h a n d i r
regu l a r thi s method i s n a t u ra lly a n d properly a dopted
I n ma n y c a s e s of exc a va tio n the work i s c a rried t o a fi n i s hed
s u rf a ce s o meti m e s a pl a n e s u rf a ce or s e ver a l pl a n e s o r s o me
other v ery s i mple s u rf a ce s o meti me s to a more co mplic a ted
s u rf a ce where cro ss s ectio n i n g the fi n i s hed s u rf a ce wo u ld n ot
re a dily a llow the f a cts to be s hown on the pl a n
I n either of the s e c a s e s the followi n g method s ee ms prefera ble
( a ) Cro s s s ectio n the ori g i n a l s u rfa ce a s before
( b) A ss u me a co n ve n ie n t hori z o n t a l pl a n e s li g htly lower
th a n the s u rfa ce to which the exca va tio n h a s bee n ca rried
F
n d the tot a l e a rthwork to the ori g i n a l cro ss s ectio n ed
i
0
( )
s u rfa ce a bo v e this a s s u med pl a n e a s a b a s e
( d ) F i n d the tot a l e a rthwork to the fi n i s hed s u rfa ce a bo ve
the a s su med pl a n e a s a b a s e I n ma n y c a s es thi s s u rfa ce will be
bo u n ded by o n ly a few pl a n e s a n d th u s wi l l a l low very s i mple
co mp u t a tio n s
ve
F
n d the di ffere n ce betw e e n
n
s
i
thi
will
i
a
d
d
e
0
;
g
( )
( )
( )
t h e a mo u n t of ea rthwork exc a va ted
254
-
.
.
,
.
,
,
,
,
,
-
.
.
-
.
,
.
-
,
.
,
.
.
.
ofte n h a ppe n s th a t a n exc a v a tio n i s ma de of con
o f n ot
a n d ofte n
s i d e r a b l e le n gth a n d n ot gre a t bre a dth
re
t
depth
I
n s trippi n g s oil u n der a propo s ed e mb a n kme n t
a
g
the se co n ditio n s pre va il The exc a va tio n ca n the n be s t be
h a n dled very mu ch a s exca v a tio n is h a n dl ed on r a ilro a d s A
li n e wi ll be r u n a n d a s erie s of cro s s s ectio n s t a ke n the li n e
a n d cro ss s ectio n s bei n g t a ke n a t
s er vin g a s a ce n ter li n e
s ta tio n s a lo n g t h e li n e a s ofte n a s req u ired by the s u rf a ce con
The cro ss s ectio n s will be very irre g u l a r n ot h a v
d it i o n s
i n g a n y u n iform b a s e b u t mu ch a s repre s e n ted i n the fig u re
below
255
It
.
,
.
.
.
-
,
,
-
,
.
-
,
,
.
.
Sp ec i a l P r o blems
17 7
.
To fin d the a re a of the s e irr eg u l a r s ectio ns it ma y fr e
e
tly
h
ppe
th
a t the be s t method ma y be o n e s i mil a r t o t h a t
n
a
u
n
q
de s cribed for cro s s s ectio n i n g on the precedi n g p a g e
i
d
ele
v a tio ns o n ori g i n a l su rf a ce AB C D E
n
a
F
( )
n d ele va t io ns o n e x c a v a ted s u rf a ce FG HIKLE
i
F
0
( )
u me a hori z o n t a l li n e a t a co n ve n ie n t ele v a tio n MN
c
A
s
s
)
(
a
t
a
C
D
d
a
l
c
u
l
e
a
re
B
E
N
C
M
F
( )
( 3 ) Ca lcu l a te a re a MFGHIKLEN
(f ) A re a re qu ired i s th e differe n ce betwee n ( 07) a n d ( e
256
,
.
-
.
.
.
.
.
.
‘
1
,
This method i s s i mple i n pri n ciple a n d de s ir a bl e i n ma ny
c a s es Where there a r e few s ectio n s t o be co mp u ted it ma y be
eco n o mic a l t o u s e a n y method a lre a dy well u n ders tood ; r a ther
th a n look u p a method le ss f a mili a r W here ma n y s ectio n s a r e
to be co mpu ted the r u le of p a g e 153 will pro v e eco n o mic a l
,
.
.
.
,
It i s frequ e n tly n eces s a ry to fin d the exc a va tio n made
by d iggi n g i n to the s ide of a hi gh b a n k C ro s s se cti on p oi n ts
o n a s teep s l o pe
of te n i n loo s e s a n d c a n n ot be expe c ted to
y ield g ood re s u lt s for co mp u ti n g exc a va tio n
I n s u ch c a s e s the followi n g method ma y pro ve va l u a ble
257
.
'
-
.
,
,
.
.
D e termi n e w i t h c a r e bo th
the po s itio n a n d ele va ti o n of
poi n t A a t ed g e of to p of b a n k a l s o of B n e a r bottom of l ope
( b) S i ght fro m A to bott o m of s ta ke a t B a n d re a d on le vel
i n g r od CC’ D D ’ etc a t th e s a me ti me me a su ri n AC AD et c
g
( 0 ) A fter the exc a va tio n h a s been ma de fin d the po s itio n s of
’ n
A a d B ; a l s o the di s t a n ce s HH’ LL’ et c ; a l so A’ H A’ L etc
,
( d ) P lo t on cro s s s ectio n p a per a n d me a su re a re a betwee n
ori g i n a l s u rfa ce a n d exc a va ted s u rf a ce Thi s ca n prob a bly be
do n e to b s t a d va n t a ge by pl a n i meter
(a )
s
.
.
.
,
,
.
,
,
,
,
,
’
,
.
,
-
.
e
.
,
.
CH AP T ER
XIV
.
EAR T HWOR K T ABLE S
ca l c l a ti o
.
q u a n titie s ca n be mu ch f a cil i t a ted
”
“
E a r thwork T a ble s
by the u se of su it a bly a rr a n g ed
258
.
Th e
u
of
n
.
re gul a r Th ree L e v el Sectio n s very co n ve n ie n t t a bles
ca n be c a lc u l a ted u po n the follow i n g p r i n c i ple s o r fo r mu l a s
“
F or
-
Us e n o ta ti o n a s befo r e for
ca
T he n
h t, h r ) db d"
4
41 S
OKE
ABKL
A
la
81
OK >
< EM
2
c( b
K
O
—
)+
sc
g
g
g
-
+ sc
E
M
(
—
h
( r
8 :
50 A ( cu
.
ft ) :
22
g
A ( cu
.
+ sc
c — 0
+ h 3)
— 2c
)
+ sc
For a pr is m of b a s e A a n d l
N D)
5 0 , the s olidity
y ds )
.
-
2 c)
1
8
6
( )
R a i lr oa d Cu r ves a n d E a r thwork
1 80
.
Ta ble s ma y be fo u n d i n A l le n s T a ble s XXXI I for
va rio u s b as e s for
’
2 61
.
A n ex a mple will ill u s tr a te t he i r u s e ,
0
14
8
l ii t o 1
No te s
c
C a l u l a tio n s :
h, + h ,
E :
L
20 :
S1
E :
L
SO
If I OO
SI
SO
There i s a ls o i n A lle n s Ta ble s XXX I a
P ri s moid a l C orrectio n
c a lc u l a ted by the for mu l a
2 62
’
.
1
(Co
c1
I n the ex a mple a bo v e
cO — CI
Z
Do
F ro m T a ble fin d opp , 7 5 for
.
.
1
2
Vi oo
V.
0
2 -8
Vp = 2 34 9
) ( Do
D1)
“
T a ble
of
E a r thwork Ta bl es
263
1 81
.
Whe n the s ect i o n i s le ss th a n 100 ft
.
pris moid a l correct io n
is
.
th a t i s
,
2 64
.
ma de
in
le n g th the
,
before mul tiplyi n g b y
c
Equ i va le n t Le v e l S e t i on s
1 00
( 19 0 )
C)
( So + 3 1
s
.
.
The T a ble of p 1 7 9 ( or Ta ble XXXI I A lle n s T a ble s ) s hows
’
,
.
i n the L
gg
colu mn the va l u e of S
— -A
for va l u es of ce n ter
hei ght 6 C o n ver s ely if there be g i v e n t h e S of a n y s ect io n
”
”
“
irre gu l a r or reg u l a r three le vel the va lu e of c for a le v el
s ectio n o f the s a me a re a ma y be fo u n d fro m the L col u mn
,
.
,
.
Exa mp le
.
F ro m p
.
180 , B a s e 1 4, S lope l l to 1 for
SI
fro m Ta ble XXXI I
6
The n o t es of thi s s ectio n will be
Three
”
”
“
a n d to
L e vel S ectio n s
Irregu l a r S ectio n s t a ble s ca n be ca l
cu l a t ed u po n the followi n g pri n ciple s a n d for mu l a s
”
The s e ta ble s a r e i n e ffect t a ble s of Tri a n g u l a r P ri sms i n
which h a v i n g g i ve n ( i n feet ) th e b a s e B a n d a ltit u de a o f a n y
tri a n gle the t a ble s g i v e the s olidity ( i n c u bic y a rd s ) for a
pri s m of le n gth t z 50 t h a t i s
265
F or g e n er a l c a lc u l a tio n s a d a pted both to re gu l a r
.
,
,
,
,
,
,
,
aB
2
‘
50
50
27
54
1
( 9 1)
Whe n e ver the c a lc u l a tio n s ca n be bro u ght i n t o the for m
,
i
gg
a B,
the re s u l t ca n be t a ke n directly fro m the t a ble
ield a n d Office T a ble s
Three L e vel
”
S ectio n s
a r e pro v ided for i n T a ble XXXII for s lope of 1
to
4
“
”
n
d
1
4
1 a
ba s e s
to 3 0
P ri s moid a l C orrectio n s a r e fo u n d i n
T a ble X XXI a n d Tri a n gu l a r Pris ms i n T a ble XX X
266
.
In
A lle n s
.
’
F
-
,
.
.
Ra i lr oa d Cu r ves a n d E ar thwork
.
22
a B t a ke s for m th u s
the ta bles the fo r mu l a S
9
£ x wi dth x h ei g h t a n d the t a ble s a r e a rr a n ge d as below
2 67
In
.
,
54
.
H E IG H T S
W I DT H S
w i dth X h ei gh t
.
t
Th e a pplic a io n to
“
We h a ve for mu l a
Three L e vel S ectio n s
p 15 1
”
-
i s a s f ollows
,
.
A
+
b
D
52
23
2
4s
pri s m 50 ft i n le n gt h ( l
a n d for a
.
.
50 )
12 3
or
S i s th e s u m of two q u a n ti t ie s , e a
c h of whi c h i s i n prope r
for m for the u s e of t h e t a ble s
F or cro ss s ectio n s of a g i ve n b a s e a n d s lope ( b a n d 3 con
50 1?
13 i s a co n s ta n t a n d a l s o
b i s c o n s ta n t
s ta n ts )
.
-
,
.
.
,
23
23
54
We ma y the n c a lcu l a te o n ce for a ll 2
'
on s t a n t
Al s o
)
23
’
and
c a ll this B ( a
.
é -Q
b
—
54
2s
b, a n d c a ll thi s a co n s ta n t E
’
50
The n
I n u s i n g th e ta bles ,
0
—
B
.
E
hei ght
D 1: Wi dth
A s i n t h e pre vi o u s ta ble s , h a v i n g fo u n d S o a n d 8 1 ,
R a i lr oa d Cu rves
2 69
.
and
Irr e g u l a r S e c t i on s
FIG 2
.
.
l
The s che me of co mp u t a tio n s ho u ld be the s a me a s th a t u s ed
with p a irs of tri a n g le s i n F i g 2 § 2 29 or a s s how n by the
”
“
r u le of p a g e 1 53
etc ] of F i g 2 the T a ble s will
I n s te a d of £[h D( dA d L)
d
4
3
+
g i v e sw a m
o
.
t
S o th a t the s u mma tio n will g i v e the re s u lt i n c u bic y a rd s
I n a s i mil a r w a y the
D i a g r a ms
t o be de s cribed i n the n ext
d L)
C h a pter will g i v e gg[h D( dA
S i mil a r co mp u t a tio n s ma y be ma de by S lide R u le s e t i n s u ch
a ke n o ff fro m the S lide
etc
will
be
t
a w a y th a t
h
d
d
L)
?g [ D( A
?
R u le a s the re s u l t of the co mp u t a tio n
If s o me co mpu ters prefer to pl a t cro ss s e ctio n s a n d co mp u te by
pl a ni meter the pla n i meter a r m ma y be s o a dj u s ted a s to record
A
r
a ther th a n A
”9
R e su lt s by D i a g r a m by S li de R u le o r b y P l a n i meter will
re a di n gs
a l l be s u bj ect to the l a ck o f preci s io n i n v ol v ed i n
A n y s u ch l a ck of preci s io n will be fa r le ss th a n the l a ck of pre
ci s i on d u e t o de t ermi n i n g the rod re a di n g s o n the s u rf a ce of
the g ro u n d fro m which c u ts a n d fills a r e co mp u ted a n d there
fore n ot ob j ectio n a ble
.
,
,
.
.
.
,
.
.
)
.
-
,
.
'
,
,
,
.
,
.
CH AP T ER
XV.
EART HWORK DI AGRAM S
270
.
Co mp u t a tio n s of e a rthwork ma y a l s o be ma de by me a n s
.
di a gr a ms fro m wh ich re su lts ma y be re a d b y i n s pec t io n
merely
The pri n ciple of their co n s tr u ction i s expl a i n ed a s follows
Gi ve n a n e qu a t io n c o n t a i n i n g three va ri a ble q u a n t iti e s a s
of
.
x
If we
( 1 94 )
zy
me s o me va l u e o f z ( ma ki n g z a co n s t a n t ) , the
e qu a tio n the n be c o me s the equ a tio n of a ri ght li n e
If thi s li n e be pl a tted , u s i n g rect a n gu l a r c o ordi n a te s ( a s the
a s su
.
li n e z
1 i n th e fi gu re ) , the n h a v i n g g i ve n a n y va l u e of y, the
corre s po n di n g v a l u e of a: ma y be
t a ke n off by s c a le If a n ew va l u e
of z be a ss u med the equ a tio n i s
obt a i n ed of a n ew li n e which ma y
a l s o be pl a tted ( a s 2:
1 i n the
fig u re ) a n d fro m which a l s o h a v
i n g g i ve n a n y va l u e o f y the cor
re s po n di n g va l u e o f x ma y be
deter min ed by s c a le A s s u min g
a s erie s of v a l u e s o f z a n d pl a tti ng
we h a v e a s eries of li n e s e a ch
repre s e n ti n g a di ffere n t va l u e o f z
a n d fro m a n y on e of which h a vi n g gi v e n a v a l u e of y we ma y
by s ca le determi n e the va lu e of it
T hu s g i ven va l u e s of z a n d y 5 r equ i r ed at we ma y fin d
.
,
,
,
,
'
.
,
,
,
,
,
.
'
,
,
,
,
,
The l i n e corre s po n di n g to the g i ve n va l u e of z a n d
2
U po n thi s lin e we ma y fin d the va l u e of ac co r re s pon din g
te th e g i ve n va l u e of y
1
.
,
.
.
,
1 86
R a i lr oa d Cu r ves
Next,
E a r th work
a nd
.
i n s te a d of pl a tti n g u po n l i n es a s coord i n a te
a xe s
we pl a t u po n cro ss s ectio n
p a per the cro s s s ectio n li n e s form
a s c a le s o th a t the va l u e s of x a n d
y n eed n ot be s c a l ed b u t ma y be
r ea d by s i mple i n s pectio n a s i n th e
fig u re
-
,
-
,
,
,
.
272:
If the equ a ti o n be i n the
for m
1
x
a zy
( 19 5 )
e proced u re i s equ a lly po s s ible a n d th e li n e repres en t
i n g a n y va l u e of z will s till be a ri ght li n e
If the e qu a tio n be i n the for m
th e s a m
,
.
1
96
(
)
wh i ch a b c d a r e co n s t a n ts the s a me proced u re i s s till
po ss ible a n d th e li n e repre s e n ti n g a g i ve n va l u e of z i s a ri ght
li n e a s before
The u s e of di a gr a ms of thi s s o r t is therefore po ss ible for th e
s ol u tio n o f equ a tio n s i n the for m of
in
,
,
,
,
,
,
,
or
.
i n s i mpler mo di fi c
a tio n s of thi s form
.
273 : R
eferri n g a ga in to the figu re a bo ve we ma y c o n s ider
the horiz o n t a l li n e s to repre s e n t su cce ss i ve va l u e s of x a n d refe r
to the m a s the li n e s
,
etc
a n d si
.
mil a rl y we ma y refer to v ert ic a l li n e s a s the lin es
=
2,
y
j u s t a s we refer to the i cl i ed li e
n
n
n
et c
.
s
etc
.
Ha v i n g g i ve n an y tw o of the q u a ntitie s a , y, z , th e thir d ma y
be fo u n d by i n spectio n fro m the di a gra m by a proce s s s i mila r
to th a t de s cribed
.
R a i lr oa d Cu r ve s
1 88
275
c.
.
and
E a r th wor k
I n l i ke ma n n er a ta ble ma y be co n s tr u cte d
.
.
- c
1
27 6
.
I t will be n oticed th a t whe n D o
D1
0 , 0 z: 0
.
Therefore for a ll va lu e s of co 6 1 the li n e s p a ss thro u gh th e
ori g i n
We ma y proceed to pl a t the li n e s co 0 1 1 Co 0 1 2
3 etc fro m d a t a s how n i n the a bo ve t a ble pl a tti n g
co
01
etc the poi n ts
u po n the li n e s D o
Do
D1
D1
s hown with circle s a ro u n d the m i n the cro s s-s ectio n s heet
,
.
,
,
,
.
,
,
.
,
,
p 1 89
.
.
Ha v i n g t h e
li n es co cl 1 co cl 2 3 pl a tted i n ter
medi a t e li n e s a r e i n terpol a ted mech a n ic a lly u po n the pri n
ci pl e th a t ver t i ca l li n e s wo u ld be proportio n a lly di vided ( a s
ML i s proportio n a lly di vi ded i n t o 5 e qu a l p a rts ) a n d po i n t s
a r e ma rked for the lin e s
,
,
,
,
,
00
61
the mo s t co n ve n ie n t u s e the va l u e s of co 0 1 a r e ta ke n
to e very s eco n d te n th of a foot i n i n t erpol a ti n g a s i s s hown on
the di a gr a m p 189 bet ween 1 a n d 2 th a t i s
F or
,
,
,
.
,
,
A compl e t e di a g r a m i s s how n a t the b a ck of th e book
.
E a r thwor k
D i a g ra ms
.
1 89
R a i lr oa d Cu rves a n d E a r thwork
1 90
277
.
For Us e
.
.
the di a g o n a l li n e c orr e s po n di n g to the gi ve n v a l u e 0 1
co
C1 ; follow thi s u p u n t il t h e v ertic a l li n e r epre s e n ti n
the
g
D 1 i s re a c hed a n d the i n ter s ectio n i s th u s
g i v e n va l u e of D O
fo u n d The n re a d off the va l u e of C corre s po n d in g to thi s
i n ter s ectio n
F in d
'
,
'
.
.
E xa mp l e
co
.
cl
Do — D1 =
Co
01
12.
278
.
Di a g r a m for T r i a n gu l a r P ri s ms
F ro m for mu l a
S
a
ta
.
ble ma y be co n st ru cted
.
di a g r a m ca n be c o n stru c t ed s imil a r i n form to
t h a t for P ri s moid a l C orrec t io n
The li n e s for a l l va l u es of c p a s s thro u g h the ori g i n
I n co n s tru cti n g thi s t a ble a n y va l u e s of D mi g ht h a ve bee n
Tho s e u s ed were s elected
t a ke n i n s te a d of tho s e u s ed here
be c a u s e they gi ve re s u lts s i mple i n va l u e e a s ily ob ta i n ed a n d
r e a di ly pl a tte d
F ro m t h is
a
.
.
,
.
,
.
,
R a i lr oa d Cu rves a n d E ar th work
oz
gg g m
-l
s z
o
-
.
coa s
D = 14
0
Whe n
D
we ma y a ga i n c a l cu l a t e di re ctly
— 0 14 0
5
gg
S :
bu t a be tter method i s to fin d h o w mu ch gre a t e r S wi l l be for
D
t
h a n for D
We h a ve
T he n for a n y n e w va l u e D
-
%
'
a
D
'
' —
—
=
D
D
S
s
>
<
a s
D
’
D
D
S
'
D
'
S
8
'
0
r:
c
wh i h is e n t e r e d i n ta b le
.
D i a g r a ms
E a r th work
Si mi l a rl
y
D
s
,
"
S
l
s
'
D
’
"
1 93
.
'
D )
gg
”
D
"
D
’
x
”
5
1
s
Si mi l a r l
y
S
,
"
”
S
’"
S
’"
m
S
”
8
”
75
C o n s t a n t i n cre me n t for D
28 1
’
D
is
Ea ch re s u lt i s e n t ered i n the t a ble i n i ts proper pl a ce
.
The fi n a l r e s u l t for c 0 a n d D
i n depe n de n t l y a s a che c k
s
.
ho u ld be c a l cu l a t e d
.
Whe n
c= 0
Wh en
D
S :
32
—
1 4
3
x
50 x 134
x
S :
ch ecks exa ctly a d ll i ter med i t e va l u e s a r e c hecke d
Th i s
,
n
a
n
a
by thi s proce s s , which i s a l s o more r a pid t h a n a n
c a lcu l a tio n for e a ch va l u e of D
i n depe n de n t
.
We n ow h a ve va l u e s of S for th e va r i o u s va lu e s of
etc whe n 0 0
D
Next fin d h ow mu ch th es e will be i n c re a s ed whe n c
1
282
.
.
,
.
,
.
F or mu l a
for a n y n e w va lu e d
S
'
S
S
’
S
ggo
1 4
3
)D
ggw
14
3
)D
c
(
%g
c
:
:
’
)D
1 94
R a i lr oa d Cu rves a n d E a r thwork
When
c = 1 and
'
’ -
c
V—
S
”
S
=2
"
c
S z
S
a nd
S
’
'
o
"
S
z
1
ggz)
2:
g
z
1,
’
o
.
’
c
)D
" -
c
'
c
=1
2
—1)
T h a t is , for a n y i n cr ea s e of 1 ft i n th e va l u e of c,
.
S
'
Wh en
D = 14
fi
Thi s we e n te r a s the
c o sta n t di fi e ce f c ol u m D
n
er
n
or
n
We h a ve a lre a d y fou n d
Th is gi ves
c ol u m
n
Whe n
14
.
D
S
( 2 0 1)
'
15
E n t e r 1 5 a s th e
c ons ta t di ffe en ce in c olu mn
n
r
We a lre a d y h a ve
Thi s a llows u s t o co mple t e
S i mil a rly for
c ol mn
u
S
D
’
S
20
a n d co mplete
ter 2 0 a s co n s t a n t differe n ce i n colu mn
col u mn a s s how n i n t a ble
S imil a rly fil l ou t a ll the col u mn s s hown i n the t a b l e
En
,
.
,
.
R a i lr oa d Cu r ve s a n d E a r th wor k
196
.
c u r ve of le vel s ectio n h a s been pl a tt ed on this di a gr a m
i n the fo l lowi n g ma n n er
F or le vel s ectio n s whe n
A
.
6
0
D 2
C
1
D 2
,
c
etc
D
“
The li n e p a s s i n g t hro u gh the s e poi n ts g i ve s th e cu rv e of
”
le v el s ectio n
A s ide fro m the direct u s e o f thi s c u rv e o f le vel s ectio n ( for
preli mi n a ry e s ti ma te s or otherwi s e ) it i s very u s ef u l i n ten di n g
to prev en t a n y gro s s error s i n the u s e of th e t a ble s i n ce i n
g e n er a l the poi n ts ( i n ters ectio n s ) u s ed i n the di a gr a m w i ll l i e
n o t fa r fro m the c u r ve o f le vel s e c t i on
.
.
,
,
,
,
,
.
28 6
.
Us e of Di a gr a m
.
F i n d the di a g o n a l li n e co r re s po n di n g to th e gi ve n va l u e of 0 ;
fo llow thi s u p u n til the vertic a l li n e repre s e n ti n g the g i ve n va lu e
of D i s re a ched a n d thi s i n ter s ectio n fo u n d
The n re a d ofi th e
va l u e of S corre s po n di n g t o t hi s i n ter s ecti on
Note s
E xa mp l e
.
,
.
.
Sta
.
l
1 60
S]
Sta 0
.
l
D = 28 4
and
c
i s the middle o f the s p a c e be t wee n
is r e a c he d
F ollow thi s u p u n til the v ert ic a l li n e
The in ters ectio n lie s u po n the li n e S 1 160
E n ter t hi s a bo ve oppo s ite S t a 1
F or
Sta
.
.
'
.
.
F or
c
St a
.
.
D
0
i s th e middle of s p a ce be t wee n
an d
ollow thi s u p u n t il th e middle of s p a ce be t wee n
i s re a ched
Th e i n ters ec t io n lie s j u s t a bov e t h e lin e
F
.
S0
E n t er th i s
78
O ppo ite St a 0
s
.
1 60
.
7 8 z 2 3 8 cu y ds
.
.
.
.
E a r th work D i a g r a ms
1 97
.
The pr i smoid a l correcti o n ma y be a pplied if de s ired
.
be co n s tru c ted i n thi s w a y th a t will
s
n
a
n
o
f
i
e
re
lt
to
re
a
ter
de
ree
reci
io
th
i
s
w
a
rr
n ted
u
a
s
s
a
v
p
g
g
g
by th e precis io n re a ched i n t a k i n g the me a su re me n ts on th e
n
ro
u
d
g
I n poi n t of r a pidity d i a g r a ms a r e mu c h mo r e r ap i d th a n ta bl es
for the c o mpu t a tio n o f Th r ee L evel S ecti on s
”
“
a n d for P r i s mo i d a l Cor r ecti o n
F or
Tr i a n g u l a r P r i s ms
the d i a gr a ms a r e s o mewha t mo r e r a p i d
F or L eve l S ecti o n s the t a ble s fer Three L e vel S e c t io n s a r e
a t l ea s t eq u a l l y r a p i d
28 7
D i a gr a ms ma y
.
.
-
.
,
.
-
,
.
288
T h e u s e of a pproxi ma te method s for a pplyi n g the pri s
.
to irre gul a r s ectio n s will n ow be re n dered
“
D i a g r a ms for Three L e vel
v ery pr a ctic a ble by the u s e o f the s e
”
S ectio n s
No u s e of di a g r a ms i s n ece ss a ry
M ethod 1
H a vi n g fo u n d for a n y irre gu l a r s ectio n s ( by tr i
M etho d 2
a n g u l a r pri s ms or a n y other method ) the s olidity S for 50 ft
i n le n g th fin d u po n the di a gr a m the li n e corre s po n di n g to
thi s va l u e of S follo w thi s li n e to t h e c u r ve of le vel s ectio n
a n d re a d o ff the va l u e of c ( for a le v el s e c tio n ) which corre
s p on ds a n d a l s o the va lu e of D for t h e s a me s ectio n
M ethod 3
H a v i n g fo u n d i n a n y w a y the v a l u e of S ; i f c i s
gi v e n fin d the va l u e o f D to corre s po n d ; if D i s g i ve n fin d the
va l u e of c to corre s po n d
M et hod 4 The u s e of di a gr a ms i s n o t n eeded
The di a gr a ms s hown a t the b a ck of the book a r e g i ve n p a rtly
to s how a g ood s che me or a rr a n g e men t a n d p a rtly to a llo w
pr a ctice i n their u se F o r re gu l a r work the s c a le i s t oo s ma ll
t o be de s ir a ble a n d tryi n g to t h e eye s
T hey a r e n ot s u ffi
ci en t l y exte n s i ve
I n office s where there i s mu ch e a rthwork
co mpu t a tio n to be do n e di a g r a ms s ho u ld b e con s tr u cte d on
do u ble the s c a le a n d exte n di n g t o hi g her n u mbers S e vera l
s heets ma y be requ ired for e a ch ki n d of di a g r a m
I t ma y
s ee m th a t
s u fficie n tly preci s e va l u e s
c a n n ot be rea d fro m
the s e di a gr a ms b u t the di a gr a ms a r e mu ch more preci s e th a n
the field work where a ce n ter cu t i s n ot s u re to on e te n th of a
foot
mo i d a l correctio n
-
.
.
.
.
.
,
,
.
,
.
,
,
.
.
.
,
.
,
.
.
'
,
.
.
,
-
.
,
CH AP T ER XVI
HAUL
.
.
Whe n ma teri a l fro m exc a va ti o n i s h a u led t o b e pl a c ed
i n e mb a n kme n t i t i s c u s t o ma ry t o p a y t o the co n tr a ctor a
certa i n su m for e very c u bic y a rd h a u led Often ti me s it i s pr o
vi d e d t h a t n o p a y me n t s h a ll be ma de for ma teri a l h a u led le s s
“
n
s
n
I
n
h
e
mm
m
f
th a a s pecified di t a ce
e a s t a co
o n li it o fre e
t
”
h a u l i s 1000 ft Ofte n i n the we s t 5 00 ft i s the li mi t of
”
“
free h a u l S o meti me s 100 ft i s the li mit
A co mmo n c u s to m i s to ma ke th e u n it for p a y me n t of h a u l
the price p a id will ofte n be fro m 1 t o
o n e y a rd h a u led 100 ft
2 ce n t s per c u bic y a rd h a u led 100 ft
”
h a u l i s s ma ll a n d therefore th e s ta n d
Th e price p a id for
a r d o f p r e ci s i on i n c a lc u l a tio n n eed n ot b e q u ite a s fin e a s i n
the c a lcu l a t io n of the qu a n titie s of e a rthwork Th e tota l
”
h a u l will be the produ c t of
289
.
,
.
.
.
.
.
.
.
,
.
.
,
»
.
1
( )
the t ota l a mou n t of exc a va ti o n h a u l ed a n d
2
( )
the a ver a g e len gth of h a u l
290
.
,
.
T h e a ve r a g e le n gth o f h a u l i s the di s t a n
c e be t ween th
e
cen te r of gr a vity of th e ma te ri a l a s fo u n d i n exc a va t i o n a n d
the ce n t er of gr a vity a s depo s ited It wou ld n ot i n g e n er a l b e
s i mple t o fin d the ce n ter of g r a v ity of the e n tire ma ss o f e x ca
a n d the mo s t co n ve n ie n t w a y i s t o t a ke e a ch
va tio n h a u led
”
“
The h a u l for e a ch se ct ion
s ectio n of e a r thwork b y i tse lf
i s the prod u c t of th e
,
,
.
,
,
.
mbe r of c u b i c y a rds i n th a t s ection
and
1
( )
nu
(2 )
di s t a nce b etwee n the ce n ter of gra vi ty i n exc a va t io n
a n d the ce n ter of g r a v ity a s depo s ited
,
.
190
,
R a i lr oa d Cu r ves
200
E a rthw or k
and
.
I
fig u r e b elow a n d fo l lowi n g the s a me
23 7
g e n er a l method of de mo n s tr a tio n u s ed pre vio u s ly i n
293
.
R eferri n g t o the
,
,
50
b a s e AB ;
bl
base D E ;
ho
a
ltit u de of ABC
h1
a
ltitu de of DEF
A5
a
re a of ABC ;
A1
a
re a of DEF ;
dis t a n ce betwee n
a llel e n d pl a n e s
l
.
A l s o u s e n ot a tio n bx , h z , A x , for a s ectio n di s ta n t x fro m ABC
.
dis t a n ce of ce n t er of g r a vi ty fro m ABC for e n tire s ectio n
o f e a rthwork
x,
2
,
.
dis t a n ce of ce n ter of gr a v ity fro m mids ectio n
x0
.
Then for a n y ele me n t a ry se ctio n of thi ckn ess dx a n d dist a n ce
x fro m ABC i ts mo me n t will be
,
,
[
to
[
bo h l — h o
fl
g
be)
b
( l
be
boh ol 2
[
ho + ( h 1
( bo
?’
ho
]
h o) i
f
l
l
( 1
l
(i t
8 l2
6 boho
4 boh i
4 bi ko
4 boho
3 boh l
3 bl ho
3 bl h l
3 boho
24
X ( boh o
boh l
bi ko
3 b1 h 1)
boh o
boh 1
hi ho
3 bl h l
x
it dx
4
— b
— h
b
h
l
l
o
1
0
)
g
(
2
4 hell o
12
fl
x dx
i
—
l
hogbl
bo l
(H
4
fl[
{
h o)
V
'
Ha u l
201
.
W h a t i s wa n ted i s as, r a ther th a n x
,
V
x,
V
Vz
é
2
b
h
o
< o
2 at,
bot ,
bl ho)
2
h
b
o
o
(
2 bl h l
boh i
bi ho)
b
h
o
( o
3 bl h l
h
b
o
o
(
bl h l )
boh 1
fr om ( 164)
bl hO)
A 1)
A
( c
( V i n cu
‘4
2
12 x 2 7
°
A1
“
V 1 n cu
.
'
V
.
ft )
.
yds )
.
for mu l a a pplie s dire c tly to s dlids wi th tri a n g u l a r e n d s
a n d wi t h t w o of the s u r f a ce s either pl a n e o r w a rped i n the
ma n n er s u gg es t ed i n 2 3 6 R eg u l a r T hree L e v el S ectio n s ma y
be di vi ded i n to p a rts of t ri a n gu l a r s ectio n s o th a t the a bo v e
formu l a wi ll a pply i n th a t c a s e I n a s i mil a r w a y it wi ll a ppl y
22 9 or t o s ectio n s e ve n more irre g
to I rr e gu l a r S ectio n s a s i n
u l a r a s on p a g e s 1 7 6 1 7 7
T hi s
.
,
.
,
,
.
ad
R a i lr o
2 02
295
Cu rves a n d Ea r th work
T h e for mu l a
.
Z
AI
2
.
AO
V
12 x 2 7
for m con ve n ien t for u s e bec a u s e we h a ve n ot fo u nd
the va l u es of A 1 a n d A O b u t i n s te a d h a ve ca lcu l a te d di rectly
from the t a ble s or di a gr a ms the va l u es of S I a n d S 0 for 50 ft
i n le n g th where
i s n ot i n
,
,
.
,
A I , or A 1
Su b s t itu ti n g ,
x,
100
S0
SI
100 x 1 00
V
12 x 2 7
27
50
where V i s the c orrect v olu me i n cu y d s
Thi s for mu l a 18 i n s h a pe co n ve n ie n t for u s e a n d resu lts
c o r rect to th e n e a re s t fo ot ca n be c a lcu l a ted with r a pidity
.
.
,
.
29 6
.
c
F or a s e t io n of le n gth l le s s th a n 100 ft
12 x 2 7
100
x 100
'
12 x 2 7
V1 00
AI
AO
V1 00
.
C H AP T E R
XVII
M AS S DI AGRAM
.
.
Ma n y q u e s tio n s of h a u l ma y be u s ef u lly t re a t ed by
a
M a ss
gra phic a l method which will be de s i g n a ted the
”
D ia g ra m
The co n s tru ctio n of the M a s s D i a g r a m will be more cle a rly
u n der s to od fro m a n ex a mple th a n fro m a g e n era l de s criptio n
“
298
.
.
.
.
C o n s ider
the e a rthwork s how n by the profile on p 206 con
”
“
“
s i s ti n g of a lter n a te
fill
To s how the work of
cu t
a nd
”
“
co n s tr u cti n g the di a gr a m i n fu ll the qu a n titie s a r e ca lcu
”
“
l a t ed thro u g ho u t b u t for co n ve n ie n ce
lev el s ectio n s a r e
u s ed a n d pri s moid a l correctio n di s re g a rded
In a c
tu a l pra ctice
the s oliditie s will h a ve been c a lcul a ted for the a ct u a l n ote s
t a ke n A llowa n ce s ho u ld be ma de for the f a ct th a t e a rth
pl a ced i n fill s hri n k s The a llow a n ce to be ma de i n col u mn 5
I n col
of t a ble wi ll depe n d o n h o w the work i s to be h a n dled
u mn 5 oppo s ite
it i s a ss u med th a t witho u t ch a n gi n g the n otes
a dditio n a l ma teri a l i s pl a ced i n the fil l to pro v ide for s hri n k a g e
or s ettle me n t which a ccord s with co mmo n pr a ctice ; a n d 5 per
ce n t s hri n kag e i s u sed here
I n the t a ble p 2 05 col u mn s 1 a n d 2 expl a i n th e ms el ve s
299
3 d col u mn g i ve s va l u e s of S fro m t a ble s
4 th col u mn g i ve s va l u e s o f S 100 or S , fo r e a ch s ectio n a n d
with s i g n
for fill
for ou t o r
5t h col u mn s hows fill s a f ter 5 per ce n t s hri n ka g e
6 th col u mn g i ve s th e tot a l or the s u m of s oliditie s u p to e a ch
s olidity i s added a n d
s t a tio n ; a n d i n g etti n g thi s tota l e a ch
e a ch s olidity i s s u btr a cted a s a ppe a r s i n the t a ble fro m t h e
re s u lts obt a i n ed
H a vi n g co mpleted the ta ble the n e xt s tep i s the co n s tr u ctio n
”
“
M a ss D i a g r a m p a g e 2 06 I n the fi gu re s hown there
of the
e a ch st a tio n li n e i s projected down a n d the va l u e fro m col u mn
6 corre s po n di n g to e a ch s t a tio n i s pl a tt ed t o sca le a s a n offs et
fro m the b a s e li n e a t th a t s t a tio n a l l
qu a n tities a bo ve the
li n e a n d a ll qu a n tit ie s below the li n e The poi n ts th u s fo u n d
”
M a s s D i a gr a m
a r e joi n ed a n d the re s u lt i s the
.
,
.
,
'
,
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'
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"
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,
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'
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,
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,
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,
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,
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,
2 04
2 08
R a i lr oa d Cu rves a n d E a r th wor k
.
Ma s s Di a g r a m
209
.
30 2
.
In a n e n tirely s i mil a r w a y , the a re a ABC ( p 2 08) repr o
.
e t the h a u l o f e a rthwork ( i n cu y ds mo ved 1 00 ft )
betwee n A a n d C a n d thi s a re a ma y be c a lc u l a ted by dividi n g
it b y a s erie s of vertic a l li n e s repre s e n t i n g s oliditie s a s s hown
Th a t thi s a re a repres e n ts th e h a u l bet ween
a bo ve G a n d F
A a n d C ma y be s how n a s follow s
T a ke a n y ele men t a ry s olidity d S a t D P ro j ect th i s dow n
u po n the di a g r a m a t F a n d dr a w t h e hori z o n t a l li n e s FG
B e twee n the poi n ts F a n d G ( or be t wee n D a n d I) ther e
fore exc a va tio n e q u a l s e mb a n k me n t a n d the ma s s (18 mu s t be
”
“
h a u l on d S will
ha u led a dis ta n ce FG a n d the a mo u n t of
be (28 x F G me a s u red by the tr a pe z oid FG S i mil a rl y with a n y
othe r ele me n ta ry ( 18
”
The to t a l h a u l betwee n A a n d C will be me a s u red by t h e
Thi s a re a
s u m of the s erie s of tr a pe z oid s or by the a re a ABC
i s prob a b ly mo s t co n ve n ie n tly me a su red by the tr a pe z oid s
for med by the vertic a l li n e s repres e n ti n g s oliditie s The a v er
a g e le n gth of h a u l will be thi s a re a di v ide d by t h e tot a l s olidity
( repre s e n ted i n t hi s c a s e on p 2 0 6 b y th e l o n g e s t verti ca l li n e
”
“
s n s
.
.
.
,
,
.
!
.
,
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,
,
,
,
.
,
.
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,
'
.
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,
.
The c o n s tr u c ti o n of
D i a gr a m a s a s eries
of tr a pe z oid s i n v ol ve s the a s s u mptio n th a t the c en ter o f gr a v ity
o f a s e c tio n o f e a rthwork lie s a t i t s mid—
s ectio n
which i s o n ly
a pproxi ma tely correct s i n ce S for the fir s t 50 ft will s eldo m be
ex a ctly the s a me a s S for the s eco n d 50 ft of a s ectio n 100 ft
lo n g I f th e li n e s j oi n in g the e n d s of the vertic a l li n e s h e ma de
a c u r ved l i n e the a s s u mp t io n beco me s more clo s ely a c c u r a t e
”
“
a n d if the a re a be c a lcu l a ted by
S i mp s o n s R u le
or by
pl a n i meter re su lts clo s ely a cc u r a te will be re a ched
It wi ll be f u rther n oticed th a t hill s e c tio n s i n the di a gra m
repre s en t h a u l forwa r d on the profile a n d va lley s ectio n s h a u l
‘
b a ckwa rd The ma s s di a gr a m ma y therefore be u sed to i n di
c a te the method s by which the work s h a ll be perfor med ;
whether exc a va t io n a t a n y po in t s h a ll be h a u led forwa rd o r
b a ckw a rd ; a n d more p a rt icu l a rly to s how the poi n t where
b a ckwa rd h a u l sh a l l ce a s e a n d forw a rd h a u l begi n a s i n dic a te d
i n the fi gu re p 2 0 8 which s hows a v ery s i mple c a s e the c u ts
a n d fill s bei n g e ve n ly b a l a n ced a n d n o h a u l o ver 900 fee t w i th
n o n e c e s s i t y for either borrowi n g or w a s ti n g
th e “ M a ss
303 :
”
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,
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'
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,
,
’
,
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,
,
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,
,
,
,
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,
,
,
,
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R a i lr oa d Cu rve s a n d E a r thwor k
.
2 12
R a i lr oa d Cu rves a n d E a r thwork
.
Ma s s Di a g r a m
21 3
.
c a s e ( 12 0 0 ft h a u l ) we s h ou ld dr a w i n
ma s s di a gr a m ( p 2 1 2) the li n e KG L Here KG i s le s s th a n
The li n e s ho u ld n o t be lower th a n G for i n t h a t c a s e
12 00 ft
the h a u l wo u ld be n e a rly a s gre a t a s KL or more th a n 1200 ft
I n the l a tter c a s e ( 800 ft h a u l ) t h e li n e wo u ld be c a rried u p
800 ft
The ma s s e s betwee n N a n d A
t o a poi n t wh ere NM
ca n better be w a s t ed th a n h a u led a n d the ma s s e s
a ls o C a n d 0
betwee n M a n d G a l s o L a n d 2 ca n b et ter be borr owed th a n
h a u led ( a lwa y s pro vided th a t t here a r e su it a ble pl a ce s a t h a n d
for borrowi n g a n d w a s ti n g )
Next prod u ce NM to R The n u mber o f y a rd s borr owed
will be the s a me whether t a ke n a t RZ
o r a t MG
LZ Th a t
“
h a u l ( prod u ct
a rr a n g e me n t o f work which g i v e s the s ma lle s t
n
o f cu
d
di
t
n
ce
h
a
u
led
i
s
the
be
s
t
a
rr
n
e
m
e
t The
a
s
s
a
x
y
g
)
“
h a u l i n on e c a s e i s me a s u red by G LRYG a n d i n the other
by MGU UYR If MGU i s le s s th a n GLRU the n it i s che a per
to borrow ( a ) RZ
r a ther t h a n ( 6) MO + LZ The mo s t e co
n o mi ca l po s i ti o n for the li n e i s whe n Q J
F o r a n y ch a n g e
J P
fro m thi s po s ition will s how a n i n cre a s e of n et a re a repre s e n ti n g
ha u l
I n a s i mil a r w a y NT a n d 8 0 ca n be more e co n o mic a lly w a s ted
t h a n NA a n d CO
H ere ma ke SV VT
306
I n t h e former
.
.
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,
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,
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,
,
,
,
,
,
.
.
,
.
'
.
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.
°
,
.
,
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.
.
The c a s e i s ofte n n ot a s s i mple a s th a t here g iv e n
Ve r y ofte n th e ma teri a l borrowed or w a s ted h a s to be h a u led
”
beyo n d th e li mi t of free h a u l
The li mit beyo n d which it i s
u n profit a ble to h a u l will v a ry a ccordi n g t o the le n gth o f h a u l
o n the borrowed o r w a s ted ma teri a l
the li mit will i n ge n era l
be i n cre a s ed by th e le n g th of h a u l on the borrowed or wa s ted
ma ter i a l The h a u l on wa s ted or borrowed ma teri a l a s NT
ma y be s how n g ra phic a lly by NTXW where NW TX s how s
”
“
the le n g th of h a u l a n d NTXW the h a u l ( ma s s di s t a n ce )
The ma s s di a gra m ca n be u s ed al s o for fi n di n g the li mit of
”
“
free h a u l on the profile a n d va rio u s a pplic a tio n s w ill s u g
g e s t the ms el ve s to tho s e who beco me f a mili a r with i t s u s e a n d
t h e pri n ciple s of i ts co n s tr u ctio n
C erta i n ly on e of i ts mo s t
”
”
“
n
i
s
i n a llowi n g
i mpor ta t u s e s
h a u l a n d borrow a n d w a s te
to be s tu died by a di a gr a m g i vi n g a co mprehe n s i ve v iew of the
whole s i tu a tio n There a r e few if a n y other a va il a ble method s
of a c c o mpl i s hi n g t hi s res u lt
30 7
.
.
.
,
,
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,
,
,
,
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,
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.
.
R a i lr oa d Cu r ve s
2 14
and
E a r thw ork
W he n ma teri a l i s firs t t a ke n
.
exc a va tio n i t
g e n er a lly occ u pie s more s p a ce th a n w a s ori g i n a lly the c a s e
When pl a ced i n emb a n kme n t it co mmo n ly s hri n ks s o mewh a t
Where ver
a n d e v e n t u a lly occ u pie s le ss s p a ce th a n ori g i n a lly
fro m a n y c a u s e the ma teri a l p u t i n to e mb a n kme n t will occu py
more s p a ce or le s s s p a ce th a n it did i n exc a v a tio n the qu a n titie s
i n e mb a n kme n t s ho u ld be corrected before fig u ri n g h a u l o r con
s tru ct i ng a M a s s D i a g r a m a n d a col u mn s ho u ld b e s hown for
thi s a s i s do n e i n T a ble p 2 05
M a n y e n g i n eers write their co n tr a ct s a n d s pecific a tio n s
309
”
“
”
“
witho u t a cl a u s e a llowi n g p a y me n t for h a u l or o verh a u l
Ne v erthele s s it a ppe a r s th a t it i s the more co mmo n pr a ctice to
i n s ert a cl a u s e pro vidi n g for p a y me n t for o verh a u l A ca n
va ss o n thi s s u bject by the A meric a n R a il wa y E n g i n eeri n g a n d
M a i n te n a n ce of W a y A s s oci a tio n i n 19 0 5 s howed t hi s pr a ctice
to pre va il i n the proportio n of 7 3 to 3 7 The free h a u l l i mit of
500 ft s ee med to meet with g re a ter f a vor th a n a n y other
”
Where a n o verh a u l cl a u s e i s i n s erted i n a co n tra ct the
b a s i s of p a y me n t h a s va ried o n di ffere n t ra ilro a d s I n o n e
method n o t reco mme n ded the tot a l h a u l i s to be co mp u ted ;
fro m thi s s h a ll be ded u cted for free h a u l the tota l “ ya rd a ge ”
mu ltiplied by the le n gth of the free h a u l li mit Un der thi s s y s
t e m wit h a 5 00 i t free h a u l li mi t there mig h t be
cu y d s
of e a rth h a u led ( a ll of it ) more th a n 50 0 ft or a n a ver a g e of
6 00 ft ; yet if there were a n other
on
yd s h a u led a n
there wo u ld be n o p a y me n t Wh a te ver for
a v er a g e of 3 00 ft
o verh a u l ; th e a ver a g e h a u l wo u ld be le s s th a n 500 ft Un
l e s s the s pecific a tio n s cle a rly s how th at thi s method i s to be
u s ed it i s u n f a ir a s well a s u n s a ti s f a ctory to the co n t r a ctor
Wh a t s ee ms a lo g ic a l a n d s a ti s fa ctory pro v i s io n i s th a t
reco mme n ded by the A meric a n R a il wa y E n g i n eeri n g a n d
M a i n te n a n ce of Wa y A s s oci a tio n by a let ter b a llo t v ote of 13 4
to 2 3 ( a n n o u n ced i n M a rch
Thi s i s a s follows
No p a y me n t will be ma de for h a u li n g ma teri a l whe n the
le n g th of h a u l doe s n ot exceed the li mit of free h a u l which
feet
s h a ll be
“
The li mits of free h a u l s h a ll be deter m
i n ed by fixi n g on the
profile two poi n ts o n e on e a ch s ide of the n e u tra l g ra de poi n t
on e i n exc a va tio n a n d the other i n e mb a n k me n t s u ch th a t the
di s ta n ce betwee n the m eq u a l s the s pecified free h a u l li mit a n d
3 08
.
in
ou t
,
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,
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,
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,
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,
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,
,
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’
,
,
,
,
R a i lr oa d Cu rve s a n d E a r th work
2 16
.
The di a g ra m o n the p a g e oppo s ite s hows a s ketch of a
profile a n d the corres pon di n g ma s s di a g r a m illu s tr a ti n g fu rther
the method of s tu dyi n g q u es tio ns of h a u l borrow a n d wa s te
F o r thi s p u rpo s e it i s a ss u med th a t the li mit of eco n o mic a l h a u l
i s 100 0 ft a n d the li n e s on the ma ss di a g r a m a r e a dj u s t ed a o
311
.
.
,
,
,
.
cor d i n gl y
.
.
( a ) L i n e AB
1 000 f t
.
g o n o lower be
a n d ca n
c a e the limit
us
of 100 0 ft wou ld be exceeded n or hi gher bec a u s e the w a s te
n e a r A a n d the borrow n e a r 8 wo u l d be i n cre a s ed
( b) L i n e C DE i s pl a ced so th a t C D D E ; th e s u m o f the t wo
borrow s ( be t wee n 8 a n d C a n d betwee n E a n d F) i s the s a me
for a n y pr a ctic a l po s it i o n of C DE ; the s u m of the two a re a s
DE
C RD a n d OS E i s a mi n i mu m whe n C D
L
n e FG
0
i
1
00
ft a n d if hi gher will exceed 1 00 0 ft
0
)
(
a n d if lower will i n cre a s e borrow n e a r F a n d w a s te n e a r G
( d ) If the li n e HM i s lowered the borrow n e a r M a n d the
wa s te n e a r H a re decre a s ed b u t the h a u l i s i n cre a s ed by tra pe
zo i d a l a re a s o f which HI J K a n d LM a r e their s ma ller ba s e s
while it i s decre a s ed by tr a pe z oi da l a re a s o f which IJ a n d KL
The n e t re su lt i s the eq u i va len t of i n
a r e their l a r g er b a s e s
cre a s i n g the h a u l by a tra pe z oid a l a re a which h a s a n u pper
b a s e of 1000 ft a n d a lower b a s e gre a ter th a n 1000 ft s o th a t
If the lin e i s r a i s ed
t h e li mit of eco n o mic a l h a u l i s exceeded
by s i mil a r re a s o n i n g the co s t of the a dditio n a l borrow a n d w a s te
will be g re a ter th a n the s a vi n g i n the h a u l ite m
1
000
ft
L
n
i
a
O
i
e
O
s
pl
a
ced
o
th
t
N
O
P
s
N
P
P
6
Q
Q
( )
A ch a n g e u p or down will i n cre a s e the co s t
n ti n u ed beyo n d s t a tio n 80 it i s q u ite
If
the
profile
were
co
(f)
po s s ible th a t the ma te ri a l i n dic a ted a s w a s te co u ld be u tili zed i n
fill or p a rt of it s o u tili z ed
m
u n t of cu t
m
a
a
the
profile
i
how
there
i
s
a
s
ll
o
s
s
n
s
A
(9 )
c a rried i n to fill clo s e to s t a tio n 80
u p to the pro
The
proj
ectio
n
of
the
po
i
n
t
B
0
etc
s
s
C
h
A
)
(
file s er ve to s how where ma teri a l s ho u ld be w a s ted where bor
rowed ; wh a t ma teri a l s ho u ld be c a rried for wa rd wh a t b a ckwa rd
The s tu dy of the ma ss di a gr a m h a s s hown th a t the a rr a n g e me n t
a dop t ed i s the mo s t eco n o mic a l
The exa ct s t a tio n s of the poi n ts A B C 0 etc ca n be deter
mi n ed a cc u r a tely fro m the cro s s s ectio n n ote s a n d the v ol u me s
of e a rt hwork a lre a dy co mp u ted if thi s s ho u ld s ee m de s ira bl e
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Ma s s Di a g r a m
2 17
.
Wa s t e
Bo r row
N OW
N OW
R a i lr oa d Cu r ves
2 18
a nd
E a r th wo r k
.
Three c a s e s of a dj u s tme n t of li n e s on the ma ss di a g r a m
t o s ec u r e eco n o my de s er ve e s peci a l a tte n tio n
I n n o n e of the s e
c a s e s s ho u ld either of the s i n g le li n e s be g re a ter t h a n the l imit
of eco n o mic a l h a u l
3 12
.
,
,
.
.
In F i g
dj us t li n e s s o th a t AB BC a n d D E EF a s pre
3 06
If either li n e ABC or DEF be either r a is ed
vi ou s l y n oted i n
”
“
or lowered the a re a s me a s u ri n g h a u l will s how a n et i n
cre as e I n thi s c a s e of F i g 1 ma teri a l mu s t be w a s ted a t SA
A c h a n g e i n the po s itio n of ABC a n d DEF e ve n to
C D a n d FT
a po s itio n where they for m o n e s tr a i g ht li n e
will n ot i n cre a s e
o r decre a s e the to t a l a mo u n t of w a s te
.
l, a
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FIG 2
.
.
I n F i g 2 , the proper a dj u s tme n t i s s how n by the li n e G HKLM
.
where GH
KL
BC a n d D E
HK
I f the li n e s a r e a dj u s ted s o th a t AB
LM
.
EF, the n it s ho u ld be n oted th a t the ma s s i n di
by DC will be i n v olv ed twice both i n co n n ectio n wi th BC
Thi s s che me the n ca n be c a rried ou t o n ly by
a n d with D E
s peci a lly borrowi n g a ma ss of e a rthwork of the s i z e i n dic a ted by
”
“
h a u l is
DC
U n der s u ch a n a rr a n g e me n t the s a v i n g i n
me a s u red by o n ly the di ffere n ce betwee n the a re a GABH a n d
HBC K a n d b et w e en DKLE a n d ELMF a n d e v ide n tly i s s ma ll
co mp a red with the co st of th e extra borrow a t DC
ca t ed
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Dia g ra m
fO i
T
H RE E LE V E L S E CTIO N S
B a s e 14
Slo pe i%
C e n te r H e i g h t s o n
t c
Su m o f Di s a n
to 1
O bl q Li
i
u e
n es
es Ou t on Ve r t i
c l Li
a
n es
Q u a n t i t i es o n Ho r i zo n t a l Li n e s i n
c bi c y d f
u
ar
s
or
50 ft . o f
L g th
en
Di a g a m
r
fo r
T HREE LEVE L S ECT i O NS
B a s e 20
Slo pe l
.
Ce n t e r H e i g h ts on
t
O
/to 1
'
bl i q u e
z
Li
n es
S u m o f Di s a n ce s o u t o n Ve r t i
Q u a n t i t i e s on H or i z o n t a l
y a r d s f o r 50 ft
.
of
L g th
en
Li
c l Li
a
n es
n e s i n cu bi
c
la
g
ra
m
fo r
PRIS MO ID AL C O RRE CT IO
Di ff er e n ce s b e t w e e n S u m o f Di st a n
ou t o n
f
Ve r t i
Di f e r e n
bl i q u e
O
c l Li
a
c b tw
Li
es
e
n es
e en
es
C e n t er He i g h t s on
n es
Q u a n ti t i e s o n H o r i z on ta l
y d s f o r 100 f t
.
c
N
.
of
L g
en
th
Li
n es i n en .