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In highway or railroad construction, the curves most generally used presently' are circular curves although parabolic and other curves are sometimes used. These types of curves are classified as Simple, Compound, Reversed or Spiral curves.
A simple curve is a circular are, extending from one tangent to the next. The point where the curve leaves the first tangent is called the "point of curvature" (P.C.) and the point where the curve joins the second tangent is called the "point of tangency" (P.T.). The P.C. and P. T. are often called the tangent points. If the tangent be produced, they will meet in a point of intersection called the "vertex". The distance from the vertex to the P.C. or P.T. is called the "tangent distance". The distance from the vertex to the curve is called the "external distance" (measured towards the center of curvature). While the line joining the middle of the curve and the middle of the chord line joining the P.C. and P.T. is called the "middle ordinate".
Geometry of the Circular Curves:
In the study of curves, the following geometric principles should be emphasized:
LACB=~ ~AOB
@
.• B
D e C
l'
B~AC
D F
SIMPLE CURVES
Sharpness of the curve is expressed in any of the three ways:
1. Degree of Curve: (Arc Basis) Degree of curve is the angle at the center subtended by an arc of 20 m. is the Metric system or 100 ft. in the English system. This is the method generally used in Highway practice.
By ratio and proportion:
2nR
D
b. English System:
100 2nR D= 360 D = 360(100) 2nR D _ 1145.916(5)
2. Degree of Curve:. (Chord Basis) Degree of curve is the angle subtended by a chord of 20 meters in Metric System or 100 ft. in English System.
b. English System:
In-=- 2 R
rn-
P. C. = poi nt of curvature P.T. = point oftangency
R = radius of the curve
M = middle ordinate Lc = length of curve C = long chord C1 and C2 =sl,lb-chord
. d 1 and d 2 = sub-angle
® Distance from mid point of CUNe to the mid point of long chord:
M= R (1 - Cos ~) M= 190.99 (1- Cos 18') M=9.35m.
I ,
, I ,
:R I
'"
, (^) "', ,I / " (^) '.. 1 I II I , (^) "';-1118",: / X"',r/ ex 111 ) • .....-:.,":~.1,~, .A »'t$ '<:."I
ill Distance from mid point of CUNe to P.I.: R- 1145.
E=R(Sec~ -1) E = 190.99 (Sec 18' -1)
R R
Sjn~=~·
Sin Q.=~ 2 2R
In- 2
. d C, Sin~ Sin~=--
d C 1 Sin~. Sin ~ 2 =--- 20 (Metric I' (^) Il
d C 1 Sin~ Sin T= 100 (English)
20Sin~ C 1 =--- (Metric) S
10-
100 Sin~
7. Sub-chords: (Chord b, ;is)
@ Stationing of B: S=R
® Stationing of D: S=R
180 S= 180 m.
.
.^ \ ,
"'-_ \ \ IR
R"<:i .. 30" , ffl
@ Distance DE:
OE=354.11 m.
DE=67.63m.
CD External distance:
·.rdTJtJI¥~.I~II-I~ii~fti~nt~:··
·.(1).••••. C9IllP~f~ ••• t~~E!xlE?rn~I.·9fsl<trce •• Qf.the 9J1'Y~.> •
•••••••• ,.~~~~,i~~~~&®.~f··ot6' •• frQrn
w.
j""- ' R <Y-,_ "-_ "- %n '-'''~- \
_A . 2}i1:~o PoC R 20+130.
CD Long chord:
R= 1145.
!:= RSin 25'
L = 2 (286.48) Sin 25' L = 242.14m.
SIMPlE CURIES
a=90·24'4O'
CD=34.80m,
@ Stationing of D:
R=229.18m.
n 229.
LC1 =Re
L
LCl =98.68 m.
® Distance CD:
.
'', ' / '\ \1 ,I' R' 'I ', , /
o
\ ' , I ', ' / ~'\ \ \ /
'\ '\ ' '\ / '\V' o
Rxn
L~ =148.24 m.
Station ofD =(2 +040) +(148.24)
@ Length of curve from PC. to A: S=R
® Length of long chord:
\ I ,I \ ' / , I ' _R_ ' / ! / \34" /
\V
..a4..n1..·Jfthe;d;@l(L~·frO#lilheiP·C;itijQOllffie •• .~~~H~.ZOOfu{>··· ....
.1•••• ·E~lr"..I·lrt!I~·.I~I·· ®•• lqh~~~9J~9fli'ltMli9fm~@W4~ ..·.•.•••••• ·.·§4·f@miwt~m~'laogthml§ng~@ff9@.eP.ri>p:tt)·· .. ... ....
CD Radius of CUNe:
I,
!R I,
! / \201 /
\V
lan8=-
R· Cos2T39'=-
R= 560.13m.
@ Area bounded by the tangents and outside the central cUrve: T=Rtan24' T= 286.48 tan 24' T= 127. A 1R (2) 'It R2 I ",.... - 2 - 360 A_ _ 127.55 (286.48)(2) 'It (286.48)2(48) ",...a - 2 - 360
® External distance:
E=R(sec~-1) E= 336.49 (sec 25' -1) E=34.79m. @ Length of long chord:
~=RSin25'
e=T3T a= 90' -12'- a=70'23'
A
<D Radius of CUNe:
Sin 50' =12~
T= 156.91 m. T=Rtan 25' 156.91 =Rtan25' R=336.49m. D= 1145.
D= 3'24'
, / " ''., ' /~
Ii'. 25' \ 25' /
<D Degree of curve:
OB = R+ R(sec~.1)
OB=R+R(SeC~.1)
In /1 OPB
5iii'"'8 =Sin 70'23' LJ = 105'39' 0=180-a.-1J
R= 286.36m..
® Length of chord:
o
@ Area bounded by the CUNe and the tangent lines:
A =. RT(2L 1t ~ (24') 2 360' T= Rtan 12'
A = 256.26m
11lSillitillll; .!lPOl'dill~t*19tg91OQN~IJ~.2<l1l)Q·pV'1W~polllt ·~ri~I~~~~.h~ ••~9()r~i#t~~ .• §f.~Q~.~~ •• ~••
q)FjfJpth~di$tal1¥ofline~P') ~••• ®IY~J°rt'md~r~~Rfs.b!lple@f'I~tb~t
AM~P·>
CD Distance of line BD:
A
Sin 4 '01' = 2 (2~.36) x= 40.12 m.
@ Chord distance:
C=2Rsin l
C=2(311) Sin 2" C= 139.92m.
@ Length of CUNe: 1t Lc =RI 180 1t
Lc = 141.13 m.
.li~'I~~I!~~~~~':~r:~i
Willilli~tM9¢l\tfbffi@IHfjElgp;<
CD Distance from mid point of CUNe to P.I.
- D
R
E= R(sec 112· 1)
E=29,55m.
® Distance of mid point of CUNe to mid point of long chord:
M=25.59m,
® Stationing of B:
S=Re
t~~ •• tans~~ltrr(ll~ep.g .• h~$.a(jlte¢ll«@ll~
b~rlng.()f.N,50· .•~, ....• lt.nll,$a.ra,dl~~pf2QOffi; •• lJ~in$ •• ·.·,;jrc ••• b<!llis.·•••..·.$tliltiQilitJ9 •••• pl••• P.q, ••• i§.
SIMPlE CURVES
(j) Tangent distance:
(j) Middle ordinate: T= Rtan 25'
T=93.26m.
® Long chord:
'Sin 25'=1:... 2R
L = 169,05m.
@ Stationing of B:
T=o 210 20
M= R (1- Cos 112)
. R= 11~.916 =286.
M = 286.48 (1- Cos 21') M= 19.03m.
1
\
R\ *" '* I R-60" \ ,I
28.0~
10+
® Degree of CUNe:
@ Stationing of 8: S=RS
Sta. of8= 10+ 155.
Solution: CD Radius of CUNe:
In 2= 95.
~ =31.5' 1=63' @ Tangent distance:
Solution: CD Tangent dsistance:
tane=-
e =14.04' 2e =28.08'
0.11nR= R=509,70m.
® Tangent distance: T=Rtan31' T= 509.70 tan 31' T=306.26m.
@ Stationing of painf x: S=Re
<D Deflection angle at the P.C.:
Cos 2e = 219.
2e= 16.988'
@ Stationing at B: S= R(2e) S= 229.18 (16.988)' 1t 180 S=67.95m.
Sta. ofB= 10+ 188.
Sin 8.49' = ~~ AB= 67.73 m.
.tqt1~A~··)<·······.
CD Central angle of 10' center curve:
OA=Ra SinQ=~ 2 2Ra
S,n-=- 2 Ra Ra= 143.47 m. '. 6' 10 SIn---
R 6 = 191.07 m.
2 RlO R 10 = 114.74 m.
OC=Re- Ra
Using Sine Law: 47.70 76.
S
. !.m. = Sin 136' In 2
ful=25'44'
@ Central angle of 6' end curves:
16 +~+ 136' = 180' 16 + 25'44' + 136' = 180'
1 L
cl- 6'
LOL = 102.93 m.
P. T. = (10 + 185.42) + 60.89 + 102.93 + 60.
mounth of tunn~
I :7OmI II
CD Stationing of the point of deviation:
j I , rt----...-.---,/.-------------....~ .. - .r.
al ,"^ ,/ ~11I2,' / ~ r;,' ,/;=163.
U
~,,', r~ilway in ~y rhe runnel
Sin Q= 10 2 R
Sin 3.5' =~ R=163.80m.
as = 163. 1= 55'04'
tan 55'04' = 70 x x=48.89m. I T tan-=- 2 R T = 163.80 tan 27'32' T=85.39 m.
Sta. of point of deviation (P. C.)
® Stationing of mouth of tunnel:
4= 157.33m.
Sta. of mouth of tunnel = (7 + 677.72) + (157.33) Sta. of mouth of tunnel = 7 + 835,
@ Direction of railway in the tunnel:
(j)yompu~ ••• th~ •••f#ntral.lln~l~of ••lt(~ •••• #~ ... (;O~e.··· ®... 9()~let!)El$diusPfIt¥fl~WWW~·. @·VVhat1s!heStallonio96HMheWf;'P,