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Formulae to find out radius of
a given curve or curve section
By
Jayesh [email protected]
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8/4/2019 Formulae to Find Out Radius of a Given Curve or Curve Section
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Formulae to find out radius of a given curve or curvesection
Formulae to find out radius of a given curve or
curve section
By
Jayesh Jaiswal
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8/4/2019 Formulae to Find Out Radius of a Given Curve or Curve Section
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Formulae to find out radius of a given curve or curvesection
Figure 1
Here, r = radius of curve
c = chord passing through curve ( 0 < c 2r )p = perpendicular distance between curve and chord c ( 0 < p r )
k = angle inscribed inside the curve ( 90 k 180 )
[A] k in terms of c and r
From graphical observations:
For 0 c r or c (0, r]
k = 180 - [(c/r)30] -------------------------------------------------(1)
For r c 7r/4 or c [r, 7r/4]
k = 180 - [(c/r) (30 + 10)] + 10
k = 190 - [(c/r)40] -------------------------------------------------(2)
For 7r/4 c 2r or c [7r/4, 2r]
k = 180 - [(c/r) (30 + 90)] + 150k = 330 - [(c/r)120] -----------------------------------------------(3)
Considering figure (2)
k1 = 180 - ()
Therefore, k1 = 180 - k2 --------------------------------------------------(4)
Hence,
for c (0, r]
() = (c/r)30 ------------------------------------------------------(5)
for c [r, 7r/4]
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Formulae to find out radius of a given curve or curvesection
() = [(c/r)40] - 10 ---------------------------------------------(6)
for c [7r/4, 2r]
() = [(c/r)120] - 150 ------------------------------------------(7)
If,
For 0 c r or c (0, r]k2 = [(c/r)30] or [c(30) r(0)]/r ------------------------------(8)
For r c 7r/4 or c [r, 7r/4]k2 = [c(40) r(10)]/r --------------------------------------------(9)
For 7r/4 c 2r or c [7r/4, 2r]k2 = [c(120) r(150)]/r ----------------------------------------(10)
Deviation degree [of (2), (3) from (1)]
For 0 c r or c (0, r]
= 0 ----------------------------------------------------------------(11)
For r c 7r/4 or c [r, 7r/4]
= 180 - [(c/r)30] - 190 + [(c/r)40]
therefore, = 10[(c/r) 1] --------------------------------------(12)
For 7r/4 c 2r or c [7r/4, 2r]
= 180 - [(c/r)30] - 330 + [(c/r)120]therefore, = [(c/r)90] 150therefore, = 30[(3c/r) 5] ------------------------------------(13)
Deviation degree [of (3) from (2)]
' = 190 - [(c/r)40] - 330 + [(c/r)120]
therefore, = [(c/r)80] 140therefore, = 20[(4c/r) 7] --------------------------------------------(14)
[B] k in terms of p and r
for p (0, r/8]
k = 180 - [(p/r)240] ---------------------------------------------(15)
for p [r/8, r/2]
k = 160 - [(p/r)80] ----------------------------------------------(16)
for p [r/2, r]
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Formulae to find out radius of a given curve or curvesection
k = 150 - [(p/r)60] ----------------------------------------------(17)
Deviation degree [of (16), (17) from (15)]
For 0 p r/8 or p (0, r/8] = 0 ---------------------------------------------------------------(18)
For r/8 p r/2 or p [r/8, r/2]
= 160 - [(p/r)80] - 180 + [(p/r)240]
therefore, = [(p/r)160] 20therefore, = 20[(8p/r) 1] -----------------------------------(19)
For r/2 p r or p [r/2, r]
= 150 - [(p/r)60] - 180 + [(p/r)240]therefore, = [(p/r)180] 30
therefore, = 30[(6p/r) 1] -----------------------------------(20)
Deviation degree [of (17) from (16)]
' = 150 - [(p/r)60] - 160 + [(p/r)80]therefore, = [(p/r)20] 10
therefore, = 10[(2p/r) 1] -------------------------------------------(21)
Correlating c and p:
For 0 c r or c (0, r] & for p (0, r/8]180 - [(p/r)240] = 180 - [(c/r)30]
therefore [(p/r)240] [(c/r)30] = 0
therefore (8p/r) (c/r) = 0
c/p = 8 -------------------------------------------------------------(22)
For r c 7r/4 or c [r, 7r/4] & for p [r/8, r/2]
160 - [(p/r)80] = 190 - [(c/r)40]therefore, [(c/r)40] [(p/r)80] = 30
therefore, (c/r) (2p/r) =
therefore, c-2p = 3r/4
therefore, c = (3r+8p)/4 -----------------------------------------(23)
For 7r/4 c 2r or c [7r/4, 2r] & for p [r/2, r]150 - [(p/r)60] = 330 - [(c/r)120]
therefore, [(c/r)120] [(p/r)60] = 180
therefore, (2c/r) (p/r) = 3
therefore, 2c = 3r + ptherefore, c = (3r+p)/2 ------------------------------------------(24)
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Formulae to find out radius of a given curve or curvesection
[D] Clarifying (22) and Finding r:
For 7r/4 c 2r or c [7r/4, 2r] & for p [r/2, r]r = (2c p)/3 -----------------------------------------------------(25)
For r c 7r/4 or c [r, 7r/4] & for p [r/8, r/2]r = (4c 8p)/3 ---------------------------------------------------(26)
For 0 c r or c (0, r] & for p (0, r/8]Assuming, r = (xc yp)/3 -------------------------------------(27)
Putting c = r and p = r/8
y 8x + 24 = 0 --------------------------------------(27a)
Putting c = 3r/4 and p = r/16
y 12x + 48 = 0 -------------------------------------(27b)Multiplying (27b) by (-1)
-y + 12x 48 = 0 ------------------------------------(27c)Now, adding (27b) and (27c)
4x 24 = 0
therefore, x = 6 ---------------------------------------(27d)Putting (27c) in (27a)
y = 24 -------------------------------------------------(27e)
Putting (27d) and (27e) in (27)r = (6c 24p)/3
therefore, r = 2(c-4p), for c (0, r] and p (0, r/8]-------(28)
For c = 0 and p = 0, value of r is indefinite.
Here, c/p [2, )
Hence one can always select c and p for a given curve or curve section such thatc/p 2
Some Interesting Observations
Suppose,
r = radius of curve,
c = chord passing through curvep = perpendicular distance between curve and curve section
k = angle inscribed inside the curved = r p
-r d r0 k 180
0 p 2r
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Formulae to find out radius of a given curve or curvesection
0 c 2r
d = 0 p = r k = 90
d = r p = 0 k = 180
d = -r p = 2r k = 0
If p increases by r k decreases by 90If p decreases by r k increases by 90
If d increases by r k increases by 90If d decreases by r k decreases by 90
If p increases by r/p k decreases by 90/p
If p decreases by r/p k increases by 90/p
p = r k = 90p = (r + r) k = 0p = (r r) k = 180
c = 2p k = 90c = (0)p k = 180
c = 3p k = 120
c = 6p k = 150
c = (0)r k = 180
c = r/2 k = 165
c = r k = 150
k = 180- (c/r)30, [r c 2r]
Deviation ,
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c
r 0
5r/4 2.5
3r/2 5
7r/4 7.5
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Formulae to find out radius of a given curve or curvesection
Double Circle Concept
Figure 2
Some Interesting Graphs
C K P K
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Formulae to find out radius of a given curve or curvesection
C deviation P C
Some Interesting Data
K1 C P
0 0 2r 360
30 r 15r/8 300
60 7r/4 3r/2 240
90 2r r 180
120 7r/4 r/2 120
150 r r/8 60
180 0 0 0
210 r r/8
240 7r/4 r/2
270 2r r
300 7r/4 3r/2
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Figure 3
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Formulae to find out radius of a given curve or curvesection
330 r 15r/8
360 0 2r
Figure 4
C 0 r 7r/4 2r
K P Q P Q P Q P Q
0 2r 0
30 37r/20 2r 59r/20 7r/2 3r 4r
60 15r/20 9r/10 41r/40 63r/40 4r/5 9r/5
90 7r/20 r/2 13r/40 7r/8 0 r
12
0
3r/20 3r/10 r/40 21r/40 2r/5 3r/5
15
0
0 3r/20 23r/80 21r/80 7r/10 3r/10
18
0
0 0 3r/20 0 11r/20 0 r 0
21
0
3r/10 3r/20 13r/16 21r/80 13r/10 3r/10
24
0
9r/20 3r/10 43r/40 21r/40 8r/5 3r/5
27
0
13r/20 r/2 57r/40 7r/8 2r r
30
0
21r/20 9r/10 17r/8 63r/40 14r/5 9r/5
33
0
43r/20 2r 81r/20 7r/2 5r 4r
360
2r 0
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