1) The document discusses spiral curves which are used to join circular curves to tangent sections of roads or railways. It defines various parameters used to design spiral curves like spiral length, spiral angle, tangent distance, and radius of curvature. 2) Equations are provided to calculate the design parameters of spiral curves like spiral angle, x-coordinate, y-coordinate, and tangent lengths based on the spiral length and radius of curvature. 3) Steps for laying out a spiral curve are outlined which involve selecting the spiral length and computing design parameters, then staking out the curve similar to a circular curve. Approximate solutions for spiral curves using basic assumptions are also presented.

1. 20/01/2013 1

2. 20/01/2013 2

3. Figure in slide 4, equal spiral curves join a circular curve
to the main tangents;
• T.S. tangent to spiral point.
• LS the length of the spiral curve.
• S.C. spiral to circular curve point.
• C.S. curve to spiral point.
• S.T. spiral to tangent point.
• TS tangent distance from the T.S. or the S.T to the P.I.
• R the radius of the circular curve.
20/01/2013 3

4. P.I.

TS
TS
Circular curve
T.S. S.T.
C.S.
S.C.
LS LS
R
R
1st spiral curve 2nd spiral
curve
20/01/2013 4

5. S.P.I. S.C.
Circular curve
T.S.
R
Enlargement of Spiral Curve
20/01/2013 5

6. Figure in slide 5, the spiral curve on the left side redrawn
to a much larger scale.
• S.P.I. the spiral point of intersection.
• S the spiral angle.
• S.T. and L.T. are the short and long tangents of
the spiral curve.
• c the degrees of curvature of the circular curve.
• XS the distance measured from the T.S. along the
main tangent to a point where a perpendicular line
to the tangent hits the S.C.
• YS the distance measured perpendicular from the
XS coordinate to the S.C.
• L.C. the long chord from the T.S. to the S.C.
20/01/2013 6

7. • P.C. the point at which the circular curve becomes
parallel to the spiral. The curves will be a distance p
apart.
• S the deflection angle from the T.S. to the S.C.
• CS the correction factor, negligible when   15.
20/01/2013 7

8. Spiral Curve Equations
5729.58
R

LS  C
S 
200
 S 2 S 4 
X S  L S 1 
  
 10 216 

 S S3 
YS  LS 
 3  42 

 
L.C.  X S  YS
2 2
YS
S .T . 
sin  S
L.T .  X S  S .T . cos  S
S
S 
3
p  YS  R (1  cos  S )

TS  X S  R sin  S  ( R  p ) tan
2
20/01/2013 8

9. Steps of Laying Out A Spiral Curve
• LS is selected considering: traffic design, speed, No. of
lanes, c and the length needed for super-elevation.
• The values for R, S, XS, YS, L.C., S.T., L.T., S, p and TS
are computed.
• The chord lengths are assumed and the deflection angle
L
is determined from   ( L ) 2
S
S
• The curve will be staked out in identical manner used for
circular curves.
20/01/2013 9

10. Approximate Solution for Spiral Problems
LS
S.C.
(2/3) S
T.S.
S = (1/3) S
Basic Assumption: LS  Long Chord, LS  L.C.
20/01/2013 10

11. Approximate Equations
Y  LS sin  S
X  LS  Y 2
2
1
q X
2
1
p Y
4
20/01/2013 11

12. Approximate Equations
Using the sine law, we obtain the following:
L.T . LS

2 sin (180   S )
sin (  S )
3
2 LS
L.T .  sin (  S ) 
3 sin  S
Using the sine law, we obtain:
S .T . LS

S sin  S
sin
3
S LS
S .T .  sin 
3 sin  S
20/01/2013 12

13. An Example of Compared Values
Referring to previous solved problem:
Precise Methods Approximate Methods
Parameters (ft) Parameters (ft)
Y 10.46 Y 10.47
X 299.67 X 299.82
q 149.94 q 149.91
p 2.61 p 2.62
L.T. 200.15 L.T. 200.20
S.T. 100.10 S.T. 100.16
20/01/2013 13