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Spiral Curves

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Spiral Curves

  1. 1. 20/01/2013 1
  2. 2. 20/01/2013 2
  3. 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. 4. P.I.  TS TS Circular curve T.S. S.T. C.S. S.C. LS LS R R 1st spiral curve 2nd spiral curve20/01/2013 4
  5. 5. S.P.I. S.C. Circular curve T.S. REnlargement of Spiral Curve 20/01/2013 5
  6. 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. 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. 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 220/01/2013 8
  9. 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. 10. Approximate Solution for Spiral Problems LS S.C. (2/3) S T.S. S = (1/3) SBasic Assumption: LS  Long Chord, LS  L.C.20/01/2013 10
  11. 11. Approximate Equations Y  LS sin  S X  LS  Y 2 2 1 q X 2 1 p Y 420/01/2013 11
  12. 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  S20/01/2013 12
  13. 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. 100.1620/01/2013 13

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