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Circular Curves

Circular Curves. TYPES OF CURVES:. Simple Curve. Compound Curve. R. R. R. r. Reverse Curve. spiral. R. R. R. R. Easement or Transitional Curve. spiral. Definitions. Arc Definition. 100 ft. “Degree of Curve” Central angle subtended by a circular ARC of 100 ft (highways). R.

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Circular Curves

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  1. Circular Curves TYPES OF CURVES: Simple Curve Compound Curve R R R r Reverse Curve spiral R R R R Easement or Transitional Curve spiral

  2. Definitions Arc Definition 100 ft “Degree of Curve” Central angle subtended by a circular ARC of 100 ft (highways) R R D D/100’ = 360/ 2p r = full circle angle / circumference So R = 5729.58 / D Chord Definition “Degree of Curve” Central angle subtended by a circular CHORD of 100 ft (railways) 100 ft R R D R = 50 / SinD/2

  3. Standard Route Terminology PI PC L LC PT PI = Point of Intersection PC = Point of Curvature PT = Point of Tangency L = Length of Curve LC = Long Chord

  4. Formulae I LC = Long Chord M = Middle Ordinate E = External Distance T = Tangent Distance I = Intersection Angle T E T L M PC PT LC R T = R Tan I/2 R I/2 L = 100 I0/D0= R I rads I LC = 2 R Sin I/2 R/ (R+E) = Cos I/2 => E = R [(1/Cos (I/2)) - 1] (R - M)/R = Cos I/2 => M = R [1 - (Cos (I/2)]

  5. Stationing(usually every 100 feet) PC 4+86.75 4+00.00 0+00.00 1+00.00 2+00.00 3+00.00 5+00.00 T PI L 6+00.00 7+00.00 PC sta = PI sta – T PT sta = PC sta + L PT 7+27.87 8+00.00 9+00.00 10+00.00 11+00.00

  6. Curve Layout Need to stake at “full stations” (XX+00.00) Set up on PC, backsight PI, turn deflection angle (d), measure chord distance (c) PI d PC chord

  7. 100’ 100’ 100’ Sub-chord 100’ Sub-chord PC PT D D d2 D D d1

  8. Vertical Curves “Crest” Curve “Sag” Curve Provides a smooth transition between different grades Parabola - constant rate of change of grade GRADE: Grade = +4.00% + rising grad - falling grade 4.00’ 100’

  9. Vertical Curve Geometry Y V Back tangent (g1) Forward tangent (g2) BVC EVC Xp Yp X L/2 L/2 L = curve length

  10. Constant rate of change of Grade r r = (g2 – g1) / L R should be low (long L) for rider comfort and sight distance Equation of Curve (parabola): Y = YBVC + g1 X + ((g2 – g1)/2L) X2 Units: g in %, L and X in stations, Y in ft/meters Or G in fractions (0.04), L, X, Y in ft/meters

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