Horizontal curves

SIGHT DISTANCE &SPIRAL and COMPOUND CURVES Prepared by: Dr. Osama Ibrahim

Sight Distance on Horizontal Curves • Sight distance can also be a problem on horizontal curves (buildings, embankments, tree growth, etc.) • The line of sight is a chord of the curve. The sight distance should be measured along the centerline of the inside lane of the curve (not the centerline of the roadway)

Sight Distance on Horizontal Curves

Elements of Design: Sight Distance Horizontal Alignment

Stopping Sight Distance (SSD) refers to both horizontal and vertical views.

HSO= R[(1- cos 28.65S)] R Where, HSO= Horizontal sightline offset (ft) R= Radius of Curve (ft) S= Stopping Sight Distance (ft)

Sight Distance Example A horizontal curve with R = 800 ft is part of a 2-lane highway with a posted speed limit of 35 mph. What is the minimum distance that a large billboard can be placed from the centerline of the inside lane of the curve without reducing required SSD? Assume p/r =2.5 and a = 11.2 ft/sec2 SSD = 1.47vt + _____v2__ _ 30(__a___  G) 32.2 5

Sight Distance Example SSD = 1.47(35 mph)(2.5 sec) + _____(35 mph)2____ = 246 feet 30(__11.2___  0) 32.2

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Sight Distance Example m = R(1 – cos [28.65 S]) R m = 800 (1 – cos [28.65 {246}]) = 9.43 feet 800

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Problem No.2

Compound curves R1 and R2 are usually known.

  1   2  t1  R1 tan 1 2 2 t2  R2 tan 2 VG VH t1  t2 t t    1 2 sin  2 sin 1 sin( 180  ) sin  T1  V G  t1 T2  V H  t2

R1 L1  180

R2 L2  180

Reverse curves Reverse curves are seldom recommended. They are absolutely NOT recommended for high-speed roads. For high-speed roads, we must provide a tangent section that will allow full development of super elevation at both ends.

Reverse curves Reverse curves usually consist of two simple curves with equal radii turning in opposite directions with a common tangent.

  1   2

 d tan    2 D angleOWX  1  2 2 d  R  R cos 1  R  R cos  2 1  2  2R(1  cos ) angleOYZ   2

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Hence,

d R 2(1  cos ) d cos   1  2R  D  d * cot 2

Suggested Steps in Horizontal Design 1. Select tangents, PIs, and general curves making sure you meet minimum radius criteria 2. Select specific curve radii/spiral and calculate important points using formula or table (those needed for design, plans, and lab requirements) 3. Station alignment (as curves are encountered) 4. Determine super and runoff for curves 5. Add information to plans

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Spiral Curves {This topic will not be covered} • Used to provide gradual transition in horizontal curvature, and hence superelevation.

• Definitions: – – – – – –

Back and forward tangents. Entrance and exit spirals. Geometrically identical. TS, SC, CS, ST. What is in between? SPI: the angle beteen the tangents at TS and SC. Spiral Angle S: the angle between the two tangents. Spiral Length LS: the arc length of the spiral.

Spiral Geometry

• Basic spiral properties:

– Radius changes uniformly from infinity at TS to the radius of the circular curve at the SC. So, it’s degree of curve DS changes uniformly from 0o to D at the SC. • Average degree of curve is D/2. • In circular curves, L = (I/D) 100 ft, or I = LD stations • similarly, S= Ls (D/2) S and D in deg, L in stations

– Spiral angles at any point is proportional to the square of the distance Lp from TS to the point. P = (LLP )2 S S

• In Fig 25-15, M is the mid point of the spiral, Lp = Ls/2 but M is not = (S /2).Since D changes uniformly, degree of the curve = D/2 at M. But D changes uniformly, so the average degree of curvature between TS and M is (D/2)/2 = D/4 • Then, M = ( Ls/2) (D/4) = (Ls D/8) = S /4