LEC16 Horizontal Curves Notes

Horizontal Curve

• Curves are arcs connecting two straight lines (tangents) separated by a deflection angle.
• Straight sections of transportation routes (highways, railroads, pipelines) are connected by curves in horizontal and vertical planes.
• Curves are used when the alignment of a roadway or railway changes direction due to obstacles or conditions (hills, lakes, temples).

Horizontal Curves: Circular Curves

• Horizontal curves connect two straight tangent sections in horizontal planes.
• Two types:
  • Circular arcs
  • Spiral
• A simple curve is a circular arc connecting two tangents; most often used.
• A compound curve consists of two or more circular arcs of different radii tangent to each other, with their centers on the same side of the alignment.
• Broken back curves include a short tangent (less than 100 ft) connecting two circular arcs with centers on the same side.
• A reverse curve consists of two circular arcs tangent to each other, with their centers on opposite sides of the alignment.
• Compound, broken-back, and reverse curves are unsuitable for modern high-speed traffic and should be avoided.
• However, they are sometimes necessary in mountainous terrain to avoid excessive grades or deep cuts and fills.
• Compound curves are often used on exit and entrance ramps of highways and expressways.
• Easement curves are desirable, especially for railroads and rapid transit systems, to lessen the sudden change in curvature at the junction of a tangent and a circular curve.
• A spiral makes an excellent easement curve because its radius decreases uniformly from infinity at the tangent to that of the curve it meets.
• Spirals are used to connect:
  • A tangent with a circular curve
  • A tangent with a tangent (double spiral)
  • A circular curve with a circular curve

Superelevation

• The effect of centrifugal force on a vehicle passing around a curve can be balanced by superelevation, which raises the outer rail of a track or outer edge of a highway pavement.
• Properly superelevated spirals ensure smooth and safe riding with less wear on equipment.
• Spirals are used for railroads and rapid-transit systems because trains are constrained to follow the tracks.
• On highways, spirals are less frequently used because drivers can compensate for abrupt directional changes at circular curves by steering a spiraled path.

Degree of Curve and Chord Length

• The rate of curvature of circular curves can be designated by:
  • Radius (R) (e.g., a 1500-m curve or a 1000-ft curve)
  • Degree of curve (D)
• Two designations for degree of curve:
  • Arc definition: Degree of curve is the central angle subtended by a circular arc of 100 ft (preferred for highway work).
  • Chord definition: Degree of curve is the angle at the center of a circular arc subtended by a chord of 100 ft (convenient for very gentle curves, such as railroads).

Circular Curve Elements

• PI: Point of intersection of the two tangents, also called the vertex V.
• Back tangent precedes the PI; the forward tangent follows it.
• PC: Beginning of the curve or point of curvature, also called BC or tangent to curve TC.
• PT: End of the curve or point of tangency, also called EC or curve to tangent CT.
• R: Curve radius.
• The radii at the PC and PT are perpendicular to the back tangent and forward tangent, respectively.
• T: Tangent distance, the distance from PC to PI and from PI to PT.
• LC: Long chord, the line connecting the PC and PT.
• L: Length of the curve, the distance from PC to PT, measured along the curve for the arc definition, or by 100-ft chords for the chord definition.
• E: External distance, the length from the PI to the curve midpoint on a radial line.
• M: Middle ordinate, the (radial) distance from the midpoint of the long chord to the curve’s midpoint.
• POC: Any point on the curve.
• POT: Any point on the tangent.
• D(a): Degree of any curve (arc definition).
• D(c): Degree of any curve (chord definition).
• I: Intersection angle, the change in direction of two tangents, equal to the central angle subtended by the curve.

Formulas for Curve Element Calculations

• $L = RI$ (I in radians)
• $L = \frac{100I}{D°}$ (ft)
• $L = \frac{I}{D°}5729.58$ (sta)
• $E = T \tan(\frac{I}{4})$
• $M = E \cos(\frac{I}{2})$
• $R = \frac{50}{\sin(D/2)}$ (ft)
• $R = \frac{5729.58}{D}$
• $T = R \tan(\frac{I}{2})$
• $LC = 2R \sin(\frac{I}{2})$
• $E = R(\frac{1}{\cos (I/2)} - 1)$
• $M = R(1-\cos(\frac{I}{2}))$

Circular Curve Stationing

• An initial route survey consists of establishing all the PIs, laying out the tangents, and establishing continuous stationing along them from the start of the project, through each PI, to the end of the job.
• English system: Full stationing (100 ft).
• The station locations of points on any curve are based upon the stationing of the curve’s PI. To compute the PC station, tangent distance T is subtracted from the PI station, and to calculate the PT station, curve length L is added to the PC station.
• Note that the stationing of the PT cannot be obtained by adding the tangent distance to the station of the PI, although the location of the PT on the ground is determined by measuring the tangent distance from the PI.
• Station equations:
  • $STA PC = STA PI – T$
  • $STA PT = STA PC + L = STA PI – T + L$

Example

• Given: $I = 80°24'$, PI station is 64+27.46, minimum radius $R = 2864.79 ft$ (arc definition).
• Objective: Calculate the PC and PT stationing and the external and middle ordinate distances for this curve.