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Unformatted text preview: The Pennsylvania State University
Department of Civil and Environmental Engineering CE 321: Highway Engineering Horizontal Alignment (Part 1) Spring 2009 Overview Vehicle Cornering Horizontal Curve Fundamentals Stopping Sight Distance Force on a Curve Uniform Circular Motion – Speed is constant – Velocity is constantly changing because direction of motion is constantly changing (Vector  magnitude and direction) – Consequently the body has an acceleration which produces change in direction Force on a Curve Vehicle on curve has centripetal acceleration (“centerseeking”). Radius, r, is associated with curve. mV Fc = Rv 2 Fc = centripetal force (lb) m = mass (slugs) V = vehicle speed (ft/sec) Rv = radius of curve (ft) Wheel Friction
Vehicle wheels exhibit static friction. When vehicles accelerate, tire pushes at point of contact. Ground pushes back – Vehicle accelerates. Curves and Friction On a curve, the static friction force (Ff) provides the centripetal acceleration.
– Fc = Ff – Fc is centripetal force and Ff is static friction force. Static friction force, Ff = fsmg
– fs = coefficient of side friction – m = vehicle mass (slugs) – g = gravitational constant (32.2 ft/sec2) Vehicle Cornering Forces ft Horizontal Curve Design Without static friction, a curve banked at an angle α supplies centripetal force:
Fc = mg tanα Relationship between radius of curve and speed
– Centripetal Force must equal friction and superelevation forces: Fcp = Ff + Wp (3.31) Fcp is centripetal force parallel to roadway (lb) Ff is static (side) friction force (lb) Wp is vehicle weight parallel to roadway (lb) Horizontal Curve Design WV 2 WV 2 W sin α + f s W cos α + sin α = gR cos α gRv v W = vehicle weight (lb) α = roadway incline angle (degrees) fs = coefficient of side friction (unitless) V = vehicle speed (ft/sec) g = gravitational constant (32.2 ft/sec2) Rv = curve radius (ft) By substituting Ff = fs (Wn + Fcn ) into previous equation Horizontal Curve Design
V2 tan α + f s = (1 − f s tan α)(3.33) gRv tan α = roadway superelevation or banking fs = coefficient of side friction (unitless) V = vehicle speed (ft/sec) g = gravitational constant (32.2 ft/sec2) Rv = curve radius (ft) Horizontal Curve Design Superelevation or banking
– Denoted e (e = 100 tan α) – Vertical feet rise per 100 feet horizontal distance Because of small values:
– Set fstan α = 0 Radius of Curve
Rv = V
2 e g fs + 100 (3.34) Rv = curve radius (ft) V = vehicle speed (ft/sec) g = gravitational constant (32.2 ft/sec2) fs = coefficient of side friction (unitless) e = roadway superelevation or banking (ft/ft) Superelevation
Values = 0.02 to 0.12 (2 to 12%) Pennsylvania Max = 0.08 (8%) Florida = 0.12 (12%) WHY??? “e + fs” is Lateral Acceleration What Values Do We Assign to “ fs“ and “ e “ ?? “Selecting” e and fs Relationship between radius of curve and speed
V (mph) 20 25 30 35 40 45 50 55 60 65 70 75 80 emax 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 fs 0.170 0.165 0.160 0.155 0.150 0.145 0.140 0.130 0.120 0.110 0.100 0.090 0.080 *Table 3.5 in MKW Why are “ fs “ and “ e “ so low?? What do “ f s” and “ e “ represent?? General Limits of Lateral Acceleration
Driver Comfort = 0.1 to 0.2 g’s Driver Uncomfortable = 0.2 to 0.3 g’s Driver Critical = > 0.4 g’s Horizontal Curve Design Horizontal Alignment
– All measures are made on the horizontal plane. Horizontal Curve Notation
D = Degree of curve R = Radius of curve (ft) D = Central angle (degrees) PC = Point of curvature PI = Point of intersection PT = Point of tangent T = Tangent length (ft) M = Middle ordinate (ft) E = External distance (ft) L = Length of curve (ft) Horizontal Curve Example Horizontal Curve Design Degree of Curve (D)  measure of the sharpness of the curve.
» The curvature of a circular arc is defined by the radius; however, in the field, for survey purposes, it is often inaccessible. Therefore, it is replaced by the degree of curve. Angle subtended by a 100foot arc along curve. Horizontal Curve Design The degree of curve is:
– Chord Definition  Dc Angle subtended by a 100 ft chord. (Used in RR Engineering)
Sin (Dc / 2) = 100 / r => Dc = 2 arcsin (50 / R) – Arc Definition  Da Angle subtended by a 100 ft arc. (Used in Highway Engineering)
Da / 360o = 100 / 2π R R => Da = 5729.58 / Horizontal Curve Design Comparison: For a 3000foot radius curve: Dc = 1.90995 º Da = 1.90986 º Note: a station = 100 ft (e.g., sta 1+00, 2+58, 7+42.97) Example Problem 1 A horizontal curve is designed with a 2000foot radius. The curve has a tangent of 400 ft and the PI is at station 103+00. Determine the stationing of the PT. Example 1 (con’t)
Solution: T = R tan Δ/2 400 = 2000 tan Δ/2 Δ = 22.62 ° Example 1 (con’t)
Solution (cont): L = (π /180) R Δ L = (3.1416/180) 2000 (22.62) = 789.58 ft Example 1 (con’t)
Solution (cont):
Given that the tangent is 400 ft,
Stationing PC = 103 + 00  4+00 = 99+00 Stationing PT = stationing PC + L Stationing PT = 99+00 + 7+89.58 = 106+89.58 Example Problem 2
Design Speed = 50 mi /hr e = 8% fs = 0.10 Find Rv Example 2 (con’t)
Solution:
Rv = V2 e g fs + 100 (50 × 5280 / 3600) 2 Rv = 8 32.2 0.10 + 100 R = 927.8 ft Variable Summary Fc = centripetal acceleration (lb) Ff is static (side) friction force m = mass (slugs) V = vehicle speed (ft/sec) Rv = radius of curve (ft) fs = coefficient of side friction g = gravitational constant (32.2 ft/sec2) Fcp is centripetal force parallel to roadway (lb) Ff is static (side) friction force (lb) Wp is vehicle weight parallel to roadway (lb) W = vehicle weight (lb) α = roadway incline angle (degrees) e = roadway superelevation or banking (ft/ft) D = Degree of curve R = Radius of curve (ft) D = Central angle (degrees) PC = Point of curvature PI = Point of intersection PT = Point of tangent T = Tangent length (ft) M = Middle ordinate (ft) E = External distance (ft) L = Length of curve (ft) ...
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 Spring '02
 Petrucha
 Force, Friction, Environmental Engineering, Horizontal curve design

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