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Vertical Alignment CE 2710 Spring 2009 Chris McCahill

Vertical Alignment CE 2710 Spring 2009 Chris McCahill. www.Exchange3D.com. Components of Highway Design. Horizontal Alignment. Plan View. Vertical Alignment. Profile View. Vertical Alignment & Topography. Texas DOT. Today’s Class. Properties of vertical curves (parabolic)

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Vertical Alignment CE 2710 Spring 2009 Chris McCahill

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  1. Vertical Alignment CE 2710Spring 2009Chris McCahill www.Exchange3D.com

  2. Components of Highway Design Horizontal Alignment Plan View Vertical Alignment Profile View

  3. Vertical Alignment & Topography Texas DOT

  4. Today’s Class • Properties of vertical curves (parabolic) • Design of vertical curves

  5. Vertical Alignment - Overview Crest Curve G2 G3 G1 Sag Curve

  6. Parabolic Curves Vertical Curve Horizontal Curve Radius Parabola

  7. Properties of Vertical Curves BVC G1 G2 EVC PI L/2 L/2 L Change in grade: A = G2 - G1 where G is expressed as % (positive /, negative \) For a crest curve, A is negative. For a sag curve, A is positive.

  8. Properties of Vertical Curves BVC G1 G2 EVC PI L/2 L/2 L Characterizing the curve: Rate of change of curvature: K = L / |A| Which is a gentler curve - small K or large K?

  9. Properties of Vertical Curves BVC G1 G2 EVC PI L/2 L/2 L Characterizing the curve: Rate of change of grade: r = (g2 - g1) / L where, g is expressed as a ratio (positive /, negative \) L is expressed in feet or meters

  10. Properties of Vertical Curves BVC G1 Elevation = y G2 EVC Point T PI L Point elevation (meters or feet): y = y0 + g1x + 1/2 rx2 where, y0 = elevation at the BVC (meters or feet) g = grade expressed as a ratio (positive /, negative \) x = horizontal distance from BVC (meters or feet) r = rate of change of grade expressed as ratio (+ sag, - crest)

  11. Properties of Vertical Curves Distance to turning point (high/low point) (xt): Given y = y0 + g1x + 1/2 rx2 Slope: dy/dx = g1 + rx At turning point, dy/dx = 0 0 = g1 + rxt xt = -(g1/r) where, r is negative for crest, positive for sag

  12. Properties of Vertical Curves BVC G1 G2 EVC PI Example: G1 = -1% G2 = +2% Elevation of PI = 125.00 m Station of EVC = 25+00 Station of PI = 24+00 Length of curve? L/2 = 2500 m - 2400 m = 100 m Sta. BVC = Sta. PI - L/2 Sta. BVC = [24+00] - 100 m Sta. BVC = 23+00 L = 200 m

  13. Properties of Vertical Curves BVC G1 G2 EVC PI Example: G1 = -1% G2 = +2% Elevation of PI = 125.00 m Station of EVC = 25+00 Station of PI = 24+00 r - value? r = (g2 - g1)/L r = (0.02 - [-0.01])/200 m r = 0.00015 / meter

  14. Properties of Vertical Curves BVC G1 G2 EVC PI Example: G1 = -1% G2 = +2% Elevation of PI = 125.00 m Station of EVC = 25+00 Station of PI = 24+00 Station of low point? x = -(g1/r) x = -([-0.01] / [0.00015/m]) x = 66.67 m Station = [23+00] + 67.67 m Station 23+67

  15. Properties of Vertical Curves BVC G1 G2 EVC PI Example: G1 = -1% G2 = +2% Elevation of PI = 125.00 m Station of EVC = 25+00 Station of PI = 24+00 Elevation at low point? y = y0 + g1x + 1/2 rx2 y0 = Elev. BVC Elev. BVC = Elev. PI - g1L/2 Elev. BVC = 125 m - [-0.01][100 m] Elev. BVC = 126 m

  16. Properties of Vertical Curves BVC G1 G2 EVC PI Example: G1 = -1% G2 = +2% Elevation of PI = 125.00 m Station of EVC = 25+00 Station of PI = 24+00 Elevation at low point? y = y0 + g1x + 1/2 rx2 y = 126 m + [-0.01][66.67 m] + 1/2 [0.00015/m][66.67 m]2 y = 125.67 m

  17. Properties of Vertical Curves BVC G1 G2 EVC PI Elevation at station 23+50? y = 126 m + [-0.01][50 m] + 1/2 [0.00015/m][50 m]2 y = 125.69 m Elevation at station 24+50? y = 126 m + [-0.01][150 m] + 1/2 [0.00015/m][150 m]2 y = 126.19 m Example: G1 = -1% G2 = +2% Elevation of PI = 125.00 m Station of EVC = 25+00 Station of PI = 24+00

  18. Design of Vertical Curves

  19. Design of Vertical Curves • Determine the minimum length (or minimum K) for a given design speed. • Sufficient sight distance • Driver comfort • Appearance

  20. Design of Vertical Curves • Crest Vertical Curve • If sight distance requirements are satisfied then safety, comfort, and appearance will not be a problem. h1 = height of driver’s eyes, in ft h2 = height of object, in ft

  21. Design of Vertical Curves Crest Vertical Curve Sample Equation: From AASHTO: h1 ≈ 3.5 ft h2 ≈ 0.5 ft (stopping sight distance) h3 ≈ 4.25 ft (passing sight distance)

  22. Design of Vertical Curves • Sag Vertical Curve • Stopping sight distance not an issue. What are the criteria? • Headlight sight distance • Rider comfort • Drainage • Appearance

  23. Design of Vertical Curves • Sag Vertical Curve • Check also: • Comfort • Change in grade, A • Design Speed • Appearance • Change in grade, A

  24. Maximum Grade • Context • Rural • Urban • Design Speed • Terrain • Level • Rolling • Mountainous 0.5% to 3.0%

  25. Maximum Grade www.geograph.org.uk Harlech, Gwynedd, UK (G = 34%)

  26. Maximum Grade www.nebraskaweatherphotos.org

  27. Maximum Grade Dee747 at picasaweb.google.com

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