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Unformatted text preview: The Pennsylvania State University
Department of Civil and Environmental Engineering Lecture 11 Vertical Curvature (Part 1) CE 321: Highway Engineering Fall 2007 Vertical Curves Crest Curve Design Sag Curve Design Types of Vertical Curves Vertical Curve Fundamentals
y = ax2 + bx + c
assuming a constant rate of change of slope and equal tangent lengths. When x = 0, y = c = elevation on curve dy / dx = 2ax + b = slope (rise / run) at x = 0 => slope is dy/dx = b = G1 (the initial slope) Vertical Curve Fundamentals
d2y/dx2 = 2a = rate of change of slope 2a = (G2  G1) / L a = (G2  G1) / 2L therefore: yx = [(G2  G1) / 2L] x2 + G1x + Elevation at PVC
Where: L (sta.), x (sta.) High / Low Point on Curve
Set dy/dx = 0 Then, if a = (G2  G1) / 2L and dy/dx = 2ax + b = 0 2ax + G1 = 0 2ax =  G1 2(G2  G1)(x)/2L =  G1 then: x =  G1L / (G2  G1) Vertical Curve Design Vertical Curves maximum grades depend on: Design Speed Type of Terrain Type of Highway Length of Grade Vertical Curve Design Grades also affect: fuel consumption speed safety or crashes (speed differential) Vertical Curve Design Design Speed 50 mi/hr 60 mi/hr 70 mi/hr Flat / Rolling 5% 4% 3% Mountainous 7% 6% 5% (+ 2 % for secondary / local roads) Example Problem 1
A 600 ft equal tangent sag vertical curve has the PVC at station 170+00 and elevation 1000 ft. The initial grade is 3.5 % and the final grade is 0.5 %. Determine the elevation and stationing of the PVI, PVT, and the lowest point on the curve. Example 1 (con't)
Solution:
Station PVI = 170+00 + 300 = 173+00 Station PVT = 170+00 + 600 = 176+00 Elev. PVI = Elev. PVC + (initial grade) * ( L / 2 ) Elev. PVI = 1000 + ( 0.035 ft/ft) (300 ft) = 989.5 ft Elev. PVT = Elev. PVI + (final grade) * ( L / 2 ) Elev. PVT = 989.5 + (0.005 ft/ft) (300 ft) = 991.0 ft Example 1 (con't)
Solution: Determining Low point on Curve
Case 1: Case 2: If the initial and final grades are opposite in sign, the low point on the curve will occur when the first derivative of the parabolic function is zero. When the initial and final grades are not opposite in sign, the low point on the curve will be at the PVC or PVT This problem is Case 1 dx = 2ax + b = 0 dy Example 1 (con't)
Solution: b = G1 =  3.5
a = (G2  G1) / 2L a = [0.5  ( 3.5)] / 2(6) = 0.3333
Substituting a and b in dx = 2ax + b = 0 dy dx dy = 2 (0.3333)x + (3.5) = 0 x = 5.25 stations Example 1 (con't)
Solution:
Station at low point = station at PVC + x = 170+00 + (5+25) = 175+25 To find elevation at lowest point: yx = a x2 + bx + Elevation of PVC y = 0.33333(5.25)2 + (3.5)(5.25) + 1000 y = 990.81 ft Other Vertical Curve Relationships
AL A 2 Y = x Ym = 800 200 L
figure 3.4 AL Yf = 200 Y = the offset at any distance, x, from the PVC (ft) Ym = the midcurve offset (ft) Yf = offset at the end of the vertical curve (ft) A = the absolute value of the difference in the grades ( G1  G2 ) (%) L = the length of the vertical curve (ft) x = the distance from the PVC (ft) Other Vertical Curve Relationships 2
"K" is the horizontal distance, in feet, to affect a 1% change in the slope. K=L/A L = length of curve (ft) A = absolute value of difference in grades (%) The high or low point can also be calculated by: xhl = KG1 (where "x" is the high/low point) xhl = distance from PVC to high/low point (ft) Crest Curve Design and Stopping Sight Distance H1 = height of driver's eye above roadway surface (ft) H2 = height of object above roadway surface (ft) S = sight distance (ft) L = length of vertical curve (ft) Crest Curve Design:
Sight Distance greater than length of vertical curve: ( S > L )
Lm = 2 S  200 ( H1 + H 2 A ) 2 Lm= Minimum length of vertical curve (ft) S = Sight Distance (ft) A = Algebraic difference in grades (%) H1 = Height of eye above roadway surface (ft) H2 = Height of object above roadway surface (ft) Crest Curve Design:
Sight Distance less than length of vertical curve: ( S < L )
Lm = 200 ( AS 2 H1 + H 2 ) 2 Lm = Minimum length of vertical curve (ft) H1 = Height of eye above roadway surface (ft) S = Sight Distance (ft) H2 = Height of object above roadway surface (ft) A = Algebraic difference in grades (%) Crest Curve Design
When you assume AASHTO guidelines:
H1 = 3.5 ft (driver eye height) S = SSD H2 = 2 ft (object height) SSD > L SSD < L Lm = 2 SSD  2158 A Lm = A SSD2 2158 Lm = Minimum length of vertical curve (ft) SSD = StoppingSight Distance (ft) A = Algebraic difference in grades (%) StoppingSight Distance Source: AASHTO Green Book (2001) Crest Curve Design (Kvalues) Source: AASHTO Green Book (2001) Example Problem 2 A +1.0% grade intersects a 2.0% grade on a highway with a design speed of 70 mi/hr. Determine the length of curve required assuming provisions are to be made for minimum SSD. Example Problem 2 (con't) First, ignore the effects of grades: SSD can be read directly from AASHTO Assume a design speed = 70 mi/hr (min speed) The corresponding SSD = 730 ft Example Problem 2 (con't) Assume: SSD < L
Lm = A SSD2 2158 Lm = 3 (730)2 2158 = 740.82 For SSD Since 740.82 > 730, the assumption L > SSD was correct Example Problem 2 (con't) Alternative Solution: Use Kvalues (Lm = KA) A = G1  G2 = 1.0  ( 2.0) = 3.0
Lm = 247(3.0) = 741.0 ft Vertical Curve Equation Variables PVC = Point of vertical curvature (initial point of curve) PVT = Point of vertical tangent (point where curve returns to final tangent) PVI = Point of vertical intersection (intersection of initial and final grades) L = Length of vertical curve in stations or feet (measured on horizontal plane) G1 = Initial roadway grade in percent or ft/ft G2 = Final roadway grade in percent or ft/ft A = absolute difference in grades (G1 G2, percent) y = roadway elevation at distance x from beginning of vertical curve in ft x = distance from beginning of vertical curve in stations or ft a, b = coefficients c = elevation of PVC in ft Vertical Curve Equation Variables 2 Y = the offset at any distance, x, from the PVC (ft) Ym = the midcurve offset (ft) Yf = offset at the end of the vertical curve (ft) x = the distance from the PVC (ft) xhl = distance from PVC to high/low point (ft) H1 = height of driver's eye above roadway surface (ft) H2 = height of object above roadway surface (ft) S = sight distance (ft) Lm= Minimum length of vertical curve (ft) SSD = stopping sight distance (ft) ...
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 Spring '07
 SCANLON,ANDREW

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