Vertical Curves

1
Vertical Curves

• Sometimes required when two gradients meet
• Transition too abrupt
• Sight Distances
• Note terms
• Distances measured horizontally
• Tangent offsets indicate cut or fill from flat
  grade to curve

2
Parabolic Curve

• Form y ax2 bx c
• Three important relationships
• Curve at midpoint is always halfway between PVI
  and a line connecting PVC and PVT.
• Tangent offsets vary with the square of the
  distance from PVC.
• Second differences between equal points are equal.

3
Parabolic Example

• Extend g1, divide into 6 equal parts
• If offset at 1 a
• Offset2 4a
• Offset3 9a
• ACV similar to ABB
• CV BB/2 18a
• CM MV 9a

4
Vertical Curve by Proportions

• Problem 21-3
• g1 4, g2 -2.1
• PVT 4650, E 682.24
• L 8 sta (800 ft)
• Sta PVC Sta PVI L/2 4250, EPVC 682.24
  4(4 Sta) 666.24
• Sta PVT Sta PVI L/2 5050, EPVT 682.24
  4 Sta(-2.1) 673.84

5
Problem 21-3 (Cont.)

• Find yPVT, yMid
• yPVT EPVC g1L EPVT 666.24 (4)(8
  Sta) 673.84 24.40
• yMid yPVT(4/8)2 24.40(.5)2 6.10, or

6
Problem 21-3, Cont.

• Ascending E EPVC g1x y
• y43 (-6.10)(0.5)2/(4)2 -0.10
• E43 666.24 (4)(0.5 sta) 0.10 668.14
• y44 (-6.10)(1.5)2/(4)2 -0.86
• E43 666.24 (4)(1.5 sta) 0.86 671.38
• Descending E EPVT g2x y
• Sta 4700, x from PVT 3.5 sta
• y47 (-6.10)(3.5)2/(4)2 -4.67
• E47 673.84 - (-2.1)(3.5 sta) 4.67 676.52

7
Curve Equation

• Elev c bx ax2
• c ElevA
• b g1 (in )
• x in Sta
• ElevX ElevAg1xax2
• Rate of Change
• d2y/dx2 2a
• a (g2-g1)/2L

8
Vertical Curve Equation

• Change in grade A g2-g1
• Rate of change r A/L

• High or Low Point
• Occurs when grade 0
• Grade changes from g1 to 0 at rate r

9
Problem 21-3 by Equation

• A (-2.1 4) -6.1
• r A/L -6.1/8 Sta -.7625 /Sta

• E43 666.24 4(0.5) - 0.38125(.5)2 668.14
• E44 666.24 4(1.5) - 0.38125(1.5)2 671.38
• E4650 666.24 4(4.0) - 0.38125(4.0)2 676.14
• E47 666.24 4(4.5) - 0.38125(4.5)2 676.52

• High Point X -g1/r -4/-0.7625 5.2469 Sta
• E4774.59 666.24 4(5.2469) - .38125(5.2469)2
  676.73
• E48 666.24 4(5.5) - 0.38125(5.5)2 676.71

10
Curve Thru a Point

• Say you want the curve to pass through a point of
  known Sta, Elev
• Select L to force curve through point
• Two possible solutions
• Tangent Offsets Author
• Equation

11
Thru Point - Tangent Offset

• xP Distance from PVI to Point, in Sta
• yB offset from the back tangent
• yF offset from forward tangent

12
Thru Point Tangent Offset
13
Thru Point Equation

• yB yF (g2 - g1)xP
• yB yF 2(EP - EPVI) - (g2 g1)xP
• Recall A g2 - g1
• xP Distance from PVI to point, in Sta

14
Superelevation

• Crown 0.015 0.02 ft/ft
• Superelevation
• Transverse Slope
• Depends on Side friction, speed, D
• Max 8- 10
• Runoff transition from flat to banked
• Linear change
• Min 100-250 ft
• Max 500-600 ft

15
Superelevation Example

• Road conditions
• Curve, D 4 , I 53
• PC at Sta 2445, Elev 299.75
• Centerline grade from PC to PT is 3.5
• Road top is 40 wide, crown is 0.015 ft/ft
• Superelevation 4, Runoff 400 feet
• Design
• PC Centerline at 299.75
• At PC need full superelevation, ? .04(20)
  0.80
• Out shoulder at 300.55, In shoulder at 298.95

16
Runoff

• PC at Sta 2445, Elev 299.75
• Out shoulder at 300.55, In shoulder at 298.95
• 400 foot runoff start at Sta 2045
• Centerline at 313.75
• Crown at 0.01520 0.30
• Out and In Shoulders at 313.45
• Linear transition
• Roll outside crown up to 1.5 (0.015 ft/ft)
• Roll both shoulders to 4