1. Vertical Curves Sometimes required when two gradients meet Note terms
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 = 46+50, E =
L = 8 sta (800 ft)
Sta PVC = Sta PVI – L/2 = 42+50, EPVC = – 4%(4 Sta) =
Sta PVT Sta PVI + L/2 = 50+50, EPVT = Sta(-2.1%) =

5. Problem 21-3 (Cont.) Find yPVT, yMid
yPVT = EPVC + g1L – EPVT = (4%)(8 Sta) – = 24.40’
yMid = yPVT(4/8)2 = 24.40’(.5)2 = 6.10’, or

6. Problem 21-3, Cont. Ascending: E = EPVC + g1x + y
E43 = (4%)(0.5 sta) – 0.10 =
y44 = (-6.10)(1.5)2/(4)2 = -0.86
E43 = (4%)(1.5 sta) – 0.86 =
Descending: E = EPVT – g2x + y
Sta 47+00, x from PVT = 3.5 sta
y47 = (-6.10)(3.5)2/(4)2 = -4.67
E47 = (-2.1%)(3.5 sta) – 4.67 =

7. Curve Equation Elev = c + bx +ax2 ElevX = ElevA+g1x+ax2 Rate of Change
c = ElevA
b = g1 (in %)
x in Sta
ElevX = ElevA+g1x+ax2
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 = %/Sta
E43 = (0.5) (.5)2 =
E44 = (1.5) (1.5)2 =
E46+50 = (4.0) (4.0)2 =
E47 = (4.5) (4.5)2 =
High Point: X = -g1/r = -4/ = Sta
E = (5.2469) (5.2469)2 =
E48 = (5.5) (5.5)2 =

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: ft
Max: ft

15. Superelevation Example
Road conditions
Curve, D = 4° , I = 53°
PC at Sta 24+45, Elev = ’
Centerline grade from PC to PT is –3.5%
Road top is 40’ wide, crown is ft/ft
Superelevation = 4%, Runoff = 400 feet
Design
PC Centerline at ’
At PC need full superelevation,  .04(20’) = 0.80’
Out shoulder at ’, In shoulder at ’

16. Runoff PC at Sta 24+45, Elev = 299.75’
Out shoulder at ’, In shoulder at ’
400 foot runoff – start at Sta 20+45
Centerline at
Crown at 0.015*20’ = 0.30’
Out and In Shoulders at
Linear transition
Roll outside crown up to 1.5% (0.015 ft/ft)
Roll both shoulders to 4%