Vertical Curves Introduction

1
Vertical Curves Introduction

• Issue
• Defining curves

2
Vertical Curve Geometry

• Vertical curves use parabolas
• Field practice

3
Geometry of Vertical Curves

• Notation
• BVC
• EVC
• V
• L
• g1
• g2

4
Equal-Tangent Vertical Curves
EVC
Long chord
BVC
g2
g1
V
L/2
L/2
L
Datum
5
Mathematics and Notation

• Rate of change of grade (r)
• Total change of grade g2 g1
• r total change/curve length

g1, g2 in L in stations

• Constant for each curve
• Sign of r shows sag curve () or crest curve
  (-)
• Length of curve (L)
• Same equation used

6
Types of Curves
ve g2
-ve g1
ve g2
ve g1
-ve g1
ve g1
-ve g2
-ve g2
7
Specification of Vertical Curves

• Given
• Specification depends on
• Design process involves

8
Computation of BVC and EVC

• For equal-tangent parabolic curves
• Compute location and elevation of BVC and EVC
  with

g1, g2 in L in stations
9
Computing Elevation on Vertical Curves

• Two principal methods
• Offsets method

10
Analytical Method

• Most common method
• Uses equation for parabolas

• For equal-tangent parabolas

• Know r, g1, BVC elevation, and x
• Compute elevation of any point on curve

11
Analytical Method (contd)

• Computation of high or low point on curve
• Find first derivative and set equal to zero

• Solve for x

• Plug x into original equation to compute elevation

12
Example 1

• Given
• g1 3.00
• g2 - 1.00
• Station V 2000
• Elevation V 200.00'
• L 800.00'

• Find
• Equation of vertical curve
• Station and elevation of high point

V
EVC
BVC
13
Example 1 Solution

• Curve parameters
• StaBVC StaV L/2 2000 400 1600
• ElevBVC ElevV L/2 ? g1 200.00' 4 ? 3.00
  188.00'
• r (g2 g1) / L (-1.00 3.00) / 8 -0.5
• Curve equation

• y -0.25 x2 3x 188.00'

14
Example 1 Solution

• Station of high point
• Turning point (first derivative 0)

• Station of high point StaBVC 6.00 1600
  600 2200

• Elevation of high point
• Consider curve equation where x 6.00
• y -0.25 x2 3x 188.00'
• y -0.25 (6)2 3(6) 188.00' 197.00'

15
Example 2

• Given
• g1 -2.00
• g2 3.00
• Station V 1800
• Elevation V 100.00'
• L 600.00'

• Find
• Elevation of all stations
• Station and elevation of low point

EVC
q
BVC
m
p
n
o
V
16
Example 2 Solution

• Curve parameters
• StaBVC StaV L/2 1800 300 1500
• ElevBVC ElevV L/2 ? g1 100.00' 3 ?
  (-2.00) 106.00'
• r (g2 g1) / L (3.00 -2.00) / 6 0.833
• Curve equation

• y 0.417 x2 2x 106.00'
• End point
• StaEVC StaV L/2 1800 300 2100
• ElevEVC ElevV L/2 ? g2 100.00' 3 ? (3.00)
  109.00'

17
Example 2 Solution

• Station of low point
• Turning point (first derivative 0)

• Station of low point StaBVC 2.40 1500
  240 1740

• Elevation of low point
• Consider curve equation where x 2.40
• y 0.417 x2 2x 106.00'
• y 0.417 (2.4)2 2(2.4) 106.00' 103.60'

18
Example 2 Solution

• Elevation of all stations

19
Sight Distance on Vertical Curves

• Issue
• Two situations
• Stopping Sight Distance (SSD)
• Passing Sight Distance (PSD)

20
Sight Distances Sag Curves
Sight obstructions
Headlight beam distance
21
Sight Distances Crest Curves
22
Estimating SSD

• Two equations might apply (feet)
• When sight distance is less than L

• When sight distance is greater than L

• Notes
• Equations assume
• Driver height 3.50 ft and obstruction height
  0.50 ft
• g1 and g2 in when L and s are in stations
• g1 and g2 in decimal form when L and s are in feet

23
Estimating PSD (Crest Curves)

• Two equations might apply (feet)
• When sight distance is less than L

• When sight distance is greater than L

• Notes
• Equations assume
• Driver height 3.50 ft and obstruction height
  4.25 ft
• g1 and g2 in when L and s are in stations
• g1 and g2 in decimal form when L and s are in feet

24
Example Sight Distance Analysis

• Given
• Design speed 70 mph
• SSD 600 feet
• PSD 2500 feet
• g1 3.00 and g2 -2.00
• Find
• Length of vertical curve for SSD
• Length of vertical curve for PSD

25
Example Sight Distance Analysis

• For SSD

• Which condition is true?

26
Example Sight Distance Analysis

• For PSD

• Which condition is true?

27
Sight Distance on Vertical Curves

• Defining curve length
• Practical issues

28
Readings

• Chapter 25 sections 25.1 25.4, 25.9, 25.11
  25.13