1. Geometric Design of Highways
Dr. Eman Magdy Ibrahim Youssef
Assistant professor, Civil Engineering Department, Delta Higher Institute of Engineering and Technology
March- 2022
1
2. Vertical Alignment
• Definition:
• It is the elevation or the profile of the center line of the road
• Objective:
Determine elevation to ensure
• Proper drainage
• Acceptable level of safety
• Primary Challenge
• Transition between two grades
• Vertical curves
G1 G2
Crest Vertical Curve
G1
G2
Sag Vertical Curve
3. • Gradients
Vertical Alignment
α
• % = tan α
• G = +ve (
الصاعد الميل
)
• G = -ve (
الهابط الميل
)
• Maximum Grades
• Passenger vehicles can easily negotiate 4 to 5% grade
without appreciable loss in speed
• Upgrades: trucks average 7% decrease in speed
• Downgrades: trucks average speed increase 5%
4. Vertical Curves
• Parabolic shape
• VPI, VPC, VPT, +/- grade, L
• Types of crest and sag curves
G1 G2
G1
G2
G2
G1
Crest Vertical Curve
G1
G2
G1
G2
G1
G2
Sag Vertical Curve
5. Vertical Curves
• Crest – stopping, or passing sight distance controls
• Sag – headlight/SSD distance, comfort, drainage and appearance control
• Green Book vertical curves defined by K = L/A = length of vertical curve/difference in
grades (in percent) = length to change one percent in grade
Parabola
y = ax2 + bx + c
Where:
y = roadway elevation at distance x
x = distance from beginning of vertical curve
• Vertical Curve Equations
a=
𝐺1−𝐺2
𝐿 b = G1
c = elevation of PVC
A
L
K
6. G1
G2
PVI
PVT
PVC
L
L/2
δ
x
y e
Crest Curve Fundamentals
c
bx
ax
y
2
Choose Either:
• G1, G2 in decimal form, L in feet
• G1, G2 in percent, L in stations
L
AX
y
200
2
Location of Highest Point
A
LG
xhigh
100
1
A
LG
yhigh
200
2
1
7. Vertical Curve AASHTO Controls (Crest)
• Minimum length must provide stopping sight distance S
• Two situations (both assume h1=3.5’ and h2=2.0’(
• Crest Curves consider only stopping, or passing sight distance controls
8. Assistant with Target Rod (2ft object height)
Observer with Sighting
Rod (3.5 ft)
2
2
1
2
200 h
h
SSD
A
L
For SSD < L
A
h
h
SSD
L
2
2
1
200
2
For SSD > L
1329
2
SSD
A
L
A
SSD
L
1329
2
Assumptions for design
h1 = driver’s eye height = 3.75 ft.
h2 = object height = 0.5 ft.
9. Example: Try SSD > L, Design speed is 60 mph, G1 = 3% and G2 = -1%, what is L?
(Assume grade = 0% for SSD)
SSD = 570feet ( see: Lec 2)
Lmin = 2 (570’( –
2158’
|)−1−3(|
= 600.5’
S < L, so it doesn’t match condition
Vertical Curves
10. Example: Assume SSD < L, Design speed is 60 mph, G1 = 3% and G2 = -1%, what is L?
Assuming average grade = 0%
SSD = 570 feet - ( see Lec 2)
Lmin =
|)−3 − 1(|× )570 ft(2
2158
= 602 ft
SSD < L, equation matches condition
Vertical Curves
11. Example: A crest vertical curve joins a +3% and –4% grade. Design speed is 75 mph.
Length = 2184.0 ft. Station at PVI is 345+ 60.00, elevation at PVI = 250 feet. Find
elevations and station for PVC and PVT. L/2 = 1092.0 ft
PVI
PVT
PVI: STA 345+60
EL 250 ft.
Station at PVC = [345 + 60.00] - [10 + 92.00] = 334 + 68.00
Distance to PVC: 0.03 x (2184/2) = 32.76 feet
ElevationPVC = 250 – 32.76 = 217.24 feet
Station at PVT = [345 + 60.00] + [10 + 92.00] = 357 + 52.00
Distance (vertical) to PVT = 0.04 x (2184/2) = 43.68 feet
Elevation PVT = 250 – 43.68 = 206.32 feet
12. Example: A crest vertical curve joins a +3% and –4% grade. Design speed is 75 mph.
Length = 2184.0 ft. Station at VPI is 345+ 60.00, elevation at VPI = 250 feet. Station at
VPC (BVC) is 334 + 60.00, Elevation at VPC: 217.24 feet.
Calculate points along the vertical curve.
X = distance from VPC
Y =
Ax2
200 L
Elevationtangent = elevation at VPC + distance x grade
Elevationcurve = Elevationtangent - Y
14. • Sight distance is governed by nighttime conditions
• Distance of curve illuminated by headlights need to be considered
• Driver comfort
• Drainage
• General appearance
Sag Vertical Curves
G1
G2
PVI
PVT
PVC
h2=0
h1
L
Light Beam Distance (SSD)
headlight beam (diverging from LOS by β degrees)
tan
200 1
2
S
h
SSD
A
L
A
SSD
h
SSD
L
tan
200
2 1
For SSD < L
For SSD > L
15. • Assumptions for design
• h1 = headlight height = 2.0 ft.
• β = 1 degree
• Simplified Equations
Sag Vertical Curves
SSD
SSD
A
L
5
.
3
400
2
A
SSD
SSD
L
5
.
3
400
2
For SSD < L
For SSD > L
16. Example: A sag vertical curve is to be designed to join a –3% to a +3% grade. Design speed is
40 mph. What is L?
Skipping steps: SSD = 313.67 feet S > L
Determine whether S<L or S>L
L = 2(313.67 ft) –
)400 + 2.5 x 313.67(
[3 – )−3(] = 377.70 ft
313.67 < 377.70, so condition does not apply
17. Example: A sag vertical curve is to be designed to join a –3% to a +3% grade. Design speed is
40 mph. What is L?
Skipping steps: SSD = 313.67 feet
L = 6 x (313.67)2 = 394.12 ft
400 + 3.5 x 313.67
313.67 < 394.12, so condition applies
18. Example: A sag vertical curve is to be designed to join a –3% to a +3% grade. Design speed is
40 mph. What is L?
Skipping steps: SSD = 313.67 feet
Testing for comfort:
L =
AV2
46.5 =
)6 x [40 mph]2(
46.5 = 206.5 feet
Testing for appearance:
L = 100A = (100 x 6) = 600 feet
