Geometric Design of Highway Vertical Curves and Criteria

 
CE 34500 – Transportation
Engineering
 
Chapter 15: 
Geometric
Design of Highway Facilities
 
1
 
Vertical Alignment
 
Vertical Curves
 
Main Design Criteria
o
Crest & Sag Vertical Curves
Provision of minimum stopping
sight distance
o
Sag Vertical Curves
Adequate drainage
Comfortable in operation
Apprearance
 
Vertical Curves
 
Two cases for minimum length based on
sight distance
o
Sight distance is less than the length of the vertical
curve (S < L)
o
Sight distance is greater than the length of the
vertical curve (S > L)
Must make assumption about which case
is true and then check your assumption
o
Use of S < L criteria for design results in
conservative (longer) vertical curve lengths
 
Crest Vertical Curve
 
One criteria for length
o
Sight distance:  Vertical curve 
must
 be
long enough so that driver can see far
enough ahead (stopping sight distance)
to react and stop before striking an
object in roadway
Eye height = 3.5 ft.
Object height = 2.0 ft.
 
Crest Vertical Curves
 
Crest Vertical Curves
 
Sag Vertical Curve
 
Four criteria for length
o
Sight distance provided by the headlight beam is >=
Stopping sight distance
Headlight height = 2.0 ft.
Upward divergence of headlight beam = 1
o
o
Driver comfort: Radial acceleration           <= 1 ft/sec
2
o
Control of drainage:  “Almost flat spot” not too long
o
General appearance:  Vertical curve doesn’t “look like a
kink”
 
Sag Vertical Curve
 
H=2’
β
 = 1 degree
 
Sag Vertical Curve
 
Comfort Criterion
 
Sag Vertical Curve
 
Drainage Criterion
o
Use when road has curbs
o
Maximum length instead of minimum
“Almost flat spot”:  Minimum grade of
0.35% provided within 50 ft of the level
point
K <= 167
 
Sag Vertical Curve
 
Appearance Criterion
 
 
 
o
Per AASHTO Green Book for Sag and
Crest Vertical Curves:
   L
MIN
 (ft) = 3 x Design Speed (mph)
 
 
 
 
 
 
Crest  & Sag Vertical Curve
 
K is length of vertical curve per %
change in A
 
 
 
K values for 
crest
 vertical curves for a
given design speed found in 
Table 15.4
K values for 
sag
 vertical curves for a
given design speed found in 
Table 15.5
 
 
Crest  & Sag Vertical
Curve
 
 
Crest  & Sag Vertical Curve
 
 
Vertical Curve Elevations
 
Vertical Curve Elevations
 
Vertical curves are parabolas with
origin at BVC
o
  General form of a parabola:
     y = c + bx + ax
2
 Vertical curve fomula:
o
  y = y
BVC 
+ G
1
 * x + a * x
2
o
  a = ( G
2
 - G
1 
) / ( 2 * L)
o
 High or low point: dy/dx = 0
 
Vertical Curve Elevations
 
1.
Determine the minimum length of curve to
satisfy sight distance requirements  (and
other requirements, if desired).
2.
Determine from the plan sheet the station and
elevation of the PVI—that is, the point where
the grades intersect and the grades on either
side of the PVI are given,
3.
Compute the elevations of the BVC and EVC.
4.
Compute elevations on the curve, usually 100
ft. apart at even stations, beginning with the
first whole station after the BVC.
5.
If necessary, compute station and elevation of
high or low point.
 
Vertical Curves
Offset from tangent
elevation
Distance
from BVC
Offset from BVC
Elevation
 
Example 15.3
 
A sag vertical curve is to be designed to join a -5% grade to
a +2% grade. If the design speed is 40 mph, determine the
minimum length of the curve that will satisfy all criteria.
Assume 
a
 = 11.2 ft/sec
2
 and perception-reaction time = 2.5
sec.
 
Example 15.4
 
A crest vertical curve joining a +3% and a -4% grade is to be
designed for 75 mph. If the tangents intersect at station
(345+60.00) at an elevation of 250 ft, determine the stations and
elevations of the BVC and EVC. Also, calculate the elevations of
intermediate points on the curve at the whole stations. A sketch of
the curve is shown in Figure 15.16.
 
A sag vertical curve joins a -3% grade and a +3% grade. If the PVI
of the grades is at station (345+50) and has an elevation of 235 ft,
determine the station and elevation of the BVC and EVC for a
design speed of 70 mph. Also, compute the elevation on the curve
at 100-ft intervals. Figure 15.17 shows a layout of the curve.
 
Example 15.5
Slide Note

Faculty Candidate: Promothes Saha, Ph.D., P.E.

Dr. Saha - University of Wyoming

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This content covers the vertical alignment in transportation engineering, focusing on the geometric design of highway facilities, specifically vertical curves like crest and sag curves. It explains the main design criteria for vertical curves, including minimum stopping sight distance provision, drainage considerations, and driver comfort. The content also details criteria for crest and sag vertical curves lengths based on sight distances, with specifications for driver visibility and comfort, control of drainage, and overall appearance of the highway. Detailed formulas and considerations are provided for calculating and designing vertical curves according to safety and operational standards.

  • Transportation Engineering
  • Highway Design
  • Vertical Curves
  • Geometric Design
  • Sight Distance

Uploaded on Jul 22, 2024 | 2 Views


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  1. CE 34500 Transportation Engineering Chapter 15: Geometric Design of Highway Facilities 1

  2. Vertical Alignment

  3. Vertical Curves Main Design Criteria o Crest & Sag Vertical Curves Provision of minimum stopping sight distance o Sag Vertical Curves Adequate drainage Comfortable in operation Apprearance

  4. Vertical Curves Two cases for minimum length based on sight distance o Sight distance is less than the length of the vertical curve (S < L) o Sight distance is greater than the length of the vertical curve (S > L) Must make assumption about which case is true and then check your assumption o Use of S < L criteria for design results in conservative (longer) vertical curve lengths

  5. Crest Vertical Curve One criteria for length o Sight distance: Vertical curve must be long enough so that driver can see far enough ahead (stopping sight distance) to react and stop before striking an object in roadway Eye height = 3.5 ft. Object height = 2.0 ft.

  6. Crest Vertical Curves 2 AS = (for S L) L ( ) min 2 + 200 H H 1 2

  7. Crest Vertical Curves ( ) 2 + 200 H H = 1 2 2 (for S L) L S min A

  8. Sag Vertical Curve Four criteria for length o Sight distance provided by the headlight beam is >= Stopping sight distance Headlight height = 2.0 ft. Upward divergence of headlight beam = 1o o Driver comfort: Radial acceleration <= 1 ft/sec2 o Control of drainage: Almost flat spot not too long o General appearance: Vertical curve doesn t look like a kink

  9. Sag Vertical Curve + + 200( tan ) (400 3.5 ) A H S S = = 2 2 (for S L) L S S A H=2 = 1 degree 2 2 AS + AS + = = (for S L) L 200( tan ) (400 3.5 ) H S S

  10. Sag Vertical Curve Comfort Criterion 2 Au L = 46 5 .

  11. Sag Vertical Curve Drainage Criterion o Use when road has curbs o Maximum length instead of minimum Almost flat spot : Minimum grade of 0.35% provided within 50 ft of the level point K <= 167

  12. Sag Vertical Curve Appearance Criterion = 100 L A o Per AASHTO Green Book for Sag and Crest Vertical Curves: LMIN (ft) = 3 x Design Speed (mph)

  13. Crest & Sag Vertical Curve K is length of vertical curve per % change in A L =KA min K values for crest vertical curves for a given design speed found in Table 15.4 K values for sag vertical curves for a given design speed found in Table 15.5

  14. Crest & Sag Vertical Curve

  15. Crest & Sag Vertical Curve

  16. Vertical Curve Elevations

  17. Vertical Curve Elevations Vertical curves are parabolas with origin at BVC o General form of a parabola: y = c + bx + ax2 Vertical curve fomula: o y = yBVC + G1 * x + a * x2 o a = ( G2 - G1 ) / ( 2 * L) o High or low point: dy/dx = 0

  18. Vertical Curve Elevations Determine the minimum length of curve to satisfy sight distance requirements (and other requirements, if desired). Determine from the plan sheet the station and elevation of the PVI that is, the point where the grades intersect and the grades on either side of the PVI are given, Compute the elevations of the BVC and EVC. Compute elevations on the curve, usually 100 ft. apart at even stations, beginning with the first whole station after the BVC. If necessary, compute station and elevation of high or low point. 1. 2. 3. 4. 5.

  19. Vertical Curves A Offset from tangent elevation Y = 2 x 200 L 100 G G LG L G = = x 1 1 G Distance from BVC ( ) ( ) (or l ow) high 100 G 1 2 1 2 2 LG G = 1 high Y 1 Offset from BVC Elevation ( ) (or low) 200 G 1 2

  20. Example 15.3 A sag vertical curve is to be designed to join a -5% grade to a +2% grade. If the design speed is 40 mph, determine the minimum length of the curve that will satisfy all criteria. Assume a = 11.2 ft/sec2 and perception-reaction time = 2.5 sec.

  21. Example 15.4 A crest vertical curve joining a +3% and a -4% grade is to be designed for 75 mph. If the tangents intersect at station (345+60.00) at an elevation of 250 ft, determine the stations and elevations of the BVC and EVC. Also, calculate the elevations of intermediate points on the curve at the whole stations. A sketch of the curve is shown in Figure 15.16.

  22. Example 15.5 A sag vertical curve joins a -3% grade and a +3% grade. If the PVI of the grades is at station (345+50) and has an elevation of 235 ft, determine the station and elevation of the BVC and EVC for a design speed of 70 mph. Also, compute the elevation on the curve at 100-ft intervals. Figure 15.17 shows a layout of the curve.

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