04 Vertical Alignment

Vertical AlignmentSee: http://www.fhwa.dot.gov/environment/flex/ch05.htm

(Chapter 5 from FHWA’s Flexibility in Highway Design)

Source: A Policy on Geometric Design of Highways and Streets (The Green Book). Washington, DC. American Association of State Highway and Transportation Officials, 2001 4th Ed. P. 284

Coordination of Vertical and Horizontal Alignment Curvature and grade should be in proper

balance Avoid

Excessive curvature to achieve flat grades

Excessive grades to achieve flat curvature

Vertical curvature should be coordinated with horizontal

Sharp horizontal curvature should not be introduced at or near the top of a pronounced crest vertical curve Drivers may not perceive change in

horizontal alignment esp. at night

Image source: http://www.webs1.uidaho.edu/niatt_labmanual/Chapters/geometricdesign/theoryandconcepts/DescendingGrades.htm

Source: A Policy on Geometric Design of Highways and Streets (The Green Book). Washington, DC. American Association of State Highway and Transportation Officials, 2001 4th Ed.

Coordination of Vertical and Horizontal Alignment

Sharp horizontal curvature should not be introduced near bottom of steep grade near the low point of a pronounced sag vertical curve Horizontal curves appear distorted Vehicle speeds (esp. trucks) are highest at the

bottom of a sag vertical curve Can result in erratic motion

On two-lane roads when passing is allowed, need to consider provision of passing lanes Difficult to accommodate with certain

arrangements of horizontal and vertical curvature

need long tangent sections to assure sufficient passing sight distance

At intersections where sight distance needs to be accommodated, both horizontal and vertical curves should be as flat as practical

In residential areas, alignment should minimize nuisance to neighborhood Depressed highways are less visible Depressed highways produce less noise Horizontal alignments can increase the buffer zone

between roadway and cluster of homes

When possible alignment should enhance scenic views of the natural and manmade environment Highway should lead into not away from

outstanding views Fall towards features of interest at low elevation Rise towards features best seen from below or

in silhouette against the sky

•Coordination of horizontal and vertical alignment should begin with preliminary design•Easier to make adjustments at this stage•Designer should study long, continuous stretches of

Coordination of Horizontal and Vertical Alignment

highway in both plan and profile and visualize the whole in three dimensions (FHWA, Chapter 5)

Source: FHWA, Chapter 5

Should be consistent with the topography

Preserve developed properties along the road

Incorporate community valuesFollow natural contours of the land

Good Coordination of Horizontal and Vertical Alignment

Does not affect aesthetic, scenic, historic, and cultural resources along the way

Enhances attractive scenic views Rivers Rock formations Parks Historic sites Outstanding buildings

There are 2 problems with this alignment. What are they?

There are 2 problems with this alignment.

What are they?

Maybe we want this if we are trying to slow people down???

Vertical Alignment

Equations

Curve/grade tradeoff

a 3% grade causes a reduction in speed of 10 mph after 1400 feet

Vertical Alignment - General1. Parabolic shape

2. VPI, VPC, VPT, +/- grade, L

3. Types of crest and

sag curves – see

Exhibit 3-73 p. 269

Vertical Alignment – General (Cont.)

4. Crest – stopping, or passing sight distance controls

5. Sag – headlight/SSD distance, comfort, drainage and appearance control

6. 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

Vertical Alignment - GeneralParabolic shape as applied to vertical curves

y = ax2 + bx + c

Where:

y = roadway elevation at distance x

x = distance from beginnning of vertical curve

a, b = coefficients that define shape

c = elevation of PVC

Vertical Alignment - GeneralParabolic shape as applied to vertical curves

a = G2 – G1

b = G1

Vertical Curve AASHTO Controls (Crest)

1. Based on stopping sight distance

2. Minimum length must provide sight distance S

3. Two situations (Crest, assumes 3.5 and 2.0 ft. heights)

Source: Transportation Engineering On-line Lab Manual, http://www.its.uidaho.edu/niatt_labmanual/

Assistant with Target Rod (2ft object height)

Observer with Sighting Rod (3.5 ft)

Note: for passing site distance, use 2800 instead of 2158

Example: Assume SSD < L,

Design speed is 60 mph

G1 = 3% and G2 = -1%,

what is L?

SSD = 525 feet

Lmin = |(-3 - 1)| (525 ft)2 = 510.9 ft

S > L, so try other equation

Example: Next try SSD > L,

Design speed is 60 mph

G1 = 3% and G2 = -1%,

what is L?

SSD = 525 feet

Lmin = 2 (525’) – 2158’ = 510.5’

S > L, so equation works

|(-3 - 1)|

Can also use

K = L / A

K = length of curve per percent algebraic difference in intersecting grade

Charts from Green Book

38From Green book

39From Green book

Since you do not at first know L, try one of these equations and compare to requirement, or use L = KA (see tables and graphs in Green Book for a given A and design speed)

Note min. L(ft) = 3V(mph) – Why?

Sag Vertical Curves

Sight distance is governed by nighttime conditions Distance of curve illuminated by headlights

need to be considered

Driver comfortDrainageGeneral appearance

Vertical Curve AASHTO Controls (Sag) Headlight Illumination sight distance with S < L

L = AS2

L = 2S – (400 + 3.5S)

AHeadlight Illumination sight distance with S > L

400 + (3.5 * S)

Vertical Curve AASHTO Controls (Sag) For driver comfort use:

L = AV2

(limits g force to 1 fps/s)

To consider general appearance use:

L = 100 A

Sag Vertical Curve: 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) = 377.70 ft

[3 – (-3)]

313.67 < 377.70, so condition does not apply

Skipping steps: SSD = 313.67 feet

L = 6 x (313.67 ft)2 = 394.12 ft

400 + 3.5 x 313.67 ft

313.67 < 394.12, so condition applies

Testing for comfort:

L = AV2 = (6 x [40 mph]2) = 206.5 feet 46.5 46.5

Testing for appearance:

L = 100A = (100 x 6) = 600 feet

Vertical Curve AASHTO Controls (Sag)

For curb drainage, want minimum of 0.3 percent grade within 50’ of low point = need Kmax = 167 (US units)

For appearance on high-type roads, use minimum design speed of 50 mph (K = 100)

As in crest, use minimum L = 3V

Other important issues:

Use lighting if need to use shorter L than headlight requirements

Sight distance at under crossings

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

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

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. Station at PVC is 334 + 68.00, Elevation at PVC: 217.24 feet.

Calculate points along the vertical curve.

X = distance from PVC

Y = Ax2

Elevationtangent = elevation at PVC + distance x grade

Elevationcurve = Elevationtangent - Y

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 elevation on the curve at a point 400 feet from PVC.

Y = A x 2 = 6 x (400 ft)2 = 4.40 feet

200L 200 (2814)

Elevation at tangent = 217.24 + (400 x 0.03) = 229.24

Elevation on curve = 229.24 – 4.40 feet = 224.84

53Source: Iowa DOT Design Manual

Source: Iowa DOT Design Manual

55Source: Iowa DOT Design Manual

Note: L is measured from here to here

Not here

04 Vertical Alignment

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