1 Vertical Alignment CE 453 Lecture 20 Sources: A Policy on Geometric Design of Highways and Streets (The Green Book). Washington, DC. American Association.

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Vertical AlignmentVertical Alignment

CE 453 Lecture 20

Sources: A Policy on Geometric Design of Highways and Streets (The Green Book). Washington, DC. American Association of State Highway and Transportation Officials, 2001 4th Edition, and FHWA’s Flexibility in Highway Design

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Coordination of Vertical and Coordination of Vertical and Horizontal AlignmentHorizontal 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

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Coordination of Vertical and Coordination of Vertical and Horizontal AlignmentHorizontal 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

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Coordination of Vertical and Coordination of Vertical and Horizontal AlignmentHorizontal Alignment

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

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Coordination of Vertical and Coordination of Vertical and Horizontal AlignmentHorizontal Alignment

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

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Coordination of Vertical and Coordination of Vertical and Horizontal AlignmentHorizontal Alignment

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

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Coordination of Horizontal Coordination of Horizontal and Vertical Alignmentand Vertical Alignment

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 highway in both plan and profile and visualize the whole in three dimensions

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Coordination of Horizontal Coordination of Horizontal and Vertical Alignmentand Vertical Alignment

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Should be consistent with the topography

Preserve developed properties along the road

Incorporate community valuesFollow natural contours of the land

Coordination of Horizontal Coordination of Horizontal and Vertical Alignmentand Vertical Alignment

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Good Coordination of Good Coordination of Horizontal and Vertical Horizontal and Vertical

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

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

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Vertical CurvesVertical Curves

Connect roadway grades (tangents)

Grade (rise over run) – 10% grade increases 10’ vertically for

every 100’ horizontal

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Vertical CurvesVertical Curves

Ascending grade:– Frequency of

collisions increases significantly when vehicles traveling more than 10 mph below the average traffic speed are present in the traffic stream

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ExampleExample If a highway with

traffic normally running at 65 mph has an inclined section with a 3% grade, what is the maximum length of grade that can be used before the speed of the larger vehicles is reduced to 55 mph?

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ExampleExample

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

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Climbing lanesClimbing lanes

When flatter grades cannot be accommodated, consider climbing lane when all 3 of the following criteria are met (AASHTO):– Upgrade traffic flow rate in excess of 200 vehicles

per hour.– Upgrade truck flow rate in excess of 20 vehicles per

hour.– One of the following conditions exists:

A 15 km/h or greater speed reduction is expected for a typical heavy truck.

Level-of-service E or F exists on the grade. A reduction of two or more levels of service is experienced

when moving from the approach segment to the grade.

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Descending GradesDescending Grades

Problem is increased speeds and loss of control for heavy trucks

Runaway vehicle ramps are often designed and included at critical locations along the grade

Ramps placed before each turn that cannot be negotiated at runaway speeds

Ramps should also be placed along straight stretches of roadway, wherever unreasonable speeds might be obtained

Ramps located on the right side of the road when possible

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Maximum GradesMaximum 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%

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Vertical CurvesVertical Curves Parabolic shape VPI, VPC, VPT, +/- grade, L Types of crest and sag curves

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Vertical CurvesVertical 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

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Parabola y = ax2 + bx + cWhere:

y = roadway elevation at distance x x = distance from beginning of

vertical curve a = G2 – G1 L b = G1 c = elevation of PVC

Vertical Curve EquationsVertical Curve Equations

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Vertical Curve AASHTO Controls Vertical Curve AASHTO Controls

(Crest)(Crest) Minimum length must provide

stopping sight distance S Two situations (both assume h1=3.5’

and h2=2.0’)

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

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Assistant with Target Rod (2ft object height)

Observer with Sighting Rod (3.5 ft)

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Vertical Curve AASHTO Controls Vertical Curve AASHTO Controls (Crest)(Crest)

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

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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: Table 3.4 of text)

Lmin = 2 (570’) – 2158’ = 600.5’

|(-1-3)|

S < L, so it doesn’t match condition

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Example: Assume SSD < L,

Design speed is 60 mph

G1 = 3% and G2 = -1%,

what is L?

Assuming average grade = 0%

SSD = 570 feet - ( Table 3.4 of text)

Lmin = |(-3 - 1)| (570 ft)2 = 602 ft

2158

SSD < L, equation matches condition

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Evaluation of example:

The AASHTO SSD distance equations provided the same design length from either equation in this special case. (600 compared to 602 - this is not typical)

Garber and Hoel recommend using the most critical grade of - 1% for SSD computation.– Resulting SSD would be: d = 573 ft– Resulting minimum curve: L = 608 ft

Difference between 602 and 608 is too small to worry about

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Text example : g1 = + 3% g2 = -3%

Design speed of 60 mph

If SSD = 570’ (AASHTO – no grade consideration)

Resulting minimum curve: L = 903 ft (S < L)

Consider grade per Garber and Hoel (p 693-694)

SSD, using - 3% grade, 598’

Resulting minimum curve L = 994 ft

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Assessment of grade Assessment of grade adjustmentadjustment

If sight distance is less than curve length, the driver will be on an upgrade a greater portion of the distance than on a down grade

(for eye ht = 3.5’ and object ht = 2.0 ft, 68% of the distance between eye and object will be on + grade.)

For crest vertical curve, selecting a curve length based on down grade SSD may produce an overly conservative design length.

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AASHTO design tablesAASHTO design tables

Vertical curve length can also be found in design tables

L = K *AWhere

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

Charts from Green Book

35From Green book

36From Green book

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Vertical Curve AASHTO Controls Vertical Curve AASHTO Controls (Crest)(Crest)

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)

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Chart vs computedChart vs computed

From chart V = 60 mph K = 151 ft / %

change

For g1 = 3 g2 = - 1

A = |g2 – g1| = |-1 – 3| = 4

L = ( K * A) = 151 * 4 = 604

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Sag Vertical CurvesSag Vertical Curves

Sight distance is governed by night- time conditions– Distance on curve illuminated by

headlights need to be consideredDriver comfortDrainageGeneral appearance

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Vertical Curve AASHTO Controls Vertical Curve AASHTO Controls

(Sag)(Sag) Headlight Illumination sight distance

S < L: L = AS2

400 + (3.5 * S)

S > L: L = 2S – (400 + 3.5S) A

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Vertical Curve AASHTO Controls Vertical Curve AASHTO Controls

(Sag)(Sag)

For driver comfort use: L > AV2

46.5 (limits g force to 1 fps/s)

To consider general appearance use:

L > 100 A

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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

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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

L = 6 x (313.67)2 = 394.12 ft

400 + 3.5 x 313.67

313.67 < 394.12, so condition applies

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Sag Vertical Curve: ExampleA 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 = (6 x [40 mph]2) = 206.5 feet 46.5 46.5

Testing for appearance:

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

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Vertical Curve AASHTO Controls Vertical Curve AASHTO Controls (Sag)(Sag)

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

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

As in crest, use min L = 3V

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Other important issues:Other important issues:

Use lighting if need to use shorter L than headlight requirements

Sight distance at under crossings

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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. Find elevations and station for VPC (BVC) and VPT (EVC).

L/2 = 1092.0 ft

Station at VPC = [345 + 60.00] - [10 + 92.00] = 334 + 68.00

Vertical Diff VPI to VPC: -0.03 x (2184/2) = - 32.76 feet

ElevationVPC = 250 – 32.76 = 217.24 feet

Station at VPT = [345 + 60.00] + [10 + 92.00] = 357 + 52.00

Vertical Diff VPI to VPT = -0.04 x (2184/2) = - 43.68 feet

Elevation VPT = 250 – 43.68 = 206.32 feet

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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

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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. Find elevation on the curve at a point 400 feet from VPC.

Y = A x 2 = - 7 x (400 ft)2 = - 2.56 feet

200L 200 (2814)

Elevation at tangent = 206.32 + (400 x 0.03) = 218.32

Elevation on curve = 218.32 – 2.56 feet = 226.68’

51Calculating x from VPC, calculating tangent elevation along +3% tangent

Y

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53Calculating x from VPT, calculating tangent elevation along +4% tangent

Y

54Source: Iowa DOT Design Manual

55Source: Iowa DOT Design Manual

56Source: Iowa DOT Design Manual

Note: L is measured from here to here

Not here