Vertical Alignment

1
Vertical Alignment

• CTC 440

2
Objectives

• Understand the basics of profiles
• Understand the basics of vertical curves

3
Grades and Elevations

• Grade-A change in elevation per unit horizontal
  length ( or - in direction of stationing)
• Elevations are usually computed to the nearest
  one-hundredth of a foot or to the nearest
  one-thousandth of a meter (43.32 ft 21.262m)

4
Examples

• Computing a grade between two stations with known
  elevations
• Computing an unknown elevation at a station given
  the known elevation at another station and given
  the grade between the stations

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Compute a grade between two pts w/ known
elevations

• Sta 100 Elev.198.30
• Sta 200 Elev.203.80
• Gradechange in elev/change in length
• Answer Grade5.50

6
Compute an unknown elevation at a station

• Sta 530 Elev.270.00
• Grade between 530 770-2.5
• Calculate elev at 770
• Answer 264.00 feet

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Vertical Curves-Definitions

• Vertical curves are parabolic curves instead of
  circular curves
• Crest Curves (3 types)
• Sag Curves (3 types)

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Other Vertical Curve Parameters

• PVI-Point of Vertical Intersection-intersection
  of the two grades
• PVC-Point of Vertical Curvature-beginning of the
  vertical curve
• PVT-Point of Vertical Tangency-end of the
  vertical curve

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Other Vertical Curve Parameters

• M,middle ordinate-A computed correction, which
  must be applied to the PVI to determine the
  elevation at the midpoint of the vertical curve
• G1grade before the PVC
• G2grade after the PVT
• Llength of the vertical curve

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Vertical Curve Elevations

• Need G1, G2, and L
• AG2-G1 ()
• rA/100L
• Elevx(r/2)x2g1xElevPVC
• g1 is in decimal form
• X is distance measured from PVC

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Calculating M and min/max elevations

• MAL/800
• Max. or min. elev. Occurs _at_ x-g1/r
• g1 is in decimal form
• To get the actual elevation substitute x into the
  elevation equation
• Elevx(r/2)x2g1xElevPVC

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Vertical Curve Examples

• English (crest) ---see below
• Metric (sag) ---on board if needed
• Comprehensive Curve (sag)---see below

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English-Crest

• A vertical crest curve with a length of 400 is
  to connect grades of 1 and -1.75. The PVI is
  located at station 35 and has an elevation of
  549.20.
• What are the elevations of the PVC, PVT and at
  all full stations on the curve?

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English-Crest

• Elev at PVC549.2-(1200)547.20
• Elev at PVT549.2-(1.75200)545.70
• Find A-1.75-1-2.75 (in )
• Find rA/(100L)-.0000688
• Find r/2-.0000344
• Find Elevations at even stations
• Elev(x)r/2x2 g1x Elev(PVC)

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English-Crest
X Sta r/2 X2 G1 X PVC elev Elev
0 3300 -.0000344 0 .01 0 547.20 547.20
100 3400 -.0000344 1002 .01 100 547.20 547.86
200 3500 -.0000344 2002 .01 200 547.20 547.82
300 3600 -.0000344 3002 .01 300 547.20 547.11
400 3700 -.0000344 4002 .01 400 547.20 545.70 (Check)
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English-Crest

• Highest Elev occurs _at_ x-g1/r145.35
• Plug x in elevation equation to get
• Highest elevation is 547.93 at Sta 3445.35

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Comprehensive Curve Example

• Finding a PVI
• Fitting a curve
• Finding elevations on the curve
• Finding M
• Finding low point

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Fitting a Curve Between Two Set Tangents (1/6)

• Given
• G1-2 G23
• Two Fixed Points (sta/elev is set)
• Sta 100 Elev450.00
• Sta 700 Elev460.00

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Fitting a Curve Between Two Set Tangents (2/6)

• Find PVI Station and Elevation
• 450-.02L1460-.03L2
• L1L2600
• Solve for L1 and L2
• L1160 ft
• L2440 ft
• PVI Sta260 (Sta 100 160)
• PVI Elev446.80

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Fitting a Curve Between Two Set Tangents (3/6)

• If were constrained at Sta 1 and 7 then the
  maximum vertical curve length we can fit is
  160160320
• L320
• A5
• rA/100L.0001563
• r/2.0000781

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Fitting a Curve Between Two Set Tangents (4/6)

• Elev (x)r/2X2g1XEl PVC

X STA r/2 X2 G1 X Elev PVC Elev
0 100 .0000781 0 -.02 0 450.00 450.00
100 200 .0000781 1002 -.02 100 450.00 448.78
160 260 .0000781 1602 -.02 160 450.00 448.80
200 300 .0000781 2002 -.02 200 450.00 449.12
300 400 .0000781 3002 -.02 300 450.00 451.03
320 420 .0000781 3202 -.02 320 450.00 451.60
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Fitting a Curve Between Two Set Tangents (5/6)

• Determine Curve Elevation _at_ PVI
• MAL/8002
• PVI STA 260 Elev 446.8
• Curve Elev _at_ PVI446.802448.80

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Fitting a Curve Between Two Set Tangents (6/6)

• Determine Low Point
• X-g1/r.02/0001563127.96
• Sta227.96
• Elev _at_ x127.96 448.72

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General Rules for Establishing Vertical Alignment

• Goal-Provide a uniform, comfortable ride and safe
  vehicle operation
• Balance cut/fills
• Gradesgt0.5 to prevent drainage problems
• Check SSD/HSD
• Check driveway and intersecting road tie-ins
• Keep simple (few curves, flat curves, gradual
  grades)
• Check clearances (over/under bridge, over
  culverts)

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Next lecture

• Checking sight distances on horizontal curves
• Checking sight distances on vertical curves
• Crest-Stopping sight distance
• Sag-Headlight sight distance