Vertical Curve Design - University of Idaho

[Pages:6]9/17/2009

Vertical Alignment Fundamentals

CE 322 Transportation Engineering Dr. Ahmed Abdel-Rahim, Ph.D., P.E.

Vertical Curve Profile Views

Fig. 3.3

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Offsets

Offsets are vertical distances from initial tangent to the curve

Fig. 3.4

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Offset Formulas

For an equal tangent parabola,

Y A x2

OR

Y ax2 (G2 G1) x2

200 L

2L

Y = offset (ft) at any distance, x, from the PVC x, A, and L are as previously defined careful with units...

1st equation:

if A is in %...x and L should be in feet If A is in ft/ft...L should be in stations and x in feet

2nd equation:

If grade is in %...x and L should be in station If grade is in ft/ft...x and L should be in feet

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`K' Values

Rate of change in grade at successive points on the curve is

Constant = L/A in percent per ft

L/A ...distance required per 1% change in gradient The quantity L/A is termed `K'

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Distance to Zero Grade

(low or high point)

Computing high/low points for curves (provided the high/low point is not at a curve end) by,

xhl = K |G1|

or

xhl

G1L G2 G1

Where xhl = distance from the PVC to the high/low point in feet

Careful of units... 1st equation: if G1 is in % then xhl is in feet if G1 is in ft/ft then xhl is in stations 2nd equation:

L can be in feet or stations and you will have xhl in similar units.

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Example Problem

A 1600-ft-long sag vertical curve (equal tangent) has a PVI at station 200+00 and elevation 1472 ft. The initial grade is ?3.5% and the final grade is +6.5%. Determine the elevation and stationing of the low point, PVC, and PVT.

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Vertical Curve Through a Point

Illustration Solution steps

Use Equation 3.1 Determine parameters a, b, and c Substitute parameters into Equation 3.1 Solve for L as a quadratic equation

Review problem Culvert clearance

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SSD and Crest Vertical Curve Design

SSD and Curve Design

Design vertical curves, to provide adequate stopping-sight distance (SSD)

Minimize costs by minimizing curve length

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SSD and Curve Design

SSD formulation was given in Chapter 2 and 3 Eq. 2.50 ds = d + dr (Eq. 2.50)

Eq.

3.12

SSD

V12

2g

a g

G

V1 tr

SSD given in Table 3.1 using AASHTO values of a = 11.2 ft/s2 and tr = 2.5 sec

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SSD Factors

Important for crest curves

Required sight distance Curve length Initial and final grades (which grade??) Eye and object heights

Fig. 3.6

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Minimum Curve Length

Minimum curve length, based on parabola

Using the equations

Eq. 3.13 Eq. 3.14

A SSD2

Lm

2 for SSD L

200 H1 H2

2

200 Lm 2 SSD

H1 A

H2

for SSD L

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Minimum Curve Length

For adequate SSD use the following specifications: H1 (driver's eye height) = 3.5 ft (1080 mm) H2 (object height) = 2.0 ft (600 mm)

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Minimum Curve Length

Substituting these values into previous two equations yields:

US Customary

Metric

For SSD < L

Lm

A SSD 2 2158

Lm

A SSD 2 658

For SSD > L

Lm

2 SSD

2158 A

Lm

2 SSD

658 A

(3.15) (3.16)

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Example Problem 3.5

A highway is being designed to AASHTO guidelines with a 70-mph design speed and, at one section, an equal tangent vertical curve must be designed to connect grades of +1.0% and ?2.0%. Determine the minimum length of vertical curve necessary to meet SSD requirements.

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Example Problem 3.5

Eq. 3.15, SSD < L Eq. 3.16, SSD > L

Both are very similar, but choose 740.82 because > SSD (730)

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Curve Where SSD > L

Speed = 70 mph SSD = 730 ft Just reduce the final grade

G1 = 1% G2 = -1%

So...A = 2

(See Mathcad worksheet)

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K Values for Adequate SSD

Table 3.2

Design Controls for Crest Vertical Curves Based on SSD

US Customary

Metric

Design speed (mi/h)

Stopping sight

distance (ft)

Rate of vertical curvature, Ka Calculated Design

Design speed (km/h)

Stopping sight

distance (m)

Rate of vertical curvature, Ka Calculated Design

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80

3.0

3

20

20

0.6

1

20

115

6.1

7

30

35

1.9

2

25

155

11.1

12

40

50

3.8

4

30

200

18.5

19

50

65

6.4

7

35

250

29.0

29

60

85

11.0

11

40

305

43.1

44

70

105

16.8

17

45

360

60.1

61

80

130

25.7

26

50

425

83.7

84

90

160

38.9

39

55

495

113.5

114

100

185

52.0

52

60

570

150.6

151

110

220

73.6

74

65

645

192.8

193

120

250

95.0

95

70

730

246.9

247

130

285 123.4

124

75

820

311.6

312

80

910

383.7

384

a Rate of vertical curvature, K, is the length of curve per percent algebraic difference in

intersecting grades (A). K = L/A

Source: American Association of State Highway and Transportation Officials, "A Policy on Geometric Design of Highways and Streets," Washington, D.C., 2001.

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Notes About Both K-values Tables

Because K = L/A...L = SSD2/2158 (US Customary)

For SSD < L

Table 3.2,

SSD calculations use G = 0

For larger grades > 3% calculate SSD

Assume SSD < L

Effects of assumption

Equation for Lm with SSD > L ... equal to or larger than other equation

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Check SSD < L Assumption

How would you test this assumption?

(see equations 3.15 and 3.16)

(see Matchcad worksheet)

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