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3.4.6 Vertical Curves, p.3- 149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed in CE361 Intro to Transportation Engineering. If you forgot the concept, please review Chapter 21 of your survey book (by McCormac?). Be familiar with the steps for determining the length of the vertical curve Know that the criteria for determining the length of crest and sag curves are different Know how to lay out a vertical curve
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3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Jan 08, 2018

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Elmer Cain

Criteria for determining the length of the vertical curve, p.3-149 The main criteria are: Provision of stopping sight distance (as defined by AASHTO) Adequate drainage Comfort in operation Pleasant appearance For both crest and sag vertical curves For sag curves only Roller coaster and hidden dip effects need to be avoided. When a vertical curve ahead is too small, the road may appear sharply bent.
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Page 1: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

3.4.6 Vertical Curves, p.3-149 ~ p.3-163

You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed in CE361 Intro to Transportation Engineering. If you forgot the concept, please review Chapter 21 of your survey book (by McCormac?).

Be familiar with the steps for determining the length of the vertical curve

Know that the criteria for determining the length of crest and sag curves are different

Know how to lay out a vertical curve (Review it by yourself)

Page 2: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Criteria for determining the length of the vertical curve, p.3-149

The main criteria are:

Provision of stopping sight distance (as defined by AASHTO)

Adequate drainage

Comfort in operation

Pleasant appearance

For both crest and sag vertical curves

For sag curves only

Roller coaster and hidden dip effects need to be avoided.

When a vertical curve ahead is too small, the road may appear sharply bent.

Page 3: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Crest vertical curves,

Fig. 3-41

Page 4: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Two cases of SSD on crest vertical curvesSSD < Length of V-curveSSD > Length of V-curve

AS

AHH

SL

21582

2002

2

21min

2158

2002

2

21

2

min

AS

HH

ASL

ftHftH

0.25.3

2

1

Eq. 3-43Eq. 3-44

Fig. 3-42

Page 5: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Derivation of crest vertical curve length formulas: S > L

Let g represent the difference between the gradient of the sight line and the gradient G1. Then, A – g will be the difference between the gradient of the sight line and the gradient G2.

gAHL

gHS

21 100

2100

To find the slope of the sight line that will make S a minimum, set dS/dg = 0.

12

121

22

21 0

)(100100

HHAHHHA

g

gAH

gH

dgdS

Substitute g in S equation above and get

A

HHSL

2

21min

2002

g Ag

(A - g)

See Appendix of this course notes file.

Page 6: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Derivation of crest vertical curve length formulas: S < L

Use the basic offset property of the parabolic curve, y = ax2. As long as the point of interest is within the parabola, we can use this. We know the mid-curve offset E = AL/800 and this happens when x = L/2. So if you have the curve offset y = H1 with x = S1, we have:

2

211

)2/(800/ LS

ALH

and

2

222

)2/(800/ LS

ALH

Solve for S1 and S2, and sum them to get S.

221

2

min200 HH

ASL

E

Page 7: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Graphical representation of minimum crest vertical lengths (Fig. 3-43: Design Controls for Crest VC)

Page 8: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Sag vertical curves (p.3-157)The minimum length of sag vertical curves is controlled by (1) sight distance provided by the headlight (at night: during the day you can see the vehicles in the opposite direction), (2) rider comfort, (3) control of drainage, and (4) general appearance.

H = 2 ft (headlight height), β = 1-degree (divergence angle)

Page 9: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Two cases of SSD on sag vertical curves(p.3-158)

For S > L For S < L

A

SS

ASHSL

5.34002

tan2002min

H = 2 ft, the height of the headlight above the ground

SAS

SHASL

5.3400

tan2002

2

min

Page 10: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Graphical representation of minimum sag vertical lengths (Fig. 3-44: Design Controls for Sag VC, p.3-159)

Page 11: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

3 other criteria for sag vertical curves(p.3-160)

Comfort criteria (minimum length), usually 50% of the SSD requirement:

5.46

2AuL u = design speed, mph

Drainage criteria (maximum length within which a grade must be established) when curbs are used (Know how to read the Drainage maximum line in Figure 3-44):

A minimum grade of 0.3% must be provided within 50 ft of the level point of the curve. (K=167)

General appearance (minimum length):

L = 100A

Page 12: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Another K value…

The minimum lengths of the crest and sag curves which are computed based on stopping sight distance (S < L cases) can be expressed like:

L = KA

To make it easier to get the value from the minimum curve length tables or charts.

Crest vertical curves: Sag vertical curves:

1329

2002

2

21

2

min

AS

HH

ASL

SAS

SHASL

5.3400

tan2002

2

min

Page 13: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

K-value for crest vertical curves(p.3-155)

a. Rate of vertical curvature, K, is the length of curve per percent algebraic difference in intersecting grades (A). K=L/A

Table 3-34

Page 14: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

K-value for PSD (rare case, though) (p.3-156)

Page 15: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Change in GB2011, because PSD criteria changed (p.3-157)

Page 16: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

K-value for sag vertical curves, (p.3-161)

a. Rate of vertical curvature, K, is the length of curve per percent algebraic difference in intersecting grades (A). K=L/A

Tab 3-36

Page 17: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Sight Distance at Undercrossings, p.3-161

Sight distance on the highway through a grade separation should be at least as long as the minimum stopping sight distance and preferably longer.

(Fig. 3-45)

Page 18: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

h1 = 8 ft for a truck driver, h2 = 2.0 ft for the taillights of a vehicle.

Why do we use the truck driver’s eye height here? p.3-162 through p.3-163

Page 19: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

See p.3-162 through p.3-163

Page 20: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

General controls for vertical alignment, p.3-163 ~p.3-164

A smooth gradeline with gradual changes should be sought.The “roller-coaster” or “hidden-dip” type of profile should be avoided.Undulating gradelines should be avoided.A “broken-back” gradelines should be avoided.On longer grades, it is recommended to have the steepest grade at the bottom and flatten the grades near the top of the ascent or to break the sustained grade by short intervals of flatter grade instead of providing a uniform sustained grade.Where at-grade intersections occur, reduce the grade through the intersection.Sag vertical curves should be avoided in cuts unless adequate drainage can be provided.

Page 21: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.

Appendix

Page 22: 3.4.6 Vertical Curves, p.3-149 ~ p.3-163 You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed.