Ch3: Vertical alignment, p.265-286You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying. If you forgot, please review Chapter 21 of your survey book by McCormac.

Explain why the criteria for determining the length of a crest and a sag curve are different

Identify a crest and a sag curve

Describe the steps for determining the length of a vertical curve

Lay out a vertical curve (Self review)

After this lecture you will be able to:

Criteria for determining the length of the vertical curve

The main criteria are:

Provision of stopping sight distance (as defined by AASHTO)

Adequate drainage

Comfortable 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.

Crest vertical curves

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

AS

A

HHSL

21582

2002

2

21min

2158

2002

2

21

2

min

AS

HH

ASL

ftH

ftH

0.2

5.3

2

1

Eq.3-41Eq.3-42

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.

gA

HL

g

HS

21 100

2

100

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

12

121

22

21 0

)(

100100

HH

AHHHAg

gA

H

g

H

dg

dS

Substitute g in S equation above and get

A

HHSL

2

21min

2002

g Ag

(A - g)

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 (eq. 16-16) 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/ L

S

AL

H and

2

222

)2/(800/ L

S

AL

H

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

221

2

min200 HH

ASL

E

Graphical representation of minimum crest vertical lengths (Exhibit 3-71: Design Controls for Crest VC)

Sag vertical curvesThe 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 = 1degree (divergence angle)

Two cases of SSD on sag vertical curves

For S > L For S < L

A

SS

A

SHSL

5.34002

tan2002min

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

S

AS

SH

ASL

5.3400

tan2002

2

min

Graphical representation of minimum sag vertical lengths (Exhibit 3-74: Design Controls for Sag VC)

3 other criteria for sag vertical curves

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 Exhibit 3-74):

A minimum grade of 0.3% must be provided within 50 ft of the level point of the curve.

General appearance (minimum length):

L = 100A

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

S

AS

SH

ASL

5.3400

tan2002

2

min

K-value for crest vertical curves

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

K-value for sag vertical curves

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

Sight Distance at Undercrossings, p.277

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

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?

General controls for vertical alignment, p.279

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 proving 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.