Lec 22, Ch.16, pp.688-704: Vertical alignment (objectives)

You learned how to lay out a vertical curve, given grades, PVC, PVI, and PVT in CE113 Surveying. If you forgot, please review Chapter 21 of your survey book by McCormac.

Understand maximum vertical grades are set by the operating characteristics of the design vehicle on the highway

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

Know that the criteria for determining the length of the sag curve and the vertical curve are different

Know how to lay out a vertical curve (Read through pp.701-705.)

What we cover today in class…

Max grade – a function of the operating characteristics of the design vehicle

Main criteria of vertical curves

How to determine the minimum length of crest vertical curve

How to determine the minimum length of sag vertical curve

Recommended grades

Facts:

Grades of 4 to 5 % have little or no effect on passenger cars except for those with high weight/horsepower ratios (small cars)

With grades greater 5%, speeds of passenger cars decrease on upgrades and increase on downgrades

Truck speed may increase up to 5% on downgrades and decrease by 7% on upgrades, depending on the percent and length of the grade

Recommended grades (cont)

Maximum grades: Based on the operating characteristics of the design vehicle on the highway. See Table 16.4 for details. Use maximum grades sparingly.

Minimum grades: Based on the drainage conditions of the highway. Zero-percent grades may be used on uncurbed pavements with adequate cross slopes to laterally drain the surface water. When curbed, facilitate the longitudinal flow of the surface water. Use a minimum of 0.5%. May be reduced to 0.3% on high-type pavement.

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 effect

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

(Review Examples 16-1 and 16-2)

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

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.

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

3 other criteria for sag vertical curves

Comfort criteria (minimum length), usually 75% 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:

A minimum grade of 0.35% must be provided within 50 ft of the level point of the curve.

General appearance (minimum length):

L = 100A(Review Example 16-3.)

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 is 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