Lecture 9

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**Vertical Alignment Design**

The vertical alignment is a combination of tangent and curved components. The curve sections in this case are PARABOLIC CURVES

The main design elements for vertical alignment

Design of Vertical Curve

Determination of maximum grade

Determination of suitable length of grade (and design of climbing lanes where appropriate)

This class

Properties of parabolic curve

Design of vertical curves

**Parabolic Curve**

CharacteristicsGrade changes gradually from G1 at VPC to -G2 at VPT

Total change in grade is A (the algebraic difference in grade in %)

A = |G1 - G2|

Curve characterized by K, the rate of change of curvature (given as the length of curve for a 1% change in grade)

K = L/A

Which is the gentler curve - small K or large K?

**Formulas for Parabolic Curves**

The Offset from the Tangent(y)The offset is proportional to square of the horizontal distance from the VPC

y = Ax^{2}/200L

The Distance to the turning point(xt)What is the grade at the turning point?

What is the change in grade from VPC to tp?

K is length for 1% change in grade. What is formula for xt?

x_{t}=G_{1}K

Elevation of turning pointElevation at turning point = (Tangent elevation at xt) - yt

yt is the offset at the turning point. What is formula?

yt = Ax^{2}_{t}/200L

**Example Calculations for
Parabolic Curve**

G1 = -1% G2 = +2.2%

Elevation of VPC = 125.230 m

Station of VPC = 2+500 (== 2500 m)

Station of VPI = 2+600

What is length of curve? K-value?

Find a) Station and Elevation of Low Point, b) Elevation at station 2+550 and 2+650.

**Design of Vertical Curves**

The main design issue is the determination of the minimum length (or minimum K) for a given design speed

The criterion that determine the minimum differs depending on the curve type (crest or sag)

Four basic curve types

Type I and II (crest curves)

Type III and IV (Sag Curves)

Design of Crest Vertical Curves

What practical factors are affected by length of curve?

Most stringent factor is the need for sufficient sight distance

**Crest Vertical Curves**

If the sight distance requirements are met then safety, comfort and appearance will not be a problem

Based on sight distance, formulas for minimum curve length

if S<L,

if S>L,

Try the first equation, if the result is inconsistent with S<L then try the second

for SSD, h1 = 1070 (1.07 m) , h2 = 150 (0.15 m)

if S<L,

if S>L,

**Design Charts for Crest
Vertical Curve**

Design values are given in

FIII-39 and 40(III-39 is based on upper limit for SSD and III-40 on the lower limit)

Figure is a plot of

A vs Lminfor different design speeds

Minimum K value is also given at each design speed - this K can be used as the design control (expect at low A values)

At low A, a limiting value of L is specified (this limiting value of L is approximately 0.6 V)

The curve length should always be greater than the value in FIII-39 or 40

**Drainage Criteria**

For proper drainage on type I crest curve (with curbs) the AASHTO recommend that the minimum G (slope) 15m from the low point should be at least 0.3%

What is the maximum K value that correspond to this criteria?

K = 51 (m per %)

This value of K is NOT considered an absolute maximum for design - the guide states that if this value is exceeded attention should be paid to ensure that the site will drain adequately

**Sag Vertical Curves**

Obviously SSD is not a problem for sag curves

What are the physical issues precluding the use of very short curves?

Four criteria considered

Headlight sight distance

Rider comfort

Drainage

Appearance

The AASHTO design is based on headlight sight distance

Minimum Length for Headlight Sight Distance

S<L

S>L

**Sag Vertical Curves**

What is the HLSD? HLSD == SSD

Drainage criteria is exactly the same as for crest vertical curves - if section is curbed then we need special attention to drainage if K>51

Design ChartsFigure III-41 and 42

Table III-37

Lecture 10

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**Vertical Alignment Design -
Maximum Grade**

The maximum grade is determined based on vehicle operating characteristics

In general, passenger cars have

Little or no loss of speed on 4-5% upgrades

Somewhat higher speeds for downgrades

No noticeable change on grades < 3%

For trucks,

Operating speeds on the level is approximately the same as for passenger cars

Significant loss of speed on up-grades

Increase in speed of up to 7% on downgrades

**Maximum Grade and Length of
Grade**

Maximum GradeAASHTO recommendation for maximum grade

5% at DS of 120 kph

7-12% at DS of 50 kph

Maximum should only be used were unavoidable

<Glenwood Canyon>

Minimum GradeNo curb - 0%

With curb - 0.5% (0.3% on high type pavements)

Length of GradeHow long should the slope be for a given grade?

The main issue is that the longer the slope the more the reduction in truck speed

If the reduction in truck speed is too great then we have a safety problem and a reduction in highway capacity

**Critical Length of Grade**

Accident rates have been show to increase dramatically as the speed differential between PC and trucks become greater than 15kph

Speed of truck on upgrade function of

Steepness

Length of grade

Weight/horsepower ratio

Entering speed

The AASHTO guide is based on wgt/hp ratio of 180 kg/KW but charts are also given for recreational vehicle

The designer need to determine if 180 kg/KW is appropriate for a given situation

**Critical Length of Grade**

AASHTO guide provides charts for determining the speed reduction - Figure III - 25A, III - 25B

Using the chartsa) Truck enter 6% upgrade at 50 kph. What is distance for 15kph reduction in speed? (III-25A)

b) Truck enter 3% downgrade at 30 kph. What is the speed after 500 m?

Measuring Length of SlopeThe critical length determined from above is the length on a tangent section - length on a vertical curve must be converted to an equivalent tangent grade length

For type I and III curves - 1/4 of the curve length is taken as part of the grade under consideration

For type II and IV curves - the length is measured from the VPI

**Measuring Equivalent Length
of Grade**

ExampleWhat is the equivalent tangent length for the 5% grade

L = 1/4 of type III curve + distance from start of tangent to the PVI of type IV curve

L = 1/4*200 + 1000 = 1050 m

**Climbing Lanes**

If the equivalent tangent length of curve exceeds the critical length then CLIMBING lane may be warranted

Aside from the grade length, factors that are considered in evaluating the need for climbing lane include i) traffic volume and ii) traffic composition

Climbing lane should

Begin where truck speed falls to below 15 kph of the overall running speed

End where truck speed is again within 15 kph of the overall running speed (a practical compromise design is to end the lane where the truck can return to the normal lane without undue hazard, i.e., where there is sufficient SD for safe passing)

Lecture 11

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**Continuity in Vertical
Alignment**

From an esthetic point of view, one of the goal in alignment design is to achieve a continuous alignment. Continuity is desirable so that the road fit the terrain, does not have a jarring aspect and present a path which is easy to follow

To achieve continuity two different elements are considered

Continuity of Form

Continuity of Scale

Continuity of FormFor vertical curves continuity of form is not a problem. There is no sudden break in grade at the PC or PT since the parabolic curve provides a gradual transition in grade

**Continuity of Scale**

Continuity of scale relates to the relative lengths of the tangent and curves

If the curve is very short, it is seen as a sudden break in the alignment

Common flaws from very short curves

Board Effect, Hump, Break

Length that is adequate from the point of view of function is too short for appearance

Function Design: DS = 110 kphCrest - kmin = 80

Sag - kmin = 43

Minimum desirable for appearance is 300 mThis minimum is particularly important for Sag and crest with small ‘A’

For crest with large ‘A’ minimum length for function is adequate for appearance

The recommended design in Man-made America is a

curvilinear profile

Curvilinear profileis defined as one with greater than 50% of the alignment on curve (traditionally only 25% of alignment is on curve)

This curvilinear alignment can be achieved by

reducing number of curves

increasing length of remaining curves

using compound curves

**Coordination of Vertical and
Horizontal Alignment**

Continuity in plan and profile does not guarantee overall continuity of alignment in 3-D

Vertical and horizontal alignment must be designed to be in sync with each other

Two main issuesfor coordination

Relative scaleof vertical and horizontal elements

Relative locationof horizontal and vertical curves

**General Rules for Proper
Coordination**

Curves in vertical and horizontal alignment should be about the same length

Horizontal and vertical curves should not start and end at the same point. Horizontal curve should

leadand generally remainlonger

Horizontal and vertical curves should generally coincide in location (location of vertices). A shift of up to a quarter phase is OK. A shift of half a phase is not desirable (appear as break in alignment)

**Plots for Evaluating
Coordination**

1/R DiagramVisual and numerical tool for evaluating the continuity of the horizontal alignment

On 1/R plot

Tangents - plot as zero

Curves - plot as straight horizontal lines

Spirals - plot as sloping straight lines

Area under plot proportional to degree of curve

1/100K DiagramVisual tool for evaluating vertical alignment

Vertical curve is roughly equivalent to a circular curve with radius of 100K

Area under plot is proportional to ‘A’

These two plots can be used together to assess coordination of vertical and horizontal alignment

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