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


Grade 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 = Ax2/200L


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?




Elevation of turning point

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

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

yt = Ax2t/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 Lmin for 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




The AASHTO design is based on headlight sight distance

Minimum Length for Headlight Sight Distance






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 Charts

Figure 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 Grade

AASHTO 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 Grade

No curb - 0%

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


Length of Grade

How 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


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 charts

a) 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 Slope

The 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


What 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 Form

For 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 kph

Crest - kmin = 80

Sag - kmin = 43


Minimum desirable for appearance is 300 m

This 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 profile is 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 issues for coordination

Relative scale of vertical and horizontal elements

Relative location of 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 lead and generally remain longer


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 Diagram

Visual 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 Diagram

Visual 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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