Lecture 5

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Horizontal Alignment

The horizontal alignment is a combination of tangent and curved components. The curves are most commonly circular curves but spiral transitions are sometimes used.


Today’s class

Properties of the horizontal curve

Determining maximum degree of curvature


Components of Circular Curves

PC - point of curvature

PI - point of intersection

PT - point of tangency

T - length of tangent (PC - PI or PI - PC)

M - Middle ordinate

L - length of curvature

D - external angle (in degrees)


Circular Curves

Traditionally, the steepness of the curvature is defined by either the ‘radius’ (R) or the ‘degree of curvature’ (D)


The degree of curvature is not used in the metric version of the policy because D is defined in terms of feet


The degree of curvature is defined as the angle subtended by an arc of length 100 ft

(this is referred to as the arc definition of D. There is also a chord definition but the arc definition is the one used for highway design applications)


Derive the relationship between D and R!


R = 5730/D


Length of Curve

Derive the relationship between L, external angle and R!


L = R D / 57.3


Note that for a given external angle the length of curve is directly related to the radius


In other words, the longer the curve, the larger the radius of curvature


How do we determine (for the following parameters) appropriate design values? external angle, curve length, curve radius


The starting point is to determine the minimum radius for design

Once Rmin is determined, the designer can use any value of R > Rmin


Minimum Radius

Equation for Minimum Radius


Derived by considering the forces acting on the occupants of a vehicle negotiating a curve and the resulting comfort level of the occupants


The minimum radius is a function of the velocity, the allowable side friction and the degree of superelevation


Derive the equation relating V, f, e, R!


f+e = V2/gR

dimensionally consistent


f+e/100 = V2 /127R

where V is kph, R is meters, g=9.809 m/s2



Maximum Superelevation

The minimum radius is a function of speed, maximum superelevation and maximum allowable side friction


What are the factors affecting the maximum superelevation and maximum allowable side friction?


Maximum Superelevation

What are the practical factors limiting side friction?


Low Speed

High CG/Loose suspension of some cars


The emax is selected based on the climate, terrain, and the likelihood of slow moving traffic


Recommended Values for emax

Absolute maximum value recommended (expect on gravel roads) is 12%


Maximum value in areas with snow and ice is 8%


Maximum value in areas where slow traffic is likely (urban areas) is from 4% to 6%


No superelevation is recommended in urban areas where congestion is expected



Side Friction Factor

A friction force is developed between the tires and the road surface to counteract forces developed during cornering


The amount of side friction that will develop is given by


f = V2 /127R + e/100


The allowable side friction (fallow) for design is much less than the impending friction that would be developed to counteract sliding


As with braking friction, maximum side friction is a function of speed, road type and condition, and tire condition


The fallow is based on the level that is comfortable and safe for car occupants

Typical values: 0.16 at 100 kph, 0.10 at 110


Lecture 6

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Horizontal Alignment Design

Last class we looked at the procedure for determining minimum radius of curvature for our design


This class we look at the overall procedure for the design of each curve in the alignment

Also we will look at the procedure for determining superelevation runoff


Elements in Horizontal Alignment Design

Determine Rmin

Determine R for each curve (R>Rmin)

Determine emax

Determine e for each curve

Determine method of superelevation runoff

Determine the length of superelevation runoff



Determination of ‘e’ for each Curve

The degree of superelevation for each curve depends on the radius - ‘e’ decreases as R increases (this is given as the relationship between e and 1/R)


The AASHTO guide a parabolic (as oppose to a simple straight line relationship) between e and 1/R



Fig 1



Determination of ‘e’ for each Curve

The method used in distributing ‘e’ is important because the choice of ‘e’ determines the amount of side friction which will develop

(f + e/100) proportional to 1/R


The amount of side-friction the comfort-level of the driver, and hence, the driving pattern


The justification for using the parabolic distribution is that drivers tend to overdrive on gentler curve so a little higher ‘e’ is needed on such curves; therefore, very high ‘f’ values are not developed even with overdriving


Figures III-10 to 14 give design ‘e’ as a function of R and V -- each figure is for a different emax



Design Values for Horizontal Curve

Design values are given in design tables


Tables III-7 to 11 (each table contain data for different emax)


Information in Tables

Rmin vs Design Speed

L - min length of superelevation runoff

Smallest R for normal crown (NC)

Smallest R adverse crown removed

Required ‘e’ for each value of R



Class Design Project - Based on Example project in the ASCE book, pg. 59


Develop two alternative routes and give the details of the curves for each route


Lecture 7

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Superelevation Runoff

On curves where superelevation is needed the roadway cross-section must be changed from a normal crown to a superelevated section - this change must be effected gradually over a relatively long distance


This distance consists of two stages: tangent runout and superelevation runoff



NC -----------RC----------------Full Superelevation


Transition both at beginning and end of curve


For design we need

To determine our method of superelevation

The length of the superelevation runoff


Length of Superelevation Runoff

The length of superelevation runoff is determined based on appearance - we want to try to avoid sharp breaks in the appearance of the profile at the pavement edge and to avoid sharp differences in the profile of the center-line and of the pavement edge


General Guideline: difference in centerline profile and edge profile should be < 0.5%


In Class Assignment

Calculate the length of run-off if the differences in slope between edge and centerline is to be <0.5%


Assume: Two-way, two-lane road, 3.6m wide lanes, nc = 0.015, e = 10%


Table III-14 gives minimum recommended L for different design speed, e, lane width

Table based partly on profile guidelines in Table III-13

Methods for Attaining Superelevation

Different approaches may be used in changing the cross-section from normal crown to superelevation


Three common methods

Revolve about centerline

Revolve about inside edge of curve

Revolve about outside edge of curve


Method 1 (about centerline) is most commonly used - the main advantage is that total change needed for each edge is not as large as with other methods

Method 3 (about outside) this has some advantages from an appearance point of view since the changes (and resulting distortions occurs on the lower side and is not as noticeable to drivers


The choice depends to some extent on need to accommodate drainage


Location of Runoff

Ideally the runoff should take place on a transitions section (such as we have with a spiral curve). If no spiral is available then we compromise: some of the runoff on tangent and some on curve


Typical Design (no spiral)

Tangent runout on tangent

2/3 of superelevation runoff on tangent

1/3 superelevation runoff on curve


Divided Highways

For divided highways, there is the additional question of whether to treat the whole cross-section as one unit for superelevation


Three approaches


Case 1: For narrow medians - whole cross-section (including median) treated as a whole unit


Case 2: Median width < 30ft - median held plane and pavement in each direction rotated separately


Case 3: Wide median - pavement in each direction treated separately (with different elevations at the median edges)


In some cases, lane widening is used on curves


Sight Distance on Horizontal Curves

Object on the inside of a curve limits the available sight distance - this formula is used to determine the relationship between sight distance and object offset


Derive the formula for the middle ordinate


M = R (1-cos (SD/200))


Spiral Curves

A transition curve is sometimes used in horizontal alignment design

It is used to provide a gradual transition between tangent sections and circular curve sections. Different types of transition curve may be used but the most common is the Euler Spiral


Advantages of Spiral Transition

Provides a smooth transition

Provides place for superelevation runoff


Properties of Euler Spiral

(reference: Route Surveying and Design, 5thEd by Meyer and Gibson]





Euler Spiral

Characteristics of Euler Spiral

Radius of spiral at any point is proportional to its length at that point

The spiral is defined by ‘k’ the rate of increase in degree of curvature per station (100 ft)




Central Angle

As with circular curve also important for spiral

Definition is a little different





compare for circular curves


Example for Euler Spiral

Note: the total length of curve (circular plus spirals) is longer than the original circular curve by one spiral leg


Example: The central angle for a curve is 24 degrees - the radius of the circular curve selected for the location is 1000 ft.


a) Determine the length of the curve (with no spiral)


b) If a spiral with central angel of 4 degrees is selected for use, determine the i) k for the spiral, ii) length of each spiral leg, iii) total length of curve


c) Verify that the condition given in the note above is correct

Lecture 8

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Horizontal Alignment Design: Esthetic Factors

Functional considerations in the AASHTO method of design afford the designer a great deal of discretion in selecting the design elements - the main restriction is that the curve radius should be greater than the minimum radius.


In this case, form does not necessarily follow function. The designer must be aware of the factors affecting the form (or appearance) of their design and must work to incorporate form into the design


Why is good form important?

Highways are prominent and are a relatively permanent part of the man-made environment


Good form may enhance the function of highways - particularly from a safety stand point


Continuity of Alignment

Some of the factors that the designer must consider are i) appropriate scale for the various elements, ii) appropriate sequence, iii) appropriate transition from one element to the other


‘Man-made America’ provide some guidelines for incorporating form into the highway design process


The fundamental consideration is the need to ensure continuity of the alignment

Continuity is considered to be desirable because

Continuous alignment matches the path of vehicles (promote safety)

Continuous alignment is a better match to the natural landscape (promote esthetics and perhaps economy)


Continuity of Alignment

Continuity refers to the overall 3-D form of the roadway


Road might be continuous in horizontal alignment but overall might be disjointed due to a lack of coordination between the horizontal and vertical alignment


To achieve continuity we must consider



Coordination of horizontal and vertical


Continuity of Horizontal Alignment

The horizontal alignment may consist of the following elements: tangent, spiral curves and circular curve


Continuity is determined by

Continuity of Form - how the various elements fit together

Continuity of Scale - the relative scale of the various elements


A continuous alignment should

appear smooth, free flowing

have no kinks or breaks obvious to the eye

have elements that appear to be part of a whole not individual pieces


Continuity of Form

Form - combination of elements to provide continuity


Two major concerns

The discontinuity at the point where a tangent is connected to a circular curve

The sequence of elements


Tangent-Curve Discontinuity

Point of contact appears as a sudden break in the path (pg 166 - Merritt]





Use of spiral curve removes this discontinuity

[pg 180]


Length of Spiral

How long should the spiral be?


One criteria - spiral should be as long as needed to accommodate superelevation runoff

This is considered to be insufficient from the point of view of appearance


Rule of thumb for appearance

S:C:S should be 1:2:1

(this ratio would be 1:7:1 if the superelevation criteria is used]


In addition, it is recommended that length of spiral should NOT be greater than the length of circular curve (to avoid the appearance of a sharp bend in the middle of the curve)


Spirals are also recommended for use with compound curves (C-S-C) is R2 > 1.5 R1 (or vice versa)


Sequence of Horizontal Elements

The alignment should be continuous in terms of the relationship of successive elements


For example, a tangent should not be put in between two curves that are in the same direction (broken back curve)


Also a single curve should not be significantly sharper than others on a given section of roadway

(For Example: if significantly sharper curves are to be used in a city then there should be a gradual change from very flat curves to steeper curves as the city center is approached)


Continuity of Scale

Scale - relative length of the different elements


Two main issues

Length of curve

Ratio of tangent to curve


Desirable Length of Curve

Short curves results in a discontinuous alignment - they look like kinks in the alignment (they may also be unsafe if they are so short that drivers over look them)


At freeway speeds - the eye focus is at 1000 - 2000 ft. Curves should be at least that long to be visually significant


Recommended desirable range for freeways

L = 1500 to 5500


At a design speed of 40 mph a range of 1000 to 2000 ft is more appropriate


Ratio of Tangent to Curve

Most common design - long tangent to short curve


For example on Merritt - alignment is 80% on tangent

This results in a discontinuous alignment where each element appear to be separate


One alternative - short tangent to long curve (Spline Alignment)

Example: GS Parkway where only about 20% of alignment is on tangent

This results in a visually continuous alignment in which the tangent appears to be part of a continuous compound curve


One difficult with this design is that it is difficult to incorporate spiral curves when the tangent is so short


Curvilinear Alignment

Second alternative - Curvilinear Alignment

Long, flat circular curves (simple or compound) connected by long spiral

Two-thirds of length on circular curve


Requires differ approach to setting out alignment

Road set out on the basis of arcs (circular) which are as long and flat as practical and are then joined by spiral transition


Tangents used in cities or in areas of flat terrain where they would be a better fit than flat curves


1/R Diagrams

1/R Diagram is a 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


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