Geometric Design Lecture5 Prof. Garrick

Geometric Design Lecture5 Prof. Garrick

http://www.engr.uconn.edu/~garrick/ce371/l05-08.htm[2/24/2015 7:47:33 PM]

Geometric Design of Transportation FacilitiesProfessor Norman W. Garrick

Lecture 5 6 7 8

Lecture 5 return to CE371 Mainpage, Top

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

1. Properties of the horizontal curve2. Determining minimum radius of curvature

Components of Circular Curves

BC - point of curvaturePI - point of intersectionEC - point of tangencyT - length of tangent (BC - PI)M - Middle ordinateL - length of curvatureD - external angle (in degrees)

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Click here for Figure showing the geometric relationships

Properties of 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. The arc definition is the one used for highway design applications. The chord definition is used in railroad design)

Based on the arc definition the relationship between R and D is as follows:

R = 5730/D

Formula for Length of Curve

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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

One design issue is to determine appropriate values for the follow: external angle, curve length or curve radius

The starting point is to determine the minimum radius for designOnce Rmin is determined, the designer can use any value of R > Rmin

Determining the Minimum Radius

The Minimum Radius is derived by considering the forces acting on the occupants of a vehicle negotiating a curve and the resulting comfort level of the occupants.

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

(0.01e+f)/(1-0.01ef) = V2/gR

note: f is in fraction, e is in %

The above formula is dimensionally consistent - the one below is used when each parameter is in the typical unit of measurement

(0.01e+f)/(1-0.01ef) = V2 /127R

where

V is kphR is meters g=9.809 m/s2

Maximum Superelevation and Maximum Allowable Side Friction

From the equation above, we can see that the minimum radius is a function of



speed, maximum superelevation and maximum allowable side friction.

In order to determine the minimum radius we must determine maximum allowable values for superelevation and side friction.

Maximum Superelevation

The practical factors limiting superelevation are

WeatherLow SpeedHigh CG/Loose suspension of some cars

Therefore, the emax is selected based on the climate, 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 - 0.01e

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 allowable side friction is selected to be comfortable and safe for all car occupants.Typical values: 0.17 at 30 kph, 0.10 at 110 kph

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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

1. Determine Rmin2. Determine R for each curve (R>Rmin)3. Determine emax4. Determine e for each curve5. Determine method of superelevation runoff6. 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

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

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

The amount of side-friction affect 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

Exhibit 3-16 to 20 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 the AASHTO design tables

Exhibit 3-21 to 25 (each table contain data for different emax)

Information in Tables

Rmin vs Design SpeedL - min length of superelevation runoffSmallest R for normal crown (NC)Smallest R adverse crown removedRequired ‘e’ for each value of R

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


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

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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%

Exhibit 3-29 gives minimum recommended L for different design speed, e, lane width

Table based partly on profile guidelines in Exhibit 3-27

Methods for Attaining Superelevation



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

Three common methods

Revolve about centerlineRevolve about inside edge of curveRevolve 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 of the cross-section and thus, 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 with some of the runoff on tangent and some on curve.

Typical Design (no spiral)

Tangent runout on tangentApproximately 80% of superelevation runoff on tangent (see Exhibit 3-30)

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))

where S is the sight distance, D is degree of curvature

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 transitionProvides place for superelevation runoff

Properties of Euler Spiral

(reference: Surveying: Principles and Applications, Kavanagh and Bird, Prentice Hall]



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)

ie

k = 100 D/ Ls

Central (or Deflection) Angle

As with circular curve the central angle is also important for spiral

Recall for circular curve

Delta (c) = Lc D / 100

But for spiral

Delta (s) = Ls D / 200

Equation for Total Deflection Angle

The total deflection angle for a spiral/circular curve system is

Delta (t) = Delta (c) + 2 Delta (s)

See figure below for more details on Spiral Geometry



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.

1. Determine the length of the curve (with no spiral)2. 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 curve3. Verify that the condition given in the note above is correct




This section is based largely on "Man-made America: Chaos or Control" by Tunnard and Pushkarev

The following discussion is applicably primarily to rural roadways and is especially important for freeways and other high speed facilities. In the urban context other factors, such as the existing road pattern, maybe be much more important in determining the fit of the alignment.

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 a continuos alignment






matches the path of vehicles (promote safety)match the natural landscape (promote esthetics and perhaps economy)

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

Continuity of Horizontal AlignmentContinuity of Vertical AlignmentCoordination 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 togetherContinuity of Scale - the relative scale of the various elements

A continuous alignment should

appear smooth, free flowing with 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 refers to how the elements (tangents and curves) of the alignment are combined.

Two major concerns

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

2. The sequence of elements

Tangent-Curve Discontinuity



Point of contact appears as a sudden break in the path

(pg 166 - Merritt, Man-made America]

Use of spiral curve removes this discontinuity.

[pg 180 - Man-made America]



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 - refers to the relative length of the different elements

Two main issues must be considered

1. Length of curve2. 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.

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 zeroCurves - plot as straight horizontal linesSpirals - plot as sloping straight lines

Lecture 5 6 7 8

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Geometric Design Lecture5 Prof. Garrick

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