Chapter 2 - Note

CHAPTER 2: Geometric Design

LECTURE 3-6 CVE3202: HIGHWAY AND TRAFFIC ENGINEERING Page 1

1.0 Design of the alignment

The alignment of a highway is composed of vertical and horizontal elements.

The vertical alignment includes straight (tangent) highway grades and the parabolic curves that connect these grades.

The horizontal alignment includes the straight (tangent) sections of the roadway and the circular curves that connect their change in direction.

The design of the alignment depends primarily on the design speed selected for the highway.

The least costly alignment is one that takes the form of the natural topography.

It is not always possible to select the lowest cost alternative because the designer must adhere to certain standards that may not exist on the natural topography.

It is important that the alignment of a given section has consistent standards to avoid sudden changes in the vertical and horizontal layout of the highway.

It is also important that both horizontal and vertical alignments be designed to complement each other,



since this will result in a safer and more attractive highway.

One factor that should be considered to achieve compatibility is the proper balancing of the grades of tangents with curvatures of horizontal curves and the location of horizontal and vertical curves with respect to each other.

For example, a design that achieves horizontal curves with large radii at the expense of steep or long grades is a poor design.

Similarly, if sharp horizontal curves are placed at or near the top of pronounced crest vertical curves at or near the bottom of a pronounced sag vertical curve, a hazardous condition will be created at these sections of the highway.

Thus, it is important that coordination of the vertical and horizontal alignments be considered at the early stages of preliminary design.



2.0 Horizontal Alignment

2.1 Introduction

The horizontal alignment consists of straight sections of the road (known as tangents) connected by curves.

The curves are usually segments of circles, which have radii that will provide for a smooth flow of traffic.

The design of the horizontal alignment entails the determination of the minimum radius, determination of the length of the curve, and the computation of the horizontal offsets from the tangents to the curve to facilitate locating the curve in the field.

In some cases, to avoid a sudden change from a tangent with infinite radius to a curve of finite radius, a curve with radii varying from infinite to the radius of the circular curve is placed between the circular curve and the tangent.

Such a curve is known as a spiral or transition curve.

There are four types of horizontal curves: simple, compound, reversed, and spiral.



2.2 Simple Curves

When stop sight distance (SSD) is unobstructed,

Where = minimum radius (m)

= design speed (km/h)

= super elevation (m/m)

= coefficient of side friction

When there are obstruction in the roadway that limit SSD on the curve, R is determined as below:



The point at which the curve begins (A) is known as the point of curve (PC).

The point at which the curve ends (B) is known as the point of tangent (PT).

The point at which the two tangents intersect is known as the point of intersection (PI) or vertex (V).

A simple circular curve is described either by its radius or by the degree of the curve.

There are two ways to define degree of the curve, which is based on 30m of arc length (highway practice) or on 30m of chord length (railroad practice).



The angle subtended at the center of a circular arc 30m in length is the degree of curve, D.

If is the angle in radians subtended at the center by an arc of a circle, the length of that arc is given by

If Da is the angle in degrees subtended at the center by an arc of length L, then

Chord definition for R in term of Dc is based on a chord of 30m,



2.2.1 Formulas for Simple Curve



(

)

(

)



2.2.2 Field Location of a Simple Horizontal Curve

Simple horizontal curves are usually located in the field by staking out points on the curve using angles measured from the tangent at the point of curve (PC) and the lengths of the chords joining consecutive whole stations.

The angles are also called defection angles because they are the angle that is deflected when the direction of the tangent changes direction to that of the chord.



To set out the horizontal curve, it is necessary to determine and .

From

,



Example 1: Design of a Simple Horizontal Curve

The intersection angle of a 4 curve is 5525, and the PC is located at station 238+13.43. Determine the length of the curve, the station of the PT, the deflection angles and the chord lengths for setting out the curve at whole stations from the PC.

Solution 1:

Given D=4, Intersection angle, =5525, PC=238+13.43



Computation of

and

for whole station

Station Deflection Angle Chord Length (m)

PC 238+13.43 0 0

PC 239 1 16.57

PC 240 3 30

PC 241 5 30

PC 242 7 30

PC 243 9 30

PC 244 11 30

PC 245 13 30

PC 246 15 30

PC 247 17 30

PC 248 19 30

PC 249 21 30

PC 250 23 30

PC 251 25 30

PC 252 27 30

PT 252 + 9.06 27 9.05



2.3 Compound Curves

Compound curves consist of two or more simple curves in succession, turning in the same direction, with any two successive curves having a comment tangent point.

These curves are used mainly to obtain desirable shapes of the horizontal alignment, particularly for at-grade intersections, ramps of interchanges, and highway sections in difficult topographic areas.

To avoid abrupt changes in the alignment, the radii of any two consecutive simple curves that form a compound curve should not be widely different.

AASHTO recommends that the ratio of the flatter radius to the sharper radius at intersection should not be greater than 2:1 so drivers can adjust to sudden changes in curvature and speed.

The maximum desirable ratio recommended for interchanges is 1.75:1, although 2:1 may be used.

To provide a smooth transition from a flat curve to sharp curve, and to facilitate a reasonable deceleration rate on a series of curves of decreasing radii, the length of each curve should observe minimum length requirements, based on the radius of curve as below recommended by AASHTO.



Seven variables, in the figure, six of which are independent, since .

In order to lay out the curve, the intersection angles and chord lengths for both curves must be determined.

Usually and can be obtained from the layout plan.

and are usually known.

& = radii of simple curves forming the compound curve

& = intersection angles of simple curves

& = tangent lengths of simple curves

& = tangent lengths of compound curves

= intersection angle of compound curve



Example 2: Design of a Compound Curve

Figure below illustrates a compound curve that is to be set out at highway intersection. If the point of compound curve (PCC) is located at station (565+10.5), determine the deflection angles for setting out the curve.

Solution 2

Given , ,

Station



For length of horizontal curve of 150m radius,

(2+29.01) station

For length of horizontal curve of 105m radius,

(1+17.65) station

Station PC = (565+10.50) (2+29.01) = 562+11.49

Station PT = (565+10.50) + (1+17.65) = 566+28.15

For curve of 150m radius,



For curve of 105m radius,



Computation of deflection angle for setting out the curve,

1. 150m Radius Curve


PC 562+11.49 0 0

563 18.51

564 30

565 30

PCC 565+10.5 10.5

2. 105m Radius Curve


PCC 565+10.5 0

566+00 19.5

PT 566+28.15 28.15



2.4 Reverse Curves

Reverse curves usually consist of two simple curve with

equal radii turning in opposite directions with a

common tangent.

Reverse curves are generally used to change the

alignment of a highway. But they are seldom

recommended because sudden change to the alignment

may result in drivers finding it difficult to keep in their

lane.

Geometry of a Reverse Curve with Parallel Tangents



If d and D are known, the following variables can be determined.

Therefore, WOY is a straight line, and hence,

If d and R are known,



2.5 Transition (Spiral) Curves

Transition curves are placed between tangents and circular curves or between two adjacent circular curves having substantially different radii.

The use of transition curves provides a vehicle path that gradually increases or decreases the radial force as the vehicle enters or leaves a circular curve.

2.5.1 Length of Spiral Curves

If the transition curve is a spiral, the degree of curve between the tangent and the circular curve varies from 0 at the tangent end to the degree of the circular curve, Da at the curve end.

When the transition is placed between two circular curves, the degree of curve varies from that of the first circular curve to that of the second circular curve.

The minimum length of a spiral transition curve should be the larger of the values obtained from below equation:



Where

= minimum length of curve (m)

= speed (km/h)

R = radius of curve (m)

C = rate of increase of radial acceleration (m/s2/s)

= 1 to 3

Pmin = minimum lateral offset between the tangent and the circular curve (0.2m)

Under operational conditions, the most desirable length of a spiral curve is approximately the length of the natural spiral path used by drivers as they traverse the curve.

2.5.2 Length of Superelevation Runoff when Spiral Curves Are Not Used

Many highway agencies do not use spiral transition curves since drivers will usually guide their vehicles into circular curves gradually.

Under these conditions, the tangent is joined directly with the main circular curve (called tangent-to-curve transition).



However, if the curve is superelevated at a rate of e m/m, an appropriate transition length must be provided.

This superelevation transition length is comprised of superelevation runoff and tangent runout.

= minimum length of superelevation runoff

= maximum relative gradient (%)

= number of lanes rotated

= adjustment factor for number of lanes rotated

= width of one traffic lane (m)

= design superelevation rate (%)

= minimum length of tangent runoff (m)

= normal cross slope rate (%)

= design superelevation rate (%)

= minimum length of superelevation runoff (m)



2.6 Attainment of Superelevation

It is essential that the change from a crowned cross section to a superelevated one be achieved without causing any discomfort to motorists or creating unsafe conditions.

One of four methods can be used to achieve this change on undivided highways:

i. A crowned pavement is rotated about the profile of the centerline.

ii. A crowned pavement is rotated about the profile of the inside edge.

iii. A crowned pavement is rotated about the profile of the outside edge.

iv. A straight cross-slope pavement is rotated about the profile of the outside edge.

Superelevation is achieved on divided highways by using one of 1st to 3rd methods.

Method 1 involves superelevating the whole cross section, including the median, as a plane section.

The rotation in most cases is done about the centerline of the median.



This method is used only for highways with narrow medians and moderate superelevation rates, since large differences in elevation can occur between the extreme pavement edges if the median is wide.

Method 2 involved rotating each pavement separately around the median edges, while keeping the median in a horizontal plane.

This method is used mainly for pavement with median widths of 10m or less, although it can be used for any median, because by keeping the median in the horizontal plane, the difference in elevation between the extreme pavement edges does not exceed the pavement superelevation.

Method 3 treats the two pavements separately, resulting in variable elevation difference between the median edges.

This method generally is used on pavements with median widths of 12m or greater.

The large difference in elevation between the extreme pavement edges is avoided by providing a compensatory slope across the median.



The procedure used of method 1:

1st to raise the outside edge of the pavement relative to the centerline, until the outer half of the cross section is horizontal (point B).

The outer edge is then raised by an additional amount to obtain a straight cross section.

Note that the inside edge is still at its original elevation, as indicated at point C.



The whole cross section is then rotated as a unit about the centerline profile until the full superelevation is achieved at point E.


The centerline profile is raised with respect to the inside pavement edge to obtain half the required changed, while the remaining half is achieved by raising the outside pavement edge with respect to the profile of the centerline.




Is similar to method 2 with the only difference being a change effected below the outside edge profile.




Which is used for sections of straight cross slopes.



3.0 Vertical Alignment

3.1 Introduction

The vertical alignment of a highway consists of straight sections known as grades, (or tangents) connected by vertical curves.

The design of the vertical alignment involves the selection of suitable grades for the tangent sections and the appropriate length of vertical curves.

The topography of the area through which the road traverses has a significant impact on the design of the vertical alignment.

The topography of the profile of a road can generally be divided into three groups, namely, FLAT, ROLLING and MOUNTAINOUS.

FLAT terrain means the topographical condition where highway sight distances, as governed by both horizontal and vertical restrictions are generally long or could be made to be so without construction difficulty or expertise. The natural ground, cross section (i.e. perpendicular to natural ground contours) in a flat terrain are generally below 3%.



ROLLING terrain means the topographical condition where the natural slopes consistently rise above and fall below the road or street grade and where occasional steep slopes offer some restrictions to normal horizontal and vertical roadway alignment. The natural ground cross slopes in a rolling terrain are generally between 3% to 25%.

MOUNTAINOUS terrain means the topographical condition where longitudinal and transverse changes in the elevation of the ground with respect to the road or street are abrupt and where benching and side hill excavation are frequently required to obtain acceptable horizontal and vertical alignment. The natural ground cross-slopes in a mountainous terrain are generally above 25%.

3.2 Highway gradient

The vertical profile of road affects the performance of vehicles.

The effect of grades on trucks which have weight power ratio of about 300 lb/hp, is considered.

The maximum grade controls in terms of design speed is summarized as below:



Design Speed (km/hr)

Desirable Maximum Grade (%)

Maximum Grade (%)

120 2 5

100 3 6

80 4 7

60 5 8

50 6 9

40 7 10

30 8 12

20 9 15

The desirable maximum grades should be aimed at in most cases. The maximum grades should be used infrequently.

The total upgrade for any section of road should not exceed 3000m, unless the grade is less than 4%.

A desirable minimum grade or 0.5% should be used.

A grade of 0.35% may be allowable where a high type pavement accurately crowned is used.



On straight stretches traversing across wide areas of low lying swamps use of even flatter grades may be allowable with prior approval.

3.3 Critical gradient length

The term critical grade length indicates the maximum length of a designated upgrade upon which a loaded truck can operate without an unreasonable reduction in speed.

To establish the design values for critical grade lengths, for which grade ability of trucks is the determining factor, the following assumptions are made:

i. The weight-power ratio of a loaded truck is about 300 lb/hp.

ii. The average running speed as related to design speed is used to approximate the speed of vehicles beginning and uphill climb.

iii. A maximum reduction in speed to half the design speed is allowed for design speeds of 80km/hr or above.

iv. For design speeds of 50 and 40km/hr where the allowable minimum speeds are 30 and 25km/hr respectively.



3.4 Vertical curves

Vertical curves are used to provide a gradual change from one tangent grade to another so that vehicles may run smoothly as they traverse the highway.

These curves are usually parabolic in shape.

The expressions developed for minimum lengths of vertical curves are therefore based on the properties of a parabola.

Figure below illustrates vertical curves that are classified as crest or sag:



3.4.1 Length of Crest Vertical Curves

Provision of a minimum stopping sight distance (SSD) is the only criterion used for design of a crest vertical curve.

Stopping sight distance, S (m),

in km/h, in second, in m/s2, = 9.81 m/s2, = gradient

There are two possible scenarios that could control the design length:

1. The SSD is greater than the length of the vertical curve.

2. The SSD is less than the length of the vertical curve.

Assume:

Height of driver, H1 = 1.1m

Height of object, H2 = 0.15m

( )



S = X1 + X3 + X2 = X1 +

+ X2

( )

(For S L)



S = S1 + S2

(for S L)



Example 3: (Minimum length of a crest vertical curve)

A crest vertical curve is to be designed to join a +3% grade with a -2% grade at a section of a two-lane highway. Determine the minimum length of the curve if the design

speed of the highway is 96 km/h, S L, and a perception-reaction time of 2.5 second. The deceleration rate for braking (a) is 3.41 m/s2.

Solution 3

Since the grade changes constantly on a vertical curve, the worst-case value for G of 3% is used to determine the braking distance.

Given G = 3% to -2%, u = 96 km/h, t = 2.5 s, a = 3.41 m/s2,

A = 3% (-2%) = 5%, S L

( )

( )

SSD = 66.72 + 114.24 = 180.96m



Example 4: (Maximum safe speed on a crest vertical curve) An existing vertical curve on a highway joins a +4.4% grade with a -4.4% grade. If the length of the curve is 82m, what is the maximum safe speed on this curve? What speed should be posted if 8 km/h increments are used?

Assume a = 3.41 m/s2, perception-reaction time = 2.5 sec,

and that S L.

Solution 4

Given G = +4.4% to -4.4%, Lmin= 82m, a = 3.41m/s2, t = 2.5s,

S L, A = +4.4% (-4.4%) = 8.8%

( )

(maximum safe speed)

the speed should be posted = 55.4 - 8 = 48.4 km/h

use a conservative value of 48 km/h.



3.4.2 Length of Sag Vertical Curves

The selection of the minimum length of a sag vertical curve is controlled by the following four criteria:

1. SSD provided by the headlight,

2. Comfort while driving on the curve,

3. General appearance of the curve, and

4. Adequate control of drainage at the low point of the curve.



3.4.2.1 Minimum length based on SSD criterion

The headlight SSD requirement is based on the fact that sight distance will be restricted during periods of darkness by the sag curve.

As a vehicle is driven on a sag vertical at night, the position of the headlight and the direction of the headlight beam will dictate the stretch of highway ahead that is lighted. Therefore the distance that can be seen by the driver is controlled by the headlight beam.

H = height headlight above the ground = 0.6 m

= angle of headlight beam is inclined upward = 1



Using the properties of the parabola,

For ,

For ,



3.4.2.2 Minimum length based on comfort criterion

The comfort criterion is based on the fact that when a vehicle travels on a sag vertical curve, both the gravitational and centrifugal forces act in combination, resulting in a greater effect than on a crest vertical curve where these forces act in opposition to each other.

Several factors such as weight carried, body suspension of the vehicle, and tire flexibility affect comfort due to change in vertical direction, making it difficult for comfort to be measured directly.

It is generally accepted that a comfortable ride will be provided if the radial acceleration is not greater than 0.3 m/s2.

u = design speed in km/h

L = minimum length based on comfort

A = algebraic difference in grades



3.4.2.3 Minimum length of curve based on appearance criterion

The criterion for acceptable appearance is usually satisfied by assuring that the minimum length of the sag curve is not less than expressed by the following equation:

3.4.2.4 Minimum length based on drainage criterion

The drainage criterion for sag vertical curves must be considered when the road is curbed.

This criterion is different from the others in that there is a maximum length requirement rather than a minimum length.

The maximum length requirement to satisfy the drainage criterion is that a minimum slope of 0.35% be provided within 15m of the lowest point of the curve.



Example 5: (Minimum length of a sag vertical curve)

A sag vertical curve is to be designed to join a -5% grade to a +2% grade. If the design speed is 64 km/h, determine the minimum length of the curve that will satisfy all criteria.

Assume a = 3.41 m/s2 and perception-reaction time = 2.5 s.

Solution 5

Given G = -5% to +2%, u = 64km/h, a = 3.41m/s2, t = 2.5s

Find the stopping sight distance.

( )

( )

Determine whether or for the headlight sight distance criteria. For ,



(This condition is not appropriate since 98.23 130.20. Therefore ).

For ,

(This condition is satisfied since 98.23 145.63.)

Determine minimum length for the comfort criterion.

Determine minimum length for the general appearance criterion

The minimum length to satisfy all criteria is 210m.

(Note: In order to check the maximum length drainage requirement, it is necessary to use procedure for calculation curve elevations)



3.4.3 Elevation of Crest and Sag Vertical Curves

The minimum length of a crest and sag vertical curve must be known if the elevations are to be determined.

As was the case for formulas to compute length, the method for computing elevations relies on the properties of the parabola.



3.4.4 Design Procedure for Crest and Sag Vertical Curves

Step 1. Determine the minimum length of curve to satisfy sight distance requirements and other criteria for sag curves (comfort, appearance, drainage).

Step 2. Determine from the layout plans the station and elevation of the point where the grades intersect (PVI).

Step 3. Compute the elevations of the beginning of vertical curve (BVC) and the end of vertical curve(EVC).

Step 4. Compute the offsets Y, as the distance between the tangent and the curve. Usually equal distance of 30m (1 station) are used, beginning with the first whole station after the BVC.

Step 5. Compute elevations on the curve for each

station as: elevation of the tangent offset from the tangent Y. For crest curves the offset is (-) and for sag curves the offset is (+)

Step 6. Compute the location and elevation of the highest (crest) or lowest (sag) point on the curve.



Example 6: Design of Crest Vertical Curves

A crest vertical curve joining a +3% and a -4% grade is to be designed for 120km/h. If the tangents intersect at station (345+18) at an elevation of 76.2m, determine the stations and elevations of the BVC and EVC. Also, calculate the elevations of intermediate points on the curve at the whole stations. A sketch of the curve is shown in below figure:

Solution 6



Example 7: Design of Sag Vertical Curves



4.0 Cross Sections

The principal elements of a highway cross section consist of the travel lanes, shoulders, and medians (for some multilane highways).

Marginal elements include median and roadside barriers, curbs, gutters, guard rails, sidewalks, and side slopes.

Typical cross section for a two-lane highway:

Typical cross section for a multilane highway:



4.1 Width of Travel Lanes

Usually vary from 2.7m to 3.6m.

Most arterials have 3.6m travel lanes.

On two-lane, two-way rural roads, lane widths of 3m or 3.3m may be used.

Lane widths of 3m are used only on low-speed facilities.

Lanes that are 2.7m wide are used occasionally in urban areas if traffic volume is low and there are extreme right-of-way constraints.



4.2 Shoulders

The shoulder of a pavement cross section is always contiguous with the traveled lane so as to provide an area along the highway for vehicles to stop when necessary.

In some cases, bicycles are permitted to use a highway shoulder particularly on rural and collector roads.

Shoulder surfaces range in width from 0.6m on minor roads to 3.6m on major arterials.

Shoulders are also used to laterally support the pavement structure.

All shoulders should be flush with the edge of the traveled lane and sloped to facilitate drainage of surface water on the traveled lanes.

Rumble strips may be used on paved shoulder along arterials as a safety measure to warm motorists that they are leaving the traffic lane.



4.3 Medians

A median is the section of a divided highway that separates the lanes in opposing directions.

Width of a median is the distance between the edges of the inside lanes, including the median shoulders.

Median widths vary from a minimum of 1.2m to 24m or more.

The wider the median, the more effective it is in providing safe operating conditions and a recovery area for out-of-control vehicles.

The functions of a median include:

1. Providing a recovery area for out-of-control vehicles

2. Separating opposing traffic

3. Providing stopping areas during emergencies

4. Providing storage areas for left-turning and U-turning vehicles

5. Providing refuge for pedestrians

6. Reducing the effect of headlight glare

7. Providing temporary lanes and cross-overs during maintenance operations.



4.4 Roadside and Median Barriers

A median barrier is defined as a longitudinal system used to prevent an errant vehicle from crossing the portion of a divided highway separating the traveled ways for traffic in opposite directions.

Roadside barriers, on the other hand, protect vehicles from obstacles or slopes on the roadside.

They also may be used to shield pedestrians and property from the traffic stream.

The provision of median barriers must be considered when traffic volumes are high and when access to multilane highway and other highway is only partially controlled.

Roadside barriers should be provided whenever conditions exist requiring the protection for vehicles along the side of the road.

There are a wide variety of roadside barriers and the selection of the most desirable system should provide the required degree of shielding at the lowest cost for the specific application.

Median barriers can be composed of cable or post and beam systems or concrete.



4.5 Curbs and Gutters

Curbs are raised structures made of either Portland cement concrete or bituminous concrete (rolled asphalt curbs) that are used mainly on urban highways to delineate pavement edges and pedestrian walkways.

Curbs are also used to control drainage, improve aesthetics, and reduce right of way.

Curbs can be generally classified as either vertical or sloping.

Gutters or drainage ditches are usually located on the pavement side of a curb to provide the principal drainage facility for the highway.



4.6 Guard rails

Guard rails are longitudinal barriers placed on the outside of sharp curves and at sections with high fills.

Their main function is to prevent vehicles from leaving the roadway.

They are installed at embankments higher than 2.4m and when shoulder slopes are greater than 4:1.

Shapes commonly used include the W beam and the box beam.

The weak post system provides for the post to collapse on impact, with the rail deflecting and absorbing the energy due to impact.

4.7 Sidewalks

Sidewalks are usually provided on roads in urban areas,

but are uncommon in rural areas.

Generally, sidewalks should be provided when

pedestrian traffic is high along main or high-speed roads

in either rural or urban areas.



4.8 Cross Slopes

Pavements on straight sections of two-lane and

multilane highways without medians are sloped from

the middle downward to both sides of the highway,

resulting in a transverse or cross slope, with a cross

section shape that can be curved, plane or a

combination of the two.

A parabola is generally used for curved cross sections,

and the highest point of the pavement (called the crown)

is slightly rounded, with the cross slope increasing

toward the pavement edge.

Plane cross slopes consist of uniform slopes at both

sides of the crown.

Cross slope on divided highways are provided by either

crowning the pavement in each direction, or by sloping

the entire pavement in one direction.



4.9 Side Slopes

Side slopes are provided on embankments and fills to

provide stability for earthworks.

They also serve as a safety feature by providing a

recovery area for out-of-control vehicles.

The hinge point should be rounded since it is potentially

hazardous and may cause vehicle to become airborne

while crossing it, resulting in loss of control of the

vehicle.

The foreslope serve principally as recovery area.



4.10 Right of Way

The right of way is the total land area acquired for the construction of a highway.

The width should be sufficient to accommodate all the elements of the highway cross section, any planned widening of the highway, and public-utility facilities that will be installed along the highway.

In some cases, the side slopes may be located outside the right of way on easement areas.

The right of way widths for

1. two-lane urban collector streets between 12m to 18m,

2. two-lane arterials more than 25m,

3. undivided four-lane arterials vary from 19m to 33m,

4. divided arterials from 36m to 90m, depending on the numbers of lanes and whether frontage roads are included.

5. Freeways depend on the number of lanes and the existence of a frontage road.

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Chapter 2 - Note