lec 06 highway engineering - vertical alignment

7/25/2019 Lec 06 Highway Engineering - Vertical Alignment

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Lecture 06 59

Highway Eng. Vertical Alignment 14 15

Dr. Firas Asad

In this lecture;

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A- Terrain and Grades.

B- Control Grades for Design.

C- Length of Vertical Curves.

D- Elevations of Vertical Curves.

Vertical Alignment

The information listed in this lecture is mainly taken from the Policy on Geometric Design of

Highways and Streets (AASHTO, 2011), Iraqi Highway Design Manual (SORB, 2005) and

Traffic and Highway Engineering (Garber and Hoel, 2009).

A- Terrain and Grades

Land topography has an influence on the alignment of roads and streets.

Topography affects both horizontal and vertical alignments. To characterize

variations in topography, engineers generally separate it into three classifications

according to terrain; Level, Rolling and Mountainous. In general, rolling terrain

generates steeper grades than level terrain, causing trucks to reduce speeds below

those of passenger cars; mountainous terrain has even greater effects, causing some

trucks to operate at crawl speeds.

Roads and streets should be designed to encourage uniform operation. It is

accepted that nearly all passenger cars can readily negotiate longitudinal grades as

steep as 4% to 5% without an appreciable loss in speed below that normally

maintained on level roadways. The effect of grades on trucks and recreational

vehicles speeds is much more pronounced than on speeds of passenger cars. That is

because they generally have higher weight/power ratios.


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B- Control Grades for Design.

Maximum grades.Maximum longitudinal grades of about 5 percent are considered

appropriate for a design speed of 110 km/h [70 mph]. For a design speed of 50 km/h

[30 mph], maximum grades generally are in the range of 7 to 12 percent, depending

on terrain. The maximum design grade should be used only infrequently; in most

cases, grades should be less than the maximum design grade. Table below shows

maximum recommended grades according to type of terrain and design speed.

Short grades less than 150 m in length and for one-way downgrades can have

maximum grade may be about 1 percent steeper than other locations; for low-

volume rural highways, the maximum grade may be 2 percent steeper.

Minimum grades.Flat grades can typically be used without problem on uncurbed

highways where the cross slope is adequate to drain the pavement surface laterally.

With curbed highways or streets, longitudinal grades should be provided to facilitate

surface drainage. An appropriate minimum grade is typically 0.5 percent.


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Critical Lengths of Grade for Design

The term critical length of grade is used to indicate the maximum length of a

designated upgrade on which a loaded truck can operate without an unreasonable

reduction in speed. For a given grade, lengths less than critical result in acceptable

operation in the desired range of speeds. If the desired freedom of operation is to

be maintained on grades longer than critical, design adjustments such as changes in

location to reduce grades or addition of extra lanes should be considered.

A common basis for determining critical length of grade is based on a reduction inspeed of trucks below the average running speed of traffic. It is recommended that a

15-km/h [10-mph] reduction criterion be used as the general guide for determining

critical lengths of grade.

The length of any given grade that will cause the speed of a representative truck

(120 kg/kW) entering the grade at 110 km/h [70 mph] to be reduced by various

amounts below the average running speed of all traffic is shown graphically in the

exhibit below.


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Climbing Lanes

A climbing lane is an extra lane in the upgrade direction (uphill lane) for use by

heavy vehicles whose speeds are significantly reduced by the grade. A climbing lane

eliminates the need for drivers of light vehicles to reduce their speed when they

encounter a heavy slow-moving vehicle. Because of the increasing rate of crashes

directly associated with the reduction in speed of heavy vehicles on steep sections

of two-lane highways and the significant reduction of the capacity of these sections

when heavy vehicles are present, the provision of climbing lanes should be

considered.

The need for a climbing lane is evident when a grade is longer than its critical length,

defined as the length that will cause a speed reduction of the heavy vehicle by at

least 10 mph. The amount by which a trucks speed is reduced when climbing a

steep grade depends on the length of the grade.


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C- Vertical Curves (Crest & Sag).

The alignment of a highway is composed of vertical and horizontal elements; see the

picture below of the Marquam Bridge as a nice example.

The vertical alignment of a transportation facility consists of tangent grades (straight

lines in the vertical plane) and the parabolic vertical curves that connect these

grades. Vertical alignment is documented by the profile. The profile is a graph that

has elevation as its vertical axis and distance, measured in stations along the

centreline or other horizontal reference line of the facility, as its horizontal axis.

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. Thefigure below illustrates vertical curves that are classified as Crestor Sag.


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The design of the vertical alignment therefore involves the selection of suitable

grades for the tangent sections and the appropriate length

C-1 Length of Crest Vertical Curves

of vertical curves.

Minimum lengths of crest vertical curves based on sight distance

The figure shows that there are two possible scenarios that could control the design

length: (1) the sight distance S is greater than the length of the vertical curve L, and

(2) the sight distance is less than the length of the vertical curve L.

criteria generally

are satisfactory for safety, comfort, and appearance. The figure below illustrates the

parameters used in determining the length of a parabolic crest vertical curve L

needed to provide any specified value of sight distance S.


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The basic equations for minimum length of a crest vertical curve as follow (AASHTO):

Design control - Stopping sight distance ------> h1= 1080 mm and h2= 600 mm


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According to AASHTO, for

convenience a short

formula L=K.A is used

as shown in this figure.

The design controls for

SSD and PSD for crest and

sag curves can be shown

in the following tables.

Design control - Passing sight distance------> h1= 1080 mm and h2= 1080 mm

Design values of crest vertical curves for passing sight distance differ from those for

SSD because of the different sight distance and object height criteria. The previousgeneral equations apply, but with 1,080 mm height of object (h2):

- Computing SSD and PSD and choosing the largest. PSD only for 2-lane 2-way hwys.

Hints regarding the application of the curve length formulas:

- For the SSD, choose the largest road grade with negative sign.

- The start of solution can be by initially assuming that SL and then applying

the corresponding equation and finally validating the initial assumption. Or it could

be simply by applying both equations and them choosing the largest L value.


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C-2 Length of Sag Vertical Curves

The AASHTO policy mentions four different criteria for establishing lengths of sag

vertical curves are recognized to some extent. These are (1) headlight sight distance,

(2) passenger comfort, (3) drainage control, and (4) general appearance.

1) Minimum Length based on SSD Criterion.

Headlight sight distance has been used directly as the basis for determining the

minimum recommended length of sag vertical curves. When a vehicle traverses a

sag vertical curve at night, the portion of highway lighted ahead is dependent on the

position of the headlights and the direction of the light beam. A headlight height of

600 mm and a 1-degree upward divergence of the light beam from the longitudinal

axis of the vehicle is commonly assumed.

The following equations show the relationships between S, L, and A, using S as the

distance between the vehicle and point where the 1-degree upward angle of thelight beam intersects the surface of the roadway.

For overall safety on highways, a sag vertical curve should be long enough that the

light beam distance is nearly the same as the stopping sight distance. Accordingly, it

is appropriate to use stopping sight distances SSDfor different design speeds as the

value of S in the above equations.


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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.

3) Minimum Length of Curve based on Appearance Criterion.

The criterion for acceptable appearance is usually satisfied by assuring that the

minimum length (m) of the sag curve is not less than expressed by the following

equation:

Lmin.(m) = 30 A

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.

Lmax.(m) = 51 A


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Sight Distance at Undercrossings

Sight distance on the highway through a grade separation should be at least as long

as the minimum stopping sight distance and preferably longer. Design of the verticalalignment is the same as at any other point on the highway except in some cases of

sag vertical curves underpassing a structure illustrated in the exhibit below. The

structure fascia may cut the line of sight and limit the sight distance to less that

otherwise is attainable.

Using an eye height of 2.4 m (h1) for a truck driver and an object height of 0.6 m (h2)

for the taillights of a vehicle, the following equations can be derived for computing

the minimum length for sag vertical curve at undercrossings:


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According to Iraqi specification listed in the Highway design Manual (SORB, 2005),

the minimum vertical clearance for highways should be at least 5.20m.

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D-Elevation of Crest and Sag Vertical Curves

Vertical tangents with different grades are joined by vertical curves such as the one

shown in the figure below. Vertical curves are normally parabolas centered about

the point of intersection (PVI) of the vertical tangents they join. Hence, the method

used for computing elevations of points on the vertical curve relies on the

properties of the parabola (2nd

order algebraic equation).

Thus, the symmetrical crest vertical curves in the figure above can be

mathematically represented as follows:

2

)()(2

1

rxxgpvcElevxElev ++=

Where:

Elev(x): elevation of a point on the curve at a distance x from the PVC (m).

Elev(pvc): elevation of the PVC (m)

g1: grade just prior to the curve (%)

x: horizontal distance from the PVC to the point on the curve (station)

r: rate of change of grade (percent per station)


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The rate of change of grade is given by:L

A

L

ggr =

=12

Where g2 (in percent) is the grade just beyond the end of the vertical curve (EVC)

and L is the length of the curve (in stations). For example, if a curve has a length of

700 m (L = 7 sta.) and grade g1at PVC was 2.25% upward and grade g2at PVT was

1.25% downward, then the rate of change would be r=(1.25 2.25) / 7 = 0.50%

per station.

Vertical curves are classified as sags (like valley) where g2 > g1 and crests (like hill)

otherwise. Therefore, r (and hence the term rx2/2) will be positive for sags and

negative for crests. It is useful to mention here that the length of the vertical curve

is the horizontal projection of the curve and not the length along the curve.

The last term of the equation rx2/2 represents y on the figure which is the external

distance from the tangent to the curve and is known as the offset. If x is always

measured from the PVC, the offset given by rx2

/2 will be measured from the g1

tangent. To determine offsets from the g2tangent, x should be measured backward

from the PVT. Since the curve is symmetrical about its center, the offsets from the g1

and g2 tangents, respectively, are also symmetrical about the centre of the curve,

which occurs at the station of its PVI.

In addition, note that vertical distances in the vertical curve formulas are the

product of grade times a horizontal distance (gx). Regarding units, if vertical

distances are to be in meters, horizontal distances should also be in meters, and

grades should be dimensionless ratios. In many cases, however, it is more

convenient to represent grades in percent and horizontal distance in stations. If

grades are in percent, horizontal distances must be in stations; likewise, if grades

are dimensionless ratios, horizontal distances must be in meters.


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The parabola is selected as the vertical curve so that the rate of change of grade (r),

which is the second derivative of the curve (d2y/dx

2), will be constant with distance.

Note that the first derivative is the grade itself (dy/dx=g), and since the rate ofchange of grade is constant, the grade of any point in the vertical curve is a linear

function of the distance from the PVC to the point. That is,

rxgdx

dyg +== 1

(when x = 0 -----> g = g1 & x = L -----> g = g2)

A key point on a vertical curve is the turning point, where the minimum (lowest) or

maximum (highest) elevation on a vertical curve occurs. The station at this point

may be computed by finding the first derivative and setting it to zero. This yields:

01

=+= rxgdx

dy ----- > XT = - g1/ r (XTin stations and g in percent %)

Finally, the mid offset (middle ordinate distance) e, the vertical distance from the

PVIto the vertical curve, can be computed using the general offset formula but with

substituting L/2 for the distance x:

Offset (external distance) y= r x2/2 ----- >

Mid Offset e (at x=L/2) = r (L/2)2/2 = rL

2/8 ; Since r = (g2- g1)/L = A/L ----- >

e = A L / 8 (e in meters; A in % and L in stations)

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lec 06 highway engineering - vertical alignment

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