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Title: Geometric Design

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Geometric Design Vertical Alignment
Transportation Engineering - I
3
Geometric Design
4
Outline

Concepts
Vertical Alignment
Fundamentals
Crest Vertical Curves
Sag Vertical Curves
Examples
Horizontal Alignment
Fundamentals
Super elevation
Other Non-Testable Stuff

5
Concepts

Alignment is a 3D problem broken down into two 2D
problems
Horizontal Alignment (plan view)
Vertical Alignment (profile view)
Stationing
Along horizontal alignment
1200 1,200 ft.

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Stationing
Horizontal Alignment
Vertical Alignment
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From Perteet Engineering
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Vertical Alignment
9
Vertical Alignment

Objective
Determine elevation to ensure
Proper drainage
Acceptable level of safety
Primary challenge
Transition between two grades
Vertical curves

Sag Vertical Curve
G1
G2
G2
G1
Crest Vertical Curve
10
Vertical Curve Fundamentals

Parabolic function
Constant rate of change of slope
Implies equal curve tangents
y is the roadway elevation x stations (or feet)
from the beginning of the curve

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Vertical Curve Fundamentals

Choose Either
G1, G2 in decimal form, L in feet
G1, G2 in percent, L in stations

PVI
G1
d
PVC
G2
PVT
L/2
L
x
Where G1 Initial roadway grade( initial tangent
grade) G2 Final roadway grade A Absolute value
of the difference in grades L Length of
vertical curve measured in a horizontal
plane PVC Initial point of the vertical
curve PVI Point of vertical intersection (
intersection of initial and final grades) PVT
Final point of the vertical curve
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Vertical curves are almost arranged such that
half of the curve length is positioned before the
PVI and half after and are referred as equal
tangent vertical curves.
A circular curve is used to connect the
horizontal straight stretches of road, a
parabolic curve is usually used to connect
gradients in the profile alignment.

It provides a constant rate of change of slope
and implies equal curve lengths.

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Level

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CREST VERTICAL CURVES
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-

-

-
Level

SAG VERTICAL CURVES
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Vertical Curve

For a vertical curve, the general form of the
parabolic equation is
Y ax2 bx c
where, y is the roadway elevation of the curve
at a point x along the curve from the beginning
of the vertical curve (PVC).
C is the elevation of the PVC since x0
corresponds the PVC

1
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Slope of Curve

To define a and b, first derivative of
equation 1 gives the slope.
At PVC, x0

2
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or
Where G1 is the initial slope.

3
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Taking second derivative of equation1, i.e. rate
of change of slope
The rate of change of slope can also be written
as

4
5
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Equating equations 4 and 5
or

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7
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Fundamentals of Vertical Curves

For vertical curve design and construction,
offsets which are vertical distances from initial
tangent to the curve are important for vertical
curve design.

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PVI
PVC
PVT
PVC
PVT
PVI
PVC
PVC
PVT
PVI
PVC
PVT
PVT
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A vertical curve also simplifies the computation
of the high and low points or crest and sag
vertical curves respectively, since high or low
point does not occur at the curve ends PVC or
PVT.
Let Y is the offset at any distance x from
PVC.

Ym is the mid curve offset Yt is the offset at
the end of the vertical curve.
From an equal tangent parabola, it can be written
as
where y is the offset in feet and A is the
absolute value of the difference in grades(G2-G1,
in ), L is length of vertical curve in feet
and x is distance from the PVC in feet.

8
26

Putting the value of xL in eq. 8

First derivative can be used to determine the
location of the low point, the alternative to
this is to use a k-value which is defined as
where L is in feet and A is in .

This value k can be used directly to compute
the high / low points for crest/ sag vertical
curves by
xkG1
where x is the distance from the PVC to the
high/ low point. k can also be defined as the
horizontal distance in feet required to affect a
1 change in the slope.

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A 500-meter equal-tangent sag vertical curve has
the PVC at station 10000 with an elevation of
1000 m. The initial grade is -4 and the final
grade is 2. Determine the stationing and
elevation of the PVI, the PVT, and the lowest
point on the

Solution The curve length is stated to be 500
meters. Therefore, the PVT is at station 10500
(10000 500) and the PVI is in the very middle
at 10250, since it is an equal tangent curve.
For the parabolic formulation, equals the
elevation at the PVC, which is stated as 1000 m.
The value of b equals the initial grade, which in
decimal is -0.04. The value of a can then be
found as 0.00006.
Basic Equation of the parabola y ax2bxc
At x 0 y C1000 b G1 -0.04 , a
G2-G1/2L (0.02-0.04))/2x5000.00006
Using the general parabolic formula, the
elevation of the PVT can be found
y 0.00006x2 (-0.04x)c 0.00006(500)(-0.0450
0)1000 995m
Since the PVI is the intersect of the two
tangents, the slope of either tangent and the
elevation of the PVC or PVT, depending, can be
used as reference. The elevation of the PVI can
then be found as y -0.04(250)1000 990 m
To find the lowest part of the curve, the first
derivative of the parabolic formula
can be found. The lowest point has a slope of
zero, and thus the low point location can be
found dy/dx 0.00012x-0.04 0 x
0.04/0.00012 333.333 m
Using the parabolic formula, the elevation can be
computed for that location. It
turns out to be at an elevation of 993.33 m,
which is the lowest point along the curve.

30
Sight Distances

Sight Distance is a length of road surface which
a particular driver can see with an acceptable
level of clarity. Sight distance plays an
important role in geometric highway design
because it establishes an acceptable design
speed, based on a driver's ability to visually
identify and stop for a particular, unforeseen
roadway hazard or pass a slower vehicle without
being in conflict with opposing traffic.
As velocities on a roadway are increased, the
design must be catered to allowing additional
viewing distances to allow for adequate time to
stop. The two types of sight distance are
(1) stopping sight distance and (2) passing
sight distance.

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Stopping Sight Distance

At every point on the roadway, the minimum sight
distance provided should be sufficient to enable
a vehicle traveling at the design speed to stop
before reaching a stationary object in its path.
Stopping sight distance is the aggregate of two
distances
brake reaction distance and braking distance.
Brake reaction time is the interval between the
instant that the driver recognizes the existence
of an object or hazard ahead and the instant that
the brakes are actually applied. Extensive
studies have been conducted to ascertain brake
reaction time. Minimum reaction times can be as
little as 1.64 seconds 0.64 for alerted drivers
plus 1 second for the unexpected signal.
Some drivers may take over 3.5 seconds to respond
under similar circumstances. For approximately
90 of drivers, including older drivers, a
reaction time of 2.5 see is considered adequate.
This value is therefore used in Table on next
page

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Vehicle Stopping Distance

Vehicle stopping distance is calculated by the
following formula
where V1 initial speed of vehicle
f friction
G percent grade

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Distance Traveled During Perception/ Reaction Time

It is calculated by the following formula
dr V1 tr
where V1 Initial Velocity of vehicle
tr time required to perceive and react to the
need to stop

Hence formula for the Stopping sight distance
will be

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SSD and Crest Vertical Curve

In providing the sufficient SSD on a vertical
curve, the length of curve L is the critical
concern.
Longer lengths of curve provide more SSD, all
else being equal, but are most costly to
construct.
Shorter curve lengths are relatively inexpensive
to construct but may not provide adequate SSD.
In developing such an expression, crest and sag
vertical curves are considered separately.
For the crest vertical curve case, consider the
diagram.

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SSD and Crest Vertical Curve
H1

S Sight distance (ft)
H1 height of drivers eye above road-way surface
(ft)
L length of the curve (ft), H2 height of
roadway object (ft) ,
A difference in grade
Lm Minimum length required for sight distance.

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Minimum Length of the Curve

For a required sight distance S is calculated as
follows
If the sight distance is found to be less than
the curve length (SgtL)
for sight distances that are greater than the
curve length (SltL)

For the sight distance required to provide
adequate SSD, standard define driver eye height
H1 is 3.5 ft and object height H2 is 0.5 ft. S is
assumed is equal to SSD. We get

SSD gt L
SSD lt L
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Working with the above equations can be
cumbersome.
To simplify matters on crest curves computations,
K- values, are used.
L KA
where k is the horizontal distance in feet,
required to affect 1 percent change in slope.

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SSD and Sag Vertical Curve

Sag vertical curve design differs from crest
vertical curve design in the sense that sight
distance is governed by night time conditions,
because in daylight, sight distance on a sag
vertical curve is unrestricted.
The critical concern for sag vertical curve is
the headlight sight distance which is a function
of the height of the head light above the road
way, H, and the inclined upward angle of the head
light beam, relative to the horizontal plane of
the car, ß.

The sag vertical curve sight distance problem is
illustrated in the following figure.

By using the properties of parabola for an equal
tangent curve, it can be shown that minimum
length of the curve, Lm for a required sight
distance is
For SgtL
For SltL

For the sight distance required to provide
adequate SSD, use a head light height of 2.0 ft
and an upward angle of 1 degree.
Substituting these design standards and S SSD
in the above equations
For SSDgtL
For SSDltL

As was the case for crest vertical curves,
K-values can also be computed for sag vertical
curves.
Caution should be exercised in using the k-values
in this table since the assumption of G0 percent
is used for SSD computations.

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Engineering vs. Politics

A current roadway has a design speed of 100
km/hr, a coefficient of friction of 0.1, and
carries drivers with perception-reaction times of
2.5 seconds. The drivers use cars that allows
their eyes to be 1 meter above the road. Because
of ample road kill in the area, the road has been
designed for car cases that are 0.5 meters in
height. All curves along that road have been
designed accordingly.
The local government, seeing the potential of
tourism in the area and the boost to the local
economy, wants to increase the speed limit to 110
km/hr to attract summer drivers. Residents along
the route claim that this is a horrible idea, as
a particular curve called "Dead Man's Hill" would
earn its name because of sight distance problems.
"Dead Man's Hill" is a crest curve that is
roughly 600 meters in length. It starts with a
grade of 1.0 and ends with (-1.0). There has
never been an accident on "Dead Man's Hill" as of
yet, but residents truly believe one will come
about in the near future.

A local politician who knows little to nothing
about engineering (but thinks he does) states
that the 600-meter length is a long distance and
more than sufficient to handle the transition of
eager big-city drivers. Still, the residents push
back, saying that 600 meters is not nearly the
distance required for the speed. The politician
begins a lengthy campaign to "Bring Tourism to
Town", saying that the residents are trying to
stop "progress". As an engineer, determine if
these residents are indeed making a valid point
or if they are simply trying to stop progress?

Using sight distance formulas from other
sections, it is found that 100 km/hr has an SSD
of 465 meters and 110 km/hr has an SSD of 555
meters,
Given the criteria stated above. Since both 465
meters and 555 meters are less than the 600-meter
curve length, the correct formula to use would
be

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For flash animation please visit the following
site
http//street.umn.edu/flash_curve_crest.html

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Problem for Pondering

Problem
To help prevent future collisions between cars
and trains, an at-grade crossing of a rail road
by a country road is being redesigned so that the
county road will pass underneath the tracks.
Currently the vertical alignment of the county
road consists of an equal tangents crest vertical
curve joining a 4 upgrade to a 3 downgrade. The
existing vertical curve is 450 feet long, the PVC
of this curve is at station 4824.00, and the
elevation of the PVC is 1591.00 feet. The
centerline of the train tracks is at station
5150.00. Your job is to find the shortest
vertical curve that provides 20 feet of clearance
between the new county road and the train tracks,
and to make a preliminary estimate of the cut at
the PVI that will be needed to construct the new
curve.