CIVL 2030: Introduction to
Transportation Engineering
Chapter 3: Geometric
Design of Highways
Satish Ukkusuri , Ph.D.
Assistant Professor
JEC 4032
Email: [email protected]
Web: http://www.rpi.edu/~ukkuss
2
Today’s Outline
Introduction
Principles of Highway Alignment
Vehicle Alignment
Vertical Curve Fundamentals
Stopping Sight Distance
Stopping Sight Distance and Vertical Curve Design
Stopping Sight Distance and Sag Vertical Curve Design
Passing Sight Distance and Crest Vertical Curve Design
Underpass Sight Distance and Sag Vertical Curve Design
Today’s Outline
Horizontal Alignment
Vehicle Cornering
Horizontal Curve Fundamentals
Stopping Sight Distance and Horizontal Curve Design
3
Introduction
Design of Highways necessitates specific design
elements:
Number of lanes
Lane width
Median type and width
Length of lanes for acceleration and deceleration
Truck climbing lanes
Curve radii
Alignment to provide adequate stopping and passing
sight distances.
4
Introduction
Variation of infrastructure
depending on their use
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5
Introduction
Factors that affect the design process:
Vehicle performance
Physical dimensions
Design guidelines should evolve over changes in
vehicle performance and dimensions as well as to
evidence collected related to the effectiveness of
existing highway design practices.
i.e. crash rates, roadway design characteristics, etc…
Current guidelines for highway design:
A Policy on Geometric Design of Highways and Streets
by the AASHTO 2001
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Introduction
7
For scope purposes, class focuses on the key
elements of highway alignment, which are the most
important components of geometric design.
The best topic for demonstrating the effect of
vehicles performance and dimension on highway
design is roadway alignment.
By concentrating on this topics, students will
develop and understanding of the procedures in
the design of highway geometric elements.
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Principles of Highway Alignment
Elements of surveying are converted from 3-D
problems into 2-D problems. (x,y,z x,z)
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Other problem simplifications are avoidance of x
and z coordinates for positioning and
measurements of roads.
Principles of Highway Alignment
Instead, distance is measured in terms of stations,
with each station consisting of 100 ft (1000 m) of
highway alignment distance.
i.e. Location of point 350 ft on roadway surface on a
blue print
1 station = 100 ft, then
3 stations = 300 ft
Residual is used with the same units
3 stations + 50 ft
Not to scale
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Principles of Highway Alignment
10
The stationing concept, combined with highway’s
alignment given in a plan view (horizontal
alignment) and the corresponding elevation of the
station (vertical alignment), provides a unique
identification system of all points on a highway in a
manner that it is equivalent to using x, y and z
coordinates.
Plan and profile view
Vertical Alignment
Specifies the elevation of points along a roadway
The elevation of a road has to be determined for:
Drivers safety
Drivers comfort
Proper drainage
One of the main concerns in vertical alignment is
the transition between two roadway grades.
This is achieved by means of a vertical curve
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Vertical Alignment
12
Vertical curves can be classified in two different
groups:
Crest curves
Sag curves
G1 = initial roadway grade in percent or ft/ft (m/m)
this grade is also referred to as the initial tangent
grade
G2 = final roadway (tangent) grade in percent or
ft/ft (m/m)
Vertical Alignment
13
A = absolute value of the difference in grades, initial minus
final, usually expressed in percent
PVC = point of the vertical curve, the initial point of the curve
PVI = point of vertical intersection, intersection of initial and
final grades
PVT = point of vertical tangent, which is the final point of the
vertical curve, the point where the curve returns to the final
grade or, equivalently, the final tangent
L = length of the curve in stations or ft (m) measured in a
constant-elevation horizontal plane.
14
Vertical Curve Fundamentals
In order to connect roadway grades with an
appropriate vertical curve, a mathematical
relationship defining elevations at all points along
the vertical curve is needed.
A parabolic function is the best fit for this scenario
since it provides a constant rate of change of slope
and implies equal curve tangents
2
y = ax + bx + c
(3.1)
Vertical Curve Fundamentals
15
Where:
y = roadway elevation at distance x from the
beginning of the vertical curve (the PVC) in stations
or ft (m),
x = distance from the beginning of the vertical curve
in stations or ft (m)
a, b = coefficients defined below, and
c = elevation of the PVC (because x = 0 corresponds
to the PVC) in ft (m).
16
Vertical Curve Fundamentals
In defining a and b, note that the first derivative of
Eq. 3.1 gives the slope and is
dy
= 2ax + b
dx
(3.2)
At the PVC, x = 0, so, using Eq. 3.2,
b =
dy
dx
= G1
(3.3)
where G1 is the initial slope in ft/ft (m/m),
17
Vertical Curve Fundamentals
Also note that the second derivative of Eq. 3.1 is
the rate of change of slope and is
d2y
= 2a
2
dx
(3.4)
However, the average rate of change of slope, by
observation of Fig. 3.3, can also be written as
G 2 − G1
d2y
=
2
L
dx
(3.5)
18
Vertical Curve Fundamentals
Equating Eqs. 3.4 and 3.5 gives
(3.6)
Example 3.1
A 600 ft (182.88 m) equal tangent sag vertical curve
has the PVC at station 170 + 00 (5 + 181.6) and
elevation 1000 ft (304.8 m). The initial grade is 3.5% and the final grade is +0.5%. Determine the
stationing and elevation of the PVI, the PVT, and
the lowest point on the curve.
Vertical Curve Fundamentals
19
Example 3.2
An equal-tangent vertical curve is to be constructed
between grades of -2.0% (initial) and +1.0%
(final). The PVI is at station 110 + 00 (3 + 352.8)
and at elevation 420 ft (128.016 m). Due to a
street crossing the roadway, the elevation of the
roadway at station 112 + 00 (3 + 413.76) must be
at 424.5 ft (129.388 m). Design the curve.
Vertical Curve Fundamentals
20
Additional properties of vertical curves:
G1 = initial roadway grade in percent or ft/ft (m/m) (this
grade is also referred to as the initial tangent grade, viewing
Fig. 3.4 from left to right)
Vertical Curve Fundamentals
G2 = final roadway (tangent) grade in percent or ft/ft (m/m)
PVC = point of the vertical curve (the initial point of the curve)
PVI = point of vertical intersection (intersection of initial and final
grades)
PVT = point of vertical tangent, which is the final point of the vertical
curve (the point where the curve returns to the final grade or,
equivalently, the final tangent)
L = length of the curve in stations or ft (m) measured in a constantelevation horizontal plane
x = distance from the PVC in ft (m)
Y = offset at any distance x from the PVC in ft (m)
Ym = midcurve offset in ft (m), and
Yf = offset at the end of the vertical curve in ft (m).
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22
Vertical Curve Fundamentals
Based on Fig. 3.4, the properties of an equal
tangent parabola can be used to give
A
2
Y =
x
200 L
(3.7)
Where:
A = absolute value of the difference in grades (|G1 –
G2|) expressed in percent, and
Other terms are as defined in Fig. 3.4.
23
Vertical Curve Fundamentals
Note that in this equation, 200 is used in the
denominator instead of 2 because A is expressed in
percent instead of ft/ft (m/m)
This division by 100 also applies to Eq. 3.8 and 3.9
below. It follows from Fig. 3.4 that:
AL
Ym =
800
(3.8)
Yf =
AL
200
(3.9)
24
Vertical Curve Fundamentals
The horizontal distance required to change the
slope by 1% is:
K =
L
A
(3.10)
Where:
K = value that is the horizontal distance, in ft (m), required to
affect a 1% change in the slope of the vertical curve
A = absolute value of the difference in grades (|G1 − G2|
expressed in percent)
L = length of curve in ft (m).
25
Vertical Curve Fundamentals
K-value can also be used to compute the high and
low point locations of a curve
xhl = K × G1
(3.11)
Where:
xhl = distance from the PVC to the high/low point in
ft (m)
K = value that is the horizontal distance, in ft (m),
required to affect a 1% change in the slope of
the vertical curve
G1 = initial grade in percent.
Vertical Curve Fundamentals
26
Example 3.4
A vertical curve crosses a 4-ft (1.219 m) diameter
pipe at right angles. The pipe is located at station
110 + 85 (3 + 378.708) and its centerline is at
elevation 1091.60 ft (332.720 m). The PVI of the
vertical curve is at station 110 + 00 (3 + 352.8)
and elevation 1098.4 ft (334.792 m). The vertical
curve is equal tangent, 600 ft (182.88 m) long, and
connects and initial grade of +1.2% and a final
grade of -1.08%. Using offsets, determine the
depth, below surface of the curve and of the top of
the pipe. Also determine the highest station.
27
Stopping Sight Distance
Design of highways involves two objective:
Minimization of cost
Adequate safety levels
Appropriate levels of safety is usually define as the
level of safety that provides drivers with sufficient
sight distance to allow them to safely stop their
vehicles to avoid collisions.
SSD =
V12
 a 


2 g    ± G 
 g 

+ V1 × t r
(3.12)
Stopping Sight Distance
Where:
SSD = stopping sight distance in ft (m),
a = deceleration rate in ft/s2 (m/s2),
V1 = initial vehicle speed in ft/s (m/s),
g = gravitational constant in ft/s2 (m/s2),
tr = perception/reaction time in sec, and
G = roadway grade (+ for uphill and – for downhill)
in percent/100.
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Stopping Sight Distance
29
With a = 11.2 ft/s2 (3.4 m/s2), tr =2.5s, the
application of Eq. 3.12 (assuming G = 0) produces
the stopping sight distances presented in Table 3.1.
Stopping Sight Distance and Crest Vertical
Curve Design
The length of the curve is the critical element in
providing sufficient SSD on a vertical curve but
longer curves are more costly.
Therefore, an expression for minimum required
length for a given SSD.
For a crest vertical curve this expressions are:
For S<L
Lm =
For S>L
AS 2
200
Lm = 2S −
(
H1 +
(
H2
200 H1 + H 2
A
(3.13)
)
2
)
2
(3.14)
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Stopping Sight Distance and Crest Vertical
Curve Design
31
Where:
S = sight distance in ft (m)
H1 = height of driver’s eye above roadway surface in ft (m)
H2 = height of object above roadway surface in ft (m)
PVC = point of the vertical curve (the initial point of the
curve)
PVI = point of vertical intersection (intersection of initial and
final grades)
PVT = point of vertical tangent, which is the final point of the
vertical curve (the point where the curve returns to the final
grade or, equivalently, the final tangent)
L = length of the curve in ft (m)
Stopping Sight Distance and Crest Vertical
Curve Design
32
AASHTO design guidelines:
H1, = 3.5 ft (1080 mm)
H2, = 2.0 ft (600 mm)
Substituting these guidelines on Eqs. 3.13 and 3.14
For SSD<L
A × SSD 2
Lm =
2158
A × SSD 2
Lm =
658
US Customary
Metric
(3.15)
Stopping Sight Distance and Crest Vertical
Curve Design
For SSD > L
2158
L m = 2 × SSD −
A
658
Lm = 2 × SSD −
A
US Customary
Metric
(3.16)
Where:
SSD = stopping sight distance in ft (m), and
A = algebraic difference in grades in percent.
33
Stopping Sight Distance and Crest Vertical
Curve Design
Example 3.5
A highway is being designed to AASHTO guidelines
with a 70 mi/h (113 km/h) design speed, and at
one section, an equal tangent vertical curve must
be designed to connect grades of +1% and -2%.
Determine the minimum length of the curve
necessary to meet SSD requirements.
34
Stopping Sight Distance and Crest Vertical
Curve Design
35
For simplification purposes the assumption of L >
SSD can be used providing a linear relationship
between A and Lm
SSD 2
K =
2158
US Customary
SSD2
K=
658
(3.18)
Metric
Where:
K = horizontal distance, in ft (m), required to affect a
1% change in the slope (as in Eq. 3.10), and is
defined as
Stopping Sight Distance and Crest Vertical
Curve Design
36
Assuming L > SSD instead of SSD > L.
If SSD > L the relationship between A and Lm is
not linear so K-values cannot be used in the L = KA
formula (Eq. 3.10).
At low values of A, it is possible to get negative
minimum curve lengths (see Eq. 3.16).
Stopping Sight Distance and Crest Vertical
Curve Design
37
L > SSD (upon which Eqs. 3.17 and 3.18 are made)
is a good one because, in many cases
L is greater than SSD and
When it is not (SSD > L), the use of the L > SSD
formula (Eq. 3.15 instead of Eq. 3.16) gives longer
curve lengths and thus the error is on the
conservative, safe side.
Example 3.6
Solve Example 3.5 using the K-values in Table 3.2
Stopping Sight Distance and Crest Vertical
Curve Design
38
Stopping Sight Distance and Crest Vertical
Curve Design
Note: Tables 3.2 and 3.2 assume G = 0.
Difficult to get exact effect of grade unless you know
exactly where on the curve the brakes were applied.
Assuming initial or final could be too conservative or too
risky.
Practices vary, but most agencies do not correct if G <
3%, then may add a fixed-distance correction
Minimum curve lengths:
100 to 325 ft (30 to 100 m) depending on individual
jurisdictional guidelines.
Or 3 times the design speed (with speed in mi/h and
length in feet), or 0.6 times the design speed (with
speed in km/h and length in meters).
39
Stopping Sight Distance and Sag Vertical
Curve Design
40
S=
sight distance in ft (m)
H = height of headlight in ft (m)
β=
inclined angle of headlight beam in degrees,
PVC = point of the vertical curve (the initial point of the curve)
PVI = point of vertical intersection (intersection of initial and final
grades)
PVT = point of vertical tangent, which is the final point of the
vertical curve (the point where the curve returns to the final
grade or, equivalently, the final tangent)
L=
length of the curve in ft (m)
Stopping Sight Distance and Sag Vertical
Curve Design
Equation to determine the minimum length of a
curve for a required sight distance:
For S < L
For S > L
AS 2
Lm =
200(H + S tan β )
(3.19)
200(H + S tanβ)
Lm = 2S −
A
(3.20)
Where:
Lm = minimum length of vertical curve in ft (m), and
Other terms are as defined in Fig. 3.7.
41
Stopping Sight Distance and Sag Vertical
Curve Design
AASHTO design guidelines suggest a headlight
height of 2.0 ft (600 mm) and an upward angle of
1 degree. Substituting this:
For SSD < L
2
A
×
SSD
A × SSD
Lm =
Lm =
120 + 3.5 × SSD
400 + 3.5 × SSD
US Customary
Metric
(3.21)
2
42
Stopping Sight Distance and Sag Vertical
Curve Design
For SSD > L
400 + 3.5 × SSD
Lm = 2 × SSD −
A
Metric
US Customary
120 + 3.5 × SSD
Lm = 2 × SSD −
A
Where:
SSD = stopping sight distance in ft (m)
A = algebraic difference in grades in percent
43
Stopping Sight Distance and Sag Vertical
Curve Design
K-values can be computed by assuming L > SSD,
which gives us the linear relationship between Lm
and A as shown in Eq. 3.21.
2
2
SSD
K =
400 + 3.5 SSD
K=
US Customary
SSD
120 + 3.5 SSD
Metric
(3.23)
Where:
K = horizontal distance, in ft (m), required to affect a 1%
change in the slope (as in Eq. 3.10), and
SSD = stopping sight distance in ft (m).
44
Stopping Sight Distance and Sag Vertical
Curve Design
45
Stopping Sight Distance and Sag Vertical
Curve Design
46
Example 3.8
An existing tunnel needs to be connected to a newly
constructed bridge with sag and crest vertical curves.
Develop a vertical alignment to connect the tunnel and
bridge by determining the highest possible common design
speed for the sag and crest vertical curves needed.
Compute the stationing and elevations of PVC, PVI, and PVT
curve points.