Springer Tracts on Transportation and Traffic

Andrzej Kobryń

Transition Curves
for Highway
Geometric
Design
Springer Tracts on Transportation and Traffic

Volume 14

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New York, USA
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Andrzej Kobryń

Transition Curves
for Highway Geometric
Design

123
Andrzej Kobryń
Faculty of Civil and Environmental
Engineering
Bialystok University of Technology
Białystok
Poland

ISSN 2194-8119 ISSN 2194-8127 (electronic)

Springer Tracts on Transportation and Traffic
ISBN 978-3-319-53726-9 ISBN 978-3-319-53727-6 (eBook)
DOI 10.1007/978-3-319-53727-6
Library of Congress Control Number: 2017931572

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Simple Horizontal and Vertical Curves . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Circular Horizontal Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Parabolic Vertical Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Mathematical Methods for Defining of Transition Curves . . . . . . . . . 15
3.1 Transition Curves Described Using Curvature Function . . . . . . . . . 15
3.2 Transition Curves Described Using Explicit Function. . . . . . . . . . . 19
3.3 Transition Curves Defined in the Polar Coordinate System . . . . . . 21
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Transition Curves Described Using Curvature Function . . . . . . . . . . 25
4.1 Classical Transition Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1.1 Spiral Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1.2 Bloss Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1.3 Grabowski Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.4 Other Transition Curves Described Using Curvature
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 32
4.1.5 Two-Parameter Spiral Curves . . . . . . . . . . . . . . . . . . . .... 36
4.2 Vertical Transition Curve Described Using Curvature
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 39
4.3 General Transition Curves Described Using Curvature
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 42
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 47

v
vi Contents

5 Transition Curves Described Using Explicit Function . . . . . . . . . . . . 49

5.1 Parabolic Transition Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Sinusoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Transition Curves Defined in the Polar Coordinate System . . . . . . . . 59
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7 Polynomial Description of Transition Curves . . . . . . . . . . . . . . . . . . . 63
7.1 Categories of Polynomial Transition Curves . . . . . . . . . . . . . . . . . . 63
7.2 Boundary Conditions for Polynomial Transition Curves . . . . . . . . . 68
7.3 Generalized Solutions of Polynomial Transition Curves . . . . . . . . . 69
7.3.1 First Generalized Solution of Polynomial Transition
Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.3.2 Second Generalized Solution of Polynomial Transition
Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.3.3 Third Generalized Solution of Polynomial Transition
Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.3.4 Fourth Generalized Solution of Polynomial Transition
Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.4 Different Solutions of Polynomial Transition Curves . . . . . . . . . . . 73
7.4.1 Polynomial Transition Curves Based on the First
Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.4.2 Polynomial Transition Curves Based on the Second
Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.4.3 Polynomial Transition Curves Based on the Third
Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.4.4 Polynomial Transition Curves Based on the Fourth
Generalized Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.5 Selection of Design Parameters for Universal and Oval
Transition Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8 Sample Applications of Transition Curves in Horizontal
Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 89
8.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 89
8.2 Parabolic Transition Curve as a Connecting Element
Between Straight Line and Circular Arc . . . . . . . . . . . . . . . . . .... 92
8.3 Sinusoid as Transition Curve Between a Straight Line
and Circular Arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 93
8.4 Polynomial Transition Curve as Connecting Element
Between a Straight Line and Circular Arc . . . . . . . . . . . . . . . .... 95
8.5 General Transition Curves as Connecting Element Between
Two Straight Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 97
8.5.1 Designing Horizontal Curves Using Sinusoid
as a General Transition Curve . . . . . . . . . . . . . . . . . . . .... 97
Contents vii

8.5.2 Designing Horizontal Curves Using Polynomial

General Transition Curves . . . . . . . . . . . . . . . . . . . . . . .... 99
8.6 Universal Transition Curves in Horizontal Alignment . . . . . . .... 100
8.6.1 Designing of Curvilinear Transitions Using First
Solution of Universal Transition Curves . . . . . . . . . . . .... 101
8.6.2 Designing of Curvilinear Transitions Using Second
Solution of Universal Transition Curves . . . . . . . . . . . .... 104
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 107
9 Sample Applications of Transition Curves in Vertical
Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.1 Optimization of Vertical Alignment Using Polynomial
Transition Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.1.1 Optimization of Vertical Alignment Using Polynomial
Transition Curves with Horizontal Tangent
at End Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.1.2 Optimization of Vertical Alignment Using General
Transition Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.2 Designing Vertical Arcs Using Polynomial Transition Curves . . . . 121
9.2.1 Designing Vertical Arcs Using Transition Curves
with Horizontal Tangent at End Point . . . . . . . . . . . . . . . . . 121
9.2.2 Designing Vertical Arcs Using General Transition
Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Chapter 1
Introduction

A designing of highways involves establishing the design details of the selected

route, including final horizontal and vertical alignments, drainage facilities, and all
items of construction. The design process of a highway involves preliminary
location study, environmental impact evaluation, and final design. This process
normally relies on a team of professionals, including engineers, planners, econo-
mists, sociologists, ecologists, and lawyers. Such a team may have responsibility
for addressing social, environmental, land-use, and community issues associated
with highway development. An important part of the highway design is a geo-
metrical design.
The geometric design of roads and highways is a very complicated engineering
task. It requires very often to take into account different terrain limitations, espe-
cially in mountainous and densely built areas (Figs. 1.1 and 1.2). Correct route
design is associated with the use of appropriate optimization methods and the use of
appropriate geometric elements that make it easy to adjust the route to those
limitations.
The basic elements of geometric design of highways are horizontal alignment,
vertical alignment, cross section, and intersection (Easa 2003). The horizontal
alignment consists of straight sections (i.e. tangents) connected by horizontal
curves, which are normally circular curves with or without transition curves. The
basic design features of horizontal alignment include minimum radius, transition
curves, superelevation and sight distance. The vertical alignment consists of straight
roadway sections (grades or tangents) connected by vertical curves. The grade line
is laid out in the preliminary location study to reduce the amount of earthwork and
to satisfy other constraints such as minimum and maximum grades.

© Springer International Publishing AG 2017 1

A. Kobryń, Transition Curves for Highway Geometric Design, Springer Tracts
on Transportation and Traffic 14, DOI 10.1007/978-3-319-53727-6_1
2 1 Introduction

Fig. 1.1 Route in the mountainous terrain

Fig. 1.2 Route in the mountainous and densely built area

Two basic curves are used for connecting straight roadway sections in geometric
design (Meyer and Gibson 1980, Lamm et al. 1999, Easa 2003, Rogers 2008,
Brockenbrough 2009, Wolhuter 2015):
• a simple circular curve for horizontal alignment (Fig. 1.3)
and
• a simple parabolic curve for vertical alignment (Fig. 1.4).
1 Introduction 3

Fig. 1.3 Simple circular

curve for horizontal
alignment

Fig. 1.4 Simple parabolic

curve for vertical alignment

Fig. 1.5 Transition curve

Other options include transition curves (Fig. 1.5), compound curves, reverse
curves and combined curves for horizontal alignment. In the case of vertical
alignment possible options include unsymmetrical crest curves (vertical curves
where the total change in gradient is negative) and sag curves (vertical curves where
the total change in gradient is positive).
Transition curves fulfill a special role in the highway design. These types of
curves are used to connect curved and straight sections of highway. They can also
be used to make easier the change between two circular curves where the difference
in radius is large. The purpose of transition curves is to permit the gradual intro-
duction of centrifugal forces. The radius of curvature of a transition curve gradually
decreases from infinity at the intersection of the tangent and the transition curve to
the designated radius at the intersection of the transition curve with the circular
curve.
4 1 Introduction

A horizontal transition curve is a curve which radius continuously changes. It

provides a transition between a tangent and a circular curve (simple transition
curve) or between two circular curves with different radii (segmental transition
curve). For simple transition curves, the radius varies from infinity at the tangent
end to the radius of the circular curve at the curve end. For segmental transition
curves, the radius varies from that of the first circular curve to that of the second
circular curve.
The objectives for providing a horizontal transition curve are given below:
• to introduce gradually the centrifugal force between the tangent point and the
beginning of the circular curve, avoiding sudden jerk on the vehicle. This
increases the comfort of passengers.
• to enable the driver to turn the steering wheel gradually for his own comfort and
security,
• to provide gradual introduction of super elevation, and
• to provide gradual introduction of extra widening.
• to enhance the aesthetic appearance of the road.
Vertical curves are traditionally designed as parabolic curves that are connected
directly to the tangents (without transitions). A vertical transition curve has been
developed for use before and after a parabolic curve (Easa and Hassan 2000a, b).
The vertical transition curve consists of transition–parabolic–transition segments.
Similar to the horizontal transition curve, the vertical transition curve is especially
useful for sharp vertical alignments.
Different, classical types of transition curves are spiral or clothoid (spiral curve),
cubic parabola and lemniscate. The spiral curve is recommended as the transition
curve because it fulfills the requirements of an ideal transition curve, that are:
• rate of change or centrifugal acceleration is consistent (smooth)
and
• radius changes linear at the any curve point.
Also other solutions of this type of curves were presented in the literature. Some
of these curves will be presented in subsequent chapters of this book. A practical
interesting group of curves are so-called polynomial transition curves. The purpose
of this book is to present different transition curves in an orderly manner. In the first
place it includes the presentation of possible ways to mathematical description of
transition curves.
Different solutions of transition curves will be subsequently presented. Some of
these curves have a geometry which is characteristic for the conventionally
understood transition curves. In many cases, the nature of these solutions is more
general than the classically understood transition curves. This approach bases on a
broader definition of transition curves. The transition curves will be understood as
such curves that connect any two points with the specified directions of tangents
and radii of curvature. So understood transition curves can be used for routing of
1 Introduction 5

various counterparts geometric systems, which are listed earlier in this chapter.
Their major advantage is that the entire geometrical transition between any points
can be described by a single equation.
The book presents purely geometrical aspects related to the design described
transition curves. In the book other aspects, such as dynamic aspects, shaping of
superelevation, sight distance analysis or coordination of the horizontal and vertical
alignment are omitted. Depending on the area of application of curves presented in
the following sections of this work, these questions require separate analysis and
research.

References

Brockenbrough RL (ed) (2009) Highway engineering handbook, 3rd edn. McGraw-Hill,

Professional Book Group, New York
Easa SM, Hassan Y (2000a) Development of transitioned vertical curve. I Prop Transp Res Part A
34(6):481–486
Easa SM, Hassan Y (2000b) Development of transitioned vertical curve. II. Sight distance. Transp
Res Part A 34(7):565–584
Easa SM (2003) Geometric design. In: The civil engineering handbook. Chen W.F, Liew J.Y.R,
(ed), CRC Press, Taylor and Francis Group, Boca Raton
Lamm R, Psarianos B, Mailänder T (1999) Highway design and traffic safety engineering
handbook. McGraw-Hill, Professional Book Group, New York
Meyer CF, Gibson DW (1980) Route surveying and design. Harper and Row, New York
Rogers M (2008) Highway engineering, 2nd edn. Wiley-Blackwell, Chichester-Oxford
Wolhuter KM (2015) Geometric design of roads handbook. CRC Press, Taylor and Francis Group,
Boca Raton
Chapter 2
Simple Horizontal and Vertical Curves

In the geometric design of highways, circular curves as horizontal curves (Sect. 2.1)
and parabolic curves as vertical curves (Sect. 2.2) are the most widespread. Apart
from this type of curves, so-called transition curves are traditionally used as a
geometric elements between the straight and the circular arc or between two circular
arcs with different radii. The most popular transition curve is a clothoid (also known
as spiral curve). Apart from the clothoid other solutions of transition curves are also
known. They will be presented in the following sections of this work.

2.1 Circular Horizontal Curve

In the geometric design of horizontal curves a circular curves are very widespread
(Brockenbrough 2009, Easa 2003, Lamm et al. 1999, Meyer and Gibson 1980,
Rogers 2008, Wolhuter 2015). Figure 2.1 shows a circular curve (with radius R and
centre O) joining two straights P’P and K’K with intersect at point W, where:
P and K tangent points,
U angle of intersection of straights P’P and K’K.
The individual geometric elements occupy the following location:
• point S is the mid-point of the circular arc and the mid-point of the tangent line
LM,
• point S lies on the line OW,
• Q is the mid-point of the chord PK and lies on the line OW,
• radii OA and OB intersect the straights P’P and K’K at right angles,

© Springer International Publishing AG 2017 7

A. Kobryń, Transition Curves for Highway Geometric Design, Springer Tracts
on Transportation and Traffic 14, DOI 10.1007/978-3-319-53727-6_2
8 2 Simple Horizontal and Vertical Curves

Fig. 2.1 Circular curve

• tangent line LM and the chord PK are parallel,

• the chord PK is perpendicular to the straight line OW.
The following formulae may be deduced from Fig. 2.1:
• tangent length (PW = WK)

U
T ¼ R tan ð2:1Þ
2

• arc length ( PK )

a¼RU ð2:2Þ

• chord length (PK)

U
c ¼ 2R sin ð2:3Þ
2
2.1 Circular Horizontal Curve 9

• mid ordinate distance (QS)

 
U
b ¼ R 1  cos ð2:4Þ
2

• secant distance (SW)

 
U
w ¼ R sec  1 ð2:5Þ
2

Circular curves can be used not only to design curvilinear transitions between
two straights, as shown in Fig. 2.1. They can also be used for designing complex
geometric systems, such as: compound circular curves, reverse circular curves and
combined curves. Such systems should be understood as follows:
• compound circular curves—two or more consecutive circular curves with dif-
ferent radii (Fig. 2.2),
• reverse circular curves—two or more consecutive circular curves, with the same
or different radii which centres lie on different sides of a common tangent point
(Fig. 2.3),
• combined curves—geometric systems consisting of consecutive transition and
circular curves (Fig. 2.4).

Fig. 2.2 Compound circular

curves (with permission from
ASCE)
10 2 Simple Horizontal and Vertical Curves

Fig. 2.3 Reverse circular curves (with permission from ASCE)

Fig. 2.4 Combined curves (with permission from ASCE)

2.2 Parabolic Vertical Curve 11

2.2 Parabolic Vertical Curve

Vertical curves are used for to smoothly connection two straight lines with different
gradients in the longitudinal profile (Brockenbrough 2009, Easa 2003, Lamm et al.
1999, Meyer and Gibson 1980, Rogers 2008, Wolhuter 2015). In sectional view
(Figs. 2.5 and 2.6), the gradient to the left of the vertical curve will be denoted by
p[%] and the gradient to the right will be denoted by q[%]. The vertical curves are
generally unsymmetrical and can be crest or sag. It depends on the total change in
gradient of two consecutive straight lines:
• crest curves—vertical curves where the total change in gradient is negative,
• sag curves—vertical curves where the total change in gradient is positive.
For some roads (high-speed roads), a cubic parabola is sometimes used as the
vertical curve whose rate of change of gradient increases or decreases with
the length of the curve. In other cases, a quadratic parabola is generally used as the
vertical curve.
In Fig. 2.7 the vertical parabolic curve between two grades p and q which
intersect at point W is shown. In this figure are adopted following designations:
P and Q tangent points,
H the reduced level of P,
L the horizontal length of the curve,
l distance of the highest point of the curve from the point P
The x-y coordinate origin is vertically below P with the x-axis being the datum
for reduced levels y.

Fig. 2.5 Vertical crest curves

12 2 Simple Horizontal and Vertical Curves

Fig. 2.6 Vertical sag curves

Fig. 2.7 Parabolic vertical

curve

The basic requirement for the vertical curve is that the rate of change of gradient
(with respect to horizontal distance) should be constant. The equation of the vertical
curve is
 q  p
y¼ x2 þ px þ H ð2:6Þ
2L

The distance of the point W from the point P is

1
LW ¼ L ð2:7Þ
2

The horizontal distance to the high point (for crest curve) or low point (for sag
curve) is
2.2 Parabolic Vertical Curve 13

p
l¼ L ð2:8Þ
qp

The reduced level of point Q is

Hq ¼ H þ pLW þ qðL  LW Þ ð2:9Þ

References

Brockenbrough RL (ed) (2009) Highway engineering handbook, 3rd edn. McGraw-Hill,

Professional Book Group, New York
Easa SM (2003) Geometric design. In: Chen WF, Liew JYR (eds) The civil engineering handbook.
CRC Press, Taylor & Francis Group, Boca Raton
Lamm R, Psarianos B, Mailänder T (1999) Highway design and traffic safety engineering
handbook. McGraw-Hill, Professional Book Group, New York
Meyer CF, Gibson DW (1980) Route surveying and design. Harper & Row, New York
Rogers M (2008) Highway engineering, 2nd edn. Wiley-Blackwell, Chichester-Oxford
Wolhuter KM (2015) Geometric design of roads handbook. CRC Press, Taylor & Francis Group,
Boca Raton