GTA 307 Minimum horizontal radius (rev.05)

Global Transport Atlas
Series 3 - Comparative Geometrics
307 Minimum horizontal radius (Revision .05)
GTA series 3 / 307 rev. 05 January 2016 Page 307 / 1
What it is
There are different types of minimum horizontal radius. These include:
1. Minimum radius whilst maintaining normal (negative) crossfall (= adverse camber)
2. Minimum radius to avoid need for transition curves
3. Minimum radius where superelevation equals normal crossfall
4. Minimum radius for an upper limit of superelevation
5. Minimum radius for aesthetics
This note refers to the minimum radius for an upper limit of superelevation.
These minimum values are usually associated with circular curves. There are other types of horizontal curve.
The conventional theory
The most accepted theory relates the horizontal radius to speed, superelevation and side friction. It imagines
a vehicle travelling along a curve on an inclined slope. It assumes that the horizontal element of the
centrifugal force equals the resisting force provided by friction at the road surface between the vehicle tyres
and the road. The formula is:
R = V2 / 127 * (e + f)
Easa and Dabbour (Ref. 1643) say this is one type of vehicle stability model, and call it the "point-mass" (PM)
model.
Where the conventional theory is applied
This formula can be found in reference works from a number of countries, such as the UK (168 / 1986), the
USA (ref. 831 /2011), Nigeria (ref. 1505 / 2013), and New Zealand (80 / 2003), Ecuador (1746 / 2013), and
e.g. for bikeways in USA ((ref. 1919).
Parameter Symbol Units
R horizontal radius metres
V
speed (usually, design
speed)
km/hr
e superelevation metres / m
f lateral friction (no units)

307 - Minimum horizontal radius for maximum superelevation
Series 3- Comparative Geometrics
row country year ref.no. e 40 50 60 70 80 90 100 110 120 note
a Switzerland 1991 732 7% 45 75 120 175 240 320 420 525 650
b Multi-country 2010 1887 7% 34 53 91 148 219 319 414 des. min.
c Multi-country 2010 1887 7% 30 47 71 102 153 236 342 abs. min.
d Argentina 2010 1860 6% 210 290 395 515 645 785 935 1095 1270 des. min.
e Germany 2008 1615 6% 280 370 470 720 900
f UK 1974 167 7% 130 230 350 510 min.
g UK 1974 167 7% 120 150 300 normal
Radius and vehicle type:
h Multi-country 2002 1968 7% 70 116 181 258 363 trucks
i Multi-country 2014 2182 6% 41 73 bicycles
 e = superelevation
 Row a, Swiss values taken as benchmark values
 Row b, c: from Austroads
 Row e: usage factor 0.4
 Row f: for rural roads
 Row g: for urban roads
 Row h: desirable minimum for trucks
Some values for minimum horizontal radius
Notes
DISCUSSION
The following sections cover:
 Where the conventional theory is modified
 Where the conventional theory may be weak
 Alternative theories
 Health warning
 Comment
 References
 Cover note and disclaimer
 History and change log for this note

Permitted degree of use
In Germany (e.g. ref. 1615) a "permitted degree of use" factor is applied to values for side friction. This factor
varies with road type.
Desirable and minimum values
The Austroads multi-country standard (ref. 1887) refers to desirable and absolute minimum values, which
are calculated from values of desirable and maximum side friction.
In Argentina (ref. 1860) the theory is modified in the sense that it refers to four different values for minimum
radius:
 Rmin (abs) - absolute minimum radius, where e, f are at their maximum values and V is guideline speed
 Rmin (des) - desirable minimum radius where e, is at maximum value, f is zero and V := VMM (average
free flow speed)
 Rmin (BR) - minimum radius with crossfall removed (e+f = 0,02 - I assume f is zero - and V := VMM)
 Rmin (BN) - minimum radius with normal crossfall (e+f = 0,015 and V is guideline speed
Permitted values of side friction and superelevation
Countries often have different ideas about which values of lateral friction and superelevation are to be used in
design (and so in the conventional formula).
Steep downgrades
Austroads (ref. 1887) says that "On steep downgrades there is a greater chance of some drivers tending to
overdrive horizontal curves. Therefore, the minimum curve radius .... should be increased by 10% for each 1%
increase in grade over 3%, (using a particular formula)
Road type
Some countries provide separate recommendations for regional, urban and rural roads. For urban roads for
example, permitted superelevation is often lower than for other roads.
Consideration of comfort
Austroads says that "the desirable maximum values (for side friction) should be used on intermediate and
high-speed roads with uniform traffic flow, on which drivers are not tolerant of discomfort” and that “In ....
mountainous terrain, drivers are more tolerant of discomfort. This permits the absolute maximum values of
side friction to be safely used in the design of horizontal curves".
Type of terrain
Same comment as above under "considerations of comfort"
Type of vehicle
Side friction varies with type of vehicle ( see e.g. table 2.6 in ref. 80). Speed limits (and therefore perhaps
design speed) also vary with type of vehicle.
Type of road surface
Where the conventional theory is modified

Side friction (and so horizontal radius) also varies with type of road surface, see(again). table 2.6 in ref. 80.
Also AASHTO 2011 (ref. 831) says that friction values for gravel surfaces are less than those for paved
surfaces, and gives a chart which shows minimum radius and superelevation for gravel-surfaced roads.
Where the minimum radius is measured
Ref. 831 says (e.g.) that “For consistency with the radius defined for turning roadways and to consider the
motorist operating within the innermost travel lane, the radius used to design horizontal curves should be
measured to the inside edge of the innermost travel lane, particularly for wide roadways with sharp
horizontal curvature"
Aesthetics
Ref. 1887 says:
“In flat terrain, aesthetics may be improved if curves of double the minimum length are provided.
Some road authorities specify that curves on more important two-lane roads should be at least 120 –
150 m long, but curves on mountain roads may be as short as 30 m. On divided roads of high standard,
curves less than 300 m long look too short”.
Where the conventional theory may be weak
The conventional theory may be wrong
In this context, E. Hauer (ref. 765; 1999) says "it is by now clear that there is no premeditated connection
between the reality of crash occurrence on horizontal curves and the procedure used for their design". On the
use of design speed as a parameter, a 1994 paper from the Dutch SWOV Institute (ref. 264) says "Regarding
especially the safety at bends, one could say that the definition of a minimal radius depending on the design
speed is both insufficient and unnecessarily constraining".
The values used to develop horizontal radius may be out of date
Some guidelines use side friction values which are 40 years old or more. For example, Asea and Dabbour
note that "The required increase in minimum radius presented in this paper is based on current design
values of side friction. These values were developed for passenger cars many years ago and should be revised
to account for the characteristics of modern passenger cars and trucks"
The explanation of what is being offered may be misleading
Users of design standards have to be vary careful in how they understand the suggested values. For example
Austroads (ref. 1887) table 7.5 refers to "minimum radius of horizontal curves..." - but these may only be
minimum radius for cars (and not, for example, for trucks)
Other weaknesses
The conventional theory does not work or may be misused:
 On low-speed roads, where the vehicle geometrics become a limiting factor
 For trucks, where high loads can lead to a higher risk of overturning than of side slipping
 On three-dimensional roads, where (for example) a combination of gradient and superelevation can lead
to excessive crossfall (note: all roads are three-dimensional objects).

 On all roads where the design values are calculated for cars and where the roads are also used by other
types of vehicle
 On unsurfaced and gravel-surfaced roads, where the design values are calculated for asphalt surfaced
roads
Vehicles may follow a sharper alignment than the curve radius
In "Safer Curves on Multiple Lane Roads" (Ref. 1996) , Johan Granlund points out that vehicles do not
always follow the alignment of a curved road, but often change lanes at the same time, and here "when
shifting lane quickly, the vehicle experience a transient “curve radius” much sharper than indicated by the
road curve radius".
The conventional theory only applies to cars and not (e.g.) to trucks
In "Lowered crash risk with banked curves designed for heavy trucks" , (Ref.1997) Mr. Granlund and his
co-authors conclude that (amongst other points) :
 For roads where same speed limits apply for both passenger cars and HGV´s, (....) the need for road
superelevation is given by HGV’s rather than by cars. Hence road design codes should use models of
HGV´s rather than of passenger cars.
 A conclusion was that the traditional point-mass “car model” can underestimate the superelevation
needed for safe HGV operations.
The position of design control lines can vary, even within one road design project
(Ref. 2133) says that "The horizontal and vertical elements of a road are described in terms of control lines.
Control lines are lines mathematically defined in the horizontal and vertical planes”. The blog post (link 1)
argues that, if you break the location of vertical, superelevation and horizontal (radius) control lines you are
perhaps breaking the theory behind road design, with the possibility that in practice one or other design
parameter will be below minimum requirements.
Different standards may refer to different measures of speed.
For example, both (Ref. 1088, SIECA 2004) and (Ref. 1887 Austroads, 2010) have tables for speed and
minimum horizontal radius - but (Ref. 1088 table 4.10) refers to design speed whereas (1887) refers to
operating speed. Perhaps the theory behind the two tables is different, but at least it makes any comparison
of the values of minimum radius a bit doubtful.
Some engineers apply different design speeds for vertical and horizontal alignment design for
the same section of road
Wolhuter (Ref. 2247) in his discussion on vertical alignment, quotes Transit New Zealand to say that:
“It is good design practice to make the vertical alignment design speed 10 to 15 km/h greater than the
horizontal alignment design speed to provide an additional safety margin (Transit New Zealand,
2002)”

Alternative theories
Bicycle model
Easa and Dabbour (Ref. 1643) describe two other vehicle stability models. One, the "bicycle model" adds
consideration of front and rear tyres (and so two different side friction factors.
Vehicle dynamics model
The second, a vehicle dynamics model from the University of Michigan, " accurately simulates a vehicle
traveling through a user-defined alignment, taking into account vehicle characteristics such as body roll,
pitch, yaw, and lateral weight distribution".
3D design approach
The same authors recommend a 3D approach. In their concluding remarks they say that "current geometric
design guides do not account for 3-D alignments in the calculation of the required minimum radius for
horizontal curves".
A recent paper by Amiridis and Psarianos on 3D road design (ref. 2271) includes an extensive list of
references on 3D design.
Rate of change of acceleration
Kilinc and Taybura (ref. 810 / 1999) describe a method for calculating minimum horizontal radius based on
the limiting values of rate of change of acceleration ("jerk").
Vehicle geometry
On roads with low design speeds, the physical limitations of the road vehicle can be the deciding factor in
determining minimum radius. For example (ref. 80) has a table which gives minimum design radius for
different types of design vehicle.
Risk of vehicles overturning
For example, on motorway slip roads with quite small radius curves and inclined vertical alignment the
minimum radius may be determined by calculation of the risk of high vehicles overturning.
Australia (or rather, Austroads) practice says that (ref. 1968):
"Trucks have a higher centre of gravity than cars. Consequently, the limiting condition for trucks
negotiating a circular curve tends to be rollover rather than skidding".
Austroads then developed a method of calculating horizontal radius for trucks which is based on a measure of
truck rollover which they call "static roll threshold".
Maximum level of centrifugal acceleration
(Ref. 1038) says that “British design practice is based on the fundamental assumption that at absolute
minimum radius the 99th percentile vehicle should not experience more than the maximum level of
centrifugal acceleration acceptable for comfort and safety. This was established at about 0.22 some 70 years
ago and has not been changed since”
….And that centrifugal acceleration is given by : v2 / R where
Parameter Symbol Units
R horizontal radius metres
v
speed (usually, design
speed)
m/sec

Health warning
Users have to be vary careful in how they understand the suggested values in any particular standard.
Preferably, each standard should attach a health warning to its suggested values, along the lines of:
“The values for minimum horizontal radius quoted in these guidelines are for cars driving on wet
regional roads in level terrain and with an asphalt or concrete surface, for the quoted values of
superelevation and side friction, and with the usual secondary provisos (car brakes and tyres in good
condition, 90th %ile driver etc.).
The values do not apply to exceptional circumstances; for example, they do not apply to roads which
are on steep gradients, or on structures. They do not apply to three-dimensional roads, to other types
of vehicle, or to other types of road surface (e.g. gravel or high-friction).
It is not sure what the relationship is betwen these values for minimum radius and road safety.
It is also not sure what the relationship is between the historical values for side friction used in
calculating the minimum radius, and the values which might be found today with modern cars, tyres
and road surfaces. In particular the values do not apply to roads with high friction surfaces.”
Comment
The use of different terms for what might be the same thing can lead to confusion. For example the
references have terms such as: normal, desirable, minimum, desirable minimum and absolute minimum.
They could be reduced in number and carefully defined or explained.
It seems likely that collecting the values for minimum horizontal radius quoted in different design standards
provides a set of results which is both over-complicated and over-simple. Over-complicated in the sense that
there should be no need for what appears to be a wide range of values in different countries. For example,
only a few values are used for maximum superelevation, and for similar road surfaces and vehicle types there
need probably only be one set of friction values. And over-simplified in that design standards usually only
refer to cars and surfaced roads.
In a tech blog post on the simplification of standards (see here) I wrote:
1. Standards should be prepared on a modular basis, with each module describing a specific parameter
2. There should be one main document for each parameter, with -"modifications" notes issued by
responsible authorities where they feel local circumstances are different
3. Standards should not be duplicated
4. Standards should be related to climate and terrain conditions rather than to administrative boundaries
5. Each module should provide details in terms of the primary influencer, design speed
6. Each module should include comments and / or values in terms of the main secondary influencers, such
as
 Road type
 Vehicle type
 Road surface
 Terrain
 Climate

80 - New Zealand, shgdm-part-2 Basic design criteria, Transit, 2003
167 - UK, Hobbs, “Traffic planning and engineering” Pergammon Press, 1974
168- UK, O’Flaherty, Traffic planning and engineering; Edward Arnold, 1986
732 - Switzerland, “VSS 640-080 Projektierung, Grundlagen; VSS, 1991
765 - Canada, Ezra Hauer, “Safety in geometric design standards”, 1999
810 - Turkey, Ahmet Sami KILINÇ and Tamer BAYBURA, “Determination of Minimum Horizontal Curve
Radius”; FIG Working Week, 2012
831 - USA, “A policy on the geometric design of highways and streets”; AASHTO, 2011
1038 - UK, C.A. O’Flaherty, “Transport planning and traffic engineering”; Elsevier, 2006
1088 - Multi-country, “Manual Centroamericano De Normas Para El Diseño Geométrico De Las Carreteras
Regionales; SIECA, 2004
1505 - Nigeria, Highway manual part 1 Design / vol. I: geometric design; Federal Ministry of Works, 2013
1615 - Germany, RAA Richtlinien für die Anlage von Autobahnen; fgsv, 2008
1643 - Canada, Easa and Dabbour, “Design radius requirements for simple horizontal curves on 3D
alignments”; Can. J. Civ. Eng.30: 1022–1033, 2003
1746 - Ecuador, NEVI 12 volume 2A norma para estudios y disenos viales ; MTOP, 2013
1860 - Argentina, Normas y Recomendaciones de Diseño Geométrico y Seguridad Vial; DNV, 2010
1887 - multi-country / Austroads, AGRD part 3: Geometric design; Austroads, 2010
1968 - multi-country / Austroads, “Geometric design for trucks - when, where and how?”; Austroads, 2002
1996 - Sweden, Granlund, "Safer Curves on Multiple Lane Roads," Transport Research Arena Europe 2010
1997 - Sweden/Norway, Granlund, Haskanes and Ibrahim, "Lowered crash risk with banked curves designed
for heavy trucks"; HVTT13, San Luis, Argentina, 2014
2133 - Canada, "Geometric Design Manual Part 2", Middlesex County, Ontario, Canada, 2015 (?)
2182 - Multi-country / Austroads, “Cycling aspects of Austroads guides”, Austroads 2014
2271 - Greece, Amiridis and Psarianos, “Three dimensional road design by applying differential geometry and
conventional design approach criteria”, Mathematical Design & Technical Aesthetics, ISSN 2310-2179
(Online) Volume 3, Issue 1 (2015)
Links
Blog post 1 https://comparativegeometrics.wordpress.com/2015/10/01/road-geometric-design-control-
lines/
Blog post 2 https://comparativegeometrics.wordpress.com/2015/11/14/simplification-of-standards-3/
References

Cover notes and Disclaimer
This is a research document. The best efforts have been made to make sure the figures are correct. However no liability
can be taken for any of the details, information or analysis in this document.
The layout, look and feel of this document is copyright.
The photos are generally copyright of REB.
This work is licensed under the Creative Commons Attribution-NoDerivs 3.0 Unported License. To view a copy of this
license, visit http://creativecommons.org/licenses/by-nd/3.0/
History and Change log for this note
First version published December 2014.
Rev. 05 (January 2016) - Added notes on different measures of speed and different speeds for horizontal and vertical
design. Added row h to values table. Modified layout. Added extra note and reference on 3D road design.
Rev. 04 -/-
Rev. 03 (November 2015) - Added note about control lines
Rev. 02 (February 2015) - Added notes on papers by Johan Granlund et al (refs. 1996, 1997).
Contact
We welcome comments on this paper, and also on new
developments in other countries in this field.
Email: roadnotes2@gmail.com
Web: http://comparativegeometrics.wordpress.com/
About the contributor
Robert Bartlett, Germany - is an experienced
transportation and urban development studies engineer
with over 25 years of professional experience. Current
engineering work: includes technical research in highway
design standards and applications in areas such as urban
planning and highway engineering. Interests include
applied GIS.

GTA 307 Minimum horizontal radius (rev.05)

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