## Basic formula for calculating horizontal radius

There are various, equally correct, methods to calculate horizontal radius. The first relates radius to speed, surface friction and superelevation. A second considers the physical limitations of a long vehicle negotiating a tight bend. This is more appropriate for low design speed roads.

The first method 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.

R =        V2 / 127 * (e + f)

where

R = horizontal radius (in metres)

V = speed (usually, design speed) in km/hr

e = superelevation, in metres / m

f = coefficient of lateral friction (no units)

This formula can be found in reference works from a number of countries, such as the UK (1), the USA (2), Germany (3), New Zealand (4).

Radius

The design radius for a circular curve is a function of the design speed V, the superelevation of the road surface, and the coefficient of lateral friction between tyre and road surface. The minimum value for R is calculated from the design speed and from maximum permitted values of e and f for that speed. Values for R range from infinity (for a straight line) to as low as 30 m (for an urban road design speed of 30 km/hr). At the lower speeds the ability of a long vehicle to negotiate a tight corner may be more important than a consideration of superelevation and surface friction.

#### Speed

Values for V range between 10 km/hr and 160 km/hr, at intervals of 10 km/hr. Highway engineers do not design for intermediate values for V (such as 108.913 km/hr).

Superelevation

Design standards place restrictions on the value of e (how steeply a road may be inclined). If it is too steep the vehicle may either overturn, or (when stopped) slide down towards the lower edge of the road.

Lateral friction

Some standards also place restrictions on f (how much of the potential coefficient of friction can be used). There are also different scenarios for f : values may apply to dry roads or to wet or icy roads. Values for f vary with speed. In some countries there are different values for f depending on the geography (for example, flat or hilly; urban or rural) or depending on the type of road. I cannot see the reason for these however.

Notes

Theoretically the recommendations on minimum radius would be identical from country to country, or at least where the countries use the same design formulae, where the same vehicles use the roads, and where the road surfaces are the same.  If the recommendations vary a lot between countries, then perhaps there is an argument for one of them to represent “best practice”values.

References

1. C.A. O’Flaherty, “Traffic planning and engineering” page 378 (Edward Arnold 1986)
2. Wolfgang S. Homburger et al, “Fundamentals of traffic engineering, 13th edition” page 19-2 (University of California, 1992)
3. Weiss, Durth u.a., “Strassenbau, Planung und Entwurf” page 170 (Verlag für Bauwesen 1997)
4. .., “State highway geometric design manual, section 2 – basic design criteria (draft)” page 2-19 (Transit, April 2003)

January 2015 update – see also this post from the 14th January.

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## About roadnotes

Robert Bartlett is an international consultant with over 30 years of professional experience as a highway and traffic engineer with leading companies and organisations in several countries, including Germany, China (Hong Kong), Qatar and the UK. Specialised in urban studies, transport and the use of GIS, research has included new ideas on subjects such as the study of social justice using GIS, the dimensions of vehicles, and comparative geometrics (highways and transport).
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### 4 Responses to Basic formula for calculating horizontal radius

1. Chris says:

This is not used in the US. We use mph and feet. Formula is e + f = v^2 / 15*R

2. sudarsan chanda says:

very good…