## Is the basic formula for horizontal radius wrong?

In an earlier blog post (here) I said:”there are various, equally correct, methods to calculate horizontal radius”. However, after further research, it seems that these methods are not equally correct, and perhaps even not correct at all. For example, I referred to a conventional formula for calculating horizontal radius.The formula (for metric units) is:

Rmin = V2 / 127 (e+f)

Where

• Rmin = minimum radius (in metres)
• V = speed (km/hr)
• e = superelevation
• f = side (or radial) friction

You can find this formula in the design standards of many countries, including the USA, Germany,Switzerland, Italy,etc.  However there are problems with the formula. For example:

1. It isn’t the only way to calculate horizontal radius

Another way relates side friction and driver comfort

2. The formula quoted is an unproven hypothesis

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”.

1. The formula may be basically wrong

Theoretically, if friction varies with speed, then the formula should show the f element (friction) varying with speed. However a paper describing the new Austroads standards (ref. 1539) suggests that friction does not vary with speed:

“The previous guides used a longitudinal friction factor (coefficient of deceleration) that reduced as the speed increased …. Newer research …., showed that modern vehicle braking performance could be considered to be uniform across a range of speeds” (ok, I know that in the formula “f” represents side friction).

E. Hauer again (ref. 765) says that

“Since the design speed has no clear relationship to either the speed limit or the speed expected to be exceeded by only a very small proportion of drivers, it is entirely unclear what it represents or why it ought to be relevant to curve design”

4. Different countries tinker with the formula in different ways

For example Norway applies a factor of safety which is a function of speed and traffic flow

5. The basic formula does not include a factor of safety

– although some manuals may openly include one (see above) or hide one in the values they suggest for side friction (theoretically, without a factor of safety any vehicle travelling even 0.1 km/hr faster than the specified speed will fly off the outer edge of the road)

1. The formula does not distinguish other variables which influence friction and superelevation

For example, side friction varies with the type of vehicle and the road surface (so a road which is designed using friction for cars is not a safe road for trucks).

Comment

Perhaps, in the same way that packets of cigarettes are often now stamped with a health warning, any use of the basic formula quoted above should begin with a preamble along the lines of:

“The hypothetical relationship between speed, side friction and superelevation given in this formula is valid only for a particular definition of speed, a particular type of vehicle and road surface, one particular theoretical and unproven understanding of the relationship between radius, friction, speed and safety, and a particular range of traffic flow and road classification. Any relationship with reality (in terms of accidents and of vehicles skidding off roads) may be largely fortuitous”.

Where things get even stranger is in the concept of consistent (safe) design. If we don’t even know if the basic formulae are correct, consistent designs (based on erroneous formulae) will never be correct.

References

264 – Netherlands, “SWOV R-94-07 Safety effects of road design standards”, SWOV; 1994

765 – Canada, Ezra Hauer “Safety in geometric design standards”, 1999

1539 – Australia, Barton and O’Callaghan, “Australia: New National Guide to Road Design”; 4th International Symposium on Highway Geometric Design; 2010