# Desired radius of a curve

## Description

The equation for the desired radius of a curve, takes into account the factors of speed and superelevation (e). This equation can be algebraically rearranged to obtain desired rate of superelevation, using input of the roadway’s designated speed and curve radius.

In curved sections, the outside edge of the road is superelevated above the centerline. Since the road is sloped down toward the inside of the curve, gravity draws the vehicle toward the inside of the curve. This causes a greater proportion of centripetal force to supplant the tyre friction that would otherwise be needed to negotiate the curve.

Superelevation slopes of 4 to 10% are applied in order to aid motorists in safely traversing these sections, while maintaining vehicle speed throughout the length of the curve. An upper bound of 12% was chosen to meet the demands of construction and maintenance practices, as well as to limit the difficulty of driving a steeply cross-sloped curve at low speeds. In areas that receive significant snow and ice, most agencies use a maximum cross slope of 6 to 8%. While steeper cross slope makes it difficult to traverse the slope at low speed when the surface is icy, and when accelerating from zero with warm tyres on the ice, lower cross slope increases the risk of loss-of-control at high speeds, especially when the surface is icy. Since the consequence of high speed skidding is much worse than that of sliding inward at a low speed, sharp curves have the benefit of greater net safety when designers select up to 8% superelevation, instead of 4%. A lower slope of 4% is commonly used on urban roadways where speeds are lower, and where a steeper slope would raise the outside road edge above adjacent terrain.

The American Association of State Highway and Transportation officials (AASHTO) provides a table from which desired superelevation rates can be interpolated, based on the designated speed and radius of a curved section of roadway. This table can also be found in many state roadway design guides and manuals in the U.S.

Recent research has shown that, considering rollover risk for heavy vehicles (semitrailers & buses), which have a relatively high centre-of-gravity, the above equation yields cross slope values which are too low.

Related formulas## Variables

R | desired radius of a curve (m) (dimensionless) |

v | speed of the vehicle (m/s) (dimensionless) |

e | superelevation (dimensionless) (dimensionless) |

f_{s} | factors of speed or coefficient of friction (dimensionless) (dimensionless) |