Bank Angle Calculator

Curves are not flat most of the time. If you look closely at highway ramps or race tracks, you will notice that the outer edge is higher than the inner edge. This incline is called banking. Its sole purpose is to keep vehicles on their path while going around a curve.

This calculator allows you to calculate the banking angle by simply entering speed and radius. If two of three values (speed, radius, banking angle) are entered this will calculate the remaining value. It can be used for both roads with banking and also aircraft bank in order to maintain a level turn. In addition it shows important values for both road and aircraft, namely safe speed range where tires have grip, felt acceleration and rate of turn.

What is a bank angle?

An object requires a force to move in a circular path. This is called the centripetal force and its magnitude is equal to mass times velocity squared divided by radius.

On a flat road only the friction between the tires and the surface can provide this inward force. If you increase your speed or decrease the radius, the required force quickly becomes larger than the available friction, and the vehicle will skid outward. Banking of the road solves this problem because the incline of the road provides an upward component of the normal force (the force that the road exerts on the car) which points inward, effectively replacing some of the friction.

Frictionless banking: design speed.

Imagine a banked curve that is completely frozen and offers no grip. Two forces act on the car: gravity pulling down, and normal force acting perpendicular to the inclined surface of the road, pushing outwards. If you break up the normal force into a vertical component equal to gravity, and a horizontal component creating the centripetal force, then the masses cancel each other out.

This equation can be adapted to problems of any kind.

where θ is the banking angle, v is the speed, r is the radius and g is the gravitational acceleration, which is about 9.81 meters per second squared. The speed at which a particular bank angle is exactly optimal, i.e. the speed where no frictional force is required, is usually referred to as the design speed of that curve.

If you take friction into account, this is the safe speed range.

A real road is not completely frozen. Because tires need grip, the safe state of a banked turn extends over a range of speeds and not just one speed. If the speed is too high, the vehicle will slide outward and up, exiting the curve. On very steep roads, if the speed is too low, the vehicle may slide inward and down. The friction force opposes both types of sliding.

If μ is used to denote the coefficient of friction, then the maximum and minimum safe speeds are given by:

If the tangent of the banking angle is less than the coefficient of friction, then the formula for minimum speed will give a negative value, which means that there is no realistic minimum speed. Put simply, if the bank is shallow enough, friction alone can prevent sliding inwards, regardless of how fast the vehicle is going. Thus gentle motorway bends have a maximum safe speed but not a minimum one, whereas steep oval track corners actually have speed limits.

Example: Highway curve

Consider a curve with a radius of 500 meters and an angle of inclination of 4 degrees. This is a typical 7 percent grade that would be used in areas where snowfall is not likely. Plugging these values into the design speed formula yields a speed of approximately 18.5 meters per second, which equates to about 67 kilometers per hour.

Assuming the tire's coefficient of friction on dry asphalt is 0.7, then the maximum safe speed would be about 63 meters per second, or nearly 227 kilometers per hour. The tangent of 4 degrees is approximately 0.07, which is much less than 0.7, so there is no minimum speed. This is because at such shallow angles, a vehicle cannot skid inwardly. Engineers wisely set the speed limit well below this upper bound.

Banking angles in the air: airplane maneuvers

The principle is similar to that of turning an airplane, except the plane itself is tilted rather than the road. When a plane banks into a turn, the lift force which normally acts perpendicular to the direction of travel becomes tilted inward. This inward component of the lift provides the centripetal force while the vertical component continues to support the weight of the aircraft.

When these two components are balanced, they result in the same geometric relationship as a frictionless track.

Lift increases because it simultaneously requires weight and power to overcome the turn. This factor of increase is called load factor which indicates the number of g-forces experienced by the aircraft and its crew.

At a 60-degree bank angle the load factor is 2. So when flying a horizontal turn at this position there are 2g acting on it, which is twice its weight. Also, the stall speed increases by the square root of the load factor, so tight turns at low altitude need to be executed with particular care.

Note on the definition of angle: some reference material gives the aircraft's bank angle as Arctan of the value obtained by dividing "g" times the radius by the square of the speed. This will give the complementary angle to that given here. The bank angle of this tool is with respect to the horizontal, and the usual relationship between load factor "n" and cosine of the angle (cos theta) is the inverse of cos theta.

Example: light aircraft flying in a curve

A light aircraft flying at a speed of about 77 metres per second (almost 150 knots) and banked at an angle of 30 degrees will describe a horizontal curve with a radius of about 1,050 metres. The load factor is the reciprocal of the cosine of 30 degrees which is approximately 1.15 g. The rate of turn is speed divided by the radius which gives about 4.2 degrees per second. This well exceeds the normal rate of turn of 3 degrees per second so a full 360-degree turn would take only one and a half minutes.

Is vehicle weight important?

No it doesn't matter. This is surprising to many people. The mass is considered on both sides of the equation and therefore completely cancels out. A fully loaded car and a small car will need the same amount of angle to take the same curve at the same speed. Mass does change the forces that the tires or wings have to apply, but it doesn't change the geometry of the curve.

Where is the bank?

Curves on motorways and regular roads are slightly banked, usually only a few percent. This allows vehicles to take the curve without relying solely on friction when driving in wet conditions.

The inclines on racetracks are much steeper. The steep inclines of high-speed ovals or the walls of velodromes allow cars and bikes to reach speeds far in excess of what would be allowed on a flat track. At the same time, there is also a "minimum speed". If this speed is not reached, the rider will slide down along the wall.

Railways also have a tilt in curves called cant. By raising the outer rail, it reduces the lateral force felt by passengers. Aircraft also change direction through a combination of roll and yaw. This creates additional lift and load factor to allow for turning.

Used symbols:

This tool is for educational purposes and initial estimates. Actual planning of roads or racetracks must also take into account aspects such as drainage, driving comfort, visibility, and legal requirements. Even in real flights, aircraft must comply with published restrictions. These values are to be understood as guidelines and may not serve as a basis for technical approvals.