Home Physics Banking of roads – The angle of banking- formula|definiton

Physics

Banking of roads – The angle of banking- formula|definiton

Have you ever seen roads that rise up at an angle, especially in turns and bends? This is called the banking of roads. This is an important technique to stop accidents due to sliding off of cars during high-speed turns.

The outer edges along the turn are at an angle of incline compared to the inner edge. In other words, the outer edge of the road is slightly raised. This is done to provide the vehicle with centripetal force to take a safer turn.

It is difficult to build a straight road always. There might be some natural barriers like a river, coastline, rocks, or mountains due to which roads bend. A straight road is usually flat but a twisted road is usually banked at an angle.

Max velocity for a safe turn in a non-banked curved road:

In a non-banked road, the centripetal force in a turn is provided by kinetic frictional force.

mv²/r = μ_k*N [N=mg]

v_max = sqrt(μ_k*r*g) [sqrt-square root]

Why banking of roads is important?

During the turn inroads, the inertia of the vehicle does not allow the vehicle to turn. The road will turn but the vehicle doesn’t. So, to overcome this inertia the road has to be designed in such a way that it generates this extra force on the opposite side.
In some cases, the centripetal force provided by the friction between road and car might not be enough to turn. Conditions like rain, poor tire grip, etc may lead to the skidding of cars off the road.
Turning at high speed on highways may lead to the skidding off of vehicles. The vehicle has to reduce speed to make such turns. But in hilly regions and coastlines, there will be plenty of turns and bends which will reduce the average speed of the vehicle. Thus banking of roads can increase the average speed of the vehicle thus reducing the travel time in a safer way.
In circular tracks where there are continuous turns, banking is very important for a safe ride.
It will reduce the wear and tear on the tires. Along with friction, the component of normal force also provides the much-needed centripetal force.

What is the mechanism behind the banking of roads?

Banking of roads provides the much-needed centripetal force which pulls you towards the center of the turn. Thus you can take a safe turn without skidding off.

Centripetal force is the force that helps a body to continue in a circular motion and pulls or pushes it towards the center.

Resolution of forces acting upon a vehicle on a banked road.

The angle of banking: The angle at which the outer edge is inclined with respect to the inner edge of the road.

The forces acting upon the vehicle:

The weight equal to mg acting in the downward direction
The vertical component of normal force balancing the weight
The horizontal component of normal force plus the friction makes up the centripetal force.

Factors to be considered while deciding banking angle:

The centripetal force of the vehicle. This is directly dependent on the speed of the vehicle. So, the average speed of the vehicle in the turns has to be studied first. The speed of vehicles in hilly regions will already be low whereas, the speed of the vehicle on broad highways will be higher.
The natural conditions in the road like rainwater, condition of the road, etc to estimate the friction provided by the road.
The type of vehicles traveling the road. Need to consider the kind of tires used by the vehicles can also be a small factor. On snow-covered roads, people use chains to increase friction.
Bank turn: The total change in direction of the vehicle’s path during the turn.

The angle of banking formula:

Let us derive the formula for the banking angle.

First, equate the forces in the vertical axis.

Weight(downward) + component of kinetic friction(downward) = component of normal force(upward)

mg + f*sinθ= N*cosθ

mg = N*cosθ- f*sinθ = N*cosθ- μ_kN*sinθ = N(cosθ- μ_k*sinθ)

mg = N(cosθ- μ_k*sinθ) ———–(1)

Now, equate the forces in horizontal direction

centripetal force(right-side) = component of kinetic friction(left-side)+ component of normal force(left-side)

mv²/r = N*sinθ + f*cosθ = N*sinθ + μ_kN*cosθ = N(sinθ + μ_k*cosθ)

mv²/r = N(sinθ + μ_k*cosθ)————–(2)

Dividing eq (1) and (2) we get,

rg/v² = (cosθ- μ_k*sinθ)/(sinθ + μ_k*cosθ) [divide by cosθ on numerator and denominator]

rg/v² = (1- μ_k*tanθ)/(tanθ + μ_k)

For a worst case scenario of μ_k = 0;

rg/v_max² = 1/(tanθ)

Hence tanθ = sqrt(v²/rg) and θ = tan^-1sqr(v²/rg)

v_max = sqrt(rg*tanθ)

Research

In this article, the authors have developed a Linear Quadratic Regulator (LQR) lateral stability and rollover controller by calculating the bank angle. The authors have studied using various simulations and analyzed key factors like Lateral load transfer, yaw rate, roll angle, and side-slip angle. Link: https://doi.org/10.3390/s17102318