14.4. Analysis of super-elevation

Super-elevation or cant or banking is the transverse slope provided at horizontal curve to counteract the centrifugal force, by raising the outer edge of the pavement with respect to the inner edge, throughout the length of the horizontal curve.

Super-elevation refers to the banking of the road on curves. When a vehicle goes around a curve, it experiences a centrifugal force pushing it outward. To counteract this force and prevent skidding, the outer edge of the road is raised relative to the inner edge. This banking helps to balance the forces acting on the vehicle, allowing it to navigate the curve safely.

Imagine riding a bicycle around a curved track. If the track is flat, you might lean inward to avoid falling outward. However, if the track is sloped on the outside (banked), you can ride around the curve more easily without needing to lean too much. The slope helps you maintain your balance.

When the outer edge is raised, a component of the curve weight will be complimented in counteracting the effect of centrifugal force. The forces acting on a vehicle while taking a horizontal curve with superelevation is shown in figure 14:2. Forces acting on a vehicle on horizontal curve of radius R m at a speed of v m/sec2 are: P the centrifugal force acting horizontally outwards through the center of gravity, W the weight of the vehicle acting downwards through the center of gravity, and F the friction force between the wheels and the pavement, along the surface inward.

As a vehicle travels around a curve with super-elevation, three main forces come into play: the centrifugal force (P) acts outward, attempting to push the vehicle away from the curve; the weight of the vehicle (W) pulls it downwards; and friction force (F) acts inward, resisting the centrifugal force. Understanding these forces is crucial for ensuring vehicles can navigate turns safely without skidding or overturning.

Think of a toy car on a track with a curve. If you push the toy car too fast, it might tumble off the track due to the centrifugal force. However, if the track is tilted outward (like super-elevated sections of roads), the car is less likely to fall off because the tilt helps keep it grounded on the track, just as a real vehicle benefits from banking when navigating a curve.

At equilibrium, by resolving the forces parallel to the surface of the pavement we get, P cosθ = W sinθ + F + F = W sinθ + f(R + R) = W sinθ + f(W cosθ + P sinθ).

In the context of super-elevation, equilibrium means that all the forces acting on the vehicle balance each other. When we break down the forces acting on the vehicle, we find that the horizontal component of centrifugal force must equal the sum of the forces due to the vehicle's weight, friction, and the effects of super-elevation. This balance ensures the vehicle stays on the path of the curve without tipping over or skidding off.

Imagine balancing a seesaw. If one side is heavier, it tips over. In this case, the seesaw is like the forces on a vehicle; if the forces (weight, centrifugal, friction) are balanced, the vehicle stays on track. Super-elevation helps manage this balance by ensuring forces are properly aligned.

We have already derived an expression for P/W equation 14.2. By substituting this in equation 14.5, we get: v2 tanθ + f = (14.6) gR 1 - f tanθ. This is an exact expression for superelevation.

The expression for super-elevation relates the speed of the vehicle (v), the radius of the curve (R), and the angle of cant (θ). This mathematical relationship helps engineers determine how much to raise the roadway on curves to achieve safe and effective navigation at various speeds. The formula incorporates the coefficient of friction as well, which affects how much grip the tires have on the pavement.

Think of this mathematical expression like a recipe for baking a cake. Just as you need the right amounts of ingredients (like flour and sugar) to make a cake rise and taste good, engineers need specific measurements for speed and curve radius to ensure cars can safely make turns without skidding off the road.

Three specific cases that can arise from equation 14.7 are as follows: 1. If there is no friction due to some practical reasons, then f =0 and equation 14.7 becomes e=v2/gR. This results in the situation where the pressure on the outer and inner wheels are the same; requiring very high super-elevation e. 2. If there is no super-elevation provided due to some practical reasons, then e=0 and equation 14.7 becomes f = v2/gR. This results in a very high coefficient of friction. 3. If e=0 and f =0.15 then for safe traveling speed from equation 14.7 is given by v = √fgR where v is the restricted speed.

These three cases highlight the critical balance needed for safe vehicle operation on curves. If there is no friction (first case), super-elevation must be very high, which isn't practical. Conversely, without super-elevation (second case), the friction required to stay on the curve would be excessive. Finally, if both factors are at their minimum, we derive a maximum safe speed for navigating curves, emphasizing the importance of both super-elevation and friction for roadway safety.

Imagine walking on ice. If it's perfectly slippery (no friction), you'd need a slope (super-elevation) to reach the other side safely before you fall. Without that slope, you'd need an incredible grip on the ice to avoid slipping. In practical terms, this is why roads need both a banking angle and appropriate pavement materials to keep cars safe when they turn.

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