Drive fast into a bend and one of two bad things can happen. The vehicle can slide sideways off the road, or it can tip over onto its side. Which of the two you get is not luck — it is decided by the shape of the vehicle and the grip of the road, and it can be predicted with two short inequalities.
This post builds those inequalities from scratch, and covers the design speed values that everything else in horizontal alignment depends on.
What Horizontal Alignment Covers
Horizontal alignment is the plan view of a road — its straights and its bends as you would see them from above.
Get it wrong and the consequences pile up: drivers slow down, comfort and safety suffer, vehicle operating costs rise, and the road carries less traffic than it should.
Five elements make up the design:
- Radius of the circular curve
- Superelevation
- Extra widening at curves
- Transition curve
- Set back distance
Design Speed Comes First
Nothing in horizontal alignment can be calculated until the design speed is fixed. It depends on two things — what class of road it is and what kind of country it crosses.
Terrain Classification
| Terrain | Cross Slope |
|---|---|
| Plain | 0 – 10 % |
| Rolling | 10 – 25 % |
| Mountainous | 25 – 60 % |
| Steep | Above 60 % |
IRC Design Speeds in km/hr (Ruling and Minimum)
| Road Class | Plain | Rolling | Hilly | Steep |
|---|---|---|---|---|
| NH & SH | 100 – 80 | 80 – 65 | 50 – 40 | 40 – 30 |
| MDR | 80 – 65 | 65 – 30 | 40 – 30 | 30 – 20 |
| ODR | 65 – 50 | 50 – 40 | 30 – 25 | 25 – 20 |
| VR | 50 – 40 | 40 – 35 | 25 – 20 | 25 – 20 |
In every cell the first figure is the ruling design speed — what the designer aims for — and the second is the minimum permitted where conditions force a compromise. Read across any row and the speed drops as the land gets harder; read down any column and it drops as the road class gets lower.
The Force That Causes All the Trouble
When a vehicle travels in a straight line, nothing pushes it sideways. The moment it enters a curve, it is being forced to change direction continuously, and it resists that change. The resistance shows up as an outward force called the centrifugal force.
You feel it every time a bus turns and you lean towards the outside of the bend.
P = W v2 / (g R)
Two features of this expression matter more than anything else in this chapter:
- It goes up with the square of speed. Double the speed and the force quadruples. This is why speed matters so much more on curves than on straights.
- It goes down as radius increases. A gentle sweeping bend produces far less force than a tight one at the same speed.
Some of this force is resisted by sideways friction between tyre and road. But friction has a limit, and past that limit something has to give.
Centrifugal Ratio
It is more useful to compare the centrifugal force to the vehicle’s own weight than to talk about it in absolute terms. That comparison is called the centrifugal ratio or impact factor:
P / W = v2 / (g R)
Weight has vanished from the right-hand side. That means the ratio depends only on speed and radius — a fully loaded truck and an empty car on the same curve at the same speed experience the same ratio.
The Forces in Play
| Force | Where It Acts |
|---|---|
| Centrifugal force P | Horizontally outward, through the centre of gravity |
| Weight W | Vertically down, through the centre of gravity |
| Ground reactions RA, RB | Upward at the two wheels |
Both P and W act through the centre of gravity, which sits at a height h above the ground. The wheel base — the distance between the wheels — is b. These two dimensions decide whether the vehicle topples.
Condition 1: Overturning
Picture the vehicle beginning to tip about its outer wheels. The moment it is on the point of tipping, the inner wheels have stopped pressing on the road — the reaction there has fallen to zero.
Take moments about the outer wheel at that instant. The centrifugal force acting at height h is trying to rotate the vehicle over; the weight acting at a horizontal distance b/2 is trying to hold it down:
P h = W (b/2)
P / W = b / (2h)
That is the tipping point. To stay safely below it:
P / W ≤ b / (2h)
v2 / (g R) ≤ b / (2h) — no overturning
Read the right-hand side as a property of the vehicle. A wide vehicle sitting low (large b, small h) has a big value and is hard to tip. A narrow vehicle with a high load has a small value and tips easily. This is exactly why an overloaded truck with a tall load is dangerous on bends while a sports car is not.
Condition 2: Transverse Skidding
The other failure is sliding. It happens when the centrifugal force exceeds the sideways grip the tyres can supply.
The maximum sideways resistance is the friction coefficient multiplied by the total normal reaction, which equals the weight:
F = FA + FB = f (RA + RB) = f W
where f is the lateral coefficient of friction. At the point of sliding:
P = f W → P / W = f = v2/(gR)
And to stay safely below it:
P / W ≤ f
v2 / (g R) ≤ f — no skidding
Putting Both Conditions Together
A curve is safe only if both conditions hold. Combining them gives the single most useful result in this topic:
P / W ≤ b / (2h) ≤ f
And this arrangement tells you something the individual conditions do not — which failure happens first.
| If | Then | Typical Vehicle |
|---|---|---|
| b/(2h) < f | The vehicle overturns before it skids | High centre of gravity, narrow track — a laden truck or bus |
| b/(2h) > f | The vehicle skids before it overturns | Low, wide vehicle — a car |
Skidding, unpleasant as it is, is the safer of the two failures — the driver may recover, and the vehicle stays upright. Overturning almost never ends well. This is why vehicle loading rules restrict load height, and why tall vehicles are given lower speed limits on bends.
Formula Summary
| Quantity | Expression |
|---|---|
| Centrifugal force | P = W v2/(gR) |
| Centrifugal ratio / impact factor | P/W = v2/(gR) |
| No overturning | v2/(gR) ≤ b/(2h) |
| No skidding | v2/(gR) ≤ f |
| Combined safe condition | P/W ≤ b/(2h) ≤ f |
| Maximum skid resistance | F = f W |
Quick Revision Notes
- Five design elements: curve radius, superelevation, extra widening, transition curve, set back distance.
- Design speed is the starting point for all of them.
- Terrain by cross slope: plain 0–10 %, rolling 10–25 %, mountainous 25–60 %, steep above 60 %.
- Centrifugal force P = Wv2/(gR), acting outward, growing with the square of speed.
- Centrifugal ratio = impact factor = v2/(gR), independent of weight.
- Overturning is about the outer wheels; the inner wheel reaction becomes zero at the tipping point.
- Skidding is resisted by the lateral coefficient of friction.
- Combined condition: P/W ≤ b/(2h) ≤ f.
- b/(2h) smaller than f means the vehicle topples first; larger means it skids first.
Mistakes Students Commonly Make
- Using longitudinal friction (0.35–0.40) here. Curves need lateral friction, 0.15.
- Writing the overturning condition as b/h. It is b/(2h), because the weight acts at half the wheel base from the outer wheel.
- Taking h as the height of the vehicle. It is the height of the centre of gravity.
- Leaving speed in kmph in P = Wv2/(gR). Convert to m/s first.
- Reading the design speed table backwards — the first number in each cell is the ruling speed.
- Assuming the heavier vehicle always overturns first. What matters is the ratio b/(2h), not the weight.
Conclusion
Everything in horizontal alignment traces back to one outward force that appears the moment a vehicle stops travelling straight. Compare that force to the vehicle’s weight and you get the centrifugal ratio; compare the ratio to the vehicle’s shape and you learn whether it will topple; compare it to the road’s grip and you learn whether it will slide. The chain P/W ≤ b/(2h) ≤ f captures all of that in one line, and the topics that follow — superelevation, extra widening, transition curves — are all engineering answers to the same force.
Frequently Asked Questions
What is horizontal alignment?
It is the layout of a road in plan. Its design elements are the radius of the circular curve, superelevation, extra widening, the transition curve and set back distance.
What is centrifugal force on a curve?
The outward force experienced by a vehicle negotiating a bend, given by P = Wv2/(gR).
What is the centrifugal ratio?
The ratio of centrifugal force to the weight of the vehicle, P/W = v2/(gR). It is also called the impact factor.
Why does weight not appear in the centrifugal ratio?
Because both the centrifugal force and the weight are proportional to mass, so mass cancels when you take their ratio.
What is the condition for no overturning?
v2/(gR) ≤ b/(2h), where b is the wheel base and h the height of the centre of gravity.
What is the condition for no skidding?
v2/(gR) ≤ f, where f is the lateral coefficient of friction.
How do you tell whether a vehicle will skid or overturn first?
Compare b/(2h) with f. If b/(2h) is smaller, the vehicle overturns first. If it is larger, the vehicle skids first.
What is the ruling design speed for a national highway in plain terrain?
100 kmph, with 80 kmph as the permitted minimum.