Superelevation, widening and transition on a 120 m highway curve
A two-lane undivided state highway has a 7.0 m carriageway and a horizontal curve of radius 120 m in plain terrain. The design speed is 65 km/h, the maximum superelevation permitted in plain terrain is 7%, the lateral friction coefficient may be taken as 0.15, the wheelbase of the design vehicle is 6.0 m, and the straight-section camber is 2.5%. The curve currently has full superelevation but no extra widening, and only 25 m of length before the curve is available to change from camber to full superelevation. Residents report that at night two-wheelers drift towards the outer edge, and that after rain water stands along the inner edge of the curve near a culvert headwall; the inner edge is 0.15 m below shoulder level. The executive engineer asks whether the 65 km/h speed can be retained or must be reduced to 50 km/h, and whether widening and a longer transition are required.
1. Calculate the superelevation demanded at 65 km/h and decide what should actually be provided.
Read the worked answer
Demand. e = V²/(225R) = 65²/(225 × 120) = 4225/27 000 = 0.156, i.e. 15.6%. That exceeds the 7% cap for plain terrain and cannot be built: it would strand slow and stopped vehicles, be hazardous when the surface is polished or wet, and make the outer edge rise unmanageably.
Correct procedure - check with friction. The equation V²/(225R) = e + f is the real requirement. Provide e = 0.07 (the maximum) and let friction supply the rest: f_required = 0.156 − 0.07 = 0.086 < 0.15 available.
Decision. 65 km/h can be retained. The curve is safe at the design speed, with a friction demand of only about 57% of the permitted value. Reducing to 50 km/h is unnecessary and would be ignored by drivers anyway.
Why the 15.6% figure is not an error. It is the superelevation that would be needed if friction did nothing. Design practice caps e for practical and safety reasons, then checks the balance against f - exactly as done above. The examinable point is that the cap is a design limit, not a failure of the curve.
Condition attached. The friction reserve exists only if the surface is textured, the aggregate is not polished, and the curve is drained. A wet, smooth surface at this radius will produce skidding even at 65 km/h - which links to the residents' complaint.
2. Compute the extra widening required and state where and over what length it should be applied.
Read the worked answer
Mechanical widening (vehicle cannot follow the curve within its own width): W_m = n L²/(2R) = 2 × 6²/(2 × 120) = 72/240 = 0.30 m.
Psychological widening (drivers keep away from the edge, more overtaking shyness on a curve): W_p = V/(9.5√R) = 65/(9.5 × √120) = 65/(9.5 × 10.95) = 65/104.1 = 0.625 m.
Total = 0.30 + 0.625 = 0.925 m, adopt 1.0 m as a constructible value.
Where. On a two-lane undivided carriageway in plain terrain, apply the widening on the inner side of the curve, so the outer edge, which already carries the superelevation rise, is not pushed further out into the shoulder or drain.
Over what length. Widening must taper from zero on the straight to 1.0 m at the start of the circular curve, over the same transition length used for superelevation - about 37 m here (next task). Starting it abruptly at the tangent point is a common site fault and produces exactly the 'drifting outward' complaint.
Observation from the numbers. At R = 120 m the mechanical component is small; the psychological term dominates. At R = 60 m the mechanical term would double to 0.60 m and the balance would change - worth stating to show the formula is understood, not memorised.
3. Assess the 25 m transition and the standing water at the inner edge, and relate both to the residents' complaint.
Read the worked answer
Transition length needed. Rotating the carriageway about the centre line, the outer edge rises by e × half-width = 0.07 × 3.5 = 0.245 m. At the permitted rate of change of 1 in 150 longitudinally, length required = 0.245 × 150 = 36.8 m, say 37 m. Only 25 m is available - the superelevation is being introduced about 1.5 times too steeply, and extra widening would have to be crammed into the same short length. That is a direct cause of vehicles being thrown outwards at the entry to the curve, and it is worse at night when the driver cannot see the curvature ahead.
Standing water at the inner edge. With rotation about the centre line, the inner edge falls as the outer edge rises. Here the inner edge is 0.15 m below the shoulder next to a culvert headwall, so the kerb line forms a sump: water ponds, the surface stays wet, and the friction reserve assumed in the 0.086 demand disappears - the skidding risk on the curve becomes real.
Recommendations.
- Lengthen the transition to at least 37 m by realigning the tangent, or reduce the superelevation rate within the available length and accept a lower design speed locally.
- Rotate about the inner edge on this stretch, or re-profile the inner edge and shoulder, so that the edge drains to the culvert instead of ponding.
- Provide a longitudinal drain or kerb channel at the headwall, and check that the raised outer edge (0.245 m) does not dam the shoulder on the other side.
- Add curve warning signs, delineators and edge-line markings; the night complaint is partly a visibility problem, not only a geometry problem.