Highway Curve: Superelevation Deficit, Sideswipe Accidents and a Speed Restriction
A two-lane, 7.0 m wide National Highway in plain terrain contains a horizontal curve of radius 150 m. The design speed is 80 km/h, the maximum superelevation is limited to 7%, and the lateral friction coefficient is taken as 0.15. The curve was built with the full 7% superelevation applied abruptly through the whole curve, with no transition length and no extra widening. In the last two years there have been several sideswipe and off-tracking crashes involving trucks, mostly in wet weather, and the district police have demanded that the curve be reconstructed or that the speed be restricted to 40 km/h. The typical truck wheelbase is 6.0 m. The Executive Engineer must reply with a technical note covering whether the existing geometry is adequate at 80 km/h, the maximum safe speed on the curve as it stands, and the minimum works needed if realignment is not possible this financial year.
1. Check whether e + f is adequate at the 80 km/h design speed, and compute the maximum safe speed on the curve with the existing 7% superelevation.
Read the worked answer
Demand: e + f = V²/(127 R) = 80²/(127 × 150) = 6400/19,050 = 0.336
Supply available: e_max + f = 0.07 + 0.15 = 0.22
The demand exceeds what the curve can supply by about 53% of the available friction margin. The geometry is not adequate at 80 km/h: at that speed the lateral friction required is 0.266, which is 78% greater than the 0.15 assumed. In wet weather the available friction falls further, which is fully consistent with the crash pattern - the trucks are losing the rear end and sideswiping, exactly the signature of a friction deficit, not of a driver error.
Maximum safe speed with the existing 7%:
V² = 127 R (e + f) = 127 × 150 × 0.22 = 4,191 → V = 64.7 km/h, say 65 km/h.
Two points must be recorded. First, 0.15 is a limit-equilibrium friction value; a speed limit set at the friction limit is unsafe for wet conditions and for loaded trucks with high centre of gravity, so an operating limit of about 50–55 km/h is more defensible. Second, the police demand of 40 km/h is over-restrictive - it would not be defensible in a technical note, though it is at least safe.
2. Compute the extra widening and the transition length that should have been provided, and comment on the consequences of omitting them.
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Extra widening (two lanes, n = 2, wheelbase l = 6.0 m, radius 150 m):
Mechanical: W_m = n l²/(2R) = 2 × 36/(2 × 150) = 0.24 m
Psychological: W_ps = V/(9.5 √R) = 65/(9.5 × √150) = 65/(9.5 × 12.25) = 0.56 m
Total W_e = 0.80 m, normally provided on the inner side of the curve. The curve has zero widening, so a truck tracking the inner edge overhangs the outer edge by about 0.8 m - which by itself explains the sideswipe crashes in the two-lane width of 7.0 m.
Transition length (plain terrain, C = 80/(75 + V) = 80/155 = 0.516):
L_s = 0.0215 V³/(C R) = 0.0215 × 512,000/(0.516 × 150) = 11,008/77.4 = 142 m
Cross-check against the rate of introduction of superelevation (1 in 150): L_s = e × W/2 × 150 = 0.07 × 3.5 × 150 = 37 m. The centrifugal criterion governs at 142 m.
Omitting the transition means the 7% superelevation is applied as a step. At 80 km/h the vehicle crosses the curve entry in a fraction of a second, encountering a sudden 0.07 lateral tilt plus a rapidly increasing centrifugal demand - the classic cause of loss of control at curve entry. In practice the 142 m transition cannot be fitted into a curve of radius 150 m on the existing alignment, which is itself the key finding: the curve is short and the alignment, not just the cross-section, is the problem.
3. Realignment is not funded this year. Recommend the minimum works to be executed now and justify the priority order.
Read the worked answer
The physics is the constraint: no signing or lining can raise e + f above 0.22. The engineering options must therefore reduce demand or raise the available friction.
Immediate (within weeks, low cost):
- Restrict speed to 50 km/h through the curve and enforce it; at 50 km/h, e + f required = 2500/19,050 = 0.13, comfortably within 0.22. This is the single measure that immediately closes the gap and should not be traded away for cosmetic works.
- Advance warning signs, chevron alignment markers, 'wet weather - reduce speed' plates, and transverse rumble strips on the approach.
- High-friction surfacing (polymer-modified or epoxy asphalt) on the curve and on the braking length before it, to lift the wet friction coefficient well above 0.15.
Short term (this financial year): 4. Provide the 0.8 m extra widening on the inner side, which can usually be done by re-forming the shoulder rather than rebuilding the carriageway. 5. Provide a transition as long as the site permits, even if it is a partial length, and a superelevation runoff on the approach; do not leave the 7% as a step. 6. Remove or shield fixed obstructions within the clear zone on the outer side.
Justification of the order: speed and friction act directly on the failure mechanism and can be delivered before the next monsoon; widening and transition are permanent but take a season to build. Realignment to a larger radius (for 80 km/h, R should not be less than about 240 m with e = 7% and f = 0.15) should be programmed in the next plan cycle, since the existing curve cannot be made compliant at 80 km/h by any combination of local measures.