The Parabolic Tunnel: Clearance at the Road Edge, Not the Crown
The Kochi prefectural road office is planning a 50 m road tunnel through a low ridge. The cross-section is a parabolic arch: the crown is 8.0 m above the road surface and the arch meets the road at two points 12.0 m apart, with the road surface horizontal. The carriageway is the central 8.0 m of the floor (two 3.5 m lanes plus 0.5 m shoulders on each side), and the design standard demands at least 5.0 m of vertical clearance above every point of the carriageway, not merely at the centre line. The office has budgeted on the basis of the profile area multiplied by the tunnel length, at ¥9,000 per cubic metre of excavation, and the draft drawing shows the 8.0 m crown as fixed. A reviewer notices that the draft never checks the clearance at the edge of the carriageway and asks for the cheapest compliant revision.
1. Taking the centre line as x = 0 and the road surface as y = 0, find the equation of the parabola, the cross-sectional area of the opening, and the excavation volume over the 50 m length.
Read the worked answer
The arch passes through (-6, 0), (0, 8) and (6, 0), so y = H(1 - x²/36) with H = 8:
y = 8 - (2/9)x².
Area = ∫₋₆⁶ (8 - 2x²/9) dx = 2[8x - 2x³/27]₀⁶ = 2(48 - 16) = 64 m².
Cross-check with the standard result for a parabolic segment, (2/3) × base × height = (2/3)(12)(8) = 64.
Volume of void over 50 m = 64 × 50 = 3200 m³, costing 3200 × ¥9,000 = ¥28,800,000 on the office's own basis.
2. Check the 5.0 m clearance rule at the edge of the carriageway. If the 8.0 m crown is kept, what is the widest carriageway that complies?
Read the worked answer
The carriageway edge is at x = 4, where
y = 8 - (2/9)(16) = 8 - 3.556 = 4.44 m < 5.0 m.
The draft fails: the crown height is not the clearance that matters, because the arch falls away towards the shoulders.
For an 8.0 m crown, compliance needs 8 - 2x²/9 ≥ 5, i.e. x² ≤ 13.5, |x| ≤ 3.674, so the carriageway width is at most 7.35 m. Two 3.5 m lanes would still fit, but the 0.5 m shoulders would shrink to about 0.18 m each.
Judgement: keeping the 8.0 m crown means defending a reduced-shoulder standard; it is a policy decision dressed up as geometry, and the reviewer is right to refuse to sign the draft as it stands.
3. Keep the 8.0 m carriageway and raise the crown instead. Find the minimum crown height, the new cross-sectional area, and the resulting extra excavation volume and cost over the 50 m length.
Read the worked answer
With crown H, the height at x = 4 is H(1 - 16/36) = 5H/9. Requiring 5H/9 ≥ 5 gives H ≥ 9.0 m.
Area = (2/3) × 12 × H = 8H, so at H = 9 the area is 72 m² (against 64 m²).
Extra volume = (72 - 64) × 50 = 400 m³; extra cost = 400 × ¥9,000 = ¥3,600,000.
Because the area is linear in H, each additional metre of crown costs 8 m² × 50 m × ¥9,000 = ¥3.6 million as well, so there is no hidden minimum near 9.0 m: 9.0 m is exactly the cheapest compliant crown.
The real choice is therefore ¥3.6 million of extra rock against about 0.65 m of lost carriageway width - state it in those terms rather than comparing areas.