When a railway meets a hill that is too steep to climb, the engineer has two ways to go straight on: dig the hill away from the top, or dig through it from the inside. Both keep the gradient gentle. They differ in how the amount of digging grows, and that difference decides most of the choices on a real route.
Cutting: depth is squared
A cutting cannot have vertical walls, because loose ground would slide into the track. The sides are cut back at a slope. Take a simple, assumed shape: a flat floor 8 m wide for the track, and side slopes of 2 m across for every 1 m of height. For a cutting of depth d the cross-section is a rectangle plus two triangles:
Area = 8d + 2d² square metres.
The first term grows in proportion to depth. The second grows with its square, and it takes over as the cutting deepens. Each metre of length multiplies that area by one metre, so the area is also the volume of earth to remove per metre of line.
| Depth | Width at top | Area per metre | Over 100 m |
|---|---|---|---|
| 5 m | 28 m | 90 m² | 9,000 m³ |
| 10 m | 48 m | 280 m² | 28,000 m³ |
| 15 m | 68 m | 570 m² | 57,000 m³ |
| 20 m | 88 m | 960 m² | 96,000 m³ |
Going from 10 m to 20 m deep does not double the work. It multiplies it by 960 / 280, about 3.4. A cutting that was cheap at 5 m becomes a major project at 20 m, and it also takes a strip 88 m wide out of the hillside.
Tunnel: length is all that counts
A tunnel's cross-section is set by the trains it must carry, not by how much hill lies above. Assume a double-track bore of 60 m² excavated. Then every metre of tunnel is 60 m³ whether the hill above is 5 m or 100 m thick. A 100 m tunnel is 6,000 m³, and a 1,000 m tunnel is 60,000 m³: volume is linear in length.
Volume is not cost
By volume alone a tunnel wins at almost any depth: 8d + 2d² passes 60 at only about 3.8 m. But a cubic metre from a tunnel costs more. It is dug in the dark, usually needs a lining, and the spoil must be hauled out through the portal. A cutting is open to the sky, and its spoil can often build an embankment close by.
So include a cost factor k: a cubic metre of tunnel costs k times a cubic metre of cutting. The tunnel is cheaper once the cutting area exceeds 60 × k. Solving 2d² + 8d = 60k gives d = −2 + √(4 + 30k).
| Tunnel cost per m³ | Cutting area at break-even | Break-even depth |
|---|---|---|
| 2 × cutting | 120 m² | 6.0 m |
| 4 × cutting | 240 m² | 9.1 m |
| 6 × cutting | 360 m² | 11.6 m |
The factors 2, 4 and 6 are assumptions chosen to show the pattern, not measured costs. Even a tunnel that costs six times as much per cubic metre wins once the hill is about 12 m deep. At 20 m, a 100 m stretch of cutting is 96,000 m³; the tunnel is 6,000 m³, and at four times the unit cost that is the equivalent of 24,000 m³.
A real hill is not a constant depth. The cutting is deepest at the summit and shallow at its ends, so engineers often combine the two: a tunnel under the top, with cuttings leading to each portal.
What the early builders chose
The Liverpool and Manchester Railway, opened in 1830, cut through sandstone at Olive Mount, a cutting of about two miles. Robert Stephenson's London and Birmingham Railway, opened in 1838, took the Chiltern ridge in the Tring cutting, about two and a half miles long, dug largely with picks, shovels and barrows. Brunel's Great Western went the other way at Box, near Bath, where the hill was too thick to cut. Box Tunnel opened in 1841, is about 1.8 miles (2.9 km) long, and carries the line on a falling gradient of 1 in 100, the ruling gradient discussed in Why railways follow rivers. At the time it was the longest railway tunnel in the world.
These were the two answers to the same problem: a cutting at Tring and Olive Mount where the ground was shallow enough, a tunnel at Box where it was not.
A rule to carry away
Doubling a cutting's depth multiplies its volume by up to four. Doubling a tunnel's length multiplies it by two. For any shape and cost factor there is a depth where the lines cross, and past it the tunnel is the cheaper way through.
In Valley Rail: Train Tycoon you drag a track from town to town, and cross the river on a bridge in one of five styles. For the structural side of those crossings, see How railway bridges carry a load.
Valley Rail: Train Tycoon has six train models for passengers and freight, from a one-coach commuter to a streamlined express. The money each trip earns is in-game only and has no cash value. Valley Rail is in closed testing and not yet publicly available.