(Disclaimer: To make things clear I'm talking about adhesion railways and excluding rack railways as those are an entirely different technology and requires specially equipped vehicles. Also I'm looking for examples as how this is engineered.)

I was surprised recently as I visited the Wutachtal railway in southern Germany. A particular section of this railway makes extremes detours with multiple bridges and tunnels in order to climb 231m while having a railway steep of only 1%.

According to the museum's explaination, the steep of 1% was required because they wanted to allow heavy military freight trains to circulate on this railway. However, steeper railways are very common, including rails which allows heavy freight trains. For instance, according to German language Wikipedia:

  • The Gotthard railway has a max steepness of 2.8%
  • The Simplon railway as well as the Neuchâtel–Pontarlier railway a max steepness of 2.5%
  • The Arlberg railway in Austria has a steepness of 3%.
  • The line between Vevey and Chexbres has a steepness of 4% and light freight trains circulates on this line.

Some trains which only allows passenger train and have narrow gauge have much steepers rails:

  • The Uetli train in Zürich has a steepness of almost 8%
  • The Pöstlingbergbahn has a steepness of 11.6%, (narrow gauge)
  • Lisbon tramway has a steepness of 13.5%

Since the maximum allowed steepness is a major factor when designing a train line, I really wonder what factors determined it.

  • What kind of problems appears when the rail is too steep for a train, is it a problem when going uphill (train slip and can't advance) or when going downhill (train slips and can't break)
  • Does the weight of the train, or the power of the locomotive play a role
  • Did technology allow steeper trains as time passed, or is there a hard physical limit that can't be overcome
  • Why could Switzerland and Austria allows freight trains on 3% or even 4% steep tracks while Germany needed to limit itself 1%

EDIT: Since I asked the question, I've learn about a very interesting thing on the subject : In Geneva, 1904 a tramway line with a steepness of 11.8% was ready for usage but an accident when doing tests, where the tramway slipped and was unable to break, cancelled the line. So apparently the major problem is being able to break when going down. But the Lisbon tramway seems even steeper and didn't have similar accident as far as I'm aware.

  • Some tracks have a toothed portion where it is very steep... – Solar Mike Aug 8 at 12:51
  • @SolarMike This needs specially equiped trains, and this technology is exclusively for passenger trains as far as I know, so I'm purposely ruling that out. – Bregalad Aug 8 at 13:07
  • 1
    Would have helped if you had been clear with your question re rack railways before people made an effort. But you should check out something called "friction" as this will be the limiting factor. – Solar Mike Aug 8 at 13:45

The actual maximum attainable slope is determined by the weight of the locomotive, the total weight of the train, the rolling friction of the carriages, and the kinetic(sliding) friction between two materials; the locomotive's wheels and road surface(rails).

The static friction(present when things aren't slipping yet) allows for steeper hills, but it'd be unsafe to go that far, one slip of the wheel and the train won't be able to regain grip.

The friction coefficient of steel on steel(dry) is around 0.6. That means that a locomotive of 100 ton is able to give 60 ton of tractive force on a flat rails. It is less so on a hill, since gravity is not working perpendicular on the train, here its 'weight' is cos(4.6deg)*100 = 99.7 ton, so it can pull 59.8 ton. At a slope of 8%(4.6deg), that means the total weight of the train can be 59.8/sin(4.6deg) = 746 ton. No rolling friction is assumed here.

Of course, there is rolling friction, safety margins have to be used, and in bad weather traction will be less, so the total weight of the train must be less. If a locomotive only has to drive itself up the hill, then the hill could be as steep as 30 degrees and it wouldn't slip yet.

The slope of the rails is just a choice, steeper just means less load can be hauled by one locomotive.

A combination of the pulled weight, the weight of the movers (the locomotives) and the coefficient of friction between wheel and rail.

The more weight on each driven wheel the more force it can exert before it starts slipping. Being able to start uphill on the slope is also important especially when considering the military needs.

The more weight you need to tow the more force you need to get it moving and keep it moving.

  • So basically in some cases they could allow steep sections between stations (where the train never stops), and flatter stations ? – Bregalad Aug 8 at 13:11
  • also note that newer locomotives are more powerful than older ones. so the older the line, the less steep the slope. – niels nielsen Aug 8 at 17:34
  • @nielsnielsen power is irrelevant if there's insufficient friction between locomotive wheels and the rails. – Carl Witthoft Aug 8 at 17:56
  • I know that, but I also know there's a long stretch of railroad up here converted to a bike path, about 8 miles of it at exactly 1% grade maximum because that's the most that a loaded steam locomotive could manage 100 years ago. – niels nielsen Aug 8 at 19:05
  • which is still more likely a weight/traction issue of the loc than a power/torque issue. – ratchet freak Aug 8 at 19:31

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