What determines the maximum possible slope of rail transportation (without rack)?

Question

(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 viaducs and tunnels (including a circular tunnel) in order to climb 231m while having a gradient of only 1%.

According to the museum's explanation, the gradient 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%.

Some trains which only allows passenger train and have narrow gauge have much steeper 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 given train, is it a problem when going uphill (train slip and can't advance) or when going downhill (train slips and can't brake).
• Does the weight of the train, or the power of the locomotive play a role
• Did technology (i.e. more powerful motors, motors and/or braking on more wheels) 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% steep tracks while Germany needed to limit itself 1% ?
• When an international freight train circulates, how do we know which routes it can or cannot take due to its weight ?

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 brake, cancelled the line. So apparently the major problem is being able to brake when going down. But the Lisbon tramway seems even steeper and didn't have similar accident as far as I'm aware.

Answer 1

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(4.6°)*100 = 99.7 ton, so it can pull 59.8 ton. At a slope of 8%(4.6°), that means the total weight of the train can be 59.8/sin(4.6°) = 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.

Answer 2

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.

Answer 3

Late answer for your 4th bullet point: This specific German railway was designed for military trasport in case of a war with France (what later was WW I) to bypass Switzerland (which was expected, as a neutral country, to forbid transport of military goods during a war).

The military wanted free usage of locomotives in a war, as one has to expect that there are serious locomotive shortages at such times. Therefore, those military were limited to 1 percent of incline.

Re the other items: Both going up and going down a steep railway is difficult and dangerous - couplings and brakes had to be invented for both. But with short trains, in the 1840s, the Americans in the Allghenies and then Carl Ghega with his Semmering railway in 1848 proved that inclines of about 3% could be safely operated.

And yes, both power and load play a role.

There is a hard technical limit, which is the angle related to the friction coefficient. No adhesion railway can be steeper than that. It's around 1 in 7 or the like.

There is lots of (not too hard) mathematics for all this; and even more lots of regulations and tables to be adhered to; which allow the railwaymen to decide on the consist and power of each train in hill or mountain territory.

H.M.

Answer 4

I think you have to take into account the proportion of driven wheels to un-driven wheels. If each wheel set can be powered then a higher slope can be climbed.