TRACTIVE EFFORT IN STEAM LOCOMOTIVES

Tractive Effort (TE) or occasionally also called Tractive Force, is often used to compare the relative power of steam locomotives. I was aware that there was a formula, but only recently came across this one:

"The force exerted at the edge of the driving wheel of a locomotive expressed in pounds.

Calculated as: D² x S x P,

where D is cylinder diameter (inches), S is piston stroke (inches) and P is 85% of boiler pressure (psi)."

OK as far as it goes. But I know that number of drivers, weight on drivers and size of drivers also affects engine power. How are they applied?

Ask and you shall be told. John Stutz kindly expanded on this in an email:

"The correct nominal average Tractive Effort (TE) formula is D²x S x .85 p / d, where d is the driving wheel diameter and p the boiler pressure.

This derives from equating the work done in the cylinders over 4 piston strokes, to the work done at the drivers over one revolution. The .85 p is the pressure across the piston averaged for a full stroke. The .85 p is a very rough approximation, generally only achievable at a walking pace. British builders tended to use .75 p, perhaps because they tended to operate at higher speeds. At anything much over 10 mph, a steam locomotive's power (TE * speed) is limited by the boiler steaming capacity. The product is then constant, and TE drops off in proportion as speed increases.

TE is limited by the total weight on drivers (WoD). This limit is usually expressed by the Adhesion Ratio: AR = WoD/TE. An AR of 4 was usually achievable on clean dry rail, or clean wet sanded rail. This is for the usual two cylinder locomotive. Three cylinder and geared locomotives could do slightly better. This is because a two cylinder locomotive's instantaneous TE actually varies by about 1/3 of the average, and it is the peak TE that determines when the locomotive starts slipping. A 3-cylinder engine tends to raise the average TE closer to the peak, lowering the AR. Grease, ice, mud & etc. on the rail will all cause the locomotive to slip before the nominal TE is reached, effectively increasing the AR.

The permissable Axle Load (AL) for drivers is determined by the track. The weight and composition of the rail are the primary factors limiting AL. The tie spacing, use of tie plates, type and depth of ballast, and quality of maintainance also affect permissable AL. Thus the AL and number of driving axles limit the WoD and achievable TE.

The total permissable locomotive and tender weight is limited by what the bridges can carry, and to some extent by how that weight is distributed. A 2-12-2T and a 2-6-2+2-6-2 Garratt with the same driver AL could have the same WoD and TE, with slightly larger total weight for the Garrett. But the concentrated weight of the tank would cause significantly more stress than the Garrett, on bridge spans of less than twice the Garratt's length.

Driver diameter is mainly a matter of intended speed. Piston speed increases with driver RPM. At excessive piston speed, the valves cannot deliver steam efficiently, and considerable power is lost in the valves. So higher speeds require larger drivers in order to maintain power to the drivers. On the other hand, larger drivers mean longer driving wheel bases, which limit the ability to take curves. So increasing the number of drivers requires either smaller drivers to limit the driving wheel base, or alternate means to allow the locomotive to take curves. Thus we get blind drivers, articulated locomotives and driver lateral motion devices.

See the BLW catalogue pages at http://narrowmind.railfan.net/ for specifics."

John Stutz