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# Tractive effort

Tractive Effort (abbr. TE) is the pulling force exerted, normally by a locomotive, though the term could also be used for anything else that pulls a load. It is normally understood to be the actual force on the locomotive's drawbar or rear coupler. When a bare figure for tractive effort is quoted without a speed qualification, this is normally for starting tractive effort, i.e. at a dead start with the wheels not turning.

In most cases, especially for steam locomotives, this figure is a calculated, not measured one.

## Steam locomotives

The normal formula used (for a 2 cylinder locomotive) is:

$t = frac \left\{cPd^2s\right\} \left\{D\right\}$

where

• t is tractive effort
• c is a constant representing losses in pressure and friction; normally 0.85 is used
• P is the boiler pressure
• d is the piston diameter (bore)
• s is the piston stroke
• D is the driving wheel diameter

As with any physical formula, consistent units of measurement are required: pressure in psi and lengths in inches give tractive effort in lbf, while pressure in Pa and lengths in metres give tractive effort in N.

The constant 0.85 was the Association of American Railroads (AAR) standard for such calculations, and certainly over-estimated the efficiency of some locomotives and underestimated that of others. Modern, roller bearing fitted locomotives were probably underestimated in this calculation.

European designers used a constant of 0.6 instead of 0.85, so the two cannot be directly compared without a conversion factor. In Britain, the main-line railways generally used a constant of 0.85 but builders of industrial locomotives often used a lower figure, typically 0.75.

The actual starting tractive effort depends on the position in which the wheels have stopped; the above formula can give the average, maximum or minimum over a wheel revolution depending on the choice of constant c.

Tractive effort is the figure most often quoted when people are comparing the power of different steam locomotives, but the use can be misleading, because tractive effort shows the ability to start a train, not the ability to do work by hauling it. Possibly the highest figure for starting tractive effort ever recorded was for the Virginian Railway's 2-8-8-8-4 Triplex locomotive, which in simple expansion mode had a starting T.E. of 199,560 lbf (888 kN) — but this did not translate into power, for the boiler was undersized and could not produce enough steam to haul at speeds over 5 mph (8 km/h).

Of more successful large steam locomotives, those with the highest rated starting tractive effort were the Virginian Railway AE-class 2-10-10-2s, at 176,000 lbf (783 kN) in simple-expansion mode. The Union Pacific's famous Big Boys had a starting T.E. of 135,375 lbf (602 kN); the Norfolk & Western's Y5, Y6, Y6a, and Y6b class 2-8-8-2s had a starting T.E. of 152,206 lbf (677 kN) in simple expansion mode (later modified, resulting in a claimed T.E. of 170,000 lbf (756 kN)); and the Pennsylvania Railroad's freight Duplex Q2 attained 114,860 lbf (511 kN) — the highest for a rigid framed locomotive. Later two cylinder passenger locomotives were generally 70,000 to 80,000 lbf (300 to 350 kN) of T.E.

## Diesel and electric locomotives

For a diesel-electric locomotive or electric locomotive, starting tractive effort can be calculated from the stall torque of the traction motors (the turning force it can produce while at a dead stop), the gearing, and the wheel diameter. For a diesel-hydraulic locomotive the starting tractive effort depends on the stall torque of the torque converter, which can be very large.

## Related statistics

A related statistic is a locomotive's factor of adhesion, which is simply the weight on the locomotive's driving wheels divided by the starting tractive effort.

For a locomotive to accelerate from a stationary position, it must apply a force to overcome the inertia of the train, along with the frictional forces in the form of mechanical friction, and wind resistance as the train accelerates. In order for this to occur a particularly high tractive effort is required, usually the maximum tractive effort of the engine is applied. This means that the engine works to produce the highest possible force that it can exert onto the wheels to cause movement or motion. Few engines can maintain work at the maximum tractive effort for very long, but neither is it usually necessary for an engine to do this. Once the train is running at a constant velocity the train no longer needs to overcome its inertia to remain at the same velocity, and hence must only provide power to compensate for frictional forces. This leads to one potential upper limit on the speed a locomotive can haul a train at, once the force due to wind resistance becomes greater than the tractive effort the locomotive can supply (fluid drag increases with the square of velocity), the locomotive cannot accelerate the train anymore (in reality the situation is more complicated than this due to a number of mechanical considerations).

## Examples

A table to illustrate the speed the maximum tractive effort, continuous tractive effort and the speed at which the tractive effort should be reduced on a selection of trains operating in the United Kingdom:

Class   Type   Top speed   Maximum
tractive
effort
Speed to
reduce
tractive effort
Continuous
tractive
effort
Maximum
power
at rail
Mass
mph   km/h
Class 08 Shunter 15 156 kN 8.8 mph 49 kN 194 kW 49.6 - 50.4 t
Class 33 Passenger 85 200 kN 17.5 mph 116 kN 906 kW 77.7 t
Class 56 Light freight 80 275 kN 16.8 mph 240 kN 1790kW 125.2 t
Class 58 Light freight 80 275 kN 17.4 mph 240 kN 1780 kW 130 t
Class 59 Heavy freight 60 or 75 506 kN 14.3 mph 291 kN 1889 kW 121 t
Class 60 Heavy freight 60 500 kN 17.4 mph 336 kN 1800 kW 129-131 t
Class 66 Heavy freight 75 409 kN 15.9 mph 260 kN 1850 kW 126 t
Class 67 Light freight 125 200 141 kN ?? mph 90 kN 1860 kW 90 t

The power at rail of a train follows the equation power (kW) = force (kN) x speed (m/s)

In general, it is more common for heavy freight trains (such as Class 59, Class 60 and Class 66 locomotives) to have a high maximum tractive effort due to the mass which they haul. Light freight trains (such as Class 56, Class 58 and Class 67 locomotives) and passenger trains (such as Class 33 and Class 43 / Intercity High Speed Train locomotives) usually have much lower maximum tractive efforts.

## Special cases

The tractive effort for steam locomotives is multiplied by 1.5 for 3-cylinder engines and by 2 for 4-cylinder engines.

In the case of compound locomotives the tractive effort is calculated using the dimensions of the low-pressure cylinder(s) with a constant of 0.80 instead of 0.85.