Tractive force

An estimate for the tractive effort of a single cylinder steam locomotive can be obtained from the cylinder pressure, cylinder bore, stroke of the piston[note 2] and the diameter of the wheel. The torque developed by the linear motion of the piston depends on the angle that the driving rod makes with the tangent of the radius on the driving wheel.[note 3] For a more useful value an average value over the rotation of the wheel is used. The driving force is the torque divided by the wheel radius.

As an approximation, the following formula can be used (for a two-cylinder locomotive):[note 4]

${\displaystyle t={\frac {d^{2}sp}{w}}\times 0.85,}$ [8]

where

• t is tractive effort
• d is the piston diameter in inches (bore)
• s is the piston stroke in inches
• p is the working pressure in pounds per square inch
• w is the diameter of the driving wheels in inches

The constant 0.85 was the Association of American Railroads (AAR) standard for such calculations, and overestimated the efficiency of some locomotives and underestimated that of others. Modern locomotives with roller bearings were probably underestimated.

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

The constant c also depends on the cylinder dimensions and the time at which the steam inlet valves are open; if the steam inlet valves are closed immediately after obtaining full cylinder pressure the piston force can be expected to have dropped to less than half the initial force.[note 5] giving a low c value. If the cylinder valves are left open for longer the value of c will rise nearer to one.

Three or four cylinders (simple)

The result should be multiplied by 1.5 for a three-cylinder locomotive and by two for a four-cylinder locomotive.[9]

Alternatively, tractive effort of all 'simple' (i.e. non-compound) locomotives can be calculated thus:

${\displaystyle t={\frac {0.85d^{2}nsp}{2w}},}$ [10]

where

• t is tractive effort
• n is the number of cylinders
• d is the piston diameter in inches
• s is the piston stroke in inches
• p is the maximum rated boiler pressure in psi
• w is the diameter of the driving wheels in inches

Multiple cylinders (compound)

For other numbers and combinations of cylinders, including double and triple expansion engines the tractive effort can be estimated by adding the tractive efforts due to the individual cylinders at their respective pressures and cylinder strokes.[note 6]

Tractive effort is the figure often quoted when comparing the powers of steam locomotives, but is misleading because tractive effort shows the ability to start a train, not the ability to haul it. Possibly the highest tractive effort ever claimed was for the Virginian Railway's 2-8-8-8-4 Triplex locomotive, which in simple expansion mode had a calculated starting T.E. of 199,560 lbf (887.7 kN)—but the boiler could not produce enough steam to haul at speeds over 5 mph (8 km/h).

Of more successful 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 (or 162,200 lb if calculated by the usual formula). The Union Pacific 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 to 170,000 lbf (756 kN), claim some enthusiasts); and the Pennsylvania Railroad's freight Duplex Q2 attained 114,860 lbf (510.9 kN, including booster)—the highest for a rigid framed locomotive. Later two-cylinder passenger locomotives were generally 40,000 to 80,000 lbf (170 to 350 kN) of T.E.

For an electric locomotive or a diesel-electric locomotive, starting tractive effort can be calculated from the amount of weight on the driving wheels (which may be less than the total locomotive weight in some cases), combined stall torque of the traction motors, the gear ratio between the traction motors and axles, and driving wheel diameter. For a diesel-hydraulic locomotive, the starting tractive effort is affected by the stall torque of the torque converter, as well as gearing, wheel diameter and locomotive weight.

The relationship between power and tractive effort was expressed by Hay (1978) as

${\displaystyle t={\frac {PE}{v}},}$ [11]

where

• t is tractive effort, in newtons (N)
• P is the power in watts (W)
• E is the efficiency, with a suggested value of 0.82 to account for losses between the motor and the rail, as well as power diverted to auxiliary systems such as lighting
• v is the speed in metres per second (m/s)

Freight locomotives are designed to produce higher maximum tractive effort than passenger units of equivalent power, necessitated by the much higher weight that is typical of a freight train. In modern locomotives, the gearing between the traction motors and axles is selected to suit the type of service in which the unit will be operated. As traction motors have a maximum speed at which they can rotate without incurring damage, gearing for higher tractive effort is at the expense of top speed. Conversely, the gearing used with passenger locomotives favors speed over maximum tractive effort.

Electric locomotives with monomotor bogies are sometimes fitted with two-speed gearing. This allows higher tractive effort for hauling freight trains but at reduced speed. Examples include the SNCF classes BB 8500 and BB 25500.