# Tractive Force - Steam locomotives

## Description

As used in mechanical engineering, the term tractive force can either refer to the total traction a vehicle exerts on a surface, or the amount of the total traction that is parallel to the direction of motion.

In railway engineering, the term tractive effort is often used synonymously with tractive force to describe the pulling or pushing capability of a locomotive. In automotive engineering, the terms are distinctive: tractive effort is generally higher than tractive force by the amount of rolling resistance present, and both terms are higher than the amount of drawbar pull by the total resistance present (including air resistance and grade). The published tractive force value for any vehicle may be theoretical—that is, calculated from known or implied mechanical properties—or obtained via testing under controlled conditions. The discussion herein covers the term’s usage in mechanical applications in which the final stage of the power transmission system is one or more wheels in frictional contact with a roadway or railroad track.

Defining tractive effort

The term tractive effort is often qualified as starting tractive effort, continuous tractive effort and maximum tractive effort. These terms apply to different operating conditions, but are related by common mechanical factors:

- input torque to the driving wheels – the wheel diameter – coefficient of friction (μ) between the driving wheels and supporting surface and – the weight applied to the driving wheels (m).

The product of μ and m is the factor of adhesion, which determines the maximum torque that can be applied before the onset of wheelspin or wheelslip.

Rail vehicles

In order to start a train and accelerate it to a given speed, the locomotive(s) must develop sufficient tractive force to overcome the train’s drag (resistance to motion), which is a combination of inertia, axle bearing friction, the friction of the wheels on the rails (which is substantially greater on curved track than on tangent track), and the force of gravity if on a grade. Once in motion, the train will develop additional drag as it accelerates due to aerodynamic forces, which increase with the square of the speed. Drag may also be produced at speed due to truck (bogie) hunting, which will increase the rolling friction between wheels and rails. If acceleration continues, the train will eventually attain a speed at which the available tractive force of the locomotive(s) will exactly offset the total drag, causing acceleration to cease. This top speed will be increased on a downgrade due to gravity assisting the motive power, and will be decreased on an upgrade due to gravity opposing the motive power.

Tractive effort can be theoretically calculated from a locomotive’s mechanical characteristics (e.g., steam pressure, weight, etc.), or by actual testing with drawbar strain sensors and a dynamometer car. Power at rail is a railway term for the available power for traction, that is, the power that is available to propel the train.

Steam locomotives

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 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. 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) as shown here.

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, giving a low c value. If the cylinder valves are left open for longer the value of c will rise nearer to one.

Related formulas## Variables

t | tractive effort (pound*force) |

d | piston diameter in inches (bore) (inch) |

s | piston stroke in inches (inch) |

p | working pressure in Pounds per square inch (ψ) |

w | diameter of driving wheel in inches (inch) |