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
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.
1 Defining tractive effort
Tractive effort curves
3 Rail vehicles
3.1 Steam locomotives
3.1.1 Values and comparisons for steam locomotives
3.2 Diesel and electric locomotives
4 See also
5 References and notes
6 Further reading
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.
Starting tractive effort: Starting tractive effort is the tractive
force that can be generated at a standstill. This figure is important
on railways because it determines the maximum train weight that a
locomotive can set into motion.
Maximum tractive effort: Maximum tractive effort is defined as the
highest tractive force that can be generated under any condition that
is not injurious to the vehicle or machine. In most cases, maximum
tractive effort is developed at low speed and may be the same as the
starting tractive effort.
Continuous tractive effort: Continuous tractive effort is the tractive
force that can be maintained indefinitely, as distinct from the higher
tractive effort that can be maintained for a limited period of time
before the power transmission system overheats. Due to the
relationship between power (P), velocity (v) and force (F), described
P = vF or P/v = F
tractive effort inversely varies with speed at any given level of
available power. Continuous tractive effort is often shown in graph
form at a range of speeds as part of a tractive effort curve.
Vehicles having a hydrodynamic coupling, hydrodynamic torque
multiplier or electric motor as part of the power transmission system
may also have a maximum continuous tractive effort rating, which is
the highest tractive force that can be produced for a short period of
time without causing component harm. The period of time for which the
maximum continuous tractive effort may be safely generated is usually
limited by thermal considerations. such as temperature rise in a
Tractive effort curves
Specifications of locomotives often include tractive effort
curves, showing the relationship between tractive effort
Diagram of tractive effort versus speed for a hypothetical locomotive
with power at rail of ~7000 kW
The shape of the graph is shown at right. The line AB shows operation
at the maximum tractive effort, the line BC shows continuous tractive
effort that is inversely proportional to speed (constant power).
Tractive effort curves often have graphs of rolling resistance
superimposed on them—the intersection of the rolling resistance
graph[note 1] and tractive effort graph gives the maximum velocity
(when net tractive effort is zero).
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
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.
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 85d^ 2 sp 100w
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 driving wheel in inches
The constant 0.85 was the
Association of American Railroads
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.
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]
Values and comparisons for steam locomotives
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.
Diesel and electric locomotives
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.
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.
Factor of adhesion, which is simply the weight on the locomotive's
driving wheels divided by the starting tractive effort
Tractor pulling, bollard pull - articles relating to tractive effort
for other forms of vehicle
Power classification -
British Railways and London, Midland and
Scottish railway classification scheme
References and notes
^ The graphs typically show rolling resistance for standard train
lengths or weights, on the level or on an uphill gradient
^ Half the stroke distance is about the same as the radial distance
from the coupling of the driving rod to the centre of the driven wheel
^ The relationship is:
Torque = Forcepiston x R (the radial distance
to the point of connection of the driving rod) x cos(A), where A is
the angle the driving rod makes with the tangent to the radius from
wheel centre to driving rod attachment
^ As with any physical formula, units of measurement must be
consistent: 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.
Gas laws for an explanation.
^ The value of the constant c for a low-pressure cylinder is taken to
be 0.80 when the value for a high-pressure cylinder is taken to be
^ SAE J2047, Tire Performance Technology, dated February 1998.
^ Simon Iwnicki, ed. (2006). Handbook of railway vehicle dynamics.
Boca Raton: CRC Press: Taylor & Francis. p. 256.
^ XPT: Delivery, test runs and demonstration runs railpage.au.org see
^ The Gravita
Locomotive Family voithturbo.de (page 2)
^ EURO 4000 Freight Diesel-Electric Locomotives vossloh-espana.com
^ Eurorunner ER20 BF and ER20 BU, Diesel electric platform locomotives
for Europe siemens.dk (page 3)
^ Eugene A. Avallone; Theodore Baumeister; Ali Sadegh, eds. (2006).
Marks Standard Handbook for Mechanical Engineers (11th ed.).
McGraw-Hill. p. 166. ISBN 978-0-07-142867-5.
^ Allan, Ian (1957).
British Railways Locomotives Combined Volume. Ian
^ Ian Allan ABC of
British Railways Locomotives, winter 1960/61
edition, part 1, page 3
A simple guide to train physics[dead link]
Tractive effort, acceleration and