Tractive effort

Since time immorial the formula has been:-

T.E (in lbs) = Cyl dia x cyl dia x stroke (all in inches) x boiler pressure (in lbs) x 85% divided by the driving wheel dia (in inches).

However this assumes that the loco has square cylinders and gives an answer suggesting that the loco has a far greater tractive effort that it can actually achieve.

Assuming 20" dia cylinders - 20" x 20" = 400 sq. ins area, against round cylinders pi x r squared = 10 x 10 x 3.142 = 314.2 sq. ins. Thus the generally accepted formula over-estimates the tractive effort by more than 27%.

Is there a flaw in my thinking, or how have countless locomotive engineers got it so wrong for such a long time?

 

Two things are relevant here -
Firstly, that TE is a calculated rather than measured figure.
Secondly, that the figure represents a calculation of the likely effort available at the rail at starting and only at starting.
It can only be an estimate since other factors have a bearing, for example, adhesion. The position of the wheel in its rotation and the resulting position of the pistons in the cylinders also causes a variation in the power available.
Tractive Effort as a guide to the power of a loco has always seemed wrong to me since, once on the move it doesn't mean a great deal. Remember TE is an indication of starting effort, once moving, various internal and external resistance factors come into play.
Compare a GWR King and an LMS Coronation. The former has the higher tractive effort but would you expect similar performance from the two at high speed?
(I will now don my flame proof suit, retire to my Stanier designed bunker and sit out the onslaught!)
To give another example that won't annoy the GW fans; compare the TE figures for an LNER A3 and a K1... They'll both start a similar size train but will generate quite different figures for power, horse power that is, at 50 mph and at 90 the K1 just isn't in it.
But before I go, I'll answer your question; (in my opinion) no, locomotive engineers have not got it so wrong for such a long time. TE is a good way of providing a comparative estimate of the power available to start a train. Beyond that, you need to build one, run it, do indicator diagrams and then you see if you designed it right!

 

In the equation to calculate the Tractive Force the 'pi' components of the cylinders and the driving wheels cancel each other out. Please see below an extract from the "Locomotive Engineers' Pocket Book 1922".

Code:

Tractive Force. Applying the principle of "Work" to the Locomotve engine:- The work done by the Steam on the Pistons must (friction neglected) be equal to the work done on the Rail by frictional contact with the wheel. Expressing this equation in general terms:-
Let D = Diameter of Cylinder in Inches.
    p = Mean Steam pressure per square inch of Piston.
    D^2 x .7854 x p = Total Pressure on Piston in pounds,
    S = Stroke of Piston in inches.
    4 = Number of strokes (2 cylinder engine) per revoluton of axle.
    (D^2 x .7854 x p) x 4 S = Work done on Pistons in one revolution in inch-pounds.
    W = Diameter of Driving Wheel in inches.
    W x 3.1416 = Circumference of ditto.
    F = Frictional resistance in pounds at circumference of wheel.
    W x 3.1416 x F = Work done in one revolution at circumference of wheel in inch-pounds.
    (D^2 x .7854 x p) x 4 S = W x 3.1416 x F.
    or D^2 x p x S = W x F
    The Tractive Force (friction neglected) being equal to the resistance:-
    Tractive Force in pounds = T = D^2 x S x p
                                   -----------
                                        W

Hope this helps.

 

I once wrote an article for the Stanier Mogul Fund's newsletter entitled 'The Truth About Tractive Effort', which basically said exactly the same as Blackfive just has. The difference is that I took almost 3,000 words and several diagrams to do it...

There is no reltionship between Nominal Tractive Effort, which is a purely theoretical value ESTIMATING the force that a locomotive could exert on the track to move itself and its train, and power output. If it has any relevance, it is what the loco might be expected to start from standstill; once the train is actually moving, it loses any relevance it might have had. Its main function is as a comparison between different engine classes. Actual Tractive Effort is the force that the loco can actually exert and is highest at zero mph, then falls away as speed rises. The rate of decline is dictated by various things: wheel size; steam flow capability (including flows within steam pipes, blast pipe arrangements, valve events and port openings); the ability of the boiler to maintain pressure; and the ability of the wheels to grip the rails. With a shunter, the initial TE might be quite high but the decline would be very rapid, but an express engine would have a far more gradual decline as high speeds are a prerequisite.

Power is defined as either Torque x Angular Velocity or Force (Tractive Effort) x Velocity (not technically correct, but call it speed), depending on whether you're dealing with Circular or Linear motion; they'll both produce the same answer. But the point is that speed is fundamental to power, but not to TE (maximum TE at zero rpm, remember). Nominal TE has absolutely no bearing on power output, although many people believe them to be interchangeable from its inclusion in the Ian Allan ABCs. Look at it like this: an 8F might bring the empty stock into Euston for the two-hour Birmingham express, but it would be a 5X that took the train. The 8F has the higher Tractive Effort, but the 5X has more Power.

 

As the posters above have said, the oft quoted tractive effort of a loco is purely a simplistic comparison. It is traditional to take a figure of 85% boiler pressure but not all loco builders used this figure. The Leeds loco builders (Hunslet, Hudswell, Manning Wardle, etc) often used a figure of 75% when quoting the tractive effort of their locos and there were others.
Most people know the formula for a steam locomotive but what about diesel and electric locos? It gets much more complex with them, especially as you can have very high tractive efforts at starting which tail off quite rapidly. It is usual (but by no means universal) to qualify diesel and electrics with a terms such as 'continuous tractive effort', 'maximum tractive effort' or even base it on an adhesion ratio.

 

It is possible in principal to define Tractive Effort at any speed, although the usually quoted figure for a steam engine is the starting tractive effort at zero speed. The usefulness of this is that it is easy to estimate and that it gives some indication of the maximum load that can be started from rest on a given gradient, providing that there is adequate adhesion. This figure is frequently used in determining maximum train loads on a given route. But it gives no indication of performance once the train is moving. This will depend on many other factors such as the ability to maintain the required steaming rate, and the various losses due to internal friction etc. The GWR used starting tractive effort as a measure for its loco classification system. The LMS and later (slightly modified) BR standard power classifications (2P, 3F, 4MT, 6P5F etc) are based on tractive effort at a defined speed of (I think) 50mph for passenger engines and 25mph for freight engines. But I have never seen a proper explanation of how the figure was determined. Was it estimated theoretically or was it measured in some way ? In practice something like equivalent drawbar horsepower as a function of speed is a more useful way of specifying locomotive power but it is quite a complex parameter to determine.

David

 

I would suggest - and I'm open to correction on this - that the actual tractive effort is what was measured on the rollers at Swindon and Rugby; the wheels were in contact with these and applied a force to them to turn them against a known resistance, as they would apply that force to rails to move a train. Since the wheel rpm was also accurately known, the power output at that speed was instantly attainable.

 

I'm confused now. I always understood tractive effort was thoretical and putting an engine on a rolling road type dyno gave an actual power (drawbar horespower?) figure. Help? #-o

 

Attached below are the relevant pages copied from 'Fowlers Mechanical Engineer's Pocket Book, 1958'.

NO it is NOT for sale.

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Not necessarilly. Tractive Effort is the force that the locomotive delivers to the rails in order to move itself and its train forwards or backwards, and the same applies to a road vehicle. The actual TE is difficult to measure on the rail or road, but can be done on rollers, e.g. at Swindon and Rugby (in days gone by), while many performance cars use a rolling road to measure their TE at the wheels, although it is usually converted to Brake Horse Power (Power = Force [TE] x Velocity]. The NOMINAL Tractive Effort as is uasually quoted is a theoretical, calculated figure dependent only on certain of the loco's dimensions.

 

If you aren't an engineer or a person that understands all these terms it can be very difficult to do so. You have to come to terms with the difference between force and power for a start, then understand all the different terms used to describe such things, some being interchangeable andf some not. Tractive effort is force, plain and simple and is generally that produced at the wheel/rail interface. It is possible to calculate the theoretical tractive effort for a particular loco but it is not constant, varying quite a bit over a wheel revolution. This is one of the reasons that locos 'stick on centre' and the driver has to set back before he can start his train. The loco has stopped on a weak point in the cycle. There is also a difference between the tractive effort that is available at the immediate point of starting and the same position one revolution later as at starting it may be that only one cylinder is getting any steam whereas, once on the move, both cylinders will have steam for much of the time. All sorts of other things have to be taken into account, such as the position of steam cut off and all the other events in the steam cycle. The length of the coupling rod also has an effect and it depends on whether the piston is on its forward stroke or back stroke and the diameter of the piston rod. The list of variables that would have to be taken into account is very long. Thus, if you want to know the actual tractive effort of a loco, the only really practical way is to measure it, by means of rollers and a dynamometer. You can measure drawbar pull at the tender more easily but this does not take account of the force required to move the loco. So, what about the oft used formula quoted by Eightpot:

T.E (in lbs) = Cyl dia x cyl dia x stroke (all in inches) x boiler pressure (in lbs) x 85% divided by the driving wheel dia (in inches)

This is just a simple way of comparing locos and gives a reasonable estimate of the tractive effort of a loco at starting but that is all it is, an estimate. Why 85%? Again this is purely an arbitrary figure and is to allow for losses in boiler pressure for whatever reason and also makes it more certain that a loco will actually produce the claimed TE, which is often used in the basis of the calculations used to determine the fundamental requirements of the design. The Hunslet Engine Co for one, generally used 75% in their tractive effort calcs.

 

Not disagreeing with the above contributors, but the generally accepted formula, by thinking in terms of square cylinders rather than round ones, does result in an end amount considerably more than is attainable. Granted that crank positions and other factors tend to make the whole thing academic anyway.

 

TE should really be ESTE - Estimated Starting Tractive Effort...

 

You've missed something there.

TE = No. of cylinders x cyl dia x cyl dia x cyl stroke x boiler pressure / 2 x wheel dia in inches, then calculated as a % based on the valve travel ("full gear" being between 75% and 85% on most engines).

Alternatively some designers used a % of full boiler pressure.

Re. your comment about squared cylinders, how can this be? Diameter is measured constant to the centre point of the circle. The distance between the centre point, the sides and the corners of a square is not equal. Discuss.

 

Re. your comment about squared cylinders, how can this be? Diameter is measured constant to the centre point of the circle. The distance between the centre point, the sides and the corners of a square is not equal. Discuss.

Assumuing cylinders are cylindrical, then the area is given by pi x r x r, or diameter x diameter times pi devided by 4.

so the formula

T.E (in lbs) = Cyl dia x cyl dia x stroke (all in inches) x boiler pressure (in lbs) x 85% divided by the driving wheel dia (in inches).

over estimates the TE by a factor of ((4/pi)-1)x100 %

 

I think you're missing my point (possibly because I've further jumbled the terminology myself - apologies).

The OP stated that the formula assumes the cylinders are square. Square not squared. This is untrue because it uses cylinder diameter as one of the determining measurements, and diameter by definition cannot be constant from the centre point all the way around the edges of something that is square.

It is true that the formula squares the cylinder diameter i.e. multiplies it by itself. That doesn't mean that TE is "over-estimated" because the TE is the TE as defined by the formula. It can't be wrong against itself.

TE is an abstract concept. I'd imagine the purpose of squaring the cylinder diameter is to reflect the far greater significance cylinder diameter has on power output relative to the cylinder stroke.

 

No it doesn't: this cancels with the pi/4 wheel diameters (i.e. a quarter circumference) travelled per stroke, assuming a 2 cylinder loco.

 

Reference my previous post containing an extract from the "Locomotive Engineers' Pocket Book 1922", I've shown in a bit more detail how the formulae is derived, hope this helps.

 

I thought the formula was for a two cylinder loco and thus you multiplied the answer by 1.5 for a three cylinder one and by 2 for a four cylinder one.

 

Phillipson's "Steam Locomotive Design: Data and Formulae" shows it both ways round - the point is, number of cylinders needs to be factored in somewhere, and that hadn't been mentioned.

He also gives formulae for calculating the tractive effort of compound locomotives, but I'm not sure we really want to go there!

More interesting is the long list of caveats which follow, starting with the words "It should be remembered that the tractive force formula is strictly limited in its application and therefore liable to abuse".