(a) Steam Plant. It is clear that the wall pressure used in our models are higher than need be, even with the plain Ramsbottom type; still more so with modern heat formed rings. For stationary, marine, and road locomotive engines, all of which can be motored for initial bedding down, 68 lbf/sq.in. should be adequate. However, there is a small problem; the radial thickness is so small with "classical" gaps that parting off may be difficult. For this reason I set a minimum value of "t" at 0.035", and use a smaller gap when necessary. Fixing on g=D/10 instead of 4t simplifies the pressure formula (4) to
but this ratio should not be used for wall pressures above about 12 lbf/sq.in. without checking the "fitting" stress. (See later). For rail locos it may be prudent to use slightly higher pressures. The low end of the "Chaddock standard" provides about 13 lbf/sq.in. with "E" at 17 x 10^{6}. (i.e. D/t = 30, g = 4t), but this will vary pro rata with the value of E.
(b) I.C. Engines. For the classical horizontal gas/petrol engine model I have substituted rings at 6 1/2 lbf/sq.in. with no problems, but where the designer has called for two rings I have used three. This pressure lies outside the "Chaddock standard" (t=D/3335) so that again ring gaps of D/10 are used. Many published designs use rings which are too wide, and I suggest w=0.03B as a guide, with a minimum of 0.04".
For all other I.C. engines it is no surprise to find that adherence to Prof. Chaddock's rules
will give satisfaction, though at the high end (D/25) it is important to check stresses, as the
stresses are higher than the usually available 17ton iron can carry. For the benefit of those
not familiar with these rules the figures are:
If "E" is 17 x 10^{6} lbf/sq.in. these rules offer wall pressures from 28 down to 13 lbf/sq.in. The wall pressure will, of course, vary with the value of E in direct proportion. The lower the wall pressure the less the friction, of course!
For very high performance engines it goes without saying that experiment is always necessary, and although it may add to cost and time a special singlecylinder prototype, arranged for measuring both oil consumption and blowby (as well as output and fuel consumption) is well worth while. Even here I would not expect to find more than about 25 lbf/sq.in. to be necessary, and that only when a single pressure ring is supported by a stepped scraper between the top ring and the main oil controller.
(a) Running gap. Many contributors to the debate have suggested that the running gap can be zero in a model, usually on the grounds that temperatures are relatively low. This is a mistake, as the risk of ring butting (especially with rings that are split by "snapping off") and consequent scoring is high. True, temperatures are lower, but it must not be forgotten that some 70% of the heat reaching the piston is transferred to the cylinder walls through the rings, and that they are further heated by their own friction.
The idea that the ring gap was a serious source of leakage was exploded over half a century ago. it is, in fact, ludicrously small. Fig. 4 shows an idealized model of the gas flow through a pair of ring gaps. "A" represents the top land, a short passage with high friction loss. "B" is the orifice formed by the top of the first gap; typically of area 2/1000,000 sq.in. for a piston with 2 thou diametrical clearance and a 2 thou gap. The gas (or steam) expands through this, losing more pressure into cavity "C", the space between the ends of the ring, and that behind the ring. It then enters the second orifice at "D", and expands through this, (with pressure drop) into "E", which is the second land; a long and frictional passage round the piston to the next gap, which it enters at the orifice "F". The same procedure is followed here, into "G", through "H", and finally, to the back pressure via"I", the tortuous passage past the skirt of the piston. The point is that it is the pressure drop at the final orifice which governs the rate of leakage, and NOT the high pressure drop at "B". The minimum gap  for steam or IC should be 0.002" and a guide might well be an installed gap equal to 0.001" + 0.001"/inch of cylinder bore.
(b) Fitting gap. This is the dimension "G" in Fig. 5, when the ring is sprung over the OD of the piston when fitting. If this is excessive the ring may be over stressed, but it is the dimension (Gg) which is significant, so that, as already remarked, the risk of over stressing increases as the free gap is reduced. Unfortunately "the books" all seem to assume that the ring clasps the piston closely when fitting but this is not the case, and stresses based on this assumption will be too low. Geometric analysis is almost impossible, as the ring does not assume the shape of a pair of halfcircles, and in any case the actual direction of the loads "pp" are indeterminate. Experiment with a number of rings of various D/t ratios and values of D shows that G varies from 6.6t to 7.5t. If g = 4t as under the Chaddock rules, then the ring will not be over stressed when installed  provided the working stress is safe, of course. As a very rough approximation, the installing stress can be estimated by writing
Where F_{i} = installing stress (7) ..... f_{i} = f_{w} x (7tg)/g f_{w} = max. working stress t and g as before
The original "Ramsbottom" rings were plain circles from which a gap was cut so that the closed ring fitted the cylinder with no more than a working gap. It was realised that such a ring would not fit properly, and so could not exert a uniform pressure, even when bedded in after hours of running. The necessary shape to achieve this was known, of course, and requires the free ring to have a radius at any point which varies as the sine of the angle of the section from a point directly opposite from the gap. Lanchester devised a machine which would turn rings to this shape, but it is doubtful whether any model engineer would undertake to make one! However, with modern NC machines the process is much easier, and many large engine rings are so manufactured.
An alternative is to make the raw ring of uniform radius and thickness, and to "peen" the inside surface, the intensity of the blows being opposite the gap and low at the gap  sometimes the spacing was variable, as well. This is very effective and quite simple machines can be built to carry out the process. It is a method used by builders of larger engines, where the number of rings needed is measured in hundreds. However, it is difficult with small diameters, too slow for volume production, and not practical for making just a few.
A third method is to use the tapered ring, Fig. (6). The O.D. of the free ring is a true circle, and it will remain so if the thickness is varied around the circumference such that:
Such a ring will be truly circular when fitted, and will exert a uniform wall pressure. However, it is evedent that when (angle) = 180 (degrees) (i.e. at the gap) then Cos(angle) = 1, so that t will be zero  an impracticable state of affairs! However, a ring approaching this condition can be made by boring the inner diameter eccentric to the O.D. by an ammount which will approximate the to the figures from the Equation (8). This will not be perfect, but it will be a considerable improvement on the plain Ramsbottom. It is a method much used by amateurs. The value of D/t_{o} is determined from (9) & (10), as is the ratio between the cylinder bore (D) and the free diameter D_{o}. The degree of eccentricity is made such that t_{gap} is as thin as practicable. (See Bradley, M.E. 22OCT42, p. 402)
A near perfect ring can be achieved by heat forming, a process which is adaptableto high volume production, and which can be used for very small rings indeed. Here, a circular ring of uniform thickness is cut with a very small gap. It is then forced into or onto a shaped former, and stress relieved so that, when cooled, the ring is the correct form to provide both true circularity and a uniform wall pressure when fitted to the cylinder, though in almost all cases a final machining operation is carried out to "skim" the O.D. to allow for inevitable tolerances. The shape of the former is, of corse, the obverse of the shape used by Lanchester years ago.

Unfortunately there seems to have been a misunderstanding of the nature of the process by some, including both Mr. Trimble and Mr. Tulloch. First, the process is NOT a copy of that used in industry and cannot form a "perfect" ring. Whilst it relies on the relationship shown in expression (2), in that the wedge exerts the "tangential force" there referred to, the process does not and cannot produce exactly the correct shape. This force may induce the correct bending moment in the ring, bnut it does not reproduce the correct deflection, for the tangential force introduces an additional compressive stress. This is small, but has a devastating effect on the shape of the rings adjacent to the gap. This effect is shown in Fig. (8) where the stress condition within the ring is shown.
At (a) is shown the situation in a perfect rinig  i.e. without the compressive load due to the tangential force. The stress is tensile at one face and compressive at the other, with a "neutral axis", where streess is zero, at the center section. These stresses rise from zero at the gap and increase proportionately to Sin^{2} O to a maximum at 180°, though I have shown only the first 30° or so.
At (b) is the situation with the compressive stress due to the wedging force added. For the first 8°10° the stress is compressive only, right accross the section  there is no "neutral axis". Thereafter the tensile stress slowly increases, but the neutral axis lies well away from the geometric center of the section, and even at 30° it still lies some 6% of the halfthickness away. See Fig. 9. The result is that the ends of the ring will, when installed, bear more heavily at the gap and so exert a higher wall pressure in that region. Further, the reaction to this higher pressure at the ends causes a further high area about 120° away on either side.

The "correct" temperature is 480°520°C, with slow heating, the temperature being held for 1 hour per inch of thickness, but at least 10 minutes for very thin rings. The stack may be air cooled from this temperature, though no harm seems to arise from oilquenching. The metal has no color at this temperature, but Messrs. Levermore, 24 Endeavour Way, London SW19 8UH are importers of the "Markall Thermomelt" crayons. A mark with one of these will turn glossy at the indicated temperature and they are available from 100° up to 1200°C. Alternativley, very little degradation of properties will result from heating to 550500°C, when the metal will just be visibly red in a dim light, but on no accout must the temperature be allowed to rise any higher. (The critical temperature is 720°C.) It is preferable to use the lower temperature for the full time than to try to speed things up by going higher. Incidentally, scaling at these temperatures is minimal  it will come off with metal polish.
The inner edges of all rings should be given a slight bevel, to avoid impact should the corner of the slot not be dead sharp. Clearance behind pressure rings, by the wayt, should not exceed 0.005", and about double this for oil control rings. Side clearance should be as small as will allow the ring to turn freely in the groove, If ring pegs are needed to prevent the ends from springing into the ports of a piston valve or 2stroke cylinder the peg must be located at the gap. The practice of pegging rings to ensure that the gaps lie at 180°, very common years ago, is quite unnecessary; it serves no useful purpose.
E = 6.22Fdd (D/t1)^{3} ______________________ w x g
Envoi
This essay is, necessarly, incomplete and should not, on any account, be reguarded as an authorative dissertion! After all, there is never any single, unique, solution to any engineering problem! However, I hope it may serve to encorage those with better facilities than I have to undertake experiment and research on the subject, if only by measuring the wall pressure on rings which they know to be satisfactory. I hope, too, that it may lead to the specification of piston rings by wall pressure rather than using the same size that John Willie fitted back in 1933!