On Standard Screw Threads

 On Standard Screw Threads

(Part 3 of Nuts & Bolts)

I am still pursuing the question "Do modern metric nuts and bolts come loose spontaneously with greater facility than the nuts and bolts of my childhood; and, if that is the case, why are the modern nuts worse than the British Standard Whitworth nuts of my youth?". 

When I raised the question 15 years ago (See ref. [1])  I baulked at the problem of calculating the effect of 'groove-angle' on (a) friction, and (b) the the force causing the nut to unwind. But I pointed my finger at the increased groove-angle as possible culprit for the present casecade of falling nuts. However, buoyed by my success 4 years ago in calcultaing the packing density of uniform spheres [2], I now venture to tackle the 3-dimensional mathematics of the bolt, and its helical thread [3]. 

I suppose a bolt could have a helical groove with a square section, and indeed there may be some special applications in which such threads are cut. However, the groove of the standard bolt of 19th and early 20th century British engineering had a symmetrical, v-shaped profile, where the angle between the upward-facing surface and the downward-facing surface was 55º. Now, in the 21st century, internatinally, that angle is 60º. 

90º⊐  55º (>  60º (> 

(Image from boltscience.com)

Once tightening is complete, there will a tension in the bolt along its axis related (by Hook's law) to the elastic stretching of the bolt. That tension (F, in the sketches below) will be countered by pressure on the surface of the (upper) bolthead and the surface of the nut in contact with the lower plate contact area. But F will also be felt by the entire surface area of the bolt thread that is in contact with the mating surface of the nut, where it will generate friction. It is that friction, proportional to the surface area in contact and to the force pressing them together, that stops the nut from unwinding. 

Let me consider first the square-sectioned groove (sketched in Fig. 2 below.)  In the middle-top sketch I have tried to draw a square-bottomed grove on the right side of a right-handed bolt, trying to show that, as the thread comes towards us, it dips by a small amount below the horizontal; I have suggested 5º (A pitch of 1.27mm in circumference of 19.95mm gives a 'lead-angle' of 3.65º). In the middle-bottom sketch I have tried to draw the more normal v-bottomed groove, with a similar small dip of 5º. 

(Fig. 2)

In the left-top I have suggested that the Hook's law tension (F) operates along the axis of the bolt. It is met, of course, by the nut. I have supposed that the friction that develops between bolt and nut is proportional to the area of the two mated surfaces, and to the force between them that acts normal to the mated surfaces. I have likewise supposed that the force that unwinds the nut operates in the plane of the nut. 

From the sketch in the left-top, it would seem plausible that, in the case of the square-bottomed groove, the normal to the bearing surface is only 5º off vertical; leaving most of the force (85/90) remaining to generate friction (and a residual 5/90 of the force to unwind the nut). In the case of the v-bottomed groove it would seem that a large portion of the of the force F operates at the wrong angle to cause friction, 27.5/90 in the case of the BSW nut, 30/90 in the case of the modern ISOmetric nut. (I have not managed to see what that tangental force does; does it stretch the nut? )

Thinking along the above lines I have managed to convince myself that the v-bottomed groove does not maximise the friction between nut and bolt, because the tension in the bolt is not normal to the mated surfaces. And in this regards the modern bolts of the ISOmetric series with a 60º groove are worse than the British Standard Whitworth (BSW) with 55º groove. Incidentally the American UTS series also uses the 60º groove. 

Perhaps a more significant difference between the BSW groove and the shallower ISOmetric groove is the smaller area of the mated surfaces in the latter. To show how this could be significant, I have sketched (Fig. 3) an equilateral triangle (sides 2, 2, 2; angles 60º, 60º, 60º), and a second isosceles triangle where one of the angles has been narrowed to 55º. You will see that the friction-generating surface of the latter is 8% larger.

(Fig. 3)

These two effects combined might explain my conviction that the nuts of today are more inclined to come loose than the nuts of the 'Whitworth' era. 

(I have also shown that a small element of the elastic tension in the bolt is directed towards un-twisting (loosening) the nut. It is dependent on the ratio of the pitch to the circumference. But in that regard the nuts of the ISOmetric series are very similar to those of the BS Whitworth series, and are occasionally 'finer' (See my Nuts and Bolts1). I have not yet managed to clarify whether the tension in the bolt applied to the sloping side of the groove could have a resultant component that un-twists the nut, like the progress of a sailing boat with a beam wind. I would urge engineers like those at  boltscience.com  to investigate these questions. )

REFERENCES

[1]  https://occidentis.blogspot.com/2010/06/nuts-and-bolts.html 

[2]  https://occidentis.blogspot.com/2021/09/excluded-volume-between-close-packed.html

[3]  https://occidentis.blogspot.com/2025/06/on-helices.html

On Helices

On Helices

(Part 2 of Nuts & Bolts)

You will recall that some 15 years ago [1] I raised the question "Do modern metric nuts and bolts come loose spontaneously with greater facility than the nuts and bolts of my childhood". I was a little surprised to find no one taking up the enquiry, as it surely affect us all. (By the way, the handle is loose again on the cold tap in my new kitchen, thus defeating a handyman and two trained plumbers!). So, I decided to look further into the matter. 

One useful step forward was identifying a company (BoltScience.com) with a testing jig that will rattle a couple of bolted plates till they loosen. They have not yet (so far as I can see) asked my questions, about metrication, tribology, and groove angle; but I may be able to cajole them into a colaboration. As a second step, I decided to make a deeper study of the helix, because its geometry is fundamental to the questions I am raising. 

At the outset, let me clear up a possible confusion: a spiral staircase is a misnomer. A spiral would lie of the floor. We should talk rather of a helical staircase; c.f. Watson & Crick's double helix. Next, let us remind ourselves of some terms. 

Let h = the (vertical) height of the helix up its long axis, 

Let p = the pitch or lead up the long axis for each turn of the nut. 

Let L = the length of the helix were it unwound and laid flat.

Let c = the circumference of the helix.

Let n = the number of turns; i.e. h/p.

I next wanted to see if I could calculate the area of the two surfaces that are juxtaposed when a nut is threaded on a bolt, the two surfaces between which the friction is generated that prevents the nut loosening; and for that I must calculate L, the length of the thread. 

Not trusting my theoretical powers, I sought to measure L, to check theory. I found my wooden rolling pin (h=400mm), milled to a uniform diameter of 39.8mm which I determined by wrapping a paper strip round the cylinder and cutting it exactly to equal  the circumference (c=125mm). I found that a dressmaker's measuring tape would lie adequately flat against the rolling pin, even when wound round like a ribbon on a May-pole.   

Two extremes are easy. If the tape runs vertically up the pin, L simply equals h. And if the tape went round and round the pin without progressing up the pin at all, L could be infinitely long. I measured the length of tape needed to wind it round one full circle in the 400mm height of the rolling pin. And again with 2 full circles; etc.  

No. of complete circles.  Measured L (mm)

1 417

2 472

3 552

4 646

5 750

With string I could have explored further, but the tape was more convenient; and, lying flat against the rolling pin suggested the next step. With the tape making one complete cycle in the height of the pin, it was easy to see that the rather stiff tape made a precise angle (𝛉) against the vertical axis, which it maintained all the way round the pin. I could find L by applying Pythagoras's theorem to the little right-angled triangle: base = 𝛿c; height = 𝛿h; hypotenuse = 𝛿L.; and assume that the relations held all the way up. Furthermore, the angle off the axis (𝛉) can be calculated by applying trigonometric functions.  For a single circle of the rolling pin, the cos of the angle (𝛉) of the thread to the axis is 'adjacent/hypotenuse' = 400/417, so 𝛉 = 17º; for two circles of the pin the angle between thread and axis (𝛉) is 32º [cos(32º)=400/472]. Likewise, the related small angle (90º - 𝛉) is arctan (pitch/circumference). 

For each circle of the pin:

L² = c² + p²

But n (the number of turns) = h/p so, for a multi-turn helix we can predict: 

L = h/p x √(c² + p²)

I add these predictions to my table. 

No. of  circles.  Measured L (mm) Calculated L (mm)

1 417 419

2 472 470

3 552 548

4 646 640

5 750 742

That is all very satisfying. And is a step towards calculating the area of the two mated surfaces between nut and bolt where friction prevents loosening. 

REFERENCE

[1] https://occidentis.blogspot.com/2010/06/nuts-and-bolts.html

Nuts (and bolts)

I often wonder why all the tap handles in my new bathroom and kitchen are loose; and why the nut falls off the door of my new wood-burning stove every other day. I am of an age at which new problems seem obviously due to slipping standards, or to the forgetting by callow youth of the wisdom of their elders. So I assume that these loose nuts are symptomatic of some failure of modern workmanship, or of modern materials. I suppose there are several possible explanations, besides 'bad luck'.

Perhaps the coefficient of friction (µ) of common materials has changed? For example, teflon is 10 times 'slippier' than steel, while de-greased steel is some 30% 'grippier' than normal steel [1]. Aluminium grips on aluminium, but not on steel. The oxide layer on materials like aluminium and steel makes a considerable difference to the grip. There is a plethora of data [1] on coefficients of friction, but all-in-all I could not identify in this area a convincing explanation for my loose nuts.

The nuts and bolts that held our bicycles and cars together for half a century before the second world war were defined (pitch, mean diameter, depth of groove) in factions of an inch. Metrication has, of course, minutely changed these linear dimensions to make them fit a metric scheme, and so I wondered if the engineers making the conversions from inches to millimetres had 'rounded' in such a way as to degrade the gripping power. If we define "lead angle" as arctan (lead/(π x mean diameter)) [2], first of all we find lead angle of metric standard bolts curiously variable. A 7 mm diameter bolt (pitch = 1 mm) should, on the face of it, grip appreciable better than the similar sized 6 mm diameter bolt (pitch = 1 mm), and the 4 mm diameter bolt appreciably worse (pitch = 0.7 mm). However, when we compare standard metric bolts with British Standard Whitworth, or American Screw Thread, we find that the new lead angle is significantly less, and should therefore be less prone to work loose.

(Bicycle threads are different. They were (and are) deliberately made with a finer thread, so even with 'amateur' servicing they are less prone to wriggle loose.)

So why then the loose tap handles, fire-door, etc. There is one measurement in the standard metric bolt that is less friendly to security than found in older standard threads, and that is the angle formed by the two sides of the groove. It is now 60º. It was 55 º in the BSW (and was 47.5º in the British Association fine instrument threads). This means that the load, which [a] generates friction and [b] generates the resultant force that causes loosening, is far from 'normal' to the interacting surfaces, but is applied at a glancing angle. I wonder if this is significant.

Wouldn't it be champion if my ruminations on this matter ultimately led to a new international thread standard, and a generation of tap handles that stayed firm indefinitely!

REFERENCES
[1] http://www.roymech.co.uk/Useful_Tables/Tribology/co_of_frict.htm
[2] "Lead" is the distance of advance up the axis of the bolt for each turn of the helix. See http://en.wikipedia.org/wiki/Lead_(engineering)

Occidentis, MORPETH, UK.