"I wrote this essay on the fundamental geometry of modern screwthreads a few years back, and this looks to be a good time to repost it:
There are three different forms of screwthreads that have been in widespread use in the industrialized world in the last century or so that we should talk about here, and probably several dozen more forms that I'll tactlessly ignore. All three of these threadforms -- the Sellers (aka Franklin Institute, National, American National, and US Standard), the Unified, and the ISO Metric -- are descended from an earlier form, the "60 degree Sharp V".
The 60 degree Sharp V threadform has been obsolete since the 1800s, but it provides a good place to begin our discussion.
Let’s start by imagining a bolt with a 60 degree Sharp V thread. Now imagine that we cut that bolt lengthwise, so that the plane of the cut contains the central axis of the screwthread, and then look closely at the profile of the thread in the plane of the sectioning cut.
If we were having this discussion over a cup of coffee in the breakroom, I'd be drawing sketches to show you what I'm trying to explain. Since we aren't, though, you might want to get out a pencil and paper and try to make your own sketches while I talk.
The Sharp V screwthread profile looks like a row of equilateral triangles, each with one side resting on a straight edge with their points pushed together. All sides of these triangles are the same length, so the “points” of successive triangle away from the straight edge are separated by the length of the triangle side . . . we'll call this distance The Pitch of the screwthread.
There is another row of these little triangles on the opposite side of the sectioned bolt, offset along the length of the bolt by a half Pitch, with their not-on-straightedge points pointing in the opposite direction from the first row's not-on-straightedge points.
Following so far? Ok, now lightly draw two parallel lines, one connecting the away-from-straightedge triangle points on one side of the bolt and the other connecting the away-from-straightedge triangle points on the other side of the bolt. These two line are separated by the Major Diameter of the screwthread.
The on-straightedge sides of the two rows of triangles are separated by the Minor Diameter of the screwthread. The away-from-straightedge points of each row of triangles are off of the straightedge by the Single Depth of Thread, which is The Pitch x Cosine 30 degrees.
The Double Depth of Thread is twice the Single Depth of Thread, 2 x The Pitch x Cosine 30 degrees. The Double Depth of Thread is also the difference between the Major Diameter and Minor Diameter.
If you were going to cut this thread on a lathe using a single-point toolbit with the compound rest slewed to feed along the flank of the screwthread, and assuming that you zero the compound when the sharp point of the toolbit just touches the already-cut-to-Major-Diameter workpiece, you'd have a complete threadform when you'd fed the tool into the workpiece by a distance equal to The Pitch. After all, all three sides of each triangle are the same length.
While the geometry of the 60 degree Sharp V screwthread is nice and simple, it has practical problems. The sharp point on the toolbit breaks or wears very quickly, the sharp ridges at the Major Diameter of bolts and Minor Diameter of nuts get banged up very easily, and the sharp grooves at the Minor Diameter of the bolts are "stress risers" weakening the bolt. By the 1860s the American industrialist and machine tool builder William Sellers proposed a modified version of the earlier threadform, one with 1/8 Pitch flats at both the Major Diameter and Minor Diameter, as a new standard.
This new threadform, the Sellers threadform, was fairly well accepted, but it didn't actually become the official US Standard threadform until well into the first half of the 20th century.
So let's modify those sketches. The general spacing and angles stay the same, but the new profiles have flats instead of sharp points at the Major Diameter and same-size flats instead of sharp grooves at the Minor Diameter. Both flats need to be 1/8 Pitch long, which reduces the length of the angled flanks AS MEASURED ALONG THE AXIS OF THE SCREWTHREAD to (Pitch - 1/8 Pitch at the Minor Diamter - 1/8 Pitch at the Major Diameter) = 6/8 Pitch = 3/4 Pitch.
The length of the flank is reduced by the same ratio, and the other Pitch-dependent calculations are adjusted accordingly.
Flank length (along-flank infeed using slewed-to-feed-along-flank compound rest) for Sellers threadform = 3/4 Pitch.
Single Depth of Sellers Screwthread = 3/4 Pitch x Cosine 30 degrees.
Double Depth of Sellers Screwthread = 2 x 3/4 Pitch x Cosine 30 Degrees.
The Sellers threadform served the US's needs well enough until World War II, when the difference between the United States' and the British Standard threadforms created major logistical headaches . . . British equipment could only be repaired with British-standard hardware while US equipment could only be repaired with US-standard hardware.
Once WWII had been won, the US, Great Britain, and Canada (which, interestingly enough, had fifty years of experience struggling to supply the appropriate British Standard and US Standard hardware when and where needed) put their collective heads together to develop a single standard that all three nations would use. To "share the pain" of forsaking a traditional standard screwthread, a new threadform was developed that both the US and Britain would need to learn to use. This new threadform was called the "Unified" threadform, and it incorporates the easier-to-tool 60-degree angle with flats at the Major and Minor Diameters of the Sellers screwthread, but with different proportions.
A decade later, the fundamental geometry of the Unified threadform was incorporated into what could be considered a metric version of the Unified form intended to replace the various European national standard threadforms. Since the International Standards Organization developed and promoted the new metric standard, it was christened the ISO Metric threadform.
The major difference between the Sellers and Unified threadforms is that the length of the flat at the Minor Diameter of the Sellers threadform was doubled to 1/4 Pitch for the Unified threadform. The flat at the Major Diameter of the Unified and ISO Metric threadforms is the same as the Major Diameter flat of the Sellers threadform, 1/8 Pitch.
So, for both the Unified and ISO Metric threadform the axial length of the flanks is reduced still further to (Pitch - 1/4 Pitch at Minor Diamter - 1/8 Pitch at Major Diameter) = 5/8 Pitch.
Flank length (along-flank infeed using slewed-to-feed-along-flank compound rest) of Unified and ISO Metric threadforms = 5/8 Pitch.
Single Depth of Unified and ISO Metric threadforms = 5/8 Pitch x Cosine 30 degrees.
Double Depth of Unified and ISO Metric threadforms = 2 x 5/8 Pitch x Cosine 30 degrees.
Minor Diameter = Major Diameter - Double Depth of Thread.
Well, that's the basic geometry of these screwthreads.
As you've already pointed out, the dimensions of real hardware are properly perturbed by allowances and tolerances. External screwthreads cannot be larger than the dimensions derived from the basic geometry, and internal screwthreads cannot be smaller than the dimensions derived from the basic geometry if the external and internal screwthreads are to fit together. The along-flank infeed calculated from the basic geometry assumes that the Major Diameter of the to-be-externally-threaded workpiece is right at the basic Major Diameter AND the flat on the toolbit is the proper width (1/4 Pitch for Unified and ISO Metric threadforms, 1/8 Pitch for the Sellers threadform) . . . or that the Minor Diameter bored into the to-be-internally-threaded workpiece is right at the basic Minor Diameter AND the flat on the tip of the toolbit is the proper 1/8 Pitch width (any of the three threadforms we've talked about).
As for the charts . . . even though the Unified threadform replaced the Sellers threadform as the official US standard threadform a half century ago, many of the tables in various "reference works" have been carried forward from edition to edition with values appropriate to the Sellers threadform.
I realize that my posting is long-winded, but I hope it's clear enough for you to follow. If not, post back and I'll try to answer your questions."
by john garner
