Shoulder of Doubleness

This whole blog thing, when I remember to do it, seems to become devoted to the many intricacies (and vagaries, and draperies) of rotary-shouldered connections. I didn’t mean to do that; it just happened.

What’s on the menu for today? Extra—extra shoulders, extra torque, extra unprovable assumptions. Extra!

See, I get to work on API’s Task Group for Drill Stem Elements (TGDSE, as it’s fondly called), and one of the long-running projects there is to standardize and publish a set of double-shouldered connection designs. Obviously lots of individual companies make their own proprietary double-shouldered connections, but the basic design principles are out there for everyone to see. So a few intrepid members have designed and tested a set of API-compatible double-shouldered connections. The Technical Report about what they did is pending publication, so now us other members of the group are all a-twitter working on what’s needed to make it a standard.

You may be shocked to learn I volunteered to look into the equations.

(tee hee hee) Ok, let’s derive!

Do you know where the regular equation comes from? The one I’ve talked about previously? It comes from a 1957 paper where A.P. Farr describes the now-famous “modified screw-jack equation.” It results from an energy balance—essentially the energy input in the form of applied torque is equal to the energy stored in the threads and the energy dissipated by friction.

Actual numbers—er, well, no, letters with more specific meanings—turns it into our dear old friend:

Extending that to 2 shoulders should be easy enough:

I mean, more terms, but it’s still the same idea, right? Right!

But here’s the problem. Now I’ve got me one equation with two unknowns—those two “force” terms (F) representing the stress state at the outer shoulder (the normal one) and the inner shoulder (the second one at the pin tip). To solve it, Imma need another equation. To understand it, Imma need to figure out how the stresses twixt the outside and inside relate to one another.

I could give you some history. See, I have considered this very thing in the before times, and I tried to relate the two stresses (pin neck to pin nose) by assuming that the threads were rigid. I know; that’s kind of a crazy assumption. It is a conservative assumption, though (rigid threads absorb no energy, so real threads would be able to handle more torque energy without failing because they do absorb some of it), and it’s actually the same assumption that Farr made. We don’t think much about it, but Farr’s “force × distance” assumes the same distance for all the threaded region—aka, rigid threads.

Once I assumed those rigid threads, I imagined this goofy-looking model to relate the stresses at every location, essentially giving me my second equation so I can solve the system. If you’re curious, meander over to OnePetro and check out my paper. (Of course I wrote a paper about it.) There’s some fun pictures in there, too:

Remember as you peruse that paper, though, that it’s obsolete.

See, as the API work group met we decided not to use the rigid thread model, but a different one that uses a much easier second equation:

I know, I know. It sounds ridiculous to just assume that the pin nose and the pin neck will always have the same stress. What about different gaps? Different thread lengths? You know the pin is supposed to be shorter than the box, right? So the pin neck is building stress while the pin nose is still at zero? What nonsense is this?!!?

Here’s why that decision was made: it’s probably true at yield (or at least very slightly after). Whichever side goes first, pin neck or pin nose, once it’s yielding the other side will catch up pretty quickly. So the neck = nose assumption is really an end-state assumption, and it’s not a bad one.

With the end/yield torque assumed, then the makeup torque is envisioned as a fraction of yield torque, not as a particular stress state. (That’s going to come back up in a bit.)

So with our magical neck-nose stress equivalence, we can start plugging in to get the box yield (T1) and pin yield (T2) equations:

(See how they look a little different, though? We assumed the pin neck stress was the same as the pin nose stress—not the box counterbore stress. There’s an extra step in there for the box to translate pin stress to box stress.)

Ah, but our good frenemy T4—the peak of the triangle, you recall—how are we to find thee?

Um, the same way you did T1 and T2?

No, no, that won’t work. T4 is the special-snowflake makeup torque where the shoulders separate (caused by the combination of makeup prestress and tensile string loading) and the pin yields at the same time. It’s fundamentally a description of stress states prior to final yielding, which we didn’t do. We’ve assumed the end state, then worked backwards. So we will never, never know what the T4 makeup torque is. Never ever.

I found it:

You cheater; you looked in my notes.

I presented this qualitative example graph to the API workgroup. (And of course that solved everything.) A typical single-shouldered curve would be the T4'-T2' line; what we calculate assuming that nose-neck stress equivalence (above, from cheaterpants) would be T4-T2.

Real life is probably some bent in-between like T4"-T2, where the connection is, in factoid, a single-shouldered connection up to some point where it’s not, then it’s a combofied double-shouldered stress-wrangling connection up to the final yield we talked about earlier. So assuming the nose-neck stress equivalence (that’s starting to sound like a Big Bang Theory episode) is wrong, conservatively on the shoulder-leak side and non-conservatively on the pin-yield side. But if we put that disclaimer in, then we can at least approximate the performance of the connection, and we’ll tell people to be judicious with their safety factors.

Thus sayeth the API work group.

Alright, there’s one more equation we need, and that’s for the special case where you’re applying torque with string tension hanging, called T3.

I won’t bore you with the details of how T3 works … wait, yes I will. I already have. So maybe I won’t bore you again.

And with that, we now have a working way of determining the load capacities of API’s upcoming double-shouldered connection designs. Maybe somebody should put it into an app …