Can't help you with a formula I'm afraid. But am thinking the weakest point would be that part of the tube which has been threaded, but does not have the component thread inside it. So you wouldn't want to have more thread in the tube than the length of the component thread. I know tap threads are tapered, so you will have some scoring of the inside of the tube. Might be worth thinking about a sleeve on the outside of the tube covering the length of pipe which is cut into on the inside. 

Just my tuppence worth

I hadn't thought of it in that way, but I see exactly where you are coming from. The mating part screws up to a shoulder, so do have the internal thread extending beyond. I might have to look at that... Thanks!

The first thing to realise is that it depends on the material, the thread form, and possibly the tolerance of the thread cut.
For example, cheap mild steel may be taken as having an ultimate tensile strength of 25 tons per square inch, but High tensile bolts are from an alloy that manages about 3 times this.
In small fasteners a lot of the stress in the core comes from just torquing against the thread - in fact any metric thread (i.,e 60 degree iso triangle form)under about half an inch / 12mm diameter can be tightened to tensile failure - larger sizes can be tightened until the thread strips out but the core remains instead....

So do not over-tighten very small bolts, as that eats into the tensile strength available for load bearing. Really these are only ever fully torqued  for clamping parts together where the load is directed to shear the joint, and the purpose of the nut and bolt is only to force the mating surfaces together really firmly, and the bulk of the joint rigidity comes from the friction that this clamping action causes.

So either the force is set by the area of the core of the bolt and the UTS in the normal snap a rod way force - UTS * area, or by the tearing off the thrianges of thread - in which case the area is the base of the thread area unwound - bit that is in shear, not tensile failure, so about 4-5 times higher .

A loose fitting thread (one where the peaks of the nut and bolt threads do not meet at the 'middles of the mountains' as it were, rather only the tips overlap. have a correspondingly reduced thread shear area, and will shear strip the threads  early.)

For anything remotely critical, I would recommend a test on bolt makers samples,

Also note that in things that really matter it is important to allow for the fact that the load may not share equally among all fixings . To be avoided at all cost is a situation where one fails and then another is overloaded and fails - that can lead to catastrophic chains of failure. Such systems have very specific  tightening settings and a sequence that must be followed.

I may not have the right mental picture of what you are doing, so not all of this may apply.

M,

Another possible mode of failure that comes to mind, presuming the threads are triangular or similar, is that a pull force will tend to be translated into a crushing force on the inner threaded tube - if it's relatively thin walled or particularly malleable , I could see it collapsing allowing the internal thread to contract and pull out.

   - Andy.

True - the normal analysis assumes the 'bolt' is solid, or at least largely so. There is an identical  "bursting"  force in a nut,

However, to collapse a cylinder, even one weakened by a spiral groove, is hard - but depends on  the diameter as well as the  sidewall thickness, for a given external pressure a tube with a smaller diameter can have a thinner side-wall. The analysis is similar to that of collapsing a  vertical beam in compression, where the danger point is when if there is small assymetry, the energy gained if futher collapse occurs is greater than that needed to cause  it. 

The text book formulae (Rourke and Young) above are for perfect defect-free thin walled tubes collapsing under external pressure. This probably is not the situation and the results should be considered with caution, or at least a factor of 5 safety.....

To a good first approximation the in ward pressure may be assumed to be related to the thread tension resolved along the active ramp of the thread angle (under load one side of the thread is unloaded). For normal 60 degree triangle threads, this is ~ twice the load force on the threads, i,e,  the tension divided by the length of thread bearing load - usually only the top turn or two of a thread are doing any work, more in elastic materials. less in harder ones although the usual advice is that thread engagement of about one bolt diameter (6-8 turns) is best, in reality half that is doing about 75% of the heavy lifting....

Mike.