What the calculation gives you
A bolted joint works by clamping: the bolt is stretched at assembly so that the parts stay pressed together under the working load. The calculation gives the preload to aim for, the torque that produces it, and how much of the working load the bolt itself sees, so that you can check it against its proof load and keep a margin against separation and loosening.
Proof load and preload
Fp = Sp · At
Sp proof strength of the bolt grade (600 MPa for 8.8, 830 for 10.9, 85 ksi for SAE grade 5) · At tensile stress area of the thread, not the nominal area
For a metric thread of nominal diameter d and pitch P, At = (π/4)(d − 0.9382 P)2; an M5 × 0.8 has 14.2 mm2 against a nominal 19.6 mm2. Design handbooks (Shigley is the usual reference) set the preload at 75% of the proof load for a joint that will be taken apart and 90% for a permanent one:
Fi = 0.75 Fp (reusable) or 0.90 Fp (permanent)
Torque to reach the preload
T = K · Fi · d
K torque coefficient, about 0.20 for dry zinc-plated or black steel, 0.15 to 0.18 lubricated, 0.12 with anti-seize · d nominal bolt diameter
Most of the torque goes into friction: roughly 50% under the nut face, 40% in the threads and only about 10% stretches the bolt. That is why K matters more than anything else, and why a torque wrench gives a preload scatter of about ±25%. When the preload matters, measure it another way: turn-of-nut, a stretch measurement, or load-indicating washers.
What the working load does
When an external tension P is applied, the bolt stretches a little more and the clamped parts relax a little; the load is shared in proportion to their stiffnesses:
Fb = Fi + C P, with C = kb / (kb + km)
Fb bolt force in service · kb, km stiffness of the bolt and of the clamped members · C is typically 0.2 to 0.3 for steel parts, higher with a soft gasket
Two margins follow. Against yielding the bolt: n = (Sp At − Fi) / (C P). Against the parts separating, which is when the joint has failed in practice: n0 = Fi / [P (1 − C)]. A well-preloaded joint barely notices the working load, which is also why it survives fatigue: the bolt sees only the small share CP of the alternating load.
Worked example
The worksheet below is a small, real case: an M16 × 1.5 oil drain plug tightened to the 32 ft·lbf the service manual gives. It takes the torque coefficient K and a lubrication reduction, turns the torque into the clamping force with F = T / (K d), and then follows that force into the parts: the stretch of the plug over its engaged length, the compression of the aluminium crush washer, and how much the washer is crushed. Change K from 0.2 to 0.15 and see how much more clamping force the same torque produces once the thread is oiled.
Common mistakes
- Nominal area instead of the tensile stress area. It overstates the capacity of a small bolt by 30 to 40%.
- A torque table for the wrong surface. The same torque on a lubricated bolt gives 30% more preload than dry, enough to yield it.
- No allowance for relaxation. Embedment of the surfaces loses 5 to 10% of the preload in the first hours; gaskets and soft materials lose more. Retorque or design for it.
- Treating the working load as all on the bolt. It is not; only the share CP is. Conversely, a joint with too little preload separates, and then the bolt does see the full load, in fatigue.
- Tightening to the proof load. 75 to 90% is the target; the rest is the margin for torque scatter and the working load.
Questions
- Why do bolts loosen under vibration?
- Transverse slip between the parts lets the nut rotate a fraction of a turn at each cycle. Enough preload stops the slip, which is the real cure; thread locker, prevailing-torque nuts and flanged heads are the back-up.
- Is the torque formula good for stainless bolts?
- Use a higher K, about 0.3, and lubricate: stainless on stainless galls, and the friction is both high and erratic.
- How many bolts share the load evenly?
- In a stiff flange, all of them, approximately; in a flexible one the bolts nearest the load take more. The worksheet's even split is the usual first pass; check the flange stiffness when the margins are thin.
